[ { "id": "Astronomy_1007", "problem": "The event horizon of a non-spinning, uncharged black hole can be thought of as a sphere with a radius equal to the Schwarzschild radius, $r_{S}=2 G M / c^{2}$ where $M$ is the mass of the black hole and $c$ is the speed of light. If the black hole at the centre of the Milky Way has a mass of $4.15 \\times 10^{6} M_{\\odot}$, what is the approximate average density within the event horizon?\nA: $\\sim 1 \\mathrm{~kg} \\mathrm{~m}^{-3}$\nB: $\\sim 10^{3} \\mathrm{~kg} \\mathrm{~m}^{-3}$\nC: $\\sim 10^{6} \\mathrm{~kg} \\mathrm{~m}^{-3}$\nD: $\\sim 10^{9} \\mathrm{~kg} \\mathrm{~m}^{-3}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe event horizon of a non-spinning, uncharged black hole can be thought of as a sphere with a radius equal to the Schwarzschild radius, $r_{S}=2 G M / c^{2}$ where $M$ is the mass of the black hole and $c$ is the speed of light. If the black hole at the centre of the Milky Way has a mass of $4.15 \\times 10^{6} M_{\\odot}$, what is the approximate average density within the event horizon?\n\nA: $\\sim 1 \\mathrm{~kg} \\mathrm{~m}^{-3}$\nB: $\\sim 10^{3} \\mathrm{~kg} \\mathrm{~m}^{-3}$\nC: $\\sim 10^{6} \\mathrm{~kg} \\mathrm{~m}^{-3}$\nD: $\\sim 10^{9} \\mathrm{~kg} \\mathrm{~m}^{-3}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_951", "problem": "The very first image released by the James Webb Space Telescope (JWST) was of a galaxy cluster called SMACS 0723. The image is considered to be Webb's first deep field, since a long exposure time of 12.5 hours was used to allow the light from very faint and distant galaxies to be seen. The spectrum of one such galaxy is shown in Figure 2.\n\n[figure1]\n\nFigure 2: Highly redshifted emission lines in the spectrum of a galaxy that is 13.1 billion years old, captured using the JWST's near-infrared spectrometer (NIRSpec). Credit: NASA, ESA, CSA, STScI.\n\nThe spectrum shows four bright hydrogen lines, which are part of the Balmer series (some of which are normally seen in the visible). The rest frame wavelengths of the longest four lines in the series are $410 \\mathrm{~nm}, 434 \\mathrm{~nm}, 486 \\mathrm{~nm}$ and $656 \\mathrm{~nm}$ (not all of which are visible in the spectrum).\n\nOnce a redshift is known, its recessional velocity can be calculated. At very high redshifts, such as these, General Relativity must be used. A conversion from redshift to recessional velocity is shown in Figure 3.\n\n[figure2]\n\nFigure 3: Conversion from redshift to recessional velocity for a linear approximation $(v=z c)$, using Special Relativity $\\left(v=c \\frac{(1+z)^{2}-1}{(1+z)^{2}+1}\\right)$, and using General Relativity $\\left(v=\\dot{a}(z) \\int_{0}^{z} \\frac{c d z^{\\prime}}{H\\left(z^{\\prime}\\right)}\\right)$. The grey area corresponds to a variety of values for cosmological parameters. The solid line corresponds to values approximately the same as the current measured cosmological parameters. Credit: Davis \\& Lineweaver (2001).\n\nTaking measurements from the spectrum, estimate the redshift of the galaxy. [Hint: you should measure more than one line to ensure you correctly identify which rest frame wavelength corresponds to which line.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe very first image released by the James Webb Space Telescope (JWST) was of a galaxy cluster called SMACS 0723. The image is considered to be Webb's first deep field, since a long exposure time of 12.5 hours was used to allow the light from very faint and distant galaxies to be seen. The spectrum of one such galaxy is shown in Figure 2.\n\n[figure1]\n\nFigure 2: Highly redshifted emission lines in the spectrum of a galaxy that is 13.1 billion years old, captured using the JWST's near-infrared spectrometer (NIRSpec). Credit: NASA, ESA, CSA, STScI.\n\nThe spectrum shows four bright hydrogen lines, which are part of the Balmer series (some of which are normally seen in the visible). The rest frame wavelengths of the longest four lines in the series are $410 \\mathrm{~nm}, 434 \\mathrm{~nm}, 486 \\mathrm{~nm}$ and $656 \\mathrm{~nm}$ (not all of which are visible in the spectrum).\n\nOnce a redshift is known, its recessional velocity can be calculated. At very high redshifts, such as these, General Relativity must be used. A conversion from redshift to recessional velocity is shown in Figure 3.\n\n[figure2]\n\nFigure 3: Conversion from redshift to recessional velocity for a linear approximation $(v=z c)$, using Special Relativity $\\left(v=c \\frac{(1+z)^{2}-1}{(1+z)^{2}+1}\\right)$, and using General Relativity $\\left(v=\\dot{a}(z) \\int_{0}^{z} \\frac{c d z^{\\prime}}{H\\left(z^{\\prime}\\right)}\\right)$. The grey area corresponds to a variety of values for cosmological parameters. The solid line corresponds to values approximately the same as the current measured cosmological parameters. Credit: Davis \\& Lineweaver (2001).\n\nTaking measurements from the spectrum, estimate the redshift of the galaxy. [Hint: you should measure more than one line to ensure you correctly identify which rest frame wavelength corresponds to which line.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-06.jpg?height=985&width=1588&top_left_y=547&top_left_x=240", "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-07.jpg?height=997&width=1334&top_left_y=187&top_left_x=361" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1031", "problem": "The Chandra X-ray Observatory celebrates 20 years of operation this year, and was 100 times more sensitive than previous X-ray telescopes when it was launched. All X-ray telescopes have either been space-borne or operate in near-space environments. This is because:\nA: X-rays cannot penetrate the Earth's atmosphere all the way to the ground\nB: on the ground there is too much interference from medical X-rays\nC: it is dangerous to be close to an X-ray telescope so it must be highly remote from human life\nD: the resolution of the telescope would be too poor for astronomical observations from the ground\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe Chandra X-ray Observatory celebrates 20 years of operation this year, and was 100 times more sensitive than previous X-ray telescopes when it was launched. All X-ray telescopes have either been space-borne or operate in near-space environments. This is because:\n\nA: X-rays cannot penetrate the Earth's atmosphere all the way to the ground\nB: on the ground there is too much interference from medical X-rays\nC: it is dangerous to be close to an X-ray telescope so it must be highly remote from human life\nD: the resolution of the telescope would be too poor for astronomical observations from the ground\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_159", "problem": "双中子星合并会产生引力波, 探测到引力波的存在, 可以证明爱因斯坦义相对论引力波预言的正确性。两颗中子星合并前, 绕二者连线上的某点转动, 将两颗中子星都看作是质量分布均匀的球体。若产生引力波的双中子星到地球的距离为 13.4 亿光年, 引力波的速度等于真空中的光速, 则下列关于双中子星合并产生引力波的描述正确的是\nA: 该引力波传到地球的时间约为 13.4 年\nB: 该引力波传到地球的时间约为 13.4 亿年\nC: 双中子星合并过程中, 其运行周期逐渐变大\nD: 双中子星合并过程中, 其运行周期逐渐变小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n双中子星合并会产生引力波, 探测到引力波的存在, 可以证明爱因斯坦义相对论引力波预言的正确性。两颗中子星合并前, 绕二者连线上的某点转动, 将两颗中子星都看作是质量分布均匀的球体。若产生引力波的双中子星到地球的距离为 13.4 亿光年, 引力波的速度等于真空中的光速, 则下列关于双中子星合并产生引力波的描述正确的是\n\nA: 该引力波传到地球的时间约为 13.4 年\nB: 该引力波传到地球的时间约为 13.4 亿年\nC: 双中子星合并过程中, 其运行周期逐渐变大\nD: 双中子星合并过程中, 其运行周期逐渐变小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_917", "problem": "The three stars that make up Orion's belt are Alnitak, Alnilam and Mintaka, with apparent magnitudes, $m$, of $1.77,1.69$ and 2.23 respectively. What is the ratio of the apparent brightnesses of the two brightest stars?\n\n[figure1]\nA: 0.655\nB: 1.076\nC: 1.528\nD: 1.644\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe three stars that make up Orion's belt are Alnitak, Alnilam and Mintaka, with apparent magnitudes, $m$, of $1.77,1.69$ and 2.23 respectively. What is the ratio of the apparent brightnesses of the two brightest stars?\n\n[figure1]\n\nA: 0.655\nB: 1.076\nC: 1.528\nD: 1.644\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-05.jpg?height=713&width=648&top_left_y=366&top_left_x=704" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_882", "problem": "Take a look at the following image:\n\n[figure1]\n\nThree Messier objects are circled in the image. Select the alternative that correctly matches each object with its type.\nA: 1 - Open cluster; 2 - Open cluster; 3 - Nebula.\nB: 1 - Open Cluster; 2 - Nebula; 3 - Galaxy.\nC: 1 - Galaxy; 2 - Nebula; 3 - Globular cluster.\nD: 1 - Open cluster; 2 - Galaxy; 3 - Globular cluster.\nE: 1 - Open cluster; 2 - Nebula; 3 - Open cluster.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTake a look at the following image:\n\n[figure1]\n\nThree Messier objects are circled in the image. Select the alternative that correctly matches each object with its type.\n\nA: 1 - Open cluster; 2 - Open cluster; 3 - Nebula.\nB: 1 - Open Cluster; 2 - Nebula; 3 - Galaxy.\nC: 1 - Galaxy; 2 - Nebula; 3 - Globular cluster.\nD: 1 - Open cluster; 2 - Galaxy; 3 - Globular cluster.\nE: 1 - Open cluster; 2 - Nebula; 3 - Open cluster.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8e66d119cc171ba269b3g-08.jpg?height=868&width=1525&top_left_y=1493&top_left_x=300" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1140", "problem": "The Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$d. Determine risco for a non-spinning black hole. Give your answer in units of $\\mathrm{r}_{\\mathrm{g}}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$\n\nproblem:\nd. Determine risco for a non-spinning black hole. Give your answer in units of $\\mathrm{r}_{\\mathrm{g}}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-07.jpg?height=704&width=1414&top_left_y=698&top_left_x=331", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-08.jpg?height=374&width=1562&top_left_y=1698&top_left_x=263", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-09.jpg?height=468&width=686&top_left_y=1388&top_left_x=705" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_112", "problem": "2019 年 4 月 10 日晚 9 时许，人类史上首张黑洞照片面世，有望证实广义相对论在极端条件下仍然成立！某同学查阅资料发现黑洞的半径 $R$ 和质量 $M$ 满足关系式\n\n$R=\\frac{2 G M}{c^{2}}$ (其中 $\\mathrm{G}$ 为引力常量, 真空中的光速 $\\mathrm{c}=3.0 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$ ), 他借助太阳发出的光传播到地球需要大约 8 分钟和地球公转的周期 1 年, 估算太阳“浓缩”为黑洞时, 对应的半径约为 ( )\nA: $3000 \\mathrm{~m}$\nB: $300 \\mathrm{~m}$\nC: $30 \\mathrm{~m}$\nD: $3 \\mathrm{~m}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2019 年 4 月 10 日晚 9 时许，人类史上首张黑洞照片面世，有望证实广义相对论在极端条件下仍然成立！某同学查阅资料发现黑洞的半径 $R$ 和质量 $M$ 满足关系式\n\n$R=\\frac{2 G M}{c^{2}}$ (其中 $\\mathrm{G}$ 为引力常量, 真空中的光速 $\\mathrm{c}=3.0 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$ ), 他借助太阳发出的光传播到地球需要大约 8 分钟和地球公转的周期 1 年, 估算太阳“浓缩”为黑洞时, 对应的半径约为 ( )\n\nA: $3000 \\mathrm{~m}$\nB: $300 \\mathrm{~m}$\nC: $30 \\mathrm{~m}$\nD: $3 \\mathrm{~m}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_277", "problem": "公元 2032 年, 某位 2022 年进入大学的航天爱好者身着飞行机甲, 实现了年少时伴飞中国天宫空间站的梦想, 在伴飞天宫前, 该航天爱好者在近地圆轨道上做无动力飞行,然后先后通过两次的瞬间加力, 成功转移到天宫所在轨道, 图中 $\\alpha$ 角为该航天爱好者第一次加力时, 天宫一号和航天爱好者相对地球球心张开的夹角, 已知, 地球半径为 $6400 \\mathrm{~km}$, 天宫距地面高度约为 $400 \\mathrm{~km}$, 为了以最短的时间到达天宫附近, $\\alpha$ 角约为\n\n[图1]\nA: $1^{\\circ}$\nB: $8^{\\circ}$\nC: $16^{\\circ}$\nD: $20^{\\circ}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n公元 2032 年, 某位 2022 年进入大学的航天爱好者身着飞行机甲, 实现了年少时伴飞中国天宫空间站的梦想, 在伴飞天宫前, 该航天爱好者在近地圆轨道上做无动力飞行,然后先后通过两次的瞬间加力, 成功转移到天宫所在轨道, 图中 $\\alpha$ 角为该航天爱好者第一次加力时, 天宫一号和航天爱好者相对地球球心张开的夹角, 已知, 地球半径为 $6400 \\mathrm{~km}$, 天宫距地面高度约为 $400 \\mathrm{~km}$, 为了以最短的时间到达天宫附近, $\\alpha$ 角约为\n\n[图1]\n\nA: $1^{\\circ}$\nB: $8^{\\circ}$\nC: $16^{\\circ}$\nD: $20^{\\circ}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-098.jpg?height=348&width=437&top_left_y=1682&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-099.jpg?height=400&width=505&top_left_y=168&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_569", "problem": "哈雷彗星是人一生中唯一可以裸眼看能看见两次的彗星, 其绕日运行的周期为 $T$ 年,若测得它在近日点距太阳中心的距离是地球公转轨道半长轴的 $N$ 倍, 则由此估算出哈雷彗星在近日点时受到太阳的引力是在远日点受太阳引力的\nA: $N^{2}$\nB: $\\left(2 T^{\\frac{2}{3}}-N\\right)^{2} N^{-2}$\nC: $\\left(2 T^{\\frac{2}{3}} N^{-1}-1\\right)$ 倍\nD: $T^{\\frac{4}{3}} N^{2}$ 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n哈雷彗星是人一生中唯一可以裸眼看能看见两次的彗星, 其绕日运行的周期为 $T$ 年,若测得它在近日点距太阳中心的距离是地球公转轨道半长轴的 $N$ 倍, 则由此估算出哈雷彗星在近日点时受到太阳的引力是在远日点受太阳引力的\n\nA: $N^{2}$\nB: $\\left(2 T^{\\frac{2}{3}}-N\\right)^{2} N^{-2}$\nC: $\\left(2 T^{\\frac{2}{3}} N^{-1}-1\\right)$ 倍\nD: $T^{\\frac{4}{3}} N^{2}$ 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_241", "problem": "2020 年 12 月 17 日嫦娥五号从月球采集月壤成功返回地球。嫦娥五号绕月运行的轨迹如图所示, 在近月点 $O$ 制动成功后被月球捕获进入近月点 $200 \\mathrm{~km}$, 远月点 $5500 \\mathrm{~km}$的粗圆轨道I, 在近月点再次制动进入椭圆轨道II, 第三次制动后进入离月面 $200 \\mathrm{~km}$ 的环月圆轨道III, 已知月球半径约为 $1700 \\mathrm{~km}$, 则 ( )\n\n[图1]\nA: 嫦娥五号在I、II、III三个轨道上运行的周期相等\nB: 嫦娥五号在I、II、III三个轨道上运行的机械能相等\nC: 嫦娥五号在 $P 、 Q$ 两点的速度与它们到月心的距离成反比\nD: $O$ 点的重力加速度约为月球表面的重力加速度的 0.8 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年 12 月 17 日嫦娥五号从月球采集月壤成功返回地球。嫦娥五号绕月运行的轨迹如图所示, 在近月点 $O$ 制动成功后被月球捕获进入近月点 $200 \\mathrm{~km}$, 远月点 $5500 \\mathrm{~km}$的粗圆轨道I, 在近月点再次制动进入椭圆轨道II, 第三次制动后进入离月面 $200 \\mathrm{~km}$ 的环月圆轨道III, 已知月球半径约为 $1700 \\mathrm{~km}$, 则 ( )\n\n[图1]\n\nA: 嫦娥五号在I、II、III三个轨道上运行的周期相等\nB: 嫦娥五号在I、II、III三个轨道上运行的机械能相等\nC: 嫦娥五号在 $P 、 Q$ 两点的速度与它们到月心的距离成反比\nD: $O$ 点的重力加速度约为月球表面的重力加速度的 0.8 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-095.jpg?height=414&width=485&top_left_y=1798&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_618", "problem": "中国在西昌卫星发射中心成功发射“亚太九号”通信卫星, 该卫星运行的轨道示意图如图所示, 卫星先沿陏圆轨道 1 运行, 近地点为 $Q$, 远地点为 $P$ 。当卫星经过 $P$ 点时点火加速, 使卫星由粗圆轨道 1 转移到地球同步轨道 2 上运行, 下列说法正确的是 ( )\n\n[图1]\nA: 卫星在轨道 1 和轨道 2 上运动时的机械能相等\nB: 卫星在轨道 1 上运行经过 $P$ 点的速度大于经过 $Q$ 点的速度\nC: 卫星在轨道 2 上时处于超重状态\nD: 卫星在轨道 1 上运行经过 $P$ 点的加速度等于在轨道 2 上运行经过 $P$ 点的加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n中国在西昌卫星发射中心成功发射“亚太九号”通信卫星, 该卫星运行的轨道示意图如图所示, 卫星先沿陏圆轨道 1 运行, 近地点为 $Q$, 远地点为 $P$ 。当卫星经过 $P$ 点时点火加速, 使卫星由粗圆轨道 1 转移到地球同步轨道 2 上运行, 下列说法正确的是 ( )\n\n[图1]\n\nA: 卫星在轨道 1 和轨道 2 上运动时的机械能相等\nB: 卫星在轨道 1 上运行经过 $P$ 点的速度大于经过 $Q$ 点的速度\nC: 卫星在轨道 2 上时处于超重状态\nD: 卫星在轨道 1 上运行经过 $P$ 点的加速度等于在轨道 2 上运行经过 $P$ 点的加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-06.jpg?height=497&width=596&top_left_y=397&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_675", "problem": "宇航员在地球表面一斜坡上 $P$ 点, 沿水平方向以初速度 $v_{0}$ 抛出一个小球, 测得小球经时间 $t$ 落到斜坡另一点 $Q$ 上现宇航员站在某质量分布均匀的星球表面相同的斜坡上 $P$点, 沿水平方向以相同的初速度 $v_{0}$ 抛出一个小球, 小球落在 $P Q$ 的中点. 已知该星球的半径为 $R$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 球的体积公式是 $V=\\frac{4}{3} \\pi R^{3}$ 。求:\n该星球的密度 $\\rho$ \n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n宇航员在地球表面一斜坡上 $P$ 点, 沿水平方向以初速度 $v_{0}$ 抛出一个小球, 测得小球经时间 $t$ 落到斜坡另一点 $Q$ 上现宇航员站在某质量分布均匀的星球表面相同的斜坡上 $P$点, 沿水平方向以相同的初速度 $v_{0}$ 抛出一个小球, 小球落在 $P Q$ 的中点. 已知该星球的半径为 $R$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 球的体积公式是 $V=\\frac{4}{3} \\pi R^{3}$ 。求:\n该星球的密度 $\\rho$ \n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-051.jpg?height=271&width=443&top_left_y=827&top_left_x=335" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_35", "problem": "如图所示, 航天器 $\\mathrm{A}$ 和卫星 $\\mathrm{B}$ 均在赤道面内, 它通过一根金属长绳相连, 在各自的轨道上绕地球做自西向东的匀速圆周运动, 不考虑绳系卫星与航天器之间的万有引力,忽略空气阻力, 不计金属长绳的质量, 则（）\n\n[图1]\nA: 正常运行时, 金属长绳中拉力为零\nB: 绳系卫星 $\\mathrm{B}$ 的线速度大于航天器 $\\mathrm{A}$ 的线速度\nC: 由于存在地磁场, 金属长绳上绳系卫星 $\\mathrm{B}$ 端的电势高于航天器 $\\mathrm{A}$ 端的电势\nD: 若在绳系卫星 B 的轨道上存在另一颗独立卫星 C, 其角速度大于绳系卫星 $\\mathrm{B}$ 的角速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 航天器 $\\mathrm{A}$ 和卫星 $\\mathrm{B}$ 均在赤道面内, 它通过一根金属长绳相连, 在各自的轨道上绕地球做自西向东的匀速圆周运动, 不考虑绳系卫星与航天器之间的万有引力,忽略空气阻力, 不计金属长绳的质量, 则（）\n\n[图1]\n\nA: 正常运行时, 金属长绳中拉力为零\nB: 绳系卫星 $\\mathrm{B}$ 的线速度大于航天器 $\\mathrm{A}$ 的线速度\nC: 由于存在地磁场, 金属长绳上绳系卫星 $\\mathrm{B}$ 端的电势高于航天器 $\\mathrm{A}$ 端的电势\nD: 若在绳系卫星 B 的轨道上存在另一颗独立卫星 C, 其角速度大于绳系卫星 $\\mathrm{B}$ 的角速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-100.jpg?height=429&width=436&top_left_y=2187&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_179", "problem": "2022 年 11 月 29 日, 神舟十五号飞行任务是中国空间站建造阶段的最后一棒, 也是空间站应用与发展阶段的第一棒. 已知空间站 $\\mathrm{Q}$ 和同步卫星 $\\mathrm{P}$ 环绕地球运行的轨道均可视为匀速圆周运动。如图所示, 已知 $\\mathrm{P}, \\mathrm{Q}$ 运动方向均沿逆时针方向, $P Q$ 与 $O P$连线的夹角最大值为 $\\alpha$ 。求:\n同步卫星 $\\mathrm{P}$ 、空间站 $\\mathrm{Q}$ 的角速度之比;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2022 年 11 月 29 日, 神舟十五号飞行任务是中国空间站建造阶段的最后一棒, 也是空间站应用与发展阶段的第一棒. 已知空间站 $\\mathrm{Q}$ 和同步卫星 $\\mathrm{P}$ 环绕地球运行的轨道均可视为匀速圆周运动。如图所示, 已知 $\\mathrm{P}, \\mathrm{Q}$ 运动方向均沿逆时针方向, $P Q$ 与 $O P$连线的夹角最大值为 $\\alpha$ 。求:\n同步卫星 $\\mathrm{P}$ 、空间站 $\\mathrm{Q}$ 的角速度之比;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-063.jpg?height=314&width=345&top_left_y=160&top_left_x=336", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-063.jpg?height=351&width=379&top_left_y=818&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_177", "problem": "如图所示, 是月亮女神、嫦娥 1 号绕月做圆周运行时某时刻的图片, 用 $R_{1} 、 R_{2} 、$ $T_{1} 、 T_{2}$ 分别表示月亮女神和嫦娥 1 号的轨道半径及周期, 用 $R$ 表示月亮的半径。此时二者的连线通过月心, 轨道半径之比为 $1: 4$ 。若不考虑月亮女神、嫦娥 1 号之间的引力,则下列说法正确的是（ ）\n\n[图1]\nA: 在图示轨道上, 月亮女神的速度小于嫦娥 1 号\nB: 在图示轨道上, 嫦娥 1 号的加速度大小是月亮女神的 4 倍\nC: 在图示轨道上, 且从图示位置开始经 $t=\\frac{T_{1} T_{2}}{2 T_{2}-2 T_{1}}$ 二者第二次相距最近\nD: 若月亮女神从图示轨道上加速, 可与嫦娥 1 号对接\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 是月亮女神、嫦娥 1 号绕月做圆周运行时某时刻的图片, 用 $R_{1} 、 R_{2} 、$ $T_{1} 、 T_{2}$ 分别表示月亮女神和嫦娥 1 号的轨道半径及周期, 用 $R$ 表示月亮的半径。此时二者的连线通过月心, 轨道半径之比为 $1: 4$ 。若不考虑月亮女神、嫦娥 1 号之间的引力,则下列说法正确的是（ ）\n\n[图1]\n\nA: 在图示轨道上, 月亮女神的速度小于嫦娥 1 号\nB: 在图示轨道上, 嫦娥 1 号的加速度大小是月亮女神的 4 倍\nC: 在图示轨道上, 且从图示位置开始经 $t=\\frac{T_{1} T_{2}}{2 T_{2}-2 T_{1}}$ 二者第二次相距最近\nD: 若月亮女神从图示轨道上加速, 可与嫦娥 1 号对接\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-052.jpg?height=483&width=537&top_left_y=158&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_214", "problem": "太阳帆飞船是依靠太阳的光压来加速飞船的。太阳帆一般由反射率极高的韧性薄膜制成的, 面积大, 质量小。某太空探测飞船运行在地球公转轨道上, 绕太阳做匀速圆周运动, 某时刻飞船的太阳帆打开，展开的面积为 $S$, 飞船能控制帆面始终垂直太阳光线。已知太阳的总辐射功率为 $P_{0}$, 日地距离为 $r_{0}$, 光速为 $c$, 引力常量为 $G$, 太阳帆反射率 $100 \\%$ 。下列说法正确的是（ ）\nA: 太阳帆受到的太阳光压力为 $F=\\frac{2}{c} P_{0}$\nB: 太阳帆受到的太阳光压力为 $F=\\frac{S}{4 \\pi c r_{0}^{2}} P_{0}$\nC: 打开太阳帆后, 飞船将沿径向远离太阳\nD: 打开太阳帆后, 飞船在以后运动中受到的光压力与太阳的引力之比恒定\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n太阳帆飞船是依靠太阳的光压来加速飞船的。太阳帆一般由反射率极高的韧性薄膜制成的, 面积大, 质量小。某太空探测飞船运行在地球公转轨道上, 绕太阳做匀速圆周运动, 某时刻飞船的太阳帆打开，展开的面积为 $S$, 飞船能控制帆面始终垂直太阳光线。已知太阳的总辐射功率为 $P_{0}$, 日地距离为 $r_{0}$, 光速为 $c$, 引力常量为 $G$, 太阳帆反射率 $100 \\%$ 。下列说法正确的是（ ）\n\nA: 太阳帆受到的太阳光压力为 $F=\\frac{2}{c} P_{0}$\nB: 太阳帆受到的太阳光压力为 $F=\\frac{S}{4 \\pi c r_{0}^{2}} P_{0}$\nC: 打开太阳帆后, 飞船将沿径向远离太阳\nD: 打开太阳帆后, 飞船在以后运动中受到的光压力与太阳的引力之比恒定\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_827", "problem": "Choose the values $(x, y, z)$ that would best complete the descriptions below for the 3 different types of twilights.\n\ni) civil twilight, when the Sun is $x^{\\circ}$ below the horizon. We can start to see the brightest stars and the sea horizon can be clearly seen. At this point it becomes hard to read outdoors without artificial light.\n\nii) nautical twilight, when the Sun is $y^{\\circ}$ below the horizon. It is too dark to see the sea horizon and you can no longer make altitude measurements for navigation using the horizon as a reference.\n\niii) astronomical twilight, when the Sun is $z^{\\circ}$ below the horizon. Scattered sunlight becomes less than the average starlight and it is about the same brightness as the aurora or zodiacal light.\nA: Civil twilight $-5^{\\circ}$, nautical twilight $-10^{\\circ}$, astronomical twilight - $15^{\\circ}$\nB: Civil twilight $-6^{\\circ}$, nautical twilight $-12^{\\circ}$, astronomical twilight - $18^{\\circ}$ (Answer)\nC: Civil twilight $-3^{\\circ}$, nautical twilight $-6^{\\circ}$, astronomical twilight - $9^{\\circ}$\nD: Civil twilight $-12^{\\circ}$, nautical twilight $-6^{\\circ}$, astronomical twilight $-18^{\\circ}$\nE: Civil twilight $-10^{\\circ}$, nautical twilight $-20^{\\circ}$, astronomical twilight $-30^{\\circ}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nChoose the values $(x, y, z)$ that would best complete the descriptions below for the 3 different types of twilights.\n\ni) civil twilight, when the Sun is $x^{\\circ}$ below the horizon. We can start to see the brightest stars and the sea horizon can be clearly seen. At this point it becomes hard to read outdoors without artificial light.\n\nii) nautical twilight, when the Sun is $y^{\\circ}$ below the horizon. It is too dark to see the sea horizon and you can no longer make altitude measurements for navigation using the horizon as a reference.\n\niii) astronomical twilight, when the Sun is $z^{\\circ}$ below the horizon. Scattered sunlight becomes less than the average starlight and it is about the same brightness as the aurora or zodiacal light.\n\nA: Civil twilight $-5^{\\circ}$, nautical twilight $-10^{\\circ}$, astronomical twilight - $15^{\\circ}$\nB: Civil twilight $-6^{\\circ}$, nautical twilight $-12^{\\circ}$, astronomical twilight - $18^{\\circ}$ (Answer)\nC: Civil twilight $-3^{\\circ}$, nautical twilight $-6^{\\circ}$, astronomical twilight - $9^{\\circ}$\nD: Civil twilight $-12^{\\circ}$, nautical twilight $-6^{\\circ}$, astronomical twilight $-18^{\\circ}$\nE: Civil twilight $-10^{\\circ}$, nautical twilight $-20^{\\circ}$, astronomical twilight $-30^{\\circ}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_899", "problem": "In January 2020, NASA's Transiting Exoplanet Survey Satellite (TESS) discovered an Earth-sized exoplanet, called TOI $700 \\mathrm{~d}$, in its star's habitable zone. This is the range of distances a planet can orbit a star so that liquid water can exist on the surface, given sufficient atmospheric pressure. It was discovered using the transit method, where the planet passes directly between the observer and the star, causing a drop in brightness.\n\n[figure1]\n\nFigure 1: The three planets of the TOI 700 system, illustrated here, orbit a small, cool M dwarf star. TOI $700 \\mathrm{~d}$ is the first Earth-size habitable-zone world discovered by TESS. Credit: NASA's Goddard Space Flight Center.\n\nTOI $700 \\mathrm{~d}$ has radius $R_{P}=1.19 R_{E}$ orbiting a star with luminosity $0.0233 L_{\\odot}$ at a distance of 0.163 au. Assume that the planet absorbs all the light that hits the surface, and that the orbit is circular.\n\nFind the value of $T_{P}$. Give your answer in ${ }^{\\circ} \\mathrm{C}$.\n\n[Hint: you may find it is colder than expected.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn January 2020, NASA's Transiting Exoplanet Survey Satellite (TESS) discovered an Earth-sized exoplanet, called TOI $700 \\mathrm{~d}$, in its star's habitable zone. This is the range of distances a planet can orbit a star so that liquid water can exist on the surface, given sufficient atmospheric pressure. It was discovered using the transit method, where the planet passes directly between the observer and the star, causing a drop in brightness.\n\n[figure1]\n\nFigure 1: The three planets of the TOI 700 system, illustrated here, orbit a small, cool M dwarf star. TOI $700 \\mathrm{~d}$ is the first Earth-size habitable-zone world discovered by TESS. Credit: NASA's Goddard Space Flight Center.\n\nTOI $700 \\mathrm{~d}$ has radius $R_{P}=1.19 R_{E}$ orbiting a star with luminosity $0.0233 L_{\\odot}$ at a distance of 0.163 au. Assume that the planet absorbs all the light that hits the surface, and that the orbit is circular.\n\nFind the value of $T_{P}$. Give your answer in ${ }^{\\circ} \\mathrm{C}$.\n\n[Hint: you may find it is colder than expected.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of C, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c3e3c992a9c51eb3e471g-06.jpg?height=405&width=1280&top_left_y=780&top_left_x=388" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "C" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_920", "problem": "Why do hurricanes rotate anti-clockwise in the northern hemisphere and clockwise in the southern hemisphere?\nA: Due to the Earth rotating from East to West\nB: Due to the different ratios of land to water area between the two hemispheres\nC: Due to the Moon's orbit being inclined by $5^{\\circ}$ above the ecliptic giving it more influence on the northern hemisphere\nD: Due to the Coriolis Effect causing paths of particles to curve as they travel over the Earth's surface\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhy do hurricanes rotate anti-clockwise in the northern hemisphere and clockwise in the southern hemisphere?\n\nA: Due to the Earth rotating from East to West\nB: Due to the different ratios of land to water area between the two hemispheres\nC: Due to the Moon's orbit being inclined by $5^{\\circ}$ above the ecliptic giving it more influence on the northern hemisphere\nD: Due to the Coriolis Effect causing paths of particles to curve as they travel over the Earth's surface\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_654", "problem": "当某一地外行星 (火星、木星、土星、天王星、海王星) 于绕日公转过程中运行到试卷第 48 页，共 150 页\n与地球、太阳成一直线的状态, 且地球恰好位于太阳和外行星之间的这种天文现象叫“冲日”, 冲日前后是观测地外行星的好时机。如图所示是土星冲日示意图, 已知地球质量为 $M$, 半径为 $R$, 公转周期是 1 年, 公转半径为 $r$, 土星质量是地球的 95 倍, 土星半径是地球的 9.5 倍, 土星的公转半径是地球的 9.5 倍。求: $\\left(\\sqrt{9.5^{3}} \\approx 29\\right)$\n地球和太阳间的万有引力是土星和太阳间的几倍？\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n当某一地外行星 (火星、木星、土星、天王星、海王星) 于绕日公转过程中运行到试卷第 48 页，共 150 页\n与地球、太阳成一直线的状态, 且地球恰好位于太阳和外行星之间的这种天文现象叫“冲日”, 冲日前后是观测地外行星的好时机。如图所示是土星冲日示意图, 已知地球质量为 $M$, 半径为 $R$, 公转周期是 1 年, 公转半径为 $r$, 土星质量是地球的 95 倍, 土星半径是地球的 9.5 倍, 土星的公转半径是地球的 9.5 倍。求: $\\left(\\sqrt{9.5^{3}} \\approx 29\\right)$\n地球和太阳间的万有引力是土星和太阳间的几倍？\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-049.jpg?height=286&width=491&top_left_y=848&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_642", "problem": "有人提出了一种不用火箭发射人造地球卫星的设想。其设想如下: 沿地球的一条弦挖一通道, 如图乙所示. 在通道的两个出口处 $A$ 和 $B$, 分别将质量为 $M$ 的物体和质量为 $m$ 的待发射卫星同时自由释放, 只要 $M$ 比 $m$ 足够大, 碰撞后, 质量为 $m$ 的物体, 即待发射的卫星就会从通道口 $B$ 冲出通道。(已知地球表面的重力加速度为 $g$, 地球半径为 $\\left.R_{0}\\right)$\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\n如果在 $A$ 处释放一个质量很大的物体 $M$, 在 $B$ 处同时释放一个质量远小于 $M$ 的物体，同时达到 $O^{\\prime}$ 处发生弹性正碰(由于大物体质量很大，可以认为碰后速度不变)，那么小物体返回从 $B$ 飞出, 为使飞出的速度达到地球的第一宇宙速度, $h$ 应为多大?", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n有人提出了一种不用火箭发射人造地球卫星的设想。其设想如下: 沿地球的一条弦挖一通道, 如图乙所示. 在通道的两个出口处 $A$ 和 $B$, 分别将质量为 $M$ 的物体和质量为 $m$ 的待发射卫星同时自由释放, 只要 $M$ 比 $m$ 足够大, 碰撞后, 质量为 $m$ 的物体, 即待发射的卫星就会从通道口 $B$ 冲出通道。(已知地球表面的重力加速度为 $g$, 地球半径为 $\\left.R_{0}\\right)$\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\n如果在 $A$ 处释放一个质量很大的物体 $M$, 在 $B$ 处同时释放一个质量远小于 $M$ 的物体，同时达到 $O^{\\prime}$ 处发生弹性正碰(由于大物体质量很大，可以认为碰后速度不变)，那么小物体返回从 $B$ 飞出, 为使飞出的速度达到地球的第一宇宙速度, $h$ 应为多大?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-024.jpg?height=229&width=232&top_left_y=682&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-024.jpg?height=237&width=368&top_left_y=681&top_left_x=570" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_333", "problem": "“天问一号”是中国自主设计的火星探测器, 已于 2021 年 3 月到达火星。已知火星\n直径约为地球直径的 50\\%, 火星质量约为地球质量的 10\\%, 下列说法正确的是（）\n\n[图1]\nA: 火星表面的重力加速度大于 $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$\nB: “天问一号”的发射速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$ 小于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nC: “天问一号”在火星表面圆轨道上的绕行速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nD: “天问一号”在火星表面圆轨道上的环绕周期小于 24 小时\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n“天问一号”是中国自主设计的火星探测器, 已于 2021 年 3 月到达火星。已知火星\n直径约为地球直径的 50\\%, 火星质量约为地球质量的 10\\%, 下列说法正确的是（）\n\n[图1]\n\nA: 火星表面的重力加速度大于 $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$\nB: “天问一号”的发射速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$ 小于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nC: “天问一号”在火星表面圆轨道上的绕行速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nD: “天问一号”在火星表面圆轨道上的环绕周期小于 24 小时\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-058.jpg?height=269&width=468&top_left_y=234&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_32", "problem": "小行星带, 是位于火星和木星轨道之间的小行星的密集区域, 在太阳系中除了九颗大行星以外, 还有成千上万颗我们肉眼看不到的小天体, 它们沿着粗圆形的轨道不停地围绕太阳公转。如图所示, 在火星与木星轨道之间有一小行星带, 假设该带中的小行星只受到太阳的引力，并绕太阳做匀速圆周运动。下列说法正确的是（ ）\nA: 若知道某一小行星绕太阳运转的周期和轨道半径可求出太阳的质量\nB: 若知道地球和某一小行星绕太阳运转的轨道半径, 可求出该小行星的质量\nC: 小行星带内侧小行星的向心加速度值小于外侧小行星的向心加速度值\nD: 在相同时间内, 地球和太阳连线扫过的面积比内侧小行星和太阳连线扫过的面积要小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n小行星带, 是位于火星和木星轨道之间的小行星的密集区域, 在太阳系中除了九颗大行星以外, 还有成千上万颗我们肉眼看不到的小天体, 它们沿着粗圆形的轨道不停地围绕太阳公转。如图所示, 在火星与木星轨道之间有一小行星带, 假设该带中的小行星只受到太阳的引力，并绕太阳做匀速圆周运动。下列说法正确的是（ ）\n\nA: 若知道某一小行星绕太阳运转的周期和轨道半径可求出太阳的质量\nB: 若知道地球和某一小行星绕太阳运转的轨道半径, 可求出该小行星的质量\nC: 小行星带内侧小行星的向心加速度值小于外侧小行星的向心加速度值\nD: 在相同时间内, 地球和太阳连线扫过的面积比内侧小行星和太阳连线扫过的面积要小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_934", "problem": "From the upper parts of the Sun's corona comes a stream of charged particles called the solar wind, meaning the Sun is slowly losing some of its mass (although the effect is negligible compared to the mass loss in nuclear fusion). The particles travel at supersonic speeds until the pressure from interstellar space causes their speed to drop to subsonic speeds instead - this transition is called the termination shock, and has been explored by the two Voyager probes as they leave the solar system.\n[figure1]\n\nFigure 3: Left: A demonstration of a termination shock formed with water flowing from a tap into a sink. Right: The Voyager spacecraft crossing the termination shock of the Solar System.\n\nAt the termination shock boundary the particles slow down from their (supersonic) $u_{\\infty}$ to a much slower (subsonic) speed. Another way of defining this boundary is that it is where the pressure of the interstellar medium, $P_{\\mathrm{ISM}}$, is equal to the kinetic energy density (i.e. KE per unit volume) of the gas. If $P_{\\mathrm{ISM}}$ is estimated to be $10^{-13} \\mathrm{~Pa}$, use your values of $u_{\\infty}$ and $\\Delta M / \\Delta t$ to calculate an estimate for the distance to the termination shock. Give your answer in $\\mathrm{AU}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nFrom the upper parts of the Sun's corona comes a stream of charged particles called the solar wind, meaning the Sun is slowly losing some of its mass (although the effect is negligible compared to the mass loss in nuclear fusion). The particles travel at supersonic speeds until the pressure from interstellar space causes their speed to drop to subsonic speeds instead - this transition is called the termination shock, and has been explored by the two Voyager probes as they leave the solar system.\n[figure1]\n\nFigure 3: Left: A demonstration of a termination shock formed with water flowing from a tap into a sink. Right: The Voyager spacecraft crossing the termination shock of the Solar System.\n\nAt the termination shock boundary the particles slow down from their (supersonic) $u_{\\infty}$ to a much slower (subsonic) speed. Another way of defining this boundary is that it is where the pressure of the interstellar medium, $P_{\\mathrm{ISM}}$, is equal to the kinetic energy density (i.e. KE per unit volume) of the gas. If $P_{\\mathrm{ISM}}$ is estimated to be $10^{-13} \\mathrm{~Pa}$, use your values of $u_{\\infty}$ and $\\Delta M / \\Delta t$ to calculate an estimate for the distance to the termination shock. Give your answer in $\\mathrm{AU}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of AU, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c744602885fab54c0985g-8.jpg?height=454&width=1280&top_left_y=835&top_left_x=386" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "AU" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_815", "problem": ". An astronomer observes an eclipsing binary star system from Earth, and he plots the following light curve.\n\n[figure1]\n\nSuppose that both stars have circular orbits and the distance between the stars is 14.8 AU. What is the total mass of the binary star system in terms of solar masses?\nA: $2.3 M_{\\odot}$\nB: $5.7 M_{\\odot}$\nC: $6.8 M_{\\odot}$\nD: $23 M_{\\odot}$\nE: $46 M_{\\odot}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\n. An astronomer observes an eclipsing binary star system from Earth, and he plots the following light curve.\n\n[figure1]\n\nSuppose that both stars have circular orbits and the distance between the stars is 14.8 AU. What is the total mass of the binary star system in terms of solar masses?\n\nA: $2.3 M_{\\odot}$\nB: $5.7 M_{\\odot}$\nC: $6.8 M_{\\odot}$\nD: $23 M_{\\odot}$\nE: $46 M_{\\odot}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ea07af8330da280030dbg-16.jpg?height=927&width=1193&top_left_y=932&top_left_x=431" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1054", "problem": "On $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.c. By empirically fitting a sinusoidal function (which is assumed to be the same for each track, just with a fixed phase difference between them) and assuming all conjunctions are separated by the average synodic period, we can give rough estimations for the separations of any given great conjunction. Note: be careful as your calculations will be very sensitive to rounding errors.\n\niii. State which track the 'Star of Bethlehem' great conjunction is on, and hence use the relevant equation to predict its separation. How does it compare to the 2020 great conjunction?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.\n\nproblem:\nc. By empirically fitting a sinusoidal function (which is assumed to be the same for each track, just with a fixed phase difference between them) and assuming all conjunctions are separated by the average synodic period, we can give rough estimations for the separations of any given great conjunction. Note: be careful as your calculations will be very sensitive to rounding errors.\n\niii. State which track the 'Star of Bethlehem' great conjunction is on, and hence use the relevant equation to predict its separation. How does it compare to the 2020 great conjunction?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\circ, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-07.jpg?height=706&width=1564&top_left_y=834&top_left_x=244", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-08.jpg?height=578&width=1566&top_left_y=196&top_left_x=242", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-09.jpg?height=1072&width=1564&top_left_y=1191&top_left_x=246" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\circ" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_107", "problem": "2017 年, 人类第一次直接探测到来自双中子星合并的引力波。根据科学家们复原的过程, 在两颗中子星合并前约 $100 \\mathrm{~s}$ 时, 它们相距约 $400 \\mathrm{~km}$, 绕二者连线上的某点每秒转动 12 圈, 将两颗中子星都看作是质量均匀分布的球体。由这些数据、万有引力常量并利用牛顿力学知识, 估算一下, 当两颗中子星相距约 $1600 \\mathrm{~km}$ 时, 绕二者连线上的某点每秒转动（ ）\nA: 0.5 圈\nB: 1.5 圈\nC: 2.5 圈\nD: 3.5 圈\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2017 年, 人类第一次直接探测到来自双中子星合并的引力波。根据科学家们复原的过程, 在两颗中子星合并前约 $100 \\mathrm{~s}$ 时, 它们相距约 $400 \\mathrm{~km}$, 绕二者连线上的某点每秒转动 12 圈, 将两颗中子星都看作是质量均匀分布的球体。由这些数据、万有引力常量并利用牛顿力学知识, 估算一下, 当两颗中子星相距约 $1600 \\mathrm{~km}$ 时, 绕二者连线上的某点每秒转动（ ）\n\nA: 0.5 圈\nB: 1.5 圈\nC: 2.5 圈\nD: 3.5 圈\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_948", "problem": "Special Relativity (SR) tells us that two observers will disagree about the duration of a time interval measured by each one's clock if one is moving at speed $v$ relative to the other, a phenomenon called time dilation. General Relativity (GR) tells us that gravitational fields dilate time too. This has an impact on satellites, since they travel at high orbital speeds (slowing down their clocks relative to the surface) but due to their altitude they are in a weaker gravitational field (speeding up their clocks relative to the surface). Which effect is dominant varies with orbital radius. Global Positioning System (GPS) satellites must compensate for this effect, since the satellites rely on accurate measurements of the time between sending and receiving a radio signal.\n\n[figure1]\n\nFigure 4: A scale diagram of the positions of the orbits for the International Space Station (ISS), GPS satellites and geostationary satellites, along with their orbital periods\n\nIn $\\mathrm{SR}$, time dilation can be calculated with\n\n$$\nt^{\\prime}=\\gamma t_{0} \\quad \\text { where } \\quad \\gamma=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}} \\quad \\text { so } \\quad \\Delta t_{\\mathrm{SR}}=t_{0}-t^{\\prime}=(1-\\gamma) t_{0}\n$$\n\nwhere $t_{0}$ is the time measured by the moving clock, $t^{\\prime}$ is the time measured by the observer, $c$ is the speed of light and $v$ is the speed of the object. A negative $\\Delta t$ indicates that the clocks are passing time slower relative to the observer, whilst a positive indicates they are passing quicker.\n\nHence calculate the daily offset in GPS positions if relativity was not taken into account.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nSpecial Relativity (SR) tells us that two observers will disagree about the duration of a time interval measured by each one's clock if one is moving at speed $v$ relative to the other, a phenomenon called time dilation. General Relativity (GR) tells us that gravitational fields dilate time too. This has an impact on satellites, since they travel at high orbital speeds (slowing down their clocks relative to the surface) but due to their altitude they are in a weaker gravitational field (speeding up their clocks relative to the surface). Which effect is dominant varies with orbital radius. Global Positioning System (GPS) satellites must compensate for this effect, since the satellites rely on accurate measurements of the time between sending and receiving a radio signal.\n\n[figure1]\n\nFigure 4: A scale diagram of the positions of the orbits for the International Space Station (ISS), GPS satellites and geostationary satellites, along with their orbital periods\n\nIn $\\mathrm{SR}$, time dilation can be calculated with\n\n$$\nt^{\\prime}=\\gamma t_{0} \\quad \\text { where } \\quad \\gamma=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}} \\quad \\text { so } \\quad \\Delta t_{\\mathrm{SR}}=t_{0}-t^{\\prime}=(1-\\gamma) t_{0}\n$$\n\nwhere $t_{0}$ is the time measured by the moving clock, $t^{\\prime}$ is the time measured by the observer, $c$ is the speed of light and $v$ is the speed of the object. A negative $\\Delta t$ indicates that the clocks are passing time slower relative to the observer, whilst a positive indicates they are passing quicker.\n\nHence calculate the daily offset in GPS positions if relativity was not taken into account.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of m, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-10.jpg?height=742&width=1236&top_left_y=791&top_left_x=410" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1069", "problem": "In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}{r}}$.e. Hence calculate the direct distance between Earth and Mars at the moment the spacecraft reaches Mars. How long would it take a radio message from the spacecraft to reach Earth?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}{r}}$.\n\nproblem:\ne. Hence calculate the direct distance between Earth and Mars at the moment the spacecraft reaches Mars. How long would it take a radio message from the spacecraft to reach Earth?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0c9b1562981df78a2b9dg-04.jpg?height=379&width=517&top_left_y=1341&top_left_x=310" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_455", "problem": "宇宙空间存在两颗质量分布均匀的球体未知星球, 经过发射绕表面运行的卫星发现,两个星球的近地卫星周期相等, 同学们据此做出如下判断, 则正确的是 ( )\nA: 这两个未知星球的体积一定相等\nB: 这两个未知星球的密度一定相等\nC: 这两个未知星球的质量若不等, 则表面的重力加速度一定不等\nD: 这两个未知星球质量大的，则其表面的重力加速度较小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇宙空间存在两颗质量分布均匀的球体未知星球, 经过发射绕表面运行的卫星发现,两个星球的近地卫星周期相等, 同学们据此做出如下判断, 则正确的是 ( )\n\nA: 这两个未知星球的体积一定相等\nB: 这两个未知星球的密度一定相等\nC: 这两个未知星球的质量若不等, 则表面的重力加速度一定不等\nD: 这两个未知星球质量大的，则其表面的重力加速度较小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1005", "problem": "Main sequence stars fuse hydrogen atoms to form helium in their cores. About $90 \\%$ of the stars in the Universe, including the Sun, are main sequence stars. These stars can range from about a tenth of the mass of the Sun to up to 200 times as massive. The main source of energy in main sequence stars is from nuclear fusion. The mass of one hydrogen nucleus is $m_{\\mathrm{H}}=1.674 \\times 10^{-27} \\mathrm{~kg}$, and the mass of one helium nucleus is $m_{\\mathrm{He}}=6.649 \\times 10^{-27} \\mathrm{~kg}$.\n[figure1]\n\nFigure 3: Left: The proton-proton chain reaction is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates in stars with masses less than or equal to that of the Sun's, and involves a net change of four hydrogen nuclei becoming one helium nucleus.\n\nRight: Only the core of a main sequence star will undergo nuclear fusion due to the higher temperature than the surrounding hydrogen shell.\nThe Sun is composed of about $71 \\%$ hydrogen, $27 \\%$ helium and some heavier elements. However, only $13 \\%$ of the hydrogen is available for hydrogen fusion in the core. The rest remains in layers of the Sun where the temperature is too low for fusion to occur. Use these figures and the answers to previous questions to calculate the lifetime in years of hydrogen fusion in the Sun (called its main sequence lifetime), assuming the Sun's luminosity remains constant.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nMain sequence stars fuse hydrogen atoms to form helium in their cores. About $90 \\%$ of the stars in the Universe, including the Sun, are main sequence stars. These stars can range from about a tenth of the mass of the Sun to up to 200 times as massive. The main source of energy in main sequence stars is from nuclear fusion. The mass of one hydrogen nucleus is $m_{\\mathrm{H}}=1.674 \\times 10^{-27} \\mathrm{~kg}$, and the mass of one helium nucleus is $m_{\\mathrm{He}}=6.649 \\times 10^{-27} \\mathrm{~kg}$.\n[figure1]\n\nFigure 3: Left: The proton-proton chain reaction is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates in stars with masses less than or equal to that of the Sun's, and involves a net change of four hydrogen nuclei becoming one helium nucleus.\n\nRight: Only the core of a main sequence star will undergo nuclear fusion due to the higher temperature than the surrounding hydrogen shell.\nThe Sun is composed of about $71 \\%$ hydrogen, $27 \\%$ helium and some heavier elements. However, only $13 \\%$ of the hydrogen is available for hydrogen fusion in the core. The rest remains in layers of the Sun where the temperature is too low for fusion to occur. Use these figures and the answers to previous questions to calculate the lifetime in years of hydrogen fusion in the Sun (called its main sequence lifetime), assuming the Sun's luminosity remains constant.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of years, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_13148c5721a741e30941g-08.jpg?height=606&width=1400&top_left_y=785&top_left_x=356" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "years" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_833", "problem": "A supernova is triggered largely by neutrinos. In fact, $99 \\%$ of the energy coming from the supernova is released in form of neutrinos. Over a time span of about three months, the supernova outputs visible light with power equivalent to 10 billion Suns. Assuming supernova neutrinos have mean energy of around $10 \\mathrm{MeV}$, that all the power of the supernova is released during the time it is visible, and that all of the power released is released in the form of either visible light or neutrinos, estimate the number of neutrinos released.\nA: $10^{54}$\nB: $10^{55}$\nC: $10^{50}$\nD: $10^{57}$\nE: $10^{60}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA supernova is triggered largely by neutrinos. In fact, $99 \\%$ of the energy coming from the supernova is released in form of neutrinos. Over a time span of about three months, the supernova outputs visible light with power equivalent to 10 billion Suns. Assuming supernova neutrinos have mean energy of around $10 \\mathrm{MeV}$, that all the power of the supernova is released during the time it is visible, and that all of the power released is released in the form of either visible light or neutrinos, estimate the number of neutrinos released.\n\nA: $10^{54}$\nB: $10^{55}$\nC: $10^{50}$\nD: $10^{57}$\nE: $10^{60}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_7205fccc557018644b5cg-05.jpg?height=637&width=1098&top_left_y=1018&top_left_x=492" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_415", "problem": "2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为第一个首次探测火星就实现“绕、落、巡”任务的国家。为了简化问题，可认为地球和火星在同一平面上绕太阳做匀速圆周运动, 如图 1 所示。已知地球的公转周期为 $T_{1}$, 公转轨道半径为 $r_{1}$, 火星的公转周期为 $T_{2}$, 火星质量为 $M$ 。如图 2 所示, 以火星为参考系, 质量为 $m_{1}$ 的探测器沿 1 号轨道到达 $B$ 点时速度为 $v_{1}, B$ 点到火星球心的距离为 $r_{3}$, 此时启动发动机, 在极短时间内一次性喷出部分气体, 喷气后探测器质量变为 $m_{2}$ 、速度变为与 $v_{1}$ 垂直的 $v_{2}$,然后进入以 $B$ 点为远火点的椭圆轨道 2 。已知万有引力势能公式 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $M$为中心天体的质量, $m$ 为卫星的质量, $G$ 为引力常量, $r$ 为卫星到中心天体球心的距离。求\n火星公转轨道半径 $r_{2}$;[图1]\n\n图1\n\n[图2]\n\n图2", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为第一个首次探测火星就实现“绕、落、巡”任务的国家。为了简化问题，可认为地球和火星在同一平面上绕太阳做匀速圆周运动, 如图 1 所示。已知地球的公转周期为 $T_{1}$, 公转轨道半径为 $r_{1}$, 火星的公转周期为 $T_{2}$, 火星质量为 $M$ 。如图 2 所示, 以火星为参考系, 质量为 $m_{1}$ 的探测器沿 1 号轨道到达 $B$ 点时速度为 $v_{1}, B$ 点到火星球心的距离为 $r_{3}$, 此时启动发动机, 在极短时间内一次性喷出部分气体, 喷气后探测器质量变为 $m_{2}$ 、速度变为与 $v_{1}$ 垂直的 $v_{2}$,然后进入以 $B$ 点为远火点的椭圆轨道 2 。已知万有引力势能公式 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $M$为中心天体的质量, $m$ 为卫星的质量, $G$ 为引力常量, $r$ 为卫星到中心天体球心的距离。求\n火星公转轨道半径 $r_{2}$;[图1]\n\n图1\n\n[图2]\n\n图2\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-006.jpg?height=452&width=534&top_left_y=1493&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-006.jpg?height=451&width=911&top_left_y=1488&top_left_x=881" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_985", "problem": "On the 1st January 2019 the New Horizons probe (having successfully flown by Pluto in 2015) had an encounter with the Kuiper belt object Ultima Thule, making it the most distant object ever visited by a spacecraft. At the time it was 43.4 au from the Sun. Given the apparent magnitude of the Sun from the Earth is - 26.74, what is the apparent magnitude of the Sun from Ultima Thule?\nA: -18.25\nB: -18.35\nC: -18.45\nD: -18.55\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOn the 1st January 2019 the New Horizons probe (having successfully flown by Pluto in 2015) had an encounter with the Kuiper belt object Ultima Thule, making it the most distant object ever visited by a spacecraft. At the time it was 43.4 au from the Sun. Given the apparent magnitude of the Sun from the Earth is - 26.74, what is the apparent magnitude of the Sun from Ultima Thule?\n\nA: -18.25\nB: -18.35\nC: -18.45\nD: -18.55\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_601", "problem": "如图甲所示, 小明在地球表面进行了物体在坚直方向做直线运动的实验, 弹簧原长时, 小球由静止释放, 在弹簧弹力与重力作用下, 测得小球的加速度 $a$ 与位移 $x$ 的关系图像如图乙所示。已知弹簧的劲度系数为 $k$, 地球的半径为 $R$, 万有引力常量为 $G$, 不考虑地球自转影响, 忽略空气阻力, 下列说法正确的是（ )\n[图1]\n\n甲\n\n[图2]\n\n乙\nA: 小球的位移为 $x_{0}$ 时, 小球正好处于完全失重状态\nB: 小球的最大速度为 $\\sqrt{a_{0} x_{0}}$\nC: 小球的质量为 $\\frac{k x_{0}}{2 a_{0}}$\nD: 地球的密度为 $\\frac{3 a_{0}}{2 \\pi G R}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图甲所示, 小明在地球表面进行了物体在坚直方向做直线运动的实验, 弹簧原长时, 小球由静止释放, 在弹簧弹力与重力作用下, 测得小球的加速度 $a$ 与位移 $x$ 的关系图像如图乙所示。已知弹簧的劲度系数为 $k$, 地球的半径为 $R$, 万有引力常量为 $G$, 不考虑地球自转影响, 忽略空气阻力, 下列说法正确的是（ )\n[图1]\n\n甲\n\n[图2]\n\n乙\n\nA: 小球的位移为 $x_{0}$ 时, 小球正好处于完全失重状态\nB: 小球的最大速度为 $\\sqrt{a_{0} x_{0}}$\nC: 小球的质量为 $\\frac{k x_{0}}{2 a_{0}}$\nD: 地球的密度为 $\\frac{3 a_{0}}{2 \\pi G R}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-018.jpg?height=257&width=222&top_left_y=203&top_left_x=323", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-018.jpg?height=303&width=346&top_left_y=157&top_left_x=615" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1034", "problem": "Why are there two tides every day?\nA: The Moon causes the one at night and the Sun causes the one during the day\nB: The Earth and Moon orbit a common centre of mass\nC: The Moon is tidally locked to the Earth\nD: Water waves can only travel around the Earth in 12 hours\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhy are there two tides every day?\n\nA: The Moon causes the one at night and the Sun causes the one during the day\nB: The Earth and Moon orbit a common centre of mass\nC: The Moon is tidally locked to the Earth\nD: Water waves can only travel around the Earth in 12 hours\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_315", "problem": "如图所示, 横截面积为 $A$ 、质量为 $m$ 的柱状飞行器沿半径为 $R$ 的圆形轨道在高空绕地球做无动力运行。将地球看作质量为 $M$ 的均匀球体。万有引力常量为 $G$ 。\n在飞行器运行轨道附近范围内有密度为 $\\rho$ (恒量) 的稀薄空气。稀薄空气可看成是由彼此没有相互作用的均匀小颗粒组成，所有小颗粒原来都静止。假设每个小颗粒与飞行器碰撞后具有与飞行器相同的速度, 且碰撞时间很短。频繁碰撞会对飞行器产生持续阻力, 飞行器的轨道高度会逐渐降低。观察发现飞行器绕地球运行很多圈之后, 其轨道高度下降了 $\\Delta H$ 。由于 $\\Delta H \\ll R$, 可将飞行器绕地球运动的每一圈运动均视为匀速圆周运动。已知当飞行器到地球球心距离为 $r$ 时, 飞行器与地球组成的系统具有的引力势能 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$ 。请根据上述条件推导:\n\n飞行器在半径为 $R$ 轨道上运行时, 所受空气阻力大小 $F$ 的表达式;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 横截面积为 $A$ 、质量为 $m$ 的柱状飞行器沿半径为 $R$ 的圆形轨道在高空绕地球做无动力运行。将地球看作质量为 $M$ 的均匀球体。万有引力常量为 $G$ 。\n在飞行器运行轨道附近范围内有密度为 $\\rho$ (恒量) 的稀薄空气。稀薄空气可看成是由彼此没有相互作用的均匀小颗粒组成，所有小颗粒原来都静止。假设每个小颗粒与飞行器碰撞后具有与飞行器相同的速度, 且碰撞时间很短。频繁碰撞会对飞行器产生持续阻力, 飞行器的轨道高度会逐渐降低。观察发现飞行器绕地球运行很多圈之后, 其轨道高度下降了 $\\Delta H$ 。由于 $\\Delta H \\ll R$, 可将飞行器绕地球运动的每一圈运动均视为匀速圆周运动。已知当飞行器到地球球心距离为 $r$ 时, 飞行器与地球组成的系统具有的引力势能 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$ 。请根据上述条件推导:\n\n飞行器在半径为 $R$ 轨道上运行时, 所受空气阻力大小 $F$ 的表达式;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-078.jpg?height=651&width=468&top_left_y=177&top_left_x=360" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_678", "problem": "我国首次发射的火星探测器“天问一号”自 2020 年 7 月 23 日成功发射入轨后, 2021 年 2 月 10 日成功被火星捕获，顺利进入环火轨道; 5 月 15 日, “天问一号”着陆巡视器顺利软着陆于火星表面。关于“天问一号”的运行，可以简化为如图所示的模型: “天问一号”先绕火星做半径为 $R_{1}$ 、周期为 $T$ 的匀速圆周运动, 在某一位置 $A$ 点改变速度,使其轨道变为粗圆, 椭圆轨道在 $B$ 点与火星表面相切, 设法使着陆巡视器落在火星上。若火星的半径为 $R_{2}$, 则下列说法正确的是 ( )\n\n[图1]\nA: “天问一号”从圆轨道变为椭圆轨道, 机械能增加\nB: “天问一号”在圆轨道和椭圆轨道上 $A$ 点的加速度相同\nC: “天问一号”从椭圆轨道的 $A$ 点运动到 $B$ 点所需的时间为 $\\frac{T}{2} \\sqrt{\\left(\\frac{R_{1}+R_{2}}{2 R_{1}}\\right)^{3}}$\nD: “天问一号”在椭圆轨道 $B$ 点的速度等于火星的第一宇宙速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n我国首次发射的火星探测器“天问一号”自 2020 年 7 月 23 日成功发射入轨后, 2021 年 2 月 10 日成功被火星捕获，顺利进入环火轨道; 5 月 15 日, “天问一号”着陆巡视器顺利软着陆于火星表面。关于“天问一号”的运行，可以简化为如图所示的模型: “天问一号”先绕火星做半径为 $R_{1}$ 、周期为 $T$ 的匀速圆周运动, 在某一位置 $A$ 点改变速度,使其轨道变为粗圆, 椭圆轨道在 $B$ 点与火星表面相切, 设法使着陆巡视器落在火星上。若火星的半径为 $R_{2}$, 则下列说法正确的是 ( )\n\n[图1]\n\nA: “天问一号”从圆轨道变为椭圆轨道, 机械能增加\nB: “天问一号”在圆轨道和椭圆轨道上 $A$ 点的加速度相同\nC: “天问一号”从椭圆轨道的 $A$ 点运动到 $B$ 点所需的时间为 $\\frac{T}{2} \\sqrt{\\left(\\frac{R_{1}+R_{2}}{2 R_{1}}\\right)^{3}}$\nD: “天问一号”在椭圆轨道 $B$ 点的速度等于火星的第一宇宙速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-017.jpg?height=394&width=371&top_left_y=177&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_787", "problem": "What is the name of the JWST component highlighted below?\n\n[figure1]\nA: Primary mirror\nB: Secondary mirror\nC: Optics subsystem\nD: Antenna\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the name of the JWST component highlighted below?\n\n[figure1]\n\nA: Primary mirror\nB: Secondary mirror\nC: Optics subsystem\nD: Antenna\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_620a57bf13ecc39e0534g-2.jpg?height=403&width=514&top_left_y=278&top_left_x=791" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_291", "problem": "“神舟八号”与“天宫一号”对接前各自绕地球运动, 设“天宫一号”在半径为 $r_{I}$ 的圆轨道上运动, 周期为 $T_{1}$, “神舟八号”在半径为 $r_{2}$ 的圆轨道上运动, $r_{1}>r_{2}$, 则 ( )\nA: “神舟八号”的周期 $T_{2}=T_{I} \\sqrt{\\frac{r_{2}{ }^{3}}{r_{1}^{3}}}$\nB: “天宫一号”的运行速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 地球表面的重力加速度 $\\mathrm{g}=\\frac{4 \\pi^{2} r_{1}}{T_{1}^{2}}$\nD: 地球的质量 $\\mathrm{M}=\\frac{4 \\pi^{2} r_{1}^{3}}{G T_{1}^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n“神舟八号”与“天宫一号”对接前各自绕地球运动, 设“天宫一号”在半径为 $r_{I}$ 的圆轨道上运动, 周期为 $T_{1}$, “神舟八号”在半径为 $r_{2}$ 的圆轨道上运动, $r_{1}>r_{2}$, 则 ( )\n\nA: “神舟八号”的周期 $T_{2}=T_{I} \\sqrt{\\frac{r_{2}{ }^{3}}{r_{1}^{3}}}$\nB: “天宫一号”的运行速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 地球表面的重力加速度 $\\mathrm{g}=\\frac{4 \\pi^{2} r_{1}}{T_{1}^{2}}$\nD: 地球的质量 $\\mathrm{M}=\\frac{4 \\pi^{2} r_{1}^{3}}{G T_{1}^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_964", "problem": "When two objects of unequal mass orbit around each other, they both orbit around a barycentre - this is the name given to the location of the centre of mass of the system. The masses of both objects, and the distance between their centres, affects the position of their barycentre. Imagine two objects, Object 1 and Object 2, with masses $m_{1}$ and $m_{2}$ respectively, and the average distance between the centre of both objects is $a$, then the average distance from the centre of Object 1 to the barycentre, $r$, is given by the formula:\n\n$$\nr=a \\frac{m_{2}}{m_{1}+m_{2}}\n$$\n\nSome have claimed that a double planet should be distinguished from a planet and large moon when a system fulfils the criterion $r>R_{1}$. The Earth-Moon system does not currently satisfy that condition for a double planet despite the Moon being rather large relative to the Earth, but the Moon is slowly moving away from the Earth at roughly $4 \\mathrm{~cm}$ per year. Assuming this rate stays constant, calculate the number of years until $r=\\mathrm{R}_{\\mathrm{E}}$. [Average distance between centres of Earth and Moon $=384400 \\mathrm{~km}$, and the Earth has 83.1 times the mass of the Moon.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nWhen two objects of unequal mass orbit around each other, they both orbit around a barycentre - this is the name given to the location of the centre of mass of the system. The masses of both objects, and the distance between their centres, affects the position of their barycentre. Imagine two objects, Object 1 and Object 2, with masses $m_{1}$ and $m_{2}$ respectively, and the average distance between the centre of both objects is $a$, then the average distance from the centre of Object 1 to the barycentre, $r$, is given by the formula:\n\n$$\nr=a \\frac{m_{2}}{m_{1}+m_{2}}\n$$\n\nSome have claimed that a double planet should be distinguished from a planet and large moon when a system fulfils the criterion $r>R_{1}$. The Earth-Moon system does not currently satisfy that condition for a double planet despite the Moon being rather large relative to the Earth, but the Moon is slowly moving away from the Earth at roughly $4 \\mathrm{~cm}$ per year. Assuming this rate stays constant, calculate the number of years until $r=\\mathrm{R}_{\\mathrm{E}}$. [Average distance between centres of Earth and Moon $=384400 \\mathrm{~km}$, and the Earth has 83.1 times the mass of the Moon.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of years, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "years" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_368", "problem": "卫星携带一探测器在半径为 $4 R$ 的圆轨道 $I$ 上绕地球做匀速圆周运动。在 $\\mathrm{A}$ 点, 卫星上的辅助动力装置短暂工作, 将探测器沿运动方向射出 (设辅助动力装置喷出的气体质量可忽略)。若探测器恰能完全脱离地球的引力范围, 即到达距地球无限远时的速度恰好为零, 而卫星沿新的椭圆轨道II运动, 如图所示, $A 、 B$ 两点分别是其椭圆轨道II的远地点和近地点（卫星通过 $A 、 B$ 两点时的线速度大小与其距地心距离的乘积相等）。地球质量为 $M$, 探测器的质量为 $m$, 卫星的质量为 $\\sqrt{2} m$, 地球半径为 $R$, 引力常量为 $G$, 已知质量分别为 $m_{1} 、 m_{2}$ 的两个质点相距为 $r$ 时, 它们之间的引力势能为 $E_{p}=-\\frac{G m_{1} m_{2}}{r}$, 求:\n\n卫星运行到近地点 $B$ 时距地心的距离 $a$ 。\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n卫星携带一探测器在半径为 $4 R$ 的圆轨道 $I$ 上绕地球做匀速圆周运动。在 $\\mathrm{A}$ 点, 卫星上的辅助动力装置短暂工作, 将探测器沿运动方向射出 (设辅助动力装置喷出的气体质量可忽略)。若探测器恰能完全脱离地球的引力范围, 即到达距地球无限远时的速度恰好为零, 而卫星沿新的椭圆轨道II运动, 如图所示, $A 、 B$ 两点分别是其椭圆轨道II的远地点和近地点（卫星通过 $A 、 B$ 两点时的线速度大小与其距地心距离的乘积相等）。地球质量为 $M$, 探测器的质量为 $m$, 卫星的质量为 $\\sqrt{2} m$, 地球半径为 $R$, 引力常量为 $G$, 已知质量分别为 $m_{1} 、 m_{2}$ 的两个质点相距为 $r$ 时, 它们之间的引力势能为 $E_{p}=-\\frac{G m_{1} m_{2}}{r}$, 求:\n\n卫星运行到近地点 $B$ 时距地心的距离 $a$ 。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-105.jpg?height=409&width=411&top_left_y=1683&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1027", "problem": "Figure 4 below is a composite image which depicts a transit of the International Space Station (ISS) across the disc of the Sun. The image comprises 26 individual photographs which were taken at regular time intervals during the transit. The total duration of the transit was less than one second. In this question we will ignore any effects caused by the rotation of the Earth.\n\n[figure1]\n\nFigure 4: A composite of a selection of the frames taken with a high-speed camera of a transit of the ISS in front of the Sun, taken from Northamptonshire at 10:22 BST on $17^{\\text {th }}$ June 2022. Credit: Jamie Cooper Photography\n\nThe ISS maintained a mean height of $415 \\mathrm{~km}$ above the surface of the Earth during the transit. What is the angle $\\theta_{2}$ subtended by the ISS between the first and last photographs, as viewed from the centre of the Earth? [Assume that the Sun was at the zenith (directly overhead) during the transit.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nFigure 4 below is a composite image which depicts a transit of the International Space Station (ISS) across the disc of the Sun. The image comprises 26 individual photographs which were taken at regular time intervals during the transit. The total duration of the transit was less than one second. In this question we will ignore any effects caused by the rotation of the Earth.\n\n[figure1]\n\nFigure 4: A composite of a selection of the frames taken with a high-speed camera of a transit of the ISS in front of the Sun, taken from Northamptonshire at 10:22 BST on $17^{\\text {th }}$ June 2022. Credit: Jamie Cooper Photography\n\nThe ISS maintained a mean height of $415 \\mathrm{~km}$ above the surface of the Earth during the transit. What is the angle $\\theta_{2}$ subtended by the ISS between the first and last photographs, as viewed from the centre of the Earth? [Assume that the Sun was at the zenith (directly overhead) during the transit.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-08.jpg?height=831&width=1588&top_left_y=738&top_left_x=240", "https://i.postimg.cc/BbpCRYJm/Screenshot-2024-04-06-at-19-58-05.png" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_652", "problem": "一宇航员到达半径为 $R$ 、密度均匀的某星球表面, 做了如下实验: 用不可伸长的轻绳拴一质量为 $m$ 的小球, 上端固定于 $O$ 点, 如图甲所示, 在最低点给小球某一初速度,使其绕 $O$ 点的坚直面内做圆周运动, 测得绳的拉力 $F$ 大小随时间 $t$ 的变化规律如图乙所示, $F_{1}=3 F_{2}$, 设 $R 、 m$ 、引力常量 $G$ 和 $F_{1}$ 为已知量, 忽略各种阻力。下列说法正确的是\n\n[图1]\n\n甲\n\n[图2]\n\n乙\nA: 该星球表面的重力加速度为 $\\frac{F_{1}}{9 m}$\nB: 星球的密度为 $\\frac{F_{1}}{6 \\pi G R m}$\nC: 卫星绕该星球运行的最小周期为 $2 \\pi \\sqrt{\\frac{R m}{F_{1}}}$\nD: 该星球的第一宇宙速度为 $\\sqrt{\\frac{2 F_{1} R}{9 m}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n一宇航员到达半径为 $R$ 、密度均匀的某星球表面, 做了如下实验: 用不可伸长的轻绳拴一质量为 $m$ 的小球, 上端固定于 $O$ 点, 如图甲所示, 在最低点给小球某一初速度,使其绕 $O$ 点的坚直面内做圆周运动, 测得绳的拉力 $F$ 大小随时间 $t$ 的变化规律如图乙所示, $F_{1}=3 F_{2}$, 设 $R 、 m$ 、引力常量 $G$ 和 $F_{1}$ 为已知量, 忽略各种阻力。下列说法正确的是\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\nA: 该星球表面的重力加速度为 $\\frac{F_{1}}{9 m}$\nB: 星球的密度为 $\\frac{F_{1}}{6 \\pi G R m}$\nC: 卫星绕该星球运行的最小周期为 $2 \\pi \\sqrt{\\frac{R m}{F_{1}}}$\nD: 该星球的第一宇宙速度为 $\\sqrt{\\frac{2 F_{1} R}{9 m}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-025.jpg?height=243&width=232&top_left_y=2157&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-025.jpg?height=352&width=689&top_left_y=2094&top_left_x=612" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_405", "problem": "2021 年 6 月 17 日, 神舟十二号载人飞船与天和核心舱完成对接, 航天员聂海胜、刘伯明、汤洪波进入天和核心舱，标志着中国人首次进入了自己的空间站。对接过程的示意图如图所示, 天和核心舱处于半径为 $r_{3}$ 的圆轨道III; 神舟十二号飞船处于半径为 $r_{I}$的圆轨道I, 运行周期为 $T_{I}$, 通过变轨操作后, 沿粗圆轨道II运动到 $B$ 点与天和核心舱对接。则下列说法正确的是（）\n[图1]\nA: 神舟十二号飞船在轨道II上运动时经 $A 、 B$ 两点速率 $v_{A}: v_{B}=r_{1}: r_{3}$\nB: 神舟十二号飞船沿轨道II运行的周期为 $T_{2}=T_{1} \\sqrt{\\left(\\frac{r_{1}+r_{3}}{2 r_{1}}\\right)^{3}}$\nC: 神舟十二号飞船沿轨道I运行的周期大于天和核心舱沿轨道III运行的周期\nD: 正常运行时, 神舟十二号飞船在轨道II上经过 $B$ 点的加速度大于在轨道III上经过 $B$ 点的加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2021 年 6 月 17 日, 神舟十二号载人飞船与天和核心舱完成对接, 航天员聂海胜、刘伯明、汤洪波进入天和核心舱，标志着中国人首次进入了自己的空间站。对接过程的示意图如图所示, 天和核心舱处于半径为 $r_{3}$ 的圆轨道III; 神舟十二号飞船处于半径为 $r_{I}$的圆轨道I, 运行周期为 $T_{I}$, 通过变轨操作后, 沿粗圆轨道II运动到 $B$ 点与天和核心舱对接。则下列说法正确的是（）\n[图1]\n\nA: 神舟十二号飞船在轨道II上运动时经 $A 、 B$ 两点速率 $v_{A}: v_{B}=r_{1}: r_{3}$\nB: 神舟十二号飞船沿轨道II运行的周期为 $T_{2}=T_{1} \\sqrt{\\left(\\frac{r_{1}+r_{3}}{2 r_{1}}\\right)^{3}}$\nC: 神舟十二号飞船沿轨道I运行的周期大于天和核心舱沿轨道III运行的周期\nD: 正常运行时, 神舟十二号飞船在轨道II上经过 $B$ 点的加速度大于在轨道III上经过 $B$ 点的加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-069.jpg?height=534&width=1144&top_left_y=1321&top_left_x=342" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_839", "problem": "In a certain day, when it is 0h UT, the sidereal time of Prime Meridian is 5h 56min 9.4s. For this day, with start and end based on UT, find the civil time of Chicago, whose longitude and time zone are respectively, $87.65004722^{\\circ} \\mathrm{W}$ and UT-6, when the sidereal time there is $20 \\mathrm{~h}$. The difference between solar time and sidereal time SHOULD be accounted for.\nA: $14 \\mathrm{~h} 1 \\mathrm{~min} 32 \\mathrm{~s}$\nB: $13 \\mathrm{~h} 26 \\mathrm{~min} 17 \\mathrm{~s}$\nC: $14 \\mathrm{~h} 36 \\mathrm{~min} 47 \\mathrm{~s}$\nD: $14 \\mathrm{~h} 0 \\mathrm{~min} 43 \\mathrm{~s}$\nE: $13 \\mathrm{~h} 51 \\mathrm{~min} 11 \\mathrm{~s}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn a certain day, when it is 0h UT, the sidereal time of Prime Meridian is 5h 56min 9.4s. For this day, with start and end based on UT, find the civil time of Chicago, whose longitude and time zone are respectively, $87.65004722^{\\circ} \\mathrm{W}$ and UT-6, when the sidereal time there is $20 \\mathrm{~h}$. The difference between solar time and sidereal time SHOULD be accounted for.\n\nA: $14 \\mathrm{~h} 1 \\mathrm{~min} 32 \\mathrm{~s}$\nB: $13 \\mathrm{~h} 26 \\mathrm{~min} 17 \\mathrm{~s}$\nC: $14 \\mathrm{~h} 36 \\mathrm{~min} 47 \\mathrm{~s}$\nD: $14 \\mathrm{~h} 0 \\mathrm{~min} 43 \\mathrm{~s}$\nE: $13 \\mathrm{~h} 51 \\mathrm{~min} 11 \\mathrm{~s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1092", "problem": "In July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.b. In reality, the effects of air resistance and the weight of the rocket are substantial. Once in the parking orbit it is travelling at $7.79 \\mathrm{~km} \\mathrm{~s}^{-1}$.\n\ni. What is its height above the Earth's surface (measured from sea level)? Give it in km.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.\n\nproblem:\nb. In reality, the effects of air resistance and the weight of the rocket are substantial. Once in the parking orbit it is travelling at $7.79 \\mathrm{~km} \\mathrm{~s}^{-1}$.\n\ni. What is its height above the Earth's surface (measured from sea level)? Give it in km.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~km}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-04.jpg?height=1010&width=1508&top_left_y=543&top_left_x=271", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-06.jpg?height=800&width=1586&top_left_y=518&top_left_x=240" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~km}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_14", "problem": "如图所示, $A$ 为地球赤道表面的物体, $B$ 为环绕地球运行的卫星, 此卫星在距离地球表面 $\\frac{R}{2}$ 的高度处做匀速圆周运动, 且向心加速度的大小为 $a$, 地球的半径为 $R$,引力常量为 $G$. 则下列说法正确的是 ( )\n\n[图1]\nA: 物体 $A$ 的向心加速度大于 $a$\nB: 物体 $A$ 的线速度比卫星 $B$ 的线速度大\nC: 地球的质量为 $\\frac{R^{2} a}{G}$\nD: 地球两极的重力加速度大小为 $\\frac{9}{4} a$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, $A$ 为地球赤道表面的物体, $B$ 为环绕地球运行的卫星, 此卫星在距离地球表面 $\\frac{R}{2}$ 的高度处做匀速圆周运动, 且向心加速度的大小为 $a$, 地球的半径为 $R$,引力常量为 $G$. 则下列说法正确的是 ( )\n\n[图1]\n\nA: 物体 $A$ 的向心加速度大于 $a$\nB: 物体 $A$ 的线速度比卫星 $B$ 的线速度大\nC: 地球的质量为 $\\frac{R^{2} a}{G}$\nD: 地球两极的重力加速度大小为 $\\frac{9}{4} a$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-86.jpg?height=328&width=377&top_left_y=1178&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1146", "problem": "Plotting the position of the Sun in the sky at the same time every day, you get an interesting figure-ofeight shape known as an analemma (see Figure 1). For observers in the Northern hemisphere, you might expect to always see the Sun due South at midday, however on some days the Sun has already passed through that bearing and on others it needs a few more minutes before it gets there. This is due to two effects: the axial tilt of the Earth, and the fact the Earth's orbit is not perfectly circular\n\n[figure1]\n\nFigure 1: The analemma above was composed from images taken every few days at noon near the village of Callanish in the Outer Hebrides in Scotland. In the foreground are the Callanish Stones and the main photo was taken on the winter solstice (when the maximum angle the Sun reaches above the horizon is the lowest of the year, so is at the bottom of the analemma). Credit: Giuseppe Petricca.\n\nThe vertical co-ordinate of a point in the analemma is entirely determined by the Earth's axial tilt. This is known as the solar declination, $\\delta$, and varies sinusoidally throughout the year. The horizontal coordinate of a point in the analemma is determined by a combination of the Earth's axial tilt and the eccentricity of the Earth's orbit. Both of these individually vary sinusoidally, but the superposition of the two is no longer sinusoidal.\n\nWe will define $\\alpha$ as the angle between due South and the Sun at local midday as seen from Oxford, where a positive value means the Sun has already passed through due South (so is on the right of the figure above) whilst a negative value means the Sun has yet to pass through due South. If $\\alpha_{\\text {tilt }}$ is the contribution due to the axial tilt and $\\alpha_{\\text {ecc }}$ is the contribution due to the Earth's orbital eccentricity, then\n\n$$\n\\alpha=\\alpha_{\\text {tilt }}+\\alpha_{\\text {ecc }}\n$$\n\nIf the angle of the axial tilt is $\\varepsilon$ and the eccentricity of the Earth's orbit is $e$, and we assume that both are small enough that the sinusoidal approximation of $\\delta, \\alpha_{\\text {tilt }}$, and $\\alpha_{\\text {ecc }}$ apply, then we find the following boundary conditions:\n\n- $\\delta$ has a period of 1 year, an amplitude of $\\varepsilon$, is maximum at the summer solstice (21 $21^{\\text {st }}$ June) and minimum at the winter solstice $\\left(21^{\\text {st }}\\right.$ December $)$\n- $\\alpha_{\\text {tilt }}$ has a period of 0.5 years, an amplitude (in radians) of $\\tan ^{2}(\\varepsilon / 2)$, is zero at the solstices and the equinoxes (vernal equinox $=21^{\\text {st }}$ March, autumnal equinox $=21^{\\text {st }}$ September), and (using our sign convention) positive just after the vernal equinox\n- $\\alpha_{\\text {ecc }}$ has a period of 1 year, an amplitude (in radians) of $2 e$, is zero at the perihelion (4 $4^{\\text {th }}$ January) and the aphelion ( $6^{\\text {th }}$ July), and (using our sign convention) negative just after the perihelion\n\nGiven the $n^{\\text {th }}$ day of the year, a value can be calculated for $\\delta$ and $\\alpha$, and these are the co-ordinates for the analemma (it is drawn by these parametric equations). For the Earth, $\\varepsilon=23.44^{\\circ}$ and $e=0.0167$.\n\nConsider an alternative version of Earth, known as Earth 2.0. On this planet, the year is unchanged and the perihelion and aphelion are at the same time, but it has a different axial tilt, a different orbital eccentricity, and a different month for the vernal equinox (although it is still on the $21^{\\text {st }}$ day of that month). The analemma as viewed from Earth 2.0 is show in Figure 2 below.\n\n[figure2]\n\nFigure 2: The analemma of the Sun at midday as seen by an observer on Earth 2.0. In this situation, $\\alpha$ ranges from -26 mins 47 secs to 18 mins 56 secs. The circled letters correspond to the same (unknown) day of each month (for example $5^{\\text {th }}$ Jan, $5^{\\text {th }}$ Feb, $5^{\\text {th }}$ March etc.). Credit: Bob Urschel.a. Although $\\alpha$ is really an angle in radians (where $2 \\pi$ radians $=360^{\\circ}$ ), it is normally more useful to convert it into time units (essentially the time since the Sun was due South, or the time until the Sun reaches due South). Taking the mean solar day to be exactly 24 hours:\n\ni. Convert the amplitude of $\\alpha_{\\text {tilt }}$ and $\\alpha_{\\text {ecc }}$ for the Earth into minutes.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nPlotting the position of the Sun in the sky at the same time every day, you get an interesting figure-ofeight shape known as an analemma (see Figure 1). For observers in the Northern hemisphere, you might expect to always see the Sun due South at midday, however on some days the Sun has already passed through that bearing and on others it needs a few more minutes before it gets there. This is due to two effects: the axial tilt of the Earth, and the fact the Earth's orbit is not perfectly circular\n\n[figure1]\n\nFigure 1: The analemma above was composed from images taken every few days at noon near the village of Callanish in the Outer Hebrides in Scotland. In the foreground are the Callanish Stones and the main photo was taken on the winter solstice (when the maximum angle the Sun reaches above the horizon is the lowest of the year, so is at the bottom of the analemma). Credit: Giuseppe Petricca.\n\nThe vertical co-ordinate of a point in the analemma is entirely determined by the Earth's axial tilt. This is known as the solar declination, $\\delta$, and varies sinusoidally throughout the year. The horizontal coordinate of a point in the analemma is determined by a combination of the Earth's axial tilt and the eccentricity of the Earth's orbit. Both of these individually vary sinusoidally, but the superposition of the two is no longer sinusoidal.\n\nWe will define $\\alpha$ as the angle between due South and the Sun at local midday as seen from Oxford, where a positive value means the Sun has already passed through due South (so is on the right of the figure above) whilst a negative value means the Sun has yet to pass through due South. If $\\alpha_{\\text {tilt }}$ is the contribution due to the axial tilt and $\\alpha_{\\text {ecc }}$ is the contribution due to the Earth's orbital eccentricity, then\n\n$$\n\\alpha=\\alpha_{\\text {tilt }}+\\alpha_{\\text {ecc }}\n$$\n\nIf the angle of the axial tilt is $\\varepsilon$ and the eccentricity of the Earth's orbit is $e$, and we assume that both are small enough that the sinusoidal approximation of $\\delta, \\alpha_{\\text {tilt }}$, and $\\alpha_{\\text {ecc }}$ apply, then we find the following boundary conditions:\n\n- $\\delta$ has a period of 1 year, an amplitude of $\\varepsilon$, is maximum at the summer solstice (21 $21^{\\text {st }}$ June) and minimum at the winter solstice $\\left(21^{\\text {st }}\\right.$ December $)$\n- $\\alpha_{\\text {tilt }}$ has a period of 0.5 years, an amplitude (in radians) of $\\tan ^{2}(\\varepsilon / 2)$, is zero at the solstices and the equinoxes (vernal equinox $=21^{\\text {st }}$ March, autumnal equinox $=21^{\\text {st }}$ September), and (using our sign convention) positive just after the vernal equinox\n- $\\alpha_{\\text {ecc }}$ has a period of 1 year, an amplitude (in radians) of $2 e$, is zero at the perihelion (4 $4^{\\text {th }}$ January) and the aphelion ( $6^{\\text {th }}$ July), and (using our sign convention) negative just after the perihelion\n\nGiven the $n^{\\text {th }}$ day of the year, a value can be calculated for $\\delta$ and $\\alpha$, and these are the co-ordinates for the analemma (it is drawn by these parametric equations). For the Earth, $\\varepsilon=23.44^{\\circ}$ and $e=0.0167$.\n\nConsider an alternative version of Earth, known as Earth 2.0. On this planet, the year is unchanged and the perihelion and aphelion are at the same time, but it has a different axial tilt, a different orbital eccentricity, and a different month for the vernal equinox (although it is still on the $21^{\\text {st }}$ day of that month). The analemma as viewed from Earth 2.0 is show in Figure 2 below.\n\n[figure2]\n\nFigure 2: The analemma of the Sun at midday as seen by an observer on Earth 2.0. In this situation, $\\alpha$ ranges from -26 mins 47 secs to 18 mins 56 secs. The circled letters correspond to the same (unknown) day of each month (for example $5^{\\text {th }}$ Jan, $5^{\\text {th }}$ Feb, $5^{\\text {th }}$ March etc.). Credit: Bob Urschel.\n\nproblem:\na. Although $\\alpha$ is really an angle in radians (where $2 \\pi$ radians $=360^{\\circ}$ ), it is normally more useful to convert it into time units (essentially the time since the Sun was due South, or the time until the Sun reaches due South). Taking the mean solar day to be exactly 24 hours:\n\ni. Convert the amplitude of $\\alpha_{\\text {tilt }}$ and $\\alpha_{\\text {ecc }}$ for the Earth into minutes.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\text { minutes }, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_9bba4f2e5c10ed29bb97g-04.jpg?height=1693&width=1470&top_left_y=550&top_left_x=293", "https://cdn.mathpix.com/cropped/2024_03_14_9bba4f2e5c10ed29bb97g-06.jpg?height=1207&width=1388&top_left_y=413&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\text { minutes }" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_348", "problem": "已知地球的半径为 $R$, 地球的自转周期为 $T$, 地表的重力加速度为 $g$, 要在地球赤道上发射一颗近地的人造地球卫星, 使其轨道在赤道的正上方, 若不计空气的阻力, 那么 ( )\nA: 向东发射与向西发射耗能相同, 均为 $\\frac{1}{2} m g R-\\frac{1}{2} m\\left(\\frac{2 \\pi R}{T}\\right)^{2}$\nB: 向东发射耗能多, 比向西发射耗能多 $\\frac{1}{2} m\\left(\\sqrt{g R}-\\frac{2 \\pi R}{T}\\right)^{2}$\nC: 向东发射与向西发射耗能相同, 均为 $\\frac{1}{2} m\\left(\\sqrt{g R}+\\frac{2 \\pi R}{T}\\right)^{2}$\nD: 向西发射耗能为 $\\frac{1}{2} m\\left(\\sqrt{g R}+\\frac{2 \\pi R}{T}\\right)^{2}$, 比向东发射耗能多\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n已知地球的半径为 $R$, 地球的自转周期为 $T$, 地表的重力加速度为 $g$, 要在地球赤道上发射一颗近地的人造地球卫星, 使其轨道在赤道的正上方, 若不计空气的阻力, 那么 ( )\n\nA: 向东发射与向西发射耗能相同, 均为 $\\frac{1}{2} m g R-\\frac{1}{2} m\\left(\\frac{2 \\pi R}{T}\\right)^{2}$\nB: 向东发射耗能多, 比向西发射耗能多 $\\frac{1}{2} m\\left(\\sqrt{g R}-\\frac{2 \\pi R}{T}\\right)^{2}$\nC: 向东发射与向西发射耗能相同, 均为 $\\frac{1}{2} m\\left(\\sqrt{g R}+\\frac{2 \\pi R}{T}\\right)^{2}$\nD: 向西发射耗能为 $\\frac{1}{2} m\\left(\\sqrt{g R}+\\frac{2 \\pi R}{T}\\right)^{2}$, 比向东发射耗能多\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_145", "problem": "我国天文学家通过 FAST, 在武仙座球状星团 $\\mathrm{M}_{1} 3$ 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A} 收到反射回来的溦光信号 | |\nA: $s=c t$\nB: $s=\\frac{V T}{2 \\pi}-R-r$\nC: $s=\\frac{V^{2}}{g}-R-r$\nD: $s=\\sqrt[3]{\\frac{g_{0} R^{2} T^{2}}{4 \\pi^{2}}}-R-r$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n某同学在学习中记录了一些与地球、月球有关的数据资料如表中所示, 利用这些数据来计算地球表面与月球表面之间的最近距离, 则下列表达式中正确的是（ ）\n\n| 地球半径 | $R=6.4 \\times 10^{6} \\mathrm{~m}$ |\n| :--- | :--- |\n| 月球半径 | $t=1.74 \\times 10^{6} \\mathrm{~m}$ |\n| 地球表面重力加速度 | $g_{0}=9.80 \\mathrm{~m} / \\mathrm{s}^{2}$ |\n| 月球表面重力加速度 | $g=1.56 \\mathrm{~m} / \\mathrm{s}^{2}$ |\n| 月球绕地球转动的线速度 | $V=1 \\mathrm{~km} / \\mathrm{s}$ |\n| 月球绕地球转动的周期 | $T=27.3$ 天 |\n| 光速 | $c=3.0 \\times 10^{3} \\mathrm{~m} / \\mathrm{s}$ |\n| 用溦光器从地球表面上正对月球表面处向月球表面发射溦光束, 经过 $t=2.56 s$ 接
收到反射回来的溦光信号 | |\n\nA: $s=c t$\nB: $s=\\frac{V T}{2 \\pi}-R-r$\nC: $s=\\frac{V^{2}}{g}-R-r$\nD: $s=\\sqrt[3]{\\frac{g_{0} R^{2} T^{2}}{4 \\pi^{2}}}-R-r$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_329", "problem": "宇宙中, 两颗靠得比较近的恒星, 只受到彼此之间的万有引力作用互相绕转, 称之为双星系统. 在浩瀚的银河系中, 多数恒星都是双星系统. 设某双星系统 $A 、 B$ 绕其连线上的 $O$ 点做匀速圆周运动, 如图所示, 若 $A O>O B$, 则 ( )\n\n[图1]\nA: 星球 $A$ 的质量一定大于 $B$ 的质量\nB: 星球 $A$ 的线速度一定大于 $B$ 的线速度\nC: 双星间距离一定, 双星的总质量越大, 其转动周期越小\nD: 双星的总质量一定, 双星之间的距离越大，其转动周期越小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇宙中, 两颗靠得比较近的恒星, 只受到彼此之间的万有引力作用互相绕转, 称之为双星系统. 在浩瀚的银河系中, 多数恒星都是双星系统. 设某双星系统 $A 、 B$ 绕其连线上的 $O$ 点做匀速圆周运动, 如图所示, 若 $A O>O B$, 则 ( )\n\n[图1]\n\nA: 星球 $A$ 的质量一定大于 $B$ 的质量\nB: 星球 $A$ 的线速度一定大于 $B$ 的线速度\nC: 双星间距离一定, 双星的总质量越大, 其转动周期越小\nD: 双星的总质量一定, 双星之间的距离越大，其转动周期越小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-09.jpg?height=117&width=457&top_left_y=2643&top_left_x=408" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_821", "problem": "Consider the binary system Kepler-16, which has the primary star Kepler-16A and the secondary star Kepler-16B. It has an orbital period $P=41.08$ days and the measured parallax is $p=$ 13.29 mas. Calculate the total mass of the stars, using the fact that their maximum angular separation measured from Earth is $\\theta=2.98$ mas and they are on an edge-on orbit.\nA: $0.756 M_{\\odot}$\nB: $0.803 M_{\\odot}$\nC: $0.891 M_{\\odot}$\nD: $0.987 M_{\\odot}$\nE: $1.326 M_{\\odot}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nConsider the binary system Kepler-16, which has the primary star Kepler-16A and the secondary star Kepler-16B. It has an orbital period $P=41.08$ days and the measured parallax is $p=$ 13.29 mas. Calculate the total mass of the stars, using the fact that their maximum angular separation measured from Earth is $\\theta=2.98$ mas and they are on an edge-on orbit.\n\nA: $0.756 M_{\\odot}$\nB: $0.803 M_{\\odot}$\nC: $0.891 M_{\\odot}$\nD: $0.987 M_{\\odot}$\nE: $1.326 M_{\\odot}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1196", "problem": "The James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.a. The telescope will spend its expected 10-20 year mission in a halo orbit about the second Lagrangian point, L2 (see Figure 5). This is one of five special points in the Sun-Earth system where the gravitational forces from the two bodies provide the centripetal force required to have a (small mass) object there have an orbital period identical to the Earth. At the L2 point, this means it orbits quicker than you would expect for an object that distance from the Sun.\n\nii. The JWST is on an orbit that will take it to within $200000 \\mathrm{~km}$ of $\\mathrm{L2}$, where it will then do a final large burn of the rockets to insert it into the halo orbit around L2. Assuming it is on a simple elliptical transfer orbit ignoring the influence of the Sun, and had a perigee at an altitude of $2100 \\mathrm{~km}$ above the surface of the Earth, how long will it take JWST to get to the L2 orbital insertion phase of its mission? Give your answer in days.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.\n\nproblem:\na. The telescope will spend its expected 10-20 year mission in a halo orbit about the second Lagrangian point, L2 (see Figure 5). This is one of five special points in the Sun-Earth system where the gravitational forces from the two bodies provide the centripetal force required to have a (small mass) object there have an orbital period identical to the Earth. At the L2 point, this means it orbits quicker than you would expect for an object that distance from the Sun.\n\nii. The JWST is on an orbit that will take it to within $200000 \\mathrm{~km}$ of $\\mathrm{L2}$, where it will then do a final large burn of the rockets to insert it into the halo orbit around L2. Assuming it is on a simple elliptical transfer orbit ignoring the influence of the Sun, and had a perigee at an altitude of $2100 \\mathrm{~km}$ above the surface of the Earth, how long will it take JWST to get to the L2 orbital insertion phase of its mission? Give your answer in days.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-09.jpg?height=618&width=1466&top_left_y=596&top_left_x=296", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-10.jpg?height=482&width=1536&top_left_y=1118&top_left_x=267", "https://cdn.mathpix.com/cropped/2024_03_14_e9aa0a135004f2f4a278g-10.jpg?height=979&width=1243&top_left_y=413&top_left_x=475" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_337", "problem": "三颗人造卫星 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 都在赤道正上方同方向绕地球做匀速圆周运动, $\\mathrm{A} 、 \\mathrm{C}$ 为地球同步卫星, 某时刻 $\\mathrm{A} 、 \\mathrm{~B}$ 相距最近, 如图所示。已知地球自转周期为 $T_{1}, \\mathrm{~B}$ 的运行周期为 $T_{2}$, 则下列说法正确的是（ ）\n\n[图1]\nA: C 加速可追上同一轨道上的 A\nB: 经过时间 $\\frac{T_{1} T_{2}}{2\\left(T_{1}-T_{2}\\right)}, \\mathrm{A} 、 \\mathrm{~B}$ 相距最远\nC: A、C 向心加速度大小相等, 且小于 $\\mathrm{B}$ 的向心加速度\nD: 在相同时间内, $\\mathrm{A}$ 与地心连线扫过的面积大于 $\\mathrm{B}$ 与地心连线扫过的面积\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n三颗人造卫星 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 都在赤道正上方同方向绕地球做匀速圆周运动, $\\mathrm{A} 、 \\mathrm{C}$ 为地球同步卫星, 某时刻 $\\mathrm{A} 、 \\mathrm{~B}$ 相距最近, 如图所示。已知地球自转周期为 $T_{1}, \\mathrm{~B}$ 的运行周期为 $T_{2}$, 则下列说法正确的是（ ）\n\n[图1]\n\nA: C 加速可追上同一轨道上的 A\nB: 经过时间 $\\frac{T_{1} T_{2}}{2\\left(T_{1}-T_{2}\\right)}, \\mathrm{A} 、 \\mathrm{~B}$ 相距最远\nC: A、C 向心加速度大小相等, 且小于 $\\mathrm{B}$ 的向心加速度\nD: 在相同时间内, $\\mathrm{A}$ 与地心连线扫过的面积大于 $\\mathrm{B}$ 与地心连线扫过的面积\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-074.jpg?height=382&width=371&top_left_y=2239&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_258", "problem": "三体问题是天体力学中的基本模型, 即探究三个质量、初始位置和初始速度都任意的可视为质点的天体，在相互之间万有引力的作用下的运动规律。三体问题同时也是一个著名的数学难题, 1772 年, 拉格朗日在“平面限制性三体问题”条件下找到了 5 个特解,它就是著名的拉格朗日点。在该点上，小天体在两个大天体的引力作用下能基本保持相对静止。如图是日地系统的 5 个拉格朗日点 $\\left(L_{1} 、 L_{2} 、 L_{3} 、 L_{4} 、 L_{5}\\right)$, 设想未来人类在这五个点上都建立了太空站, 若不考虑其它天体对太空站的引力, 则下列说法正确的是\n\n[图1]\nA: 位于 $L_{l}$ 点的太空站处于受力平衡状态\nB: 位于 $L_{2}$ 点的太空站的线速度小于地球的线速度\nC: 位于 $L_{3}$ 点的太空站的向心加速度大于位于 $L_{1}$ 点的太空站的向心加速度\nD: 位于 $L_{4}$ 点的太空站向心力大小一定等于位于 $L_{5}$ 点的太空站向心力大小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n三体问题是天体力学中的基本模型, 即探究三个质量、初始位置和初始速度都任意的可视为质点的天体，在相互之间万有引力的作用下的运动规律。三体问题同时也是一个著名的数学难题, 1772 年, 拉格朗日在“平面限制性三体问题”条件下找到了 5 个特解,它就是著名的拉格朗日点。在该点上，小天体在两个大天体的引力作用下能基本保持相对静止。如图是日地系统的 5 个拉格朗日点 $\\left(L_{1} 、 L_{2} 、 L_{3} 、 L_{4} 、 L_{5}\\right)$, 设想未来人类在这五个点上都建立了太空站, 若不考虑其它天体对太空站的引力, 则下列说法正确的是\n\n[图1]\n\nA: 位于 $L_{l}$ 点的太空站处于受力平衡状态\nB: 位于 $L_{2}$ 点的太空站的线速度小于地球的线速度\nC: 位于 $L_{3}$ 点的太空站的向心加速度大于位于 $L_{1}$ 点的太空站的向心加速度\nD: 位于 $L_{4}$ 点的太空站向心力大小一定等于位于 $L_{5}$ 点的太空站向心力大小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-043.jpg?height=614&width=805&top_left_y=1338&top_left_x=360" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_801", "problem": "The habitable zone of a star is defined as the one where water in the liquid state can exist in the surface of a planet. Therefore, considering that the planets are ideal black bodies with fast rotation, determine the maximum eccentricity that the orbit of a planet can have so that it can be home to life. Ignore any thermodynamic effects that might happen in the atmosphere or the sidereal space. Consider that the melting point of water is $273 \\mathrm{~K}$ and the boiling point is $373 \\mathrm{~K}$.\nA: 0.274\nB: 0.302\nC: 0.316\nD: 0.328\nE: 0.540\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe habitable zone of a star is defined as the one where water in the liquid state can exist in the surface of a planet. Therefore, considering that the planets are ideal black bodies with fast rotation, determine the maximum eccentricity that the orbit of a planet can have so that it can be home to life. Ignore any thermodynamic effects that might happen in the atmosphere or the sidereal space. Consider that the melting point of water is $273 \\mathrm{~K}$ and the boiling point is $373 \\mathrm{~K}$.\n\nA: 0.274\nB: 0.302\nC: 0.316\nD: 0.328\nE: 0.540\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_432", "problem": "万有引力定律 $F_{\\text {引 }}=G \\frac{m_{1} m_{2}}{r^{2}}$ 和库仑定律 $F_{\\text {电 }}=k \\frac{q_{1} q_{2}}{r^{2}}$ 都满足力与距离平方成反比关系。如图所示, 计算物体从距离地球球心 $r_{1}$ 处, 远离至与地心距离 $r_{2}$ 处, 万有引力对物体做功时, 由于力的大小随距离而变化, 一般需采用微元法。也可采用从 $r_{1}$ 到 $r_{2}$ 过程的平均力, 即 $\\overline{F_{\\text {引 }}}=G \\frac{m_{1} m_{2}}{r_{1} \\cdot r_{2}}$ 计算做功。已知物体质量为 $m$, 地球质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。\n氢原子是最简单的原子, 电子绕原子核做匀速圆周运动与人造卫星绕地球做匀速圆周运动类似。已知电子质量为 $m$, 带电量为 $-e$, 氢原子核带电量为 $+e$, 电子绕核运动半径为 $r$, 静电力常量为 $k$, 求电子绕核运动的速度 $v_{l}$ 大小; 若要使氢原子电离 (使核外电子运动至无穷远, 逃出原子核的电场范围), 则至少额外需要提供多大的能量 $\\Delta E$ 。\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n万有引力定律 $F_{\\text {引 }}=G \\frac{m_{1} m_{2}}{r^{2}}$ 和库仑定律 $F_{\\text {电 }}=k \\frac{q_{1} q_{2}}{r^{2}}$ 都满足力与距离平方成反比关系。如图所示, 计算物体从距离地球球心 $r_{1}$ 处, 远离至与地心距离 $r_{2}$ 处, 万有引力对物体做功时, 由于力的大小随距离而变化, 一般需采用微元法。也可采用从 $r_{1}$ 到 $r_{2}$ 过程的平均力, 即 $\\overline{F_{\\text {引 }}}=G \\frac{m_{1} m_{2}}{r_{1} \\cdot r_{2}}$ 计算做功。已知物体质量为 $m$, 地球质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。\n氢原子是最简单的原子, 电子绕原子核做匀速圆周运动与人造卫星绕地球做匀速圆周运动类似。已知电子质量为 $m$, 带电量为 $-e$, 氢原子核带电量为 $+e$, 电子绕核运动半径为 $r$, 静电力常量为 $k$, 求电子绕核运动的速度 $v_{l}$ 大小; 若要使氢原子电离 (使核外电子运动至无穷远, 逃出原子核的电场范围), 则至少额外需要提供多大的能量 $\\Delta E$ 。\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[电子绕核运动的速度, 至少额外提供多大的能量]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-057.jpg?height=268&width=574&top_left_y=203&top_left_x=410" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "电子绕核运动的速度", "至少额外提供多大的能量" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1136", "problem": "Which of the following planets has the longest day, defined as the period of a complete\nrotation about its axis?\nA: Venus([figure1])\nB: Earth([figure2])\nC: Mars([figure3])\nD: Jupiter([figure4])\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhich of the following planets has the longest day, defined as the period of a complete\nrotation about its axis?\n\nA: Venus([figure1])\nB: Earth([figure2])\nC: Mars([figure3])\nD: Jupiter([figure4])\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_bb6f165f9dbff6083d7cg-03.jpg?height=311&width=297&top_left_y=2220&top_left_x=388", "https://cdn.mathpix.com/cropped/2024_03_14_bb6f165f9dbff6083d7cg-03.jpg?height=311&width=400&top_left_y=2220&top_left_x=677", "https://cdn.mathpix.com/cropped/2024_03_14_bb6f165f9dbff6083d7cg-03.jpg?height=297&width=303&top_left_y=2233&top_left_x=1068", "https://cdn.mathpix.com/cropped/2024_03_14_bb6f165f9dbff6083d7cg-03.jpg?height=297&width=303&top_left_y=2233&top_left_x=1365" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_613", "problem": "$\\mathrm{A} 、 \\mathrm{~B}$ 两颗卫星在同一平面内沿同一方向绕地球做匀速圆周运动, 它们之间的距离 $\\triangle r$ 随时间变化的关系如图所示。已知地球的半径为 $0.8 r$, 万有引力常量为 $G$, 卫星 A\n的线速度大于卫星 $\\mathrm{B}$ 的线速度, 不考虑 $\\mathrm{A} 、 \\mathrm{~B}$ 之间的万有引力, 则下列说法正确的是\n\n[图1]\nA: 卫星 A 的加速度大于卫星 $\\mathrm{B}$ 的加速度\nB: 卫星 A 的发射速度可能大于第二宇宙速度\nC: 地球的质量为 $\\frac{256 \\pi^{2} r^{3}}{49 G T^{2}}$\nD: 地球的第一宇宙速度为 $\\frac{8 \\sqrt{5} \\pi r}{7 T}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n$\\mathrm{A} 、 \\mathrm{~B}$ 两颗卫星在同一平面内沿同一方向绕地球做匀速圆周运动, 它们之间的距离 $\\triangle r$ 随时间变化的关系如图所示。已知地球的半径为 $0.8 r$, 万有引力常量为 $G$, 卫星 A\n的线速度大于卫星 $\\mathrm{B}$ 的线速度, 不考虑 $\\mathrm{A} 、 \\mathrm{~B}$ 之间的万有引力, 则下列说法正确的是\n\n[图1]\n\nA: 卫星 A 的加速度大于卫星 $\\mathrm{B}$ 的加速度\nB: 卫星 A 的发射速度可能大于第二宇宙速度\nC: 地球的质量为 $\\frac{256 \\pi^{2} r^{3}}{49 G T^{2}}$\nD: 地球的第一宇宙速度为 $\\frac{8 \\sqrt{5} \\pi r}{7 T}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-081.jpg?height=317&width=508&top_left_y=321&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_267", "problem": "2017 年, 人类第一次直接探测到来自双中子星合并的引力波。根据科学家们复原的过程, 在两颗中子星合并前约 $100 \\mathrm{~s}$ 时, 它们相距约 $r=400 \\mathrm{~km}$, 绕二者连线上的某点每秒转动 12 圈, 将两颗中子星都看作是质量均匀分布的球体, 由这些数据、万有引力常量并利用牛顿力学知识, 可以估算出这一时刻两颗中子星 ( )\nA: 各自的速率为 $2 \\pi r f$\nB: 各自的自转角速度为 $2 \\pi f$\nC: 质量之和为 $\\frac{(2 \\pi f)^{2} r^{3}}{G}$\nD: 速率之积为 $2 \\pi r f$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2017 年, 人类第一次直接探测到来自双中子星合并的引力波。根据科学家们复原的过程, 在两颗中子星合并前约 $100 \\mathrm{~s}$ 时, 它们相距约 $r=400 \\mathrm{~km}$, 绕二者连线上的某点每秒转动 12 圈, 将两颗中子星都看作是质量均匀分布的球体, 由这些数据、万有引力常量并利用牛顿力学知识, 可以估算出这一时刻两颗中子星 ( )\n\nA: 各自的速率为 $2 \\pi r f$\nB: 各自的自转角速度为 $2 \\pi f$\nC: 质量之和为 $\\frac{(2 \\pi f)^{2} r^{3}}{G}$\nD: 速率之积为 $2 \\pi r f$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_608", "problem": "在力学发展的过程中, 许多物理学家的科学发现推动了物理学的进步。对以下几位物理学家所作科学贡献的表述中, 与事实不相符的是 ( )\nA: 伽利略首先建立平均速度、瞬时速度和加速度等描述运动的概念\nB: 胡克提出如果行星的轨道是圆形, 太阳与行星间的引力与距离的平方成反比\nC: 卡文迪许是测量地球质量的第一人\nD: 伽利略根据理想斜面实验, 得出自由落体运动是匀变速直线运动\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n在力学发展的过程中, 许多物理学家的科学发现推动了物理学的进步。对以下几位物理学家所作科学贡献的表述中, 与事实不相符的是 ( )\n\nA: 伽利略首先建立平均速度、瞬时速度和加速度等描述运动的概念\nB: 胡克提出如果行星的轨道是圆形, 太阳与行星间的引力与距离的平方成反比\nC: 卡文迪许是测量地球质量的第一人\nD: 伽利略根据理想斜面实验, 得出自由落体运动是匀变速直线运动\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_655", "problem": "《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求:\n配重空间站受到缆绳的力大小为多少;\n\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求:\n配重空间站受到缆绳的力大小为多少;\n\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-064.jpg?height=388&width=1082&top_left_y=1345&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_36", "problem": "2021 年 2 月 10 日, 在历经近 7 个月的太空飞行后, 我国首个火星探测器“天问一号”成功“太空刹车”, 顺利被火星捕获，进入环火星轨道。物体在万有引力场中具有的势能叫作引力势能, 若取两物体相距无穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力源中心为 $r$ 时, 其引力势能 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$ (式中 $G$ 为引力常量)。已知地球半径约为 $6400 \\mathrm{~km}$, 地球的第一宇宙速度为 $7.9 \\mathrm{~km} / \\mathrm{s}$, 火星半径约为地球半径的 $\\frac{1}{2}$, 火星质量约为球质量的 $\\frac{1}{9}$ 。则“天问一号”刹车后相对于火星的速度不可能为 ( )\nA: $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: $5.5 \\mathrm{~km} / \\mathrm{s}$\nC: $4.0 \\mathrm{~km} / \\mathrm{s}$\nD: $3.2 \\mathrm{~km} / \\mathrm{s}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2021 年 2 月 10 日, 在历经近 7 个月的太空飞行后, 我国首个火星探测器“天问一号”成功“太空刹车”, 顺利被火星捕获，进入环火星轨道。物体在万有引力场中具有的势能叫作引力势能, 若取两物体相距无穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力源中心为 $r$ 时, 其引力势能 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$ (式中 $G$ 为引力常量)。已知地球半径约为 $6400 \\mathrm{~km}$, 地球的第一宇宙速度为 $7.9 \\mathrm{~km} / \\mathrm{s}$, 火星半径约为地球半径的 $\\frac{1}{2}$, 火星质量约为球质量的 $\\frac{1}{9}$ 。则“天问一号”刹车后相对于火星的速度不可能为 ( )\n\nA: $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: $5.5 \\mathrm{~km} / \\mathrm{s}$\nC: $4.0 \\mathrm{~km} / \\mathrm{s}$\nD: $3.2 \\mathrm{~km} / \\mathrm{s}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_845", "problem": "Knowing that the following image was taken at at $11: 59 \\mathrm{pm}$, determine the name of which constellation was the sun passing in front of in that same day.\n\n[figure1]\nA: Scorpius\nB: Virgo\nC: Big Dipper\nD: Cancer\nE: Taurus\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nKnowing that the following image was taken at at $11: 59 \\mathrm{pm}$, determine the name of which constellation was the sun passing in front of in that same day.\n\n[figure1]\n\nA: Scorpius\nB: Virgo\nC: Big Dipper\nD: Cancer\nE: Taurus\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ea07af8330da280030dbg-22.jpg?height=1041&width=1312&top_left_y=344&top_left_x=404", "https://cdn.mathpix.com/cropped/2024_03_06_ea07af8330da280030dbg-23.jpg?height=2136&width=1464&top_left_y=233&top_left_x=363", "https://cdn.mathpix.com/cropped/2024_03_06_ea07af8330da280030dbg-24.jpg?height=965&width=1022&top_left_y=420&top_left_x=579" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_804", "problem": "Taking the radius of the black hole to be the Schwarzschild radius (the radius at which the escape velocity of an object would be equal to the speed of light), what is the surface area of a black hole of mass M?\nA: $A=\\frac{16 \\pi}{c^{4}} G^{2} M^{2}$ \nB: $A=\\frac{4 \\pi}{c^{4}} G^{2} M^{2}$\nC: $A=\\frac{4 \\pi}{3 c^{4}} G^{2} M^{2}$\nD: $A=16 \\pi G^{2} M^{2}$\nE: $A=\\frac{16 \\pi}{c^{2}} G^{2} M^{2}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTaking the radius of the black hole to be the Schwarzschild radius (the radius at which the escape velocity of an object would be equal to the speed of light), what is the surface area of a black hole of mass M?\n\nA: $A=\\frac{16 \\pi}{c^{4}} G^{2} M^{2}$ \nB: $A=\\frac{4 \\pi}{c^{4}} G^{2} M^{2}$\nC: $A=\\frac{4 \\pi}{3 c^{4}} G^{2} M^{2}$\nD: $A=16 \\pi G^{2} M^{2}$\nE: $A=\\frac{16 \\pi}{c^{2}} G^{2} M^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_23", "problem": "建造一条能通向太空的天梯, 是人们长期的梦想。当今在美国宇航局（NASA）支持下, 洛斯阿拉莫斯国家实验室的科学家已在进行这方面的研究。一种简单的设计是把天梯看作一条长度达千万层楼高的质量均匀分布的缆绳, 它由一种高强度、很轻的纳米碳管制成，由传统的太空飞船运到太空上，然后慢慢垂到地球表面。最后达到这样的状态和位置: 天梯本身呈直线状; 其上端指向太空, 下端刚与地面接触但与地面之间无相互作用; 整个天梯相对于地球静止不动。如果只考虑地球对天梯的万有引力, 试求此天梯的长度。已知地球半径 $R_{0}=6.37 \\times 10^{6} \\mathrm{~m}$, 地球表面处的重力加速度 $\\mathrm{g}=9.80 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n建造一条能通向太空的天梯, 是人们长期的梦想。当今在美国宇航局（NASA）支持下, 洛斯阿拉莫斯国家实验室的科学家已在进行这方面的研究。一种简单的设计是把天梯看作一条长度达千万层楼高的质量均匀分布的缆绳, 它由一种高强度、很轻的纳米碳管制成，由传统的太空飞船运到太空上，然后慢慢垂到地球表面。最后达到这样的状态和位置: 天梯本身呈直线状; 其上端指向太空, 下端刚与地面接触但与地面之间无相互作用; 整个天梯相对于地球静止不动。如果只考虑地球对天梯的万有引力, 试求此天梯的长度。已知地球半径 $R_{0}=6.37 \\times 10^{6} \\mathrm{~m}$, 地球表面处的重力加速度 $\\mathrm{g}=9.80 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以m为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-018.jpg?height=211&width=828&top_left_y=346&top_left_x=340", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-019.jpg?height=531&width=757&top_left_y=1553&top_left_x=358" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_621", "problem": "如图所示, 质量分别为 $m$ 和 $M$ 的两个星球 $A$ 和 $B$ 在引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $A$ 和 $B$ 两者中心之间距离为 $L$ 。已知 $A 、 B$ 的中心和 $O$ 三点始终共线, $A$和 $B$ 分别在 $O$ 的两侧, 引力常量为 $G$ 。求:\n\n两星球做圆周运动的周期;\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 质量分别为 $m$ 和 $M$ 的两个星球 $A$ 和 $B$ 在引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $A$ 和 $B$ 两者中心之间距离为 $L$ 。已知 $A 、 B$ 的中心和 $O$ 三点始终共线, $A$和 $B$ 分别在 $O$ 的两侧, 引力常量为 $G$ 。求:\n\n两星球做圆周运动的周期;\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-139.jpg?height=457&width=531&top_left_y=1248&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1102", "problem": "The Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$b. Assuming Sgr A* is a non-spinning black hole with mass $4.1 \\times 10^{6} \\mathrm{M}_{\\odot}$ and at a distance of $8.34 \\mathrm{kpc}$ :\n\ni. Derive the (unlensed) radius of the photon sphere, $r_{\\mathrm{ph}}$, in units of $\\mathrm{r}_{\\mathrm{g}}$, by considering a balance between the centripetal and (Newtonian) gravitational forces, but with the relativistic correction $v^{\\prime}=v \\sqrt{1-2 r_{g} / r}$ where $\\mathrm{v}_{0}$ is the classical velocity and $\\mathrm{r}=\\mathrm{r}_{\\mathrm{ph}}$ when $\\mathrm{v}=\\mathrm{c}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$\n\nproblem:\nb. Assuming Sgr A* is a non-spinning black hole with mass $4.1 \\times 10^{6} \\mathrm{M}_{\\odot}$ and at a distance of $8.34 \\mathrm{kpc}$ :\n\ni. Derive the (unlensed) radius of the photon sphere, $r_{\\mathrm{ph}}$, in units of $\\mathrm{r}_{\\mathrm{g}}$, by considering a balance between the centripetal and (Newtonian) gravitational forces, but with the relativistic correction $v^{\\prime}=v \\sqrt{1-2 r_{g} / r}$ where $\\mathrm{v}_{0}$ is the classical velocity and $\\mathrm{r}=\\mathrm{r}_{\\mathrm{ph}}$ when $\\mathrm{v}=\\mathrm{c}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of r_{g}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-07.jpg?height=704&width=1414&top_left_y=698&top_left_x=331", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-08.jpg?height=374&width=1562&top_left_y=1698&top_left_x=263", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-09.jpg?height=468&width=686&top_left_y=1388&top_left_x=705" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "r_{g}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_704", "problem": "“天问一号” 火星探测器已经被火星捕获。若探测器在距离火星表面高为 $h$ 的圆形轨道上绕火星飞行, 环绕 $n$ 周飞行总时间为 $t$, 已知引力常量为 $G$, 火星半径为 $R$, 则下列给出的火星表面重力加速度 $g$ (忽略自转) 和平均密度 $\\rho$ 的表达式正确的是 ( )\nA: $g=\\frac{4 \\pi^{2}(R+h)^{3}}{R^{2} t^{2}}, \\rho=\\frac{3 \\pi(R+h)^{3}}{G t^{2} R^{3}}$\nB: $g=\\frac{4 \\pi^{2} n^{2}(R+h)^{3}}{R^{2} t^{2}}, \\rho=\\frac{3 \\pi n^{2}(R+h)^{3}}{G t^{2} R^{3}}$\nC: $g=\\frac{4 \\pi^{2} t^{2}(R+h)^{3}}{R^{2} n^{2}}, \\quad \\rho=\\frac{3 \\pi t^{2}(R+h)^{3}}{G n^{2} R^{3}}$\nD: $g=\\frac{4 \\pi^{2} n^{2}(R+h)^{3}}{R^{2} t^{2}}, \\quad \\rho=\\frac{3 \\pi(R+h)^{3}}{G t^{2} R^{3}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n“天问一号” 火星探测器已经被火星捕获。若探测器在距离火星表面高为 $h$ 的圆形轨道上绕火星飞行, 环绕 $n$ 周飞行总时间为 $t$, 已知引力常量为 $G$, 火星半径为 $R$, 则下列给出的火星表面重力加速度 $g$ (忽略自转) 和平均密度 $\\rho$ 的表达式正确的是 ( )\n\nA: $g=\\frac{4 \\pi^{2}(R+h)^{3}}{R^{2} t^{2}}, \\rho=\\frac{3 \\pi(R+h)^{3}}{G t^{2} R^{3}}$\nB: $g=\\frac{4 \\pi^{2} n^{2}(R+h)^{3}}{R^{2} t^{2}}, \\rho=\\frac{3 \\pi n^{2}(R+h)^{3}}{G t^{2} R^{3}}$\nC: $g=\\frac{4 \\pi^{2} t^{2}(R+h)^{3}}{R^{2} n^{2}}, \\quad \\rho=\\frac{3 \\pi t^{2}(R+h)^{3}}{G n^{2} R^{3}}$\nD: $g=\\frac{4 \\pi^{2} n^{2}(R+h)^{3}}{R^{2} t^{2}}, \\quad \\rho=\\frac{3 \\pi(R+h)^{3}}{G t^{2} R^{3}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_109", "problem": "在天体运动中, 把两颗相距很近的恒星称为双星。已知组成某双星系统的两颗恒星质量分别为 $m_{l}$ 和 $m_{2}$ 相距为 $L$ 。在万有引力作用下各自绕它们连线上的某一点, 在同平面内做匀速同周运动, 运动过程中二者之间的距离始终不变。已知万有引力常量为 $G$ 。 $m_{1}$ 的动能为 $E_{k}$ 则 $m_{2}$ 的动能为\nA: $G \\frac{m_{1} m_{2}}{l}-E_{k}$\nB: $G \\frac{m_{1} m_{2}}{2 l}-E_{k}$\nC: $\\frac{m_{1}}{m_{2}} E_{k}$\nD: $\\frac{m_{2}}{m_{1}} E_{k}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n在天体运动中, 把两颗相距很近的恒星称为双星。已知组成某双星系统的两颗恒星质量分别为 $m_{l}$ 和 $m_{2}$ 相距为 $L$ 。在万有引力作用下各自绕它们连线上的某一点, 在同平面内做匀速同周运动, 运动过程中二者之间的距离始终不变。已知万有引力常量为 $G$ 。 $m_{1}$ 的动能为 $E_{k}$ 则 $m_{2}$ 的动能为\n\nA: $G \\frac{m_{1} m_{2}}{l}-E_{k}$\nB: $G \\frac{m_{1} m_{2}}{2 l}-E_{k}$\nC: $\\frac{m_{1}}{m_{2}} E_{k}$\nD: $\\frac{m_{2}}{m_{1}} E_{k}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_364", "problem": "如图所示, 在发射地球同步卫星的过程中, 卫星首先进入粗圆轨道 $\\mathrm{I}$, 然后在 $\\mathrm{Q}$ 点通过改变卫星速度, 让卫星进入地球同步轨道II, 则 ( )\n\n[图1]\nA: 该卫星的发射速度必定大于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nB: 卫星在轨道上运行不受重力\nC: 在轨道 $\\mathrm{I}$ 上, 卫星在 $\\mathrm{P}$ 点的速度大于在 $\\mathrm{Q}$ 点的速度\nD: 卫星在 $\\mathrm{Q}$ 点通过加速实现由轨道 I 进入轨道II\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 在发射地球同步卫星的过程中, 卫星首先进入粗圆轨道 $\\mathrm{I}$, 然后在 $\\mathrm{Q}$ 点通过改变卫星速度, 让卫星进入地球同步轨道II, 则 ( )\n\n[图1]\n\nA: 该卫星的发射速度必定大于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nB: 卫星在轨道上运行不受重力\nC: 在轨道 $\\mathrm{I}$ 上, 卫星在 $\\mathrm{P}$ 点的速度大于在 $\\mathrm{Q}$ 点的速度\nD: 卫星在 $\\mathrm{Q}$ 点通过加速实现由轨道 I 进入轨道II\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-12.jpg?height=334&width=391&top_left_y=181&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_940", "problem": "Forces of Nature\n\nIn the BBC programme Forces of Nature, Brian Cox uses a Eurofighter Typhoon to try and overtake the spin of the Earth such that the setting Sun appears to rise instead.\n\n[figure1]\n\nA Eurofighter sets off from the equator just as the top edge of the Sun has gone below the horizon, and rapidly accelerates due west up to a speed of $500 \\mathrm{~m} \\mathrm{~s}^{-1}$. Given that the Sun has an angular diameter of $0.5^{\\circ}$ as viewed from Earth, what is the minimum amount of time the fighter jet needs to fly for in order to see the whole of the Sun above the horizon?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nForces of Nature\n\nIn the BBC programme Forces of Nature, Brian Cox uses a Eurofighter Typhoon to try and overtake the spin of the Earth such that the setting Sun appears to rise instead.\n\n[figure1]\n\nA Eurofighter sets off from the equator just as the top edge of the Sun has gone below the horizon, and rapidly accelerates due west up to a speed of $500 \\mathrm{~m} \\mathrm{~s}^{-1}$. Given that the Sun has an angular diameter of $0.5^{\\circ}$ as viewed from Earth, what is the minimum amount of time the fighter jet needs to fly for in order to see the whole of the Sun above the horizon?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of minutes, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6d91a7785df4f4beaa9ag-06.jpg?height=611&width=1065&top_left_y=745&top_left_x=473", "https://i.postimg.cc/NFBz3QHr/Screenshot-2024-04-07-at-16-34-22.png" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "minutes" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_879", "problem": "Abhay looks at the light curves for two main sequence stars A and B, which you can assume are blackbodies. A has its peak at a frequency two times as high as that of B. By looking at the depth of spectral lines, Abhay can also determine that A has higher metallicity than B. Abhay makes the following statements:\n\n$\\mathrm{P}$ : A has higher absolute magnitude than $\\mathrm{B}$.\n\nQ: A is older than B.\n\nWhich of the following is true?\nA: P and Q are true.\nB: $\\mathrm{P}$ and $\\mathrm{Q}$ are false.\nC: $\\mathrm{P}$ is true and $\\mathrm{Q}$ is false.\nD: $\\mathrm{P}$ is false and $\\mathrm{Q}$ is true.\nE: We don't have sufficient information for one or more of these statements.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAbhay looks at the light curves for two main sequence stars A and B, which you can assume are blackbodies. A has its peak at a frequency two times as high as that of B. By looking at the depth of spectral lines, Abhay can also determine that A has higher metallicity than B. Abhay makes the following statements:\n\n$\\mathrm{P}$ : A has higher absolute magnitude than $\\mathrm{B}$.\n\nQ: A is older than B.\n\nWhich of the following is true?\n\nA: P and Q are true.\nB: $\\mathrm{P}$ and $\\mathrm{Q}$ are false.\nC: $\\mathrm{P}$ is true and $\\mathrm{Q}$ is false.\nD: $\\mathrm{P}$ is false and $\\mathrm{Q}$ is true.\nE: We don't have sufficient information for one or more of these statements.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_424", "problem": "2009 年 10 月 7 日电，美国宇航局（NASA）的斯皮策（Spitzer）太空望远镜近期发现土星外环绕着一个巨大的漫射环. 该环比已知的由太空尘埃和冰块组成的土星环要大得多. 据悉, 这个由细小冰粒及尘埃组成的土星环温度接近 $-157^{\\circ} \\mathrm{C}$, 结构非常松散,难以反射光线, 所以此前一直未被发现, 而仅能被红外探测仪检测到. 这一暗淡的土星环由微小粒子构成, 环内侧距土星中心约 600 万公里, 外侧距土星中心约 1800 万公里. 若忽略微粒间的作用力，假设土环上的微粒均绕土星做圆周运动，则土环内侧、外侧微粒的\nA: 线速度之比为 $\\sqrt{3}: 1$\nB: 角速度之比为 $1: 1$\nC: 周期之比为 $1: 1$\nD: 向心加速度之比为 9:1\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2009 年 10 月 7 日电，美国宇航局（NASA）的斯皮策（Spitzer）太空望远镜近期发现土星外环绕着一个巨大的漫射环. 该环比已知的由太空尘埃和冰块组成的土星环要大得多. 据悉, 这个由细小冰粒及尘埃组成的土星环温度接近 $-157^{\\circ} \\mathrm{C}$, 结构非常松散,难以反射光线, 所以此前一直未被发现, 而仅能被红外探测仪检测到. 这一暗淡的土星环由微小粒子构成, 环内侧距土星中心约 600 万公里, 外侧距土星中心约 1800 万公里. 若忽略微粒间的作用力，假设土环上的微粒均绕土星做圆周运动，则土环内侧、外侧微粒的\n\nA: 线速度之比为 $\\sqrt{3}: 1$\nB: 角速度之比为 $1: 1$\nC: 周期之比为 $1: 1$\nD: 向心加速度之比为 9:1\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1064", "problem": "In science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System.\n[figure1]\n\nFigure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right).\n\nThis question will look at the real distances between asteroids.b. The main part of the asteroid belt extends from $2.1 \\mathrm{AU}$ to $3.3 \\mathrm{AU}$, and has an average angular width of $16.0^{\\circ}$, as viewed from the Sun. Calculate the average thickness of the belt, and hence its total volume, $V_{\\text {belt }}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System.\n[figure1]\n\nFigure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right).\n\nThis question will look at the real distances between asteroids.\n\nproblem:\nb. The main part of the asteroid belt extends from $2.1 \\mathrm{AU}$ to $3.3 \\mathrm{AU}$, and has an average angular width of $16.0^{\\circ}$, as viewed from the Sun. Calculate the average thickness of the belt, and hence its total volume, $V_{\\text {belt }}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~km}^{3}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f87d81e0622ba23867ceg-4.jpg?height=618&width=1260&top_left_y=584&top_left_x=388", "https://cdn.mathpix.com/cropped/2024_03_14_5a1ee9e10687b1305402g-1.jpg?height=172&width=651&top_left_y=1339&top_left_x=291" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~km}^{3}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_295", "problem": "如图所示, $\\mathrm{A}$ 为地球赤道表面上的物体, $\\mathrm{B}$ 为地球赤道平面内的圆轨道卫星, $\\mathrm{C}$ 为地球同步卫星。已知卫星 $\\mathrm{B} 、 \\mathrm{C}$ 的轨道半径之比为 $1: 3$, 下列说法正确的是 ( )\n\n[图1]\nA: 卫星 B 的向心加速度大于物体 A 的向心加速度 \nB: 卫星 B 的向心力大于卫星 C 的向心力\nC: 卫星 $\\mathrm{B} 、 \\mathrm{C}$ 的线速度大小之比为 3: 1\nD: 卫星 $\\mathrm{B} 、 \\mathrm{C}$ 的运行周期之比为 1: 9\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, $\\mathrm{A}$ 为地球赤道表面上的物体, $\\mathrm{B}$ 为地球赤道平面内的圆轨道卫星, $\\mathrm{C}$ 为地球同步卫星。已知卫星 $\\mathrm{B} 、 \\mathrm{C}$ 的轨道半径之比为 $1: 3$, 下列说法正确的是 ( )\n\n[图1]\n\nA: 卫星 B 的向心加速度大于物体 A 的向心加速度 \nB: 卫星 B 的向心力大于卫星 C 的向心力\nC: 卫星 $\\mathrm{B} 、 \\mathrm{C}$ 的线速度大小之比为 3: 1\nD: 卫星 $\\mathrm{B} 、 \\mathrm{C}$ 的运行周期之比为 1: 9\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-038.jpg?height=365&width=397&top_left_y=2276&top_left_x=358" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_206", "problem": "神舟六号载人航天飞船经过 115 小时 32 分钟的太空飞行, 绕地球飞行 77 圈, 飞船返回舱终于在 2005 年 10 月 17 日凌晨 4 时 33 分成功着陆, 航天员费俊龙、聂海胜安全返回, 已知万有引力常量 $G$, 地球表面的重力加速度 $g$, 地球的半径 $R$, 神舟六号飞船太空飞行近似为圆周运动, 则下列论述正确的是( )\nA: 可以计算神舟六号飞船绕地球的太空飞行离地球表面的高度 $h$\nB: 可以计算神舟六号飞船在绕地球的太空飞行的加速度\nC: 可以计算神舟六号飞船在绕地球的太空飞行的线速度\nD: 飞船返回舱打开减速伞下降的过程中，飞船中的宇航员处于失重状态\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n神舟六号载人航天飞船经过 115 小时 32 分钟的太空飞行, 绕地球飞行 77 圈, 飞船返回舱终于在 2005 年 10 月 17 日凌晨 4 时 33 分成功着陆, 航天员费俊龙、聂海胜安全返回, 已知万有引力常量 $G$, 地球表面的重力加速度 $g$, 地球的半径 $R$, 神舟六号飞船太空飞行近似为圆周运动, 则下列论述正确的是( )\n\nA: 可以计算神舟六号飞船绕地球的太空飞行离地球表面的高度 $h$\nB: 可以计算神舟六号飞船在绕地球的太空飞行的加速度\nC: 可以计算神舟六号飞船在绕地球的太空飞行的线速度\nD: 飞船返回舱打开减速伞下降的过程中，飞船中的宇航员处于失重状态\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1040", "problem": "In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}{r}}$.d. Derive an expression for the total time spent on the transfer orbit, $t_{H}$, and calculate it for an Earth to Mars transfer. Give your answer in months. (Use 1 month = 30 days).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIn order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}{r}}$.\n\nproblem:\nd. Derive an expression for the total time spent on the transfer orbit, $t_{H}$, and calculate it for an Earth to Mars transfer. Give your answer in months. (Use 1 month = 30 days).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1053", "problem": "It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1).\n[figure1]\n\nFigure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\\sim 40^{\\circ}$. Credit: Daniel V. Schroeder / Weber State University.\n\nRight: The same idea but viewed from Iceland at a latitude of $65^{\\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland.\n\nDuring the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees.\n\nA simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\\delta$, is given as:\n\n$$\nh=-\\left(90^{\\circ}-\\phi\\right) \\cos (A)+\\delta\n$$\n\nThe solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\\delta=+23.44^{\\circ}$ at the June solstice (roughly $21^{\\text {st }}$ June) to a minimum of $\\delta=-23.44^{\\circ}$ on the December solstice (roughly $21^{\\text {st }}$ December).\n\nIt can be shown using spherical trigonometry that the precise model connecting $\\delta, h, \\phi$ and $A$ is:\n\n$$\n\\sin (\\delta)=\\sin (h) \\sin (\\phi)+\\cos (h) \\cos (\\phi) \\cos (A) .\n$$\n\nUsing the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2.\n\n[figure2]\n\nFigure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia.\n\nBy using further spherical trigonometry, we can derive a second helpful equation in the precise model:\n\n$$\n\\sin (h)=\\sin (\\phi) \\sin (\\delta)+\\cos (\\phi) \\cos (\\delta) \\cos (H)\n$$\n\nHere, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\\circ}=24^{\\mathrm{h}}$.c. Reconsider the Oxford observer at the June solstice, but this time use the two equations of the precise model. Ignore any atmospheric effects.\n\ni. Calculate the bearing of sunrise and the duration of the day (in hours and minutes), taking the Sun to be a point source.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIt is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1).\n[figure1]\n\nFigure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\\sim 40^{\\circ}$. Credit: Daniel V. Schroeder / Weber State University.\n\nRight: The same idea but viewed from Iceland at a latitude of $65^{\\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland.\n\nDuring the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees.\n\nA simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\\delta$, is given as:\n\n$$\nh=-\\left(90^{\\circ}-\\phi\\right) \\cos (A)+\\delta\n$$\n\nThe solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\\delta=+23.44^{\\circ}$ at the June solstice (roughly $21^{\\text {st }}$ June) to a minimum of $\\delta=-23.44^{\\circ}$ on the December solstice (roughly $21^{\\text {st }}$ December).\n\nIt can be shown using spherical trigonometry that the precise model connecting $\\delta, h, \\phi$ and $A$ is:\n\n$$\n\\sin (\\delta)=\\sin (h) \\sin (\\phi)+\\cos (h) \\cos (\\phi) \\cos (A) .\n$$\n\nUsing the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2.\n\n[figure2]\n\nFigure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia.\n\nBy using further spherical trigonometry, we can derive a second helpful equation in the precise model:\n\n$$\n\\sin (h)=\\sin (\\phi) \\sin (\\delta)+\\cos (\\phi) \\cos (\\delta) \\cos (H)\n$$\n\nHere, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\\circ}=24^{\\mathrm{h}}$.\n\nproblem:\nc. Reconsider the Oxford observer at the June solstice, but this time use the two equations of the precise model. Ignore any atmospheric effects.\n\ni. Calculate the bearing of sunrise and the duration of the day (in hours and minutes), taking the Sun to be a point source.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-04.jpg?height=668&width=1478&top_left_y=523&top_left_x=290", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-05.jpg?height=648&width=1234&top_left_y=738&top_left_x=385" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1030", "problem": "The Universe consists of three main components: radiation (including neutrinos), matter (both atoms and dark matter), and dark energy. The overall density of the universe has been dominated by the density of each of those in turn at different times in its history, leading to three different epochs (shown in Fig 3).\n\n[figure1]\n\nFigure 3: A outline of the three main epochs in the history of the Universe. You do not need to read any data off this graph to answer this question. Credit: Pearson Education, Inc.\n\nThe scale factor, $a$, describes how the Universe has expanded (i.e. a measure of the relative radius of the Universe), and the current value is defined as $a_{0} \\equiv 1$ where the subscript ' 0 ' indicates it is as measured today. At earlier times $a<1$ and at the Big Bang $a=0$.\n\nThe redshift of an object, $z$, is related to the scale factor as $a=(1+z)^{-1}$ and so the redshift corresponding to now is $z=0$, and far away objects have higher redshift $(z>0)$ since we observe them as they were long ago when the scale factor was smaller.\n\nFor a Universe to be flat (i.e. zero curvature), its average density must be equal to the critical density,\n\n$$\n\\rho_{\\text {crit }, 0}=\\frac{3 H_{0}^{2}}{8 \\pi G},\n$$\n\nwhere $H_{0}$ is the Hubble constant, measured in 2018 from the cosmic microwave background by the Planck spacecraft to be $67.36 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\n\nThe density of the $i^{\\text {th }}$ component of the Universe can be expressed relative to the critical density as the density parameter,\n\n$$\n\\Omega_{i}=\\frac{\\rho_{i}}{\\rho_{\\text {crit }}} .\n$$\n\nPlanck measured the current density parameters of dark energy and matter as $\\Omega_{\\Lambda, 0}=0.6847$ and $\\Omega_{m, 0}=0.3153$ respectively.\n\nIn each epoch, the scale factor increases at a different rate with time, $t$, as the density also varies differently with scale factor.\n\n- Radiation-dominated epoch: The Universe's early history, where $\\rho \\propto a^{-4}$ and so $a \\propto t^{1 / 2}$\n- Matter-dominated epoch: This represents much of the history of the Universe, where $\\rho \\propto$ $a^{-3}$ and so $a \\propto t^{2 / 3}$\n- Dark-energy-dominated epoch: This is an era we have recently entered and will remain in for the rest of time, where $\\rho$ doesn't vary with scale factor (i.e. is a constant) and so $a \\propto e^{H_{0} t}$\n\nAssuming the Universe has always been flat, find the time $t_{e q}$, corresponding to when the densities of matter and radiation were equal, given that data from Planck has allowed us to calculate the redshift of this to be $z_{e q}=3402$, and find the average density of the Universe at $t_{e q}$. Again, do not try and read any data off the graph.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nThe Universe consists of three main components: radiation (including neutrinos), matter (both atoms and dark matter), and dark energy. The overall density of the universe has been dominated by the density of each of those in turn at different times in its history, leading to three different epochs (shown in Fig 3).\n\n[figure1]\n\nFigure 3: A outline of the three main epochs in the history of the Universe. You do not need to read any data off this graph to answer this question. Credit: Pearson Education, Inc.\n\nThe scale factor, $a$, describes how the Universe has expanded (i.e. a measure of the relative radius of the Universe), and the current value is defined as $a_{0} \\equiv 1$ where the subscript ' 0 ' indicates it is as measured today. At earlier times $a<1$ and at the Big Bang $a=0$.\n\nThe redshift of an object, $z$, is related to the scale factor as $a=(1+z)^{-1}$ and so the redshift corresponding to now is $z=0$, and far away objects have higher redshift $(z>0)$ since we observe them as they were long ago when the scale factor was smaller.\n\nFor a Universe to be flat (i.e. zero curvature), its average density must be equal to the critical density,\n\n$$\n\\rho_{\\text {crit }, 0}=\\frac{3 H_{0}^{2}}{8 \\pi G},\n$$\n\nwhere $H_{0}$ is the Hubble constant, measured in 2018 from the cosmic microwave background by the Planck spacecraft to be $67.36 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\n\nThe density of the $i^{\\text {th }}$ component of the Universe can be expressed relative to the critical density as the density parameter,\n\n$$\n\\Omega_{i}=\\frac{\\rho_{i}}{\\rho_{\\text {crit }}} .\n$$\n\nPlanck measured the current density parameters of dark energy and matter as $\\Omega_{\\Lambda, 0}=0.6847$ and $\\Omega_{m, 0}=0.3153$ respectively.\n\nIn each epoch, the scale factor increases at a different rate with time, $t$, as the density also varies differently with scale factor.\n\n- Radiation-dominated epoch: The Universe's early history, where $\\rho \\propto a^{-4}$ and so $a \\propto t^{1 / 2}$\n- Matter-dominated epoch: This represents much of the history of the Universe, where $\\rho \\propto$ $a^{-3}$ and so $a \\propto t^{2 / 3}$\n- Dark-energy-dominated epoch: This is an era we have recently entered and will remain in for the rest of time, where $\\rho$ doesn't vary with scale factor (i.e. is a constant) and so $a \\propto e^{H_{0} t}$\n\nAssuming the Universe has always been flat, find the time $t_{e q}$, corresponding to when the densities of matter and radiation were equal, given that data from Planck has allowed us to calculate the redshift of this to be $z_{e q}=3402$, and find the average density of the Universe at $t_{e q}$. Again, do not try and read any data off the graph.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [when the densities of matter and radiation were equal, the average density of the Universe].\nTheir units are, in order, [years, $\\text{kg m}^{-3}$], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c3e3c992a9c51eb3e471g-08.jpg?height=1080&width=1271&top_left_y=739&top_left_x=398" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "years", "$\\text{kg m}^{-3}$" ], "answer_sequence": [ "when the densities of matter and radiation were equal", "the average density of the Universe" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_81", "problem": "2020 年 7 月 23 日, 我国“天宫一号”探测器在中国文昌航天发射场发射升空。设未来的某天, 该探测器在火星表面完成探测任务返回地球, 探测器在控制系统的指令下,离开火星表面坚直向上做加速直线运动; 探测器的内部有一固定的压力传感器, 质量为\n$m$ 的物体水平放置在压力传感器上, 当探测器上升到距火星表面高度为火星半径的 $\\frac{1}{4}$ 时,探测器的加速度为 $a$, 压力传感器的示数为 $F$, 引力常量为 $G$ 。忽略火星的自转, 则火星表面的重力加速度为 ( )\nA: $\\frac{5}{4}\\left(\\frac{F}{m}-a\\right)$\nB: $\\frac{25}{16}\\left(\\frac{F}{m}-a\\right)$\nC: $\\frac{4}{5}\\left(\\frac{F}{m}-a\\right)$\nD: $\\frac{16}{25}\\left(\\frac{F}{m}-a\\right)$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年 7 月 23 日, 我国“天宫一号”探测器在中国文昌航天发射场发射升空。设未来的某天, 该探测器在火星表面完成探测任务返回地球, 探测器在控制系统的指令下,离开火星表面坚直向上做加速直线运动; 探测器的内部有一固定的压力传感器, 质量为\n$m$ 的物体水平放置在压力传感器上, 当探测器上升到距火星表面高度为火星半径的 $\\frac{1}{4}$ 时,探测器的加速度为 $a$, 压力传感器的示数为 $F$, 引力常量为 $G$ 。忽略火星的自转, 则火星表面的重力加速度为 ( )\n\nA: $\\frac{5}{4}\\left(\\frac{F}{m}-a\\right)$\nB: $\\frac{25}{16}\\left(\\frac{F}{m}-a\\right)$\nC: $\\frac{4}{5}\\left(\\frac{F}{m}-a\\right)$\nD: $\\frac{16}{25}\\left(\\frac{F}{m}-a\\right)$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1152", "problem": "All stars lose mass during their lifetimes due to two main routes: particles escaping their surface (referred to as the stellar wind), and the mass defect of the nuclear reactions occurring in their cores.\n\nIn practice, the mass loss rate can vary quite considerably during a star's lifetime, particularly once it has left the main sequence when the stellar wind can become much more substantial. Wolf-Rayet stars are massive stars near the end of their lives, presumed to be the in the stage just before a supernova, and are losing substantial amounts of mass due to very fast stellar winds. This deposits considerable energy into the surrounding interstellar medium (ISM) and can sweep up material into a thin bubble around the star, visible as a type of planetary nebula.\n\n[figure1]\n\nFigure 1: The nebula NGC 2359 around the Wolf-Rayet star WR7. The nebula is known as Thor's Helmet due to its resemblance to the helmet worn by the character from the Marvel Comics series.\n\nCredit: Star Shadows Remote Observatory and PROMPT/UNC.c. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\\odot}$, a radius of $1.41 R_{\\odot}$ and a surface temperature of $112000 \\mathrm{~K}$, with a measured $v_{\\infty}$ of $1545 \\mathrm{~km} \\mathrm{~s}^{-1}$.\n\nUsing your new value for $\\dot{M}$, calculate the total mass expelled from the star and hence the total kinetic energy the stellar wind has so far deposited into the ISM during this stage of the star's life.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nAll stars lose mass during their lifetimes due to two main routes: particles escaping their surface (referred to as the stellar wind), and the mass defect of the nuclear reactions occurring in their cores.\n\nIn practice, the mass loss rate can vary quite considerably during a star's lifetime, particularly once it has left the main sequence when the stellar wind can become much more substantial. Wolf-Rayet stars are massive stars near the end of their lives, presumed to be the in the stage just before a supernova, and are losing substantial amounts of mass due to very fast stellar winds. This deposits considerable energy into the surrounding interstellar medium (ISM) and can sweep up material into a thin bubble around the star, visible as a type of planetary nebula.\n\n[figure1]\n\nFigure 1: The nebula NGC 2359 around the Wolf-Rayet star WR7. The nebula is known as Thor's Helmet due to its resemblance to the helmet worn by the character from the Marvel Comics series.\n\nCredit: Star Shadows Remote Observatory and PROMPT/UNC.\n\nproblem:\nc. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\\odot}$, a radius of $1.41 R_{\\odot}$ and a surface temperature of $112000 \\mathrm{~K}$, with a measured $v_{\\infty}$ of $1545 \\mathrm{~km} \\mathrm{~s}^{-1}$.\n\nUsing your new value for $\\dot{M}$, calculate the total mass expelled from the star and hence the total kinetic energy the stellar wind has so far deposited into the ISM during this stage of the star's life.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~km}^{3}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-04.jpg?height=811&width=1110&top_left_y=1236&top_left_x=473" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~km}^{3}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1151", "problem": "GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nThe rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as\n\n$$\n\\dot{f}_{\\mathrm{GW}}=\\frac{96}{5} \\pi^{8 / 3}\\left(\\frac{G \\mathcal{M}}{c^{3}}\\right)^{5 / 3} f_{\\mathrm{GW}}^{11 / 3}\n$$c. Given that the gravitational frequency, $f_{G W}$, is twice the orbital frequency (i.e. $f_{G W}=\\omega / \\pi$ ) and the 'chirp mass', $\\left.\\mathcal{M}=\\left(\\mu^{3} M_{\\text {tot }}\\right)^{2}\\right)^{1 / 5}$, express $h$ in terms of only $\\mathcal{M}, r, f_{G W}$, and various fundamental constants.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nGW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nThe rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as\n\n$$\n\\dot{f}_{\\mathrm{GW}}=\\frac{96}{5} \\pi^{8 / 3}\\left(\\frac{G \\mathcal{M}}{c^{3}}\\right)^{5 / 3} f_{\\mathrm{GW}}^{11 / 3}\n$$\n\nproblem:\nc. Given that the gravitational frequency, $f_{G W}$, is twice the orbital frequency (i.e. $f_{G W}=\\omega / \\pi$ ) and the 'chirp mass', $\\left.\\mathcal{M}=\\left(\\mu^{3} M_{\\text {tot }}\\right)^{2}\\right)^{1 / 5}$, express $h$ in terms of only $\\mathcal{M}, r, f_{G W}$, and various fundamental constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_969", "problem": "The website what 3 words splits up the Earth's surface into $3 \\mathrm{~m} \\times 3 \\mathrm{~m}$ squares and gives a coordinate of three randomly chosen words (for example the entrance to the Oxford University Physics Department is engage.proud.police). If each of the words is taken from the same list of $n$ words, what value of $n$ is needed?\nA: $\\sim 10000$\nB: $\\sim 20000$\nC: $\\sim 30000$\nD: $\\sim 40000$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe website what 3 words splits up the Earth's surface into $3 \\mathrm{~m} \\times 3 \\mathrm{~m}$ squares and gives a coordinate of three randomly chosen words (for example the entrance to the Oxford University Physics Department is engage.proud.police). If each of the words is taken from the same list of $n$ words, what value of $n$ is needed?\n\nA: $\\sim 10000$\nB: $\\sim 20000$\nC: $\\sim 30000$\nD: $\\sim 40000$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_995", "problem": "If a planet orbits the Sun with a semi-major axis of $4 \\mathrm{AU}$, what is the period of its orbit?\nA: 4 years\nB: 8 years\nC: 12 years\nD: 64 years\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIf a planet orbits the Sun with a semi-major axis of $4 \\mathrm{AU}$, what is the period of its orbit?\n\nA: 4 years\nB: 8 years\nC: 12 years\nD: 64 years\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_160", "problem": "地球和月球在长期相互作用过程中，形成了“潮汐锁定”月球总是一面正对地球，另一面背离地球, 月球绕地球的运动可看成匀速圆周运动。以下说法正确的是（）\n\n[图1]\nA: 月球的公转周期与自转周期相同\nB: 地球对月球的引力大于月球对地球的引力\nC: 月球上远地端的向心加速度大于近地端的向心加速度\nD: 若测得月球公转的周期和半径可估测月球质量\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n地球和月球在长期相互作用过程中，形成了“潮汐锁定”月球总是一面正对地球，另一面背离地球, 月球绕地球的运动可看成匀速圆周运动。以下说法正确的是（）\n\n[图1]\n\nA: 月球的公转周期与自转周期相同\nB: 地球对月球的引力大于月球对地球的引力\nC: 月球上远地端的向心加速度大于近地端的向心加速度\nD: 若测得月球公转的周期和半径可估测月球质量\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-025.jpg?height=408&width=454&top_left_y=190&top_left_x=378" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_656", "problem": "79. 关于人造卫星和宇宙飞船，下列说法正确的是（）\nA: 一艘绕地球运转的宇宙飞船, 宇航员从舱内慢慢走出, 并离开飞船, 飞船因质量减小，所受到的万有引力减小，故飞行速度减小\nB: 两颗人造卫星, 只要它们在圆形轨道的运行速度相等, 不管它们的质量、形状差别有多大，它们的运行速度相等，周期也相等\nC: 原来在同一轨道上沿同一方向运转的人造卫星一前一后, 若要后一个卫星追上前一个卫星并发生碰撞, 只要将后面一个卫星速率增大一些即可\nD: 关于航天飞机与空间站对接问题, 先让航天飞机进入较低的轨道, 然后再对其进行加速，即可实现对接\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n79. 关于人造卫星和宇宙飞船，下列说法正确的是（）\n\nA: 一艘绕地球运转的宇宙飞船, 宇航员从舱内慢慢走出, 并离开飞船, 飞船因质量减小，所受到的万有引力减小，故飞行速度减小\nB: 两颗人造卫星, 只要它们在圆形轨道的运行速度相等, 不管它们的质量、形状差别有多大，它们的运行速度相等，周期也相等\nC: 原来在同一轨道上沿同一方向运转的人造卫星一前一后, 若要后一个卫星追上前一个卫星并发生碰撞, 只要将后面一个卫星速率增大一些即可\nD: 关于航天飞机与空间站对接问题, 先让航天飞机进入较低的轨道, 然后再对其进行加速，即可实现对接\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_790", "problem": "Two asteroids with masses $m_{1}, m_{2}$ and velocities $v_{1}, v_{2}$ collide horizontally and merge into a single object. What is the velocity of the new asteroid?\n\n[figure1]\nA: $\\frac{m_{1} v_{1}+m_{2} v_{2}}{2}$\nB: $\\frac{m_{1} v_{1}+m_{2} v_{2}}{m_{1}+m_{2}}$\nC: $\\frac{m_{1} v_{1}-m_{2} v_{2}}{2}$\nD: $\\frac{m_{1} v_{1}-m_{2} v_{2}}{m_{1}+m_{2}}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo asteroids with masses $m_{1}, m_{2}$ and velocities $v_{1}, v_{2}$ collide horizontally and merge into a single object. What is the velocity of the new asteroid?\n\n[figure1]\n\nA: $\\frac{m_{1} v_{1}+m_{2} v_{2}}{2}$\nB: $\\frac{m_{1} v_{1}+m_{2} v_{2}}{m_{1}+m_{2}}$\nC: $\\frac{m_{1} v_{1}-m_{2} v_{2}}{2}$\nD: $\\frac{m_{1} v_{1}-m_{2} v_{2}}{m_{1}+m_{2}}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_620a57bf13ecc39e0534g-3.jpg?height=132&width=646&top_left_y=1416&top_left_x=705" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_580", "problem": "“天问一号”的发射为人类探索火星的奥秘打下了坚实的基础, 已知地球与火星的半\n径之比为 $a$, 地球与火星的质量之比为 $b$, 地球表面的重力加速度为 $g$ 。现将一汽缸开口向上放在火星表面的水平面上, 质量均为 $m$ 的活塞 $P 、 Q$ 将一定质量的气体密封且分成甲、乙两部分, 活塞 $P 、 Q$ 导热性能良好。已知火星表面的大气压为 $p_{0}$, 温度为 $T_{0}$,活塞的截面积为 $S$, 当系统平衡时, 甲、乙两部分气体的长度均为 $l_{0}, \\frac{m a^{2} g}{b}=p_{0} S$, 外界环境温度保持不变。现在活塞 $P$ 上缓慢地添加质量为 $2 \\mathrm{~m}$ 的砝码, 经过一段时间系统再次达到平衡。求:\n\n 系统平衡时, 将活塞 $P$ 锁定, 同时将活塞 $Q$ 换成不导热的活塞,缓慢地升高气体乙的温度, 当气柱乙的长度再次为 $l_{0}$ 时, 气体乙的温度应为多高?\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n“天问一号”的发射为人类探索火星的奥秘打下了坚实的基础, 已知地球与火星的半\n径之比为 $a$, 地球与火星的质量之比为 $b$, 地球表面的重力加速度为 $g$ 。现将一汽缸开口向上放在火星表面的水平面上, 质量均为 $m$ 的活塞 $P 、 Q$ 将一定质量的气体密封且分成甲、乙两部分, 活塞 $P 、 Q$ 导热性能良好。已知火星表面的大气压为 $p_{0}$, 温度为 $T_{0}$,活塞的截面积为 $S$, 当系统平衡时, 甲、乙两部分气体的长度均为 $l_{0}, \\frac{m a^{2} g}{b}=p_{0} S$, 外界环境温度保持不变。现在活塞 $P$ 上缓慢地添加质量为 $2 \\mathrm{~m}$ 的砝码, 经过一段时间系统再次达到平衡。求:\n\n 系统平衡时, 将活塞 $P$ 锁定, 同时将活塞 $Q$ 换成不导热的活塞,缓慢地升高气体乙的温度, 当气柱乙的长度再次为 $l_{0}$ 时, 气体乙的温度应为多高?\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-096.jpg?height=414&width=346&top_left_y=975&top_left_x=364" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_888", "problem": "The MIT students have found out about the plans of the Caltech students of sending a rocket to fly a parachute above the \"Great Dome\" of MIT. They know the rocket will follow an elliptical orbit of semiaxis a with the center of the Earth at one of its foci. To counter this prank, they will aim another rocket at the original rocket, which upon collision will transfer enough energy to the Caltech rocket to make it reach the escape velocity of the Earth at an early point in its trajectory and make it unable to reach back to the Earth's surface again. What is the energy that needs to be transferred to the Caltech rocket in order for the MIT students to reach their goal? Assume that the Earth has mass M, and the Caltech rocket has mass $\\mathrm{m}$.\nA: $G M m /(3 a)$\nB: $G M m /\\left(3 a^{\\wedge} 2\\right)$\nC: $G M m /(2 a)$\nD: $G M m / a$\nE: $G M m / a^{\\wedge} 2$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe MIT students have found out about the plans of the Caltech students of sending a rocket to fly a parachute above the \"Great Dome\" of MIT. They know the rocket will follow an elliptical orbit of semiaxis a with the center of the Earth at one of its foci. To counter this prank, they will aim another rocket at the original rocket, which upon collision will transfer enough energy to the Caltech rocket to make it reach the escape velocity of the Earth at an early point in its trajectory and make it unable to reach back to the Earth's surface again. What is the energy that needs to be transferred to the Caltech rocket in order for the MIT students to reach their goal? Assume that the Earth has mass M, and the Caltech rocket has mass $\\mathrm{m}$.\n\nA: $G M m /(3 a)$\nB: $G M m /\\left(3 a^{\\wedge} 2\\right)$\nC: $G M m /(2 a)$\nD: $G M m / a$\nE: $G M m / a^{\\wedge} 2$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_471", "problem": "如图所示, 两星球相距为 $L$, 质量比为 $\\mathrm{m}_{\\mathrm{A}}: \\mathrm{m}_{\\mathrm{B}}=1: 9$, 两星球半径远小于 $L$. 从星球 $A$ 沿 $A 、 B$ 连线向 $B$ 以某一初速度发射一探测器. 只考虑星球 $A 、 B$ 对探测器的作用,下列说法正确的是( )\n\n[图1]\nA: 探测器的速度一直减小\nB: 探测器在距星球 $A$ 为 $\\frac{L}{4}$ 处加速度为零\nC: 若探测器能到达星球 $B$, 其速度可能恰好为零\nD: 若探测器能到达星球 $B$, 其速度一定大于发射时的初速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 两星球相距为 $L$, 质量比为 $\\mathrm{m}_{\\mathrm{A}}: \\mathrm{m}_{\\mathrm{B}}=1: 9$, 两星球半径远小于 $L$. 从星球 $A$ 沿 $A 、 B$ 连线向 $B$ 以某一初速度发射一探测器. 只考虑星球 $A 、 B$ 对探测器的作用,下列说法正确的是( )\n\n[图1]\n\nA: 探测器的速度一直减小\nB: 探测器在距星球 $A$ 为 $\\frac{L}{4}$ 处加速度为零\nC: 若探测器能到达星球 $B$, 其速度可能恰好为零\nD: 若探测器能到达星球 $B$, 其速度一定大于发射时的初速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://i.postimg.cc/D0p74Vzw/image.png" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1157", "problem": "Recently a group of researchers announced that they had discovered an Earth-sized exoplanet around our nearest star, Proxima Centauri. Its closeness raises an intriguing possibility about whether or not we might be able to image it directly using telescopes. The difficulty comes from the small angular scales that need to be resolved and the extreme differences in brightness between the reflected light from the planet and the light given out by the star.\n\n[figure1]\n\nFigure 6: Artist's impression of the view from the surface of Proxima Centauri b. Credit: ESO / M. Kornmesser\n\nData about the star and the planet are summarised below:\n\n| Proxima Centauri (star) | | Proxima Centauri b (planet) | |\n| :--- | :--- | :--- | :--- |\n| Distance | $1.295 \\mathrm{pc}$ | Orbital period | 11.186 days |\n| Mass | $0.123 \\mathrm{M}_{\\odot}$ | Mass $(\\mathrm{min})$ | $\\approx 1.27 \\mathrm{M}_{\\oplus}$ |\n| Radius | $0.141 \\mathrm{R}_{\\odot}$ | Radius $(\\mathrm{min})$ | $\\approx 1.1 \\mathrm{R}_{\\oplus}$ |\n| Surface temperature | $3042 \\mathrm{~K}$ | | |\n| Apparent magnitude | 11.13 | | |\n\nThe following formulae may also be helpful:\n\n$$\nm-\\mathcal{M}=5 \\log \\left(\\frac{d}{10}\\right) \\quad \\mathcal{M}-\\mathcal{M}_{\\odot}=-2.5 \\log \\left(\\frac{L}{\\mathrm{~L}_{\\odot}}\\right) \\quad \\Delta m=2.5 \\log C R\n$$\n\nwhere $m$ is the apparent magnitude, $\\mathcal{M}$ is the absolute magnitude, $d$ is the distance in parsecs, and the contrast ratio $(C R)$ is defined as the ratio of fluxes from the star and planet, $C R=\\frac{f_{\\text {star }}}{f_{\\text {planet }}}$.a. Calculate the maximum angular separation between the star and the planet, assuming a circular orbit. Give your answer in arcseconds (where $3600 \\operatorname{arcseconds}=1^{\\circ}$ ).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nRecently a group of researchers announced that they had discovered an Earth-sized exoplanet around our nearest star, Proxima Centauri. Its closeness raises an intriguing possibility about whether or not we might be able to image it directly using telescopes. The difficulty comes from the small angular scales that need to be resolved and the extreme differences in brightness between the reflected light from the planet and the light given out by the star.\n\n[figure1]\n\nFigure 6: Artist's impression of the view from the surface of Proxima Centauri b. Credit: ESO / M. Kornmesser\n\nData about the star and the planet are summarised below:\n\n| Proxima Centauri (star) | | Proxima Centauri b (planet) | |\n| :--- | :--- | :--- | :--- |\n| Distance | $1.295 \\mathrm{pc}$ | Orbital period | 11.186 days |\n| Mass | $0.123 \\mathrm{M}_{\\odot}$ | Mass $(\\mathrm{min})$ | $\\approx 1.27 \\mathrm{M}_{\\oplus}$ |\n| Radius | $0.141 \\mathrm{R}_{\\odot}$ | Radius $(\\mathrm{min})$ | $\\approx 1.1 \\mathrm{R}_{\\oplus}$ |\n| Surface temperature | $3042 \\mathrm{~K}$ | | |\n| Apparent magnitude | 11.13 | | |\n\nThe following formulae may also be helpful:\n\n$$\nm-\\mathcal{M}=5 \\log \\left(\\frac{d}{10}\\right) \\quad \\mathcal{M}-\\mathcal{M}_{\\odot}=-2.5 \\log \\left(\\frac{L}{\\mathrm{~L}_{\\odot}}\\right) \\quad \\Delta m=2.5 \\log C R\n$$\n\nwhere $m$ is the apparent magnitude, $\\mathcal{M}$ is the absolute magnitude, $d$ is the distance in parsecs, and the contrast ratio $(C R)$ is defined as the ratio of fluxes from the star and planet, $C R=\\frac{f_{\\text {star }}}{f_{\\text {planet }}}$.\n\nproblem:\na. Calculate the maximum angular separation between the star and the planet, assuming a circular orbit. Give your answer in arcseconds (where $3600 \\operatorname{arcseconds}=1^{\\circ}$ ).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-10.jpg?height=708&width=1082&top_left_y=551&top_left_x=493" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_338", "problem": "已知某卫星在赤道上空轨道半径为 $r_{1}$ 的圆形轨道上绕地球运行的周期为 $T$, 卫星运动方向与地球自转方向相同, 赤道上某城市的人每三天恰好五次看到该卫星掠过其正上方。假设某时刻该卫星在 $A$ 点变轨进入椭圆轨道, 近地点 $B$ 到地心距离为 $r_{2}$ 。如图所示设卫星由 $A$ 到 $B$ (只经 $B$ 点一次) 运动的时间为 $t$, 地球自转周期为 $T_{0}$, 不计空气阻力,则 ( )\n\n[图1]\nA: $T=\\frac{3 T_{0}}{8}$\nB: $T=\\frac{3 T_{0}}{5}$\nC: $t=\\frac{\\left(r_{1}+r_{2}\\right) T}{4 r_{1}} \\sqrt{\\frac{r_{1}+r_{2}}{2 r_{1}}}$\nD: $t=\\frac{\\left(r_{1}+r_{2}\\right) T}{6 r_{1}} \\sqrt{\\frac{r_{1}+r_{2}}{2 r_{1}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n已知某卫星在赤道上空轨道半径为 $r_{1}$ 的圆形轨道上绕地球运行的周期为 $T$, 卫星运动方向与地球自转方向相同, 赤道上某城市的人每三天恰好五次看到该卫星掠过其正上方。假设某时刻该卫星在 $A$ 点变轨进入椭圆轨道, 近地点 $B$ 到地心距离为 $r_{2}$ 。如图所示设卫星由 $A$ 到 $B$ (只经 $B$ 点一次) 运动的时间为 $t$, 地球自转周期为 $T_{0}$, 不计空气阻力,则 ( )\n\n[图1]\n\nA: $T=\\frac{3 T_{0}}{8}$\nB: $T=\\frac{3 T_{0}}{5}$\nC: $t=\\frac{\\left(r_{1}+r_{2}\\right) T}{4 r_{1}} \\sqrt{\\frac{r_{1}+r_{2}}{2 r_{1}}}$\nD: $t=\\frac{\\left(r_{1}+r_{2}\\right) T}{6 r_{1}} \\sqrt{\\frac{r_{1}+r_{2}}{2 r_{1}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-109.jpg?height=400&width=414&top_left_y=628&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_475", "problem": "宇宙中某一质量为 $M$ 、半径为 $R$ 的星球, 有三颗卫星 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 在同一平面上沿逆时针方向做匀速圆周运动, 其位置关系如图所示。其中 $\\mathrm{A}$ 到该星球表面的高度为 $h$, 已知引力常量为 $G$, 则下列说法正确的是 ( )\n\n[图1]\nA: 三颗卫星的向心加速度大小关系为 $a_{\\mathrm{A}}v_{\\mathrm{B}}=v_{\\mathrm{C}}$\nC: 卫星 $\\mathrm{C}$ 加速后可以追到卫星 $\\mathrm{B}$\nD: 卫星 $\\mathrm{A}$ 的公转周期为 $2 \\pi \\sqrt{\\frac{h^{3}}{G M}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n宇宙中某一质量为 $M$ 、半径为 $R$ 的星球, 有三颗卫星 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 在同一平面上沿逆时针方向做匀速圆周运动, 其位置关系如图所示。其中 $\\mathrm{A}$ 到该星球表面的高度为 $h$, 已知引力常量为 $G$, 则下列说法正确的是 ( )\n\n[图1]\n\nA: 三颗卫星的向心加速度大小关系为 $a_{\\mathrm{A}}v_{\\mathrm{B}}=v_{\\mathrm{C}}$\nC: 卫星 $\\mathrm{C}$ 加速后可以追到卫星 $\\mathrm{B}$\nD: 卫星 $\\mathrm{A}$ 的公转周期为 $2 \\pi \\sqrt{\\frac{h^{3}}{G M}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-103.jpg?height=383&width=436&top_left_y=2127&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_751", "problem": "What is the name of Jupiter's moon shown in the figure below?\n\n[figure1]\nA: Io\nB: Europa\nC: Callisto\nD: Ganymede\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the name of Jupiter's moon shown in the figure below?\n\n[figure1]\n\nA: Io\nB: Europa\nC: Callisto\nD: Ganymede\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_620a57bf13ecc39e0534g-6.jpg?height=386&width=397&top_left_y=1760&top_left_x=841" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_840", "problem": "A comet passes near the Sun on a parabolic orbit. While it's passing near the Sun with orbital velocity $V$, the Sun's heat causes the comet to melt, and it shatters into many small fragments. The fragments move away uniformly in all directions (in the comet's reference frame) with velocity $v \\ll V$. What fraction of the fragments will escape the solar system? Ignore any forces other than the Sun's gravity.\nA: $0 \\%$\nB: $50 \\%$\nC: $100 \\%$\nD: $\\frac{v}{V}$\nE: $1-\\frac{v}{V}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA comet passes near the Sun on a parabolic orbit. While it's passing near the Sun with orbital velocity $V$, the Sun's heat causes the comet to melt, and it shatters into many small fragments. The fragments move away uniformly in all directions (in the comet's reference frame) with velocity $v \\ll V$. What fraction of the fragments will escape the solar system? Ignore any forces other than the Sun's gravity.\n\nA: $0 \\%$\nB: $50 \\%$\nC: $100 \\%$\nD: $\\frac{v}{V}$\nE: $1-\\frac{v}{V}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1072", "problem": "A day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4).\n\n[figure1]\n\nFigure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun.\n\nCredit: Wikipedia.\n\nThe length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time.\n\n[Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \\times 10^{-11} \\mathrm{~m}^{3} \\mathrm{~kg}^{-1} \\mathrm{~s}^{-2}, R_{\\oplus}=6371 \\mathrm{~km}, M_{\\oplus}=5.972 \\times 10^{24} \\mathrm{~kg}, M_{\\odot}=$ $1.989 \\times 10^{30} \\mathrm{~kg}$ and $1 \\mathrm{au}=1.496 \\times 10^{11} \\mathrm{~m}$.]\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\\frac{2}{5} M R^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\omega$ is $L=I \\omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.\n\nThe speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$d. Imagine creating an incredibly powerful rocket, positioned on the Earth's equator, that when fired once can apply a huge force to the Earth in a very short time period, delivering a total impulse of $\\Delta p$. Assuming the Earth's orbit is initially circular, calculate:\n\ni. The total impulse required to slow the Earth's rotation to give a year of 360 solar days, but with no change in the orbit.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nA day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4).\n\n[figure1]\n\nFigure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun.\n\nCredit: Wikipedia.\n\nThe length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time.\n\n[Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \\times 10^{-11} \\mathrm{~m}^{3} \\mathrm{~kg}^{-1} \\mathrm{~s}^{-2}, R_{\\oplus}=6371 \\mathrm{~km}, M_{\\oplus}=5.972 \\times 10^{24} \\mathrm{~kg}, M_{\\odot}=$ $1.989 \\times 10^{30} \\mathrm{~kg}$ and $1 \\mathrm{au}=1.496 \\times 10^{11} \\mathrm{~m}$.]\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\\frac{2}{5} M R^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\omega$ is $L=I \\omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.\n\nThe speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nproblem:\nd. Imagine creating an incredibly powerful rocket, positioned on the Earth's equator, that when fired once can apply a huge force to the Earth in a very short time period, delivering a total impulse of $\\Delta p$. Assuming the Earth's orbit is initially circular, calculate:\n\ni. The total impulse required to slow the Earth's rotation to give a year of 360 solar days, but with no change in the orbit.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~kg} \\mathrm{~m} \\mathrm{~s}^{-1}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-06.jpg?height=1276&width=782&top_left_y=567&top_left_x=657" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~kg} \\mathrm{~m} \\mathrm{~s}^{-1}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_864", "problem": "Suppose you are in Houston $\\left(29^{\\circ} 46^{\\prime} N 95^{\\circ} 23^{\\prime} W\\right)$ on the fall equinox and you just observed Deneb culminating (upper culmination). Knowing the data in the table of exercise 24, what is the hour angle of the Sun?\nA: 8h41min\nB: 20h41min\nC: $12 \\mathrm{~h} 00 \\mathrm{~min}$\nD: $14 \\mathrm{~h} 19 \\mathrm{~min}$\nE: $18 \\mathrm{~h} 22 \\mathrm{~min}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nSuppose you are in Houston $\\left(29^{\\circ} 46^{\\prime} N 95^{\\circ} 23^{\\prime} W\\right)$ on the fall equinox and you just observed Deneb culminating (upper culmination). Knowing the data in the table of exercise 24, what is the hour angle of the Sun?\n\nA: 8h41min\nB: 20h41min\nC: $12 \\mathrm{~h} 00 \\mathrm{~min}$\nD: $14 \\mathrm{~h} 19 \\mathrm{~min}$\nE: $18 \\mathrm{~h} 22 \\mathrm{~min}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_380", "problem": "如图所示, 质量分别为 $M$ 和 $m$ 的两个星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ (均视为质点) 在它们之间的引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 之间的距离为 $L$ 。已知星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 和 $O$三点始终共线, $\\mathrm{A}$ 和 $\\mathrm{B}$ 分别在 $O$ 点的两侧。引力常量为 $G$ 。\n\n求星球 $\\mathrm{A}$ 的周期 $T$;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 质量分别为 $M$ 和 $m$ 的两个星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ (均视为质点) 在它们之间的引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 之间的距离为 $L$ 。已知星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 和 $O$三点始终共线, $\\mathrm{A}$ 和 $\\mathrm{B}$ 分别在 $O$ 点的两侧。引力常量为 $G$ 。\n\n求星球 $\\mathrm{A}$ 的周期 $T$;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-031.jpg?height=429&width=488&top_left_y=154&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_559", "problem": "放置在水平平台上的物体, 其表观重力在数值上等于物体对平台的压力, 方向与压力的方向相同。微重力环境是指系统内物体的表观重力远小于其实际重力（万有引力）的环境。此环境下，物体的表观重力与其质量之比称为微重力加速度。\n\n如图所示, 中国科学院力学研究所微重力实验室落塔是我国微重力实验的主要设施之一, 实验中落舱可采用单舱和双舱两种模式进行。已知地球表面的重力加速度为 $g$;\n环绕地球做匀速圆周运动的人造卫星内部也存在微重力环境. 其产生原因简单来说是由于卫星实验舱不能被看作质点造成的, 只有在卫星的质心（质点系的质量中心）位置, 万有引力才恰好等于向心力. 已知某卫星绕地球做匀速圆周运动, 其质心到地心的距离为 $r$, 假设卫星实验舱中各点绕图中地球运动的角速度均与质心一致, 求 $g_{4}$ 与该卫星质心处的向心加速度 $a_{n}$ 的比值。\n[图1]\n\n落塔 落舱", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n放置在水平平台上的物体, 其表观重力在数值上等于物体对平台的压力, 方向与压力的方向相同。微重力环境是指系统内物体的表观重力远小于其实际重力（万有引力）的环境。此环境下，物体的表观重力与其质量之比称为微重力加速度。\n\n如图所示, 中国科学院力学研究所微重力实验室落塔是我国微重力实验的主要设施之一, 实验中落舱可采用单舱和双舱两种模式进行。已知地球表面的重力加速度为 $g$;\n环绕地球做匀速圆周运动的人造卫星内部也存在微重力环境. 其产生原因简单来说是由于卫星实验舱不能被看作质点造成的, 只有在卫星的质心（质点系的质量中心）位置, 万有引力才恰好等于向心力. 已知某卫星绕地球做匀速圆周运动, 其质心到地心的距离为 $r$, 假设卫星实验舱中各点绕图中地球运动的角速度均与质心一致, 求 $g_{4}$ 与该卫星质心处的向心加速度 $a_{n}$ 的比值。\n[图1]\n\n落塔 落舱\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-033.jpg?height=317&width=808&top_left_y=201&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_361", "problem": "宇宙中存在一些离其他恒星较远的三星系统, 通常可忽略其他星体对它们的引力作用, 现已观测到稳定的三星系统存在两种基本的构成形式: 一种是三颗星位于同一直线上, 两颗星围绕中央星做圆周运动, 如图甲所示; 另一种是三颗星位于等边三角形的三个顶点上, 并沿外接于等边三角形的圆形轨道运行, 如图乙所示, 设两种系统中三个星体的质量及各星间的距离如图甲、乙中所示, 已知引力常量为 $G$, 试分别求出两个系统做圆周运动的周期。\n[图1]\n\n甲\n\n乙", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n宇宙中存在一些离其他恒星较远的三星系统, 通常可忽略其他星体对它们的引力作用, 现已观测到稳定的三星系统存在两种基本的构成形式: 一种是三颗星位于同一直线上, 两颗星围绕中央星做圆周运动, 如图甲所示; 另一种是三颗星位于等边三角形的三个顶点上, 并沿外接于等边三角形的圆形轨道运行, 如图乙所示, 设两种系统中三个星体的质量及各星间的距离如图甲、乙中所示, 已知引力常量为 $G$, 试分别求出两个系统做圆周运动的周期。\n[图1]\n\n甲\n\n乙\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[甲系统做圆周运动的周期, 乙系统做圆周运动的周期]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-091.jpg?height=334&width=1004&top_left_y=1169&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "甲系统做圆周运动的周期", "乙系统做圆周运动的周期" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_176", "problem": "宇航员来到某星球表面做了如下实验: 将一小钢球由距星球表面高 $h(h$ 远小于星球半径）处由静止释放, 小钢球经过时间 $t$ 落到星球表面, 该星球为密度均匀的球体,引力常量为 $G$ 。\n\n若该星球的半径为 $R$, 有一颗卫星在距该星球表面高度为 $H$ 处的圆轨道上绕该星球做匀速圆周运动, 求该卫星的线速度大小和周期。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n宇航员来到某星球表面做了如下实验: 将一小钢球由距星球表面高 $h(h$ 远小于星球半径）处由静止释放, 小钢球经过时间 $t$ 落到星球表面, 该星球为密度均匀的球体,引力常量为 $G$ 。\n\n若该星球的半径为 $R$, 有一颗卫星在距该星球表面高度为 $H$ 处的圆轨道上绕该星球做匀速圆周运动, 求该卫星的线速度大小和周期。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[线速度大小, 周期]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "线速度大小", "周期" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_496", "problem": "假设宇宙是一团球形的密度均匀的物质, 其各物理量均具有球对称性（即只与球的半径有关)。宇宙球对称地向外膨胀, 半径为 $r$ 的位置具有速度 $v(r)$ 。不难发现, 宇宙膨胀的过程中, 其平均密度必然下降。若假设该宇宙球在膨胀过程中密度均匀（即球内各处密度相等), 则应该有 $v=\\mathrm{H} r^{\\alpha}$, 其中 $\\mathrm{H}$ 是一个可变化但与 $r$ 无关的系数, 那么 $\\alpha$ 的值应为 ( )\n\n[提示: 若 $p(t)$ 是某一物理量, 则 $p^{a}$ 对时间的导数为 $a p^{a-1} p^{\\prime}(t)$ ]\nA: 1\nB: 2\nC: 3\nD: 4\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n假设宇宙是一团球形的密度均匀的物质, 其各物理量均具有球对称性（即只与球的半径有关)。宇宙球对称地向外膨胀, 半径为 $r$ 的位置具有速度 $v(r)$ 。不难发现, 宇宙膨胀的过程中, 其平均密度必然下降。若假设该宇宙球在膨胀过程中密度均匀（即球内各处密度相等), 则应该有 $v=\\mathrm{H} r^{\\alpha}$, 其中 $\\mathrm{H}$ 是一个可变化但与 $r$ 无关的系数, 那么 $\\alpha$ 的值应为 ( )\n\n[提示: 若 $p(t)$ 是某一物理量, 则 $p^{a}$ 对时间的导数为 $a p^{a-1} p^{\\prime}(t)$ ]\n\nA: 1\nB: 2\nC: 3\nD: 4\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_327", "problem": "发射地球同步卫星时, 先将卫星发射至近地圆轨道 1, 然后经点火, 使其沿椭圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3. 轨道 $1 、 2$ 相切于 $Q$ 点, 轨道 $2 、 3$相切于 $P$ 点, 如图所示, 则当卫星分别在 $1 、 2 、 3$ 轨道上正常运行时, 以下说法不正确的是 ( )\n\n[图1]\nA: 要将卫星由圆轨道 1 送入圆轨道 3 , 需要在圆轨道 1 的 $Q$ 点和椭圆轨道 2 的远地点 $P$ 分别点火加速一次\nB: 由于卫星由圆轨道 1 送入圆轨道 3 点火加速两次, 则卫星在圆轨道 3 上正常运行速度大于卫星在圆轨道 1 上正常运行速度\nC: 卫星在椭圆轨道 2 上的近地点 $Q$ 的速度一定大于 $7.9 \\mathrm{~km} / \\mathrm{s}$, 而在远地点 $P$ 的速度一定小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nD: 卫星在椭圆轨道 2 上经过 $P$ 点时的加速度一定等于它在圆轨道 3 上经过 $P$ 点时的加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n发射地球同步卫星时, 先将卫星发射至近地圆轨道 1, 然后经点火, 使其沿椭圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3. 轨道 $1 、 2$ 相切于 $Q$ 点, 轨道 $2 、 3$相切于 $P$ 点, 如图所示, 则当卫星分别在 $1 、 2 、 3$ 轨道上正常运行时, 以下说法不正确的是 ( )\n\n[图1]\n\nA: 要将卫星由圆轨道 1 送入圆轨道 3 , 需要在圆轨道 1 的 $Q$ 点和椭圆轨道 2 的远地点 $P$ 分别点火加速一次\nB: 由于卫星由圆轨道 1 送入圆轨道 3 点火加速两次, 则卫星在圆轨道 3 上正常运行速度大于卫星在圆轨道 1 上正常运行速度\nC: 卫星在椭圆轨道 2 上的近地点 $Q$ 的速度一定大于 $7.9 \\mathrm{~km} / \\mathrm{s}$, 而在远地点 $P$ 的速度一定小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nD: 卫星在椭圆轨道 2 上经过 $P$ 点时的加速度一定等于它在圆轨道 3 上经过 $P$ 点时的加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-10.jpg?height=405&width=391&top_left_y=2359&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1137", "problem": "It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1).\n[figure1]\n\nFigure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\\sim 40^{\\circ}$. Credit: Daniel V. Schroeder / Weber State University.\n\nRight: The same idea but viewed from Iceland at a latitude of $65^{\\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland.\n\nDuring the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees.\n\nA simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\\delta$, is given as:\n\n$$\nh=-\\left(90^{\\circ}-\\phi\\right) \\cos (A)+\\delta\n$$\n\nThe solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\\delta=+23.44^{\\circ}$ at the June solstice (roughly $21^{\\text {st }}$ June) to a minimum of $\\delta=-23.44^{\\circ}$ on the December solstice (roughly $21^{\\text {st }}$ December).\n\nIt can be shown using spherical trigonometry that the precise model connecting $\\delta, h, \\phi$ and $A$ is:\n\n$$\n\\sin (\\delta)=\\sin (h) \\sin (\\phi)+\\cos (h) \\cos (\\phi) \\cos (A) .\n$$\n\nUsing the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2.\n\n[figure2]\n\nFigure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia.\n\nBy using further spherical trigonometry, we can derive a second helpful equation in the precise model:\n\n$$\n\\sin (h)=\\sin (\\phi) \\sin (\\delta)+\\cos (\\phi) \\cos (\\delta) \\cos (H)\n$$\n\nHere, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\\circ}=24^{\\mathrm{h}}$.c. Reconsider the Oxford observer at the June solstice, but this time use the two equations of the precise model. Ignore any atmospheric effects.\n\nii. Calculate the duration of sunrise (in minutes and seconds), assuming a solar angular diameter of $0.525^{\\circ}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\nHere is some context information for this question, which might assist you in solving it:\nIt is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1).\n[figure1]\n\nFigure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\\sim 40^{\\circ}$. Credit: Daniel V. Schroeder / Weber State University.\n\nRight: The same idea but viewed from Iceland at a latitude of $65^{\\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland.\n\nDuring the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees.\n\nA simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\\delta$, is given as:\n\n$$\nh=-\\left(90^{\\circ}-\\phi\\right) \\cos (A)+\\delta\n$$\n\nThe solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\\delta=+23.44^{\\circ}$ at the June solstice (roughly $21^{\\text {st }}$ June) to a minimum of $\\delta=-23.44^{\\circ}$ on the December solstice (roughly $21^{\\text {st }}$ December).\n\nIt can be shown using spherical trigonometry that the precise model connecting $\\delta, h, \\phi$ and $A$ is:\n\n$$\n\\sin (\\delta)=\\sin (h) \\sin (\\phi)+\\cos (h) \\cos (\\phi) \\cos (A) .\n$$\n\nUsing the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2.\n\n[figure2]\n\nFigure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia.\n\nBy using further spherical trigonometry, we can derive a second helpful equation in the precise model:\n\n$$\n\\sin (h)=\\sin (\\phi) \\sin (\\delta)+\\cos (\\phi) \\cos (\\delta) \\cos (H)\n$$\n\nHere, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\\circ}=24^{\\mathrm{h}}$.\n\nproblem:\nc. Reconsider the Oxford observer at the June solstice, but this time use the two equations of the precise model. Ignore any atmospheric effects.\n\nii. Calculate the duration of sunrise (in minutes and seconds), assuming a solar angular diameter of $0.525^{\\circ}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [H_{-}, H_{+}].\nTheir units are, in order, [\\circ, \\circ], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-04.jpg?height=668&width=1478&top_left_y=523&top_left_x=290", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-05.jpg?height=648&width=1234&top_left_y=738&top_left_x=385" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "\\circ", "\\circ" ], "answer_sequence": [ "H_{-}", "H_{+}" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_836", "problem": "After that slight headache, Austin is back at MIT in Boston! For his astronomy research, he is observing the LARES satellite which is a ball of diameter $36.4 \\mathrm{~cm}$ made out of THA-18N (a tungsten alloy). It orbits at a distance $1450 \\mathrm{~km}$ from the surface of the Earth and at an inclination of $69.49^{\\circ}$ relative to the equatorial plane. What is the highest altitude Austin can point his telescope if he wants to observe LARES at its highest latitude?\nA: $9.4^{\\circ}$\nB: $14.4^{\\circ}$\nC: $18.4^{\\circ}$\nD: $23.4^{\\circ}$\nE: $33.4^{\\circ}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAfter that slight headache, Austin is back at MIT in Boston! For his astronomy research, he is observing the LARES satellite which is a ball of diameter $36.4 \\mathrm{~cm}$ made out of THA-18N (a tungsten alloy). It orbits at a distance $1450 \\mathrm{~km}$ from the surface of the Earth and at an inclination of $69.49^{\\circ}$ relative to the equatorial plane. What is the highest altitude Austin can point his telescope if he wants to observe LARES at its highest latitude?\n\nA: $9.4^{\\circ}$\nB: $14.4^{\\circ}$\nC: $18.4^{\\circ}$\nD: $23.4^{\\circ}$\nE: $33.4^{\\circ}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_34", "problem": "2021 年 5 月，基于俗称“中国天眼”的 500 米口径球面射电望远镜（FAST）的观测,国家天文台李䓎、朱炜玮研究团组姚菊枚博士等首次研究发现脉冲星三维速度与自转轴共线的证据。之前的 2020 年 3 月, 我国天文学家通过 FAST, 在武仙座球状星团 （M13）中发现一个脉冲双星系统。如图所示，假设在太空中有恒星 A、B 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}<$ $R_{B}, \\mathrm{C}$ 为 $\\mathrm{B}$ 的卫星, 绕 $\\mathrm{B}$ 做逆时针匀速圆周运动, 周期为 $T_{2}$, 忽略 $\\mathrm{A}$ 与 $\\mathrm{C}$ 之间的引力,万引力常量为 $G$, 则以下说法正确的是 ( )\n\n[图1]\nA: 若知道 $\\mathrm{C}$ 的轨道半径, 则可求出 $\\mathrm{C}$ 的质量\nB: 恒星 $\\mathrm{A} 、 \\mathrm{~B}$ 的质量和为 $\\frac{4 \\pi^{2}\\left(R_{\\mathrm{A}}+R_{\\mathrm{B}}\\right)^{3}}{G T_{1}^{2}}$\nC: 若 $\\mathrm{A}$ 也有一颗运动周期为 $\\mathrm{T}_{2}$ 的卫星, 则其轨道半径大于 $\\mathrm{C}$ 的轨道半径\nD: 设 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三星由图示位置到再次共线的时间为 $t$, 则 $t=\\frac{T_{1} T_{2}}{\\left.2 T_{1}+T_{2}\\right)}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2021 年 5 月，基于俗称“中国天眼”的 500 米口径球面射电望远镜（FAST）的观测,国家天文台李䓎、朱炜玮研究团组姚菊枚博士等首次研究发现脉冲星三维速度与自转轴共线的证据。之前的 2020 年 3 月, 我国天文学家通过 FAST, 在武仙座球状星团 （M13）中发现一个脉冲双星系统。如图所示，假设在太空中有恒星 A、B 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}<$ $R_{B}, \\mathrm{C}$ 为 $\\mathrm{B}$ 的卫星, 绕 $\\mathrm{B}$ 做逆时针匀速圆周运动, 周期为 $T_{2}$, 忽略 $\\mathrm{A}$ 与 $\\mathrm{C}$ 之间的引力,万引力常量为 $G$, 则以下说法正确的是 ( )\n\n[图1]\n\nA: 若知道 $\\mathrm{C}$ 的轨道半径, 则可求出 $\\mathrm{C}$ 的质量\nB: 恒星 $\\mathrm{A} 、 \\mathrm{~B}$ 的质量和为 $\\frac{4 \\pi^{2}\\left(R_{\\mathrm{A}}+R_{\\mathrm{B}}\\right)^{3}}{G T_{1}^{2}}$\nC: 若 $\\mathrm{A}$ 也有一颗运动周期为 $\\mathrm{T}_{2}$ 的卫星, 则其轨道半径大于 $\\mathrm{C}$ 的轨道半径\nD: 设 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三星由图示位置到再次共线的时间为 $t$, 则 $t=\\frac{T_{1} T_{2}}{\\left.2 T_{1}+T_{2}\\right)}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-088.jpg?height=314&width=371&top_left_y=891&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_636", "problem": "如图所示，宇宙飞船绕地球做圆周运动时，由于地球遮挡阳光，会经历“日全食”过\n\n程, 太阳光可看作平行光, 宇航员在 $A$ 点测出地球的张角为 $\\alpha$ 。已知地球半径为 $R$, 地\n球质量为 $\\mathrm{M}$, 引力常量为 $G$, 不考虑地球公转的影响。求:\n\n飞船运行的高度 $h$;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示，宇宙飞船绕地球做圆周运动时，由于地球遮挡阳光，会经历“日全食”过\n\n程, 太阳光可看作平行光, 宇航员在 $A$ 点测出地球的张角为 $\\alpha$ 。已知地球半径为 $R$, 地\n球质量为 $\\mathrm{M}$, 引力常量为 $G$, 不考虑地球公转的影响。求:\n\n飞船运行的高度 $h$;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-086.jpg?height=277&width=514&top_left_y=393&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_945", "problem": "The One-Mile Telescope is a radio telescope just outside Cambridge that has three dishes that can be spread out up to one mile $(=1.6 \\mathrm{~km})$ apart. Two of the dishes are fixed, whilst one can move along an $800-\\mathrm{m}$ set of former railway tracks. In order for the tracks to be perfectly flat, how much did they need to raise one end to compensate for the curvature of the Earth?\nA: $5 \\mathrm{~cm}$\nB: $10 \\mathrm{~cm}$\nC: $15 \\mathrm{~cm}$\nD: $20 \\mathrm{~cm}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe One-Mile Telescope is a radio telescope just outside Cambridge that has three dishes that can be spread out up to one mile $(=1.6 \\mathrm{~km})$ apart. Two of the dishes are fixed, whilst one can move along an $800-\\mathrm{m}$ set of former railway tracks. In order for the tracks to be perfectly flat, how much did they need to raise one end to compensate for the curvature of the Earth?\n\nA: $5 \\mathrm{~cm}$\nB: $10 \\mathrm{~cm}$\nC: $15 \\mathrm{~cm}$\nD: $20 \\mathrm{~cm}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://i.postimg.cc/y6nqF3xs/Screenshot-2024-04-06-at-22-47-32.png" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_470", "problem": "在 X 星球表面, 宇航员做了一个实验：如图甲所示, 轻杆一端固定在 $O$ 点, 另一端固定一小球，现让小球在坚直平面内做半径为 $R$ 的圆周运动。小球运动到最高点时,受到的弹力为 $F$, 速度大小为 $v$, 其 $F-v^{2}$ 图像如乙图所示。已知 $\\mathrm{X}$ 星球的半径为 $R_{0}$, 万有引力常量为 $G$, 不考虑星球自转。则下列说法正确的是 ( )\n\n[图1]\n\n甲\n\n[图2]\n\n乙\nA: X 星球的第一宇宙速度 $v_{1}=\\sqrt{b}$\nB: X 星球的密度 $\\rho=\\frac{3 b}{4 \\pi G R_{0}}$\nC: $\\mathrm{X}$ 星球的质量 $M=\\frac{b R_{0}^{2}}{G R}$\nD: 环绕 $\\mathrm{X}$ 星球的轨道离星球表面高度为 $R_{0}$ 的卫星周期 $T=2 \\pi \\sqrt{\\frac{8 R R_{0}}{b}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n在 X 星球表面, 宇航员做了一个实验：如图甲所示, 轻杆一端固定在 $O$ 点, 另一端固定一小球，现让小球在坚直平面内做半径为 $R$ 的圆周运动。小球运动到最高点时,受到的弹力为 $F$, 速度大小为 $v$, 其 $F-v^{2}$ 图像如乙图所示。已知 $\\mathrm{X}$ 星球的半径为 $R_{0}$, 万有引力常量为 $G$, 不考虑星球自转。则下列说法正确的是 ( )\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\nA: X 星球的第一宇宙速度 $v_{1}=\\sqrt{b}$\nB: X 星球的密度 $\\rho=\\frac{3 b}{4 \\pi G R_{0}}$\nC: $\\mathrm{X}$ 星球的质量 $M=\\frac{b R_{0}^{2}}{G R}$\nD: 环绕 $\\mathrm{X}$ 星球的轨道离星球表面高度为 $R_{0}$ 的卫星周期 $T=2 \\pi \\sqrt{\\frac{8 R R_{0}}{b}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-067.jpg?height=248&width=237&top_left_y=207&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-067.jpg?height=272&width=323&top_left_y=178&top_left_x=661" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_714", "problem": "2018 年 10 月 20 日, 酒泉卫星发射中心迎来 60 岁生日. 作为我国航天事业的发祥地, 中心拥有我国最早的航天发射场和目前唯一的载人航天发射场. 2013 年 6 月, 我国成功实现目标飞行器“神舟十号”与轨道空间站“天宫一号” 的对接. 如图所示, 已知“神舟十号”从捕获“天宫一号”到两个飞行器实现刚性对接用时为 $t$, 这段时间内组合体绕地球转过的角度为 $\\theta$, 地球半径为 $R$, 组合体离地面的高度为 $H$, 万有引力常量为 $G$, 据以上信息可求地球的质量为\n\n[图1]\n\n“天宫一号”与 “神舟十号”成功实现自动交会对接\nA: $\\frac{(R+H)^{3} \\theta^{2}}{G t^{2}}$\nB: $\\frac{\\pi^{2}(R+H)^{3} \\theta^{2}}{G t^{2}}$\nC: $\\frac{(R+H)^{3} \\theta^{2}}{4 \\pi G t^{2}}$\nD: $\\frac{4 \\pi^{4}(R+H)^{3} \\theta^{2}}{G t^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2018 年 10 月 20 日, 酒泉卫星发射中心迎来 60 岁生日. 作为我国航天事业的发祥地, 中心拥有我国最早的航天发射场和目前唯一的载人航天发射场. 2013 年 6 月, 我国成功实现目标飞行器“神舟十号”与轨道空间站“天宫一号” 的对接. 如图所示, 已知“神舟十号”从捕获“天宫一号”到两个飞行器实现刚性对接用时为 $t$, 这段时间内组合体绕地球转过的角度为 $\\theta$, 地球半径为 $R$, 组合体离地面的高度为 $H$, 万有引力常量为 $G$, 据以上信息可求地球的质量为\n\n[图1]\n\n“天宫一号”与 “神舟十号”成功实现自动交会对接\n\nA: $\\frac{(R+H)^{3} \\theta^{2}}{G t^{2}}$\nB: $\\frac{\\pi^{2}(R+H)^{3} \\theta^{2}}{G t^{2}}$\nC: $\\frac{(R+H)^{3} \\theta^{2}}{4 \\pi G t^{2}}$\nD: $\\frac{4 \\pi^{4}(R+H)^{3} \\theta^{2}}{G t^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-59.jpg?height=340&width=594&top_left_y=1001&top_left_x=363" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_141", "problem": "2022 年 4 月 16 日神舟十三号载人飞船返回舱成功着陆, 三位航天员在空间站出差半年, 完成了两次太空行走和 20 多项科学实验, 并开展了两次“天宫课堂”活动, 刷新了中国航天新纪录。已知地球半径为 $R$, 地球表面重力加速度为 $g$ 。总质量为 $m_{0}$ 的空间站绕地球的运动可近似为匀速圆周运动, 距地球表面高度为 $h$, 阻力忽略不计。\n\n物体间由于存在万有引力而具有的势能称为引力势能。若取两物体相距无穷远时引力势能为 0 , 质点 $m_{1}$ 和 $m_{2}$ 的距离为 $r$ 时, 其引力势能为 $E_{\\mathrm{p}}=-\\frac{G m_{1} m_{2}}{r}$ (式中 $G$ 为万有引力常量)。假设空间站为避免与其它飞行物相撞, 将从原轨道转移到距地球表面高为 $1.2 h$ 的新圆周轨道上, 则该转移至少需要提供多少额外的能量;", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2022 年 4 月 16 日神舟十三号载人飞船返回舱成功着陆, 三位航天员在空间站出差半年, 完成了两次太空行走和 20 多项科学实验, 并开展了两次“天宫课堂”活动, 刷新了中国航天新纪录。已知地球半径为 $R$, 地球表面重力加速度为 $g$ 。总质量为 $m_{0}$ 的空间站绕地球的运动可近似为匀速圆周运动, 距地球表面高度为 $h$, 阻力忽略不计。\n\n物体间由于存在万有引力而具有的势能称为引力势能。若取两物体相距无穷远时引力势能为 0 , 质点 $m_{1}$ 和 $m_{2}$ 的距离为 $r$ 时, 其引力势能为 $E_{\\mathrm{p}}=-\\frac{G m_{1} m_{2}}{r}$ (式中 $G$ 为万有引力常量)。假设空间站为避免与其它飞行物相撞, 将从原轨道转移到距地球表面高为 $1.2 h$ 的新圆周轨道上, 则该转移至少需要提供多少额外的能量;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_281", "problem": "假设有一载人宇宙飞船在距地面高度为 $4200 \\mathrm{~km}$ 的赤道上空绕地球做匀速圆周运动, 地球半径约为 $6400 \\mathrm{~km}$, 地球同步卫星距地面高为 $36000 \\mathrm{~km}$ 。宇宙飞船和一地球同步卫星绕地球同向运动, 每当两者相距最近时, 宇宙飞船就向同步卫星发射信号, 然后再由同步卫星将信号发送到地面接收站。某时刻两者相距最远, 从此刻开始, 在一昼夜的时间内，接收站共接收到信号的次数为\nA: 7 次\nB: 6 次\nC: 5 次\nD: 4 次\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n假设有一载人宇宙飞船在距地面高度为 $4200 \\mathrm{~km}$ 的赤道上空绕地球做匀速圆周运动, 地球半径约为 $6400 \\mathrm{~km}$, 地球同步卫星距地面高为 $36000 \\mathrm{~km}$ 。宇宙飞船和一地球同步卫星绕地球同向运动, 每当两者相距最近时, 宇宙飞船就向同步卫星发射信号, 然后再由同步卫星将信号发送到地面接收站。某时刻两者相距最远, 从此刻开始, 在一昼夜的时间内，接收站共接收到信号的次数为\n\nA: 7 次\nB: 6 次\nC: 5 次\nD: 4 次\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1222", "problem": "The speed of light is considered to be the speed limit of the Universe, however knots of plasma in the jets from active galactic nuclei (AGN) have been observed to be moving with apparent transverse speeds in excess of this, called superluminal speeds. Some of the more extreme examples can be appearing to move at up to 6 times the speed of light (see Figure 7).\n[figure1]\n\nFigure 7: Left: The jet coming from the elliptical galaxy M87 as viewed by the Hubble Space Telescope (HST). Right: Sequence of HST images showing motion at six times the speed of light. The slanting lines track the moving features, and the speeds are given in units of the velocity of light, $c$. Credit: NASA / Space Telescope Science Institute / John Biretta.\n\nThis can be explained by understanding that the jet is offset by an angle $\\theta$ from the sightline to Earth, and that the real speed of the plasma knot, $v$, is less than $c$, and from it we can define the scaled speed $\\beta \\equiv v / c$.\n\nSuperluminal jets are not limited just to AGN, as they have also been observed from systems within our own galaxy. A particularly famous one is the 'microquasar' GRS $1915+105$, which is a low mass X-ray binary consisting of a small star orbiting a black hole. A symmetrical jet with components approaching and receding from us is observed (as expected for jets coming from the poles of the black hole), and the apparent transverse motion of material in those jets has been measured using very high resolution radio imaging. Fender et. al (1999) measure these motions to be $\\mu_{a}=23.6$ mas day $^{-1}$ and $\\mu_{r}=$ 10.0 mas day $^{-1}$ for the approaching and receding jet respectively ( 1 mas $=1$ milliarcsecond, a unit of angle, and there are 3600 arcseconds in a degree) and the distance to the system as $11 \\mathrm{kpc}$.\n\nIn practice, for a given $\\beta_{\\text {app }}$ the values of $\\beta$ and $\\theta$ are degenerate and it is unlikely that the orientation of the jet is such that $\\beta_{\\text {app }}$ has been maximised, so the value in part $\\mathrm{c}$. is just a lower limit. However, since there are two jets then if we assume that they are from the same event (and so equal in speed but opposite in direction) we can break this degeneracy.\n\nSince it is a binary system, we can gain information about the masses of the objects by looking at their period and radial velocity. Formally, the relationship is\n\n$$\n\\frac{\\left(M_{\\mathrm{BH}} \\sin i\\right)^{3}}{\\left(M_{\\mathrm{BH}}+M_{\\star}\\right)^{2}}=\\frac{P_{\\mathrm{orb}} K_{d}^{3}}{2 \\pi G}\n$$\n\nwhere $M_{\\mathrm{BH}}$ is the mass of the black hole, $M_{\\star}$ is the mass of the orbiting star, $i$ is the inclination of the orbit, $P_{\\text {orb }}$ is the orbital period, and $K_{d}$ is the amplitude of the radial velocity curve. Normally the inclination can't be measured, however if we assume that the orbit is perpendicular to the jets then $i=\\theta$ and we can measure the mass of the black hole.c. Calculate $\\beta_{\\text {app }}$ for both jets, and use your formula from part $b$. to calculate the minimum value of $\\beta$ to explain the apparent superluminal motion.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe speed of light is considered to be the speed limit of the Universe, however knots of plasma in the jets from active galactic nuclei (AGN) have been observed to be moving with apparent transverse speeds in excess of this, called superluminal speeds. Some of the more extreme examples can be appearing to move at up to 6 times the speed of light (see Figure 7).\n[figure1]\n\nFigure 7: Left: The jet coming from the elliptical galaxy M87 as viewed by the Hubble Space Telescope (HST). Right: Sequence of HST images showing motion at six times the speed of light. The slanting lines track the moving features, and the speeds are given in units of the velocity of light, $c$. Credit: NASA / Space Telescope Science Institute / John Biretta.\n\nThis can be explained by understanding that the jet is offset by an angle $\\theta$ from the sightline to Earth, and that the real speed of the plasma knot, $v$, is less than $c$, and from it we can define the scaled speed $\\beta \\equiv v / c$.\n\nSuperluminal jets are not limited just to AGN, as they have also been observed from systems within our own galaxy. A particularly famous one is the 'microquasar' GRS $1915+105$, which is a low mass X-ray binary consisting of a small star orbiting a black hole. A symmetrical jet with components approaching and receding from us is observed (as expected for jets coming from the poles of the black hole), and the apparent transverse motion of material in those jets has been measured using very high resolution radio imaging. Fender et. al (1999) measure these motions to be $\\mu_{a}=23.6$ mas day $^{-1}$ and $\\mu_{r}=$ 10.0 mas day $^{-1}$ for the approaching and receding jet respectively ( 1 mas $=1$ milliarcsecond, a unit of angle, and there are 3600 arcseconds in a degree) and the distance to the system as $11 \\mathrm{kpc}$.\n\nIn practice, for a given $\\beta_{\\text {app }}$ the values of $\\beta$ and $\\theta$ are degenerate and it is unlikely that the orientation of the jet is such that $\\beta_{\\text {app }}$ has been maximised, so the value in part $\\mathrm{c}$. is just a lower limit. However, since there are two jets then if we assume that they are from the same event (and so equal in speed but opposite in direction) we can break this degeneracy.\n\nSince it is a binary system, we can gain information about the masses of the objects by looking at their period and radial velocity. Formally, the relationship is\n\n$$\n\\frac{\\left(M_{\\mathrm{BH}} \\sin i\\right)^{3}}{\\left(M_{\\mathrm{BH}}+M_{\\star}\\right)^{2}}=\\frac{P_{\\mathrm{orb}} K_{d}^{3}}{2 \\pi G}\n$$\n\nwhere $M_{\\mathrm{BH}}$ is the mass of the black hole, $M_{\\star}$ is the mass of the orbiting star, $i$ is the inclination of the orbit, $P_{\\text {orb }}$ is the orbital period, and $K_{d}$ is the amplitude of the radial velocity curve. Normally the inclination can't be measured, however if we assume that the orbit is perpendicular to the jets then $i=\\theta$ and we can measure the mass of the black hole.\n\nproblem:\nc. Calculate $\\beta_{\\text {app }}$ for both jets, and use your formula from part $b$. to calculate the minimum value of $\\beta$ to explain the apparent superluminal motion.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-10.jpg?height=812&width=1458&top_left_y=504&top_left_x=296" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_722", "problem": "两颗人造卫星绕地球逆时针运动, 卫星 1、卫星 2 分别沿圆轨道、椭圆轨道运动,圆的半径与粗圆的半长轴相等, 两轨道相交于 $A 、 B$ 两点, 某时刻两卫星与地球在同一直线上，如图所示，下列说法中正确的是（）\n\n[图1]\nA: 两卫星在图示位置的速度 $v_{2}=v_{1}$\nB: 两卫星在 $A$ 处的加速度大小不相等\nC: 两颗卫星在 $A$ 或 $B$ 点处可能相遇\nD: 两卫星永远不可能相遇\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n两颗人造卫星绕地球逆时针运动, 卫星 1、卫星 2 分别沿圆轨道、椭圆轨道运动,圆的半径与粗圆的半长轴相等, 两轨道相交于 $A 、 B$ 两点, 某时刻两卫星与地球在同一直线上，如图所示，下列说法中正确的是（）\n\n[图1]\n\nA: 两卫星在图示位置的速度 $v_{2}=v_{1}$\nB: 两卫星在 $A$ 处的加速度大小不相等\nC: 两颗卫星在 $A$ 或 $B$ 点处可能相遇\nD: 两卫星永远不可能相遇\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-071.jpg?height=354&width=734&top_left_y=2364&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_31", "problem": "嫦娥四号”在月球背面软着陆和巡视探测, 创造了人类探月的历史. 为了实现“嫦娥四号”与地面间的太空通讯，我国于 2018 年 5 月发射了中继卫星“鹊桥”，它是运行于\n地月拉格朗日 $\\mathrm{L}_{2}$ 点的通信卫星, $\\mathrm{L}_{2}$ 点位于地球和月球连线的延长线上. 若某飞行器位于 $\\mathrm{L}_{2}$ 点, 可以在几乎不消耗燃料的情况下与月球同步绕地球做匀速圆周运动, 如图所示. 已知地球质量是月球质量的 $\\mathrm{k}$ 倍, 飞行器质量远小于月球质量, 地球与月球中心距离是 $\\mathrm{L}_{2}$ 点与月球中心距离的 $\\mathrm{n}$ 倍. 下列说法正确的是\n\n[图1]\nA: 飞行器的加速度大于月球的加速度\nB: 飞行器的运行周期大于月球的运行周期\nC: 飞行器所需的向心力由地球对其引力提供\nD: $\\mathrm{k}$ 与 $\\mathrm{n}$ 满足 $\\mathrm{k}=\\frac{n^{3}(n+1)^{2}}{3 n^{2}+3 n+1}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n嫦娥四号”在月球背面软着陆和巡视探测, 创造了人类探月的历史. 为了实现“嫦娥四号”与地面间的太空通讯，我国于 2018 年 5 月发射了中继卫星“鹊桥”，它是运行于\n地月拉格朗日 $\\mathrm{L}_{2}$ 点的通信卫星, $\\mathrm{L}_{2}$ 点位于地球和月球连线的延长线上. 若某飞行器位于 $\\mathrm{L}_{2}$ 点, 可以在几乎不消耗燃料的情况下与月球同步绕地球做匀速圆周运动, 如图所示. 已知地球质量是月球质量的 $\\mathrm{k}$ 倍, 飞行器质量远小于月球质量, 地球与月球中心距离是 $\\mathrm{L}_{2}$ 点与月球中心距离的 $\\mathrm{n}$ 倍. 下列说法正确的是\n\n[图1]\n\nA: 飞行器的加速度大于月球的加速度\nB: 飞行器的运行周期大于月球的运行周期\nC: 飞行器所需的向心力由地球对其引力提供\nD: $\\mathrm{k}$ 与 $\\mathrm{n}$ 满足 $\\mathrm{k}=\\frac{n^{3}(n+1)^{2}}{3 n^{2}+3 n+1}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-15.jpg?height=320&width=329&top_left_y=480&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-15.jpg?height=117&width=831&top_left_y=1752&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_397", "problem": "如图所示, 甲、乙分别为常见的三星系统模型和四星系统模型。甲图中三颗质量均为 $m$ 的行星都绕边长为 $L_{1}$ 的等边三角形的中心做匀速圆周运动, 周期为 $T_{1}$; 乙图中三\n颗质量均为 $m$ 的行星都绕静止于边长为 $L_{2}$ 的等边三角形中心的中央星做匀速圆周运动,周期为 $T_{2}$, 不考虑其它星系的影响。已知四星系统内中央星的质量 $M=\\sqrt{3} m$, $L_{2}=2 L_{1}$, 则两个系统的周期之比为 $(\\quad)$\n\n[图1]\n\n甲\n\n[图2]\n\n乙\nA: $T_{1}: T_{2}=1: 1$\nB: $T_{1}: T_{2}=1: \\sqrt{2}$\nC: $T_{1}: T_{2}=1: \\sqrt{3}$\nD: $T_{1}: T_{2}=1: 2$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 甲、乙分别为常见的三星系统模型和四星系统模型。甲图中三颗质量均为 $m$ 的行星都绕边长为 $L_{1}$ 的等边三角形的中心做匀速圆周运动, 周期为 $T_{1}$; 乙图中三\n颗质量均为 $m$ 的行星都绕静止于边长为 $L_{2}$ 的等边三角形中心的中央星做匀速圆周运动,周期为 $T_{2}$, 不考虑其它星系的影响。已知四星系统内中央星的质量 $M=\\sqrt{3} m$, $L_{2}=2 L_{1}$, 则两个系统的周期之比为 $(\\quad)$\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\nA: $T_{1}: T_{2}=1: 1$\nB: $T_{1}: T_{2}=1: \\sqrt{2}$\nC: $T_{1}: T_{2}=1: \\sqrt{3}$\nD: $T_{1}: T_{2}=1: 2$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-030.jpg?height=346&width=379&top_left_y=495&top_left_x=336", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-030.jpg?height=377&width=391&top_left_y=497&top_left_x=750" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_981", "problem": "Which well-known star dropped in brightness by $40 \\%$ between October 2019 and April 2020, leading to speculation it may be about to go supernova?\nA: Aldebaran\nB: Antares\nC: Arcturus\nD: Betelgeuse\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhich well-known star dropped in brightness by $40 \\%$ between October 2019 and April 2020, leading to speculation it may be about to go supernova?\n\nA: Aldebaran\nB: Antares\nC: Arcturus\nD: Betelgeuse\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_630", "problem": "一卫星绕地球做椭圆轨道运动, 近地点距地表 $h_{1}=3600 \\mathrm{~km}$, 远地点距地表 $h_{2}=23600 \\mathrm{~km}$ 。假设在近地点卫星加速, 使得椭圆轨道的远地点距离地球表面 $h_{3}=33600 \\mathrm{~km}$ 。已知地球半径 $r=6400 \\mathrm{~km}$, 则卫星变轨时的速度增量应约为（）（设地球表面重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$ )。\n\n[图1]\nA: $1000 \\mathrm{~m} / \\mathrm{s}$\nB: $500 \\mathrm{~m} / \\mathrm{s}$\nC: $250 \\mathrm{~m} / \\mathrm{s}$\nD: $100 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n一卫星绕地球做椭圆轨道运动, 近地点距地表 $h_{1}=3600 \\mathrm{~km}$, 远地点距地表 $h_{2}=23600 \\mathrm{~km}$ 。假设在近地点卫星加速, 使得椭圆轨道的远地点距离地球表面 $h_{3}=33600 \\mathrm{~km}$ 。已知地球半径 $r=6400 \\mathrm{~km}$, 则卫星变轨时的速度增量应约为（）（设地球表面重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$ )。\n\n[图1]\n\nA: $1000 \\mathrm{~m} / \\mathrm{s}$\nB: $500 \\mathrm{~m} / \\mathrm{s}$\nC: $250 \\mathrm{~m} / \\mathrm{s}$\nD: $100 \\mathrm{~m} / \\mathrm{s}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-024.jpg?height=317&width=999&top_left_y=1743&top_left_x=357" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1132", "problem": "Recent years have seen an explosion in the discovery of new exoplanets. About $85 \\%$ of transiting exoplanets discovered by the NASA Kepler telescope have radii less than Neptune ( $\\sim 4 R_{\\oplus}$ ), meaning we are improving our understanding of what the transition between rocky Earth-size planets and gaseous Neptune-size planets looks like.\n\nGiven how common these \"super-Earths\" and \"gas dwarfs\" seem to be, it was odd that we didn't have any in our own Solar System. However, Batygin \\& Brown (2016) suggested that a hypothetical ninth planet (called 'Planet Nine') could explain some of the unusual properties of the orbits of objects in the Kuiper Belt. This planet is inferred to have a mass of $10 M_{\\oplus}$, and so would be an example of a super-Earth.\n\n[figure1]\n\nFigure 5: A plot of planet density versus radius for 33 extrasolar planets (circles) and the planets in our solar system (diamonds).\n\nCredit: Marcy et al. (2014).\n\nAnalysing exoplanets discovered by Kepler, Marcy et al. (2014) used a piecewise function to describe their planetary density data such that:\n\n$$\n\\begin{aligned}\n\\text { For } R_{\\mathrm{P}} \\leq 1.5 R_{\\oplus} & \\rho & =2.32+3.18 \\frac{R_{\\mathrm{P}}}{R_{\\oplus}}\\left[\\mathrm{g} \\mathrm{cm}^{-3}\\right] \\\\\n\\text { For } 1.5 R_{\\oplus}5 \\mathrm{~s}$ ) if $\\mathrm{T}=10$ years. Hint: Using the chain rule, $\\mathrm{dn} / \\mathrm{dt}=\\mathrm{dn} / \\mathrm{da}$", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nSome of the very first exoplanets to be discovered in large surveys were dubbed 'hot Jupiters' as they were similar in mass to Jupiter (i.e. a gas giant) but were much closer to their star than Mercury is to the Sun (and hence are in a very hot environment). Planetary formation models suggest that they were unlikely to have formed there, but instead formed much further out from the star and migrated inwards, due to gravitational interactions with other planets in the system. Studies of 'hot Jupiters' show that there is an overabundance of them with periods of $\\sim 3-4$ days, and very few with periods shorter than that. Since large, close-in planets should be the easiest to detect in all of the main methods of finding exoplanets, this scarcity is likely to be a real effect and suggests that exoplanets which are that close to their star are in a relatively rapid (by astronomical standards) inspiral towards destruction by their star.\n[figure1]\n\nFigure 6: Left: The orbital radius of several 'hot Jupiters' scaled by the Roche radius of the system (where tidal forces would destroy the planet). There is an expected pile up close to radii double the Roche radius (dotted line), and very few with radii smaller than that - those that are will inevitably spiral into the star and be destroyed by the tidal forces when they get too close. Credit: Birkby et al. (2014).\n\nRight: As the planets inspiral we should see a shift in when their transits occur. This figure shows the predicted size of the shift after a period of 10 years if the tidal dissipation quality factor $Q_{\\star}^{\\prime}=10^{6}$, as well as the current detection limit of 5 seconds (dotted line). Therefore measuring if there is any shift in the transit times over the course of a decade of observations can put stringent limits on the value of $Q_{\\star}^{\\prime}$. Credit: Birkby et al. (2014).\n\nThe Roche radius, where a planet will be torn apart due to the tidal forces acting on it, is defined as\n\n$$\na_{\\text {Roche }} \\approx 2.16 R_{P}\\left(\\frac{M_{\\star}}{M_{P}}\\right)^{1 / 3}\n$$\n\nwhere $R_{P}$ is the radius of the planet, $M_{P}$ is the mass of the planet and $M_{\\star}$ is the mass of the star. If a gas giant is knocked into a highly elliptical orbit (i.e. $e \\approx 1$ ) that has a periapsis $r_{\\text {peri }}5 \\mathrm{~s}$ ) if $\\mathrm{T}=10$ years. Hint: Using the chain rule, $\\mathrm{dn} / \\mathrm{dt}=\\mathrm{dn} / \\mathrm{da}$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-10.jpg?height=600&width=1512&top_left_y=745&top_left_x=274" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_577", "problem": "如图所示, $a$ 为放在赤道上相对地球静止的物体, 随地球自转做匀速圆周运动, $b$为沿地面表面附近做匀速圆周运动的人造卫星 (轨道半径等于地球半径), $c$ 为地球的同步卫星，以下关于 $a 、 b 、 c$ 的说法中正确的是（）\n\n[图1]\nA: $a 、 b 、 c$ 做匀速圆周运动的向心加速度大小关系为 $a_{b}>a_{c}>a_{a}$\nB: $a 、 b 、 c$ 做匀速圆周运动的角速度大小关系为 $\\omega_{a}=\\omega_{c}>\\omega_{b}$\nC: $a 、 b 、 c$ 做匀速圆周运动的线速度大小关系为 $v_{a}=v_{b}>v_{c}$\nD: $a 、 b 、 c$ 做匀速圆周运动的周期关系为 $T_{a}=T_{c}>T_{b}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, $a$ 为放在赤道上相对地球静止的物体, 随地球自转做匀速圆周运动, $b$为沿地面表面附近做匀速圆周运动的人造卫星 (轨道半径等于地球半径), $c$ 为地球的同步卫星，以下关于 $a 、 b 、 c$ 的说法中正确的是（）\n\n[图1]\n\nA: $a 、 b 、 c$ 做匀速圆周运动的向心加速度大小关系为 $a_{b}>a_{c}>a_{a}$\nB: $a 、 b 、 c$ 做匀速圆周运动的角速度大小关系为 $\\omega_{a}=\\omega_{c}>\\omega_{b}$\nC: $a 、 b 、 c$ 做匀速圆周运动的线速度大小关系为 $v_{a}=v_{b}>v_{c}$\nD: $a 、 b 、 c$ 做匀速圆周运动的周期关系为 $T_{a}=T_{c}>T_{b}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-104.jpg?height=376&width=414&top_left_y=483&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_472", "problem": "2017 年 8 月 28 日, 中科院南极天文中心的巡天望远镜观测到一个由双中子星构成的孤立双星系统产生的引力波。该双星系统以引力波的形式向外辐射能量, 使得圆周运动的周期 $T$ 极其缓慢地减小, 双星的质量 $m_{1}$ 与 $m_{2}$ 均不变, 则下列关于该双星系统变化的说法正确的是 ( )\n\n[图1]\nA: 双星间的间距逐渐增大\nB: 双星间的万有引力逐渐增大\nC: 双星的线速度逐渐减小\nD: 双星的角速度减小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2017 年 8 月 28 日, 中科院南极天文中心的巡天望远镜观测到一个由双中子星构成的孤立双星系统产生的引力波。该双星系统以引力波的形式向外辐射能量, 使得圆周运动的周期 $T$ 极其缓慢地减小, 双星的质量 $m_{1}$ 与 $m_{2}$ 均不变, 则下列关于该双星系统变化的说法正确的是 ( )\n\n[图1]\n\nA: 双星间的间距逐渐增大\nB: 双星间的万有引力逐渐增大\nC: 双星的线速度逐渐减小\nD: 双星的角速度减小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-020.jpg?height=320&width=485&top_left_y=1028&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_433", "problem": "星体 $\\mathrm{P}$ (行星或彗星) 绕太阳运动的轨迹为圆锥曲线 $r=\\frac{k}{1+\\varepsilon \\cos \\theta}$ 式中, $r$ 是 $\\mathrm{P}$ 到太阳 $\\mathrm{S}$ 的距离, $\\theta$ 是矢径 $\\mathrm{SP}$ 相对于极轴 $\\mathrm{SA}$ 的夹角 (以逆时针方向为正), $k=\\frac{L^{2}}{G M m^{2}}, L$是 $\\mathrm{P}$ 相对于太阳的角动量, $G=6.67 \\times 10^{-11} \\mathrm{~m} 3 \\cdot \\mathrm{kg}^{-1} \\cdot \\mathrm{s}^{-2}$ 为引力常量, $M \\approx 1.99 \\times 10^{30} \\mathrm{~kg}$ 为太阳的质量, $\\varepsilon=\\sqrt{1+\\frac{2 E L^{2}}{G^{2} M^{2} m^{3}}}$ 为偏心率, $m$ 和 $E$ 分别为 $\\mathrm{P}$ 的质量和机械能。假设有一颗彗星绕太阳运动的轨道为抛物线, 地球绕太阳运动的轨道可近似为圆, 两轨道相交于 $C 、 D$两点, 如图所示。已知地球轨道半径 $R_{E}=1.49 \\times 10^{11} \\mathrm{~m}$, 彗星轨道近日点 $A$ 到太阳的距离为地球轨道半径的三分之一, 不考虑地球和彗星之间的相互影响。求彗星经过 $C 、 D$ 两点时速度的大小\n\n已知积分公式 $\\int \\frac{x d x}{\\sqrt{x+a}}=\\frac{2}{3}(x+a)^{\\frac{3}{2}}-2 a(x+a)^{\\frac{1}{2}}+C$, 式中 $\\mathrm{C}$ 是任意常数。\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n星体 $\\mathrm{P}$ (行星或彗星) 绕太阳运动的轨迹为圆锥曲线 $r=\\frac{k}{1+\\varepsilon \\cos \\theta}$ 式中, $r$ 是 $\\mathrm{P}$ 到太阳 $\\mathrm{S}$ 的距离, $\\theta$ 是矢径 $\\mathrm{SP}$ 相对于极轴 $\\mathrm{SA}$ 的夹角 (以逆时针方向为正), $k=\\frac{L^{2}}{G M m^{2}}, L$是 $\\mathrm{P}$ 相对于太阳的角动量, $G=6.67 \\times 10^{-11} \\mathrm{~m} 3 \\cdot \\mathrm{kg}^{-1} \\cdot \\mathrm{s}^{-2}$ 为引力常量, $M \\approx 1.99 \\times 10^{30} \\mathrm{~kg}$ 为太阳的质量, $\\varepsilon=\\sqrt{1+\\frac{2 E L^{2}}{G^{2} M^{2} m^{3}}}$ 为偏心率, $m$ 和 $E$ 分别为 $\\mathrm{P}$ 的质量和机械能。假设有一颗彗星绕太阳运动的轨道为抛物线, 地球绕太阳运动的轨道可近似为圆, 两轨道相交于 $C 、 D$两点, 如图所示。已知地球轨道半径 $R_{E}=1.49 \\times 10^{11} \\mathrm{~m}$, 彗星轨道近日点 $A$ 到太阳的距离为地球轨道半径的三分之一, 不考虑地球和彗星之间的相互影响。求彗星经过 $C 、 D$ 两点时速度的大小\n\n已知积分公式 $\\int \\frac{x d x}{\\sqrt{x+a}}=\\frac{2}{3}(x+a)^{\\frac{3}{2}}-2 a(x+a)^{\\frac{1}{2}}+C$, 式中 $\\mathrm{C}$ 是任意常数。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[彗星经过 C点时速度的大小, 彗星经过 D点时速度的大小]\n它们的单位依次是[m/s, m/s]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-016.jpg?height=363&width=577&top_left_y=161&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "m/s", "m/s" ], "answer_sequence": [ "彗星经过 C点时速度的大小", "彗星经过 D点时速度的大小" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1022", "problem": "The James Webb Space Telescope (JWST) is an exciting new space-based observatory which is capable of detecting incredibly faint objects that have never been seen before, but it is also possible to be seen from Earth if you have a large enough telescope. It has now entered a halo orbit around the second Lagrangian point, $L_{2}$, of the Sun-Earth system at a distance of about 1.5 million $\\mathrm{km}$ from Earth, directly along the Sun-Earth line.\n[figure1]\n\nFigure 5: Left: An image of NASA's James Webb Space Telescope reaching its final distance from Earth. It is a tiny speck among a sea of background stars. The stars appear smudged because the telescope was tracking the motion of JWST, which appears as a small white speck. Credit: Gianluca Masi / The Virtual Telescope Project. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe rectangular sunshield is rather large (measuring $21 \\mathrm{~m}$ by $14 \\mathrm{~m}$, roughly the same as a tennis court), very reflective (reflecting $\\sim 90 \\%$ of the incident light), and always points directly towards the Sun to protect the other parts of the telescope, especially to keep it cool enough to do infrared astronomy.\n\nOn a very dark night, your eye's pupil opens up to about $6 \\mathrm{~mm}$ in diameter and you are able to see stars as faint as $m=+6$. Estimate what diameter of telescope you would need to look through in order to see the JWST in similarly dark conditions.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe James Webb Space Telescope (JWST) is an exciting new space-based observatory which is capable of detecting incredibly faint objects that have never been seen before, but it is also possible to be seen from Earth if you have a large enough telescope. It has now entered a halo orbit around the second Lagrangian point, $L_{2}$, of the Sun-Earth system at a distance of about 1.5 million $\\mathrm{km}$ from Earth, directly along the Sun-Earth line.\n[figure1]\n\nFigure 5: Left: An image of NASA's James Webb Space Telescope reaching its final distance from Earth. It is a tiny speck among a sea of background stars. The stars appear smudged because the telescope was tracking the motion of JWST, which appears as a small white speck. Credit: Gianluca Masi / The Virtual Telescope Project. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe rectangular sunshield is rather large (measuring $21 \\mathrm{~m}$ by $14 \\mathrm{~m}$, roughly the same as a tennis court), very reflective (reflecting $\\sim 90 \\%$ of the incident light), and always points directly towards the Sun to protect the other parts of the telescope, especially to keep it cool enough to do infrared astronomy.\n\nOn a very dark night, your eye's pupil opens up to about $6 \\mathrm{~mm}$ in diameter and you are able to see stars as faint as $m=+6$. Estimate what diameter of telescope you would need to look through in order to see the JWST in similarly dark conditions.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of cm, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-09.jpg?height=514&width=1494&top_left_y=594&top_left_x=286" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "cm" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_558", "problem": "我国天文学家通过 FAST, 在武仙座球状星团 $\\mathrm{M}_{1} 3$ 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}T_{B}$\nC: $v_{A}T_{B}$\nC: $v_{A}v_{3}>v_{2}$\nB: $v_{1}>v_{2}>v_{3}$\nC: $a_{1}>a_{2}=a_{3}$\nD: $T_{1}v_{3}>v_{2}$\nB: $v_{1}>v_{2}>v_{3}$\nC: $a_{1}>a_{2}=a_{3}$\nD: $T_{1}v_{1}>v_{3}$\nB: $v_{3}>v_{1}>v_{2}$\nC: $a_{2}>a_{3}>a_{1}$\nD: $a_{1}>a_{2}>a_{3}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球同步卫星的加速度为 $a_{1}$, 运行速度为 $v_{1}$, 地面附近卫星的加速度为 $a_{2}$, 运行速度为 $v_{2}$, 地球赤道上物体随地球自转的向心加速度为 $a_{3}$, 运行速度为 $v_{3}$, 则 ( )\n\nA: $v_{2}>v_{1}>v_{3}$\nB: $v_{3}>v_{1}>v_{2}$\nC: $a_{2}>a_{3}>a_{1}$\nD: $a_{1}>a_{2}>a_{3}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_463", "problem": "宇航员在某星球表面 (无空气) 将小球从空中坚直向下抛出, 测得小球速率的二次方与其离开抛出点的距离的关系如图所示 (图中的 $b 、 c 、 d$ 均为已知量)。该星球的半径为 $R$, 引力常量为 $G$, 将该星球视为球体, 忽略该星球的自转。下列说法正确的是\n\n[图1]\nA: 该星球表面的重力加速度大小为 $\\frac{c-b}{2 d}$\nB: 该星球的质量为 $\\frac{c R^{2}}{2 G d}$\nC: 该星球的平均密度为 $\\frac{3(c-b)}{8 \\pi G d R}$\nD: 该星球的第一宇宙速度为 $\\sqrt{\\frac{(c-b) R}{d}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇航员在某星球表面 (无空气) 将小球从空中坚直向下抛出, 测得小球速率的二次方与其离开抛出点的距离的关系如图所示 (图中的 $b 、 c 、 d$ 均为已知量)。该星球的半径为 $R$, 引力常量为 $G$, 将该星球视为球体, 忽略该星球的自转。下列说法正确的是\n\n[图1]\n\nA: 该星球表面的重力加速度大小为 $\\frac{c-b}{2 d}$\nB: 该星球的质量为 $\\frac{c R^{2}}{2 G d}$\nC: 该星球的平均密度为 $\\frac{3(c-b)}{8 \\pi G d R}$\nD: 该星球的第一宇宙速度为 $\\sqrt{\\frac{(c-b) R}{d}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-059.jpg?height=594&width=628&top_left_y=822&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_539", "problem": "假设地球可视为质量均匀分布的球体。已知地球表面的重力加速度在两极的大小为 $g_{0}$, 在赤道的大小为 $g$; 地球半径为 $R$, 引力常数为 $G$, 则 ( )\nA: 地球同步卫星距地表的高度为 $\\left(\\sqrt{\\frac{g_{0}}{g_{0}-g}}-1\\right) R$\nB: 地球的质量为 $\\frac{g_{0} R^{2}}{G}$\nC: 地球的第一宇宙速度为 $\\sqrt{g R}$\nD: 地球密度为 $\\frac{3 g}{4 \\pi R G}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n假设地球可视为质量均匀分布的球体。已知地球表面的重力加速度在两极的大小为 $g_{0}$, 在赤道的大小为 $g$; 地球半径为 $R$, 引力常数为 $G$, 则 ( )\n\nA: 地球同步卫星距地表的高度为 $\\left(\\sqrt{\\frac{g_{0}}{g_{0}-g}}-1\\right) R$\nB: 地球的质量为 $\\frac{g_{0} R^{2}}{G}$\nC: 地球的第一宇宙速度为 $\\sqrt{g R}$\nD: 地球密度为 $\\frac{3 g}{4 \\pi R G}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1127", "problem": "A \"supermoon\" is a new or full moon that occurs with the Moon at or near its closest approach to Earth in a given orbit (perigee). The media commonly associates supermoons with extreme brightness and size, sometimes implying that the Moon itself will become larger and have an impact on human behaviour, but just how different is a supermoon compared to the 'normal' Moon we see each month?\n\nLunar Data:\n\nSynodic Period Anomalistic Period Semi-major axis Orbit eccentricity\n\n$$\n\\begin{aligned}\n& =29.530589 \\text { days (time between same phases e.g. full moon to full moon) } \\\\\n& =27.554550 \\text { days (time between perigees i.e. perigee to perigee) } \\\\\n& =3.844 \\times 10^{5} \\mathrm{~km} \\\\\n& =0.0549 \\\\\n& =1738.1 \\mathrm{~km}\n\\end{aligned}\n$$\n\n$$\n\\begin{array}{ll}\n\\text { Radius of the Moon } & =1738.1 \\mathrm{~km} \\\\\n\\text { Mass of the Moon } & =7.342 \\times 10^{22} \\mathrm{~kg}\n\\end{array}\n$$\n\nIn this question, we will only consider a full moon that is at perigee to be a supermoon.c). Determine the difference in the angular diameter of a supermoon and a full moon observed at apogee. Thus, determine the percentage difference in the brightness of a supermoon and a full moon observed at apogee. (Ignore the effects of the Moon's orbital tilt with respect to the Earth).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nA \"supermoon\" is a new or full moon that occurs with the Moon at or near its closest approach to Earth in a given orbit (perigee). The media commonly associates supermoons with extreme brightness and size, sometimes implying that the Moon itself will become larger and have an impact on human behaviour, but just how different is a supermoon compared to the 'normal' Moon we see each month?\n\nLunar Data:\n\nSynodic Period Anomalistic Period Semi-major axis Orbit eccentricity\n\n$$\n\\begin{aligned}\n& =29.530589 \\text { days (time between same phases e.g. full moon to full moon) } \\\\\n& =27.554550 \\text { days (time between perigees i.e. perigee to perigee) } \\\\\n& =3.844 \\times 10^{5} \\mathrm{~km} \\\\\n& =0.0549 \\\\\n& =1738.1 \\mathrm{~km}\n\\end{aligned}\n$$\n\n$$\n\\begin{array}{ll}\n\\text { Radius of the Moon } & =1738.1 \\mathrm{~km} \\\\\n\\text { Mass of the Moon } & =7.342 \\times 10^{22} \\mathrm{~kg}\n\\end{array}\n$$\n\nIn this question, we will only consider a full moon that is at perigee to be a supermoon.\n\nproblem:\nc). Determine the difference in the angular diameter of a supermoon and a full moon observed at apogee. Thus, determine the percentage difference in the brightness of a supermoon and a full moon observed at apogee. (Ignore the effects of the Moon's orbital tilt with respect to the Earth).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of % percentage, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "% percentage" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_724", "problem": "某国际研究小组观测到了一组双星系统, 它们绕二者连线上的某点做匀速圆周运动,双星系统中质量较小的星体能“吸食”质量较大的星体的表面物质，达到质量转移的目的. 根据大爆宇宙学可知, 双星间的距离在缓慢增大, 假设星体的轨道近似为圆, 则在该过程中（ )\nA: 双星做圆周运动的角速度不断减小\nB: 双星做圆周运动的角速度不断增大\nC: 质量较大的星体做圆周运动的轨道半径减小\nD: 质量较大的星体做圆周运动的轨道半径增大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n某国际研究小组观测到了一组双星系统, 它们绕二者连线上的某点做匀速圆周运动,双星系统中质量较小的星体能“吸食”质量较大的星体的表面物质，达到质量转移的目的. 根据大爆宇宙学可知, 双星间的距离在缓慢增大, 假设星体的轨道近似为圆, 则在该过程中（ )\n\nA: 双星做圆周运动的角速度不断减小\nB: 双星做圆周运动的角速度不断增大\nC: 质量较大的星体做圆周运动的轨道半径减小\nD: 质量较大的星体做圆周运动的轨道半径增大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_817", "problem": "Billions of years from now, as the Moon moves farther away from the Earth, the Earth's axial tilt may become unstable. Imagine the Earth's tilt is such that the angle between the celestial equator and the ecliptic is $60^{\\circ}$, rather than the current $23.44^{\\circ}$ - so the Arctic Circle is now as far south as $30^{\\circ}$ North. For an observer at $40^{\\circ}$ North, how many days out of the year would the Sun never set (also known as the \"polar day\")? (Ignore atmospheric refraction, and assume the Earth's orbit is circular and nothing else has changed from today.)\nA: 28\nB: 56\nC: 61\nD: 67\nE: 113\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nBillions of years from now, as the Moon moves farther away from the Earth, the Earth's axial tilt may become unstable. Imagine the Earth's tilt is such that the angle between the celestial equator and the ecliptic is $60^{\\circ}$, rather than the current $23.44^{\\circ}$ - so the Arctic Circle is now as far south as $30^{\\circ}$ North. For an observer at $40^{\\circ}$ North, how many days out of the year would the Sun never set (also known as the \"polar day\")? (Ignore atmospheric refraction, and assume the Earth's orbit is circular and nothing else has changed from today.)\n\nA: 28\nB: 56\nC: 61\nD: 67\nE: 113\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_753", "problem": "Compared to the Sun's surface temperature, sunspots are ...\nA: cooler\nB: hotter\nC: same temperature\nD: sometimes hotter and sometimes cooler\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nCompared to the Sun's surface temperature, sunspots are ...\n\nA: cooler\nB: hotter\nC: same temperature\nD: sometimes hotter and sometimes cooler\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1207", "problem": "The James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.c. Computer models suggest the first galaxies formed around $z \\sim 10-20$. One of the best ways to look for high-redshift galaxies is to try and detect the emission from the Lyman alpha (Lya) emission line at $\\lambda_{\\text {emit }}=121.6 \\mathrm{~nm}$ as it is a relatively bright line. Some of the brightest galaxies in that initial era of galaxy formation would have an absolute magnitude of $\\mathcal{M} \\sim 20$. In this question, you are given that $\\Omega_{0, \\mathrm{~m}}=0.3, \\Omega_{0, \\Lambda}=0.7, \\Omega_{0, \\mathrm{r}}=0$ and $\\mathrm{H}_{0}=70 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\niii. Calculate the luminosity distance to the galaxy and hence its apparent magnitude. Assume all emitted flux is picked up by the telescope.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.\n\nproblem:\nc. Computer models suggest the first galaxies formed around $z \\sim 10-20$. One of the best ways to look for high-redshift galaxies is to try and detect the emission from the Lyman alpha (Lya) emission line at $\\lambda_{\\text {emit }}=121.6 \\mathrm{~nm}$ as it is a relatively bright line. Some of the brightest galaxies in that initial era of galaxy formation would have an absolute magnitude of $\\mathcal{M} \\sim 20$. In this question, you are given that $\\Omega_{0, \\mathrm{~m}}=0.3, \\Omega_{0, \\Lambda}=0.7, \\Omega_{0, \\mathrm{r}}=0$ and $\\mathrm{H}_{0}=70 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\niii. Calculate the luminosity distance to the galaxy and hence its apparent magnitude. Assume all emitted flux is picked up by the telescope.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-09.jpg?height=618&width=1466&top_left_y=596&top_left_x=296", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-10.jpg?height=482&width=1536&top_left_y=1118&top_left_x=267" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_319", "problem": "学习物理知识后, 我们可以用物理的视角观察周围的世界, 思考身边的事物, 并对些媒体报道的真伪做出判断。小明在浏览一些网站时, 看到了如下一些关于发射卫星的报道，其中一定不真实的消息是（）\nA: 发射一颗轨道与地球表面上某一纬度线 (非赤道) 为共面同心圆的地球卫星\nB: 发射一颗与地球表面上某一经度线所决定的圆为共面同心圆的地球卫星\nC: 发射一颗 $1 \\mathrm{~h}$ 绕地球运转一周的地球卫星\nD: 发射一颗每天同一时间都能通过北京上空的地球卫星\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n学习物理知识后, 我们可以用物理的视角观察周围的世界, 思考身边的事物, 并对些媒体报道的真伪做出判断。小明在浏览一些网站时, 看到了如下一些关于发射卫星的报道，其中一定不真实的消息是（）\n\nA: 发射一颗轨道与地球表面上某一纬度线 (非赤道) 为共面同心圆的地球卫星\nB: 发射一颗与地球表面上某一经度线所决定的圆为共面同心圆的地球卫星\nC: 发射一颗 $1 \\mathrm{~h}$ 绕地球运转一周的地球卫星\nD: 发射一颗每天同一时间都能通过北京上空的地球卫星\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_151", "problem": "已知一质量为 $m$ 的物体静止在北极与赤道对地面的压力差为 $\\Delta N$, 假设地球是质量均匀的球体, 半径为 $R$ 。则地球的自转周期为 (设地球表面的重力加速度为 $g$ ) ( )\nA: 地球的自转周期为 $T=2 \\pi \\sqrt{\\frac{m R}{\\Delta N}}$\nB: 地球的自转周期为 $T=\\pi \\sqrt{\\frac{m R}{\\Delta N}}$\nC: 地球同步卫星的轨道半径为 $\\left(\\frac{m g}{\\Delta N}\\right)^{\\frac{1}{3}} R$\nD: 地球同步卫星的轨道半径为 $2\\left(\\frac{m g}{\\Delta N}\\right)^{\\frac{1}{3}} R$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n已知一质量为 $m$ 的物体静止在北极与赤道对地面的压力差为 $\\Delta N$, 假设地球是质量均匀的球体, 半径为 $R$ 。则地球的自转周期为 (设地球表面的重力加速度为 $g$ ) ( )\n\nA: 地球的自转周期为 $T=2 \\pi \\sqrt{\\frac{m R}{\\Delta N}}$\nB: 地球的自转周期为 $T=\\pi \\sqrt{\\frac{m R}{\\Delta N}}$\nC: 地球同步卫星的轨道半径为 $\\left(\\frac{m g}{\\Delta N}\\right)^{\\frac{1}{3}} R$\nD: 地球同步卫星的轨道半径为 $2\\left(\\frac{m g}{\\Delta N}\\right)^{\\frac{1}{3}} R$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1063", "problem": "In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.c. In practice, the gravitational binding energy of the Earth is much lower than that of the Sun, and so the First Order would not need to drain the whole star to get enough energy to destroy the Earth. Assuming the weapon is able to channel towards it all the energy being radiated from the Sun's entire surface, how long would it take them to charge the superweapon sufficiently to do this?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.\n\nproblem:\nc. In practice, the gravitational binding energy of the Earth is much lower than that of the Sun, and so the First Order would not need to drain the whole star to get enough energy to destroy the Earth. Assuming the weapon is able to channel towards it all the energy being radiated from the Sun's entire surface, how long would it take them to charge the superweapon sufficiently to do this?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-06.jpg?height=611&width=1448&top_left_y=505&top_left_x=310" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1046", "problem": "Young, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\\star}=q \\ll h^{3}$.\n[figure1]\n\nFigure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF).\n\nRight: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset.\n\nSince the planet is assumed small $\\left(q \\ll h^{3}\\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \\approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via:\n\n$$\n\\frac{\\mathrm{d} L}{\\mathrm{~d} t}=\\Gamma\n$$\n\nIt is convenient to write the torque in terms of the reference value\n\n$$\n\\Gamma_{0}=\\left(\\frac{q}{h}\\right)^{2} \\Sigma_{p} r_{p}^{4} \\Omega_{p}^{2}\n$$\nc. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\\Gamma_{C}$. For a disc of uniform entropy ( $\\left.\\mathrm{d} s=0\\right)$, and with surface density profile $\\Sigma \\propto r^{-\\alpha}$, and pressure profile $P \\propto r^{-\\delta}$, Tanaka et al. (2002) and Paardekooper \\& Papaloizou (2009) find these torques are given by:\n\n$$\n\\begin{gathered}\n\\Gamma_{L}=(-3.20+0.86 \\alpha-2.33 \\delta) \\Gamma_{0} \\\\\n\\Gamma_{C}=5.97(1.5-\\alpha) \\Gamma_{0}\n\\end{gathered}\n$$\n\nWe assume the gas in the disc obeys the ideal gas law, so that:\n\n$$\n\\frac{P}{\\Sigma T}=\\text { constant }, \\quad \\mathrm{d} s=\\text { constant } \\times\\left(\\frac{1}{\\gamma-1} \\frac{\\mathrm{d} T}{T}-\\frac{\\mathrm{d} \\Sigma}{\\Sigma}\\right),\n$$\n\nwhere $T$ is the absolute temperature and $\\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy,\n\n$$\n\\Gamma=\\Gamma_{L}+\\Gamma_{C}=(5.76-(5.11+2.33 \\gamma) \\alpha) \\Gamma_{0}\n$$\n\n[Hint: if $\\frac{\\mathrm{d} y}{y}=\\lambda \\frac{\\mathrm{d} x}{x}$, then $y \\propto x^{\\lambda}$.]\n\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\Omega$ is $L=I \\Omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.b. Consider now a rotationally symmetric disc with surface density profile $\\Sigma=\\Sigma_{0}\\left(r / r_{0}\\right)^{-3 / 2}$, and outer radius $r_{\\text {out }}=9 r_{0}$. Find the mass of the disc $M_{\\text {disc }}$ in terms of $\\Sigma_{0}$ and $r_{0}$ assuming its inner radius $r_{\\text {in }}<F_{3}$\nB: $a_{2}>a_{3}>a_{1}$\nC: $\\omega_{1}=\\omega_{3}<\\omega_{2}$\nD: $v_{1}=v_{2}>v_{3}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n地球赤道上有一物体随地球的自转而做圆周运动, 所受的向心力为 $F_{1}$, 向心加速度为 $a_{1}$, 线速度为 $v_{1}$, 角速度为 $\\omega_{1}$; 绕地球表面附近做圆周运动的人造卫星受的向心力为 $F_{2}$, 向心加速度为 $a_{2}$, 线速度为 $v_{2}$, 角速度为 $\\omega_{2}$; 地球同步卫星所受的向心力为 $F_{3}$, 向心加速度为 $a_{3}$, 线速度为 $v_{3}$, 角速度为 $\\omega_{3}$. 假设三者质量相等, 则 $(\\quad)$\n\nA: $F_{1}=F_{2}>F_{3}$\nB: $a_{2}>a_{3}>a_{1}$\nC: $\\omega_{1}=\\omega_{3}<\\omega_{2}$\nD: $v_{1}=v_{2}>v_{3}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_707", "problem": "2020 年 7 月 23 日, “天问一号”探测器成功发射, 开启了探测火星之旅。截至 2022 年 4 月，“天问一号”已依次完成了“绕、落、巡”三大目标。假设地球近地卫星的周期与\n火星近火卫星的周期比值为 $k$, 地球半径与火星半径的比值为 $n$ 。则下列说法正确的是\nA: 地球质量与火星质量之比为 $n^{3}: k^{2}$\nB: 地球密度与火星密度之比为 $1: k$\nC: 地球第一宇宙速度与火星第一宇宙速度之比为 $\\sqrt{n}: \\sqrt{k}$\nD: 如果地球的某一卫星与火星的某一卫星轨道半径相同, 则两卫星的加速度之比为 $n: k^{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年 7 月 23 日, “天问一号”探测器成功发射, 开启了探测火星之旅。截至 2022 年 4 月，“天问一号”已依次完成了“绕、落、巡”三大目标。假设地球近地卫星的周期与\n火星近火卫星的周期比值为 $k$, 地球半径与火星半径的比值为 $n$ 。则下列说法正确的是\n\nA: 地球质量与火星质量之比为 $n^{3}: k^{2}$\nB: 地球密度与火星密度之比为 $1: k$\nC: 地球第一宇宙速度与火星第一宇宙速度之比为 $\\sqrt{n}: \\sqrt{k}$\nD: 如果地球的某一卫星与火星的某一卫星轨道半径相同, 则两卫星的加速度之比为 $n: k^{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_676", "problem": "宇宙中存在一些离其他恒星较远的四颗星组成的四星系统。若某个四星系统中每个星体的质量均为 $m$, 半径均为 $R$, 忽略其他星体对它们的引力作用和忽略星体自转效应,则可能存在如下运动形式: 四颗星分别位于边长为 $L$ 的正方形的四个顶点上 ( $L$ 远大于 $R)$ ，在相互之间的万有引力作用下，绕某一共同的圆心做角速度相同的圆周运动。已知万有引力常量为 $G$, 则关于此四星系统, 下列说法正确的是 $(\\quad)$\nA: 四颗星做圆周运动的轨道半径均为 $\\frac{L}{2}$\nB: 四颗星表面的重力加速度均为 $G \\frac{m}{R^{2}}$\nC: 四颗星做圆周运动的向心力大小为 $\\frac{G m^{2}}{L^{2}}(2 \\sqrt{2}+1)$\nD: 四颗星做圆周运动的角速度均为 $\\sqrt{\\frac{(4+\\sqrt{2}) G m}{2 L^{3}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇宙中存在一些离其他恒星较远的四颗星组成的四星系统。若某个四星系统中每个星体的质量均为 $m$, 半径均为 $R$, 忽略其他星体对它们的引力作用和忽略星体自转效应,则可能存在如下运动形式: 四颗星分别位于边长为 $L$ 的正方形的四个顶点上 ( $L$ 远大于 $R)$ ，在相互之间的万有引力作用下，绕某一共同的圆心做角速度相同的圆周运动。已知万有引力常量为 $G$, 则关于此四星系统, 下列说法正确的是 $(\\quad)$\n\nA: 四颗星做圆周运动的轨道半径均为 $\\frac{L}{2}$\nB: 四颗星表面的重力加速度均为 $G \\frac{m}{R^{2}}$\nC: 四颗星做圆周运动的向心力大小为 $\\frac{G m^{2}}{L^{2}}(2 \\sqrt{2}+1)$\nD: 四颗星做圆周运动的角速度均为 $\\sqrt{\\frac{(4+\\sqrt{2}) G m}{2 L^{3}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_300", "problem": "遥远的星球。某星球完全由不可压缩的液态水组成。星球的表面重力加速度为 $g_{0}=9.8 \\mathrm{~m} / \\mathrm{s}^{2}$, 半径为 $R$, 且没有自转。水的密度 $\\rho=10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$, 万有引力常数 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} \\cdot \\mathrm{kg}^{-2}$ 。本题中可能用到如下公式: 半径为 $R$ 的球，其体积为 $V=\\frac{4}{3} \\pi R^{3}$, 表面积为 $S=4 \\pi R^{2}$ 。星球的半径 $R$ 是多少?", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n遥远的星球。某星球完全由不可压缩的液态水组成。星球的表面重力加速度为 $g_{0}=9.8 \\mathrm{~m} / \\mathrm{s}^{2}$, 半径为 $R$, 且没有自转。水的密度 $\\rho=10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$, 万有引力常数 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} \\cdot \\mathrm{kg}^{-2}$ 。本题中可能用到如下公式: 半径为 $R$ 的球，其体积为 $V=\\frac{4}{3} \\pi R^{3}$, 表面积为 $S=4 \\pi R^{2}$ 。星球的半径 $R$ 是多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以m为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_393", "problem": "宇宙中有一星球, 其半径为 $R$, 自转周期为 $T$, 引力常量为 $G$, 该天体的质量为 $M$ 。若一宇航员来到位于赤道的一斜坡前, 将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示, 小球垂直击中了斜坡, 落点为 $P$ 点, 求\n\n则 $P$ 点距水平地面的高度 $h$ 。\n\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n宇宙中有一星球, 其半径为 $R$, 自转周期为 $T$, 引力常量为 $G$, 该天体的质量为 $M$ 。若一宇航员来到位于赤道的一斜坡前, 将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示, 小球垂直击中了斜坡, 落点为 $P$ 点, 求\n\n则 $P$ 点距水平地面的高度 $h$ 。\n\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-114.jpg?height=340&width=648&top_left_y=150&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_739", "problem": "卫星围绕某行星做匀速圆周运动的轨道半径的三次方 $\\left(r^{3}\\right)$ 与周期的平方 $\\left(T^{2}\\right)$ 之间的关系如图所示。若该行星的半径 $R_{0}$ 和卫星在该行星表面运行的周期 $T_{0}$ 已知, 引力常量为 $G$, 则下列物理量中不能求出的是 ( )\n\n[图1]\nA: 该卫星的线速度\nB: 该卫星的动能\nC: 该行星的平均密度\nD: 该行星表面的重力加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n卫星围绕某行星做匀速圆周运动的轨道半径的三次方 $\\left(r^{3}\\right)$ 与周期的平方 $\\left(T^{2}\\right)$ 之间的关系如图所示。若该行星的半径 $R_{0}$ 和卫星在该行星表面运行的周期 $T_{0}$ 已知, 引力常量为 $G$, 则下列物理量中不能求出的是 ( )\n\n[图1]\n\nA: 该卫星的线速度\nB: 该卫星的动能\nC: 该行星的平均密度\nD: 该行星表面的重力加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-09.jpg?height=402&width=505&top_left_y=2009&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_65", "problem": "我国天文学家通过 FAST, 在武仙座球状星团 $\\mathrm{M}_{1} 3$ 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}0)$ since we observe them as they were long ago when the scale factor was smaller.\n\nFor a Universe to be flat (i.e. zero curvature), its average density must be equal to the critical density,\n\n$$\n\\rho_{\\text {crit }, 0}=\\frac{3 H_{0}^{2}}{8 \\pi G},\n$$\n\nwhere $H_{0}$ is the Hubble constant, measured in 2018 from the cosmic microwave background by the Planck spacecraft to be $67.36 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\n\nThe density of the $i^{\\text {th }}$ component of the Universe can be expressed relative to the critical density as the density parameter,\n\n$$\n\\Omega_{i}=\\frac{\\rho_{i}}{\\rho_{\\text {crit }}} .\n$$\n\nPlanck measured the current density parameters of dark energy and matter as $\\Omega_{\\Lambda, 0}=0.6847$ and $\\Omega_{m, 0}=0.3153$ respectively.\n\nIn each epoch, the scale factor increases at a different rate with time, $t$, as the density also varies differently with scale factor.\n\n- Radiation-dominated epoch: The Universe's early history, where $\\rho \\propto a^{-4}$ and so $a \\propto t^{1 / 2}$\n- Matter-dominated epoch: This represents much of the history of the Universe, where $\\rho \\propto$ $a^{-3}$ and so $a \\propto t^{2 / 3}$\n- Dark-energy-dominated epoch: This is an era we have recently entered and will remain in for the rest of time, where $\\rho$ doesn't vary with scale factor (i.e. is a constant) and so $a \\propto e^{H_{0} t}$\n\nAssuming our Universe is flat, calculate the current average density of the Universe.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe Universe consists of three main components: radiation (including neutrinos), matter (both atoms and dark matter), and dark energy. The overall density of the universe has been dominated by the density of each of those in turn at different times in its history, leading to three different epochs (shown in Fig 3).\n\n[figure1]\n\nFigure 3: A outline of the three main epochs in the history of the Universe. You do not need to read any data off this graph to answer this question. Credit: Pearson Education, Inc.\n\nThe scale factor, $a$, describes how the Universe has expanded (i.e. a measure of the relative radius of the Universe), and the current value is defined as $a_{0} \\equiv 1$ where the subscript ' 0 ' indicates it is as measured today. At earlier times $a<1$ and at the Big Bang $a=0$.\n\nThe redshift of an object, $z$, is related to the scale factor as $a=(1+z)^{-1}$ and so the redshift corresponding to now is $z=0$, and far away objects have higher redshift $(z>0)$ since we observe them as they were long ago when the scale factor was smaller.\n\nFor a Universe to be flat (i.e. zero curvature), its average density must be equal to the critical density,\n\n$$\n\\rho_{\\text {crit }, 0}=\\frac{3 H_{0}^{2}}{8 \\pi G},\n$$\n\nwhere $H_{0}$ is the Hubble constant, measured in 2018 from the cosmic microwave background by the Planck spacecraft to be $67.36 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\n\nThe density of the $i^{\\text {th }}$ component of the Universe can be expressed relative to the critical density as the density parameter,\n\n$$\n\\Omega_{i}=\\frac{\\rho_{i}}{\\rho_{\\text {crit }}} .\n$$\n\nPlanck measured the current density parameters of dark energy and matter as $\\Omega_{\\Lambda, 0}=0.6847$ and $\\Omega_{m, 0}=0.3153$ respectively.\n\nIn each epoch, the scale factor increases at a different rate with time, $t$, as the density also varies differently with scale factor.\n\n- Radiation-dominated epoch: The Universe's early history, where $\\rho \\propto a^{-4}$ and so $a \\propto t^{1 / 2}$\n- Matter-dominated epoch: This represents much of the history of the Universe, where $\\rho \\propto$ $a^{-3}$ and so $a \\propto t^{2 / 3}$\n- Dark-energy-dominated epoch: This is an era we have recently entered and will remain in for the rest of time, where $\\rho$ doesn't vary with scale factor (i.e. is a constant) and so $a \\propto e^{H_{0} t}$\n\nAssuming our Universe is flat, calculate the current average density of the Universe.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~kg} \\mathrm{~m}^{-3}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c3e3c992a9c51eb3e471g-08.jpg?height=1080&width=1271&top_left_y=739&top_left_x=398" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~kg} \\mathrm{~m}^{-3}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_779", "problem": "A star has the luminosity $L_{0}$. The temperature $T$ of the star doubles. How does the luminosity change?\nA: $2 \\times L_{0}$\nB: $4 \\times L_{0}$\nC: $8 \\times L_{0}$\nD: $16 \\times L_{0}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA star has the luminosity $L_{0}$. The temperature $T$ of the star doubles. How does the luminosity change?\n\nA: $2 \\times L_{0}$\nB: $4 \\times L_{0}$\nC: $8 \\times L_{0}$\nD: $16 \\times L_{0}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_823", "problem": "Now suppose that Lucas is standing still in Baia (from the previous question), and Justin is standing still on the equator. Let $P_{L 1}$ and $P_{J 1}$ be the paths of Lucas's and Justin's shadows on the summer solstice, respectively. Let $P_{L 2}$ and $P_{J 2}$ be the paths of Lucas's and Justin's shadows on the vernal equinox, respectively. Assume that the heights of Lucas and Justin are small compared to the radius of the Earth, there is no atmospheric refraction, and that the Sun is a point. Given Earth's obliquity $\\varepsilon=23.44^{\\circ}$, which of the following is the most specific accurate description of the shapes of each path?\nA: $P_{L 1}:$ Parabola, $P_{J 1}:$ Hyperbola, $P_{L 2}:$ Line, $P_{J 2}:$ Line\nB: $P_{L 1}:$ Parabola, $P_{J 1}:$ Hyperbola, $P_{L 2}:$ Hyperbola, $P_{J 2}:$ Line\nC: $P_{L 1}:$ Parabola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Line, $P_{J 2}:$ Line\nD: $P_{L 1}:$ Hyperbola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Hyperbola, $P_{J 2}:$ Line\nE: $P_{L 1}:$ Hyperbola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Line, $P_{J 2}:$ Line\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nNow suppose that Lucas is standing still in Baia (from the previous question), and Justin is standing still on the equator. Let $P_{L 1}$ and $P_{J 1}$ be the paths of Lucas's and Justin's shadows on the summer solstice, respectively. Let $P_{L 2}$ and $P_{J 2}$ be the paths of Lucas's and Justin's shadows on the vernal equinox, respectively. Assume that the heights of Lucas and Justin are small compared to the radius of the Earth, there is no atmospheric refraction, and that the Sun is a point. Given Earth's obliquity $\\varepsilon=23.44^{\\circ}$, which of the following is the most specific accurate description of the shapes of each path?\n\nA: $P_{L 1}:$ Parabola, $P_{J 1}:$ Hyperbola, $P_{L 2}:$ Line, $P_{J 2}:$ Line\nB: $P_{L 1}:$ Parabola, $P_{J 1}:$ Hyperbola, $P_{L 2}:$ Hyperbola, $P_{J 2}:$ Line\nC: $P_{L 1}:$ Parabola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Line, $P_{J 2}:$ Line\nD: $P_{L 1}:$ Hyperbola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Hyperbola, $P_{J 2}:$ Line\nE: $P_{L 1}:$ Hyperbola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Line, $P_{J 2}:$ Line\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1191", "problem": "The Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$b. Assuming Sgr A* is a non-spinning black hole with mass $4.1 \\times 10^{6} \\mathrm{M}_{\\odot}$ and at a distance of $8.34 \\mathrm{kpc}$ :\n\nii. Determine the angular diameter (in microarcseconds) of the lensed photon sphere of Sgr A* and hence verify that the EHT can resolve it.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$\n\nproblem:\nb. Assuming Sgr A* is a non-spinning black hole with mass $4.1 \\times 10^{6} \\mathrm{M}_{\\odot}$ and at a distance of $8.34 \\mathrm{kpc}$ :\n\nii. Determine the angular diameter (in microarcseconds) of the lensed photon sphere of Sgr A* and hence verify that the EHT can resolve it.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-07.jpg?height=704&width=1414&top_left_y=698&top_left_x=331", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-08.jpg?height=374&width=1562&top_left_y=1698&top_left_x=263", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-09.jpg?height=468&width=686&top_left_y=1388&top_left_x=705" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_810", "problem": "Kerbyn is a small rocky planet in a circular orbit around a $0.2 M$ star with a semimajor axis of $0.1 A U$. Kerbyn has an axial tilt of $\\epsilon=42^{\\circ}$ and a sidereal rotation period of $05^{h} 59^{m} 9.4^{s}$. On the vernal equinox, what is the length of the apparent solar day on Kerbyn? The apparent solar day is defined as the interval between successive crossings of the meridian by the sun.\nA: $05^{h} 55^{m} 39.3^{s}$\nB: $05^{h} 57^{m} 15.2^{s}$\nC: $06^{h} 00^{m} 00.0^{s}$\nD: $06^{h} 01^{m} 45.1^{s}$\nE: $06^{h} 02^{m} 39.5^{s}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nKerbyn is a small rocky planet in a circular orbit around a $0.2 M$ star with a semimajor axis of $0.1 A U$. Kerbyn has an axial tilt of $\\epsilon=42^{\\circ}$ and a sidereal rotation period of $05^{h} 59^{m} 9.4^{s}$. On the vernal equinox, what is the length of the apparent solar day on Kerbyn? The apparent solar day is defined as the interval between successive crossings of the meridian by the sun.\n\nA: $05^{h} 55^{m} 39.3^{s}$\nB: $05^{h} 57^{m} 15.2^{s}$\nC: $06^{h} 00^{m} 00.0^{s}$\nD: $06^{h} 01^{m} 45.1^{s}$\nE: $06^{h} 02^{m} 39.5^{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_617", "problem": "宇航员抵达一半径为 $R$ 的星球后，做了如下的实验 取一根细绳穿过光滑的细直管,细绳的一端拴一质量为 $m$ 的砝码, 另一端连接在一固定的拉力传感器上, 手捏细直管抢动砝码, 使它在坚直平面内做圆周运动. 若该星球表面没有空气, 不计阻力, 停止抢动细直管, 砝码可继续在同一坚直平面内做完整的圆周运动, 如图所示, 此时拉力传感器显示砝码运动到最低点与最高点两位置时读数差的绝对值为 $\\Delta F$. 已知万有引力常量为 $G$, 根据题中提供的条件和测量结果, 可知( )\n\n[图1]\nA: 该星球表面的重力加速度为 $\\frac{\\Delta F}{2 m}$\nB: 该星球表面的重力加速度为 $\\frac{\\Delta F}{6 m}$\nC: 该星球的质量为 $\\frac{\\Delta F R^{2}}{6 G m}$\nD: 该星球的质量为 $\\frac{\\Delta F R^{2}}{3 G m}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇航员抵达一半径为 $R$ 的星球后，做了如下的实验 取一根细绳穿过光滑的细直管,细绳的一端拴一质量为 $m$ 的砝码, 另一端连接在一固定的拉力传感器上, 手捏细直管抢动砝码, 使它在坚直平面内做圆周运动. 若该星球表面没有空气, 不计阻力, 停止抢动细直管, 砝码可继续在同一坚直平面内做完整的圆周运动, 如图所示, 此时拉力传感器显示砝码运动到最低点与最高点两位置时读数差的绝对值为 $\\Delta F$. 已知万有引力常量为 $G$, 根据题中提供的条件和测量结果, 可知( )\n\n[图1]\n\nA: 该星球表面的重力加速度为 $\\frac{\\Delta F}{2 m}$\nB: 该星球表面的重力加速度为 $\\frac{\\Delta F}{6 m}$\nC: 该星球的质量为 $\\frac{\\Delta F R^{2}}{6 G m}$\nD: 该星球的质量为 $\\frac{\\Delta F R^{2}}{3 G m}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://i.postimg.cc/Hkd0dL4W/image.png" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_756", "problem": "The surface temperature (in ${ }^{\\circ} \\mathrm{C}$ ) of the Sun is close to ...\nA: $5200{ }^{\\circ} \\mathrm{C}$\nB: $5500^{\\circ} \\mathrm{C}$\nC: $5800{ }^{\\circ} \\mathrm{C}$\nD: $6000^{\\circ} \\mathrm{C}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe surface temperature (in ${ }^{\\circ} \\mathrm{C}$ ) of the Sun is close to ...\n\nA: $5200{ }^{\\circ} \\mathrm{C}$\nB: $5500^{\\circ} \\mathrm{C}$\nC: $5800{ }^{\\circ} \\mathrm{C}$\nD: $6000^{\\circ} \\mathrm{C}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_210", "problem": "太空中存在一些离其他恒星很远的、由两颗星体组成的双星系统, 可忽略其他星体对它们的引力作用。如果将某双星系统简化为理想的圆周运动模型, 如图所示, 两星球在相互的万有引力作用下, 绕 $\\mathrm{O}$ 点做匀速圆周运动。由于双星间的距离减小，则 $(\\quad)$\n\n[图1]\nA: 两星的运动角速度均逐渐减小\nB: 两星的运动周期均逐渐减小\nC: 两星的向心加速度均逐渐减小\nD: 两星的运动线速度均逐渐减小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n太空中存在一些离其他恒星很远的、由两颗星体组成的双星系统, 可忽略其他星体对它们的引力作用。如果将某双星系统简化为理想的圆周运动模型, 如图所示, 两星球在相互的万有引力作用下, 绕 $\\mathrm{O}$ 点做匀速圆周运动。由于双星间的距离减小，则 $(\\quad)$\n\n[图1]\n\nA: 两星的运动角速度均逐渐减小\nB: 两星的运动周期均逐渐减小\nC: 两星的向心加速度均逐渐减小\nD: 两星的运动线速度均逐渐减小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-34.jpg?height=314&width=323&top_left_y=1619&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_669", "problem": "压强表示单位面积上压力的大小, 是物理学中的重要概念。\n\n单个粒子碰撞在某一平面上会产生一个短暂的作用力, 而大量粒子持续碰撞会产生一个持续的作用力。一束均匀粒子流持续碰撞一平面, 设该束粒子流中每个粒子的质量均为 $m$ 、速度大小均为 $v$, 方向都与该平面垂直, 单位体积内的粒子数为 $n$, 粒子与该平面碰撞后均不反弹，忽略空气阻力，不考虑粒子所受重力以及粒子间的相互作用。求粒子流对该平面所产生的压强 $p$ 。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n压强表示单位面积上压力的大小, 是物理学中的重要概念。\n\n单个粒子碰撞在某一平面上会产生一个短暂的作用力, 而大量粒子持续碰撞会产生一个持续的作用力。一束均匀粒子流持续碰撞一平面, 设该束粒子流中每个粒子的质量均为 $m$ 、速度大小均为 $v$, 方向都与该平面垂直, 单位体积内的粒子数为 $n$, 粒子与该平面碰撞后均不反弹，忽略空气阻力，不考虑粒子所受重力以及粒子间的相互作用。求粒子流对该平面所产生的压强 $p$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_997", "problem": "For a satellite in a circular orbit around the Earth, which of the arrows in the figure below describe the direction of the velocity and the resultant force?\n\n[figure1]\nA: Velocity $=1$, Resultant force $=1$\nB: Velocity $=1$, Resultant force $=2$\nC: Velocity $=2$, Resultant force $=1$\nD: Velocity $=3$, Resultant force $=4$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nFor a satellite in a circular orbit around the Earth, which of the arrows in the figure below describe the direction of the velocity and the resultant force?\n\n[figure1]\n\nA: Velocity $=1$, Resultant force $=1$\nB: Velocity $=1$, Resultant force $=2$\nC: Velocity $=2$, Resultant force $=1$\nD: Velocity $=3$, Resultant force $=4$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-04.jpg?height=377&width=500&top_left_y=617&top_left_x=778" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_740", "problem": "宇航员驾驶宇宙飞船绕一星球做匀速圆周运动, 测得飞船线速度大小的二次方与轨道半径的倒数的关系图像如图中实线所示, 该图线 (直线) 的斜率为 $k$, 图中 $r_{0}$ (该星球的半径）为已知量。引力常量为 $G$, 下列说法正确的是 ( )\n\n[图1]\nA: 该星球的密度为 $\\frac{3 k}{4 \\pi G r_{0}^{3}}$\nB: 该星球自转的周期为 $\\sqrt{\\frac{r_{0}^{3}}{k}}$\nC: 该星球表面的重力加速度大小为 $\\frac{k}{r_{0}}$\nD: 该星球的第一宇宙速度为 $\\sqrt{\\frac{2 k}{r_{0}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n宇航员驾驶宇宙飞船绕一星球做匀速圆周运动, 测得飞船线速度大小的二次方与轨道半径的倒数的关系图像如图中实线所示, 该图线 (直线) 的斜率为 $k$, 图中 $r_{0}$ (该星球的半径）为已知量。引力常量为 $G$, 下列说法正确的是 ( )\n\n[图1]\n\nA: 该星球的密度为 $\\frac{3 k}{4 \\pi G r_{0}^{3}}$\nB: 该星球自转的周期为 $\\sqrt{\\frac{r_{0}^{3}}{k}}$\nC: 该星球表面的重力加速度大小为 $\\frac{k}{r_{0}}$\nD: 该星球的第一宇宙速度为 $\\sqrt{\\frac{2 k}{r_{0}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-060.jpg?height=574&width=617&top_left_y=684&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_528", "problem": "如图所示, $A$ 为地球表面赤道上的物体, $B$ 为轨道在赤道平面内的实验卫星, $C$ 为\n在赤道上空的地球同步卫星, 已知卫星 $C$ 和卫星 $B$ 的轨道半径之比为 3: 1, 且两卫星的环绕方向相同, 下列说法正确的是 ( )\n\n[图1]\nA: 卫星 $B 、 C$ 运行速度之比为 $3: 1$\nB: 卫星 $B$ 的加速度大于物体 $A$ 的加速度\nC: 同一物体在卫星 $B$ 中对支持物的压力比在卫星 $C$ 中大\nD: 在卫星 $B$ 中一天内可看到 3 次日出\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, $A$ 为地球表面赤道上的物体, $B$ 为轨道在赤道平面内的实验卫星, $C$ 为\n在赤道上空的地球同步卫星, 已知卫星 $C$ 和卫星 $B$ 的轨道半径之比为 3: 1, 且两卫星的环绕方向相同, 下列说法正确的是 ( )\n\n[图1]\n\nA: 卫星 $B 、 C$ 运行速度之比为 $3: 1$\nB: 卫星 $B$ 的加速度大于物体 $A$ 的加速度\nC: 同一物体在卫星 $B$ 中对支持物的压力比在卫星 $C$ 中大\nD: 在卫星 $B$ 中一天内可看到 3 次日出\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-088.jpg?height=371&width=391&top_left_y=314&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_63", "problem": "在星球 $\\mathrm{M}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\\mathrm{P}$ 轻放在弹簧上端, $\\mathrm{P}$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $\\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $\\mathrm{Q}$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $\\mathrm{M}$ 的半径是星球 $\\mathrm{N}$ 的 3 倍, 则（ ）\n\n[图1]\nA: $\\mathrm{M}$ 与 $\\mathrm{N}$ 的密度相等\nB: $Q$ 的质量是 $P$ 的 3 倍\nC: $\\mathrm{Q}$ 下落过程中的最大动能是 $\\mathrm{P}$ 的 4 倍\nD: Q 下落过程中弹簧的最大压缩量是 $\\mathrm{P}$ 的 4 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n在星球 $\\mathrm{M}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\\mathrm{P}$ 轻放在弹簧上端, $\\mathrm{P}$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $\\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $\\mathrm{Q}$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $\\mathrm{M}$ 的半径是星球 $\\mathrm{N}$ 的 3 倍, 则（ ）\n\n[图1]\n\nA: $\\mathrm{M}$ 与 $\\mathrm{N}$ 的密度相等\nB: $Q$ 的质量是 $P$ 的 3 倍\nC: $\\mathrm{Q}$ 下落过程中的最大动能是 $\\mathrm{P}$ 的 4 倍\nD: Q 下落过程中弹簧的最大压缩量是 $\\mathrm{P}$ 的 4 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-04.jpg?height=334&width=488&top_left_y=1152&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_123", "problem": "科学家在夏威夷利用红外望远镜设施发现了两颗近地小行星, 这两颗小行星富含金属, 且行星表面金属含量超过了 $85 \\%$. 倘若有甲、乙两颗行星, 且各自卫星公转半径的三次方的倒数 $\\frac{1}{r^{3}}$ 与公转角速度的平方 $\\omega^{2}$ 的关系图像如图所示, 其中甲对应图线 $a$, 乙对应图线 $b$, 且甲、乙两行星的半径接近, 为方便分析认为两者相等, 下列说法正确的是 ( )\n\n[图1]\nA: 甲行星的质量比乙行星大\nB: 甲行星表面的卫星速度比乙行星小\nC: 若甲、乙分别有一颗卫星 $A 、 B$, 且 $A 、 B$ 运行周期相同, 则 $A$ 的速度较大\nD: 若甲、乙分别有一颗卫星 $C 、 D$, 且 $C 、 D$ 轨道半径相同, 则 $D$ 的向心加速度较大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n科学家在夏威夷利用红外望远镜设施发现了两颗近地小行星, 这两颗小行星富含金属, 且行星表面金属含量超过了 $85 \\%$. 倘若有甲、乙两颗行星, 且各自卫星公转半径的三次方的倒数 $\\frac{1}{r^{3}}$ 与公转角速度的平方 $\\omega^{2}$ 的关系图像如图所示, 其中甲对应图线 $a$, 乙对应图线 $b$, 且甲、乙两行星的半径接近, 为方便分析认为两者相等, 下列说法正确的是 ( )\n\n[图1]\n\nA: 甲行星的质量比乙行星大\nB: 甲行星表面的卫星速度比乙行星小\nC: 若甲、乙分别有一颗卫星 $A 、 B$, 且 $A 、 B$ 运行周期相同, 则 $A$ 的速度较大\nD: 若甲、乙分别有一颗卫星 $C 、 D$, 且 $C 、 D$ 轨道半径相同, 则 $D$ 的向心加速度较大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-109.jpg?height=300&width=374&top_left_y=684&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_480", "problem": "据中新网报道, 中国自主研发的北斗卫星导航系统“北斗三号”第 17 颗卫星已于 2018 年 11 月 2 日在西昌卫星发射中心成功发射。该卫星是北斗三号全球导航系统的首颗地球同步轨道卫星, 也是北斗三号系统中功能最强、信号最多、承载最大、寿命最长的卫星。关于该卫星, 下列说法正确的是（）\nA: 它的发射速度一定小于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nB: 它运行的线速度一定不小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 它在由过渡轨道进入运行轨道时必须减速\nD: 由于稀薄大气的影响, 如不加干预, 在运行一段时间后, 该卫星的速度可能会增加\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n据中新网报道, 中国自主研发的北斗卫星导航系统“北斗三号”第 17 颗卫星已于 2018 年 11 月 2 日在西昌卫星发射中心成功发射。该卫星是北斗三号全球导航系统的首颗地球同步轨道卫星, 也是北斗三号系统中功能最强、信号最多、承载最大、寿命最长的卫星。关于该卫星, 下列说法正确的是（）\n\nA: 它的发射速度一定小于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nB: 它运行的线速度一定不小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 它在由过渡轨道进入运行轨道时必须减速\nD: 由于稀薄大气的影响, 如不加干预, 在运行一段时间后, 该卫星的速度可能会增加\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_622", "problem": "如图所示, $a$ 为放在地球赤道上随地球表面一起转动的物体, $b$ 为处于地面附近近地轨道上的卫星, $c$ 是地球同步卫星, $d$ 是高空探测卫星, 若 $a 、 b 、 c 、 d$ 的质量相同,地球表面附近的重力加速度为 $g$. 则下列说法正确的是( )\n\n[图1]\nA: $d$ 是三颗卫星中动能最小，机械能最大的\nB: $c$ 距离地面的高度不是一确定值\nC: $a$ 和 $b$ 的向心加速度都等于重力加速度 $g$\nD: $a$ 的角速度最大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, $a$ 为放在地球赤道上随地球表面一起转动的物体, $b$ 为处于地面附近近地轨道上的卫星, $c$ 是地球同步卫星, $d$ 是高空探测卫星, 若 $a 、 b 、 c 、 d$ 的质量相同,地球表面附近的重力加速度为 $g$. 则下列说法正确的是( )\n\n[图1]\n\nA: $d$ 是三颗卫星中动能最小，机械能最大的\nB: $c$ 距离地面的高度不是一确定值\nC: $a$ 和 $b$ 的向心加速度都等于重力加速度 $g$\nD: $a$ 的角速度最大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-48.jpg?height=294&width=1096&top_left_y=1281&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_25", "problem": "2017 年 8 月,我国调制望远镜一“慧眼\"成功监测到了引力波源所在的天区。已知 A、 $\\mathrm{B}$ 两个恒星靠着相互间的引力绕二者连线上的某点做匀速圆周运动. 在环绕过程中会辐射出引力波, 该引力波的频率与两星做圆周运动的频率具有相同的数量级. 通过观察测得 A 的质量为太阳质量的 29 倍, B 的质量为太阳质量的 36 倍,两星间的距离为 $2 \\times$ $10^{5} \\mathrm{~m}$. 取太阳的质量为 $2 \\times 10^{30} \\mathrm{~kg}, \\mathrm{G}=6.7 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$, 则可估算出该引力波频率的数量级为 ( )\nA: $10^{2} \\mathrm{~Hz}$\nB: $10^{4} \\mathrm{~Hz}$\nC: $10^{6} \\mathrm{~Hz}$\nD: $10^{8} \\mathrm{~Hz}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2017 年 8 月,我国调制望远镜一“慧眼\"成功监测到了引力波源所在的天区。已知 A、 $\\mathrm{B}$ 两个恒星靠着相互间的引力绕二者连线上的某点做匀速圆周运动. 在环绕过程中会辐射出引力波, 该引力波的频率与两星做圆周运动的频率具有相同的数量级. 通过观察测得 A 的质量为太阳质量的 29 倍, B 的质量为太阳质量的 36 倍,两星间的距离为 $2 \\times$ $10^{5} \\mathrm{~m}$. 取太阳的质量为 $2 \\times 10^{30} \\mathrm{~kg}, \\mathrm{G}=6.7 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$, 则可估算出该引力波频率的数量级为 ( )\n\nA: $10^{2} \\mathrm{~Hz}$\nB: $10^{4} \\mathrm{~Hz}$\nC: $10^{6} \\mathrm{~Hz}$\nD: $10^{8} \\mathrm{~Hz}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_627", "problem": "银河系中由一颗白矮星和它的类日伴星组成的双星系统, 由于白矮星不停的吸收类日伴星抛出的物质致使其质量不断增加, 假设类日伴星所释放的物质被白矮星全部吸收,并且两星之间的距离在一段时间内不变, 两星球的总质量不变, 则下列说法正确的是\nA: 两星间的万有引力不变\nB: 两星的运动周期不变\nC: 类日伴星的轨道半径增大\nD: 白矮星的轨道半径增大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n银河系中由一颗白矮星和它的类日伴星组成的双星系统, 由于白矮星不停的吸收类日伴星抛出的物质致使其质量不断增加, 假设类日伴星所释放的物质被白矮星全部吸收,并且两星之间的距离在一段时间内不变, 两星球的总质量不变, 则下列说法正确的是\n\nA: 两星间的万有引力不变\nB: 两星的运动周期不变\nC: 类日伴星的轨道半径增大\nD: 白矮星的轨道半径增大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_226", "problem": "随着科技的发展, 人类必将揭开火星的神秘面纱.如图所示, 火星的人造卫星在火星赤道的正上方距离火星表面高度为 $R$ 处环绕火星做匀速圆周运动, 已知卫星的运行方向与火星的自转方向相同, $a$ 点为火星赤道上的点, 该点有一接收器, 可接收到卫星发出信号。已知火星的半径为 $R$, 火星同步卫星的周期为 $T$, 近火卫星的线速度为 $v$, 引力常量为 $G$ 。则下列说法正确的是 ( )\n\n[图1]\nA: 火星的质量为 $\\frac{G}{v^{2} R}$\nB: 卫星的环绕周期为 $\\frac{4 \\pi R}{v}$\nC: $a$ 点连续收到信号的最长时间为 $\\frac{4 \\sqrt{2} \\pi R}{3 v}$\nD: 火星同步卫星到火星表面的高度为 $\\sqrt[3]{\\frac{v^{2} T^{2} R}{4 \\pi^{2}}}-R$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n随着科技的发展, 人类必将揭开火星的神秘面纱.如图所示, 火星的人造卫星在火星赤道的正上方距离火星表面高度为 $R$ 处环绕火星做匀速圆周运动, 已知卫星的运行方向与火星的自转方向相同, $a$ 点为火星赤道上的点, 该点有一接收器, 可接收到卫星发出信号。已知火星的半径为 $R$, 火星同步卫星的周期为 $T$, 近火卫星的线速度为 $v$, 引力常量为 $G$ 。则下列说法正确的是 ( )\n\n[图1]\n\nA: 火星的质量为 $\\frac{G}{v^{2} R}$\nB: 卫星的环绕周期为 $\\frac{4 \\pi R}{v}$\nC: $a$ 点连续收到信号的最长时间为 $\\frac{4 \\sqrt{2} \\pi R}{3 v}$\nD: 火星同步卫星到火星表面的高度为 $\\sqrt[3]{\\frac{v^{2} T^{2} R}{4 \\pi^{2}}}-R$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-025.jpg?height=346&width=602&top_left_y=2203&top_left_x=333", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-027.jpg?height=428&width=442&top_left_y=563&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1188", "problem": "The surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.b. The energy density of black-body radiation, $u$, and number density, $n$, at temperature $T$ are given.\n\nii. Assuming the plasma of Fe ions is in thermal equilibrium with the photons, and that the average energy of the photons is equal to the ionisation energy of FeX (which is $22540 \\mathrm{~kJ}$ $\\mathrm{mol}^{-1}$ ), calculate the temperature of the plasma. Give your answer to 4 s.f.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.\n\nproblem:\nb. The energy density of black-body radiation, $u$, and number density, $n$, at temperature $T$ are given.\n\nii. Assuming the plasma of Fe ions is in thermal equilibrium with the photons, and that the average energy of the photons is equal to the ionisation energy of FeX (which is $22540 \\mathrm{~kJ}$ $\\mathrm{mol}^{-1}$ ), calculate the temperature of the plasma. Give your answer to 4 s.f.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~K}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-10.jpg?height=792&width=1572&top_left_y=598&top_left_x=241" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~K}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_402", "problem": "我国的“天链一号”卫星是地球同步卫星, 可为中低轨道卫星提供数据通讯, 如图为 “天链一号”卫星 $a$ 、赤道平面内的低轨道卫星 $b$ 、地球的位置关系示意图, $O$ 为地心，地球相对卫星 $a 、 b$ 的张角分别为 $\\theta_{1}$ 和 $\\theta_{2}\\left(\\theta_{2}\\right.$ 图中未标出), 卫星 $a$ 的轨道半径是 $b$ 的 4 倍, 已知卫星 $a 、 b$ 绕地球同向运行, 卫星 $a$ 的周期为 $T$, 在运行过程中由于地球的遮挡, 卫星 $b$ 会进入卫星 $a$ 通讯的盲区，卫星间的通讯信号视为沿直线传播，信号传输时间可忽略. 下列分析正确的是 ( )\n\n[图1]\nA: 卫星 $a, b$ 的速度之比为 $2: 1$\nB: 卫星 $b$ 的周期为 $T / 8$\nC: 卫星 $b$ 每次在盲区运行的时间为 $\\frac{\\theta_{1}+\\theta_{2}}{14 \\pi} T$\nD: 卫星 $b$ 每次在盲区运行的时间为 $\\frac{\\theta_{1}+\\theta_{2}}{16 \\pi} T$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n我国的“天链一号”卫星是地球同步卫星, 可为中低轨道卫星提供数据通讯, 如图为 “天链一号”卫星 $a$ 、赤道平面内的低轨道卫星 $b$ 、地球的位置关系示意图, $O$ 为地心，地球相对卫星 $a 、 b$ 的张角分别为 $\\theta_{1}$ 和 $\\theta_{2}\\left(\\theta_{2}\\right.$ 图中未标出), 卫星 $a$ 的轨道半径是 $b$ 的 4 倍, 已知卫星 $a 、 b$ 绕地球同向运行, 卫星 $a$ 的周期为 $T$, 在运行过程中由于地球的遮挡, 卫星 $b$ 会进入卫星 $a$ 通讯的盲区，卫星间的通讯信号视为沿直线传播，信号传输时间可忽略. 下列分析正确的是 ( )\n\n[图1]\n\nA: 卫星 $a, b$ 的速度之比为 $2: 1$\nB: 卫星 $b$ 的周期为 $T / 8$\nC: 卫星 $b$ 每次在盲区运行的时间为 $\\frac{\\theta_{1}+\\theta_{2}}{14 \\pi} T$\nD: 卫星 $b$ 每次在盲区运行的时间为 $\\frac{\\theta_{1}+\\theta_{2}}{16 \\pi} T$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-67.jpg?height=520&width=560&top_left_y=982&top_left_x=337", "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-68.jpg?height=505&width=543&top_left_y=1118&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_593", "problem": "宇宙中有两颗孤立的中子星, 它们在相互的万有引力作用下间距保持不变, 并沿半径不同的同心圆轨道做匀速圆周运动. 如果双星间距为 $L$, 质量分别为 $m_{1}$ 和 $m_{2}$, 引力常量为 $G$, 则 ( )\nA: 双星中 $m_{1}$ 的轨道半径 $r_{1}=\\frac{m_{2}}{m_{1}+m_{2}} L$\nB: 双星的运行周期 $T=2 \\pi L \\sqrt{\\frac{m_{2} L}{G\\left(m_{1}+m_{2}\\right)}}$\nC: $m_{1}$ 的线速度大小 $v_{1}=m_{1} \\sqrt{\\frac{G}{L\\left(m_{1}+m_{2}\\right)}}$\nD: 若周期为 $T$, 则总质量 $m_{1}+m_{2}=\\frac{4 \\pi^{2} L^{3}}{G T^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇宙中有两颗孤立的中子星, 它们在相互的万有引力作用下间距保持不变, 并沿半径不同的同心圆轨道做匀速圆周运动. 如果双星间距为 $L$, 质量分别为 $m_{1}$ 和 $m_{2}$, 引力常量为 $G$, 则 ( )\n\nA: 双星中 $m_{1}$ 的轨道半径 $r_{1}=\\frac{m_{2}}{m_{1}+m_{2}} L$\nB: 双星的运行周期 $T=2 \\pi L \\sqrt{\\frac{m_{2} L}{G\\left(m_{1}+m_{2}\\right)}}$\nC: $m_{1}$ 的线速度大小 $v_{1}=m_{1} \\sqrt{\\frac{G}{L\\left(m_{1}+m_{2}\\right)}}$\nD: 若周期为 $T$, 则总质量 $m_{1}+m_{2}=\\frac{4 \\pi^{2} L^{3}}{G T^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-086.jpg?height=425&width=508&top_left_y=867&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_826", "problem": "Here is a map of MIT and the surrounding area, where North points directly upwards, as taken from https://whereis.mit.edu:\n\n[figure1]\n\nLeo is biking along the Harvard Bridge (marked as \"A\") when he stops and looks out at the river. Looking out downriver (to the right on this map) and parallel to the banks, he sees the Sun straight in front of him, peeking out from above the buildings, and has to avert his eyes to not be blinded. What part of the academic year is it?\nA: Early fall semester (late September-early October)\nB: Late fall semester (late November-early December)\nC: Independent Activities Period (January)\nD: Early spring semester (late February-early March)\nE: Late spring semester (late April-early May)\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nHere is a map of MIT and the surrounding area, where North points directly upwards, as taken from https://whereis.mit.edu:\n\n[figure1]\n\nLeo is biking along the Harvard Bridge (marked as \"A\") when he stops and looks out at the river. Looking out downriver (to the right on this map) and parallel to the banks, he sees the Sun straight in front of him, peeking out from above the buildings, and has to avert his eyes to not be blinded. What part of the academic year is it?\n\nA: Early fall semester (late September-early October)\nB: Late fall semester (late November-early December)\nC: Independent Activities Period (January)\nD: Early spring semester (late February-early March)\nE: Late spring semester (late April-early May)\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_4833241f8ea3264f9ff9g-20.jpg?height=854&width=1523&top_left_y=213&top_left_x=298" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_211", "problem": "牛顿在 1689 年出版的《自然哲学的数学原理》中设想, 物体抛出的速度很大时, 就不会落回地面, 它将绕地球运动, 成为人造地球卫星。如图所示, 将物体从一座高山上\n的 $O$ 点水平抛出, 抛出速度一次比一次大, 落地点一次比一次远, 设图中 $A 、 B 、 C 、$ $D 、 E$ 是从 $O$ 点以不同的速度抛出的物体所对应的运动轨道。已知 $B$ 是圆形轨道, $C 、 D$是椭圆轨道, 在轨道 $E$ 上运动的物体将会克服地球的引力, 永远地离开地球, 空气阻力和地球自转的影响不计, 则下列说法正确的是 ( )\n\n[图1]\nA: 物体从 $O$ 点抛出后, 沿轨道 $A$ 运动落到地面上, 物体的运动可能是平抛运动\nB: 在轨道 $B$ 上运动的物体, 抛出时的速度大小为 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 使轨道 $C 、 D$ 上物体的运动轨道变为圆轨道, 这个圆轨道可以过 $O$ 点\nD: 在轨道 $E$ 上运动的物体，抛出时的速度一定等于或大于第三宇宙速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n牛顿在 1689 年出版的《自然哲学的数学原理》中设想, 物体抛出的速度很大时, 就不会落回地面, 它将绕地球运动, 成为人造地球卫星。如图所示, 将物体从一座高山上\n的 $O$ 点水平抛出, 抛出速度一次比一次大, 落地点一次比一次远, 设图中 $A 、 B 、 C 、$ $D 、 E$ 是从 $O$ 点以不同的速度抛出的物体所对应的运动轨道。已知 $B$ 是圆形轨道, $C 、 D$是椭圆轨道, 在轨道 $E$ 上运动的物体将会克服地球的引力, 永远地离开地球, 空气阻力和地球自转的影响不计, 则下列说法正确的是 ( )\n\n[图1]\n\nA: 物体从 $O$ 点抛出后, 沿轨道 $A$ 运动落到地面上, 物体的运动可能是平抛运动\nB: 在轨道 $B$ 上运动的物体, 抛出时的速度大小为 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 使轨道 $C 、 D$ 上物体的运动轨道变为圆轨道, 这个圆轨道可以过 $O$ 点\nD: 在轨道 $E$ 上运动的物体，抛出时的速度一定等于或大于第三宇宙速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-010.jpg?height=517&width=443&top_left_y=495&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_276", "problem": "人类太空探测计划旨在探测恒星亮度以寻找适合人类居住的宜居行星。在某次探测中发现距地球数光年处有一颗相对太阳静止的质量为 $M$ 的恒星 $\\mathrm{A}$, 将恒星 $\\mathrm{A}$ 视为黑体,根据斯特藩-玻尔兹曼定律: 一个黑体表面单位面积辐射出的功率与黑体本身的热力学温度 $T$ 的四次方成正比, 即黑体表面单位面积辐射出的功率为 $\\sigma T^{4}$ (其中 $\\sigma$ 为常数), A 的表面温度为 $T_{0}$, 地球上正对 $\\mathrm{A}$ 的单位面积接收到 $\\mathrm{A}$ 辐射出的功率为 $I$ 。已知 $\\mathrm{A}$ 在地球轨道平面上，地球公转半径为 $R_{0}$, 一年内地球上的观测者测得地球与 $\\mathrm{A}$ 的连线之间的最大夹角为 $\\theta$ (角 $\\theta$ 很小, 可认为 $\\sin \\theta \\approx \\tan \\theta \\approx \\theta$ )。恒星 $\\mathrm{A}$ 有一颗绕它做匀速圆周运动的行星 $\\mathrm{B}$, 该行星也可视为黑体, 其表面的温度保持为 $T_{1}$, 恒星 $\\mathrm{A}$ 射向行星 $\\mathrm{B}$ 的光可看作平行光。已知引力常量为 $G$, 求:\n\n 行星 $\\mathrm{B}$ 的运动周期 $T_{\\mathrm{B}}$ 。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n人类太空探测计划旨在探测恒星亮度以寻找适合人类居住的宜居行星。在某次探测中发现距地球数光年处有一颗相对太阳静止的质量为 $M$ 的恒星 $\\mathrm{A}$, 将恒星 $\\mathrm{A}$ 视为黑体,根据斯特藩-玻尔兹曼定律: 一个黑体表面单位面积辐射出的功率与黑体本身的热力学温度 $T$ 的四次方成正比, 即黑体表面单位面积辐射出的功率为 $\\sigma T^{4}$ (其中 $\\sigma$ 为常数), A 的表面温度为 $T_{0}$, 地球上正对 $\\mathrm{A}$ 的单位面积接收到 $\\mathrm{A}$ 辐射出的功率为 $I$ 。已知 $\\mathrm{A}$ 在地球轨道平面上，地球公转半径为 $R_{0}$, 一年内地球上的观测者测得地球与 $\\mathrm{A}$ 的连线之间的最大夹角为 $\\theta$ (角 $\\theta$ 很小, 可认为 $\\sin \\theta \\approx \\tan \\theta \\approx \\theta$ )。恒星 $\\mathrm{A}$ 有一颗绕它做匀速圆周运动的行星 $\\mathrm{B}$, 该行星也可视为黑体, 其表面的温度保持为 $T_{1}$, 恒星 $\\mathrm{A}$ 射向行星 $\\mathrm{B}$ 的光可看作平行光。已知引力常量为 $G$, 求:\n\n 行星 $\\mathrm{B}$ 的运动周期 $T_{\\mathrm{B}}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_513", "problem": "一颗侦察卫星在通过地球两极上空的圆轨道上运行, 它的运行轨道距地面高度为 $h$ 。设地球半径为 $R$, 地面重力加速度为 $g$, 地球自转的周期为 $T$ 。要使该卫星在一天的时间内将地面上赤道各处在日照条件下的全部情况全都拍摄下来, 则卫星在通过赤道上空时, 卫星上的摄像机应拍摄地面上赤道圆周的弧长至少为（ ）\nA: $\\frac{4 \\pi^{2}(R+h)}{T} \\sqrt{\\frac{R+h}{g}}$\nB: $\\frac{2 \\pi(R+h)}{T} \\sqrt{\\frac{R+h}{g}}$\nC: $\\frac{2 \\pi R}{T} \\sqrt{g}$\nD: $\\frac{4 \\pi^{2}(R+h)}{T} \\sqrt{g}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n一颗侦察卫星在通过地球两极上空的圆轨道上运行, 它的运行轨道距地面高度为 $h$ 。设地球半径为 $R$, 地面重力加速度为 $g$, 地球自转的周期为 $T$ 。要使该卫星在一天的时间内将地面上赤道各处在日照条件下的全部情况全都拍摄下来, 则卫星在通过赤道上空时, 卫星上的摄像机应拍摄地面上赤道圆周的弧长至少为（ ）\n\nA: $\\frac{4 \\pi^{2}(R+h)}{T} \\sqrt{\\frac{R+h}{g}}$\nB: $\\frac{2 \\pi(R+h)}{T} \\sqrt{\\frac{R+h}{g}}$\nC: $\\frac{2 \\pi R}{T} \\sqrt{g}$\nD: $\\frac{4 \\pi^{2}(R+h)}{T} \\sqrt{g}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_304", "problem": "被誉为“中国天眼”的射电望远镜 $F A \\mathrm{~S} T$ 自工作以来, 已经发现 43 颗脉冲星, 为我国天文观测做出了巨大的贡献。脉冲星实质是快速自转的中子星, 中子星每自转一周,它的磁场就会扫过地球一次，地球就会接收到一个射电脉冲。若观测到某个中子星的射电脉冲周期为 $T$, 中子星两极处的重力加速度为 $g$, 密度为 $\\rho$, 引力常量为 $G$ 。下列说法正确的是（）\nA: 中子星的半径为 $\\frac{g T^{2}}{4 \\pi^{2}}$\nB: 中子星的质量为 $\\frac{9 g^{3}}{16 \\pi^{2} G^{2} \\rho^{2}}$\nC: 中子星的第一宇宙速度为 $\\sqrt{\\frac{3 g}{4 \\pi G \\rho}}$\nD: 若地球接收射电脉冲的周期变为 $\\sqrt{\\frac{3 \\pi}{G \\rho}}$, 则中子星赤道上的物体会离开中子星表面\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n被誉为“中国天眼”的射电望远镜 $F A \\mathrm{~S} T$ 自工作以来, 已经发现 43 颗脉冲星, 为我国天文观测做出了巨大的贡献。脉冲星实质是快速自转的中子星, 中子星每自转一周,它的磁场就会扫过地球一次，地球就会接收到一个射电脉冲。若观测到某个中子星的射电脉冲周期为 $T$, 中子星两极处的重力加速度为 $g$, 密度为 $\\rho$, 引力常量为 $G$ 。下列说法正确的是（）\n\nA: 中子星的半径为 $\\frac{g T^{2}}{4 \\pi^{2}}$\nB: 中子星的质量为 $\\frac{9 g^{3}}{16 \\pi^{2} G^{2} \\rho^{2}}$\nC: 中子星的第一宇宙速度为 $\\sqrt{\\frac{3 g}{4 \\pi G \\rho}}$\nD: 若地球接收射电脉冲的周期变为 $\\sqrt{\\frac{3 \\pi}{G \\rho}}$, 则中子星赤道上的物体会离开中子星表面\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_953", "problem": "Although Saturn is famous for its rings, all of the gas giants in the Solar System have ring systems. The outer ring is known as the Adams ring and is very thin. Normally such a thin structure would widen over time so there needs to be a process keeping it constrained. One hypothesis is that the Neptunian moon Galatea, with an orbit just slightly smaller than the ring, acts as a 'shepherd moon' by having a 42 : 43 orbital resonance with particles in the ring, in terms of the period of their orbits. The ring and the moon are shown in Figure 1.\n[figure1]\n\nFigure 1: Left: Neptune as seen by the Voyager 2 mission in August 1989, a few days before its flyby. Credit: NASA / JPL / Voyager-ISS / Justin Cowart.\n\nRight: Neptune and its ring system as imaged in the infrared by the NIRCam instrument on the James Webb Space Telescope in July 2022. Multiple moons and rings are visible, with Galatea and the Adams ring labelled. Credit: NASA / ESA / CSA / STScI / Joseph DePasquale.\n\nThe semi-major axis of Galatea is $61953 \\mathrm{~km}$. Assume the moon and the ring particles travel in circular orbits.\n\nCalculate the semi-major axis of a particle in the centre of the ring.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nAlthough Saturn is famous for its rings, all of the gas giants in the Solar System have ring systems. The outer ring is known as the Adams ring and is very thin. Normally such a thin structure would widen over time so there needs to be a process keeping it constrained. One hypothesis is that the Neptunian moon Galatea, with an orbit just slightly smaller than the ring, acts as a 'shepherd moon' by having a 42 : 43 orbital resonance with particles in the ring, in terms of the period of their orbits. The ring and the moon are shown in Figure 1.\n[figure1]\n\nFigure 1: Left: Neptune as seen by the Voyager 2 mission in August 1989, a few days before its flyby. Credit: NASA / JPL / Voyager-ISS / Justin Cowart.\n\nRight: Neptune and its ring system as imaged in the infrared by the NIRCam instrument on the James Webb Space Telescope in July 2022. Multiple moons and rings are visible, with Galatea and the Adams ring labelled. Credit: NASA / ESA / CSA / STScI / Joseph DePasquale.\n\nThe semi-major axis of Galatea is $61953 \\mathrm{~km}$. Assume the moon and the ring particles travel in circular orbits.\n\nCalculate the semi-major axis of a particle in the centre of the ring.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of km, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-05.jpg?height=574&width=1420&top_left_y=838&top_left_x=318" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "km" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1080", "problem": "In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}{r}}$.f. How long would any astronauts on board the spacecraft need to wait until they could use a Hohmann transfer orbit to return to Earth? Hence calculate the total duration of the mission.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}{r}}$.\n\nproblem:\nf. How long would any astronauts on board the spacecraft need to wait until they could use a Hohmann transfer orbit to return to Earth? Hence calculate the total duration of the mission.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of days, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0c9b1562981df78a2b9dg-05.jpg?height=383&width=517&top_left_y=297&top_left_x=313" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "days" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1134", "problem": "The James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.a. The telescope will spend its expected 10-20 year mission in a halo orbit about the second Lagrangian point, L2 (see Figure 5). This is one of five special points in the Sun-Earth system where the gravitational forces from the two bodies provide the centripetal force required to have a (small mass) object there have an orbital period identical to the Earth. At the L2 point, this means it orbits quicker than you would expect for an object that distance from the Sun.\n\ni. Taking 1 year as 365.25 days and 1 au as $1.496 \\times 10^{11} \\mathrm{~m}$, using numerical methods show that the distance between the Earth and L2 is $~ 1.5 \\times 10^{6} \\mathrm{~km}$. Give your answer to 4 s.f.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.\n\nproblem:\na. The telescope will spend its expected 10-20 year mission in a halo orbit about the second Lagrangian point, L2 (see Figure 5). This is one of five special points in the Sun-Earth system where the gravitational forces from the two bodies provide the centripetal force required to have a (small mass) object there have an orbital period identical to the Earth. At the L2 point, this means it orbits quicker than you would expect for an object that distance from the Sun.\n\ni. Taking 1 year as 365.25 days and 1 au as $1.496 \\times 10^{11} \\mathrm{~m}$, using numerical methods show that the distance between the Earth and L2 is $~ 1.5 \\times 10^{6} \\mathrm{~km}$. Give your answer to 4 s.f.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~m}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-09.jpg?height=618&width=1466&top_left_y=596&top_left_x=296", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-10.jpg?height=482&width=1536&top_left_y=1118&top_left_x=267" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~m}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_272", "problem": "按照我国整个月球探测活动的计划, 在第一步“绕月”工程圆满完成各项目标和科学探测任务后, 第二步是“落月”工程。已在 2013 年以前完成。假设月球半径为 $R$, 月球表面的重力加速度为 $g_{0}$, 飞船沿距月球表面高度为 $3 R$ 的圆形轨道I运动, 到达轨道的 $A$点时点火变轨进入椭圆轨道II, 到达轨道的近月点 $B$ 时再次点火进入月球近月轨道III绕月球做圆周运动。下列判断正确的是（）\n\n[图1]\nA: 飞船在轨道 $\\mathrm{I}$ 上的运行速率 $v=\\frac{\\sqrt{g_{0} R}}{2}$\nB: 飞船在 $A$ 点处点火变轨时, 速度增大\nC: 飞船从 $A$ 到 $B$ 运行的过程中加速度增大\nD: 飞船在轨道III绕月球运动一周所需的时间 $T=2 \\pi \\sqrt{\\frac{R}{g_{0}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n按照我国整个月球探测活动的计划, 在第一步“绕月”工程圆满完成各项目标和科学探测任务后, 第二步是“落月”工程。已在 2013 年以前完成。假设月球半径为 $R$, 月球表面的重力加速度为 $g_{0}$, 飞船沿距月球表面高度为 $3 R$ 的圆形轨道I运动, 到达轨道的 $A$点时点火变轨进入椭圆轨道II, 到达轨道的近月点 $B$ 时再次点火进入月球近月轨道III绕月球做圆周运动。下列判断正确的是（）\n\n[图1]\n\nA: 飞船在轨道 $\\mathrm{I}$ 上的运行速率 $v=\\frac{\\sqrt{g_{0} R}}{2}$\nB: 飞船在 $A$ 点处点火变轨时, 速度增大\nC: 飞船从 $A$ 到 $B$ 运行的过程中加速度增大\nD: 飞船在轨道III绕月球运动一周所需的时间 $T=2 \\pi \\sqrt{\\frac{R}{g_{0}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-105.jpg?height=388&width=408&top_left_y=660&top_left_x=344" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_95", "problem": "根据图片中新闻信息, 并查阅资料获知火星的质量和万有引力常量, 则可估算出\n\n## 中国首次火星探测任务环绕火星获得成功新京报\n\n据央视新闻客户端消息记者从国家航天局获悉, 刚刚，中国首次火星探测任务天问一号探测器实施近火捕获制动, 环绕器 $3000 \\mathrm{~N}$ 轨控发动机点火工作约 15 分钟, 探测器顺利进入近火点高度约 400 千米, 周期约 10 个地球日, 倾角约 10 的大椭圆环火轨道, 成为我国第一颗人造火星卫星, 实现“绕、着、巡“第一步“绕”的目标, 环绕火星获得成功。\nA: 火星的平均密度\nB: 探测器的环火轨道半长轴\nC: 火星表面的重力加速度\nD: 探测器在近火点的加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n根据图片中新闻信息, 并查阅资料获知火星的质量和万有引力常量, 则可估算出\n\n## 中国首次火星探测任务环绕火星获得成功新京报\n\n据央视新闻客户端消息记者从国家航天局获悉, 刚刚，中国首次火星探测任务天问一号探测器实施近火捕获制动, 环绕器 $3000 \\mathrm{~N}$ 轨控发动机点火工作约 15 分钟, 探测器顺利进入近火点高度约 400 千米, 周期约 10 个地球日, 倾角约 10 的大椭圆环火轨道, 成为我国第一颗人造火星卫星, 实现“绕、着、巡“第一步“绕”的目标, 环绕火星获得成功。\n\nA: 火星的平均密度\nB: 探测器的环火轨道半长轴\nC: 火星表面的重力加速度\nD: 探测器在近火点的加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_427", "problem": "金星是从太阳向外的第二颗行星, 假设金星和地球都是围绕太阳做匀速圆周运动,现从地球发射一颗金星探测器, 可以简化为这样的过程, 选择恰当的时间窗口, 探测器先脱离地球束缚成为和地球同轨道的人造小行星, 然后通过速度调整进入制圆转移轨道,经粗圆转移轨道 (关闭发动机) 到达金星轨道, 椭圆长轴的两端一端和地球轨道相切,一端和金星轨道相切。若太阳质量为 $M$, 探测器质量为 $m$, 太阳与探测器间距离为 $r$,\n则它们之间的引力势能公式为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$ 。已知椭圆转移轨道与两圆轨道相切于 $A 、 B$两点且恰好对应椭圆的长轴, 地球轨道半径为 $r_{\\mathrm{A}}$ 、周期为 $T_{\\mathrm{A}}$, 金星轨道半径为 $r_{\\mathrm{B}}$, 周期为 $T_{\\mathrm{B}}$, 万有引力常量 $G$ 。忽略除太阳外其它星体对探测器的影响, 则 ( )\n\n[图1]\n\n地球轨道\nA: 探测器在地球轨道上的 $A$ 点和转移轨道上的 $A$ 点处向心加速度不同\nB: 探测器从地球轨道经转移轨道到达金星轨道的最短时间为 $\\frac{T_{\\mathrm{A}}+T_{\\mathrm{B}}}{2}$\nC: 探测器在转移轨道 $A$ 点的速度 $v_{\\mathrm{A}}=\\sqrt{\\frac{2 r_{\\mathrm{B}} G M}{r_{\\mathrm{A}}\\left(r_{\\mathrm{A}}+r_{\\mathrm{B}}\\right)}}$\nD: 探测器在地球正圆轨道的机械能比在金星正圆轨道的机械能小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n金星是从太阳向外的第二颗行星, 假设金星和地球都是围绕太阳做匀速圆周运动,现从地球发射一颗金星探测器, 可以简化为这样的过程, 选择恰当的时间窗口, 探测器先脱离地球束缚成为和地球同轨道的人造小行星, 然后通过速度调整进入制圆转移轨道,经粗圆转移轨道 (关闭发动机) 到达金星轨道, 椭圆长轴的两端一端和地球轨道相切,一端和金星轨道相切。若太阳质量为 $M$, 探测器质量为 $m$, 太阳与探测器间距离为 $r$,\n则它们之间的引力势能公式为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$ 。已知椭圆转移轨道与两圆轨道相切于 $A 、 B$两点且恰好对应椭圆的长轴, 地球轨道半径为 $r_{\\mathrm{A}}$ 、周期为 $T_{\\mathrm{A}}$, 金星轨道半径为 $r_{\\mathrm{B}}$, 周期为 $T_{\\mathrm{B}}$, 万有引力常量 $G$ 。忽略除太阳外其它星体对探测器的影响, 则 ( )\n\n[图1]\n\n地球轨道\n\nA: 探测器在地球轨道上的 $A$ 点和转移轨道上的 $A$ 点处向心加速度不同\nB: 探测器从地球轨道经转移轨道到达金星轨道的最短时间为 $\\frac{T_{\\mathrm{A}}+T_{\\mathrm{B}}}{2}$\nC: 探测器在转移轨道 $A$ 点的速度 $v_{\\mathrm{A}}=\\sqrt{\\frac{2 r_{\\mathrm{B}} G M}{r_{\\mathrm{A}}\\left(r_{\\mathrm{A}}+r_{\\mathrm{B}}\\right)}}$\nD: 探测器在地球正圆轨道的机械能比在金星正圆轨道的机械能小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-008.jpg?height=329&width=440&top_left_y=481&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1028", "problem": "A typical cheap handheld telescope has a diameter of $10 \\mathrm{~cm}$, whilst ones for keen amateurs can have diameters of $40 \\mathrm{~cm}$. How much greater light gathering power does the larger telescope have?\nA: 2\nB: 4\nC: 8\nD: 16\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA typical cheap handheld telescope has a diameter of $10 \\mathrm{~cm}$, whilst ones for keen amateurs can have diameters of $40 \\mathrm{~cm}$. How much greater light gathering power does the larger telescope have?\n\nA: 2\nB: 4\nC: 8\nD: 16\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_685", "problem": "总质量为 $m$ 的返回式人造地球卫星沿半径为 $R$ 的圆轨道绕地球运动到某点时, 向原来运动方向喷出气体以降低卫星的速度, 随后卫星转到与地球相切的椭圆轨道, 要使卫星相对地面的速度变为原来的 $k$ 倍 $(k<1)$, 则卫星在该点将质量为 $\\Delta m$ 的气体喷出的对地速度大小应为 (将连续喷气等效为一次性喷气, 地球半径为 $R_{0}$, 地球表面重力加速度为 $g$ )\nA: $k \\sqrt{\\frac{g R_{0}^{2}}{R}}$\nB: $\\frac{1}{k} \\sqrt{\\frac{g R_{0}^{2}}{R}}$\nC: $\\left(\\frac{m+k \\Delta m-k m}{\\Delta m}\\right) \\sqrt{\\frac{g R_{0}^{2}}{R}}$\nD: $\\left(\\frac{m-k \\Delta m+k m}{\\Delta m}\\right) \\sqrt{\\frac{g R_{0}^{2}}{R}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n总质量为 $m$ 的返回式人造地球卫星沿半径为 $R$ 的圆轨道绕地球运动到某点时, 向原来运动方向喷出气体以降低卫星的速度, 随后卫星转到与地球相切的椭圆轨道, 要使卫星相对地面的速度变为原来的 $k$ 倍 $(k<1)$, 则卫星在该点将质量为 $\\Delta m$ 的气体喷出的对地速度大小应为 (将连续喷气等效为一次性喷气, 地球半径为 $R_{0}$, 地球表面重力加速度为 $g$ )\n\nA: $k \\sqrt{\\frac{g R_{0}^{2}}{R}}$\nB: $\\frac{1}{k} \\sqrt{\\frac{g R_{0}^{2}}{R}}$\nC: $\\left(\\frac{m+k \\Delta m-k m}{\\Delta m}\\right) \\sqrt{\\frac{g R_{0}^{2}}{R}}$\nD: $\\left(\\frac{m-k \\Delta m+k m}{\\Delta m}\\right) \\sqrt{\\frac{g R_{0}^{2}}{R}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_867", "problem": "In 1974, Stephen Hawking proved that black holes emit blackbody radiation according to the Stefan-Boltzmann law (due to quantum effects near the event horizon). This radiation is called Hawking radiation and through this process, black holes slowly evaporate their mass away in the absence of new material to accrete. Assume that the Hawking temperature of a black hole is inversely proportional to its mass (i.e. $T_{H}=$ const. / M) and that our initial black hole of mass $\\mathrm{M}$ gets split into $\\mathrm{N}$ smaller black holes, each with a mass $M$ / $N$. Using the results found in problems 6 and 7, what is the relation between the final combined luminosity of the smaller black holes $(\\mathrm{L})$ and the luminosity of the initial black hole $\\left(L_{0}\\right)$ ?\nA: $L=L_{0}$\nB: $L=L_{0} / N$\nC: $L=N L_{0}$\nD: $L=N^{2} L_{0}$\nE: $L=N^{3} L_{0}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn 1974, Stephen Hawking proved that black holes emit blackbody radiation according to the Stefan-Boltzmann law (due to quantum effects near the event horizon). This radiation is called Hawking radiation and through this process, black holes slowly evaporate their mass away in the absence of new material to accrete. Assume that the Hawking temperature of a black hole is inversely proportional to its mass (i.e. $T_{H}=$ const. / M) and that our initial black hole of mass $\\mathrm{M}$ gets split into $\\mathrm{N}$ smaller black holes, each with a mass $M$ / $N$. Using the results found in problems 6 and 7, what is the relation between the final combined luminosity of the smaller black holes $(\\mathrm{L})$ and the luminosity of the initial black hole $\\left(L_{0}\\right)$ ?\n\nA: $L=L_{0}$\nB: $L=L_{0} / N$\nC: $L=N L_{0}$\nD: $L=N^{2} L_{0}$\nE: $L=N^{3} L_{0}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_139", "problem": "引力波探测于 2017 年获得诺贝尔物理学奖。双星的运动是引力波的来源之一，假设宇宙中有一双星系统由 $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星体组成, 这两颗星绕它们连线上的某一点在二者之间万有引力作用下做匀速圆周运动, 测得 $\\mathrm{P}$ 星的周期为 $T, \\mathrm{P} 、 \\mathrm{Q}$ 两颗星之间的距离为 $l, \\mathrm{P} 、 \\mathrm{Q}$ 两颗星的轨道半径之差为 $\\Delta r$ ( $\\mathrm{P}$ 星的轨道半径大于 $\\mathrm{Q}$ 星的轨道半径), 引力常量为 $G$, 则 $(\\quad)$\nA: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的向心力大小相等\nB: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的线速度之差为 $\\frac{2 \\pi \\Delta r}{G T^{2}}$\nC: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的质量之差为 $\\frac{4 \\pi^{2} l^{2} \\Delta r}{G T^{2}}$\nD: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的质量之和为 $\\frac{4 \\pi^{2} l^{2}}{G T^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n引力波探测于 2017 年获得诺贝尔物理学奖。双星的运动是引力波的来源之一，假设宇宙中有一双星系统由 $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星体组成, 这两颗星绕它们连线上的某一点在二者之间万有引力作用下做匀速圆周运动, 测得 $\\mathrm{P}$ 星的周期为 $T, \\mathrm{P} 、 \\mathrm{Q}$ 两颗星之间的距离为 $l, \\mathrm{P} 、 \\mathrm{Q}$ 两颗星的轨道半径之差为 $\\Delta r$ ( $\\mathrm{P}$ 星的轨道半径大于 $\\mathrm{Q}$ 星的轨道半径), 引力常量为 $G$, 则 $(\\quad)$\n\nA: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的向心力大小相等\nB: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的线速度之差为 $\\frac{2 \\pi \\Delta r}{G T^{2}}$\nC: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的质量之差为 $\\frac{4 \\pi^{2} l^{2} \\Delta r}{G T^{2}}$\nD: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的质量之和为 $\\frac{4 \\pi^{2} l^{2}}{G T^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1044", "problem": "Which of the following is not a zodiacal constellation?\nA: Virgo \nB: Cancer\nC: Aquila\nD: Gemini\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhich of the following is not a zodiacal constellation?\n\nA: Virgo \nB: Cancer\nC: Aquila\nD: Gemini\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_802", "problem": "Consider a satellite that has a circular orbit with a radius of $6.0 \\times 10^{8} \\mathrm{~m}$ around Venus. Due to a failure in its ignition system, the satellite's orbital velocity was suddenly decreased to zero during a maneuver. How long does the satellite take to hit the surface of the planet? Consider that the mass of Venus is $4.67 \\times 10^{24} \\mathrm{~kg}$ and neglect any gravitational effects on the satellite other than that from Venus.\nA: 15 hours.\nB: 3 days.\nC: 11 days.\nD: 25 days.\nE: 37 days.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nConsider a satellite that has a circular orbit with a radius of $6.0 \\times 10^{8} \\mathrm{~m}$ around Venus. Due to a failure in its ignition system, the satellite's orbital velocity was suddenly decreased to zero during a maneuver. How long does the satellite take to hit the surface of the planet? Consider that the mass of Venus is $4.67 \\times 10^{24} \\mathrm{~kg}$ and neglect any gravitational effects on the satellite other than that from Venus.\n\nA: 15 hours.\nB: 3 days.\nC: 11 days.\nD: 25 days.\nE: 37 days.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_232", "problem": "太阳系中各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。“行星冲日”是指某行星、地球和太阳几乎排成一直线的状态，地球位于太阳与该行星之间。已知相邻两次“冲日”的时间间隔火星约为 800 天，土星约为 378 天，则（）\nA: 火星公转周期约为 1.8 年\nB: 火星的公转周期比土星的公转周期大\nC: 火星的公转轨道半径比土星的公转轨道半径大\nD: 火星和土星的公转轨道半径之比为 $\\sqrt[3]{\\left(\\frac{800}{378}\\right)^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n太阳系中各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。“行星冲日”是指某行星、地球和太阳几乎排成一直线的状态，地球位于太阳与该行星之间。已知相邻两次“冲日”的时间间隔火星约为 800 天，土星约为 378 天，则（）\n\nA: 火星公转周期约为 1.8 年\nB: 火星的公转周期比土星的公转周期大\nC: 火星的公转轨道半径比土星的公转轨道半径大\nD: 火星和土星的公转轨道半径之比为 $\\sqrt[3]{\\left(\\frac{800}{378}\\right)^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1130", "problem": "The speed of light is considered to be the speed limit of the Universe, however knots of plasma in the jets from active galactic nuclei (AGN) have been observed to be moving with apparent transverse speeds in excess of this, called superluminal speeds. Some of the more extreme examples can be appearing to move at up to 6 times the speed of light (see Figure 7).\n[figure1]\n\nFigure 7: Left: The jet coming from the elliptical galaxy M87 as viewed by the Hubble Space Telescope (HST). Right: Sequence of HST images showing motion at six times the speed of light. The slanting lines track the moving features, and the speeds are given in units of the velocity of light, $c$. Credit: NASA / Space Telescope Science Institute / John Biretta.\n\nThis can be explained by understanding that the jet is offset by an angle $\\theta$ from the sightline to Earth, and that the real speed of the plasma knot, $v$, is less than $c$, and from it we can define the scaled speed $\\beta \\equiv v / c$.\n\nSuperluminal jets are not limited just to AGN, as they have also been observed from systems within our own galaxy. A particularly famous one is the 'microquasar' GRS $1915+105$, which is a low mass X-ray binary consisting of a small star orbiting a black hole. A symmetrical jet with components approaching and receding from us is observed (as expected for jets coming from the poles of the black hole), and the apparent transverse motion of material in those jets has been measured using very high resolution radio imaging. Fender et. al (1999) measure these motions to be $\\mu_{a}=23.6$ mas day $^{-1}$ and $\\mu_{r}=$ 10.0 mas day $^{-1}$ for the approaching and receding jet respectively ( 1 mas $=1$ milliarcsecond, a unit of angle, and there are 3600 arcseconds in a degree) and the distance to the system as $11 \\mathrm{kpc}$.\n\nIn practice, for a given $\\beta_{\\text {app }}$ the values of $\\beta$ and $\\theta$ are degenerate and it is unlikely that the orientation of the jet is such that $\\beta_{\\text {app }}$ has been maximised, so the value in part $\\mathrm{c}$. is just a lower limit. However, since there are two jets then if we assume that they are from the same event (and so equal in speed but opposite in direction) we can break this degeneracy.\n\nSince it is a binary system, we can gain information about the masses of the objects by looking at their period and radial velocity. Formally, the relationship is\n\n$$\n\\frac{\\left(M_{\\mathrm{BH}} \\sin i\\right)^{3}}{\\left(M_{\\mathrm{BH}}+M_{\\star}\\right)^{2}}=\\frac{P_{\\mathrm{orb}} K_{d}^{3}}{2 \\pi G}\n$$\n\nwhere $M_{\\mathrm{BH}}$ is the mass of the black hole, $M_{\\star}$ is the mass of the orbiting star, $i$ is the inclination of the orbit, $P_{\\text {orb }}$ is the orbital period, and $K_{d}$ is the amplitude of the radial velocity curve. Normally the inclination can't be measured, however if we assume that the orbit is perpendicular to the jets then $i=\\theta$ and we can measure the mass of the black hole.b. Determine the relationship between $\\beta$ and $\\theta$ that maximises $\\beta_{\\text {app }}$ for a given value of $\\beta$, and hence determine the minimum value of $\\beta$ needed to give rise to superluminal apparent speeds (i.e. when $\\beta_{a p p}^{\\max }>1$ ).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe speed of light is considered to be the speed limit of the Universe, however knots of plasma in the jets from active galactic nuclei (AGN) have been observed to be moving with apparent transverse speeds in excess of this, called superluminal speeds. Some of the more extreme examples can be appearing to move at up to 6 times the speed of light (see Figure 7).\n[figure1]\n\nFigure 7: Left: The jet coming from the elliptical galaxy M87 as viewed by the Hubble Space Telescope (HST). Right: Sequence of HST images showing motion at six times the speed of light. The slanting lines track the moving features, and the speeds are given in units of the velocity of light, $c$. Credit: NASA / Space Telescope Science Institute / John Biretta.\n\nThis can be explained by understanding that the jet is offset by an angle $\\theta$ from the sightline to Earth, and that the real speed of the plasma knot, $v$, is less than $c$, and from it we can define the scaled speed $\\beta \\equiv v / c$.\n\nSuperluminal jets are not limited just to AGN, as they have also been observed from systems within our own galaxy. A particularly famous one is the 'microquasar' GRS $1915+105$, which is a low mass X-ray binary consisting of a small star orbiting a black hole. A symmetrical jet with components approaching and receding from us is observed (as expected for jets coming from the poles of the black hole), and the apparent transverse motion of material in those jets has been measured using very high resolution radio imaging. Fender et. al (1999) measure these motions to be $\\mu_{a}=23.6$ mas day $^{-1}$ and $\\mu_{r}=$ 10.0 mas day $^{-1}$ for the approaching and receding jet respectively ( 1 mas $=1$ milliarcsecond, a unit of angle, and there are 3600 arcseconds in a degree) and the distance to the system as $11 \\mathrm{kpc}$.\n\nIn practice, for a given $\\beta_{\\text {app }}$ the values of $\\beta$ and $\\theta$ are degenerate and it is unlikely that the orientation of the jet is such that $\\beta_{\\text {app }}$ has been maximised, so the value in part $\\mathrm{c}$. is just a lower limit. However, since there are two jets then if we assume that they are from the same event (and so equal in speed but opposite in direction) we can break this degeneracy.\n\nSince it is a binary system, we can gain information about the masses of the objects by looking at their period and radial velocity. Formally, the relationship is\n\n$$\n\\frac{\\left(M_{\\mathrm{BH}} \\sin i\\right)^{3}}{\\left(M_{\\mathrm{BH}}+M_{\\star}\\right)^{2}}=\\frac{P_{\\mathrm{orb}} K_{d}^{3}}{2 \\pi G}\n$$\n\nwhere $M_{\\mathrm{BH}}$ is the mass of the black hole, $M_{\\star}$ is the mass of the orbiting star, $i$ is the inclination of the orbit, $P_{\\text {orb }}$ is the orbital period, and $K_{d}$ is the amplitude of the radial velocity curve. Normally the inclination can't be measured, however if we assume that the orbit is perpendicular to the jets then $i=\\theta$ and we can measure the mass of the black hole.\n\nproblem:\nb. Determine the relationship between $\\beta$ and $\\theta$ that maximises $\\beta_{\\text {app }}$ for a given value of $\\beta$, and hence determine the minimum value of $\\beta$ needed to give rise to superluminal apparent speeds (i.e. when $\\beta_{a p p}^{\\max }>1$ ).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-10.jpg?height=812&width=1458&top_left_y=504&top_left_x=296" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_84", "problem": "两颗卫星在同一轨道平面绕地球做匀速圆周运动, 地球半径为 $R, a$ 卫星离地面的高度等于 $R, b$ 卫星离地面高度为 $3 R$, 则:\n\n(2) 若某时刻两卫星正好同时通过地面同一点的正上方, 则 $a$ 至少经过多少个周期两卫星相距最远？", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题有多个正确答案，你需要包含所有。\n\n问题:\n两颗卫星在同一轨道平面绕地球做匀速圆周运动, 地球半径为 $R, a$ 卫星离地面的高度等于 $R, b$ 卫星离地面高度为 $3 R$, 则:\n\n(2) 若某时刻两卫星正好同时通过地面同一点的正上方, 则 $a$ 至少经过多少个周期两卫星相距最远？\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n它们的答案类型依次是[数值, 数值]。\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MA", "unit": [ null, null ], "answer_sequence": null, "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1170", "problem": "In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.e. The Starkiller Base wants to destroy all the planets in a stellar system on the far side of the galaxy and so drains $0.10 M_{\\odot}$ from the Sun to charge its weapon. Assuming that the $U$ per unit volume of the Sun stays approximately constant during this process, calculate:\niii. The new temperature of the surface of the Sun (current $T_{\\odot}=5780 \\mathrm{~K}$ ), and suggest (with a suitable calculation) what change will be seen in terms of its colour.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.\n\nproblem:\ne. The Starkiller Base wants to destroy all the planets in a stellar system on the far side of the galaxy and so drains $0.10 M_{\\odot}$ from the Sun to charge its weapon. Assuming that the $U$ per unit volume of the Sun stays approximately constant during this process, calculate:\niii. The new temperature of the surface of the Sun (current $T_{\\odot}=5780 \\mathrm{~K}$ ), and suggest (with a suitable calculation) what change will be seen in terms of its colour.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of m, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-06.jpg?height=611&width=1448&top_left_y=505&top_left_x=310" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_184", "problem": "拉格朗日点指在两个大天体引力作用下, 能使小物体稳定的点 (小物体质量相对两\n大天体可忽略不计)。这些点的存在由法国数学家拉格朗日于 1772 年推导证明的, 1906 年首次发现运动于木星轨道上的小行星 (见脱罗央群小行星) 在木星和太阳的作用下处于拉格朗日点上。在每个由两大天体构成的系统中, 按推论有 5 个拉格朗日点, 其中连线上有三个拉格朗日点, 分别是 $L_{1} 、 L_{2} 、 L_{3}$, 如图所示。我国发射的“鹊桥”卫星就在地月系统平衡点 $L_{2}$ 点做周期运动, 通过定期轨控保持轨道的稳定性, 可实现对着陆器和巡视器的中继通信覆盖, 首次实现地月 $L_{2}$ 点周期轨道的长期稳定运行。设某两个天体系统的中心天体质量为 $M$, 环绕天体质量为 $m$, 两天体间距离为 $L$, 万有引力常量为 $G, L_{1}$ 点到中心天体的距离为 $R_{1}, L_{2}$ 点到中心天体的距离为 $R_{2}$ 。求:\n\n处于 $L_{l}$ 点小物体的向心加速度;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n拉格朗日点指在两个大天体引力作用下, 能使小物体稳定的点 (小物体质量相对两\n大天体可忽略不计)。这些点的存在由法国数学家拉格朗日于 1772 年推导证明的, 1906 年首次发现运动于木星轨道上的小行星 (见脱罗央群小行星) 在木星和太阳的作用下处于拉格朗日点上。在每个由两大天体构成的系统中, 按推论有 5 个拉格朗日点, 其中连线上有三个拉格朗日点, 分别是 $L_{1} 、 L_{2} 、 L_{3}$, 如图所示。我国发射的“鹊桥”卫星就在地月系统平衡点 $L_{2}$ 点做周期运动, 通过定期轨控保持轨道的稳定性, 可实现对着陆器和巡视器的中继通信覆盖, 首次实现地月 $L_{2}$ 点周期轨道的长期稳定运行。设某两个天体系统的中心天体质量为 $M$, 环绕天体质量为 $m$, 两天体间距离为 $L$, 万有引力常量为 $G, L_{1}$ 点到中心天体的距离为 $R_{1}, L_{2}$ 点到中心天体的距离为 $R_{2}$ 。求:\n\n处于 $L_{l}$ 点小物体的向心加速度;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-098.jpg?height=469&width=620&top_left_y=1299&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_170", "problem": "火星各种环境与地球十分相似, 人类对未来移居火星有着强烈的期望。地球体积为火星的 7 倍。质量为火星的 11 倍。假设某天人类移居火星后, 小华同学在火星表面制造了如下装置。如图所示。半径为 $r=1 \\mathrm{~m}$ 的光滑圆弧固定在坚直平面内, 其末端与木板 $B$ 的上表面所在平面相切, 且初始时木板 $B$ 的左端刚好与圆弧末端对齐, 木板 $B$ 带电,电荷量为 $1 \\mathrm{C}$, 木板 $B$ 左端紧挨着光滑小物块 $A$, 小物块 $A$ 左侧有一橡胶墙壁, 能与 $A$发生弹性正碰, 空间内存在水平向左的匀强电场, 电场强度 $E=1 \\mathrm{~N} / \\mathrm{C}$, 开始时由圆弧轨道上端静止释放一带电小物块 $C$, 电荷量 $q_{c}=-\\frac{1}{3} C$, 当小物块 $C$ 达到圆弧最底端时,其对圆弧轨道的压力大小为 $\\frac{28}{3} \\mathrm{~N}$, 此时 $A B$ 之间存在的炸药爆炸, 给予 $A B$ 等量的动能,\n动能为 $2 \\mathrm{~J}, C$ 与 $B$ 之间的动摩擦因数为 $\\mu_{1}=0.2, B$ 与水平面间的动摩擦因数为 $\\mu_{2}=0.1$, $A C$ 质量未知。 $B$ 的质量 $m_{B}=4 \\mathrm{~kg}$, 已知地球表面的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。为简化计算取 $\\frac{\\sqrt[3]{49}}{11}=\\frac{1}{3}$ 。\n\n求 $C$ 的质量\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n火星各种环境与地球十分相似, 人类对未来移居火星有着强烈的期望。地球体积为火星的 7 倍。质量为火星的 11 倍。假设某天人类移居火星后, 小华同学在火星表面制造了如下装置。如图所示。半径为 $r=1 \\mathrm{~m}$ 的光滑圆弧固定在坚直平面内, 其末端与木板 $B$ 的上表面所在平面相切, 且初始时木板 $B$ 的左端刚好与圆弧末端对齐, 木板 $B$ 带电,电荷量为 $1 \\mathrm{C}$, 木板 $B$ 左端紧挨着光滑小物块 $A$, 小物块 $A$ 左侧有一橡胶墙壁, 能与 $A$发生弹性正碰, 空间内存在水平向左的匀强电场, 电场强度 $E=1 \\mathrm{~N} / \\mathrm{C}$, 开始时由圆弧轨道上端静止释放一带电小物块 $C$, 电荷量 $q_{c}=-\\frac{1}{3} C$, 当小物块 $C$ 达到圆弧最底端时,其对圆弧轨道的压力大小为 $\\frac{28}{3} \\mathrm{~N}$, 此时 $A B$ 之间存在的炸药爆炸, 给予 $A B$ 等量的动能,\n动能为 $2 \\mathrm{~J}, C$ 与 $B$ 之间的动摩擦因数为 $\\mu_{1}=0.2, B$ 与水平面间的动摩擦因数为 $\\mu_{2}=0.1$, $A C$ 质量未知。 $B$ 的质量 $m_{B}=4 \\mathrm{~kg}$, 已知地球表面的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。为简化计算取 $\\frac{\\sqrt[3]{49}}{11}=\\frac{1}{3}$ 。\n\n求 $C$ 的质量\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以kg为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-045.jpg?height=377&width=756&top_left_y=805&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "kg" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_761", "problem": "The astronomical unit parsec (pc) plays a crucial role in astronomy. One parsec is equal to about 3.26 light-years. How is one parsec defined in astronomy?\nA: Distance at which one astronomical unit measures one arcsecond from Earth.\nB: Orbital distance of the solar system around the center of the Milky Way in one year.\nC: Effective distance of the solar wind (i.e. the radius of the heliosphere).\nD: Historical distance to the brightest star Sirius.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe astronomical unit parsec (pc) plays a crucial role in astronomy. One parsec is equal to about 3.26 light-years. How is one parsec defined in astronomy?\n\nA: Distance at which one astronomical unit measures one arcsecond from Earth.\nB: Orbital distance of the solar system around the center of the Milky Way in one year.\nC: Effective distance of the solar wind (i.e. the radius of the heliosphere).\nD: Historical distance to the brightest star Sirius.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_240", "problem": "2019 年 9 月 12 日, 我国在太原卫星发射中心又一次“一箭三星”发射成功。现假设三颗星 $a 、 b 、 c$ 均在赤道平面上空绕地球做匀速圆周运动, 其中 $a 、 b$ 转动方向与地球自转方向相同, $c$ 转动方向与地球自转方向相反, $a 、 b 、 c$ 三颗星的周期分别为 $T_{a}=6 \\mathrm{~h}$ 、 $T_{b}=24 \\mathrm{~h} 、 T_{c}=12 \\mathrm{~h}$, 某一时刻三个卫星位置如图所示, 从该时刻起, 下列说法正确的是\n\n[图1]\nA: $a 、 b$ 每经过 $4 \\mathrm{~h}$ 相距最近一次\nB: $a 、 b$ 经过 $8 \\mathrm{~h}$ 第一次相距最远\nC: $b 、 c$ 经过 $4 \\mathrm{~h}$ 第一次相距最远\nD: $b 、 c$ 每经过 $8 \\mathrm{~h}$ 相距最近一次\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2019 年 9 月 12 日, 我国在太原卫星发射中心又一次“一箭三星”发射成功。现假设三颗星 $a 、 b 、 c$ 均在赤道平面上空绕地球做匀速圆周运动, 其中 $a 、 b$ 转动方向与地球自转方向相同, $c$ 转动方向与地球自转方向相反, $a 、 b 、 c$ 三颗星的周期分别为 $T_{a}=6 \\mathrm{~h}$ 、 $T_{b}=24 \\mathrm{~h} 、 T_{c}=12 \\mathrm{~h}$, 某一时刻三个卫星位置如图所示, 从该时刻起, 下列说法正确的是\n\n[图1]\n\nA: $a 、 b$ 每经过 $4 \\mathrm{~h}$ 相距最近一次\nB: $a 、 b$ 经过 $8 \\mathrm{~h}$ 第一次相距最远\nC: $b 、 c$ 经过 $4 \\mathrm{~h}$ 第一次相距最远\nD: $b 、 c$ 每经过 $8 \\mathrm{~h}$ 相距最近一次\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-057.jpg?height=414&width=411&top_left_y=250&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_522", "problem": "《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求:\n\n通过缆绳连接的配重空间站线速度大小为多少;\n\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求:\n\n通过缆绳连接的配重空间站线速度大小为多少;\n\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-064.jpg?height=388&width=1082&top_left_y=1345&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_554", "problem": "我国首个火星探测器—“年问一号”, 于 2020 年 7 月 23 日在海南文昌航天发射中心成功发射, 计划于 2021 年 5 月至 6 月择机实施火星着陆, 开展巡视探测, 如图为“天问一号” \"环绕火星变轨示意图。已知地球质量为 $M$, 地球半径为 $R$, 地球表面重力加速度为 $g$; 火星的质量约为地球质量的 $\\frac{1}{9}$, 半径约为地球半径的 $\\frac{1}{2}$; 着陆器质量为 $m$ 。下列说法正确的是 ( )\n\n[图1]\nA: “天问一号”探测器环绕火星运动的速度应大于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nB: “天问一号”在轨道II运行到 $Q$ 点的速度大于在圆轨道 $\\mathrm{I}$ 运行的速度\nC: 若轨道I为近火星圆轨道, 测得周期为 $T$, 则火星的密度约为 $\\frac{3 \\pi M}{T^{2} R^{2} g}$\nD: 着陆器在火星表面所受重力约为 $\\frac{m g}{4}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n我国首个火星探测器—“年问一号”, 于 2020 年 7 月 23 日在海南文昌航天发射中心成功发射, 计划于 2021 年 5 月至 6 月择机实施火星着陆, 开展巡视探测, 如图为“天问一号” \"环绕火星变轨示意图。已知地球质量为 $M$, 地球半径为 $R$, 地球表面重力加速度为 $g$; 火星的质量约为地球质量的 $\\frac{1}{9}$, 半径约为地球半径的 $\\frac{1}{2}$; 着陆器质量为 $m$ 。下列说法正确的是 ( )\n\n[图1]\n\nA: “天问一号”探测器环绕火星运动的速度应大于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nB: “天问一号”在轨道II运行到 $Q$ 点的速度大于在圆轨道 $\\mathrm{I}$ 运行的速度\nC: 若轨道I为近火星圆轨道, 测得周期为 $T$, 则火星的密度约为 $\\frac{3 \\pi M}{T^{2} R^{2} g}$\nD: 着陆器在火星表面所受重力约为 $\\frac{m g}{4}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-076.jpg?height=472&width=873&top_left_y=1740&top_left_x=363", "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-077.jpg?height=49&width=1163&top_left_y=267&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_120", "problem": "宇航员来到某星球表面做了如下实验: 将一小钢球由距星球表面高 $h(h$ 远小于星球半径）处由静止释放, 小钢球经过时间 $t$ 落到星球表面, 该星球为密度均匀的球体,引力常量为 $G$ 。\n\n若该星球的半径为 $R$, 忽略星球的自转, 求该星球的密度;", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n宇航员来到某星球表面做了如下实验: 将一小钢球由距星球表面高 $h(h$ 远小于星球半径）处由静止释放, 小钢球经过时间 $t$ 落到星球表面, 该星球为密度均匀的球体,引力常量为 $G$ 。\n\n若该星球的半径为 $R$, 忽略星球的自转, 求该星球的密度;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_484", "problem": "人造地球卫星与地心间距离为 $r$ 时, 取无穷远处为零势能点, 引力势能可以表示为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $G$ 为引力常量, $M$ 为地球质量, $m$ 为卫星质量。卫星原来在半径为 $r_{1}$的轨道上绕地球做匀速圆周运动, 由于稀薄空气等因素的影响, 飞行一段时间后其圆周运动的半径减小为 $r_{2}$ 。此过程中损失的机械能为 ( )\nA: $\\frac{G M m}{2}\\left(\\frac{1}{r_{2}}-\\frac{1}{r_{1}}\\right)$\nB: $\\frac{G M m}{2}\\left(\\frac{1}{r_{1}}-\\frac{1}{r_{2}}\\right)$\nC: $\\operatorname{GMm}\\left(\\frac{1}{r_{2}}-\\frac{1}{r_{1}}\\right)$\nD: $G M m\\left(\\frac{1}{r_{1}}-\\frac{1}{r_{2}^{2}}\\right)$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n人造地球卫星与地心间距离为 $r$ 时, 取无穷远处为零势能点, 引力势能可以表示为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $G$ 为引力常量, $M$ 为地球质量, $m$ 为卫星质量。卫星原来在半径为 $r_{1}$的轨道上绕地球做匀速圆周运动, 由于稀薄空气等因素的影响, 飞行一段时间后其圆周运动的半径减小为 $r_{2}$ 。此过程中损失的机械能为 ( )\n\nA: $\\frac{G M m}{2}\\left(\\frac{1}{r_{2}}-\\frac{1}{r_{1}}\\right)$\nB: $\\frac{G M m}{2}\\left(\\frac{1}{r_{1}}-\\frac{1}{r_{2}}\\right)$\nC: $\\operatorname{GMm}\\left(\\frac{1}{r_{2}}-\\frac{1}{r_{1}}\\right)$\nD: $G M m\\left(\\frac{1}{r_{1}}-\\frac{1}{r_{2}^{2}}\\right)$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1078", "problem": "On $21^{\\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history.\n\n## Total Solar Eclipse of 2017 Aug 21\n\n[figure1]\n\nFigure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC.\n\nThe path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse (\"GE\"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question:\n\n- The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\\prime} 48.7^{\\prime \\prime}$ and $16^{\\prime} 03.4^{\\prime \\prime}$, respectively, where the notation $x x^{\\prime} y y . y^{\\prime \\prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$\n- The latitude and longitude of the location of GE are $36^{\\circ} 58.0^{\\prime} \\mathrm{N}$ and $87^{\\circ} 40.3^{\\prime} \\mathrm{W}$, respectively\n- Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\\odot}=695700 \\mathrm{~km}, R_{\\oplus}=$ $6371 \\mathrm{~km}$ and $R_{\\text {Moon }}=1737 \\mathrm{~km}$, and a day to be 24 hours\n- Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \\mathrm{~km}$ and $384400 \\mathrm{~km}$, respectively\n- As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction\n\nFor an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as:\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nThe point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration (\"GD\") was at co-ordinates of $37^{\\circ} 35^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 07^{\\prime} \\mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \\mathrm{~s}$ longer than the value calculated in part $\\mathrm{c}$.\n\n[figure2]\n\nFigure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \\& Google Maps.d. The town of Carbondale, Illinois, is the closest big town to the point of GD, with co-ordinates $37^{\\circ} 44^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 13^{\\prime} \\mathrm{W}$ longitude. Assuming the path of maximum totality can be treated as linear as it passes through the region around GD and GE:\n\nii. Calculate the distance (in $\\mathrm{km}$ ) between Carbondale and CP.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn $21^{\\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history.\n\n## Total Solar Eclipse of 2017 Aug 21\n\n[figure1]\n\nFigure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC.\n\nThe path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse (\"GE\"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question:\n\n- The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\\prime} 48.7^{\\prime \\prime}$ and $16^{\\prime} 03.4^{\\prime \\prime}$, respectively, where the notation $x x^{\\prime} y y . y^{\\prime \\prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$\n- The latitude and longitude of the location of GE are $36^{\\circ} 58.0^{\\prime} \\mathrm{N}$ and $87^{\\circ} 40.3^{\\prime} \\mathrm{W}$, respectively\n- Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\\odot}=695700 \\mathrm{~km}, R_{\\oplus}=$ $6371 \\mathrm{~km}$ and $R_{\\text {Moon }}=1737 \\mathrm{~km}$, and a day to be 24 hours\n- Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \\mathrm{~km}$ and $384400 \\mathrm{~km}$, respectively\n- As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction\n\nFor an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as:\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nThe point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration (\"GD\") was at co-ordinates of $37^{\\circ} 35^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 07^{\\prime} \\mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \\mathrm{~s}$ longer than the value calculated in part $\\mathrm{c}$.\n\n[figure2]\n\nFigure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \\& Google Maps.\n\nproblem:\nd. The town of Carbondale, Illinois, is the closest big town to the point of GD, with co-ordinates $37^{\\circ} 44^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 13^{\\prime} \\mathrm{W}$ longitude. Assuming the path of maximum totality can be treated as linear as it passes through the region around GD and GE:\n\nii. Calculate the distance (in $\\mathrm{km}$ ) between Carbondale and CP.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~km}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-08.jpg?height=1011&width=1014&top_left_y=497&top_left_x=521", "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-09.jpg?height=859&width=1213&top_left_y=924&top_left_x=410" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~km}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1103", "problem": "Recently a group of researchers announced that they had discovered an Earth-sized exoplanet around our nearest star, Proxima Centauri. Its closeness raises an intriguing possibility about whether or not we might be able to image it directly using telescopes. The difficulty comes from the small angular scales that need to be resolved and the extreme differences in brightness between the reflected light from the planet and the light given out by the star.\n\n[figure1]\n\nFigure 6: Artist's impression of the view from the surface of Proxima Centauri b. Credit: ESO / M. Kornmesser\n\nData about the star and the planet are summarised below:\n\n| Proxima Centauri (star) | | Proxima Centauri b (planet) | |\n| :--- | :--- | :--- | :--- |\n| Distance | $1.295 \\mathrm{pc}$ | Orbital period | 11.186 days |\n| Mass | $0.123 \\mathrm{M}_{\\odot}$ | Mass $(\\mathrm{min})$ | $\\approx 1.27 \\mathrm{M}_{\\oplus}$ |\n| Radius | $0.141 \\mathrm{R}_{\\odot}$ | Radius $(\\mathrm{min})$ | $\\approx 1.1 \\mathrm{R}_{\\oplus}$ |\n| Surface temperature | $3042 \\mathrm{~K}$ | | |\n| Apparent magnitude | 11.13 | | |\n\nThe following formulae may also be helpful:\n\n$$\nm-\\mathcal{M}=5 \\log \\left(\\frac{d}{10}\\right) \\quad \\mathcal{M}-\\mathcal{M}_{\\odot}=-2.5 \\log \\left(\\frac{L}{\\mathrm{~L}_{\\odot}}\\right) \\quad \\Delta m=2.5 \\log C R\n$$\n\nwhere $m$ is the apparent magnitude, $\\mathcal{M}$ is the absolute magnitude, $d$ is the distance in parsecs, and the contrast ratio $(C R)$ is defined as the ratio of fluxes from the star and planet, $C R=\\frac{f_{\\text {star }}}{f_{\\text {planet }}}$.c. Verify that the HST (which is diffraction limited since it's in space) would be sensitive enough to image the planet in the visible, but is unable to resolve it from its host star (take $\\lambda=550 \\mathrm{~nm}$ ).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nRecently a group of researchers announced that they had discovered an Earth-sized exoplanet around our nearest star, Proxima Centauri. Its closeness raises an intriguing possibility about whether or not we might be able to image it directly using telescopes. The difficulty comes from the small angular scales that need to be resolved and the extreme differences in brightness between the reflected light from the planet and the light given out by the star.\n\n[figure1]\n\nFigure 6: Artist's impression of the view from the surface of Proxima Centauri b. Credit: ESO / M. Kornmesser\n\nData about the star and the planet are summarised below:\n\n| Proxima Centauri (star) | | Proxima Centauri b (planet) | |\n| :--- | :--- | :--- | :--- |\n| Distance | $1.295 \\mathrm{pc}$ | Orbital period | 11.186 days |\n| Mass | $0.123 \\mathrm{M}_{\\odot}$ | Mass $(\\mathrm{min})$ | $\\approx 1.27 \\mathrm{M}_{\\oplus}$ |\n| Radius | $0.141 \\mathrm{R}_{\\odot}$ | Radius $(\\mathrm{min})$ | $\\approx 1.1 \\mathrm{R}_{\\oplus}$ |\n| Surface temperature | $3042 \\mathrm{~K}$ | | |\n| Apparent magnitude | 11.13 | | |\n\nThe following formulae may also be helpful:\n\n$$\nm-\\mathcal{M}=5 \\log \\left(\\frac{d}{10}\\right) \\quad \\mathcal{M}-\\mathcal{M}_{\\odot}=-2.5 \\log \\left(\\frac{L}{\\mathrm{~L}_{\\odot}}\\right) \\quad \\Delta m=2.5 \\log C R\n$$\n\nwhere $m$ is the apparent magnitude, $\\mathcal{M}$ is the absolute magnitude, $d$ is the distance in parsecs, and the contrast ratio $(C R)$ is defined as the ratio of fluxes from the star and planet, $C R=\\frac{f_{\\text {star }}}{f_{\\text {planet }}}$.\n\nproblem:\nc. Verify that the HST (which is diffraction limited since it's in space) would be sensitive enough to image the planet in the visible, but is unable to resolve it from its host star (take $\\lambda=550 \\mathrm{~nm}$ ).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of radians, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-10.jpg?height=708&width=1082&top_left_y=551&top_left_x=493" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "radians" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_944", "problem": "In the rotating reference frame where the Earth is stationary, an asteroid orbits the Sun in 3.5 years. What is the distance between the asteroid and the Sun?\nA: 1.25 au\nB: $2.08 \\mathrm{au}$\nC: $3.95 \\mathrm{au}$\nD: 6.54 au\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn the rotating reference frame where the Earth is stationary, an asteroid orbits the Sun in 3.5 years. What is the distance between the asteroid and the Sun?\n\nA: 1.25 au\nB: $2.08 \\mathrm{au}$\nC: $3.95 \\mathrm{au}$\nD: 6.54 au\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1017", "problem": "The James Webb Space Telescope (JWST) is an exciting new space-based observatory which is capable of detecting incredibly faint objects that have never been seen before, but it is also possible to be seen from Earth if you have a large enough telescope. It has now entered a halo orbit around the second Lagrangian point, $L_{2}$, of the Sun-Earth system at a distance of about 1.5 million $\\mathrm{km}$ from Earth, directly along the Sun-Earth line.\n[figure1]\n\nFigure 5: Left: An image of NASA's James Webb Space Telescope reaching its final distance from Earth. It is a tiny speck among a sea of background stars. The stars appear smudged because the telescope was tracking the motion of JWST, which appears as a small white speck. Credit: Gianluca Masi / The Virtual Telescope Project. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe rectangular sunshield is rather large (measuring $21 \\mathrm{~m}$ by $14 \\mathrm{~m}$, roughly the same as a tennis court), very reflective (reflecting $\\sim 90 \\%$ of the incident light), and always points directly towards the Sun to protect the other parts of the telescope, especially to keep it cool enough to do infrared astronomy.\nCalculate the apparent magnitude of the JWST at $L_{2}$ due to the light reflected off its sunshield, given the apparent magnitude of the Sun is $m_{\\odot}=-26.832$ as viewed from Earth. [Hint: you may wish to calculate the intensity of light that corresponds to an apparent magnitude of zero.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe James Webb Space Telescope (JWST) is an exciting new space-based observatory which is capable of detecting incredibly faint objects that have never been seen before, but it is also possible to be seen from Earth if you have a large enough telescope. It has now entered a halo orbit around the second Lagrangian point, $L_{2}$, of the Sun-Earth system at a distance of about 1.5 million $\\mathrm{km}$ from Earth, directly along the Sun-Earth line.\n[figure1]\n\nFigure 5: Left: An image of NASA's James Webb Space Telescope reaching its final distance from Earth. It is a tiny speck among a sea of background stars. The stars appear smudged because the telescope was tracking the motion of JWST, which appears as a small white speck. Credit: Gianluca Masi / The Virtual Telescope Project. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe rectangular sunshield is rather large (measuring $21 \\mathrm{~m}$ by $14 \\mathrm{~m}$, roughly the same as a tennis court), very reflective (reflecting $\\sim 90 \\%$ of the incident light), and always points directly towards the Sun to protect the other parts of the telescope, especially to keep it cool enough to do infrared astronomy.\nCalculate the apparent magnitude of the JWST at $L_{2}$ due to the light reflected off its sunshield, given the apparent magnitude of the Sun is $m_{\\odot}=-26.832$ as viewed from Earth. [Hint: you may wish to calculate the intensity of light that corresponds to an apparent magnitude of zero.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-09.jpg?height=514&width=1494&top_left_y=594&top_left_x=286" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_764", "problem": "Scientists detect no $\\mathrm{CH}_{3} \\mathrm{OH}$ and no $\\mathrm{NH}_{3}$ in the atmosphere of a sub-Neptune planet. What type of surface does this planet probably have?\nA: Shallow surface\nB: Water oceans\nC: Dry surface\nD: Methane oceans\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nScientists detect no $\\mathrm{CH}_{3} \\mathrm{OH}$ and no $\\mathrm{NH}_{3}$ in the atmosphere of a sub-Neptune planet. What type of surface does this planet probably have?\n\nA: Shallow surface\nB: Water oceans\nC: Dry surface\nD: Methane oceans\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_326", "problem": "2020 年 1 月 7 号, 通信技术试验卫星五号发射升空, 卫星发射时一般需要先到圆轨道 1 , 然后通过变轨进入圆轨道 2 。假设卫星在两圆轨道上速率之比 $v_{1}: v_{2}=5: 3$, 卫星质量不变, 则 $(\\quad)$\n\n[图1]\nA: 卫星通过椭圆轨道进入轨道 2 时应减速\nB: 卫星在两圆轨道运行时的角速度大小之比 $\\omega_{1}: \\omega_{2}=125: 27$\nC: 卫星在 1 轨道运行时和地球之间的万有引力不变\nD: 卫星在两圆轨道运行时的动能之比 $E_{k 1}: E_{k 2}=9: 25$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年 1 月 7 号, 通信技术试验卫星五号发射升空, 卫星发射时一般需要先到圆轨道 1 , 然后通过变轨进入圆轨道 2 。假设卫星在两圆轨道上速率之比 $v_{1}: v_{2}=5: 3$, 卫星质量不变, 则 $(\\quad)$\n\n[图1]\n\nA: 卫星通过椭圆轨道进入轨道 2 时应减速\nB: 卫星在两圆轨道运行时的角速度大小之比 $\\omega_{1}: \\omega_{2}=125: 27$\nC: 卫星在 1 轨道运行时和地球之间的万有引力不变\nD: 卫星在两圆轨道运行时的动能之比 $E_{k 1}: E_{k 2}=9: 25$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-08.jpg?height=383&width=388&top_left_y=1779&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_485", "problem": "100 多年前爱因斯坦预言了引力波存在, 2015 年科学家探测到黑洞合并引起的引力波。双星的运动是产生引力波的来源之一, 在宇宙中有一双星系统由 $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星体组成, 这两颗星绕它们连线的某一点只在二者间的万有引力作用下做匀速圆周运动, 测得 $\\mathrm{P}$ 星的周期为 $\\mathrm{T}, \\mathrm{P} 、 \\mathrm{Q}$ 两颗星的距离为 $l, \\mathrm{P} 、 \\mathrm{Q}$ 两颗星的轨道半径之差为 $\\Delta r(\\mathrm{P}$ 星的轨道半径大于 $\\mathrm{Q}$ 星的轨道半径), 引力常量为 $\\mathrm{G}$, 则下列结论错误的是 ( )\nA: $\\mathrm{Q} 、 \\mathrm{P}$ 两颗星的质量差为 $\\frac{4 \\pi^{2} l^{2} \\Delta r}{G T^{2}}$\nB: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的线速度大小之差为 $\\frac{2 \\pi \\Delta r}{T}$\nC: $\\mathrm{Q} 、 \\mathrm{P}$ 两颗星的质量之比为 $\\frac{l}{l-\\Delta r}$\nD: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的运动半径之比为 $\\frac{l+\\Delta r}{l-\\Delta r}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n100 多年前爱因斯坦预言了引力波存在, 2015 年科学家探测到黑洞合并引起的引力波。双星的运动是产生引力波的来源之一, 在宇宙中有一双星系统由 $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星体组成, 这两颗星绕它们连线的某一点只在二者间的万有引力作用下做匀速圆周运动, 测得 $\\mathrm{P}$ 星的周期为 $\\mathrm{T}, \\mathrm{P} 、 \\mathrm{Q}$ 两颗星的距离为 $l, \\mathrm{P} 、 \\mathrm{Q}$ 两颗星的轨道半径之差为 $\\Delta r(\\mathrm{P}$ 星的轨道半径大于 $\\mathrm{Q}$ 星的轨道半径), 引力常量为 $\\mathrm{G}$, 则下列结论错误的是 ( )\n\nA: $\\mathrm{Q} 、 \\mathrm{P}$ 两颗星的质量差为 $\\frac{4 \\pi^{2} l^{2} \\Delta r}{G T^{2}}$\nB: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的线速度大小之差为 $\\frac{2 \\pi \\Delta r}{T}$\nC: $\\mathrm{Q} 、 \\mathrm{P}$ 两颗星的质量之比为 $\\frac{l}{l-\\Delta r}$\nD: $\\mathrm{P} 、 \\mathrm{Q}$ 两颗星的运动半径之比为 $\\frac{l+\\Delta r}{l-\\Delta r}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_421", "problem": "利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \\times 10^{3} \\mathrm{~W} / \\mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 水的密度 $\\rho=1.0 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。\n\n(1)若利用太阳能发电, 需要发射一颗卫星到地球同步轨道上, 然后通过微波持续不断地将电能输送到地面, 这样就建成了宇宙太阳能发电站。已知地球同步轨道半径约为地球半径的 $2 \\sqrt{11}$ 倍。\n\n(2) 三峡水电站发电机输出的电压为 $18 \\mathrm{kV}$ 。若采用 $500 \\mathrm{kV}$ 直流电向某地区输电 $5.0 \\times 10^{6} \\mathrm{~kW}$, 要求输电线上损耗的功率不高于输送功率的 $5 \\%$\n\n三峡水电站水库面积约 $S^{\\prime}=1.0 \\times 10^{9} \\mathrm{~m}^{2}$, 平均流量 $Q=1.5 \\times 10^{3} \\mathrm{~m}^{3} / \\mathrm{s}$, 水库水面与\n发电机所在位置的平均高度差为 $h=100 \\mathrm{~m}$, 并且在发电过程中水库水面高度保持不变。发电站将水的势能转化为电能的总效率 $\\eta=60 \\%$ 。在地球同步轨道上, 太阳光垂直照射时的辐射功率为 $10 P_{0}$ 。太阳能电池板将太阳能转化为电能的效率为 $20 \\%$, 将电能输送到地面的过程要损失 50\\%。若要使（1）中的宇宙太阳能发电站与三峡电站具有相同的发电能力, 同步卫星上太阳能电池板的面积至少为多大?", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \\times 10^{3} \\mathrm{~W} / \\mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 水的密度 $\\rho=1.0 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。\n\n(1)若利用太阳能发电, 需要发射一颗卫星到地球同步轨道上, 然后通过微波持续不断地将电能输送到地面, 这样就建成了宇宙太阳能发电站。已知地球同步轨道半径约为地球半径的 $2 \\sqrt{11}$ 倍。\n\n(2) 三峡水电站发电机输出的电压为 $18 \\mathrm{kV}$ 。若采用 $500 \\mathrm{kV}$ 直流电向某地区输电 $5.0 \\times 10^{6} \\mathrm{~kW}$, 要求输电线上损耗的功率不高于输送功率的 $5 \\%$\n\n三峡水电站水库面积约 $S^{\\prime}=1.0 \\times 10^{9} \\mathrm{~m}^{2}$, 平均流量 $Q=1.5 \\times 10^{3} \\mathrm{~m}^{3} / \\mathrm{s}$, 水库水面与\n发电机所在位置的平均高度差为 $h=100 \\mathrm{~m}$, 并且在发电过程中水库水面高度保持不变。发电站将水的势能转化为电能的总效率 $\\eta=60 \\%$ 。在地球同步轨道上, 太阳光垂直照射时的辐射功率为 $10 P_{0}$ 。太阳能电池板将太阳能转化为电能的效率为 $20 \\%$, 将电能输送到地面的过程要损失 50\\%。若要使（1）中的宇宙太阳能发电站与三峡电站具有相同的发电能力, 同步卫星上太阳能电池板的面积至少为多大?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~m}^{2}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}^{2}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_186", "problem": "我国计划在 2030 年前实现载人登陆月球开展科学探索, 其后将探索建造月球科研试验站, 开展系统、连续的月球探测和相关技术试验验证。假设质量为 $m$ 的飞船到达月球时, 在距离月面的高度等于月球半径的 $\\frac{1}{2}$ 处先绕着月球表面做匀速圆周运动, 其周期为 $T_{1}$, 已知月球的自转周期为 $T_{2}$, 月球的半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( )\nA: 月球的第一宇宙速度为 $\\frac{3 \\sqrt{3} \\pi R}{2 T_{1}}$\nB: 月球两极的重力加速度为 $\\frac{27 \\pi^{2} R}{T_{1}^{2}}$\nC: 当飞船停在月球纬度 $60^{\\circ}$ 的区域时, 其自转向心加速度为 $\\frac{\\sqrt{3} \\pi^{2} R}{2 T_{2}^{2}}$\nD: 当飞船停在月球赤道的水平面上时, 受到的支持力为 $\\pi^{2} m R\\left(\\frac{27}{2 T_{1}^{2}}-\\frac{4}{T_{2}^{2}}\\right)$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n我国计划在 2030 年前实现载人登陆月球开展科学探索, 其后将探索建造月球科研试验站, 开展系统、连续的月球探测和相关技术试验验证。假设质量为 $m$ 的飞船到达月球时, 在距离月面的高度等于月球半径的 $\\frac{1}{2}$ 处先绕着月球表面做匀速圆周运动, 其周期为 $T_{1}$, 已知月球的自转周期为 $T_{2}$, 月球的半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( )\n\nA: 月球的第一宇宙速度为 $\\frac{3 \\sqrt{3} \\pi R}{2 T_{1}}$\nB: 月球两极的重力加速度为 $\\frac{27 \\pi^{2} R}{T_{1}^{2}}$\nC: 当飞船停在月球纬度 $60^{\\circ}$ 的区域时, 其自转向心加速度为 $\\frac{\\sqrt{3} \\pi^{2} R}{2 T_{2}^{2}}$\nD: 当飞船停在月球赤道的水平面上时, 受到的支持力为 $\\pi^{2} m R\\left(\\frac{27}{2 T_{1}^{2}}-\\frac{4}{T_{2}^{2}}\\right)$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_45", "problem": "中国火星探测器于 2020 年发射, 预计 2021 年到达火星 (火星与太阳的距离大于地球与太阳的距离), 要一次性完成 “环绕、着陆、巡视”三步走。现用 $h$ 表示探测器与火星表面的距离, $a$ 表示探测器所受的火星引力产生的加速度, $a$ 随 $h$ 变化的图像如图所示, 图像中 $a_{1} 、 a_{2} 、 h_{0}$ 为已知, 引力常量为 $G$ 。下列判断正确的是 ( )\n\n[图1]\nA: 火星绕太阳做圆周运动的线速度小于地球绕太阳做圆周运动的线速度\nB: 火星表面的重力加速度大小为 $a_{2}$\nC: 火星的半径为 $\\frac{\\sqrt{a_{1}}}{\\sqrt{a_{2}}-\\sqrt{a_{1}}} h_{0}$\nD: 火星的质量为 $\\left(\\frac{\\sqrt{a_{1} a_{2}}}{\\sqrt{a_{2}}-\\sqrt{a_{1}}}\\right)^{2} \\frac{h_{0}^{2}}{2 G}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n中国火星探测器于 2020 年发射, 预计 2021 年到达火星 (火星与太阳的距离大于地球与太阳的距离), 要一次性完成 “环绕、着陆、巡视”三步走。现用 $h$ 表示探测器与火星表面的距离, $a$ 表示探测器所受的火星引力产生的加速度, $a$ 随 $h$ 变化的图像如图所示, 图像中 $a_{1} 、 a_{2} 、 h_{0}$ 为已知, 引力常量为 $G$ 。下列判断正确的是 ( )\n\n[图1]\n\nA: 火星绕太阳做圆周运动的线速度小于地球绕太阳做圆周运动的线速度\nB: 火星表面的重力加速度大小为 $a_{2}$\nC: 火星的半径为 $\\frac{\\sqrt{a_{1}}}{\\sqrt{a_{2}}-\\sqrt{a_{1}}} h_{0}$\nD: 火星的质量为 $\\left(\\frac{\\sqrt{a_{1} a_{2}}}{\\sqrt{a_{2}}-\\sqrt{a_{1}}}\\right)^{2} \\frac{h_{0}^{2}}{2 G}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-079.jpg?height=357&width=420&top_left_y=164&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_266", "problem": "2018 年 12 月 7 日是我国发射“悟空”探测卫星三周年的日子，该卫星的发射为人类对暗物质的研究做出了重大贡献. 假设两颗质量相等的星球绕其球心连线中点转动, 理论计算的周期与实际观测的周期有出入, 且 $\\frac{T_{\\text {理论 }}}{T_{\\text {观测 }}}=\\frac{\\sqrt{n}}{1}(n>1)$, 科学家推测, 在以两星球球心连线为直径的球体空间中均匀分布着暗物质, 设两星球球心连线长度为 $L$, 质量均为 $m$ ，据此推测, 暗物质的质量为( )\nA: $(n-1) m$\nB: $n m$\nC: $\\frac{n-2}{8} m$\nD: $\\frac{n-1}{4} m$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2018 年 12 月 7 日是我国发射“悟空”探测卫星三周年的日子，该卫星的发射为人类对暗物质的研究做出了重大贡献. 假设两颗质量相等的星球绕其球心连线中点转动, 理论计算的周期与实际观测的周期有出入, 且 $\\frac{T_{\\text {理论 }}}{T_{\\text {观测 }}}=\\frac{\\sqrt{n}}{1}(n>1)$, 科学家推测, 在以两星球球心连线为直径的球体空间中均匀分布着暗物质, 设两星球球心连线长度为 $L$, 质量均为 $m$ ，据此推测, 暗物质的质量为( )\n\nA: $(n-1) m$\nB: $n m$\nC: $\\frac{n-2}{8} m$\nD: $\\frac{n-1}{4} m$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_578", "problem": "假设在半径为 $R$ 的某天体上发射一颗该天体的卫星, 已知引力常量为 $G$, 忽略该天体自转。\n\n若卫星距该天体表面的高度为 $h$, 测得卫星在该处做圆周运动的周期为 $T_{l}$, 则该天体的密度是多少?", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n假设在半径为 $R$ 的某天体上发射一颗该天体的卫星, 已知引力常量为 $G$, 忽略该天体自转。\n\n若卫星距该天体表面的高度为 $h$, 测得卫星在该处做圆周运动的周期为 $T_{l}$, 则该天体的密度是多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_619", "problem": "电影《流浪地球》讲述的是面对太阳快速老化膨胀的灾难, 人类制定了“流浪地球”计划, 这首先需要使自转角速度大小为 $\\omega$ 的地球停止自转, 再将地球推移出太阳系到达距离太阳最近的恒星 (比邻星)。为了使地球停止自转, 设想的方案就是在地球赤道上均匀地安装 $N$ 台“喷气”发动机, 如图所示 ( $N$ 较大, 图中只画出了 4 个)。假设每台发动机均能沿赤道的切线方向提供大小恒为 $F$ 的推力, 该推力可阻碍地球的自转。已知描述地球转动的动力学方程与描述质点运动的牛顿第二定律方程 $F=m a$ 具有相似性, 为 $M=I \\beta$, 其中 $M$ 为外力的总力矩, 即外力与对应力臂乘积的总和, 其值为 $N F R ; I$ 为地球相对地轴的转动惯量; $\\beta$ 为单位时间内地球的角速度的改变量。将地球看成质量分布均匀的球体，下列说法中正确的是（）\n\n[图1]\nA: 在 $M=I \\beta$ 与 $F=m a$ 的类比中, 与质量 $m$ 对应的物理量是转动惯量 $I$, 其物理意义是反映改变地球绕地轴转动情况的难易程度\nB: 地球自转刹车过程中，赤道表面附近的重力加速度逐渐变小\nC: 地球停止自转后, 赤道附近比两极点附近的重力加速度大\nD: 地球自转刹车过程中, 两极点的重力加速度逐渐变大\nE: 这些行星发动机同时开始工作, 使地球停止自转所需要的时间为 $\\frac{\\omega I}{N F}$\nF: 若发动机“喷气”方向与地球上该点的自转线速度方向相反, 则地球赤道地面的人可能会“飘”起来\nG: 在 $M=I \\beta$ 与 $F=m a$ 的类比中, 力矩 $M$ 对应的物理量是 $m$, 其物理意义是反映改变地球绕地轴转动情况的难易程度\nH: $\\beta$ 的单位应为 $\\mathrm{rad} / \\mathrm{s}$\nI: $\\beta-t$ 图象中曲线与 $t$ 轴围成的面积的绝对值等于角速度的变化量的大小\nJ: 地球自转刹车过程中, 赤道表面附近的重力加速度逐渐变大\nK: 若停止自转后, 地球仍为均匀球体, 则赤道处附近与极地附近的重力加速度大小没有差异\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n电影《流浪地球》讲述的是面对太阳快速老化膨胀的灾难, 人类制定了“流浪地球”计划, 这首先需要使自转角速度大小为 $\\omega$ 的地球停止自转, 再将地球推移出太阳系到达距离太阳最近的恒星 (比邻星)。为了使地球停止自转, 设想的方案就是在地球赤道上均匀地安装 $N$ 台“喷气”发动机, 如图所示 ( $N$ 较大, 图中只画出了 4 个)。假设每台发动机均能沿赤道的切线方向提供大小恒为 $F$ 的推力, 该推力可阻碍地球的自转。已知描述地球转动的动力学方程与描述质点运动的牛顿第二定律方程 $F=m a$ 具有相似性, 为 $M=I \\beta$, 其中 $M$ 为外力的总力矩, 即外力与对应力臂乘积的总和, 其值为 $N F R ; I$ 为地球相对地轴的转动惯量; $\\beta$ 为单位时间内地球的角速度的改变量。将地球看成质量分布均匀的球体，下列说法中正确的是（）\n\n[图1]\n\nA: 在 $M=I \\beta$ 与 $F=m a$ 的类比中, 与质量 $m$ 对应的物理量是转动惯量 $I$, 其物理意义是反映改变地球绕地轴转动情况的难易程度\nB: 地球自转刹车过程中，赤道表面附近的重力加速度逐渐变小\nC: 地球停止自转后, 赤道附近比两极点附近的重力加速度大\nD: 地球自转刹车过程中, 两极点的重力加速度逐渐变大\nE: 这些行星发动机同时开始工作, 使地球停止自转所需要的时间为 $\\frac{\\omega I}{N F}$\nF: 若发动机“喷气”方向与地球上该点的自转线速度方向相反, 则地球赤道地面的人可能会“飘”起来\nG: 在 $M=I \\beta$ 与 $F=m a$ 的类比中, 力矩 $M$ 对应的物理量是 $m$, 其物理意义是反映改变地球绕地轴转动情况的难易程度\nH: $\\beta$ 的单位应为 $\\mathrm{rad} / \\mathrm{s}$\nI: $\\beta-t$ 图象中曲线与 $t$ 轴围成的面积的绝对值等于角速度的变化量的大小\nJ: 地球自转刹车过程中, 赤道表面附近的重力加速度逐渐变大\nK: 若停止自转后, 地球仍为均匀球体, 则赤道处附近与极地附近的重力加速度大小没有差异\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D, E, F, G, H, I, J, K]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-089.jpg?height=560&width=571&top_left_y=168&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_389", "problem": "地球和木星绕太阳运行的轨道可以看作是圆形的, 它们各自的卫星轨道也可看作是圆形的. 已知木星的公转轨道半径约为地球公转轨道半径的 5 倍, 木星半径约为地球半径的 11 倍, 木星质量大于地球质量。如图所示是地球和木星的不同卫星做圆周运动的半径 $r$ 的立方与周期 $T$ 的平方的关系图像, 已知万有引力常量为 $G$, 地球的半径为 $R$ 。下列说法正确的是 ( )\n\n[图1]\nA: 木星与地球的质量之比为 $\\frac{b d}{11 a c}$\nB: 木星与地球的线速度之比为 $1: 5$\nC: 地球密度为 $\\frac{3 \\pi a}{G d R^{3}}$\nD: 木星密度为 $\\frac{3 \\pi b}{125 G c R^{3}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球和木星绕太阳运行的轨道可以看作是圆形的, 它们各自的卫星轨道也可看作是圆形的. 已知木星的公转轨道半径约为地球公转轨道半径的 5 倍, 木星半径约为地球半径的 11 倍, 木星质量大于地球质量。如图所示是地球和木星的不同卫星做圆周运动的半径 $r$ 的立方与周期 $T$ 的平方的关系图像, 已知万有引力常量为 $G$, 地球的半径为 $R$ 。下列说法正确的是 ( )\n\n[图1]\n\nA: 木星与地球的质量之比为 $\\frac{b d}{11 a c}$\nB: 木星与地球的线速度之比为 $1: 5$\nC: 地球密度为 $\\frac{3 \\pi a}{G d R^{3}}$\nD: 木星密度为 $\\frac{3 \\pi b}{125 G c R^{3}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-63.jpg?height=417&width=531&top_left_y=1596&top_left_x=357" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1016", "problem": "On $24^{\\text {th }}$ August 2016, astronomers discovered a planet orbiting the closest star to the Sun, Proxima Centauri, situated 4.22 light years away, which fulfils a long-standing dream of science-fiction writers: a world that is close enough for humans to send their first interstellar spacecraft.\n\nAstronomers have noted how the motion of Proxima Centauri changed in the first months of 2016, with the star moving towards and away from the Earth, as seen in the figure below. Sometimes Proxima Centauri is approaching Earth at $5 \\mathrm{~km} \\mathrm{hour}^{-1}-$ normal human walking pace - and at times receding at the same speed. This regular pattern of changing radial velocities caused by an unseen planet, which they named Proxima Centauri B, repeats and results in tiny Doppler shifts in the star's light, making the light appear slightly redder, then bluer.\n\n[figure1]\n\nBy considering that the total linear momentum of the star-planet system in the centre of mass frame is zero, estimate the minimum mass of the planet in terms of Earth masses. Why is this a minimum for the mass of the planet?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nOn $24^{\\text {th }}$ August 2016, astronomers discovered a planet orbiting the closest star to the Sun, Proxima Centauri, situated 4.22 light years away, which fulfils a long-standing dream of science-fiction writers: a world that is close enough for humans to send their first interstellar spacecraft.\n\nAstronomers have noted how the motion of Proxima Centauri changed in the first months of 2016, with the star moving towards and away from the Earth, as seen in the figure below. Sometimes Proxima Centauri is approaching Earth at $5 \\mathrm{~km} \\mathrm{hour}^{-1}-$ normal human walking pace - and at times receding at the same speed. This regular pattern of changing radial velocities caused by an unseen planet, which they named Proxima Centauri B, repeats and results in tiny Doppler shifts in the star's light, making the light appear slightly redder, then bluer.\n\n[figure1]\n\nBy considering that the total linear momentum of the star-planet system in the centre of mass frame is zero, estimate the minimum mass of the planet in terms of Earth masses. Why is this a minimum for the mass of the planet?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6d91a7785df4f4beaa9ag-10.jpg?height=545&width=1602&top_left_y=1007&top_left_x=227" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_797", "problem": "Order the following phases of the Sun's evolution from first to last chronologically.\n13. Helium flash\n14. White dwarf\n15. Red giant branch\n16. Asymptotic giant branch\n17. End of hydrogen fusion in the core\nA: 5, 4, 1, 3, 2\nB: $5,3,1,4,2$\nC: $1,5,3,4,2$\nD: $5,2,4,1,3$\nE: $3,5,1,4,2$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOrder the following phases of the Sun's evolution from first to last chronologically.\n13. Helium flash\n14. White dwarf\n15. Red giant branch\n16. Asymptotic giant branch\n17. End of hydrogen fusion in the core\n\nA: 5, 4, 1, 3, 2\nB: $5,3,1,4,2$\nC: $1,5,3,4,2$\nD: $5,2,4,1,3$\nE: $3,5,1,4,2$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_769", "problem": "A rocket that has accelerated to the second cosmic velocity can...\nA: circle around the Earth in a stable orbit.\nB: circle around the Earth in an elliptic orbit.\nC: escape the gravitational field of the Earth.\nD: escape the gravitational field of the Sun.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA rocket that has accelerated to the second cosmic velocity can...\n\nA: circle around the Earth in a stable orbit.\nB: circle around the Earth in an elliptic orbit.\nC: escape the gravitational field of the Earth.\nD: escape the gravitational field of the Sun.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_85", "problem": "作为一种新型的多功能航天飞行器, 航天飞机集火箭、卫星和飞机的技术特点于一身。假设一航天飞机在完成某维修任务后, 在 $A$ 点从圆形轨道I进入椭圆轨道II, 如图所示, 已知 $A$ 点距地面的高度为 $2 R$ ( $R$ 为地球半径), $B$ 点为轨道II上的近地点（离地面高度忽略不计), 地表重力加速度为 $g$, 地球质量为 $M$ 。又知若物体在与星球无穷远处时其引力势能为零, 则当物体与星球球心距离为 $r$ 时, 其引力势能 $E_{p}=-\\frac{G M m}{r}$ (式中 $m$为物体的质量, $M$ 为星球的质量, $G$ 为引力常量), 不计空气阻力。则下列说法中正确的有\n\n[图1]\nA: 该航天飞机在轨道 $\\mathrm{II}$ 上运动的周期 $T_{2}$ 小于在轨道 $\\mathrm{I}$ 上运动的周期 $T_{1}$\nB: 该航天飞机在轨道II上经过 $B$ 点的速度大于轨道I上经过 $A$ 点的速度\nC: 该航天飞机在轨道II上经过 $B$ 点的速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$, 小于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nD: 该航天飞机在轨道II上从 $A$ 运动到 $B$ 的时间为 $\\pi \\sqrt{\\frac{2 R}{g}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n作为一种新型的多功能航天飞行器, 航天飞机集火箭、卫星和飞机的技术特点于一身。假设一航天飞机在完成某维修任务后, 在 $A$ 点从圆形轨道I进入椭圆轨道II, 如图所示, 已知 $A$ 点距地面的高度为 $2 R$ ( $R$ 为地球半径), $B$ 点为轨道II上的近地点（离地面高度忽略不计), 地表重力加速度为 $g$, 地球质量为 $M$ 。又知若物体在与星球无穷远处时其引力势能为零, 则当物体与星球球心距离为 $r$ 时, 其引力势能 $E_{p}=-\\frac{G M m}{r}$ (式中 $m$为物体的质量, $M$ 为星球的质量, $G$ 为引力常量), 不计空气阻力。则下列说法中正确的有\n\n[图1]\n\nA: 该航天飞机在轨道 $\\mathrm{II}$ 上运动的周期 $T_{2}$ 小于在轨道 $\\mathrm{I}$ 上运动的周期 $T_{1}$\nB: 该航天飞机在轨道II上经过 $B$ 点的速度大于轨道I上经过 $A$ 点的速度\nC: 该航天飞机在轨道II上经过 $B$ 点的速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$, 小于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nD: 该航天飞机在轨道II上从 $A$ 运动到 $B$ 的时间为 $\\pi \\sqrt{\\frac{2 R}{g}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-078.jpg?height=412&width=494&top_left_y=1356&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_993", "problem": "Light from a star is split into a line spectrum of different colours. The line spectrum from the star is shown, along with the line spectra of some individual elements. Identify the elements present in the star.\n[figure1]\nA: Helium and hydrogen\nB: Potassium and sodium and hydrogen\nC: Hydrogen and sodium\nD: Sodium and potassium\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nLight from a star is split into a line spectrum of different colours. The line spectrum from the star is shown, along with the line spectra of some individual elements. Identify the elements present in the star.\n[figure1]\n\nA: Helium and hydrogen\nB: Potassium and sodium and hydrogen\nC: Hydrogen and sodium\nD: Sodium and potassium\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-06.jpg?height=688&width=1194&top_left_y=366&top_left_x=431" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_255", "problem": "太阳系的行星几乎在同一平面内沿同一方向绕太阳做圆周运动, 当地球恰好运行到某个行星和太阳之间，且三者几乎成一条直线的现象，天文学成为“行星冲日”据报道， 2014 年各行星冲日时间分别是: 1 月 6 日，木星冲日， 4 月 9 日火星冲日， 6 月 11 日土星冲日, 8 月 29 日, 海王星冲日, 10 月 8 日, 天王星冲日, 已知地球轨道以外的行星绕太阳运动的轨道半径如下表所示，则下列判断正确的是（）\n\n| | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| 轨道半径 (AU) | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 |\nA: 各地外行星每年都会出现冲日现象\nB: 在 2015 年内一定会出现木星冲日\nC: 天王星相邻两次的冲日的时间是土星的一半\nD: 地外行星中海王星相邻两次冲日间隔时间最短\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n太阳系的行星几乎在同一平面内沿同一方向绕太阳做圆周运动, 当地球恰好运行到某个行星和太阳之间，且三者几乎成一条直线的现象，天文学成为“行星冲日”据报道， 2014 年各行星冲日时间分别是: 1 月 6 日，木星冲日， 4 月 9 日火星冲日， 6 月 11 日土星冲日, 8 月 29 日, 海王星冲日, 10 月 8 日, 天王星冲日, 已知地球轨道以外的行星绕太阳运动的轨道半径如下表所示，则下列判断正确的是（）\n\n| | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| 轨道半径 (AU) | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 |\n\nA: 各地外行星每年都会出现冲日现象\nB: 在 2015 年内一定会出现木星冲日\nC: 天王星相邻两次的冲日的时间是土星的一半\nD: 地外行星中海王星相邻两次冲日间隔时间最短\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_902", "problem": "An observer on the equator sees the Moon rise at 22:00. Ignoring the inclination and eccentricity of the Moon's orbit, when will it rise the next night? The Moon's orbital period is 27.3 days.\nA: 21:07\nB: $21: 47$\nC: $22: 13$\nD: $22: 53$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn observer on the equator sees the Moon rise at 22:00. Ignoring the inclination and eccentricity of the Moon's orbit, when will it rise the next night? The Moon's orbital period is 27.3 days.\n\nA: 21:07\nB: $21: 47$\nC: $22: 13$\nD: $22: 53$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_126", "problem": "在发射卫星时, 往往先将卫星发送到一个粗圆轨道上, 再变轨到圆轨道。已知某卫星运行的椭圆轨道的近地点 $M$ 距地面 $210 \\mathrm{~km}$, 远地点 $N$ 距地面 $345 \\mathrm{~km}$, 卫星进入该轨道正常运行时, 通过 $M$ 点和 $N$ 点时的速率分别为 $v_{1}$ 和 $v_{2}$, 当某次卫星通过 $N$ 点时, 启动卫星上的发动机, 使卫星在短时间内加速后进入离地面 $345 \\mathrm{~km}$ 的圆形轨道, 开始绕地球做匀速圆周运动, 这时卫星的速率为 $v_{3}$ 。比较卫星在 $M 、 N 、 P$ 三点正常运行时（不包括启动发动机加速阶段）的速率 $v_{1} 、 v_{2} 、 v_{3}$ 和加速度大小 $a_{1} 、 a_{2} 、 a_{3}$, 下列结论正确的是 ( )\n\n[图1]\nA: $v_{1}>v_{2}>v_{3}, a_{1}>a_{2}=a_{3}$\nB: $v_{1}>v_{2}=v_{3}, a_{1}>a_{2}>a_{3}$\nC: $v_{1}>v_{3}>v_{2}, a_{1}>a_{3}>a_{2}$\nD: $v_{1}>v_{3}>v_{2}, a_{1}>a_{2}=a_{3}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n在发射卫星时, 往往先将卫星发送到一个粗圆轨道上, 再变轨到圆轨道。已知某卫星运行的椭圆轨道的近地点 $M$ 距地面 $210 \\mathrm{~km}$, 远地点 $N$ 距地面 $345 \\mathrm{~km}$, 卫星进入该轨道正常运行时, 通过 $M$ 点和 $N$ 点时的速率分别为 $v_{1}$ 和 $v_{2}$, 当某次卫星通过 $N$ 点时, 启动卫星上的发动机, 使卫星在短时间内加速后进入离地面 $345 \\mathrm{~km}$ 的圆形轨道, 开始绕地球做匀速圆周运动, 这时卫星的速率为 $v_{3}$ 。比较卫星在 $M 、 N 、 P$ 三点正常运行时（不包括启动发动机加速阶段）的速率 $v_{1} 、 v_{2} 、 v_{3}$ 和加速度大小 $a_{1} 、 a_{2} 、 a_{3}$, 下列结论正确的是 ( )\n\n[图1]\n\nA: $v_{1}>v_{2}>v_{3}, a_{1}>a_{2}=a_{3}$\nB: $v_{1}>v_{2}=v_{3}, a_{1}>a_{2}>a_{3}$\nC: $v_{1}>v_{3}>v_{2}, a_{1}>a_{3}>a_{2}$\nD: $v_{1}>v_{3}>v_{2}, a_{1}>a_{2}=a_{3}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-055.jpg?height=320&width=331&top_left_y=1853&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_187", "problem": "某行星的卫星 $\\mathrm{A} 、 \\mathrm{~B}$ 绕以其为焦点的椭圆轨道运行, 作用于 $\\mathrm{A} 、 \\mathrm{~B}$ 的引力随时间的变化如图所示, 其中 $t_{2}=\\sqrt{2} t_{1}$, 行星到卫星 $\\mathrm{A} 、 \\mathrm{~B}$ 轨道上点的距离分别记为 $r_{A} 、 r_{B}$ 。假设 $\\mathrm{A} 、 \\mathrm{~B}$ 只受到行星的引力, 下列叙述正确的是 ( )\n\n[图1]\nA: B 与 A 的绕行周期之比为 $\\sqrt{2}: 1$\nB: $r_{B}$ 的最大值与 $r_{B}$ 的最小值之比为 $2: 1$\nC: $r_{A}$ 的最大值与 $r_{A}$ 的最小值之比为 3:1\nD: $r_{B}$ 的最小值小于 $r_{A}$ 的最大值\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n某行星的卫星 $\\mathrm{A} 、 \\mathrm{~B}$ 绕以其为焦点的椭圆轨道运行, 作用于 $\\mathrm{A} 、 \\mathrm{~B}$ 的引力随时间的变化如图所示, 其中 $t_{2}=\\sqrt{2} t_{1}$, 行星到卫星 $\\mathrm{A} 、 \\mathrm{~B}$ 轨道上点的距离分别记为 $r_{A} 、 r_{B}$ 。假设 $\\mathrm{A} 、 \\mathrm{~B}$ 只受到行星的引力, 下列叙述正确的是 ( )\n\n[图1]\n\nA: B 与 A 的绕行周期之比为 $\\sqrt{2}: 1$\nB: $r_{B}$ 的最大值与 $r_{B}$ 的最小值之比为 $2: 1$\nC: $r_{A}$ 的最大值与 $r_{A}$ 的最小值之比为 3:1\nD: $r_{B}$ 的最小值小于 $r_{A}$ 的最大值\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-058.jpg?height=405&width=874&top_left_y=1682&top_left_x=320" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_585", "problem": "2019 年 4 月 11 日 21 时黑洞视界望远镜合作组织(ETE)宣布了近邻巨粗圆星系 M87 中心捕获的首张黑洞图像, 提供了黑洞存在的直接“视觉”证据, 验证了 1915 年爱因斯坦的伟大预言。一种理论认为，整个宇宙很可能是个黑洞，如今可观测宇宙的范围膨胀到了半径 465 亿光年的规模, 也就是说, 我们的宇宙就像一个直径 930 亿光年的球体。黑洞的质量 $M$ 和半径 $R$ 的关系满足史瓦西半径公式 $\\frac{M}{R}=\\frac{c^{2}}{2 G}$ (其中 $\\mathrm{c}$ 为光速, 其值为 $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}, G$ 为引力常量, 其值为 $6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$ ) 则, 由此可估算出宇宙的总质量的数量级约为 ( )\nA: $10^{54} \\mathrm{~kg}$\nB: $10^{44} \\mathrm{~kg}$\nC: $10^{34} \\mathrm{~kg}$\nD: $10^{24} \\mathrm{~kg}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2019 年 4 月 11 日 21 时黑洞视界望远镜合作组织(ETE)宣布了近邻巨粗圆星系 M87 中心捕获的首张黑洞图像, 提供了黑洞存在的直接“视觉”证据, 验证了 1915 年爱因斯坦的伟大预言。一种理论认为，整个宇宙很可能是个黑洞，如今可观测宇宙的范围膨胀到了半径 465 亿光年的规模, 也就是说, 我们的宇宙就像一个直径 930 亿光年的球体。黑洞的质量 $M$ 和半径 $R$ 的关系满足史瓦西半径公式 $\\frac{M}{R}=\\frac{c^{2}}{2 G}$ (其中 $\\mathrm{c}$ 为光速, 其值为 $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}, G$ 为引力常量, 其值为 $6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$ ) 则, 由此可估算出宇宙的总质量的数量级约为 ( )\n\nA: $10^{54} \\mathrm{~kg}$\nB: $10^{44} \\mathrm{~kg}$\nC: $10^{34} \\mathrm{~kg}$\nD: $10^{24} \\mathrm{~kg}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_596", "problem": "“势阱” 是量子力学中的常见概念, 在经典力学中也有体现。当粒子在某力场中运动,其势能函数曲线在空间某范围内存在最小值，形如陷阶，粒子很难跑出来。各种形式的势能函数只要具有这种特点, 我们都可以称它为势阱, 比如重力势阱、引力势阱、弹力势阱等。\n\n如图甲所示, 光滑轨道 $a b c$ 固定在坚直平面内形成一重力势阱, 两侧高分别为 $k H$ 和 $H$ 。一可视为质点的质量为 $m$ 的小球, 静置于水平轨道 $b$ 处。已知重力加速度为 $g$;\n\n以 $a$ 处所在平面为重力势能面, 写出该小球在 $b$ 处机械能的表达式;\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\n[图3]\n\n丙", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n“势阱” 是量子力学中的常见概念, 在经典力学中也有体现。当粒子在某力场中运动,其势能函数曲线在空间某范围内存在最小值，形如陷阶，粒子很难跑出来。各种形式的势能函数只要具有这种特点, 我们都可以称它为势阱, 比如重力势阱、引力势阱、弹力势阱等。\n\n如图甲所示, 光滑轨道 $a b c$ 固定在坚直平面内形成一重力势阱, 两侧高分别为 $k H$ 和 $H$ 。一可视为质点的质量为 $m$ 的小球, 静置于水平轨道 $b$ 处。已知重力加速度为 $g$;\n\n以 $a$ 处所在平面为重力势能面, 写出该小球在 $b$ 处机械能的表达式;\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\n[图3]\n\n丙\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-144.jpg?height=274&width=506&top_left_y=711&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-144.jpg?height=500&width=374&top_left_y=498&top_left_x=841", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-144.jpg?height=506&width=627&top_left_y=498&top_left_x=1180" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_573", "problem": "2014 年 12 月 14 日, 北京飞行控制中心传来好消息, 嫦娥三号探测器平稳落月, 已知嫦娥三号探测器在地球表面受的重力为 $G_{1}$, 绕月球表面飞行时受到月球的引力为 $G_{2}$,地球的半径为 $R_{1}$, 月球的半径为 $R_{2}$, 地球表面处的重力加速为 $g$, 则 ( )\nA: 月球与地球的质量之比为 $\\frac{G_{1} R_{2}{ }^{2}}{G_{2} R_{1}{ }^{2}}$\nB: 月球表面处的重力加速度为 $\\frac{G_{1}}{G_{2}} g$\nC: 月球与地球的第一宇宙速度之比为 $\\sqrt{\\frac{R_{2} G_{1}}{R_{1} G_{2}}}$\nD: 探测器沿月球表面轨道上做匀速圆周运动的周期为 $2 \\pi \\sqrt{\\frac{R_{2} G_{1}}{g G_{2}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2014 年 12 月 14 日, 北京飞行控制中心传来好消息, 嫦娥三号探测器平稳落月, 已知嫦娥三号探测器在地球表面受的重力为 $G_{1}$, 绕月球表面飞行时受到月球的引力为 $G_{2}$,地球的半径为 $R_{1}$, 月球的半径为 $R_{2}$, 地球表面处的重力加速为 $g$, 则 ( )\n\nA: 月球与地球的质量之比为 $\\frac{G_{1} R_{2}{ }^{2}}{G_{2} R_{1}{ }^{2}}$\nB: 月球表面处的重力加速度为 $\\frac{G_{1}}{G_{2}} g$\nC: 月球与地球的第一宇宙速度之比为 $\\sqrt{\\frac{R_{2} G_{1}}{R_{1} G_{2}}}$\nD: 探测器沿月球表面轨道上做匀速圆周运动的周期为 $2 \\pi \\sqrt{\\frac{R_{2} G_{1}}{g G_{2}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_328", "problem": "北京时间 2019 年 4 月 10 日，人类历史上首张黑洞“照片”（如图）被正式披露，引起世界轰动; 2020 年 4 月 7 日“事件视界望远镜（EHT）”项目组公布了第二张黑洞“照片”，呈现了更多有关黑洞的信息。黑洞是质量极大的天体，引力极强。一个事件刚好能被观察到的那个时空界面称为视界。例如, 发生在黑洞里的事件不会被黑洞外的人所观察到，因此我们可以把黑洞的视界作为黑洞的“边界”。在黑洞视界范围内，连光也不能逃逸。由于黑洞质量极大，其周围时空严重变形。这样，即使是被黑洞挡着的恒星发出的光, 有一部分光会落入黑洞中, 但还有另一部分离黑洞较远的光线会绕过黑洞, 通过弯曲的路径到达地球。根据上述材料, 结合所学知识判断下列说法正确的是\n\n[图1]\nA: 黑洞“照片”明亮部分是地球上的观测者捕捉到的黑洞自身所发出的光\nB: 地球观测者看到的黑洞“正后方”的几个恒星之间的距离比实际的远\nC: 视界是真实的物质面, 只是外部观测者对它一无所知\nD: 黑洞的第二宇宙速度小于光速 $c$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n北京时间 2019 年 4 月 10 日，人类历史上首张黑洞“照片”（如图）被正式披露，引起世界轰动; 2020 年 4 月 7 日“事件视界望远镜（EHT）”项目组公布了第二张黑洞“照片”，呈现了更多有关黑洞的信息。黑洞是质量极大的天体，引力极强。一个事件刚好能被观察到的那个时空界面称为视界。例如, 发生在黑洞里的事件不会被黑洞外的人所观察到，因此我们可以把黑洞的视界作为黑洞的“边界”。在黑洞视界范围内，连光也不能逃逸。由于黑洞质量极大，其周围时空严重变形。这样，即使是被黑洞挡着的恒星发出的光, 有一部分光会落入黑洞中, 但还有另一部分离黑洞较远的光线会绕过黑洞, 通过弯曲的路径到达地球。根据上述材料, 结合所学知识判断下列说法正确的是\n\n[图1]\n\nA: 黑洞“照片”明亮部分是地球上的观测者捕捉到的黑洞自身所发出的光\nB: 地球观测者看到的黑洞“正后方”的几个恒星之间的距离比实际的远\nC: 视界是真实的物质面, 只是外部观测者对它一无所知\nD: 黑洞的第二宇宙速度小于光速 $c$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-24.jpg?height=263&width=368&top_left_y=1051&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_114", "problem": "如图所示, 地球和行星绕太阳做匀速圆周运动, 且此时行星、地球、太阳三者共线 $\\cdot$地球和行星做匀速圆周运动的半径 $r_{1} 、 r_{2}$ 之比为 $r_{1}: r_{2}=1: 4$, 不计地球和行星之间的相互影响$\\cdot$下列说法正确的是\n\n[图1]\nA: 行星绕太阳做圆周运动的周期为 8 年\nB: 地球和行星的线速度大小之比为 $1: 2$\nC: 至少经过 $\\frac{8}{7}$ 年, 地球位于太阳和行星连线之间\nD: 经过相同时间, 地球和行星半径扫过的面积之比为 1:2\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 地球和行星绕太阳做匀速圆周运动, 且此时行星、地球、太阳三者共线 $\\cdot$地球和行星做匀速圆周运动的半径 $r_{1} 、 r_{2}$ 之比为 $r_{1}: r_{2}=1: 4$, 不计地球和行星之间的相互影响$\\cdot$下列说法正确的是\n\n[图1]\n\nA: 行星绕太阳做圆周运动的周期为 8 年\nB: 地球和行星的线速度大小之比为 $1: 2$\nC: 至少经过 $\\frac{8}{7}$ 年, 地球位于太阳和行星连线之间\nD: 经过相同时间, 地球和行星半径扫过的面积之比为 1:2\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-15.jpg?height=449&width=533&top_left_y=2174&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_459", "problem": "脉冲星的本质是中子星, 具有在地面实验室无法实现的极端物理性质, 是理想的天体物理实验室, 对其进行研究, 有希望得到许多重大物理学问题的答案。譬如: 脉冲星的自转周期极棒稳定, 准确的时钟信号为强力波探测。航天器导航等重大科学及技术应用提供了理想工具。2017 年 8 月我国 FAST 天文望远镜首次发现两颗太空脉冲星, 其\n中一颗的自转周期为 $\\mathrm{T}$ (实际测量为 $1.83 \\mathrm{~s}$, 距离地球 1.6 万光年), 假设该星球恰好能维持自转而不瓦解; 地球可视为球体, 其自转周期为 $\\mathrm{T}_{0}$, 同一物体在地球赤道上用弹簧科测得的重力为两极处的 0.9 倍, 已知万有引力常量为 $\\mathrm{G}$, 则该脉冲星的平均密度 $\\rho$及其与地球的平均密度 $\\rho_{0}$ 之比正确的是 ( )\nA: $\\rho=\\frac{3 \\pi}{G T^{2}}$\nB: $\\frac{\\rho}{\\rho_{0}}=\\frac{T_{0}^{2}}{10 T^{2}}$\nC: $\\rho_{0}=\\frac{3 \\pi}{G T_{0}^{2}}$\nD: $\\frac{\\rho}{\\rho_{0}}=\\frac{10 T_{0}^{2}}{T^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n脉冲星的本质是中子星, 具有在地面实验室无法实现的极端物理性质, 是理想的天体物理实验室, 对其进行研究, 有希望得到许多重大物理学问题的答案。譬如: 脉冲星的自转周期极棒稳定, 准确的时钟信号为强力波探测。航天器导航等重大科学及技术应用提供了理想工具。2017 年 8 月我国 FAST 天文望远镜首次发现两颗太空脉冲星, 其\n中一颗的自转周期为 $\\mathrm{T}$ (实际测量为 $1.83 \\mathrm{~s}$, 距离地球 1.6 万光年), 假设该星球恰好能维持自转而不瓦解; 地球可视为球体, 其自转周期为 $\\mathrm{T}_{0}$, 同一物体在地球赤道上用弹簧科测得的重力为两极处的 0.9 倍, 已知万有引力常量为 $\\mathrm{G}$, 则该脉冲星的平均密度 $\\rho$及其与地球的平均密度 $\\rho_{0}$ 之比正确的是 ( )\n\nA: $\\rho=\\frac{3 \\pi}{G T^{2}}$\nB: $\\frac{\\rho}{\\rho_{0}}=\\frac{T_{0}^{2}}{10 T^{2}}$\nC: $\\rho_{0}=\\frac{3 \\pi}{G T_{0}^{2}}$\nD: $\\frac{\\rho}{\\rho_{0}}=\\frac{10 T_{0}^{2}}{T^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_423", "problem": "万有引力定律 $F_{\\text {引 }}=G \\frac{m_{1} m_{2}}{r^{2}}$ 和库仑定律 $F_{\\text {电 }}=k \\frac{q_{1} q_{2}}{r^{2}}$ 都满足力与距离平方成反比关系。如图所示, 计算物体从距离地球球心 $r_{1}$ 处, 远离至与地心距离 $r_{2}$ 处, 万有引力对物体做功时, 由于力的大小随距离而变化, 一般需采用微元法。也可采用从 $r_{1}$ 到 $r_{2}$ 过程的平均力, 即 $\\overline{F_{\\text {引 }}}=G \\frac{m_{1} m_{2}}{r_{1} \\cdot r_{2}}$ 计算做功。已知物体质量为 $m$, 地球质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。\n求该物体从距离地心 $r_{1}$ 处至距离地心 $r_{2}$ 处的过程中, 万有引力对物体做功 $W$;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n万有引力定律 $F_{\\text {引 }}=G \\frac{m_{1} m_{2}}{r^{2}}$ 和库仑定律 $F_{\\text {电 }}=k \\frac{q_{1} q_{2}}{r^{2}}$ 都满足力与距离平方成反比关系。如图所示, 计算物体从距离地球球心 $r_{1}$ 处, 远离至与地心距离 $r_{2}$ 处, 万有引力对物体做功时, 由于力的大小随距离而变化, 一般需采用微元法。也可采用从 $r_{1}$ 到 $r_{2}$ 过程的平均力, 即 $\\overline{F_{\\text {引 }}}=G \\frac{m_{1} m_{2}}{r_{1} \\cdot r_{2}}$ 计算做功。已知物体质量为 $m$, 地球质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。\n求该物体从距离地心 $r_{1}$ 处至距离地心 $r_{2}$ 处的过程中, 万有引力对物体做功 $W$;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-057.jpg?height=268&width=574&top_left_y=203&top_left_x=410" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_862", "problem": "How far from the Solar System would a galaxy with a redshift of 0.035 be?\nA: $150 \\mathrm{Mpc}$\nB: $200 \\mathrm{Mpc}$\nC: $250 \\mathrm{Mpc}$\nD: $300 \\mathrm{Mpc}$\nE: $350 \\mathrm{Mpc}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nHow far from the Solar System would a galaxy with a redshift of 0.035 be?\n\nA: $150 \\mathrm{Mpc}$\nB: $200 \\mathrm{Mpc}$\nC: $250 \\mathrm{Mpc}$\nD: $300 \\mathrm{Mpc}$\nE: $350 \\mathrm{Mpc}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_192", "problem": "有人设想: 如果在地球的赤道上坚直向上建一座非常高的高楼, 是否可以在楼上直接释放人造卫星呢? 高楼设想图如图所示。已知地球自转周期为 $T$, 同步卫星轨道半径为 $r$; 高楼上有 $A 、 B$ 两个可视为点的小房间, $A$ 到地心的距离为 $\\frac{r}{2}, B$ 到地心的距离为 $\\frac{3 r}{2}$ 。从 $B$ 房间窗口发射一颗小卫星, 要让这颗小卫星能在与 $B$ 点等高的圆轨道上绕地心与地球自转同向运行, 应给这颗卫星加速还是减速？ $\\Delta v$ 大小多少？[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n有人设想: 如果在地球的赤道上坚直向上建一座非常高的高楼, 是否可以在楼上直接释放人造卫星呢? 高楼设想图如图所示。已知地球自转周期为 $T$, 同步卫星轨道半径为 $r$; 高楼上有 $A 、 B$ 两个可视为点的小房间, $A$ 到地心的距离为 $\\frac{r}{2}, B$ 到地心的距离为 $\\frac{3 r}{2}$ 。从 $B$ 房间窗口发射一颗小卫星, 要让这颗小卫星能在与 $B$ 点等高的圆轨道上绕地心与地球自转同向运行, 应给这颗卫星加速还是减速？ $\\Delta v$ 大小多少？[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[加速还是减速, 速度变化大小]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-008.jpg?height=401&width=607&top_left_y=1847&top_left_x=359" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "加速还是减速", "速度变化大小" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_989", "problem": "Which of these constellations is entirely south of the ecliptic?\nA: Orion\nB: Andromeda\nC: Aquila\nD: Ophiuchus\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhich of these constellations is entirely south of the ecliptic?\n\nA: Orion\nB: Andromeda\nC: Aquila\nD: Ophiuchus\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_465", "problem": "在星球 $\\mathrm{M}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\\mathrm{P}$ 轻放在弹簧上端, $\\mathrm{P}$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $\\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $\\mathrm{Q}$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $\\mathrm{M}$ 的半径是星球 $\\mathrm{N}$ 的 3 倍, 求:\n\n星球 $\\mathrm{M}$ 和星球 $\\mathrm{N}$ 的密度之比为多少;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n在星球 $\\mathrm{M}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\\mathrm{P}$ 轻放在弹簧上端, $\\mathrm{P}$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $\\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $\\mathrm{Q}$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $\\mathrm{M}$ 的半径是星球 $\\mathrm{N}$ 的 3 倍, 求:\n\n星球 $\\mathrm{M}$ 和星球 $\\mathrm{N}$ 的密度之比为多少;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-087.jpg?height=300&width=439&top_left_y=330&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1012", "problem": "The Universe consists of three main components: radiation (including neutrinos), matter (both atoms and dark matter), and dark energy. The overall density of the universe has been dominated by the density of each of those in turn at different times in its history, leading to three different epochs (shown in Fig 3).\n\n[figure1]\n\nFigure 3: A outline of the three main epochs in the history of the Universe. You do not need to read any data off this graph to answer this question. Credit: Pearson Education, Inc.\n\nThe scale factor, $a$, describes how the Universe has expanded (i.e. a measure of the relative radius of the Universe), and the current value is defined as $a_{0} \\equiv 1$ where the subscript ' 0 ' indicates it is as measured today. At earlier times $a<1$ and at the Big Bang $a=0$.\n\nThe redshift of an object, $z$, is related to the scale factor as $a=(1+z)^{-1}$ and so the redshift corresponding to now is $z=0$, and far away objects have higher redshift $(z>0)$ since we observe them as they were long ago when the scale factor was smaller.\n\nFor a Universe to be flat (i.e. zero curvature), its average density must be equal to the critical density,\n\n$$\n\\rho_{\\text {crit }, 0}=\\frac{3 H_{0}^{2}}{8 \\pi G},\n$$\n\nwhere $H_{0}$ is the Hubble constant, measured in 2018 from the cosmic microwave background by the Planck spacecraft to be $67.36 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\n\nThe density of the $i^{\\text {th }}$ component of the Universe can be expressed relative to the critical density as the density parameter,\n\n$$\n\\Omega_{i}=\\frac{\\rho_{i}}{\\rho_{\\text {crit }}} .\n$$\n\nPlanck measured the current density parameters of dark energy and matter as $\\Omega_{\\Lambda, 0}=0.6847$ and $\\Omega_{m, 0}=0.3153$ respectively.\n\nIn each epoch, the scale factor increases at a different rate with time, $t$, as the density also varies differently with scale factor.\n\n- Radiation-dominated epoch: The Universe's early history, where $\\rho \\propto a^{-4}$ and so $a \\propto t^{1 / 2}$\n- Matter-dominated epoch: This represents much of the history of the Universe, where $\\rho \\propto$ $a^{-3}$ and so $a \\propto t^{2 / 3}$\n- Dark-energy-dominated epoch: This is an era we have recently entered and will remain in for the rest of time, where $\\rho$ doesn't vary with scale factor (i.e. is a constant) and so $a \\propto e^{H_{0} t}$\n\nFind the time, $t_{D E}$, when the current epoch began (i.e. when the densities of dark energy and matter were equal), given that the age of the Universe today is $t_{0}=13.80 \\mathrm{Gyr}$ (where 1 Gyr $=10^{9}$ years). Give you answer in Gyr. You do not need to read anything off the graph.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe Universe consists of three main components: radiation (including neutrinos), matter (both atoms and dark matter), and dark energy. The overall density of the universe has been dominated by the density of each of those in turn at different times in its history, leading to three different epochs (shown in Fig 3).\n\n[figure1]\n\nFigure 3: A outline of the three main epochs in the history of the Universe. You do not need to read any data off this graph to answer this question. Credit: Pearson Education, Inc.\n\nThe scale factor, $a$, describes how the Universe has expanded (i.e. a measure of the relative radius of the Universe), and the current value is defined as $a_{0} \\equiv 1$ where the subscript ' 0 ' indicates it is as measured today. At earlier times $a<1$ and at the Big Bang $a=0$.\n\nThe redshift of an object, $z$, is related to the scale factor as $a=(1+z)^{-1}$ and so the redshift corresponding to now is $z=0$, and far away objects have higher redshift $(z>0)$ since we observe them as they were long ago when the scale factor was smaller.\n\nFor a Universe to be flat (i.e. zero curvature), its average density must be equal to the critical density,\n\n$$\n\\rho_{\\text {crit }, 0}=\\frac{3 H_{0}^{2}}{8 \\pi G},\n$$\n\nwhere $H_{0}$ is the Hubble constant, measured in 2018 from the cosmic microwave background by the Planck spacecraft to be $67.36 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\n\nThe density of the $i^{\\text {th }}$ component of the Universe can be expressed relative to the critical density as the density parameter,\n\n$$\n\\Omega_{i}=\\frac{\\rho_{i}}{\\rho_{\\text {crit }}} .\n$$\n\nPlanck measured the current density parameters of dark energy and matter as $\\Omega_{\\Lambda, 0}=0.6847$ and $\\Omega_{m, 0}=0.3153$ respectively.\n\nIn each epoch, the scale factor increases at a different rate with time, $t$, as the density also varies differently with scale factor.\n\n- Radiation-dominated epoch: The Universe's early history, where $\\rho \\propto a^{-4}$ and so $a \\propto t^{1 / 2}$\n- Matter-dominated epoch: This represents much of the history of the Universe, where $\\rho \\propto$ $a^{-3}$ and so $a \\propto t^{2 / 3}$\n- Dark-energy-dominated epoch: This is an era we have recently entered and will remain in for the rest of time, where $\\rho$ doesn't vary with scale factor (i.e. is a constant) and so $a \\propto e^{H_{0} t}$\n\nFind the time, $t_{D E}$, when the current epoch began (i.e. when the densities of dark energy and matter were equal), given that the age of the Universe today is $t_{0}=13.80 \\mathrm{Gyr}$ (where 1 Gyr $=10^{9}$ years). Give you answer in Gyr. You do not need to read anything off the graph.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of Gyr, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c3e3c992a9c51eb3e471g-08.jpg?height=1080&width=1271&top_left_y=739&top_left_x=398" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "Gyr" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_458", "problem": "太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间，且三者几乎排成一条直线的现象，天文学称为“行星冲日”。已知地球及各地外行星绕太阳运动的轨道半径如下表所示, 天文单位用符号 AU 表示。则\n\n| 行星 | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| 轨道半径 $r / \\mathrm{AU}$ | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 |\nA: 木星相邻两次冲日的时间间隔约为 1.1 年\nB: 木星的环绕周期约为 25 年\nC: 天王星的环绕速度约为土星的两倍\nD: 地外行星中, 海王星相邻两次冲日的时间间隔最长\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间，且三者几乎排成一条直线的现象，天文学称为“行星冲日”。已知地球及各地外行星绕太阳运动的轨道半径如下表所示, 天文单位用符号 AU 表示。则\n\n| 行星 | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| 轨道半径 $r / \\mathrm{AU}$ | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 |\n\nA: 木星相邻两次冲日的时间间隔约为 1.1 年\nB: 木星的环绕周期约为 25 年\nC: 天王星的环绕速度约为土星的两倍\nD: 地外行星中, 海王星相邻两次冲日的时间间隔最长\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_906", "problem": "What is the declination of the Sun on the Winter solstice?\nA: $+45^{\\circ}$\nB: $+23.5^{\\circ}$\nC: $0^{\\circ}$\nD: $-23.5^{\\circ}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the declination of the Sun on the Winter solstice?\n\nA: $+45^{\\circ}$\nB: $+23.5^{\\circ}$\nC: $0^{\\circ}$\nD: $-23.5^{\\circ}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_536", "problem": "中国北斗卫星导航系统 (BDS) 是中国自行研制的全球卫星导航系统, 是继美国全球定位系统（GPS）、俄罗斯格洛纳斯卫星导航系统（GLONASS）之后第三个成熟的卫星导航系统。2020 年北斗卫星导航系统已形成全球覆盖能力。如图所示是北斗导航系统中部分卫星的轨道示意图, 已知 $a 、 b 、 c$ 三颗卫星均做匀速圆周运动, $a$ 是地球同步卫星，则（）\n\n[图1]\nA: 卫星 $a$ 的运行速度大于卫星 $c$ 的运行速度\nB: 卫星 $c$ 的加速度大于卫星 $b$ 的加速度\nC: 卫星 $c$ 的运行速度小于第一宇宙速度\nD: 卫星 $c$ 的周期大于 $24 \\mathrm{~h}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n中国北斗卫星导航系统 (BDS) 是中国自行研制的全球卫星导航系统, 是继美国全球定位系统（GPS）、俄罗斯格洛纳斯卫星导航系统（GLONASS）之后第三个成熟的卫星导航系统。2020 年北斗卫星导航系统已形成全球覆盖能力。如图所示是北斗导航系统中部分卫星的轨道示意图, 已知 $a 、 b 、 c$ 三颗卫星均做匀速圆周运动, $a$ 是地球同步卫星，则（）\n\n[图1]\n\nA: 卫星 $a$ 的运行速度大于卫星 $c$ 的运行速度\nB: 卫星 $c$ 的加速度大于卫星 $b$ 的加速度\nC: 卫星 $c$ 的运行速度小于第一宇宙速度\nD: 卫星 $c$ 的周期大于 $24 \\mathrm{~h}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-092.jpg?height=400&width=411&top_left_y=2173&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_662", "problem": "如图所示。甲、乙为地球赤道面内围绕地球运转的通讯卫星。已知甲是与地面相对静止的同步卫星; 乙的运转方向与地球自转方向相反, 轨道半径为地球半径的 2 倍, 周期为 $T$, 在地球赤道上的 $P$ 点有一位观测者，观测者始终相对于地面静止。若地球半径为 $R$, 地球的自转周期为 $T_{0}$ 。求:\n若甲、乙之间可进行无线信号通讯, 不计信号传输时间, 甲卫星对地球的最大视角为 $\\theta$ ，则甲、乙卫星间信号连续中断的最长时间是多少?\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示。甲、乙为地球赤道面内围绕地球运转的通讯卫星。已知甲是与地面相对静止的同步卫星; 乙的运转方向与地球自转方向相反, 轨道半径为地球半径的 2 倍, 周期为 $T$, 在地球赤道上的 $P$ 点有一位观测者，观测者始终相对于地面静止。若地球半径为 $R$, 地球的自转周期为 $T_{0}$ 。求:\n若甲、乙之间可进行无线信号通讯, 不计信号传输时间, 甲卫星对地球的最大视角为 $\\theta$ ，则甲、乙卫星间信号连续中断的最长时间是多少?\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-092.jpg?height=440&width=397&top_left_y=1096&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-093.jpg?height=674&width=1039&top_left_y=168&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_259", "problem": "2016 年 10 月 17 日 7 时 30 分, 中国在酒泉卫星发射中心使用长征二号 FY11 运载火箭将神舟十一号载人飞船送入太空, 2016 年 10 月 19 日凌晨, 神舟十一号飞船与天宫二号自动交会对接成功, 过去神舟十号与天宫一号对接时, 轨道高度是 343 公里, 而神舟十一号和天宫二号对接时的轨道高度是 393 公里, 比过去高了 50 公里. 由以上信息下列说法正确的是（ ）\nA: 天宫一号的运行速度小于天宫二号的运行速度\nB: 天宫一号的运行周期小于天宫二号的运行周期\nC: 神舟十一号飞船如果从 343 公里的轨道变轨到 393 公里的对接轨道机械能减小\nD: 天宫一号的加速度小于天宫二号的的加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2016 年 10 月 17 日 7 时 30 分, 中国在酒泉卫星发射中心使用长征二号 FY11 运载火箭将神舟十一号载人飞船送入太空, 2016 年 10 月 19 日凌晨, 神舟十一号飞船与天宫二号自动交会对接成功, 过去神舟十号与天宫一号对接时, 轨道高度是 343 公里, 而神舟十一号和天宫二号对接时的轨道高度是 393 公里, 比过去高了 50 公里. 由以上信息下列说法正确的是（ ）\n\nA: 天宫一号的运行速度小于天宫二号的运行速度\nB: 天宫一号的运行周期小于天宫二号的运行周期\nC: 神舟十一号飞船如果从 343 公里的轨道变轨到 393 公里的对接轨道机械能减小\nD: 天宫一号的加速度小于天宫二号的的加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_149", "problem": "已知地球半径为 $R$, 地球表面的重力加速度为 $g$ 。质量为 $m$ 的宇宙飞船在半径为 $2 R$的轨道 1 上绕地球中心 $O$ 做圆两运动。现飞船在轨道 1 的 $A$ 点加速到陏圆轨道 2 上,再在远地点 $B$ 点加速, 从而使飞船转移到半径为 $4 R$ 的轨道 3 上, 如图所示。若相距 $r$的两物体间引力势能为 $E_{\\mathrm{p}}=-G \\frac{M m}{r}$, 求:\n\n飞船在轨道 2 上经过近地点 $A$ 和远地点 $B$ 的速率之比 $\\left(v_{A}: v_{B}\\right)$ 。\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n已知地球半径为 $R$, 地球表面的重力加速度为 $g$ 。质量为 $m$ 的宇宙飞船在半径为 $2 R$的轨道 1 上绕地球中心 $O$ 做圆两运动。现飞船在轨道 1 的 $A$ 点加速到陏圆轨道 2 上,再在远地点 $B$ 点加速, 从而使飞船转移到半径为 $4 R$ 的轨道 3 上, 如图所示。若相距 $r$的两物体间引力势能为 $E_{\\mathrm{p}}=-G \\frac{M m}{r}$, 求:\n\n飞船在轨道 2 上经过近地点 $A$ 和远地点 $B$ 的速率之比 $\\left(v_{A}: v_{B}\\right)$ 。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-084.jpg?height=425&width=423&top_left_y=153&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1186", "problem": "The surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.c. The Rayleigh criterion and speed of sound in a plasma are given.\n\ni. Determine the theoretical minimum angular diameter of an element resolvable by this optical system. Give your answer in arcseconds (\").", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.\n\nproblem:\nc. The Rayleigh criterion and speed of sound in a plasma are given.\n\ni. Determine the theoretical minimum angular diameter of an element resolvable by this optical system. Give your answer in arcseconds (\").\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-10.jpg?height=792&width=1572&top_left_y=598&top_left_x=241" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_990", "problem": "The year 1990 held many exciting events in astronomy. Which of the following did not celebrate its $30^{\\text {th }}$ anniversary this year?\n\n[figure1]\nA: The launch of the Hubble Space Telescope, part of NASA's 'Great Observatories' programme\nB: The most distant photo ever taken of the Earth by the probe Voyager 1 from 40.5 au, nicknamed the 'Pale Blue Dot' by Carl Sagan\nC: The arrival at Venus of the Magellan probe, where it used radar to create the highest resolution maps we have of the surface of the planet\nD: ESA's Giotto probe passing within $600 \\mathrm{~km}$ of the nucleus of Halley's Comet, taking our best photos of this famous object\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe year 1990 held many exciting events in astronomy. Which of the following did not celebrate its $30^{\\text {th }}$ anniversary this year?\n\n[figure1]\n\nA: The launch of the Hubble Space Telescope, part of NASA's 'Great Observatories' programme\nB: The most distant photo ever taken of the Earth by the probe Voyager 1 from 40.5 au, nicknamed the 'Pale Blue Dot' by Carl Sagan\nC: The arrival at Venus of the Magellan probe, where it used radar to create the highest resolution maps we have of the surface of the planet\nD: ESA's Giotto probe passing within $600 \\mathrm{~km}$ of the nucleus of Halley's Comet, taking our best photos of this famous object\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c3e3c992a9c51eb3e471g-04.jpg?height=343&width=1280&top_left_y=591&top_left_x=388" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_705", "problem": "《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求:\n若配重空间站没有缆绳连接, 在该处绕地球做匀速圆周运动的线速度大小为多少?\n\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求:\n若配重空间站没有缆绳连接, 在该处绕地球做匀速圆周运动的线速度大小为多少?\n\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-064.jpg?height=388&width=1082&top_left_y=1345&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_249", "problem": "放置在水平平台上的物体, 其表观重力在数值上等于物体对平台的压力, 方向与压力的方向相同。微重力环境是指系统内物体的表观重力远小于其实际重力（万有引力）的环境。此环境下，物体的表观重力与其质量之比称为微重力加速度。\n\n如图所示, 中国科学院力学研究所微重力实验室落塔是我国微重力实验的主要设施之一, 实验中落舱可采用单舱和双舱两种模式进行。已知地球表面的重力加速度为 $g$;\n\n单舱模式是指让固定在单舱上的实验平台随单舱在落塔中自由下落实现微重力环境。若舱体下落时, 受到的阻力恒为舱体总重力的 0.01 倍, 求单舱中的微重力加速度的大小 $g_{1} ;$\n\n[图1]\n\n落塔 落舱", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n放置在水平平台上的物体, 其表观重力在数值上等于物体对平台的压力, 方向与压力的方向相同。微重力环境是指系统内物体的表观重力远小于其实际重力（万有引力）的环境。此环境下，物体的表观重力与其质量之比称为微重力加速度。\n\n如图所示, 中国科学院力学研究所微重力实验室落塔是我国微重力实验的主要设施之一, 实验中落舱可采用单舱和双舱两种模式进行。已知地球表面的重力加速度为 $g$;\n\n单舱模式是指让固定在单舱上的实验平台随单舱在落塔中自由下落实现微重力环境。若舱体下落时, 受到的阻力恒为舱体总重力的 0.01 倍, 求单舱中的微重力加速度的大小 $g_{1} ;$\n\n[图1]\n\n落塔 落舱\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-033.jpg?height=317&width=808&top_left_y=201&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_269", "problem": "如图所示为发射同步卫星的三个轨道, 轨道 I 为近地轨道, 轨道 II 为转移轨道, 轨道 III 为同步轨道, $P 、 Q$ 分别是转移轨道的近地点和远地点。假设卫星在各轨道运行时质量不变, 关于卫星在这个三个轨道上的运动, 下列说法正确的是 ( )\n\n[图1]\nA: 卫星在各个轨道上的运行速度一定都小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: 卫星在轨道 III 上 $Q$ 点的运行速度小于在轨道 II 上 $Q$ 点的运行速度\nC: 卫星在轨道 II 上从 $P$ 点运动到 $Q$ 点的过程中，运行时间一定小于 $12 \\mathrm{~h}$\nD: 卫星在各个轨道上的机械能一样大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示为发射同步卫星的三个轨道, 轨道 I 为近地轨道, 轨道 II 为转移轨道, 轨道 III 为同步轨道, $P 、 Q$ 分别是转移轨道的近地点和远地点。假设卫星在各轨道运行时质量不变, 关于卫星在这个三个轨道上的运动, 下列说法正确的是 ( )\n\n[图1]\n\nA: 卫星在各个轨道上的运行速度一定都小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: 卫星在轨道 III 上 $Q$ 点的运行速度小于在轨道 II 上 $Q$ 点的运行速度\nC: 卫星在轨道 II 上从 $P$ 点运动到 $Q$ 点的过程中，运行时间一定小于 $12 \\mathrm{~h}$\nD: 卫星在各个轨道上的机械能一样大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-092.jpg?height=377&width=392&top_left_y=2319&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_930", "problem": "From the upper parts of the Sun's corona comes a stream of charged particles called the solar wind, meaning the Sun is slowly losing some of its mass (although the effect is negligible compared to the mass loss in nuclear fusion). The particles travel at supersonic speeds until the pressure from interstellar space causes their speed to drop to subsonic speeds instead - this transition is called the termination shock, and has been explored by the two Voyager probes as they leave the solar system.\n[figure1]\n\nFigure 3: Left: A demonstration of a termination shock formed with water flowing from a tap into a sink. Right: The Voyager spacecraft crossing the termination shock of the Solar System.\n\nAs the solar wind moves further from the Sun its speed increases (at an ever decreasing rate), until it asymptotes at a speed, $u_{\\infty}$, equal to the escape velocity, $v_{\\text {esc }}$, of the Sun. Given that $v_{\\text {esc }}=\\sqrt{2 G M / R}$ for an object with radius $R$ and mass $M$, calculate the escape velocity of the Sun. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nFrom the upper parts of the Sun's corona comes a stream of charged particles called the solar wind, meaning the Sun is slowly losing some of its mass (although the effect is negligible compared to the mass loss in nuclear fusion). The particles travel at supersonic speeds until the pressure from interstellar space causes their speed to drop to subsonic speeds instead - this transition is called the termination shock, and has been explored by the two Voyager probes as they leave the solar system.\n[figure1]\n\nFigure 3: Left: A demonstration of a termination shock formed with water flowing from a tap into a sink. Right: The Voyager spacecraft crossing the termination shock of the Solar System.\n\nAs the solar wind moves further from the Sun its speed increases (at an ever decreasing rate), until it asymptotes at a speed, $u_{\\infty}$, equal to the escape velocity, $v_{\\text {esc }}$, of the Sun. Given that $v_{\\text {esc }}=\\sqrt{2 G M / R}$ for an object with radius $R$ and mass $M$, calculate the escape velocity of the Sun. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of km/s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c744602885fab54c0985g-8.jpg?height=454&width=1280&top_left_y=835&top_left_x=386" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "km/s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_679", "problem": "北京时间 2019 年 4 月 10 日晚 21 时 07 分人类首张黑洞照片面向全球同步发布（如图所示)。照片中间黑色的才是黑洞本体，直径大概 $1 \\times 10^{11} \\mathrm{~km}$, 周围是被它吸积成一圈的气体, 因湍急流动而摩擦发光。从地球上看, 这个距离我们 5000 万光年的 M87 黑洞是在顺时针旋转的。黑洞质量和半径 $R$ 的关系满足 $\\frac{M}{R}=\\frac{c^{2}}{2 G}$ (其中 $c$ 为光速, $G$ 为引力常量)，则该黑洞表面重力加速度的数量级为（）\n\n[图1]\nA: $9 \\times 10^{2} \\mathrm{~m} / \\mathrm{s}^{2}$\nB: $10^{5} \\mathrm{~m} / \\mathrm{s}^{2}$\nC: $10 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: $10^{9} \\mathrm{~m} / \\mathrm{s}^{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n北京时间 2019 年 4 月 10 日晚 21 时 07 分人类首张黑洞照片面向全球同步发布（如图所示)。照片中间黑色的才是黑洞本体，直径大概 $1 \\times 10^{11} \\mathrm{~km}$, 周围是被它吸积成一圈的气体, 因湍急流动而摩擦发光。从地球上看, 这个距离我们 5000 万光年的 M87 黑洞是在顺时针旋转的。黑洞质量和半径 $R$ 的关系满足 $\\frac{M}{R}=\\frac{c^{2}}{2 G}$ (其中 $c$ 为光速, $G$ 为引力常量)，则该黑洞表面重力加速度的数量级为（）\n\n[图1]\n\nA: $9 \\times 10^{2} \\mathrm{~m} / \\mathrm{s}^{2}$\nB: $10^{5} \\mathrm{~m} / \\mathrm{s}^{2}$\nC: $10 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: $10^{9} \\mathrm{~m} / \\mathrm{s}^{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-084.jpg?height=289&width=408&top_left_y=1900&top_left_x=344" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_647", "problem": "下列物体的加速度最大的是（ ）\nA. 加速升空阶段的火箭\nB. 月球上自由下落的物体\nC. 击发后在枪筒中的子弹\nD. 在地表随地球自转的物体\nA: 加速升空阶段的火箭\nB: 月球上自由下落的物体\nC: 击发后在枪筒中的子弹\nD: 在地表随地球自转的物体\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n下列物体的加速度最大的是（ ）\nA. 加速升空阶段的火箭\nB. 月球上自由下落的物体\nC. 击发后在枪筒中的子弹\nD. 在地表随地球自转的物体\n\nA: 加速升空阶段的火箭\nB: 月球上自由下落的物体\nC: 击发后在枪筒中的子弹\nD: 在地表随地球自转的物体\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_174", "problem": "5 月 15 日, 我国首次火星探测任务天问一号探测器在火星乌托邦平原南部预选着陆区着陆, 在火星上首次留下中国印迹, 地球上每 26 个月才有一个火星探测器发射窗口, 此时出发最节省燃料。地球和火星绕太阳公转均可视为匀速圆周运动, 已知地球公转平均半径约 1.5 亿千米, 火星公转平均半径约 2.25 亿千米, 地球与火星处于何种位置关系时从地球发射天问一号能经粗圆轨道直接被火星捕获进入环火轨道运行（）\nA: [图1]\nB: [图2]\nC: [图3]\nD: [图4]\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n5 月 15 日, 我国首次火星探测任务天问一号探测器在火星乌托邦平原南部预选着陆区着陆, 在火星上首次留下中国印迹, 地球上每 26 个月才有一个火星探测器发射窗口, 此时出发最节省燃料。地球和火星绕太阳公转均可视为匀速圆周运动, 已知地球公转平均半径约 1.5 亿千米, 火星公转平均半径约 2.25 亿千米, 地球与火星处于何种位置关系时从地球发射天问一号能经粗圆轨道直接被火星捕获进入环火轨道运行（）\n\nA: [图1]\nB: [图2]\nC: [图3]\nD: [图4]\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-042.jpg?height=363&width=425&top_left_y=1326&top_left_x=470", "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-042.jpg?height=351&width=368&top_left_y=1332&top_left_x=1118", "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-042.jpg?height=363&width=391&top_left_y=1709&top_left_x=453", "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-042.jpg?height=360&width=368&top_left_y=1710&top_left_x=1118", "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-042.jpg?height=409&width=417&top_left_y=2331&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_984", "problem": "Special Relativity (SR) tells us that two observers will disagree about the duration of a time interval measured by each one's clock if one is moving at speed $v$ relative to the other, a phenomenon called time dilation. General Relativity (GR) tells us that gravitational fields dilate time too. This has an impact on satellites, since they travel at high orbital speeds (slowing down their clocks relative to the surface) but due to their altitude they are in a weaker gravitational field (speeding up their clocks relative to the surface). Which effect is dominant varies with orbital radius. Global Positioning System (GPS) satellites must compensate for this effect, since the satellites rely on accurate measurements of the time between sending and receiving a radio signal.\n\n[figure1]\n\nFigure 4: A scale diagram of the positions of the orbits for the International Space Station (ISS), GPS satellites and geostationary satellites, along with their orbital periods\n\nIn $\\mathrm{SR}$, time dilation can be calculated with\n\n$$\nt^{\\prime}=\\gamma t_{0} \\quad \\text { where } \\quad \\gamma=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}} \\quad \\text { so } \\quad \\Delta t_{\\mathrm{SR}}=t_{0}-t^{\\prime}=(1-\\gamma) t_{0}\n$$\n\nwhere $t_{0}$ is the time measured by the moving clock, $t^{\\prime}$ is the time measured by the observer, $c$ is the speed of light and $v$ is the speed of the object. A negative $\\Delta t$ indicates that the clocks are passing time slower relative to the observer, whilst a positive indicates they are passing quicker.\n\nGiven $a_{\\text {GPS }}$ is the radius of the GPS satellite's orbit, calculate $\\Delta t_{\\text {overall }}$ when $t_{0}=1$ day. Give your answer in $\\mu$ s and state the significance of the sign.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nSpecial Relativity (SR) tells us that two observers will disagree about the duration of a time interval measured by each one's clock if one is moving at speed $v$ relative to the other, a phenomenon called time dilation. General Relativity (GR) tells us that gravitational fields dilate time too. This has an impact on satellites, since they travel at high orbital speeds (slowing down their clocks relative to the surface) but due to their altitude they are in a weaker gravitational field (speeding up their clocks relative to the surface). Which effect is dominant varies with orbital radius. Global Positioning System (GPS) satellites must compensate for this effect, since the satellites rely on accurate measurements of the time between sending and receiving a radio signal.\n\n[figure1]\n\nFigure 4: A scale diagram of the positions of the orbits for the International Space Station (ISS), GPS satellites and geostationary satellites, along with their orbital periods\n\nIn $\\mathrm{SR}$, time dilation can be calculated with\n\n$$\nt^{\\prime}=\\gamma t_{0} \\quad \\text { where } \\quad \\gamma=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}} \\quad \\text { so } \\quad \\Delta t_{\\mathrm{SR}}=t_{0}-t^{\\prime}=(1-\\gamma) t_{0}\n$$\n\nwhere $t_{0}$ is the time measured by the moving clock, $t^{\\prime}$ is the time measured by the observer, $c$ is the speed of light and $v$ is the speed of the object. A negative $\\Delta t$ indicates that the clocks are passing time slower relative to the observer, whilst a positive indicates they are passing quicker.\n\nGiven $a_{\\text {GPS }}$ is the radius of the GPS satellite's orbit, calculate $\\Delta t_{\\text {overall }}$ when $t_{0}=1$ day. Give your answer in $\\mu$ s and state the significance of the sign.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-10.jpg?height=742&width=1236&top_left_y=791&top_left_x=410" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1172", "problem": "In July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.a. Ignoring the effects of air resistance, the weight of the rocket, and assuming 1-D motion only\n\nii. Determine the constant acceleration produced by the third stage (S-IVB) of the Saturn V rocket and hence calculate the total mass carried into the parking orbit at the end of the first burn.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIn July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.\n\nproblem:\na. Ignoring the effects of air resistance, the weight of the rocket, and assuming 1-D motion only\n\nii. Determine the constant acceleration produced by the third stage (S-IVB) of the Saturn V rocket and hence calculate the total mass carried into the parking orbit at the end of the first burn.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\text { tonnes }, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without any units and equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-04.jpg?height=1010&width=1508&top_left_y=543&top_left_x=271", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-06.jpg?height=800&width=1586&top_left_y=518&top_left_x=240" ], "answer": null, "solution": null, "answer_type": "EX", "unit": [ "\\text { tonnes }" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_798", "problem": "In 1995, researchers at the University of Geneva discovered an exoplanet in the main-sequence star 51 Pegasi. This was the first-ever discovery of an exoplanet orbiting a Sun-like star! When they observed the star, a periodic Doppler shifting of its stellar spectrum indicated that its\nradial velocity was varying sinusoidally; this wobbling could be explained if the star was being pulled in a circle by the gravity of an exoplanet. The radial velocity sinusoid of 51 Pegasi was measured to have a semi-amplitude of $56 \\mathrm{~m} / \\mathrm{s}$ with a period of 4.2 days, and the mass of the star is known to be $1.1 M_{\\odot}$. Assuming that the researchers at Geneva viewed the planet's orbit edge-on and that the orbit was circular, what is the mass of the exoplanet in Jupiter masses?\nA: $0.81 M_{4}$\nB: $0.75 M_{4}$\nC: $0.69 M_{4}$\nD: $0.47 M_{4}$\nE: $0.33 M_{4}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn 1995, researchers at the University of Geneva discovered an exoplanet in the main-sequence star 51 Pegasi. This was the first-ever discovery of an exoplanet orbiting a Sun-like star! When they observed the star, a periodic Doppler shifting of its stellar spectrum indicated that its\nradial velocity was varying sinusoidally; this wobbling could be explained if the star was being pulled in a circle by the gravity of an exoplanet. The radial velocity sinusoid of 51 Pegasi was measured to have a semi-amplitude of $56 \\mathrm{~m} / \\mathrm{s}$ with a period of 4.2 days, and the mass of the star is known to be $1.1 M_{\\odot}$. Assuming that the researchers at Geneva viewed the planet's orbit edge-on and that the orbit was circular, what is the mass of the exoplanet in Jupiter masses?\n\nA: $0.81 M_{4}$\nB: $0.75 M_{4}$\nC: $0.69 M_{4}$\nD: $0.47 M_{4}$\nE: $0.33 M_{4}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_819", "problem": "After a day spent showing a visiting friend around Boston, Leo is walking back along the bridge (see diagram in previous problem) to return to Next House. The time is such that the Sun now aligns with perfectly upriver, so it is in the opposite direction compared to the morning. How high in the sky is the Sun relative to the morning?\nA: Above the horizon, at the same altitude as in the morning\nB: Above the horizon, but lower than in the morning\nC: On the horizon\nD: Below the horizon\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAfter a day spent showing a visiting friend around Boston, Leo is walking back along the bridge (see diagram in previous problem) to return to Next House. The time is such that the Sun now aligns with perfectly upriver, so it is in the opposite direction compared to the morning. How high in the sky is the Sun relative to the morning?\n\nA: Above the horizon, at the same altitude as in the morning\nB: Above the horizon, but lower than in the morning\nC: On the horizon\nD: Below the horizon\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_4833241f8ea3264f9ff9g-21.jpg?height=1347&width=1287&top_left_y=573&top_left_x=381" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_719", "problem": "如图所示, 微信启动新界面, 其画面视角从非洲大陆上空（左）变成中国上空\n\n（右），新照片由我国新一代静止轨道卫星“风云四号”拍摄，见证着科学家 15 年的辛苦和努力。下列说法正确的是（ ）\n\n[图1]\nA: “风云四号”可能经过北京正上空\nB: “风云四号”不可能经过北京正上空\nC: 与“风云四号”同轨道的卫星运动的动能都相等\nD: “风云四号”的运行速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 微信启动新界面, 其画面视角从非洲大陆上空（左）变成中国上空\n\n（右），新照片由我国新一代静止轨道卫星“风云四号”拍摄，见证着科学家 15 年的辛苦和努力。下列说法正确的是（ ）\n\n[图1]\n\nA: “风云四号”可能经过北京正上空\nB: “风云四号”不可能经过北京正上空\nC: 与“风云四号”同轨道的卫星运动的动能都相等\nD: “风云四号”的运行速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-072.jpg?height=314&width=603&top_left_y=1436&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_750", "problem": "What is the name of the JWST component highlighted below?\n\n[figure1]\nA: Antenna\nB: Sunshield\nC: Optics subsystem\nD: Stabilization flap\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the name of the JWST component highlighted below?\n\n[figure1]\n\nA: Antenna\nB: Sunshield\nC: Optics subsystem\nD: Stabilization flap\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_620a57bf13ecc39e0534g-2.jpg?height=349&width=506&top_left_y=1933&top_left_x=792" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_898", "problem": "As of September 2021, which of the following billionaires has not been into space (defined as at least $80 \\mathrm{~km}$ above the surface of the Earth)?\n\n[figure1]\nA: Jeff Bezos with the company Blue Origin\nB: Richard Branson with the company Virgin Galactic\nC: Elon Musk with the company SpaceX\nD: Dennis Tito with the company Space Adventures\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAs of September 2021, which of the following billionaires has not been into space (defined as at least $80 \\mathrm{~km}$ above the surface of the Earth)?\n\n[figure1]\n\nA: Jeff Bezos with the company Blue Origin\nB: Richard Branson with the company Virgin Galactic\nC: Elon Musk with the company SpaceX\nD: Dennis Tito with the company Space Adventures\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_2c19fdb17927c588761dg-04.jpg?height=334&width=1280&top_left_y=593&top_left_x=385" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_488", "problem": "2021 年 10 月 16 日, 神舟十三号载人飞船采用自主快速交会对接模式成功对接于天和核心舱径向端口，对接过程简化如图所示。神舟十三号先到达天和核心舱轨道正下方 $d_{l}=200$ 米的第一停泊点并保持相对静止, 完成各种测控后, 开始沿地心与天和核心舱连线 (径向) 向天和核心舱靠近, 到距离天和核心舱 $d_{2}=19$ 米的第二停泊点短暂驻留,完成各种测控后, 继续径向靠近, 以很小的相对速度完成精准的端口对接。对接技术非常复杂, 故做如下简化。假设地球是半径为 $R_{0}$ 的标准球体，地表重力加速度为 $g$, 忽略自转; 核心舱轨道是半径为 $R$ 的正圆; 神舟十三号质量为 $m_{l}$, 对接前组合体的总质量为 $m_{2}$ ；忽略对接前后神舟十三号质量的变化。\n\n虽然对接时两者相对速度很小，但如果不及时控制也会造成组合体偏离正确轨道,假设不考虑转动, 设对接靠近速度为 $v$, 求控制组合体轨道复位的火箭要对组合体做的功 $W$ 。\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2021 年 10 月 16 日, 神舟十三号载人飞船采用自主快速交会对接模式成功对接于天和核心舱径向端口，对接过程简化如图所示。神舟十三号先到达天和核心舱轨道正下方 $d_{l}=200$ 米的第一停泊点并保持相对静止, 完成各种测控后, 开始沿地心与天和核心舱连线 (径向) 向天和核心舱靠近, 到距离天和核心舱 $d_{2}=19$ 米的第二停泊点短暂驻留,完成各种测控后, 继续径向靠近, 以很小的相对速度完成精准的端口对接。对接技术非常复杂, 故做如下简化。假设地球是半径为 $R_{0}$ 的标准球体，地表重力加速度为 $g$, 忽略自转; 核心舱轨道是半径为 $R$ 的正圆; 神舟十三号质量为 $m_{l}$, 对接前组合体的总质量为 $m_{2}$ ；忽略对接前后神舟十三号质量的变化。\n\n虽然对接时两者相对速度很小，但如果不及时控制也会造成组合体偏离正确轨道,假设不考虑转动, 设对接靠近速度为 $v$, 求控制组合体轨道复位的火箭要对组合体做的功 $W$ 。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-107.jpg?height=508&width=489&top_left_y=174&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_61", "problem": "宇宙中有许多双星系统由两颗恒星组成, 两恒星在相互引力的作用下, 分别围绕其连线上的某一点做周期相同的匀速圆周运动. 研究发现, 双星系统演化过程中，两星的总质量、距离和周期均可能发生变化. 若某双星系统中两星做圆周运动的周期为 $\\mathrm{T}, \\mathrm{M}_{1}$星线速度大小为 $\\mathrm{v}_{1}, \\mathrm{M}_{2}$ 星线速度大小为 $\\mathrm{v}_{2}$, 经过一段时间演化后, 两星总质量变为原来的 $\\frac{1}{k}(\\mathrm{k}>1)$ 倍, 两星之间的距离变为原来的 $\\mathrm{n}(\\mathrm{n}>1)$ 倍, 则此时双星系统圆周运动的周期 $\\mathrm{T}^{\\prime}$ 和线速度之和 $\\mathrm{v}_{1}{ }^{\\prime}+\\mathrm{v}_{2}{ }^{\\prime}$ 是\n\n[图1]\nA: $T^{\\prime}=\\sqrt{n^{3} k} T$\nB: $T^{\\prime}=\\sqrt{\\frac{n^{3}}{k}} T$\nC: $v_{1}^{\\prime}+v_{2}^{\\prime}=\\frac{1}{\\sqrt{n k}}\\left(v_{1}+v_{2}\\right)$\nD: $v_{1}^{\\prime}+v_{2}^{\\prime}=\\sqrt{n k}\\left(v_{1}+v_{2}\\right)$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇宙中有许多双星系统由两颗恒星组成, 两恒星在相互引力的作用下, 分别围绕其连线上的某一点做周期相同的匀速圆周运动. 研究发现, 双星系统演化过程中，两星的总质量、距离和周期均可能发生变化. 若某双星系统中两星做圆周运动的周期为 $\\mathrm{T}, \\mathrm{M}_{1}$星线速度大小为 $\\mathrm{v}_{1}, \\mathrm{M}_{2}$ 星线速度大小为 $\\mathrm{v}_{2}$, 经过一段时间演化后, 两星总质量变为原来的 $\\frac{1}{k}(\\mathrm{k}>1)$ 倍, 两星之间的距离变为原来的 $\\mathrm{n}(\\mathrm{n}>1)$ 倍, 则此时双星系统圆周运动的周期 $\\mathrm{T}^{\\prime}$ 和线速度之和 $\\mathrm{v}_{1}{ }^{\\prime}+\\mathrm{v}_{2}{ }^{\\prime}$ 是\n\n[图1]\n\nA: $T^{\\prime}=\\sqrt{n^{3} k} T$\nB: $T^{\\prime}=\\sqrt{\\frac{n^{3}}{k}} T$\nC: $v_{1}^{\\prime}+v_{2}^{\\prime}=\\frac{1}{\\sqrt{n k}}\\left(v_{1}+v_{2}\\right)$\nD: $v_{1}^{\\prime}+v_{2}^{\\prime}=\\sqrt{n k}\\left(v_{1}+v_{2}\\right)$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-19.jpg?height=120&width=525&top_left_y=2556&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_42", "problem": "中国计划已经实现返回式月球软着陆器对月球进行科学探测, 如图所示, 发射一颗运动半径为 $r$ 的绕月卫星, 登月着陆器从绕月卫星出发 (不影响绕月卫星运动), 沿粗圆轨道降落到月球的表面上, 与月球表面经多次碰撞和弹跳才停下来。假设着陆器第一次弹起的最大高度为 $h$, 水平速度为 $v_{1}$, 第二次着陆时速度为 $v_{2}$, 已知月球半径为 $R$,着陆器质量为 $m$, 不计一切阻力和月球的自转。求:\n设想软着陆器完成了对月球的科学考察任务后, 再返回绕月卫星, 返回与卫星对接时, 二者具有相同的速度, 着陆器在返回过程中需克服月球引力做功 $W=m g_{\\text {月 }}\\left(1-\\frac{R}{r}\\right) R$, 则着陆器的电池应提供给着陆器多少能量, 才能使着陆器安全返回到绕月卫星。\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n中国计划已经实现返回式月球软着陆器对月球进行科学探测, 如图所示, 发射一颗运动半径为 $r$ 的绕月卫星, 登月着陆器从绕月卫星出发 (不影响绕月卫星运动), 沿粗圆轨道降落到月球的表面上, 与月球表面经多次碰撞和弹跳才停下来。假设着陆器第一次弹起的最大高度为 $h$, 水平速度为 $v_{1}$, 第二次着陆时速度为 $v_{2}$, 已知月球半径为 $R$,着陆器质量为 $m$, 不计一切阻力和月球的自转。求:\n设想软着陆器完成了对月球的科学考察任务后, 再返回绕月卫星, 返回与卫星对接时, 二者具有相同的速度, 着陆器在返回过程中需克服月球引力做功 $W=m g_{\\text {月 }}\\left(1-\\frac{R}{r}\\right) R$, 则着陆器的电池应提供给着陆器多少能量, 才能使着陆器安全返回到绕月卫星。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-111.jpg?height=408&width=512&top_left_y=1441&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_251", "problem": "人造卫星在绕地球运行时, 会遇到稀薄大气的阻力。如果不进行必要的轨道维持,稀薄大气对卫星的这种微小阻力会导致卫星轨道半径逐渐减小, 以至最终落回地球。这个过程是非常漫长的, 因此卫星每一圈的运动仍可以认为是匀速圆周运动。规定两质点相距无穷远时的引力势能为零, 理论上可以得出质量分别 $m_{1} 、 m_{2}$ 的两个物体相距 $r$ 时,系统的引力势能为 $E_{p}=\\frac{G m_{1} m_{2}}{r}$ 。已知人造卫星的质量为 $m$, 某时刻绕地球做匀速圆周运动的轨道半径为 $r$, 地球半径为 $R$, 地球表面附近的重力加速度为 $g$ 。\n由于大气阻力的影响, 卫星的轨道半径逐渐减小。求在这个过程中, 万有引力做的功 $\\mathrm{W}_{\\mathrm{G}}$ 与克服大气阻力做的功 $\\mathrm{W}_{\\mathrm{f}}$ 的比。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n人造卫星在绕地球运行时, 会遇到稀薄大气的阻力。如果不进行必要的轨道维持,稀薄大气对卫星的这种微小阻力会导致卫星轨道半径逐渐减小, 以至最终落回地球。这个过程是非常漫长的, 因此卫星每一圈的运动仍可以认为是匀速圆周运动。规定两质点相距无穷远时的引力势能为零, 理论上可以得出质量分别 $m_{1} 、 m_{2}$ 的两个物体相距 $r$ 时,系统的引力势能为 $E_{p}=\\frac{G m_{1} m_{2}}{r}$ 。已知人造卫星的质量为 $m$, 某时刻绕地球做匀速圆周运动的轨道半径为 $r$, 地球半径为 $R$, 地球表面附近的重力加速度为 $g$ 。\n由于大气阻力的影响, 卫星的轨道半径逐渐减小。求在这个过程中, 万有引力做的功 $\\mathrm{W}_{\\mathrm{G}}$ 与克服大气阻力做的功 $\\mathrm{W}_{\\mathrm{f}}$ 的比。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_615", "problem": "2022 年 4 月 16 日神舟十三号载人飞船返回舱成功着陆, 三位航天员在空间站出差半年, 完成了两次太空行走和 20 多项科学实验, 并开展了两次“天宫课堂”活动, 刷新了中国航天新纪录。已知地球半径为 $R$, 地球表面重力加速度为 $g$ 。总质量为 $m_{0}$ 的空间站绕地球的运动可近似为匀速圆周运动, 距地球表面高度为 $h$, 阻力忽略不计。\n\n求空间站绕地球匀速圆周运动的动能块 $E_{\\mathrm{k} 1}$;", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2022 年 4 月 16 日神舟十三号载人飞船返回舱成功着陆, 三位航天员在空间站出差半年, 完成了两次太空行走和 20 多项科学实验, 并开展了两次“天宫课堂”活动, 刷新了中国航天新纪录。已知地球半径为 $R$, 地球表面重力加速度为 $g$ 。总质量为 $m_{0}$ 的空间站绕地球的运动可近似为匀速圆周运动, 距地球表面高度为 $h$, 阻力忽略不计。\n\n求空间站绕地球匀速圆周运动的动能块 $E_{\\mathrm{k} 1}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_731", "problem": "某卫星绕地球做匀速圆周运动, 地球相对卫星的张角为 $\\theta$, 当卫星与地心连线扫过 $\\theta$ (弧度) 的角度时, 圆周运动通过的弧长为 $s$, 已知地球表面的重力加速度为 $g$, 则下列判断正确的是 ( )。\nA: 地球半径为 $\\frac{s}{\\theta} \\cos \\frac{\\theta}{2}$\nB: 卫星的向心加速度为 $g \\sin \\frac{\\theta}{2}$\nC: 卫星的线速度为 $\\sqrt{\\frac{2 g s}{\\theta}} \\cdot \\sin \\frac{\\theta}{2}$\nD: 卫星的角速度为 $\\sqrt{\\frac{g \\theta}{s}} \\cdot \\sin \\frac{\\theta}{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n某卫星绕地球做匀速圆周运动, 地球相对卫星的张角为 $\\theta$, 当卫星与地心连线扫过 $\\theta$ (弧度) 的角度时, 圆周运动通过的弧长为 $s$, 已知地球表面的重力加速度为 $g$, 则下列判断正确的是 ( )。\n\nA: 地球半径为 $\\frac{s}{\\theta} \\cos \\frac{\\theta}{2}$\nB: 卫星的向心加速度为 $g \\sin \\frac{\\theta}{2}$\nC: 卫星的线速度为 $\\sqrt{\\frac{2 g s}{\\theta}} \\cdot \\sin \\frac{\\theta}{2}$\nD: 卫星的角速度为 $\\sqrt{\\frac{g \\theta}{s}} \\cdot \\sin \\frac{\\theta}{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_481", "problem": "地球的公转轨道接近圆, 哈雷彗星的公转轨道则是一个非常扁的粗圆, 如图所示。天文学家哈雷成功预言了哈雷彗星的回归, 哈雷彗星最近出现的时间是 1986 年, 预测下次飞近地球将在 2061 年左右。若哈雷彗星在近日点与太阳中心的距离为 $r_{1}$, 远日点与太阳中心的距离为 $r_{2}$ 。下列说法正确的是 ( )\n\n[图1]\nA: 哈雷彗星轨道的半长轴是地球公转半径的 $\\sqrt{75^{3}}$ 倍\nB: 哈雷彗星在近日点的速度一定大于地球的公转速度\nC: 哈雷彗星在近日点和远日点的速度之比为 $\\sqrt{r_{2}}: \\sqrt{r_{1}}$\nD: 相同时间内, 哈雷彗星与太阳连线扫过的面积和地球与太阳连线扫过的面积相等\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球的公转轨道接近圆, 哈雷彗星的公转轨道则是一个非常扁的粗圆, 如图所示。天文学家哈雷成功预言了哈雷彗星的回归, 哈雷彗星最近出现的时间是 1986 年, 预测下次飞近地球将在 2061 年左右。若哈雷彗星在近日点与太阳中心的距离为 $r_{1}$, 远日点与太阳中心的距离为 $r_{2}$ 。下列说法正确的是 ( )\n\n[图1]\n\nA: 哈雷彗星轨道的半长轴是地球公转半径的 $\\sqrt{75^{3}}$ 倍\nB: 哈雷彗星在近日点的速度一定大于地球的公转速度\nC: 哈雷彗星在近日点和远日点的速度之比为 $\\sqrt{r_{2}}: \\sqrt{r_{1}}$\nD: 相同时间内, 哈雷彗星与太阳连线扫过的面积和地球与太阳连线扫过的面积相等\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-105.jpg?height=208&width=671&top_left_y=513&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_416", "problem": "2019 年 12 月 17 日, 中国科学院云南天文台研究人员在对密近双星半人马座 V752 进行观测和分析研究时, 发现了一种双星轨道变化的新模式。双星在运行时周期突然增大, 研究人员分析有可能是受到了来自其伴星双星的动力学扰动, 从而引起了双星间的物质相互交流，周期开始持续增加。若小质量的子星的物质被吸引而转移至大质量的子星上 (不考虑质量的损失), 导致周期增大为原来的 $k$ 倍, 则下列说法中正确的是 ( )\nA: 两子星间距增大为原来的 $k^{\\frac{3}{2}}$ 倍\nB: 两子星间的万有引力增大\nC: 质量小的子星轨道半径增大\nD: 质量大的子星线速度增大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2019 年 12 月 17 日, 中国科学院云南天文台研究人员在对密近双星半人马座 V752 进行观测和分析研究时, 发现了一种双星轨道变化的新模式。双星在运行时周期突然增大, 研究人员分析有可能是受到了来自其伴星双星的动力学扰动, 从而引起了双星间的物质相互交流，周期开始持续增加。若小质量的子星的物质被吸引而转移至大质量的子星上 (不考虑质量的损失), 导致周期增大为原来的 $k$ 倍, 则下列说法中正确的是 ( )\n\nA: 两子星间距增大为原来的 $k^{\\frac{3}{2}}$ 倍\nB: 两子星间的万有引力增大\nC: 质量小的子星轨道半径增大\nD: 质量大的子星线速度增大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_466", "problem": "科幻影片《流浪地球》中为了让地球逃离太阳系, 人们在地球上建造特大功率发动机, 使地球完成一系列变轨操作, 其逃离过程可设想成如图所示, 地球在椭圆轨道 I 上运行到远日点 $P$ 变轨进入圆形轨道 II, 在圆形轨道 II 上运行一段时间后在 $P$ 点时再次加速变轨, 从而最终摆脱太阳束缚。对于该过程, 下列说法正确的是( )\n\n[图1]\nA: 地球在 $P$ 点通过向前喷气减速实现由轨道 I 进入轨道 II\nB: 若地球在 I、II 轨道上运行的周期分别为 $T_{1} 、 T_{2}$, 则 $T_{1}r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$e. Taking the mass of $\\mathrm{M}_{8} 7^{*}$ as $6.5 \\times 10^{9} \\mathrm{M}_{\\odot}$ :\n\ni. Determine the period of a particle in the ISCO of $\\mathrm{M} 87^{*}$ for the $\\mathrm{a}=1, \\mathrm{a}=-1$ and $\\mathrm{a}=0$ (i.e. non-spinning) cases. Give your answer in days.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$\n\nproblem:\ne. Taking the mass of $\\mathrm{M}_{8} 7^{*}$ as $6.5 \\times 10^{9} \\mathrm{M}_{\\odot}$ :\n\ni. Determine the period of a particle in the ISCO of $\\mathrm{M} 87^{*}$ for the $\\mathrm{a}=1, \\mathrm{a}=-1$ and $\\mathrm{a}=0$ (i.e. non-spinning) cases. Give your answer in days.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of days, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-07.jpg?height=704&width=1414&top_left_y=698&top_left_x=331", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-08.jpg?height=374&width=1562&top_left_y=1698&top_left_x=263", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-09.jpg?height=468&width=686&top_left_y=1388&top_left_x=705" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "days" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_306", "problem": "已知质量分布均匀的球壳对壳内物体的万有引力为零, 假设地球是质量分布均匀的球体, 如图若在地球内挖一球形内切空腔, 有一小球自切点 $A$ 自由释放, 则小球在球形空腔内将做（）\n\n[图1]\nA: 自由落体运动\nB: 加速度越来越大的直线运动\nC: 匀加速直线运动\nD: 加速度越来越小的直线运动\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n已知质量分布均匀的球壳对壳内物体的万有引力为零, 假设地球是质量分布均匀的球体, 如图若在地球内挖一球形内切空腔, 有一小球自切点 $A$ 自由释放, 则小球在球形空腔内将做（）\n\n[图1]\n\nA: 自由落体运动\nB: 加速度越来越大的直线运动\nC: 匀加速直线运动\nD: 加速度越来越小的直线运动\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-059.jpg?height=294&width=257&top_left_y=978&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_39", "problem": "火卫一是太阳系最暗的天体之一, 假设火卫一围绕火星做匀速圆周运动的轨道半径为 $r_{1}$, 运行周期为 $T_{1}$, 火星半径为 $R$ 。已知行星与卫星间引力势能的表达式为 $E_{\\mathrm{p}}=-\\frac{G M_{0} m_{0}}{r}, r$ 为行星与卫星的中心距离, 则火星的第二宇宙速度为 $(\\quad)$\nA: $\\frac{\\pi r_{1}}{T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\nB: $\\frac{2 \\pi r_{1}}{T_{1}} \\sqrt{\\frac{r_{1}}{R}}$\nC: $\\frac{2 \\pi r_{1}}{R T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\nD: $\\frac{2 \\pi r_{1}}{T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n火卫一是太阳系最暗的天体之一, 假设火卫一围绕火星做匀速圆周运动的轨道半径为 $r_{1}$, 运行周期为 $T_{1}$, 火星半径为 $R$ 。已知行星与卫星间引力势能的表达式为 $E_{\\mathrm{p}}=-\\frac{G M_{0} m_{0}}{r}, r$ 为行星与卫星的中心距离, 则火星的第二宇宙速度为 $(\\quad)$\n\nA: $\\frac{\\pi r_{1}}{T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\nB: $\\frac{2 \\pi r_{1}}{T_{1}} \\sqrt{\\frac{r_{1}}{R}}$\nC: $\\frac{2 \\pi r_{1}}{R T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\nD: $\\frac{2 \\pi r_{1}}{T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1212", "problem": "Hanny's Voorwerp (Dutch for 'object') is a rare type of astronomical object discovered in 2007 by the school teacher Hanny van Arkel whilst participating as a volunteer in the Galaxy Zoo project. When inspecting the image of the galaxy IC 2497 in the constellation Leo Minor, she observed a bright green blob close to the galaxy.\n\n[figure1]\n\nFigure 5: HST image of galaxy IC 2497 and the glowing Voorwerp below it.\n\nCredit: Keel et al. (2012) \\& Galaxy Zoo.\n\nSubsequent observations have shown that the galaxy IC 2497 is at a redshift of $z=0.05$, with the Voorwerp at a similar distance and with a projected angular separation of 20 arcseconds from the centre of the galaxy $\\left(3600\\right.$ arcseconds $\\left.=1^{\\circ}\\right)$. Radio observations suggest that the Voorwerp is a massive cloud of gas, made of ionized hydrogen, with a size of $10 \\mathrm{kpc}$ and a mass of $10^{11} \\mathrm{M}_{\\odot}$. It is probably a cloud of gas that was stripped from the galaxy during a merger with another nearby galaxy.\n\nIn this question you will explore the cause of the 'glow' of the Voorwerp and will learn about a new type of an astronomical object; a quasar.c. The gravitational potential energy of the material falling to radius, which in this case is a black hole with radius equal to the Schwarzschild radius, $R_{S}=2 G M / c^{2}$, at a mass accretion rate $\\dot{m} \\equiv \\delta m / \\delta t$, is converted into radiation with an efficiency of. Show that the power (luminosity) output of the SMBH is given by $L=\\frac{1}{2} \\eta \\dot{m} c^{2}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nHanny's Voorwerp (Dutch for 'object') is a rare type of astronomical object discovered in 2007 by the school teacher Hanny van Arkel whilst participating as a volunteer in the Galaxy Zoo project. When inspecting the image of the galaxy IC 2497 in the constellation Leo Minor, she observed a bright green blob close to the galaxy.\n\n[figure1]\n\nFigure 5: HST image of galaxy IC 2497 and the glowing Voorwerp below it.\n\nCredit: Keel et al. (2012) \\& Galaxy Zoo.\n\nSubsequent observations have shown that the galaxy IC 2497 is at a redshift of $z=0.05$, with the Voorwerp at a similar distance and with a projected angular separation of 20 arcseconds from the centre of the galaxy $\\left(3600\\right.$ arcseconds $\\left.=1^{\\circ}\\right)$. Radio observations suggest that the Voorwerp is a massive cloud of gas, made of ionized hydrogen, with a size of $10 \\mathrm{kpc}$ and a mass of $10^{11} \\mathrm{M}_{\\odot}$. It is probably a cloud of gas that was stripped from the galaxy during a merger with another nearby galaxy.\n\nIn this question you will explore the cause of the 'glow' of the Voorwerp and will learn about a new type of an astronomical object; a quasar.\n\nproblem:\nc. The gravitational potential energy of the material falling to radius, which in this case is a black hole with radius equal to the Schwarzschild radius, $R_{S}=2 G M / c^{2}$, at a mass accretion rate $\\dot{m} \\equiv \\delta m / \\delta t$, is converted into radiation with an efficiency of. Show that the power (luminosity) output of the SMBH is given by $L=\\frac{1}{2} \\eta \\dot{m} c^{2}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-08.jpg?height=800&width=577&top_left_y=508&top_left_x=745" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_113", "problem": "宇宙中组成双星系统的甲、乙两颗恒星的质量分别为 $m 、 k m(k>1)$, 甲绕两恒星连线上一点做圆周运动的半径为 $r$, 周期为 $T$, 根据宇宙大爆炸理论, 两恒星间的距离会缓慢增大, 若干年后, 甲做圆周运动的半径增大为 $n r(n>k)$, 设甲、乙两恒星的质量保持不变, 引力常量为 $G$, 则若干年后下列说法正确的是\nA: 恒星甲做圆周运动的向心力为 $G \\frac{\\mathrm{km}^{2}}{(\\mathrm{nr})^{2}}$\nB: 恒星甲做圆周运动周期变大\nC: 恒星乙做圆周运动的半径为 $k n r$\nD: 恒星乙做圆周运动的线速度为恒星甲做圆周运动线速度的 $\\frac{1}{k}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇宙中组成双星系统的甲、乙两颗恒星的质量分别为 $m 、 k m(k>1)$, 甲绕两恒星连线上一点做圆周运动的半径为 $r$, 周期为 $T$, 根据宇宙大爆炸理论, 两恒星间的距离会缓慢增大, 若干年后, 甲做圆周运动的半径增大为 $n r(n>k)$, 设甲、乙两恒星的质量保持不变, 引力常量为 $G$, 则若干年后下列说法正确的是\n\nA: 恒星甲做圆周运动的向心力为 $G \\frac{\\mathrm{km}^{2}}{(\\mathrm{nr})^{2}}$\nB: 恒星甲做圆周运动周期变大\nC: 恒星乙做圆周运动的半径为 $k n r$\nD: 恒星乙做圆周运动的线速度为恒星甲做圆周运动线速度的 $\\frac{1}{k}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_575", "problem": "2022 年 9 月 27 日的“木星冲日”是观测木星的一次好机会。木星冲日就是指木星、\n地球和太阳依次排列大致形成一条直线时的天象。已知木星与地球几乎在同一平面内沿同一方向绕太阳近似做匀速圆周运动, 木星质量约为地球质量的 318 倍, 木星半径约为地球半径的 11 倍, 木星到太阳的距离大约是地球到太阳距离的 5 倍, 则下列说法正确的是 ( )\nA: 木星运行的加速度比地球运行的加速度大\nB: 木星表面的重力加速度比地球表面的重力加速度小\nC: 在木星表面附近发射飞行器的速度至少为 $7.9 \\mathrm{~km} / \\mathrm{s}$\nD: 上一次“木星冲日”的时间大约在 2021 年 8 月份\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2022 年 9 月 27 日的“木星冲日”是观测木星的一次好机会。木星冲日就是指木星、\n地球和太阳依次排列大致形成一条直线时的天象。已知木星与地球几乎在同一平面内沿同一方向绕太阳近似做匀速圆周运动, 木星质量约为地球质量的 318 倍, 木星半径约为地球半径的 11 倍, 木星到太阳的距离大约是地球到太阳距离的 5 倍, 则下列说法正确的是 ( )\n\nA: 木星运行的加速度比地球运行的加速度大\nB: 木星表面的重力加速度比地球表面的重力加速度小\nC: 在木星表面附近发射飞行器的速度至少为 $7.9 \\mathrm{~km} / \\mathrm{s}$\nD: 上一次“木星冲日”的时间大约在 2021 年 8 月份\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_992", "problem": "Given the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\\gtrsim 80 \\%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet.\n[figure1]\n\nFigure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\\sim 100 \\%$ iron) might look like.\n\nThe Roche limiting distance, $a_{\\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is\n\n$$\na_{\\min }=2.44 R_{\\star}\\left(\\frac{\\rho_{\\star}}{\\rho_{p}}\\right)^{1 / 3}\n$$\n\nwhere $R_{\\star}$ is the radius of the star, $\\rho_{\\star}$ is the density of the star and $\\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as\n\n$$\n\\log _{10}\\left(\\frac{R}{R_{E}}\\right)=0.295 \\log _{10}\\left(\\frac{M}{M_{E}}\\right)+\\alpha\n$$\n\nwhere $\\alpha=0.0286$ in the pure silicate case and $\\alpha=-0.1090$ in the pure iron case.\n\nThe exoplanet KOI 1843.03 is measured (from transit lightcurves) to have a radius of $0.61 R_{E}$. Calculate the minimum period for the exoplanet in the pure silicate and pure iron cases. Give your answer in hours.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nGiven the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\\gtrsim 80 \\%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet.\n[figure1]\n\nFigure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\\sim 100 \\%$ iron) might look like.\n\nThe Roche limiting distance, $a_{\\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is\n\n$$\na_{\\min }=2.44 R_{\\star}\\left(\\frac{\\rho_{\\star}}{\\rho_{p}}\\right)^{1 / 3}\n$$\n\nwhere $R_{\\star}$ is the radius of the star, $\\rho_{\\star}$ is the density of the star and $\\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as\n\n$$\n\\log _{10}\\left(\\frac{R}{R_{E}}\\right)=0.295 \\log _{10}\\left(\\frac{M}{M_{E}}\\right)+\\alpha\n$$\n\nwhere $\\alpha=0.0286$ in the pure silicate case and $\\alpha=-0.1090$ in the pure iron case.\n\nThe exoplanet KOI 1843.03 is measured (from transit lightcurves) to have a radius of $0.61 R_{E}$. Calculate the minimum period for the exoplanet in the pure silicate and pure iron cases. Give your answer in hours.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of hrs, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-09.jpg?height=620&width=1468&top_left_y=861&top_left_x=292" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "hrs" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_51", "problem": "天文观测发现, 天狼星 $\\mathrm{A}$ 与其伴星 $\\mathrm{B}$ 是一个双星系统。它们始终绕着 $O$ 点在两个不同椭圆轨道上运动, 如图所示, 实线为天狼星 $\\mathrm{A}$ 的运行轨迹, 虚线为其伴星 $\\mathrm{B}$ 的轨迹，则（ )\n\n[图1]\nA: A 的运行周期小于 B 的运行周期\nB: $\\mathrm{A}$ 的质量小于 $\\mathrm{B}$ 的质量\nC: A 的加速度总是小于 $\\mathrm{B}$ 的加速度\nD: $\\mathrm{A}$ 与 $\\mathrm{B}$ 绕 $O$ 点的旋转方向可能相同, 可能相反\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n天文观测发现, 天狼星 $\\mathrm{A}$ 与其伴星 $\\mathrm{B}$ 是一个双星系统。它们始终绕着 $O$ 点在两个不同椭圆轨道上运动, 如图所示, 实线为天狼星 $\\mathrm{A}$ 的运行轨迹, 虚线为其伴星 $\\mathrm{B}$ 的轨迹，则（ )\n\n[图1]\n\nA: A 的运行周期小于 B 的运行周期\nB: $\\mathrm{A}$ 的质量小于 $\\mathrm{B}$ 的质量\nC: A 的加速度总是小于 $\\mathrm{B}$ 的加速度\nD: $\\mathrm{A}$ 与 $\\mathrm{B}$ 绕 $O$ 点的旋转方向可能相同, 可能相反\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-051.jpg?height=320&width=600&top_left_y=1802&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1002", "problem": "When two objects of unequal mass orbit around each other, they both orbit around a barycentre - this is the name given to the location of the centre of mass of the system. The masses of both objects, and the distance between their centres, affects the position of their barycentre. Imagine two objects, Object 1 and Object 2, with masses $m_{1}$ and $m_{2}$ respectively, and the average distance between the centre of both objects is $a$, then the average distance from the centre of Object 1 to the barycentre, $r$, is given by the formula:\n\n$$\nr=a \\frac{m_{2}}{m_{1}+m_{2}}\n$$\n\nOne of the most famous examples of a system with the barycentre lying outside the larger object is the Pluto-Charon system (see Fig 4), which is why many considered it to be a double-planet long before it was reclassified as a dwarf planet. Given the barycentre of the system is at $1.83 \\mathrm{R}_{\\text {Pluto }}$ (where $\\mathrm{R}_{\\text {Pluto }}=1187 \\mathrm{~km}$ ) and the average separation between Pluto and Charon (as measured from centre to centre) is $19570 \\mathrm{~km}$, calculate the ratio of their masses (i.e. $m_{\\text {Pluto }} / m_{\\text {Charon }}$ ).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nWhen two objects of unequal mass orbit around each other, they both orbit around a barycentre - this is the name given to the location of the centre of mass of the system. The masses of both objects, and the distance between their centres, affects the position of their barycentre. Imagine two objects, Object 1 and Object 2, with masses $m_{1}$ and $m_{2}$ respectively, and the average distance between the centre of both objects is $a$, then the average distance from the centre of Object 1 to the barycentre, $r$, is given by the formula:\n\n$$\nr=a \\frac{m_{2}}{m_{1}+m_{2}}\n$$\n\nOne of the most famous examples of a system with the barycentre lying outside the larger object is the Pluto-Charon system (see Fig 4), which is why many considered it to be a double-planet long before it was reclassified as a dwarf planet. Given the barycentre of the system is at $1.83 \\mathrm{R}_{\\text {Pluto }}$ (where $\\mathrm{R}_{\\text {Pluto }}=1187 \\mathrm{~km}$ ) and the average separation between Pluto and Charon (as measured from centre to centre) is $19570 \\mathrm{~km}$, calculate the ratio of their masses (i.e. $m_{\\text {Pluto }} / m_{\\text {Charon }}$ ).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_312", "problem": "建立物理模型是解决实际问题的重要方法。\n\n如图 1 所示, 圆和椭圆是分析卫星运动时常用的模型。已知, 地球质量为 $M$, 半径为 $R$, 万有引力常量为 $G$ 。\n\n卫星在近地轨道I上围绕地球的运动, 可视做匀速圆周运动, 轨道半径近似等于地球半径。求卫星在近地轨道I上的运行速度大小 $v$ 。\n\n图 1\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n建立物理模型是解决实际问题的重要方法。\n\n如图 1 所示, 圆和椭圆是分析卫星运动时常用的模型。已知, 地球质量为 $M$, 半径为 $R$, 万有引力常量为 $G$ 。\n\n卫星在近地轨道I上围绕地球的运动, 可视做匀速圆周运动, 轨道半径近似等于地球半径。求卫星在近地轨道I上的运行速度大小 $v$ 。\n\n图 1\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-060.jpg?height=254&width=320&top_left_y=1118&top_left_x=842" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_632", "problem": "深空中, 某行星 $\\mathrm{X}$ 绕恒星 $\\mathrm{Y}$ 逆时针方向公转, 卫星 $\\mathrm{Z}$ 绕 $\\mathrm{X}$ 逆时针方向运行, $\\mathrm{X}$ 轨道与 $\\mathrm{Z}$ 轨道在同一平面内。如图, 某时刻 $\\mathrm{Z} 、 \\mathrm{X}$ 和 $\\mathrm{Y}$ 在同一直线上, 经过时间 $t, \\mathrm{Z} 、 \\mathrm{X}$和 $\\mathrm{Y}$ 再次在同一直线上 (相对位置的顺序不变)。已知 $\\mathrm{Z}$ 绕 $\\mathrm{X}$ 做匀速圆周运动的周期为 $T, \\mathrm{X}$ 绕 $\\mathrm{Y}$ 做匀速圆周运动的周期大于 $T, \\mathrm{X}$ 与 $\\mathrm{Y}$ 间的距离为 $r$, 则 $\\mathrm{Y}$ 的质量为 ( )\n\n[图1]\nA: $\\frac{4 \\pi^{2} r^{3}(t-T)^{2}}{G t^{2} T^{2}}$\nB: $\\frac{4 \\pi^{2} r^{3}(t+T)^{2}}{G t^{2} T^{2}}$\nC: $\\frac{4 \\pi^{2} r^{3}}{G t^{2}}$\nD: $\\frac{4 \\pi^{2} r^{3}}{G T^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n深空中, 某行星 $\\mathrm{X}$ 绕恒星 $\\mathrm{Y}$ 逆时针方向公转, 卫星 $\\mathrm{Z}$ 绕 $\\mathrm{X}$ 逆时针方向运行, $\\mathrm{X}$ 轨道与 $\\mathrm{Z}$ 轨道在同一平面内。如图, 某时刻 $\\mathrm{Z} 、 \\mathrm{X}$ 和 $\\mathrm{Y}$ 在同一直线上, 经过时间 $t, \\mathrm{Z} 、 \\mathrm{X}$和 $\\mathrm{Y}$ 再次在同一直线上 (相对位置的顺序不变)。已知 $\\mathrm{Z}$ 绕 $\\mathrm{X}$ 做匀速圆周运动的周期为 $T, \\mathrm{X}$ 绕 $\\mathrm{Y}$ 做匀速圆周运动的周期大于 $T, \\mathrm{X}$ 与 $\\mathrm{Y}$ 间的距离为 $r$, 则 $\\mathrm{Y}$ 的质量为 ( )\n\n[图1]\n\nA: $\\frac{4 \\pi^{2} r^{3}(t-T)^{2}}{G t^{2} T^{2}}$\nB: $\\frac{4 \\pi^{2} r^{3}(t+T)^{2}}{G t^{2} T^{2}}$\nC: $\\frac{4 \\pi^{2} r^{3}}{G t^{2}}$\nD: $\\frac{4 \\pi^{2} r^{3}}{G T^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-29.jpg?height=391&width=340&top_left_y=176&top_left_x=344" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_3", "problem": "理论研究表明地球上的物体速度达到第二宇宙速度 $11.2 \\mathrm{~km} / \\mathrm{s}$ 时, 物体就能脱离地球, 又知第二宇宙速度是第一宇宙速度的 $\\sqrt{2}$ 倍. 现有某探测器完成了对某未知星球的探测任务悬停在该星球表面. 通过探测到的数据得到该星球的有关参量（1)其密度基本与地球密度一致. (2)其半径约为地球半径的 2 倍. 若不考虑该星球自转的影响, 欲使探测器脱离该星球, 则探测器从该星球表面的起飞速度至少约为 ( )\nA: $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: $11.2 \\mathrm{~km} / \\mathrm{s}$\nC: $15.8 \\mathrm{~km} / \\mathrm{s}$\nD: $22.4 \\mathrm{~km} / \\mathrm{s}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n理论研究表明地球上的物体速度达到第二宇宙速度 $11.2 \\mathrm{~km} / \\mathrm{s}$ 时, 物体就能脱离地球, 又知第二宇宙速度是第一宇宙速度的 $\\sqrt{2}$ 倍. 现有某探测器完成了对某未知星球的探测任务悬停在该星球表面. 通过探测到的数据得到该星球的有关参量（1)其密度基本与地球密度一致. (2)其半径约为地球半径的 2 倍. 若不考虑该星球自转的影响, 欲使探测器脱离该星球, 则探测器从该星球表面的起飞速度至少约为 ( )\n\nA: $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: $11.2 \\mathrm{~km} / \\mathrm{s}$\nC: $15.8 \\mathrm{~km} / \\mathrm{s}$\nD: $22.4 \\mathrm{~km} / \\mathrm{s}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_702", "problem": "航天技术的发展是当今各国综合国力的直接体现，近年来，我国的航天技术取得了让世界瞩目的成绩，也引领科技爱好者思索航天技术的发展，有人就提出了一种不用火箭发射人造地球卫星的设想。其设想如下: 如图所示, 在地球上距地心 $h$ 处沿一条弦挖一光滑通道, 在通道的两个出口处 $A$ 和 $B$ 分别将质量为 $M$ 的物体和质量为 $m$ 的待发射卫星同时自由释放, $M \\gg m$, 在中点 $O^{\\prime}$ 弹性正撞后，质量为 $m$ 的物体，即待发射的卫星就会从通道口 $B$ 冲出通道, 设置一个装置, 卫星从 $B$ 冲出就把速度变为沿地球切线方向, 但不改变速度大小, 这样就有可能成功发射卫星。已知地球可视为质量分布均匀\n的球体, 且质量分布均匀的球壳对壳内物体的引力为零, 地球半径为 $R_{0}$, 表面的重力加速度为 $g$ 。\n求地球的第一宇宙速度\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n航天技术的发展是当今各国综合国力的直接体现，近年来，我国的航天技术取得了让世界瞩目的成绩，也引领科技爱好者思索航天技术的发展，有人就提出了一种不用火箭发射人造地球卫星的设想。其设想如下: 如图所示, 在地球上距地心 $h$ 处沿一条弦挖一光滑通道, 在通道的两个出口处 $A$ 和 $B$ 分别将质量为 $M$ 的物体和质量为 $m$ 的待发射卫星同时自由释放, $M \\gg m$, 在中点 $O^{\\prime}$ 弹性正撞后，质量为 $m$ 的物体，即待发射的卫星就会从通道口 $B$ 冲出通道, 设置一个装置, 卫星从 $B$ 冲出就把速度变为沿地球切线方向, 但不改变速度大小, 这样就有可能成功发射卫星。已知地球可视为质量分布均匀\n的球体, 且质量分布均匀的球壳对壳内物体的引力为零, 地球半径为 $R_{0}$, 表面的重力加速度为 $g$ 。\n求地球的第一宇宙速度\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-003.jpg?height=422&width=425&top_left_y=674&top_left_x=356" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_956", "problem": "On $24^{\\text {th }}$ August 2016, astronomers discovered a planet orbiting the closest star to the Sun, Proxima Centauri, situated 4.22 light years away, which fulfils a long-standing dream of science-fiction writers: a world that is close enough for humans to send their first interstellar spacecraft.\n\nAstronomers have noted how the motion of Proxima Centauri changed in the first months of 2016, with the star moving towards and away from the Earth, as seen in the figure below. Sometimes Proxima Centauri is approaching Earth at $5 \\mathrm{~km} \\mathrm{hour}^{-1}-$ normal human walking pace - and at times receding at the same speed. This regular pattern of changing radial velocities caused by an unseen planet, which they named Proxima Centauri B, repeats and results in tiny Doppler shifts in the star's light, making the light appear slightly redder, then bluer.\n\n[figure1]\n\nFrom the radial velocity curve above, determine the period of the planet around Proxima Centauri.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nOn $24^{\\text {th }}$ August 2016, astronomers discovered a planet orbiting the closest star to the Sun, Proxima Centauri, situated 4.22 light years away, which fulfils a long-standing dream of science-fiction writers: a world that is close enough for humans to send their first interstellar spacecraft.\n\nAstronomers have noted how the motion of Proxima Centauri changed in the first months of 2016, with the star moving towards and away from the Earth, as seen in the figure below. Sometimes Proxima Centauri is approaching Earth at $5 \\mathrm{~km} \\mathrm{hour}^{-1}-$ normal human walking pace - and at times receding at the same speed. This regular pattern of changing radial velocities caused by an unseen planet, which they named Proxima Centauri B, repeats and results in tiny Doppler shifts in the star's light, making the light appear slightly redder, then bluer.\n\n[figure1]\n\nFrom the radial velocity curve above, determine the period of the planet around Proxima Centauri.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of days, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6d91a7785df4f4beaa9ag-10.jpg?height=545&width=1602&top_left_y=1007&top_left_x=227" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "days" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1006", "problem": "At a latitude of $52^{\\circ} \\mathrm{N}$ what is the altitude of Polaris above the horizon?\nA: $38^{\\circ}$\nB: $48^{\\circ}$\nC: $52^{\\circ}$\nD: $90^{\\circ}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAt a latitude of $52^{\\circ} \\mathrm{N}$ what is the altitude of Polaris above the horizon?\n\nA: $38^{\\circ}$\nB: $48^{\\circ}$\nC: $52^{\\circ}$\nD: $90^{\\circ}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_254", "problem": "如图所示, 假设在太空中有恒星 A、B 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{l}$, 它们的轨道半径分别为 $r_{A} 、 r_{B}, r_{A}v$\nB: $v^{\\prime}v$\nB: $v^{\\prime}1)$ 。科学家推测在以两星球球心连线为直径的球体空间中均匀分布着暗物质, 设两星球球心连线长度为 $L$, 质量均为 $m$,据此推测,暗物质的质量为( )\nA: $(n-1) m$\nB: $(2 n-1) m$\nC: $\\frac{n-1}{4} m$\nD: $\\frac{n-2}{8} m$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2016 年 12 月 17 口是我国发射“悟空”探测卫星一周年的日子.“悟空”探测卫星的发射为人类对暗物质的研究,做出了重大贡献.假设两颗质量相等的星球绕其球心连线中心转动,理论计算的周期与实际观测周期有出入,且 $\\frac{T_{\\text {理论 }}}{T_{\\text {观测 }}}=\\frac{\\sqrt{n}}{1}(n>1)$ 。科学家推测在以两星球球心连线为直径的球体空间中均匀分布着暗物质, 设两星球球心连线长度为 $L$, 质量均为 $m$,据此推测,暗物质的质量为( )\n\nA: $(n-1) m$\nB: $(2 n-1) m$\nC: $\\frac{n-1}{4} m$\nD: $\\frac{n-2}{8} m$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_98", "problem": "2020 年 11 月 24 日, 长征五号遥五运载火箭搭载嫦娥五号探测器成功发射升空并将其送入轨道, 11 月 28 日, 嫦娥五号进入环月轨道飞行, 12 月 17 日凌晨, 嫦娥五号返回器携带月壤着陆地球。假设嫦娥五号环绕月球飞行时, 在距月球表面高度为 $h$ 处,绕月球做匀速圆周运动 (不计周围其他天体的影响), 测出其飞行周期 $T$, 已知引力常量 $G$ 和月球半径 $R$, 则下列说法正确的是 ( )\nA: 嫦娥五号绕月球飞行的线速度为 $\\frac{2 \\pi(R+h)}{T}$\nB: 月球的质量为 $\\frac{4 \\pi^{2}(R+h)^{2}}{G T^{2}}$\nC: 月球的第一宇宙速度为 $\\frac{2 \\pi(R+h)}{T} \\sqrt{\\frac{R+h}{R}}$\nD: 月球表面的重力加速度为 $\\frac{4 \\pi^{2}(R+h)^{3}}{R^{2} T^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2020 年 11 月 24 日, 长征五号遥五运载火箭搭载嫦娥五号探测器成功发射升空并将其送入轨道, 11 月 28 日, 嫦娥五号进入环月轨道飞行, 12 月 17 日凌晨, 嫦娥五号返回器携带月壤着陆地球。假设嫦娥五号环绕月球飞行时, 在距月球表面高度为 $h$ 处,绕月球做匀速圆周运动 (不计周围其他天体的影响), 测出其飞行周期 $T$, 已知引力常量 $G$ 和月球半径 $R$, 则下列说法正确的是 ( )\n\nA: 嫦娥五号绕月球飞行的线速度为 $\\frac{2 \\pi(R+h)}{T}$\nB: 月球的质量为 $\\frac{4 \\pi^{2}(R+h)^{2}}{G T^{2}}$\nC: 月球的第一宇宙速度为 $\\frac{2 \\pi(R+h)}{T} \\sqrt{\\frac{R+h}{R}}$\nD: 月球表面的重力加速度为 $\\frac{4 \\pi^{2}(R+h)^{3}}{R^{2} T^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1051", "problem": "In July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.a. Ignoring the effects of air resistance, the weight of the rocket, and assuming 1-D motion only\n\niii. Sketch an acceleration-time graph of the journey from lift-off to reaching the parking orbit. Give accelerations in units of $g_{0}$. Assume the time between one stage finishing, detaching, and ignition of the next stage is negligible (i.e. you will have discontinuities in the graph at the end of each stage).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIn July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.\n\nproblem:\na. Ignoring the effects of air resistance, the weight of the rocket, and assuming 1-D motion only\n\niii. Sketch an acceleration-time graph of the journey from lift-off to reaching the parking orbit. Give accelerations in units of $g_{0}$. Assume the time between one stage finishing, detaching, and ignition of the next stage is negligible (i.e. you will have discontinuities in the graph at the end of each stage).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-04.jpg?height=1010&width=1508&top_left_y=543&top_left_x=271", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-06.jpg?height=800&width=1586&top_left_y=518&top_left_x=240", "https://cdn.mathpix.com/cropped/2024_03_14_e83ffef93b1fee5a6c8bg-02.jpg?height=505&width=1308&top_left_y=850&top_left_x=477" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_314", "problem": "某地面卫星接收站的纬度为 $\\theta(\\theta>0)$. 已知地球半径为 $R$, 重力加速度为 $g$, 自转周期为 $T$. 光速为 $c$, 则地球同步卫星发射的电磁波到该接收站的时间不小于 ( )\nA: $\\frac{\\sqrt[3]{\\frac{R^{2} T^{2} g}{4 \\pi^{2}}}}{c}$\nB: $\\sqrt[3]{\\frac{R^{2} T^{2} g}{4 \\pi^{2}}}-R$\nC: $\\frac{\\sqrt{R^{2}+r^{2}+2 R r \\cos \\theta}}{c}\\left(\\right.$ 其中 $r=\\sqrt[3]{\\frac{g R^{2} T^{2}}{4 \\pi^{2}}}$ )\nD: $\\frac{\\sqrt{R^{2}+r^{2}-2 R r \\cos \\theta}}{c}\\left(\\right.$ 其中 $r=\\sqrt[3]{\\frac{g R^{2} T^{2}}{4 \\pi^{2}}}$ )\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n某地面卫星接收站的纬度为 $\\theta(\\theta>0)$. 已知地球半径为 $R$, 重力加速度为 $g$, 自转周期为 $T$. 光速为 $c$, 则地球同步卫星发射的电磁波到该接收站的时间不小于 ( )\n\nA: $\\frac{\\sqrt[3]{\\frac{R^{2} T^{2} g}{4 \\pi^{2}}}}{c}$\nB: $\\sqrt[3]{\\frac{R^{2} T^{2} g}{4 \\pi^{2}}}-R$\nC: $\\frac{\\sqrt{R^{2}+r^{2}+2 R r \\cos \\theta}}{c}\\left(\\right.$ 其中 $r=\\sqrt[3]{\\frac{g R^{2} T^{2}}{4 \\pi^{2}}}$ )\nD: $\\frac{\\sqrt{R^{2}+r^{2}-2 R r \\cos \\theta}}{c}\\left(\\right.$ 其中 $r=\\sqrt[3]{\\frac{g R^{2} T^{2}}{4 \\pi^{2}}}$ )\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-78.jpg?height=285&width=425&top_left_y=1448&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_534", "problem": "2016 年 10 月 19 日 3 时 31 分, 神舟十一号载人飞船与天宫二号空间实验室成功实\n现自动交会对接. 若对接前神舟十一号和天宫二号分别在图示轨道上绕地球同向 (天宫二号在前）做匀速圆周运动, 则\n\n[图1]\nA: 神舟十一号的线速度小于天宫二号的线速度\nB: 神舟十一号的周期大于天宫二号的周期\nC: 神舟十一号的角速度小于天宫二号的角速度\nD: 神舟十一号可在轨道上加速后实现与天宫二号的对接\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2016 年 10 月 19 日 3 时 31 分, 神舟十一号载人飞船与天宫二号空间实验室成功实\n现自动交会对接. 若对接前神舟十一号和天宫二号分别在图示轨道上绕地球同向 (天宫二号在前）做匀速圆周运动, 则\n\n[图1]\n\nA: 神舟十一号的线速度小于天宫二号的线速度\nB: 神舟十一号的周期大于天宫二号的周期\nC: 神舟十一号的角速度小于天宫二号的角速度\nD: 神舟十一号可在轨道上加速后实现与天宫二号的对接\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-007.jpg?height=311&width=516&top_left_y=324&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_735", "problem": "“神舟十三号”飞船开始在半径为 $r_{1}$ 的圆轨道 $\\mathrm{I}$ 上运行, 运行周期为 $T_{1}$, 在 $A$ 点通过变轨操作后进入粗圆轨道II运动, 沿轨道II运动到远地点 $B$ 时正好与处于半径为 $r_{3}$ 的圆轨道III上的核心舱对接, $A$ 为粗圆轨道II的近地点。假设飞船质量始终不变, 关于飞船的运动, 下列说法正确的是（ ）\n\n[图1]\nA: 沿轨道I运动到 $A$ 时的速率大于沿轨道II运动到 $B$ 时的速率\nB: 沿轨道I运行时的机械能等于沿轨道II运行时的机械能\nC: 沿轨道II运行的周期为 $T_{1} \\sqrt{\\left(\\frac{r_{1}+r_{3}}{2 r_{1}}\\right)^{3}}$\nD: 沿轨道I运动到 $A$ 点时的加速度小于沿轨道II运动到 $B$ 点时的加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n“神舟十三号”飞船开始在半径为 $r_{1}$ 的圆轨道 $\\mathrm{I}$ 上运行, 运行周期为 $T_{1}$, 在 $A$ 点通过变轨操作后进入粗圆轨道II运动, 沿轨道II运动到远地点 $B$ 时正好与处于半径为 $r_{3}$ 的圆轨道III上的核心舱对接, $A$ 为粗圆轨道II的近地点。假设飞船质量始终不变, 关于飞船的运动, 下列说法正确的是（ ）\n\n[图1]\n\nA: 沿轨道I运动到 $A$ 时的速率大于沿轨道II运动到 $B$ 时的速率\nB: 沿轨道I运行时的机械能等于沿轨道II运行时的机械能\nC: 沿轨道II运行的周期为 $T_{1} \\sqrt{\\left(\\frac{r_{1}+r_{3}}{2 r_{1}}\\right)^{3}}$\nD: 沿轨道I运动到 $A$ 点时的加速度小于沿轨道II运动到 $B$ 点时的加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-059.jpg?height=522&width=531&top_left_y=1004&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1203", "problem": "The Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\\text {peri }}=9.86 R_{\\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft.\n[figure1]\n\nFigure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman.\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nGiven that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.\n\nClose to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2.\n\n[figure2]\n\nFigure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL.\n\nWhen considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\\angle y c z$ in the figure, such that the shaded areas are the same).\n\n[figure3]\n\nFigure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia.\n\n\nThe eccentric anomaly can be related to the mean anomaly through Kepler's Equation,\n\n$$\nM=E-e \\sin E \\text {. }\n$$c. After the first flyby of Venus on 3rd October 2018 it was moved into an orbit with a 150 day period, and the subsequent first perihelion on 6th November 2018 was at a distance of $35.7 R_{\\odot}$. Given its mass at launch was $685 \\mathrm{~kg}$, calculate the total amount of energy that had to be lost by the probe to get from this first orbit (ignoring the orbital properties prior to the Venus flyby) to the final orbit. Ignore any change in the mass of the probe due to burning fuel.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\\text {peri }}=9.86 R_{\\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft.\n[figure1]\n\nFigure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman.\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nGiven that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.\n\nClose to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2.\n\n[figure2]\n\nFigure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL.\n\nWhen considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\\angle y c z$ in the figure, such that the shaded areas are the same).\n\n[figure3]\n\nFigure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia.\n\n\nThe eccentric anomaly can be related to the mean anomaly through Kepler's Equation,\n\n$$\nM=E-e \\sin E \\text {. }\n$$\n\nproblem:\nc. After the first flyby of Venus on 3rd October 2018 it was moved into an orbit with a 150 day period, and the subsequent first perihelion on 6th November 2018 was at a distance of $35.7 R_{\\odot}$. Given its mass at launch was $685 \\mathrm{~kg}$, calculate the total amount of energy that had to be lost by the probe to get from this first orbit (ignoring the orbital properties prior to the Venus flyby) to the final orbit. Ignore any change in the mass of the probe due to burning fuel.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~J}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-04.jpg?height=708&width=1438&top_left_y=694&top_left_x=318", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-05.jpg?height=411&width=1539&top_left_y=383&top_left_x=264", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-05.jpg?height=603&width=714&top_left_y=1429&top_left_x=677" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~J}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_828", "problem": "Consider Galaxies $\\mathrm{A}$ and $\\mathrm{B}$, both of which have radius $R$. At a distance $R$ from its center, Galaxy A's rotational velocity is equal to $v$. Meanwhile, Galaxy B's radial velocity dispersion is also equal to $v$. However, galaxy A is spiral while galaxy B is spherical elliptical and composed of uniform, evenly-spaced stars. Calculate the masses of both galaxies. (Answer choices are listed as $\\left.m_{A} ; m_{B}\\right)$.\nA: $v^{2} R / G ; v^{2} R / G$\nB: $v^{2} R / G ; 5 / 6 v^{2} R / G$\nC: $v^{2} R / G ; 5 / 4 v^{2} R / G$\nD: $v^{2} R / G ; 5 v^{2} R / G$\nE: $5 / 2 v^{2} R / G ; v^{2} R / G$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nConsider Galaxies $\\mathrm{A}$ and $\\mathrm{B}$, both of which have radius $R$. At a distance $R$ from its center, Galaxy A's rotational velocity is equal to $v$. Meanwhile, Galaxy B's radial velocity dispersion is also equal to $v$. However, galaxy A is spiral while galaxy B is spherical elliptical and composed of uniform, evenly-spaced stars. Calculate the masses of both galaxies. (Answer choices are listed as $\\left.m_{A} ; m_{B}\\right)$.\n\nA: $v^{2} R / G ; v^{2} R / G$\nB: $v^{2} R / G ; 5 / 6 v^{2} R / G$\nC: $v^{2} R / G ; 5 / 4 v^{2} R / G$\nD: $v^{2} R / G ; 5 v^{2} R / G$\nE: $5 / 2 v^{2} R / G ; v^{2} R / G$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_963", "problem": "Currently, Polaris is very close to the north celestial pole (the projection of the Earth's rotational axis on the sky) and so all other stars appear to rotate around it. However, this axis is drawing out a large circle in the sky with an angular radius of $23.44^{\\circ}$ so Polaris will only temporarily be the pole star (see Figure 2). This precession of the rotational axis is mainly driven by the gravitational pull of the Moon and the Sun.\n[figure1]\n\nFigure 2: Top left: The Earth's rotational axis itself rotates slowly (white circle), in what is known as axial precession. Credit: David Battisti / University of Washington.\n\nTop right: Due to precession, the pole star has changed over time. About 5000 years ago, the star Thuban in the constellation of Draco was the pole star. Credit: Richard W. Pogge / Ohio State University.\n\nBottom: The position of the Sun at the spring equinox (where the celestial equator meets the ecliptic) has also changed over the same period, moving from Aries to Pisces. Credit: Guy Ottewell / Universal Workshop.\n\nAnother consequence is that the position of the Sun at the equinoxes varies slightly, moving slowly westwards. This gives rise to two definitions of a year:\n\n- a sidereal year (the time taken for the Earth to orbit the Sun once with respect to the background stars) $=365.256363$ days\n- a tropical year (the time taken for the Sun to return to the same position in the cycle of the seasons) $=365.242190$ days\n\nThe Gregorian calendar is a 400-year cycle with a system of leap years. The rule is: \"every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100 , but these centurial years are leap years if they are exactly divisible by 400. .\"\n\nBy working out the average length of a year in the Gregorian calendar, is it closer to the sidereal or the tropical year?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nCurrently, Polaris is very close to the north celestial pole (the projection of the Earth's rotational axis on the sky) and so all other stars appear to rotate around it. However, this axis is drawing out a large circle in the sky with an angular radius of $23.44^{\\circ}$ so Polaris will only temporarily be the pole star (see Figure 2). This precession of the rotational axis is mainly driven by the gravitational pull of the Moon and the Sun.\n[figure1]\n\nFigure 2: Top left: The Earth's rotational axis itself rotates slowly (white circle), in what is known as axial precession. Credit: David Battisti / University of Washington.\n\nTop right: Due to precession, the pole star has changed over time. About 5000 years ago, the star Thuban in the constellation of Draco was the pole star. Credit: Richard W. Pogge / Ohio State University.\n\nBottom: The position of the Sun at the spring equinox (where the celestial equator meets the ecliptic) has also changed over the same period, moving from Aries to Pisces. Credit: Guy Ottewell / Universal Workshop.\n\nAnother consequence is that the position of the Sun at the equinoxes varies slightly, moving slowly westwards. This gives rise to two definitions of a year:\n\n- a sidereal year (the time taken for the Earth to orbit the Sun once with respect to the background stars) $=365.256363$ days\n- a tropical year (the time taken for the Sun to return to the same position in the cycle of the seasons) $=365.242190$ days\n\nThe Gregorian calendar is a 400-year cycle with a system of leap years. The rule is: \"every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100 , but these centurial years are leap years if they are exactly divisible by 400. .\"\n\nBy working out the average length of a year in the Gregorian calendar, is it closer to the sidereal or the tropical year?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_2c19fdb17927c588761dg-07.jpg?height=890&width=1144&top_left_y=574&top_left_x=456" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_353", "problem": "“亚洲一号”地球同步通讯卫星的质量是 $1.2 \\mathrm{t}$, 下列有关它的说法正确的是 ( )\nA: 若将它的质量增加为 $2.4 \\mathrm{t}$, 其同步轨道半径变为原来的 2 倍\nB: 它的运行速度小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 它可以绕过北京的正上方, 所以我国能利用其进行电视转播\nD: 它的周期是 $24 \\mathrm{~h}$, 其轨道平面与赤道平面重合且距地面高度一定\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n“亚洲一号”地球同步通讯卫星的质量是 $1.2 \\mathrm{t}$, 下列有关它的说法正确的是 ( )\n\nA: 若将它的质量增加为 $2.4 \\mathrm{t}$, 其同步轨道半径变为原来的 2 倍\nB: 它的运行速度小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 它可以绕过北京的正上方, 所以我国能利用其进行电视转播\nD: 它的周期是 $24 \\mathrm{~h}$, 其轨道平面与赤道平面重合且距地面高度一定\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_796", "problem": "Edwin Hubble published in 1929 the paper \"A Relation Between Distance and Extragalactic Nebulae\" explaining his find that there exists a linear relationship between the radial velocities and distances for extragalactic objects. In the following graph you can see his data used in the paper. The units for the velocity are in $\\mathrm{km} / \\mathrm{s}$. (notice the \"/ $\\mathrm{s}$ \" is missing from his original graph). What is the value for the Hubble constant that he derived from this data? Use the linear fit to the data marked with the continuous line to derive your estimation.\n\n[figure1]\nA: $500 \\mathrm{~km} / \\mathrm{s} / \\mathrm{pc}$\nB: $72 \\mathrm{~km} / \\mathrm{s} / \\mathrm{Mpc}$\nC: $500 \\mathrm{~km} / \\mathrm{s} / \\mathrm{Mpc}$\nD: $50 \\mathrm{~km} / \\mathrm{s} / \\mathrm{Mpc}$\nE: $67 \\mathrm{~km} / \\mathrm{s} / \\mathrm{Mpc}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nEdwin Hubble published in 1929 the paper \"A Relation Between Distance and Extragalactic Nebulae\" explaining his find that there exists a linear relationship between the radial velocities and distances for extragalactic objects. In the following graph you can see his data used in the paper. The units for the velocity are in $\\mathrm{km} / \\mathrm{s}$. (notice the \"/ $\\mathrm{s}$ \" is missing from his original graph). What is the value for the Hubble constant that he derived from this data? Use the linear fit to the data marked with the continuous line to derive your estimation.\n\n[figure1]\n\nA: $500 \\mathrm{~km} / \\mathrm{s} / \\mathrm{pc}$\nB: $72 \\mathrm{~km} / \\mathrm{s} / \\mathrm{Mpc}$\nC: $500 \\mathrm{~km} / \\mathrm{s} / \\mathrm{Mpc}$\nD: $50 \\mathrm{~km} / \\mathrm{s} / \\mathrm{Mpc}$\nE: $67 \\mathrm{~km} / \\mathrm{s} / \\mathrm{Mpc}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_56d1b5239b3c83be7aceg-15.jpg?height=1055&width=1534&top_left_y=1012&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_972", "problem": "Which of the following is not a zodiacal constellation, according to the astronomical definition?\n[figure1]\nA: Aquila\nB: Aquarius\nC: Ophiucus\nD: Pisces\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhich of the following is not a zodiacal constellation, according to the astronomical definition?\n[figure1]\n\nA: Aquila\nB: Aquarius\nC: Ophiucus\nD: Pisces\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6d91a7785df4f4beaa9ag-04.jpg?height=458&width=1722&top_left_y=2152&top_left_x=223" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_588", "problem": "2021 年 7 月和 10 月, SpaceX 公司星链卫星两次侵入中国天宫空间站轨道, 为保证空间站内工作的三名航天员生命安全, 中方不得不调整轨道高度, 紧急避碰。其中星链 -2305 号卫星, 采取连续变轨模式接近中国空间站, 中国发现且规避后, 该卫星轨道又重新回到正常轨道。已知中国空间站在高度 390 千米附近的近圆轨道，轨道倾角 $41.58^{\\circ}$, 而星链卫星在高度为 550 千米附近的近圆轨道, 倾角为 $53^{\\circ}$ 。已知地球半径为 $6400 \\mathrm{~km}$, 引力常量为 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$ 。下列说法正确的是（）\nA: 中国空间站的运行速度约为 11.68 千米/秒\nB: 由题目所给数据可以计算出地球的密度\nC: 星链卫星在正常圆轨道需减速才能降到中国空间站所在高度\nD: 星链卫星保持轨道平面不变, 降至 390 公里附近圆轨道时, 其速度与中国空间站速度接近, 不会相撞（轨道倾角是指卫星轨道面与地球赤道面之间的夹角）\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2021 年 7 月和 10 月, SpaceX 公司星链卫星两次侵入中国天宫空间站轨道, 为保证空间站内工作的三名航天员生命安全, 中方不得不调整轨道高度, 紧急避碰。其中星链 -2305 号卫星, 采取连续变轨模式接近中国空间站, 中国发现且规避后, 该卫星轨道又重新回到正常轨道。已知中国空间站在高度 390 千米附近的近圆轨道，轨道倾角 $41.58^{\\circ}$, 而星链卫星在高度为 550 千米附近的近圆轨道, 倾角为 $53^{\\circ}$ 。已知地球半径为 $6400 \\mathrm{~km}$, 引力常量为 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$ 。下列说法正确的是（）\n\nA: 中国空间站的运行速度约为 11.68 千米/秒\nB: 由题目所给数据可以计算出地球的密度\nC: 星链卫星在正常圆轨道需减速才能降到中国空间站所在高度\nD: 星链卫星保持轨道平面不变, 降至 390 公里附近圆轨道时, 其速度与中国空间站速度接近, 不会相撞（轨道倾角是指卫星轨道面与地球赤道面之间的夹角）\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_878", "problem": "Consider that the absolute magnitude of a star is $m_{0}$. Imagine that the first star gets split into $\\mathrm{N}$ smaller identical stars with the same temperature and average densities as the initial star, and that the sum of the masses of all $\\mathrm{N}$ smaller stars is equal to the initial star's mass (i.e., total mass is conserved). What is the total combined absolute magnitude $(\\mathrm{m})$ of all the $\\mathrm{N}$ stars assuming that none of the stars obstruct each other's light (i.e. their luminosities add linearly)?\nA: $m=m_{0}-\\log (N)$\nB: $m=m_{0}-2.5 \\log (N)$\nC: $m=m_{0}-\\frac{2.5}{3} \\log (N)$\nD: $m=m_{0}-\\frac{2.5}{N}$\nE: $m=m_{0}-2.5 \\mathrm{~N}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nConsider that the absolute magnitude of a star is $m_{0}$. Imagine that the first star gets split into $\\mathrm{N}$ smaller identical stars with the same temperature and average densities as the initial star, and that the sum of the masses of all $\\mathrm{N}$ smaller stars is equal to the initial star's mass (i.e., total mass is conserved). What is the total combined absolute magnitude $(\\mathrm{m})$ of all the $\\mathrm{N}$ stars assuming that none of the stars obstruct each other's light (i.e. their luminosities add linearly)?\n\nA: $m=m_{0}-\\log (N)$\nB: $m=m_{0}-2.5 \\log (N)$\nC: $m=m_{0}-\\frac{2.5}{3} \\log (N)$\nD: $m=m_{0}-\\frac{2.5}{N}$\nE: $m=m_{0}-2.5 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_863", "problem": "A comet is approaching our solar system from the depths of space with a velocity of $10000 \\mathrm{~m} / \\mathrm{s}$, and if it continues moving in a straight line on its current trajectory, it will just barely graze the surface of the Sun! What is the eccentricity of the comet's orbit?\nA: 1.00014\nB: 1.000014\nC: 1.0000014\nD: 1.00000014\nE: 1.000000014\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA comet is approaching our solar system from the depths of space with a velocity of $10000 \\mathrm{~m} / \\mathrm{s}$, and if it continues moving in a straight line on its current trajectory, it will just barely graze the surface of the Sun! What is the eccentricity of the comet's orbit?\n\nA: 1.00014\nB: 1.000014\nC: 1.0000014\nD: 1.00000014\nE: 1.000000014\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_467", "problem": "利用金星凌日现象, 我们可以估算出地球与太阳之间的平均距离。日地平均距离也被定义为 1 个天文单位 (1A.U.), 是天文学中常用的距离单位。\n\n金星轨道在地球轨道内侧, 某些特殊时刻, 地球、金星、太阳恰在一条直线上, 这时从地球上可以看到金星就像一个小黑点一样在太阳表面缓慢移动, 如图甲所示, 天文学称之为“金星凌日”。在地球上的不同地点, 比如图乙中的 $A 、 B$ 两点, 它们在同一时刻观察到的金星在日面上的位置是不同的，我们分别记为 $A^{\\prime} 、 B^{\\prime}$ 。\n\n设金星与太阳的距离为 $k$ 倍日地距离, 即 $k \\mathrm{~A}$. U. 可测得地球上 $A 、 B$ 之间的距离为 $l$, 估算 $A^{\\prime} 、 B^{\\prime}$ 在太阳表面的真实距离\n\n[图1]\n\n甲\n\n[图2]\n\n丙\n\n[图3]\n\n乙", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n利用金星凌日现象, 我们可以估算出地球与太阳之间的平均距离。日地平均距离也被定义为 1 个天文单位 (1A.U.), 是天文学中常用的距离单位。\n\n金星轨道在地球轨道内侧, 某些特殊时刻, 地球、金星、太阳恰在一条直线上, 这时从地球上可以看到金星就像一个小黑点一样在太阳表面缓慢移动, 如图甲所示, 天文学称之为“金星凌日”。在地球上的不同地点, 比如图乙中的 $A 、 B$ 两点, 它们在同一时刻观察到的金星在日面上的位置是不同的，我们分别记为 $A^{\\prime} 、 B^{\\prime}$ 。\n\n设金星与太阳的距离为 $k$ 倍日地距离, 即 $k \\mathrm{~A}$. U. 可测得地球上 $A 、 B$ 之间的距离为 $l$, 估算 $A^{\\prime} 、 B^{\\prime}$ 在太阳表面的真实距离\n\n[图1]\n\n甲\n\n[图2]\n\n丙\n\n[图3]\n\n乙\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-038.jpg?height=214&width=240&top_left_y=201&top_left_x=474", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-038.jpg?height=249&width=280&top_left_y=178&top_left_x=751", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-038.jpg?height=205&width=853&top_left_y=480&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_67", "problem": "银河系处于某超星系团的边缘, 已知银河系距离星系团中心约 2 亿光年, 绕星系团中心运行的公转周期约为 1000 亿年, 引力常量 $\\mathrm{G}=6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$, 根据上述数据可估算\nA: 银河系绕超星系团中心运动的线速度\nB: 银河系绕超星系团中心运动的加速度\nC: 银河系的质量\nD: 超星系团的质量\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n银河系处于某超星系团的边缘, 已知银河系距离星系团中心约 2 亿光年, 绕星系团中心运行的公转周期约为 1000 亿年, 引力常量 $\\mathrm{G}=6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$, 根据上述数据可估算\n\nA: 银河系绕超星系团中心运动的线速度\nB: 银河系绕超星系团中心运动的加速度\nC: 银河系的质量\nD: 超星系团的质量\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_391", "problem": "2020 年 6 月 23 日上午, 北斗三号全球卫星导航系统的“收官之星”成功发射, 标志着北斗三号全球卫星导航系统全球星座组网部署最后一步完成。中国北斗, 将点亮世界卫星导航的天空。“收官之星”最后静止在地面上空（与地面保持相对静止）。该卫星距地面的高度为 $h$, 已知地球的半径为 $R$, 地球表面的重力加速度为 $g$, 万有引力常量为 $G$ 。由此可知（）\nA: “收官之星”运动的周期为 $2 \\pi \\sqrt{\\frac{R}{g}}$\nB: “收官之星”运动的加速度为 $\\frac{g R}{R+h}$\nC: “收官之星”运动的轨道一定与赤道不共面\nD: 地球的平均密度为 $\\frac{3 g}{4 \\pi G R}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年 6 月 23 日上午, 北斗三号全球卫星导航系统的“收官之星”成功发射, 标志着北斗三号全球卫星导航系统全球星座组网部署最后一步完成。中国北斗, 将点亮世界卫星导航的天空。“收官之星”最后静止在地面上空（与地面保持相对静止）。该卫星距地面的高度为 $h$, 已知地球的半径为 $R$, 地球表面的重力加速度为 $g$, 万有引力常量为 $G$ 。由此可知（）\n\nA: “收官之星”运动的周期为 $2 \\pi \\sqrt{\\frac{R}{g}}$\nB: “收官之星”运动的加速度为 $\\frac{g R}{R+h}$\nC: “收官之星”运动的轨道一定与赤道不共面\nD: 地球的平均密度为 $\\frac{3 g}{4 \\pi G R}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1141", "problem": "In July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.a. Ignoring the effects of air resistance, the weight of the rocket, and assuming 1-D motion only\n\ni. Show that the thrust generated by the S-IC is about $3.3 \\times 10^{7} \\mathrm{~N}$ and hence calculate the acceleration experienced by the astronauts firstly at lift-off and secondly when the S-IC finishes its burn (ignore that the S-IC ignites a few seconds before lift off). Give your answer in units of $g_{0}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIn July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.\n\nproblem:\na. Ignoring the effects of air resistance, the weight of the rocket, and assuming 1-D motion only\n\ni. Show that the thrust generated by the S-IC is about $3.3 \\times 10^{7} \\mathrm{~N}$ and hence calculate the acceleration experienced by the astronauts firstly at lift-off and secondly when the S-IC finishes its burn (ignore that the S-IC ignites a few seconds before lift off). Give your answer in units of $g_{0}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-04.jpg?height=1010&width=1508&top_left_y=543&top_left_x=271", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-06.jpg?height=800&width=1586&top_left_y=518&top_left_x=240" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_9", "problem": "在如图所示的双星系统中, $A 、 B$ 两个恒星靠着相互之间的引力正在做匀速圆周运\n\n动, 已知恒星 $A$ 的质量为太阳质量的 29 倍, 恒星 $B$ 的质量为太阳质量的 36 倍, 两星之间的距离 $L=2 \\times 10^{5} \\mathrm{~m}$, 太阳质量 $M=2 \\times 10^{30} \\mathrm{~kg}$, 引力常量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}, \\pi^{2}$ $=10$. 若两星在环绕过程中会辐射出引力波, 该引力波的频率与两星做圆周运动的频率具有相同的数量级, 则根据题目所给信息估算该引力波频率的数量级是( )\n\n[图1]\nA: $10^{2} \\mathrm{~Hz}$\nB: $10^{4} \\mathrm{~Hz}$\nC: $10^{6} \\mathrm{~Hz}$\nD: $10^{8} \\mathrm{~Hz}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n在如图所示的双星系统中, $A 、 B$ 两个恒星靠着相互之间的引力正在做匀速圆周运\n\n动, 已知恒星 $A$ 的质量为太阳质量的 29 倍, 恒星 $B$ 的质量为太阳质量的 36 倍, 两星之间的距离 $L=2 \\times 10^{5} \\mathrm{~m}$, 太阳质量 $M=2 \\times 10^{30} \\mathrm{~kg}$, 引力常量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}, \\pi^{2}$ $=10$. 若两星在环绕过程中会辐射出引力波, 该引力波的频率与两星做圆周运动的频率具有相同的数量级, 则根据题目所给信息估算该引力波频率的数量级是( )\n\n[图1]\n\nA: $10^{2} \\mathrm{~Hz}$\nB: $10^{4} \\mathrm{~Hz}$\nC: $10^{6} \\mathrm{~Hz}$\nD: $10^{8} \\mathrm{~Hz}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-45.jpg?height=214&width=414&top_left_y=1703&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_932", "problem": "When two objects of unequal mass orbit around each other, they both orbit around a barycentre - this is the name given to the location of the centre of mass of the system. The masses of both objects, and the distance between their centres, affects the position of their barycentre. Imagine two objects, Object 1 and Object 2, with masses $m_{1}$ and $m_{2}$ respectively, and the average distance between the centre of both objects is $a$, then the average distance from the centre of Object 1 to the barycentre, $r$, is given by the formula:\n\n$$\nr=a \\frac{m_{2}}{m_{1}+m_{2}}\n$$\n\nVerify that the barycentre of the Sun-Jupiter system lies outside the Sun (i.e. $r>\\mathrm{R}_{\\odot}$ ), given that the mass of Jupiter is $1.90 \\times 10^{27} \\mathrm{~kg}$ and it is on average $5.20 \\mathrm{AU}$ from the Sun.\n\n[figure1]\n\nFigure 4: The relative positions of Pluto, Charon and the system's barycentre.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nWhen two objects of unequal mass orbit around each other, they both orbit around a barycentre - this is the name given to the location of the centre of mass of the system. The masses of both objects, and the distance between their centres, affects the position of their barycentre. Imagine two objects, Object 1 and Object 2, with masses $m_{1}$ and $m_{2}$ respectively, and the average distance between the centre of both objects is $a$, then the average distance from the centre of Object 1 to the barycentre, $r$, is given by the formula:\n\n$$\nr=a \\frac{m_{2}}{m_{1}+m_{2}}\n$$\n\nVerify that the barycentre of the Sun-Jupiter system lies outside the Sun (i.e. $r>\\mathrm{R}_{\\odot}$ ), given that the mass of Jupiter is $1.90 \\times 10^{27} \\mathrm{~kg}$ and it is on average $5.20 \\mathrm{AU}$ from the Sun.\n\n[figure1]\n\nFigure 4: The relative positions of Pluto, Charon and the system's barycentre.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c744602885fab54c0985g-9.jpg?height=306&width=1110&top_left_y=952&top_left_x=473" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_711", "problem": "如图甲, “星下点” 是指卫星和地心连线与地球表面的交点。图乙是航天控制中心大屏幕上显示卫星 FZ01 的“星下点”在一段时间内的轨迹, 已知地球同步卫星的轨道半径为 $r, \\mathrm{FZ} 01$ 绕行方向与地球自转方向一致, 则下列说法正确的是 ( )\n\n[图1]\n\n甲\n\n[图2]\n\n乙\nA: 卫星 FZ01 的轨道半径约为 $\\frac{r}{2}$\nB: 卫星 FZ01 的轨道半径约为 $\\frac{r}{5}$\nC: 卫星 FZ01 可以记录到北极点的气候变化\nD: 卫星 FZ01 不可以记录到北极点的气候变化\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图甲, “星下点” 是指卫星和地心连线与地球表面的交点。图乙是航天控制中心大屏幕上显示卫星 FZ01 的“星下点”在一段时间内的轨迹, 已知地球同步卫星的轨道半径为 $r, \\mathrm{FZ} 01$ 绕行方向与地球自转方向一致, 则下列说法正确的是 ( )\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\nA: 卫星 FZ01 的轨道半径约为 $\\frac{r}{2}$\nB: 卫星 FZ01 的轨道半径约为 $\\frac{r}{5}$\nC: 卫星 FZ01 可以记录到北极点的气候变化\nD: 卫星 FZ01 不可以记录到北极点的气候变化\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-033.jpg?height=359&width=354&top_left_y=777&top_left_x=337", "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-033.jpg?height=365&width=762&top_left_y=754&top_left_x=727", "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-034.jpg?height=265&width=505&top_left_y=156&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1154", "problem": "It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1).\n[figure1]\n\nFigure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\\sim 40^{\\circ}$. Credit: Daniel V. Schroeder / Weber State University.\n\nRight: The same idea but viewed from Iceland at a latitude of $65^{\\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland.\n\nDuring the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees.\n\nA simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\\delta$, is given as:\n\n$$\nh=-\\left(90^{\\circ}-\\phi\\right) \\cos (A)+\\delta\n$$\n\nThe solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\\delta=+23.44^{\\circ}$ at the June solstice (roughly $21^{\\text {st }}$ June) to a minimum of $\\delta=-23.44^{\\circ}$ on the December solstice (roughly $21^{\\text {st }}$ December).\n\nIt can be shown using spherical trigonometry that the precise model connecting $\\delta, h, \\phi$ and $A$ is:\n\n$$\n\\sin (\\delta)=\\sin (h) \\sin (\\phi)+\\cos (h) \\cos (\\phi) \\cos (A) .\n$$\n\nUsing the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2.\n\n[figure2]\n\nFigure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia.\n\nBy using further spherical trigonometry, we can derive a second helpful equation in the precise model:\n\n$$\n\\sin (h)=\\sin (\\phi) \\sin (\\delta)+\\cos (\\phi) \\cos (\\delta) \\cos (H)\n$$\n\nHere, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\\circ}=24^{\\mathrm{h}}$.d. This exam is being taken on $24^{\\text {th }}$ January and is 3 hours long.\n\ni. Estimate the solar declination on this date.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIt is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1).\n[figure1]\n\nFigure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\\sim 40^{\\circ}$. Credit: Daniel V. Schroeder / Weber State University.\n\nRight: The same idea but viewed from Iceland at a latitude of $65^{\\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland.\n\nDuring the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees.\n\nA simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\\delta$, is given as:\n\n$$\nh=-\\left(90^{\\circ}-\\phi\\right) \\cos (A)+\\delta\n$$\n\nThe solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\\delta=+23.44^{\\circ}$ at the June solstice (roughly $21^{\\text {st }}$ June) to a minimum of $\\delta=-23.44^{\\circ}$ on the December solstice (roughly $21^{\\text {st }}$ December).\n\nIt can be shown using spherical trigonometry that the precise model connecting $\\delta, h, \\phi$ and $A$ is:\n\n$$\n\\sin (\\delta)=\\sin (h) \\sin (\\phi)+\\cos (h) \\cos (\\phi) \\cos (A) .\n$$\n\nUsing the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2.\n\n[figure2]\n\nFigure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia.\n\nBy using further spherical trigonometry, we can derive a second helpful equation in the precise model:\n\n$$\n\\sin (h)=\\sin (\\phi) \\sin (\\delta)+\\cos (\\phi) \\cos (\\delta) \\cos (H)\n$$\n\nHere, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\\circ}=24^{\\mathrm{h}}$.\n\nproblem:\nd. This exam is being taken on $24^{\\text {th }}$ January and is 3 hours long.\n\ni. Estimate the solar declination on this date.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\circ, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-04.jpg?height=668&width=1478&top_left_y=523&top_left_x=290", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-05.jpg?height=648&width=1234&top_left_y=738&top_left_x=385" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\circ" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_288", "problem": "经长期观测发现, $A$ 行星运行的轨道半径为 $\\mathrm{R}_{0}$, 周期为 $\\mathrm{T}_{0}$, 但其实际运行的轨道与圆轨道总存在一些偏离, 且周期性地每隔 $\\mathrm{t}_{0}$ 时间发生一次最大的偏离. 如图所示, 天文学家认为形成这种现象的原因可能是 $\\mathrm{A}$ 行星外侧还存在着一颗未知行星 $\\mathrm{B}$, 则行星 $\\mathrm{B}$运动轨道半径表示错误的是(\n\n[图1]\nA: $\\mathrm{R}=\\mathrm{R}_{0} \\sqrt[3]{\\frac{t_{0}^{2}}{t_{0}-T_{0}}}$\nB: $\\mathrm{R}=\\mathrm{R}_{0} \\frac{t_{0}}{t_{0}-T}$\nC: $\\mathrm{R}=\\mathrm{R}_{0} \\sqrt[3]{\\frac{t_{0}}{\\left(t_{0}-T_{0}\\right)^{2}}}$\nD: $\\mathrm{R}=\\mathrm{R}_{0} \\sqrt[3]{\\frac{t_{0}^{2}}{t_{0}-T_{0}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n经长期观测发现, $A$ 行星运行的轨道半径为 $\\mathrm{R}_{0}$, 周期为 $\\mathrm{T}_{0}$, 但其实际运行的轨道与圆轨道总存在一些偏离, 且周期性地每隔 $\\mathrm{t}_{0}$ 时间发生一次最大的偏离. 如图所示, 天文学家认为形成这种现象的原因可能是 $\\mathrm{A}$ 行星外侧还存在着一颗未知行星 $\\mathrm{B}$, 则行星 $\\mathrm{B}$运动轨道半径表示错误的是(\n\n[图1]\n\nA: $\\mathrm{R}=\\mathrm{R}_{0} \\sqrt[3]{\\frac{t_{0}^{2}}{t_{0}-T_{0}}}$\nB: $\\mathrm{R}=\\mathrm{R}_{0} \\frac{t_{0}}{t_{0}-T}$\nC: $\\mathrm{R}=\\mathrm{R}_{0} \\sqrt[3]{\\frac{t_{0}}{\\left(t_{0}-T_{0}\\right)^{2}}}$\nD: $\\mathrm{R}=\\mathrm{R}_{0} \\sqrt[3]{\\frac{t_{0}^{2}}{t_{0}-T_{0}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-33.jpg?height=332&width=354&top_left_y=151&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_687", "problem": "有人设想: 如果在地球的赤道上坚直向上建一座非常高的高楼, 是否可以在楼上直接释放人造卫星呢? 高楼设想图如图所示。已知地球自转周期为 $T$, 同步卫星轨道半径为 $r$; 高楼上有 $A 、 B$ 两个可视为点的小房间, $A$ 到地心的距离为 $\\frac{r}{2}, B$ 到地心的距离为 $\\frac{3 r}{2}$ 。 $A$ 房间天花板上连一坚直轻弹簧, 弹簧下端连一质量为 $m$ 的小球, 求弹簧给小球的作用力大小和方向;[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n有人设想: 如果在地球的赤道上坚直向上建一座非常高的高楼, 是否可以在楼上直接释放人造卫星呢? 高楼设想图如图所示。已知地球自转周期为 $T$, 同步卫星轨道半径为 $r$; 高楼上有 $A 、 B$ 两个可视为点的小房间, $A$ 到地心的距离为 $\\frac{r}{2}, B$ 到地心的距离为 $\\frac{3 r}{2}$ 。 $A$ 房间天花板上连一坚直轻弹簧, 弹簧下端连一质量为 $m$ 的小球, 求弹簧给小球的作用力大小和方向;[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-008.jpg?height=401&width=607&top_left_y=1847&top_left_x=359" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1171", "problem": "In November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \\mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked.\n[figure1]\n\nFigure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC.\n\nRight: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \\theta$. Credit: OpenStax, College Physics.\n\nMercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres.\n\nThe radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \\mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \\mu \\mathrm{s}$ pulses at $430 \\mathrm{MHz}$ with a transmitted power of $2.0 \\mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1.\n\nThe signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \\mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2).\n\nPhotons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole).\n\n[figure2]\n\nFigure 2: Left: Snapshots of the reflections of a single $100 \\mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \\mathrm{c} / \\mathrm{s}=1 \\mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967).\n\nTop right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967).\n\nBottom right: The same as the snapshots, but this time summed over the first $500 \\mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\\sim 5 \\mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967).\n\nThe Doppler shift with light is given as\n\n$$\n\\frac{\\Delta f}{f}=\\frac{v}{c}\n$$\n\nwhere $\\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light.\n\nEver since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time.\n\n[figure3]\n\nFigure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966).b. Averaging over a series of pulses from August 1965, after correcting for the relative motion of the Earth and Mercury and the rotation rate of the Earth during the observations, the difference between the frequencies of photons from the extreme left and right parts of an annulus received $500 \\mu$ s after the initial echo was $\\Delta f_{\\text {total }}=4.27 \\mathrm{~Hz}$.\n\nii. Mercury has a semi-major axis of 0.387 au. Rounding slightly if necessary, express the ratio orbital period : rotational period in a simple integer form (the integers should be $<10$ ).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \\mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked.\n[figure1]\n\nFigure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC.\n\nRight: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \\theta$. Credit: OpenStax, College Physics.\n\nMercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres.\n\nThe radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \\mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \\mu \\mathrm{s}$ pulses at $430 \\mathrm{MHz}$ with a transmitted power of $2.0 \\mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1.\n\nThe signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \\mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2).\n\nPhotons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole).\n\n[figure2]\n\nFigure 2: Left: Snapshots of the reflections of a single $100 \\mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \\mathrm{c} / \\mathrm{s}=1 \\mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967).\n\nTop right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967).\n\nBottom right: The same as the snapshots, but this time summed over the first $500 \\mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\\sim 5 \\mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967).\n\nThe Doppler shift with light is given as\n\n$$\n\\frac{\\Delta f}{f}=\\frac{v}{c}\n$$\n\nwhere $\\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light.\n\nEver since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time.\n\n[figure3]\n\nFigure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966).\n\nproblem:\nb. Averaging over a series of pulses from August 1965, after correcting for the relative motion of the Earth and Mercury and the rotation rate of the Earth during the observations, the difference between the frequencies of photons from the extreme left and right parts of an annulus received $500 \\mu$ s after the initial echo was $\\Delta f_{\\text {total }}=4.27 \\mathrm{~Hz}$.\n\nii. Mercury has a semi-major axis of 0.387 au. Rounding slightly if necessary, express the ratio orbital period : rotational period in a simple integer form (the integers should be $<10$ ).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\text { days }, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-04.jpg?height=512&width=1374&top_left_y=652&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-05.jpg?height=1094&width=1560&top_left_y=218&top_left_x=248", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-06.jpg?height=994&width=897&top_left_y=1359&top_left_x=585" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\text { days }" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1185", "problem": "The surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.a. When it reached first perihelion, radio signals from the probe took $446.58 \\mathrm{~s}$ to reach Earth.\n\niii. Find the values of $\\mathrm{D}$ and $\\mathrm{E}$, given as fractions in their simplest terms, and hence calculate a new value for the distance travelled by the probe (also to 4 s.f.). Compare this to the approximation in the previous part and comment on your answer.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.\n\nproblem:\na. When it reached first perihelion, radio signals from the probe took $446.58 \\mathrm{~s}$ to reach Earth.\n\niii. Find the values of $\\mathrm{D}$ and $\\mathrm{E}$, given as fractions in their simplest terms, and hence calculate a new value for the distance travelled by the probe (also to 4 s.f.). Compare this to the approximation in the previous part and comment on your answer.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of au, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-10.jpg?height=792&width=1572&top_left_y=598&top_left_x=241" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "au" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_103", "problem": "火星表面特征非常接近地球, 适合人类居住, 近期, 我国宇航员王跃与俄罗斯宇航员一起进行“模拟登火星”实验活动, 已知火星半径是地球半径的 $\\frac{1}{2}$, 质量是地球质量的 $\\frac{1}{9}$, 自转周期与地球的基本相同, 地球表面重力加速度为 $g$, 王跃在地面上能向上跳起的最大高度是 $h$, 在忽略自转影响的条件下, 下列分析不正确的是 ( )\nA: 火星表面的重力加速度是 $\\frac{4 g}{9}$\nB: 火星的第一宇宙速度是地球第一宇宙速度的 $\\frac{\\sqrt{2}}{3}$\nC: 王跃在火星表面受的万有引力是在地球表面受万有引力的 $\\frac{2}{9}$ 倍\nD: 王跃以相同的初速度在火星上起跳时, 可跳的最大高度是 $\\frac{9 h}{4}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n火星表面特征非常接近地球, 适合人类居住, 近期, 我国宇航员王跃与俄罗斯宇航员一起进行“模拟登火星”实验活动, 已知火星半径是地球半径的 $\\frac{1}{2}$, 质量是地球质量的 $\\frac{1}{9}$, 自转周期与地球的基本相同, 地球表面重力加速度为 $g$, 王跃在地面上能向上跳起的最大高度是 $h$, 在忽略自转影响的条件下, 下列分析不正确的是 ( )\n\nA: 火星表面的重力加速度是 $\\frac{4 g}{9}$\nB: 火星的第一宇宙速度是地球第一宇宙速度的 $\\frac{\\sqrt{2}}{3}$\nC: 王跃在火星表面受的万有引力是在地球表面受万有引力的 $\\frac{2}{9}$ 倍\nD: 王跃以相同的初速度在火星上起跳时, 可跳的最大高度是 $\\frac{9 h}{4}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_252", "problem": "华为 Mate 60 Pro 通过中国自主研制的天通一号卫星通信系统实现了卫星电话功能,天通一号卫星是地球同步卫星。天通一号卫星发射首先利用火箭将卫星运载至地球附近圆形轨道 1, 通过多次变轨最终进入地球同步轨道 3。其变轨简化示意图如图所示, 轨道 1 离地面高度为 $h$ 。已知天通一号卫星质量为 $m$, 地球自转周期为 $T_{0}$, 地球半径为 $R$,地球表面的重力加速度为 $g, O$ 为地球中心，引力常量为 $G$ 。如果规定距地球无限远处为地球引力零势能点, 地球附近物体的引力势能可表示为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $M$ (未知)为地球质量, $m$ 为物体质量, $r$ 为物体到地心距离。求:\n\n假设在变轨点 $P$ 和 $Q$ 通过两次发动机加速, “天通一号”卫星正好进入地球同步轨道 3，则发动机至少做多少功?\n\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n华为 Mate 60 Pro 通过中国自主研制的天通一号卫星通信系统实现了卫星电话功能,天通一号卫星是地球同步卫星。天通一号卫星发射首先利用火箭将卫星运载至地球附近圆形轨道 1, 通过多次变轨最终进入地球同步轨道 3。其变轨简化示意图如图所示, 轨道 1 离地面高度为 $h$ 。已知天通一号卫星质量为 $m$, 地球自转周期为 $T_{0}$, 地球半径为 $R$,地球表面的重力加速度为 $g, O$ 为地球中心，引力常量为 $G$ 。如果规定距地球无限远处为地球引力零势能点, 地球附近物体的引力势能可表示为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $M$ (未知)为地球质量, $m$ 为物体质量, $r$ 为物体到地心距离。求:\n\n假设在变轨点 $P$ 和 $Q$ 通过两次发动机加速, “天通一号”卫星正好进入地球同步轨道 3，则发动机至少做多少功?\n\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-040.jpg?height=543&width=543&top_left_y=1810&top_left_x=331" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1221", "problem": "GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nThe rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as\n\n$$\n\\dot{f}_{\\mathrm{GW}}=\\frac{96}{5} \\pi^{8 / 3}\\left(\\frac{G \\mathcal{M}}{c^{3}}\\right)^{5 / 3} f_{\\mathrm{GW}}^{11 / 3}\n$$e. Typically, you measure $\\tau \\equiv f_{G W} / \\dot{f}_{G W}$, rather than $\\dot{f}_{G W}$ directly. Given that just as the merger began the detectors measured $\\tau=0.0023 \\mathrm{~s}, f_{G W}=300 \\mathrm{~Hz}$ and $h=6.0 \\times 10^{-21}$, estimate the distance to GW170817 (in Mpc) and its absolute uncertainty (assuming a percentage uncertainty of $\\pm 10 \\%$ ). How does this compare with your answers in parts $a$. and $b$.?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nGW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nThe rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as\n\n$$\n\\dot{f}_{\\mathrm{GW}}=\\frac{96}{5} \\pi^{8 / 3}\\left(\\frac{G \\mathcal{M}}{c^{3}}\\right)^{5 / 3} f_{\\mathrm{GW}}^{11 / 3}\n$$\n\nproblem:\ne. Typically, you measure $\\tau \\equiv f_{G W} / \\dot{f}_{G W}$, rather than $\\dot{f}_{G W}$ directly. Given that just as the merger began the detectors measured $\\tau=0.0023 \\mathrm{~s}, f_{G W}=300 \\mathrm{~Hz}$ and $h=6.0 \\times 10^{-21}$, estimate the distance to GW170817 (in Mpc) and its absolute uncertainty (assuming a percentage uncertainty of $\\pm 10 \\%$ ). How does this compare with your answers in parts $a$. and $b$.?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{Mpc}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{Mpc}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_135", "problem": "光电效应和康普顿效应深入地揭示了光的粒子性的一面。前者表明光子具有能量,后者表明光子除了具有能量之外还具有动量。由狭义相对论可知, 一定的质量 $m$ 与一\n定的能量 $E$ 相对应: $E=m c^{2}$, 其中 $c$ 为真空中光速。\n光照射到物体表面时, 光子被物体吸收或反射时, 光都会对物体产生压强, 这就是“光压”。已知太阳半径为 $R$, 单位时间辐射的总能量为 $P_{0}$, 光速为 $c$ 。求:科幻作品中经常以文明能够利用的能量程度来对文明进行分级。比如, 一级文明能够利用行星的全部能量, 人类处于 0.73 级; 二级文明能够利用恒星的全部能量; 三级文明能够利用整个星系的能量。科幻作家们设想了一种被称为“戴森球”的装置来收集整个恒星的能量。“戴森球”是一种包围整个恒星的球状膜, 它的内表面可以吸收恒星发出的所有光的能量。如果给太阳安装一个“戴森球”, 求: 戴森球的半径 $r$ 多大时, 戴森球可以靠着光压来抵抗太阳的万有引力, 避免其向太阳塌缩。已知引力常量为 $G$, 太阳的质量为 $M$ ，戴森球的总质量记为 $m$ 。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n光电效应和康普顿效应深入地揭示了光的粒子性的一面。前者表明光子具有能量,后者表明光子除了具有能量之外还具有动量。由狭义相对论可知, 一定的质量 $m$ 与一\n定的能量 $E$ 相对应: $E=m c^{2}$, 其中 $c$ 为真空中光速。\n光照射到物体表面时, 光子被物体吸收或反射时, 光都会对物体产生压强, 这就是“光压”。已知太阳半径为 $R$, 单位时间辐射的总能量为 $P_{0}$, 光速为 $c$ 。求:科幻作品中经常以文明能够利用的能量程度来对文明进行分级。比如, 一级文明能够利用行星的全部能量, 人类处于 0.73 级; 二级文明能够利用恒星的全部能量; 三级文明能够利用整个星系的能量。科幻作家们设想了一种被称为“戴森球”的装置来收集整个恒星的能量。“戴森球”是一种包围整个恒星的球状膜, 它的内表面可以吸收恒星发出的所有光的能量。如果给太阳安装一个“戴森球”, 求: 戴森球的半径 $r$ 多大时, 戴森球可以靠着光压来抵抗太阳的万有引力, 避免其向太阳塌缩。已知引力常量为 $G$, 太阳的质量为 $M$ ，戴森球的总质量记为 $m$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_476", "problem": "小型登月器连接在航天站上, 一起绕月球做圆周运动, 其轨道半径为月球半径的 3 倍, 某时刻, 航天站使登月器减速分离, 登月器沿如图所示的椭圆轨道登月, 在月球表面逗留一段时间完成科考工作后，经快速启动仍沿原椭圆轨道返回，当第一次回到分离点时恰与航天站对接, 登月器快速启动所用的时间可以忽略不计, 整个过程中航天站保持原轨道绕月运行, 不考虑月球自转的影响, 则下列说法正确的是 ( )\n\n[图1]\nA: 从登月器与航天站分离到对接, 航天站至少转过半个周期\nB: 从登月器与航天站分离到对接, 航天站至少转过 2 个周期\nC: 航天站做圆周运动的周期与登月器在椭圆轨道上运动的周期之比为 $\\sqrt{\\frac{27}{8}}$\nD: 航天站做圆周运动的周期与登月器在椭圆轨道上运动的周期之比为 $\\frac{27}{8}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n小型登月器连接在航天站上, 一起绕月球做圆周运动, 其轨道半径为月球半径的 3 倍, 某时刻, 航天站使登月器减速分离, 登月器沿如图所示的椭圆轨道登月, 在月球表面逗留一段时间完成科考工作后，经快速启动仍沿原椭圆轨道返回，当第一次回到分离点时恰与航天站对接, 登月器快速启动所用的时间可以忽略不计, 整个过程中航天站保持原轨道绕月运行, 不考虑月球自转的影响, 则下列说法正确的是 ( )\n\n[图1]\n\nA: 从登月器与航天站分离到对接, 航天站至少转过半个周期\nB: 从登月器与航天站分离到对接, 航天站至少转过 2 个周期\nC: 航天站做圆周运动的周期与登月器在椭圆轨道上运动的周期之比为 $\\sqrt{\\frac{27}{8}}$\nD: 航天站做圆周运动的周期与登月器在椭圆轨道上运动的周期之比为 $\\frac{27}{8}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-62.jpg?height=494&width=445&top_left_y=153&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_870", "problem": "What is the time difference between the longest day of the year and the shortest day of the year in San Francisco $\\left(37.7^{\\circ} \\mathrm{N}, 122.4^{\\circ} \\mathrm{W}\\right)$ ?\n\nNeglect atmospheric refraction.\nA: $2 \\mathrm{~h} 30 \\mathrm{~min}$\nB: $3 \\mathrm{~h} 32 \\mathrm{~min}$\nC: $4 \\mathrm{~h} 08 \\mathrm{~min}$\nD: $5 \\mathrm{~h} 12 \\mathrm{~min}$\nE: 6 h25min\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the time difference between the longest day of the year and the shortest day of the year in San Francisco $\\left(37.7^{\\circ} \\mathrm{N}, 122.4^{\\circ} \\mathrm{W}\\right)$ ?\n\nNeglect atmospheric refraction.\n\nA: $2 \\mathrm{~h} 30 \\mathrm{~min}$\nB: $3 \\mathrm{~h} 32 \\mathrm{~min}$\nC: $4 \\mathrm{~h} 08 \\mathrm{~min}$\nD: $5 \\mathrm{~h} 12 \\mathrm{~min}$\nE: 6 h25min\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_526", "problem": "$\\mathrm{A} 、 \\mathrm{~B}$ 是两颗人造地球卫星, 已知 $\\mathrm{A}$ 的环绕半径大于 $\\mathrm{B}$ 的环绕半径 $r_{A}>r_{B}$, 设 $\\mathrm{A}$ 和 $\\mathrm{B}$ 的运行速度分别为 $v_{A}$ 和 $v_{B}$, 最初的发射速度分别为 $v_{0 A}$ 和 $v_{0 B}$, 那么以下说法中正确的是( )\nA: $v_{A}>v_{B}, v_{0 A}>v_{0 B}$\nB: $v_{A}>v_{B}, v_{0 A}v_{0 B}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n$\\mathrm{A} 、 \\mathrm{~B}$ 是两颗人造地球卫星, 已知 $\\mathrm{A}$ 的环绕半径大于 $\\mathrm{B}$ 的环绕半径 $r_{A}>r_{B}$, 设 $\\mathrm{A}$ 和 $\\mathrm{B}$ 的运行速度分别为 $v_{A}$ 和 $v_{B}$, 最初的发射速度分别为 $v_{0 A}$ 和 $v_{0 B}$, 那么以下说法中正确的是( )\n\nA: $v_{A}>v_{B}, v_{0 A}>v_{0 B}$\nB: $v_{A}>v_{B}, v_{0 A}v_{0 B}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1070", "problem": "The surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.a. When it reached first perihelion, radio signals from the probe took $446.58 \\mathrm{~s}$ to reach Earth.\n\nii. Use the Ramanujan approximation to work out the distance travelled by the probe between launch and perihelion to 4 s.f.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.\n\nproblem:\na. When it reached first perihelion, radio signals from the probe took $446.58 \\mathrm{~s}$ to reach Earth.\n\nii. Use the Ramanujan approximation to work out the distance travelled by the probe between launch and perihelion to 4 s.f.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of au, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-10.jpg?height=792&width=1572&top_left_y=598&top_left_x=241" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "au" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_50", "problem": "“天宫一号”的运行圆轨道离地高度为 $350 \\mathrm{~km}$, “神舟十号”需要追赶“天宫一号”并成功与之对接, 对接开始前它们在同一平面绕地球做匀速圆周运动且运行方向相同, 要成功对接则对接前“神舟十号”应该（）\nA: 从离地高度等于 $350 \\mathrm{~km}$ 的圆轨道上加速且对接成功后运行速度比开始对接前大\nB: 从离地高度大于 $350 \\mathrm{~km}$ 的圆轨道上减速且对接成功后运行速度比开始对接前小\nC: 从离地高度小于 $350 \\mathrm{~km}$ 圆轨道上加速且对接成功后运行速度比开始对接前小\nD: 从离地高度小于 $350 \\mathrm{~km}$ 圆轨道上加速且对接成功后运行速度比开始对接前大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n“天宫一号”的运行圆轨道离地高度为 $350 \\mathrm{~km}$, “神舟十号”需要追赶“天宫一号”并成功与之对接, 对接开始前它们在同一平面绕地球做匀速圆周运动且运行方向相同, 要成功对接则对接前“神舟十号”应该（）\n\nA: 从离地高度等于 $350 \\mathrm{~km}$ 的圆轨道上加速且对接成功后运行速度比开始对接前大\nB: 从离地高度大于 $350 \\mathrm{~km}$ 的圆轨道上减速且对接成功后运行速度比开始对接前小\nC: 从离地高度小于 $350 \\mathrm{~km}$ 圆轨道上加速且对接成功后运行速度比开始对接前小\nD: 从离地高度小于 $350 \\mathrm{~km}$ 圆轨道上加速且对接成功后运行速度比开始对接前大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_907", "problem": "Star Wars Rogue One\n\nThe new Star Wars film Rogue One concentrates on the creation of the first Death Star, which in one scene causes a total eclipse on the planet Scarif, where it is being built.\n\n[figure1]\n\nAssume the Death Star is being built in orbit around the Earth instead, but still causes a very brief total solar eclipse when it passes in front of the Sun. The Death Star has a diameter of $120 \\mathrm{~km}$, and so by comparing it to the size and distance to the Sun, calculate the altitude it is being built at. How does that compare to the altitude of the International Space Station $(400 \\mathrm{~km})$ ?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nStar Wars Rogue One\n\nThe new Star Wars film Rogue One concentrates on the creation of the first Death Star, which in one scene causes a total eclipse on the planet Scarif, where it is being built.\n\n[figure1]\n\nAssume the Death Star is being built in orbit around the Earth instead, but still causes a very brief total solar eclipse when it passes in front of the Sun. The Death Star has a diameter of $120 \\mathrm{~km}$, and so by comparing it to the size and distance to the Sun, calculate the altitude it is being built at. How does that compare to the altitude of the International Space Station $(400 \\mathrm{~km})$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of km, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6d91a7785df4f4beaa9ag-07.jpg?height=554&width=1474&top_left_y=474&top_left_x=291" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "km" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_925", "problem": "The cosmic microwave background (CMB) is measured today to have an almost perfectly uniform temperature of $T_{0}=2.725 \\mathrm{~K}$. The temperature of the CMB at any redshift can be calculated using $T=T_{0}(1+z)$ where $z$ is the redshift. During the early expansion of the Universe, the temperature was high enough to ionize the hydrogen atoms filling the universe and make the universe opaque. Once the temperature dropped below $3000 \\mathrm{~K}$ the universe became transparent and the CMB was emitted in a process known as photon decoupling.\n[figure1]\n\nFigure 2: Left: The CMB has been measured to have an almost perfect black-body spectrum. Credit: NASA/WMAP Science Team. Right: The Planck satellite was launched to measure the tiny deviations from a perfect black-body. This is the map of the sky from their final data release from July 2018. Credit: ESA/Planck Collaboration.\n\nThe relative size of the Universe, called the scale factor $a$, varies with redshift as $a=a_{0}(1+z)^{-1}$, whilst in a matter dominated universe (a valid assumption after the CMB was emitted) the scale factor varies with time $t$ as $a \\propto t^{2 / 3}$.\n\nIf the scale factor today is defined to be $a_{0} \\equiv 1$, calculate the age of the Universe when the CMB was emitted. [Current age of the Universe $t_{0}=13.8$ Gyr.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe cosmic microwave background (CMB) is measured today to have an almost perfectly uniform temperature of $T_{0}=2.725 \\mathrm{~K}$. The temperature of the CMB at any redshift can be calculated using $T=T_{0}(1+z)$ where $z$ is the redshift. During the early expansion of the Universe, the temperature was high enough to ionize the hydrogen atoms filling the universe and make the universe opaque. Once the temperature dropped below $3000 \\mathrm{~K}$ the universe became transparent and the CMB was emitted in a process known as photon decoupling.\n[figure1]\n\nFigure 2: Left: The CMB has been measured to have an almost perfect black-body spectrum. Credit: NASA/WMAP Science Team. Right: The Planck satellite was launched to measure the tiny deviations from a perfect black-body. This is the map of the sky from their final data release from July 2018. Credit: ESA/Planck Collaboration.\n\nThe relative size of the Universe, called the scale factor $a$, varies with redshift as $a=a_{0}(1+z)^{-1}$, whilst in a matter dominated universe (a valid assumption after the CMB was emitted) the scale factor varies with time $t$ as $a \\propto t^{2 / 3}$.\n\nIf the scale factor today is defined to be $a_{0} \\equiv 1$, calculate the age of the Universe when the CMB was emitted. [Current age of the Universe $t_{0}=13.8$ Gyr.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-08.jpg?height=504&width=1518&top_left_y=644&top_left_x=270" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_692", "problem": "设想在地面上通过火箭将质量为 $m$ 的人造小飞船送入预定轨道, 至少需要做功 $W$ 。若预定轨道半径为 $r$, 地球半径为 $R$, 地球表面处的重力加速度为 $g$, 忽略空气阻力,\n不考虑地球自转的影响。取地面为零势能面, 则下列说法错误的是 ( )\nA: 地球的质量为 $\\frac{g r^{2}}{G}$\nB: 小飞船在预定轨道的周期为 $2 \\pi \\sqrt{\\frac{R^{3}}{g r^{2}}}$\nC: 小飞船在预定轨道的动能为 $\\frac{m g r^{2}}{2 R}$\nD: 小飞船在预定轨道的势能为 $W-\\frac{m g R^{2}}{2 r}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n设想在地面上通过火箭将质量为 $m$ 的人造小飞船送入预定轨道, 至少需要做功 $W$ 。若预定轨道半径为 $r$, 地球半径为 $R$, 地球表面处的重力加速度为 $g$, 忽略空气阻力,\n不考虑地球自转的影响。取地面为零势能面, 则下列说法错误的是 ( )\n\nA: 地球的质量为 $\\frac{g r^{2}}{G}$\nB: 小飞船在预定轨道的周期为 $2 \\pi \\sqrt{\\frac{R^{3}}{g r^{2}}}$\nC: 小飞船在预定轨道的动能为 $\\frac{m g r^{2}}{2 R}$\nD: 小飞船在预定轨道的势能为 $W-\\frac{m g R^{2}}{2 r}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_806", "problem": "The problem of magnetic monopoles - that is, the apparent absence of magnetic monopoles in the universe - arises from the fact that some modern physical theories (such as string theory) predict that the number density of magnetic monopoles at the time of their creation was $n_{M}\\left(t_{G U T}\\right) \\approx 10^{82} \\mathrm{~m}^{-3}$. The inflation theory provides a possible solution to this problem, as the exponential expansion of the primordial universe would \"dilute\" the monopoles. Calculate, approximately, how much the universe expanded during the inflationary period so that today the probability of a single magnetic monopole existing in the observational universe is $1 \\%$. Consider that the beginning of inflation coincides with the time of the creation of magnetic monopoles, and that the universe is flat (Euclidean geometry can be used on large scales). You can use that the diameter of the observational universe is $28.5 \\mathrm{Gpc}$, and that between the end of inflation and today, the universe has linearly expanded by a factor of $5 \\times 10^{26}$.\nA: $e^{40}$\nB: $e^{50}$\nC: $e^{55}$\nD: $e^{65}$\nE: $e^{85}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe problem of magnetic monopoles - that is, the apparent absence of magnetic monopoles in the universe - arises from the fact that some modern physical theories (such as string theory) predict that the number density of magnetic monopoles at the time of their creation was $n_{M}\\left(t_{G U T}\\right) \\approx 10^{82} \\mathrm{~m}^{-3}$. The inflation theory provides a possible solution to this problem, as the exponential expansion of the primordial universe would \"dilute\" the monopoles. Calculate, approximately, how much the universe expanded during the inflationary period so that today the probability of a single magnetic monopole existing in the observational universe is $1 \\%$. Consider that the beginning of inflation coincides with the time of the creation of magnetic monopoles, and that the universe is flat (Euclidean geometry can be used on large scales). You can use that the diameter of the observational universe is $28.5 \\mathrm{Gpc}$, and that between the end of inflation and today, the universe has linearly expanded by a factor of $5 \\times 10^{26}$.\n\nA: $e^{40}$\nB: $e^{50}$\nC: $e^{55}$\nD: $e^{65}$\nE: $e^{85}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_629", "problem": "如图所示, 有一质量为 $M$, 半径为 $R$, 密度均匀的球体, 在距离球心 $O$ 为 $2 R$ 的地方有一质量为 $m$ 的质点, 现从 $M$ 中挖去一半径为 $\\frac{R}{2}$ 的球体, 试求:\n 剩余部分对质点 $m$ 的引力大小;\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 有一质量为 $M$, 半径为 $R$, 密度均匀的球体, 在距离球心 $O$ 为 $2 R$ 的地方有一质量为 $m$ 的质点, 现从 $M$ 中挖去一半径为 $\\frac{R}{2}$ 的球体, 试求:\n 剩余部分对质点 $m$ 的引力大小;\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-062.jpg?height=286&width=443&top_left_y=174&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_204", "problem": "放置在水平平台上的物体, 其表观重力在数值上等于物体对平台的压力, 方向与压力的方向相同。微重力环境是指系统内物体的表观重力远小于其实际重力（万有引力）的环境。此环境下，物体的表观重力与其质量之比称为微重力加速度。\n\n如图所示, 中国科学院力学研究所微重力实验室落塔是我国微重力实验的主要设施之一, 实验中落舱可采用单舱和双舱两种模式进行。已知地球表面的重力加速度为 $g$;\n\n如图所示, 双舱模式是采用内外双舱结构, 实验平台固定在内舱中, 实验时让双舱同时下落。落体下落时受到的空气阻力可表示为 $f=k \\rho v^{2}$, 式中 $k$ 为由落体形状决定的常数, $\\rho$ 为空气密度, $v$ 为落体相对于周围空气的速率。若某次实验中, 内舱与舱内物体总质量为 $m_{1}$, 外舱与舱内物体总质量为 $m_{2}$ (不含内舱)。某时刻, 外舱相对于地面的速度为 $v_{1}$, 内舱相对于地面的速度为 $v_{2}$, 它们所受空气阻力的常数 $k$ 相同, 外舱中与外部环境的空气密度相同, 不考虑外舱内空气对外舱自身运动的影响。求此时内舱与外舱中的微重力加速度之比 $g_{2}: g_{3} ;$\n\n[图1]\n\n落塔 落舱", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n放置在水平平台上的物体, 其表观重力在数值上等于物体对平台的压力, 方向与压力的方向相同。微重力环境是指系统内物体的表观重力远小于其实际重力（万有引力）的环境。此环境下，物体的表观重力与其质量之比称为微重力加速度。\n\n如图所示, 中国科学院力学研究所微重力实验室落塔是我国微重力实验的主要设施之一, 实验中落舱可采用单舱和双舱两种模式进行。已知地球表面的重力加速度为 $g$;\n\n如图所示, 双舱模式是采用内外双舱结构, 实验平台固定在内舱中, 实验时让双舱同时下落。落体下落时受到的空气阻力可表示为 $f=k \\rho v^{2}$, 式中 $k$ 为由落体形状决定的常数, $\\rho$ 为空气密度, $v$ 为落体相对于周围空气的速率。若某次实验中, 内舱与舱内物体总质量为 $m_{1}$, 外舱与舱内物体总质量为 $m_{2}$ (不含内舱)。某时刻, 外舱相对于地面的速度为 $v_{1}$, 内舱相对于地面的速度为 $v_{2}$, 它们所受空气阻力的常数 $k$ 相同, 外舱中与外部环境的空气密度相同, 不考虑外舱内空气对外舱自身运动的影响。求此时内舱与外舱中的微重力加速度之比 $g_{2}: g_{3} ;$\n\n[图1]\n\n落塔 落舱\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-033.jpg?height=317&width=808&top_left_y=201&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1178", "problem": "A day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4).\n\n[figure1]\n\nFigure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun.\n\nCredit: Wikipedia.\n\nThe length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time.\n\n[Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \\times 10^{-11} \\mathrm{~m}^{3} \\mathrm{~kg}^{-1} \\mathrm{~s}^{-2}, R_{\\oplus}=6371 \\mathrm{~km}, M_{\\oplus}=5.972 \\times 10^{24} \\mathrm{~kg}, M_{\\odot}=$ $1.989 \\times 10^{30} \\mathrm{~kg}$ and $1 \\mathrm{au}=1.496 \\times 10^{11} \\mathrm{~m}$.]\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\\frac{2}{5} M R^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\omega$ is $L=I \\omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.\n\nThe speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$d. Imagine creating an incredibly powerful rocket, positioned on the Earth's equator, that when fired once can apply a huge force to the Earth in a very short time period, delivering a total impulse of $\\Delta p$. Assuming the Earth's orbit is initially circular, calculate:\nii. The total impulse required to change the orbit to give a year of 360 solar days, but with no change in the length of a solar day, also explaining how the rocket needs to be fired.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nA day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4).\n\n[figure1]\n\nFigure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun.\n\nCredit: Wikipedia.\n\nThe length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time.\n\n[Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \\times 10^{-11} \\mathrm{~m}^{3} \\mathrm{~kg}^{-1} \\mathrm{~s}^{-2}, R_{\\oplus}=6371 \\mathrm{~km}, M_{\\oplus}=5.972 \\times 10^{24} \\mathrm{~kg}, M_{\\odot}=$ $1.989 \\times 10^{30} \\mathrm{~kg}$ and $1 \\mathrm{au}=1.496 \\times 10^{11} \\mathrm{~m}$.]\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\\frac{2}{5} M R^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\omega$ is $L=I \\omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.\n\nThe speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nproblem:\nd. Imagine creating an incredibly powerful rocket, positioned on the Earth's equator, that when fired once can apply a huge force to the Earth in a very short time period, delivering a total impulse of $\\Delta p$. Assuming the Earth's orbit is initially circular, calculate:\nii. The total impulse required to change the orbit to give a year of 360 solar days, but with no change in the length of a solar day, also explaining how the rocket needs to be fired.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~kg} \\mathrm{~m} \\mathrm{~s}^{-1}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-06.jpg?height=1276&width=782&top_left_y=567&top_left_x=657" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~kg} \\mathrm{~m} \\mathrm{~s}^{-1}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_582", "problem": "以两天体 $\\mathrm{A} 、 \\mathrm{~B}$ 中心连线为底的等边三角形的第三个顶点被称为“三角拉格朗日点”, 如果在该点有一颗质量远小于 $\\mathrm{A} 、 \\mathrm{~B}$ 的卫星 $\\mathrm{C}$, 则三者可以组成一个稳定的三星系统, 如图所示。由于 $\\mathrm{C}$ 对 $\\mathrm{A} 、 \\mathrm{~B}$ 的影响很小, 故 $\\mathrm{A} 、 \\mathrm{~B}$ 又可视作双星系统绕连线上某定点 $P$ (未画出) 做匀速圆周运动。已知天体 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 的质量分布均匀, 且分别为 $m_{1} 、 m_{2} 、 m_{3}$, 已知 $m_{1}=2 m_{2}$, 两天体 $\\mathrm{A} 、 \\mathrm{~B}$ 中心间距为 $L$, 万有引力常量为 $G$, 则下列说法正确的是（ ）\n\n[图1]\nA: 天体 $\\mathrm{A}$ 做匀速圆周运动的轨道半径为 $\\frac{L}{2}$\nB: 天体 $A 、 B$ 所需要的向心力大小之比为 2: 1\nC: 卫星 $\\mathrm{C}$ 所受合力恰好指向 $P$ 点\nD: 卫星 $\\mathrm{C}$ 的周期为 $2 \\pi \\sqrt{\\frac{L^{3}}{G\\left(m_{1}+m_{2}\\right)}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n以两天体 $\\mathrm{A} 、 \\mathrm{~B}$ 中心连线为底的等边三角形的第三个顶点被称为“三角拉格朗日点”, 如果在该点有一颗质量远小于 $\\mathrm{A} 、 \\mathrm{~B}$ 的卫星 $\\mathrm{C}$, 则三者可以组成一个稳定的三星系统, 如图所示。由于 $\\mathrm{C}$ 对 $\\mathrm{A} 、 \\mathrm{~B}$ 的影响很小, 故 $\\mathrm{A} 、 \\mathrm{~B}$ 又可视作双星系统绕连线上某定点 $P$ (未画出) 做匀速圆周运动。已知天体 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 的质量分布均匀, 且分别为 $m_{1} 、 m_{2} 、 m_{3}$, 已知 $m_{1}=2 m_{2}$, 两天体 $\\mathrm{A} 、 \\mathrm{~B}$ 中心间距为 $L$, 万有引力常量为 $G$, 则下列说法正确的是（ ）\n\n[图1]\n\nA: 天体 $\\mathrm{A}$ 做匀速圆周运动的轨道半径为 $\\frac{L}{2}$\nB: 天体 $A 、 B$ 所需要的向心力大小之比为 2: 1\nC: 卫星 $\\mathrm{C}$ 所受合力恰好指向 $P$ 点\nD: 卫星 $\\mathrm{C}$ 的周期为 $2 \\pi \\sqrt{\\frac{L^{3}}{G\\left(m_{1}+m_{2}\\right)}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-064.jpg?height=363&width=423&top_left_y=161&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1023", "problem": "Star Wars Rogue One\n\nThe new Star Wars film Rogue One concentrates on the creation of the first Death Star, which in one scene causes a total eclipse on the planet Scarif, where it is being built.\n\n[figure1]\n\nWhat will be its orbital period, assuming it moves in a circular orbit?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nStar Wars Rogue One\n\nThe new Star Wars film Rogue One concentrates on the creation of the first Death Star, which in one scene causes a total eclipse on the planet Scarif, where it is being built.\n\n[figure1]\n\nWhat will be its orbital period, assuming it moves in a circular orbit?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of hours, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6d91a7785df4f4beaa9ag-07.jpg?height=554&width=1474&top_left_y=474&top_left_x=291" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "hours" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_131", "problem": "2019 年 2 月 15 日，一群中国学生拍摄的地月同框照, 被外媒评价为迄今为止最好的地月合影之一。如图所示, 把地球和月球看做绕同一圆心做匀速圆周运动的双星系统,质量分别为 $M 、 m$, 相距为 $L$, 周期为 $T$, 若有间距也为 $L$ 的双星 $\\mathrm{P} 、 \\mathrm{Q}, \\mathrm{P} 、 \\mathrm{Q}$ 的质量分别为 $2 M 、 2 m$ ，则 $(\\quad)$\n\n[图1]\nA: 地、月运动的轨道半径之比为 $\\frac{M}{m}$\nB: 地、月运动的加速度之比为 $\\frac{M}{m}$\nC: P 运动的速率与地球的相等\nD: $\\mathrm{P} 、 \\mathrm{Q}$ 运动的周期均为 $\\frac{\\sqrt{2}}{2} T$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2019 年 2 月 15 日，一群中国学生拍摄的地月同框照, 被外媒评价为迄今为止最好的地月合影之一。如图所示, 把地球和月球看做绕同一圆心做匀速圆周运动的双星系统,质量分别为 $M 、 m$, 相距为 $L$, 周期为 $T$, 若有间距也为 $L$ 的双星 $\\mathrm{P} 、 \\mathrm{Q}, \\mathrm{P} 、 \\mathrm{Q}$ 的质量分别为 $2 M 、 2 m$ ，则 $(\\quad)$\n\n[图1]\n\nA: 地、月运动的轨道半径之比为 $\\frac{M}{m}$\nB: 地、月运动的加速度之比为 $\\frac{M}{m}$\nC: P 运动的速率与地球的相等\nD: $\\mathrm{P} 、 \\mathrm{Q}$ 运动的周期均为 $\\frac{\\sqrt{2}}{2} T$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-41.jpg?height=348&width=400&top_left_y=820&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_937", "problem": "A telescope with a focal length of $750 \\mathrm{~mm}$ is used with an eyepiece. Which eyepiece focal length would give the greatest overall magnification?\nA: $25 \\mathrm{~mm}$\nB: $20 \\mathrm{~mm}$\nC: $15 \\mathrm{~mm}$\nD: $10 \\mathrm{~mm}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA telescope with a focal length of $750 \\mathrm{~mm}$ is used with an eyepiece. Which eyepiece focal length would give the greatest overall magnification?\n\nA: $25 \\mathrm{~mm}$\nB: $20 \\mathrm{~mm}$\nC: $15 \\mathrm{~mm}$\nD: $10 \\mathrm{~mm}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_772", "problem": "What is true for a type-la (\"type one-a\") supernova?\nA: This type occurs in binary systems.\nB: This type occurs often in young galaxies.\nC: This type produces gamma-ray bursts.\nD: This type produces high amounts of X-rays.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is true for a type-la (\"type one-a\") supernova?\n\nA: This type occurs in binary systems.\nB: This type occurs often in young galaxies.\nC: This type produces gamma-ray bursts.\nD: This type produces high amounts of X-rays.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_168", "problem": "一颗人造卫星在地球赤道平面内做匀速圆周运动, 运动方向跟地球自转方向相同,每 6 天经过赤道上同一地点上空一次, 已知地球同步卫星轨道半径为 $\\mathrm{r}$, 下列有关该卫星的说法正确的有\nA: 周期可能为 6 天, 轨道半径为 $\\sqrt[3]{36} r$\nB: 周期可能为 1.2 天, 轨道半径为 $\\sqrt[3]{\\frac{36}{25}} r$\nC: 周期可能为 $\\frac{6}{7}$ 天, 轨道半径为 $\\sqrt[3]{\\frac{36}{49} r}$\nD: 周期可能为 5 天, 轨道半径为 $\\sqrt[3]{25} r$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n一颗人造卫星在地球赤道平面内做匀速圆周运动, 运动方向跟地球自转方向相同,每 6 天经过赤道上同一地点上空一次, 已知地球同步卫星轨道半径为 $\\mathrm{r}$, 下列有关该卫星的说法正确的有\n\nA: 周期可能为 6 天, 轨道半径为 $\\sqrt[3]{36} r$\nB: 周期可能为 1.2 天, 轨道半径为 $\\sqrt[3]{\\frac{36}{25}} r$\nC: 周期可能为 $\\frac{6}{7}$ 天, 轨道半径为 $\\sqrt[3]{\\frac{36}{49} r}$\nD: 周期可能为 5 天, 轨道半径为 $\\sqrt[3]{25} r$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_529", "problem": "地球赤道上的重力加速度为 $g$, 物体在赤道上随地球自转的向心加速度为 $a$, 卫星甲、乙、丙在如图所示三个粗圆轨道上绕地球运行, 卫星甲和乙的运行轨道在 $P$ 点相切,以下说法中正确的是（ ）[图1]\nA: 如果地球自转的角速度突然变为原来的 $\\frac{g+a}{a}$ 倍, 则赤道上的物体刚好“飘”起来\nB: 卫星甲、乙经过 $P$ 点时的加速度大小相等\nC: 卫星甲的周期最小\nD: 三个卫星在远地点的速度可能大于第一宇宙速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球赤道上的重力加速度为 $g$, 物体在赤道上随地球自转的向心加速度为 $a$, 卫星甲、乙、丙在如图所示三个粗圆轨道上绕地球运行, 卫星甲和乙的运行轨道在 $P$ 点相切,以下说法中正确的是（ ）[图1]\n\nA: 如果地球自转的角速度突然变为原来的 $\\frac{g+a}{a}$ 倍, 则赤道上的物体刚好“飘”起来\nB: 卫星甲、乙经过 $P$ 点时的加速度大小相等\nC: 卫星甲的周期最小\nD: 三个卫星在远地点的速度可能大于第一宇宙速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://i.postimg.cc/tJQ1wg7K/image.png" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_767", "problem": "What does the astronomical term barycentre describe?\nA: The centre of mass of multiple each other orbiting objects.\nB: The central axis of rotation of a spinning stellar object.\nC: The central axis of the precession of a rotating stellar object.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat does the astronomical term barycentre describe?\n\nA: The centre of mass of multiple each other orbiting objects.\nB: The central axis of rotation of a spinning stellar object.\nC: The central axis of the precession of a rotating stellar object.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_546", "problem": "星体 $\\mathrm{P}$ (行星或彗星) 绕太阳运动的轨迹为圆锥曲线 $r=\\frac{k}{1+\\varepsilon \\cos \\theta}$ 式中, $r$ 是 $\\mathrm{P}$ 到太阳 $\\mathrm{S}$ 的距离, $\\theta$ 是矢径 $\\mathrm{SP}$ 相对于极轴 $\\mathrm{SA}$ 的夹角 (以逆时针方向为正), $k=\\frac{L^{2}}{G M m^{2}}, L$是 $\\mathrm{P}$ 相对于太阳的角动量, $G=6.67 \\times 10^{-11} \\mathrm{~m} 3 \\cdot \\mathrm{kg}^{-1} \\cdot \\mathrm{s}^{-2}$ 为引力常量, $M \\approx 1.99 \\times 10^{30} \\mathrm{~kg}$ 为太阳的质量, $\\varepsilon=\\sqrt{1+\\frac{2 E L^{2}}{G^{2} M^{2} m^{3}}}$ 为偏心率, $m$ 和 $E$ 分别为 $\\mathrm{P}$ 的质量和机械能。假设有一颗彗星绕太阳运动的轨道为抛物线, 地球绕太阳运动的轨道可近似为圆, 两轨道相交于 $C 、 D$两点, 如图所示。已知地球轨道半径 $R_{E}=1.49 \\times 10^{11} \\mathrm{~m}$, 彗星轨道近日点 $A$ 到太阳的距离为地球轨道半径的三分之一, 不考虑地球和彗星之间的相互影响。求彗星先后两次穿过地球轨道所用的时间;\n\n已知积分公式 $\\int \\frac{x d x}{\\sqrt{x+a}}=\\frac{2}{3}(x+a)^{\\frac{3}{2}}-2 a(x+a)^{\\frac{1}{2}}+C$, 式中 $\\mathrm{C}$ 是任意常数。\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n星体 $\\mathrm{P}$ (行星或彗星) 绕太阳运动的轨迹为圆锥曲线 $r=\\frac{k}{1+\\varepsilon \\cos \\theta}$ 式中, $r$ 是 $\\mathrm{P}$ 到太阳 $\\mathrm{S}$ 的距离, $\\theta$ 是矢径 $\\mathrm{SP}$ 相对于极轴 $\\mathrm{SA}$ 的夹角 (以逆时针方向为正), $k=\\frac{L^{2}}{G M m^{2}}, L$是 $\\mathrm{P}$ 相对于太阳的角动量, $G=6.67 \\times 10^{-11} \\mathrm{~m} 3 \\cdot \\mathrm{kg}^{-1} \\cdot \\mathrm{s}^{-2}$ 为引力常量, $M \\approx 1.99 \\times 10^{30} \\mathrm{~kg}$ 为太阳的质量, $\\varepsilon=\\sqrt{1+\\frac{2 E L^{2}}{G^{2} M^{2} m^{3}}}$ 为偏心率, $m$ 和 $E$ 分别为 $\\mathrm{P}$ 的质量和机械能。假设有一颗彗星绕太阳运动的轨道为抛物线, 地球绕太阳运动的轨道可近似为圆, 两轨道相交于 $C 、 D$两点, 如图所示。已知地球轨道半径 $R_{E}=1.49 \\times 10^{11} \\mathrm{~m}$, 彗星轨道近日点 $A$ 到太阳的距离为地球轨道半径的三分之一, 不考虑地球和彗星之间的相互影响。求彗星先后两次穿过地球轨道所用的时间;\n\n已知积分公式 $\\int \\frac{x d x}{\\sqrt{x+a}}=\\frac{2}{3}(x+a)^{\\frac{3}{2}}-2 a(x+a)^{\\frac{1}{2}}+C$, 式中 $\\mathrm{C}$ 是任意常数。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以s为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-016.jpg?height=363&width=577&top_left_y=161&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_590", "problem": "暗物质是二十世纪物理学之谜, 对该问题的研究可能带来一场物理学的革命。为了探测暗物质，我国在 2015 年 12 月 17 日成功发射了一颗被命名为“悟空”的暗物质探测卫星, 已知“悟空”在低于同步卫星的轨道上绕地球做匀速圆运动, 经过时间 $t$ ( $t$ 小于其\n运动周期), 运动的弧长为 $s$, 与地球中心连扫过的角度为 $\\beta$ (弧度), 引力常量为 $G$,则下列说法中正确的是（）\nA: “悟空”的线速度大于第一宇宙速度\nB: “悟空”的向心加速度小于地球同步卫星的向心加速度\nC: “悟空”的环绕周期为 $\\frac{2 \\pi t}{\\beta}$\nD: “悟空”的质量为 $\\frac{s^{3}}{\\left(G t^{2} \\beta\\right)}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n暗物质是二十世纪物理学之谜, 对该问题的研究可能带来一场物理学的革命。为了探测暗物质，我国在 2015 年 12 月 17 日成功发射了一颗被命名为“悟空”的暗物质探测卫星, 已知“悟空”在低于同步卫星的轨道上绕地球做匀速圆运动, 经过时间 $t$ ( $t$ 小于其\n运动周期), 运动的弧长为 $s$, 与地球中心连扫过的角度为 $\\beta$ (弧度), 引力常量为 $G$,则下列说法中正确的是（）\n\nA: “悟空”的线速度大于第一宇宙速度\nB: “悟空”的向心加速度小于地球同步卫星的向心加速度\nC: “悟空”的环绕周期为 $\\frac{2 \\pi t}{\\beta}$\nD: “悟空”的质量为 $\\frac{s^{3}}{\\left(G t^{2} \\beta\\right)}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_381", "problem": "如图所示, 质量分别为 $M$ 和 $m$ 的两个星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ (均视为质点) 在它们之间的引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 之间的距离为 $L$ 。已知星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 和 $O$三点始终共线, $\\mathrm{A}$ 和 $\\mathrm{B}$ 分别在 $O$ 点的两侧。引力常量为 $G$ 。\n\n在地月系统中, 若忽略其他星球的影响, 可以将月球和地球看成上述星球 $\\mathrm{A}$ 和 $\\mathrm{B}$,月球绕其轨道中心运行的周期记为 $T_{1}$ 。但在处理近似问题时, 常常认为月球是绕地心做圆周运动的, 这样算得的运行周期为 $T_{2}$ 。已知地球和月球的质量分别为 $m_{\\text {地 }}=6 \\times 10^{24} \\mathrm{~kg}$和 $m_{\\text {月 }}=7 \\times 10^{22} \\mathrm{~kg}$ 。求 $\\frac{T_{2}^{2}}{T_{1}^{2}}$ 。(结果保留三位有效数字)\n\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n如图所示, 质量分别为 $M$ 和 $m$ 的两个星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ (均视为质点) 在它们之间的引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 之间的距离为 $L$ 。已知星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 和 $O$三点始终共线, $\\mathrm{A}$ 和 $\\mathrm{B}$ 分别在 $O$ 点的两侧。引力常量为 $G$ 。\n\n在地月系统中, 若忽略其他星球的影响, 可以将月球和地球看成上述星球 $\\mathrm{A}$ 和 $\\mathrm{B}$,月球绕其轨道中心运行的周期记为 $T_{1}$ 。但在处理近似问题时, 常常认为月球是绕地心做圆周运动的, 这样算得的运行周期为 $T_{2}$ 。已知地球和月球的质量分别为 $m_{\\text {地 }}=6 \\times 10^{24} \\mathrm{~kg}$和 $m_{\\text {月 }}=7 \\times 10^{22} \\mathrm{~kg}$ 。求 $\\frac{T_{2}^{2}}{T_{1}^{2}}$ 。(结果保留三位有效数字)\n\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-031.jpg?height=429&width=488&top_left_y=154&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1096", "problem": "The Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\\text {peri }}=9.86 R_{\\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft.\n[figure1]\n\nFigure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman.\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nGiven that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.\n\nClose to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2.\n\n[figure2]\n\nFigure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL.\n\nWhen considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\\angle y c z$ in the figure, such that the shaded areas are the same).\n\n[figure3]\n\nFigure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia.\n\n\nThe eccentric anomaly can be related to the mean anomaly through Kepler's Equation,\n\n$$\nM=E-e \\sin E \\text {. }\n$$a. When the probe is at its closest perihelion:\n\n\nii. Calculate the temperature the heat shield must be able to survive. Assume that the heat shield of the probe absorbs all of the incident radiation, radiates as a perfect black body, and that only one side of the probe ever faces the Sun (to protect the instruments) such that the emitting (surface) area is double the absorbing (cross-sectional) area.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\\text {peri }}=9.86 R_{\\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft.\n[figure1]\n\nFigure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman.\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nGiven that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.\n\nClose to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2.\n\n[figure2]\n\nFigure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL.\n\nWhen considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\\angle y c z$ in the figure, such that the shaded areas are the same).\n\n[figure3]\n\nFigure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia.\n\n\nThe eccentric anomaly can be related to the mean anomaly through Kepler's Equation,\n\n$$\nM=E-e \\sin E \\text {. }\n$$\n\nproblem:\na. When the probe is at its closest perihelion:\n\n\nii. Calculate the temperature the heat shield must be able to survive. Assume that the heat shield of the probe absorbs all of the incident radiation, radiates as a perfect black body, and that only one side of the probe ever faces the Sun (to protect the instruments) such that the emitting (surface) area is double the absorbing (cross-sectional) area.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~K}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-04.jpg?height=708&width=1438&top_left_y=694&top_left_x=318", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-05.jpg?height=411&width=1539&top_left_y=383&top_left_x=264", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-05.jpg?height=603&width=714&top_left_y=1429&top_left_x=677" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~K}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_444", "problem": "2020 年左右我国将进行第一次火星探测, 美国已发射了“凤凰号”着陆器降落在 火星北极勘察是否有水的存在。如图所示为“凤凰号”着陆器经过多次变轨后登陆火 星的轨迹图, 轨道上的 $P 、 S 、 Q$ 三点与火星中心在同一直线上, $P 、 Q$ 两点分别是粗 圆轨道的远火星点和近火星点, 且 $P Q=2 Q S$, (已知轨道II为圆轨道) 下列说法正确的 是\n\n[图1]\nA: 着陆器在 $P$ 点由轨道I进入轨道II需要点火加速\nB: 着陆器在轨道II上 $S$ 点的速度小于在轨道III上 $Q$ 点的速度\nC: 着陆器在轨道II上 $S$ 点的加速度大小大于在轨道III上 $P$ 点的加速度大小\nD: 着陆器在轨道II上由 $P$ 点运动到 $S$ 点的时间是着陆器在轨道III上由 $P$ 点运动到 $Q$ 点的时间的 2 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年左右我国将进行第一次火星探测, 美国已发射了“凤凰号”着陆器降落在 火星北极勘察是否有水的存在。如图所示为“凤凰号”着陆器经过多次变轨后登陆火 星的轨迹图, 轨道上的 $P 、 S 、 Q$ 三点与火星中心在同一直线上, $P 、 Q$ 两点分别是粗 圆轨道的远火星点和近火星点, 且 $P Q=2 Q S$, (已知轨道II为圆轨道) 下列说法正确的 是\n\n[图1]\n\nA: 着陆器在 $P$ 点由轨道I进入轨道II需要点火加速\nB: 着陆器在轨道II上 $S$ 点的速度小于在轨道III上 $Q$ 点的速度\nC: 着陆器在轨道II上 $S$ 点的加速度大小大于在轨道III上 $P$ 点的加速度大小\nD: 着陆器在轨道II上由 $P$ 点运动到 $S$ 点的时间是着陆器在轨道III上由 $P$ 点运动到 $Q$ 点的时间的 2 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-32.jpg?height=397&width=422&top_left_y=1552&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_561", "problem": "地球赤道上的物体随地球自转的向心加速度为 $a$, 地球的同步卫星绕地球做匀速圆周运动的轨道半径为 $r_{1}$, 向心加速度为 $a_{1}$ 。已知万有引力常量为 $G$, 地球半径为 $R$, 地球赤道表面的加速度为 $g$ 。下列说法正确的是 ( )\nA: 地球质量 $M=\\frac{a R^{2}}{G}$\nB: 地球质量 $M=\\frac{a_{1} r_{1}^{2}}{G}$\nC: 加速度之比 $\\frac{a_{1}}{a}=\\frac{R^{2}}{r_{1}^{2}}$\nD: $a 、 a_{1} 、 g$ 的关系是 $aO B$, 且黑洞 $A$ 的半径大于黑洞 $B$ 的半径. 根据你所学的知识, 下列说法正确的是\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙\nA: 两黑洞质量之间的关系为 $M_{1}>M_{2}$\nB: 黑洞 $A$ 的第一宇宙速度小于黑洞 $B$ 的第一宇宙速度\nC: 若两黑洞间距离不变, 设法将黑洞 $A$ 上的一部分物质移到黑洞 $B$ 上, 则它们间的万有引力将变大\nD: 人类要将宇航器发射到距离黑洞 $A$ 或黑洞 $B$ 较近的区域进行探索, 发射速度一定大于 $16.7 \\mathrm{~km} / \\mathrm{s}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图甲所示, 河外星系中两黑洞 $A 、 B$ 的质量分别为 $M_{1}$ 和 $M_{2}$, 它们以两者连线上的某一点为圆心做匀速圆周运动. 为研究方便可简化为如图乙所示示意图, 黑洞 $A$ 和黑洞 $B$ 均可看成匀质球体, 天文学家测得 $O A>O B$, 且黑洞 $A$ 的半径大于黑洞 $B$ 的半径. 根据你所学的知识, 下列说法正确的是\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙\n\nA: 两黑洞质量之间的关系为 $M_{1}>M_{2}$\nB: 黑洞 $A$ 的第一宇宙速度小于黑洞 $B$ 的第一宇宙速度\nC: 若两黑洞间距离不变, 设法将黑洞 $A$ 上的一部分物质移到黑洞 $B$ 上, 则它们间的万有引力将变大\nD: 人类要将宇航器发射到距离黑洞 $A$ 或黑洞 $B$ 较近的区域进行探索, 发射速度一定大于 $16.7 \\mathrm{~km} / \\mathrm{s}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-40.jpg?height=225&width=348&top_left_y=1692&top_left_x=340", "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-40.jpg?height=97&width=251&top_left_y=1782&top_left_x=751" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_851", "problem": "Jupiter's deep atmosphere is very warm due to convection leading to an adiabatic temperature profile that increases with increasing pressure. Assuming (for simplicity) that this outer layer of Jupiter has a temperature of $500 \\mathrm{~K}$, perform a back-of-the-envelope estimate of the characteristic thickness (or e-folding scale) of the envelope of Jupiter (you may find that this is independent of pressure level). You may further use that the specific gas constant in Jupiter's atmosphere is $3600 \\mathrm{~J} \\mathrm{~kg}^{-1} \\mathrm{~K}^{-1}$.\nA: $20 \\mathrm{~km}$\nB: $73 \\mathrm{~km}$\nC: $568 \\mathrm{~km}$\nD: $3,120 \\mathrm{~km}$\nE: $10,233 \\mathrm{~km}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nJupiter's deep atmosphere is very warm due to convection leading to an adiabatic temperature profile that increases with increasing pressure. Assuming (for simplicity) that this outer layer of Jupiter has a temperature of $500 \\mathrm{~K}$, perform a back-of-the-envelope estimate of the characteristic thickness (or e-folding scale) of the envelope of Jupiter (you may find that this is independent of pressure level). You may further use that the specific gas constant in Jupiter's atmosphere is $3600 \\mathrm{~J} \\mathrm{~kg}^{-1} \\mathrm{~K}^{-1}$.\n\nA: $20 \\mathrm{~km}$\nB: $73 \\mathrm{~km}$\nC: $568 \\mathrm{~km}$\nD: $3,120 \\mathrm{~km}$\nE: $10,233 \\mathrm{~km}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_377", "problem": "两颗人造卫星绕地球运动周期相同, 轨道如图所示, 分别为圆轨道和椭圆轨道, $\\mathrm{AB}$为粗圆的长轴, $\\mathrm{C} 、 \\mathrm{D}$ 为两轨道交点. 已知椭圆轨道上的卫星到 $\\mathrm{C}$ 点时速度方向与 $\\mathrm{AB}$平行, 则下列说法中正确的是( )\n\n[图1]\nA: 卫星在圆轨道的速率为 $v_{0}$, 卫星椭圆轨道 $\\mathrm{A}$ 点的速率为 $v_{A}$, 则 $v_{0}>v_{A}$\nB: 卫星在圆轨道的速率为 $v_{0}$, 卫星在椭圆轨道 $\\mathrm{B}$ 点的速率为 $v_{B}$, 则 $v_{B}>v_{0}$\nC: 两个轨道上的卫星运动到 $\\mathrm{C}$ 点时的加速度相同\nD: 两个轨道上的卫星运动到 $\\mathrm{C}$ 点时的向心加速度大小相等\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n两颗人造卫星绕地球运动周期相同, 轨道如图所示, 分别为圆轨道和椭圆轨道, $\\mathrm{AB}$为粗圆的长轴, $\\mathrm{C} 、 \\mathrm{D}$ 为两轨道交点. 已知椭圆轨道上的卫星到 $\\mathrm{C}$ 点时速度方向与 $\\mathrm{AB}$平行, 则下列说法中正确的是( )\n\n[图1]\n\nA: 卫星在圆轨道的速率为 $v_{0}$, 卫星椭圆轨道 $\\mathrm{A}$ 点的速率为 $v_{A}$, 则 $v_{0}>v_{A}$\nB: 卫星在圆轨道的速率为 $v_{0}$, 卫星在椭圆轨道 $\\mathrm{B}$ 点的速率为 $v_{B}$, 则 $v_{B}>v_{0}$\nC: 两个轨道上的卫星运动到 $\\mathrm{C}$ 点时的加速度相同\nD: 两个轨道上的卫星运动到 $\\mathrm{C}$ 点时的向心加速度大小相等\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-66.jpg?height=288&width=400&top_left_y=2449&top_left_x=337", "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-67.jpg?height=54&width=1279&top_left_y=858&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_154", "problem": "宇航员在地球表面一斜坡上 $P$ 点, 沿水平方向以初速度 $v_{0}$ 抛出一个小球, 测得小球经时间 $t$ 落到斜坡另一点 $Q$ 上现宇航员站在某质量分布均匀的星球表面相同的斜坡上 $P$点, 沿水平方向以相同的初速度 $v_{0}$ 抛出一个小球, 小球落在 $P Q$ 的中点. 已知该星球的半径为 $R$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 球的体积公式是 $V=\\frac{4}{3} \\pi R^{3}$ 。求:\n 该星球的质量 $M$;\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n宇航员在地球表面一斜坡上 $P$ 点, 沿水平方向以初速度 $v_{0}$ 抛出一个小球, 测得小球经时间 $t$ 落到斜坡另一点 $Q$ 上现宇航员站在某质量分布均匀的星球表面相同的斜坡上 $P$点, 沿水平方向以相同的初速度 $v_{0}$ 抛出一个小球, 小球落在 $P Q$ 的中点. 已知该星球的半径为 $R$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 球的体积公式是 $V=\\frac{4}{3} \\pi R^{3}$ 。求:\n 该星球的质量 $M$;\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-051.jpg?height=271&width=443&top_left_y=827&top_left_x=335" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_896", "problem": "Where on the Moon did astronauts walk in July 1969 ?\nA: Mare Serenitatis\nB: Mare Tranquillitatis\nC: Mare Imbrium\nD: Mare Nubium\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhere on the Moon did astronauts walk in July 1969 ?\n\nA: Mare Serenitatis\nB: Mare Tranquillitatis\nC: Mare Imbrium\nD: Mare Nubium\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_194", "problem": "为了探知未知天体, 假如宇航员乘坐宇宙飞船到达某星球, 测得飞船在该星球表面附近做圆周运动的周期为 $T$ 。飞船降落到该星球表面后, 宇航员将小球从 $H$ 高处以初速度 $v_{0}$ 水平抛出, 落地时水平位移为 $x$, 忽略空气阻力和该星球的自转, 已知引力常量为 $G$ ，将该星球视为质量分布均匀的球体，则以下说法正确的是（）\nA: 该星球的半径为 $\\frac{H v_{0}^{2} T^{2}}{2 \\pi^{2} x^{2}}$\nB: 该星球的半径为 $\\frac{H v_{0}^{2} T^{2}}{4 \\pi^{2} x^{2}}$\nC: 该星球的质量为 $\\frac{H^{3} v_{0}^{6} T^{4}}{2 G \\pi^{4} x^{6}}$\nD: 该星球的质量为 $\\frac{H^{3} v_{0}^{6} T^{4}}{G \\pi^{4} x^{6}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n为了探知未知天体, 假如宇航员乘坐宇宙飞船到达某星球, 测得飞船在该星球表面附近做圆周运动的周期为 $T$ 。飞船降落到该星球表面后, 宇航员将小球从 $H$ 高处以初速度 $v_{0}$ 水平抛出, 落地时水平位移为 $x$, 忽略空气阻力和该星球的自转, 已知引力常量为 $G$ ，将该星球视为质量分布均匀的球体，则以下说法正确的是（）\n\nA: 该星球的半径为 $\\frac{H v_{0}^{2} T^{2}}{2 \\pi^{2} x^{2}}$\nB: 该星球的半径为 $\\frac{H v_{0}^{2} T^{2}}{4 \\pi^{2} x^{2}}$\nC: 该星球的质量为 $\\frac{H^{3} v_{0}^{6} T^{4}}{2 G \\pi^{4} x^{6}}$\nD: 该星球的质量为 $\\frac{H^{3} v_{0}^{6} T^{4}}{G \\pi^{4} x^{6}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1219", "problem": "In November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \\mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked.\n[figure1]\n\nFigure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC.\n\nRight: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \\theta$. Credit: OpenStax, College Physics.\n\nMercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres.\n\nThe radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \\mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \\mu \\mathrm{s}$ pulses at $430 \\mathrm{MHz}$ with a transmitted power of $2.0 \\mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1.\n\nThe signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \\mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2).\n\nPhotons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole).\n\n[figure2]\n\nFigure 2: Left: Snapshots of the reflections of a single $100 \\mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \\mathrm{c} / \\mathrm{s}=1 \\mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967).\n\nTop right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967).\n\nBottom right: The same as the snapshots, but this time summed over the first $500 \\mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\\sim 5 \\mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967).\n\nThe Doppler shift with light is given as\n\n$$\n\\frac{\\Delta f}{f}=\\frac{v}{c}\n$$\n\nwhere $\\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light.\n\nEver since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time.\n\n[figure3]\n\nFigure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966).c. The eccentricity of the planet's orbit became the prime suspect as to why its actual ratio would be stable over long time periods.\n\nii. Assuming the tidal torque at perihelion is the dominating factor in setting Mercury's rotation rate, predict the rotational period of Mercury if it were to behave as though it was tidally locked when passing through perihelion. Compare this to the measured value and comment on validity of the assumption.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \\mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked.\n[figure1]\n\nFigure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC.\n\nRight: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \\theta$. Credit: OpenStax, College Physics.\n\nMercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres.\n\nThe radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \\mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \\mu \\mathrm{s}$ pulses at $430 \\mathrm{MHz}$ with a transmitted power of $2.0 \\mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1.\n\nThe signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \\mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2).\n\nPhotons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole).\n\n[figure2]\n\nFigure 2: Left: Snapshots of the reflections of a single $100 \\mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \\mathrm{c} / \\mathrm{s}=1 \\mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967).\n\nTop right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967).\n\nBottom right: The same as the snapshots, but this time summed over the first $500 \\mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\\sim 5 \\mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967).\n\nThe Doppler shift with light is given as\n\n$$\n\\frac{\\Delta f}{f}=\\frac{v}{c}\n$$\n\nwhere $\\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light.\n\nEver since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time.\n\n[figure3]\n\nFigure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966).\n\nproblem:\nc. The eccentricity of the planet's orbit became the prime suspect as to why its actual ratio would be stable over long time periods.\n\nii. Assuming the tidal torque at perihelion is the dominating factor in setting Mercury's rotation rate, predict the rotational period of Mercury if it were to behave as though it was tidally locked when passing through perihelion. Compare this to the measured value and comment on validity of the assumption.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\text { days }, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-04.jpg?height=512&width=1374&top_left_y=652&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-05.jpg?height=1094&width=1560&top_left_y=218&top_left_x=248", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-06.jpg?height=994&width=897&top_left_y=1359&top_left_x=585" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\text { days }" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_563", "problem": "建造一条能通向太空的电梯 (如图甲所示), 是人们长期的梦想。材料的力学强度是材料众多性能中被人类极为看重的一种性能, 目前已发现的高强度材料碳纳米管的抗拉强度是钢的 100 倍, 密度是其 $\\frac{1}{6}$ ，这使得人们有望在赤道上建造垂直于水平面的“太空电梯”。图乙中 $r$ 为航天员到地心的距离, $R$ 为地球半径, $a-r$ 图像中的图线 $A$ 表示地球引力对航天员产生的加速度大小与 $r$ 的关系, 图线 $B$ 表示航天员由于地球自转而产生的向心加速度大小与 $r$ 的关系, 关于相对地面静止在不同高度的航天员, 地面附近重力加速度 $g$ 取 $10 \\mathrm{~m} / \\mathrm{s}^{2}$, 地球自转角速度 $\\omega=7.3 \\times 10^{-5} \\mathrm{rad} / \\mathrm{s}$, 地球半径 $R=6.4 \\times 10^{3} \\mathrm{~km}$ 。下列说法正确的有 ( )\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙\nA: 随着 $r$ 增大, 航天员受到电梯舱的弹力减小\nB: 航天员在 $r=R$ 处的线速度等于第一宇宙速度\nC: 图中 $r_{0}$ 为地球同步卫星的轨道半径\nD: 电梯舱停在距地面高度为 $6.6 R$ 的站点时, 舱内质量 $60 \\mathrm{~kg}$ 的航天员对水平地板的压力为零\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n建造一条能通向太空的电梯 (如图甲所示), 是人们长期的梦想。材料的力学强度是材料众多性能中被人类极为看重的一种性能, 目前已发现的高强度材料碳纳米管的抗拉强度是钢的 100 倍, 密度是其 $\\frac{1}{6}$ ，这使得人们有望在赤道上建造垂直于水平面的“太空电梯”。图乙中 $r$ 为航天员到地心的距离, $R$ 为地球半径, $a-r$ 图像中的图线 $A$ 表示地球引力对航天员产生的加速度大小与 $r$ 的关系, 图线 $B$ 表示航天员由于地球自转而产生的向心加速度大小与 $r$ 的关系, 关于相对地面静止在不同高度的航天员, 地面附近重力加速度 $g$ 取 $10 \\mathrm{~m} / \\mathrm{s}^{2}$, 地球自转角速度 $\\omega=7.3 \\times 10^{-5} \\mathrm{rad} / \\mathrm{s}$, 地球半径 $R=6.4 \\times 10^{3} \\mathrm{~km}$ 。下列说法正确的有 ( )\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙\n\nA: 随着 $r$ 增大, 航天员受到电梯舱的弹力减小\nB: 航天员在 $r=R$ 处的线速度等于第一宇宙速度\nC: 图中 $r_{0}$ 为地球同步卫星的轨道半径\nD: 电梯舱停在距地面高度为 $6.6 R$ 的站点时, 舱内质量 $60 \\mathrm{~kg}$ 的航天员对水平地板的压力为零\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-118.jpg?height=377&width=545&top_left_y=1839&top_left_x=344", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-118.jpg?height=409&width=488&top_left_y=1823&top_left_x=1001" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_604", "problem": "木星有众多卫星, 木卫三是其中最大的一颗, 其直径大于行星中的水星。假设木卫三绕木星做匀速圆周运动的轨道半径为 $r_{1}$, 运行周期为 $T_{1}$, 木星半径为 $R$ 。已知行星与卫星间引力势能的表达式为 $E_{P}=-\\frac{G M_{0} m_{0}}{r}, r$ 为行星与卫星的中心距离, 则木星的第二宇宙速度为 ( )\nA: $\\frac{\\pi r_{1}}{T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\nB: $\\frac{\\pi r_{1}}{T_{1}} \\sqrt{\\frac{r_{1}}{R}}$\nC: $\\frac{2 \\pi r_{1}}{R T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\nD: $\\frac{2 \\pi r_{1}}{T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n木星有众多卫星, 木卫三是其中最大的一颗, 其直径大于行星中的水星。假设木卫三绕木星做匀速圆周运动的轨道半径为 $r_{1}$, 运行周期为 $T_{1}$, 木星半径为 $R$ 。已知行星与卫星间引力势能的表达式为 $E_{P}=-\\frac{G M_{0} m_{0}}{r}, r$ 为行星与卫星的中心距离, 则木星的第二宇宙速度为 ( )\n\nA: $\\frac{\\pi r_{1}}{T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\nB: $\\frac{\\pi r_{1}}{T_{1}} \\sqrt{\\frac{r_{1}}{R}}$\nC: $\\frac{2 \\pi r_{1}}{R T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\nD: $\\frac{2 \\pi r_{1}}{T_{1}} \\sqrt{\\frac{2 r_{1}}{R}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_268", "problem": "据央视报道, 5 月 31 日搭载两名美国宇航员的美国太空探索技术公司 SpaceX 龙飞船发射成功, 乘“猎鹰 9 号”火箭飞往国际空间站。并在 31 日成功与国际空间站对接,把两位航天员送入国际空间站。龙飞船发射过程可简化为: 先让飞船进入一个近地的圆轨道 I, 然后在 $P$ 点变轨, 进入椭圆形转移轨道 II (该椭圆轨道的近地点与近地圆轨道 I 相切于 $P$ 点, 远地点与最终运行轨道 III 相切于 $Q$ 点), 到达远地点 $Q$ 时再次变轨, 进入圆形轨道 III。关于飞船的运动, 下列说法正确的是（ ）\n\n[图1]\nA: 飞船沿轨道 $\\mathrm{I}$, 经过 $P$ 点时需要的向心力小于沿轨道 II 经过 $P$ 点时需要的向心力\nB: 飞船沿转移轨道 II 运行到远地点 $Q$ 点时的速率 $v_{3}$ 等于在轨道 III 上运行的速率 $v_{4}$\nC: 飞船沿轨道 I 经过 $P$ 点时受到的万有引力小于沿轨道 II 经过 $P$ 点时受到的万有引力\nD: 飞船在轨道 I 上运行的速率 $v_{1}$ 等于在轨道 II 运行经过 $P$ 点时的速率 $v_{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n据央视报道, 5 月 31 日搭载两名美国宇航员的美国太空探索技术公司 SpaceX 龙飞船发射成功, 乘“猎鹰 9 号”火箭飞往国际空间站。并在 31 日成功与国际空间站对接,把两位航天员送入国际空间站。龙飞船发射过程可简化为: 先让飞船进入一个近地的圆轨道 I, 然后在 $P$ 点变轨, 进入椭圆形转移轨道 II (该椭圆轨道的近地点与近地圆轨道 I 相切于 $P$ 点, 远地点与最终运行轨道 III 相切于 $Q$ 点), 到达远地点 $Q$ 时再次变轨, 进入圆形轨道 III。关于飞船的运动, 下列说法正确的是（ ）\n\n[图1]\n\nA: 飞船沿轨道 $\\mathrm{I}$, 经过 $P$ 点时需要的向心力小于沿轨道 II 经过 $P$ 点时需要的向心力\nB: 飞船沿转移轨道 II 运行到远地点 $Q$ 点时的速率 $v_{3}$ 等于在轨道 III 上运行的速率 $v_{4}$\nC: 飞船沿轨道 I 经过 $P$ 点时受到的万有引力小于沿轨道 II 经过 $P$ 点时受到的万有引力\nD: 飞船在轨道 I 上运行的速率 $v_{1}$ 等于在轨道 II 运行经过 $P$ 点时的速率 $v_{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-035.jpg?height=394&width=440&top_left_y=1339&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_202", "problem": "北斗卫星导航系统组网成功后会有 5 颗静止轨道卫星。已知地球赤道上的重力加速\n度大小为 $g$, 将地球视为半径为 $R_{0}$ 的球体, 地球的自转周期为 $T_{0}$, 则地球静止轨道卫星（同步卫星）的运行半径为（）\nA: $\\sqrt{R_{0}{ }^{3}+\\frac{g R_{0}{ }^{2} T_{0}{ }^{2}}{4 \\pi^{2}}}$\nB: $\\sqrt[3]{\\frac{g R_{0}{ }^{2} T_{0}{ }^{2}-R_{0}{ }^{3}}{4 \\pi^{2}}}$\nC: $\\sqrt[3]{R_{0}{ }^{3}+\\frac{g R_{0}{ }^{2} T_{0}{ }^{2}}{4 \\pi^{2}}}$\nD: $\\sqrt{\\frac{g R_{0} T_{0}}{2 \\pi}+R_{0}^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n北斗卫星导航系统组网成功后会有 5 颗静止轨道卫星。已知地球赤道上的重力加速\n度大小为 $g$, 将地球视为半径为 $R_{0}$ 的球体, 地球的自转周期为 $T_{0}$, 则地球静止轨道卫星（同步卫星）的运行半径为（）\n\nA: $\\sqrt{R_{0}{ }^{3}+\\frac{g R_{0}{ }^{2} T_{0}{ }^{2}}{4 \\pi^{2}}}$\nB: $\\sqrt[3]{\\frac{g R_{0}{ }^{2} T_{0}{ }^{2}-R_{0}{ }^{3}}{4 \\pi^{2}}}$\nC: $\\sqrt[3]{R_{0}{ }^{3}+\\frac{g R_{0}{ }^{2} T_{0}{ }^{2}}{4 \\pi^{2}}}$\nD: $\\sqrt{\\frac{g R_{0} T_{0}}{2 \\pi}+R_{0}^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1204", "problem": "In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.d. Taking the Starkiller Base's ice planet to have a diameter of $660 \\mathrm{~km}$, show that the Sun can be safely contained, even if it was fully drained.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.\n\nproblem:\nd. Taking the Starkiller Base's ice planet to have a diameter of $660 \\mathrm{~km}$, show that the Sun can be safely contained, even if it was fully drained.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of m, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-06.jpg?height=611&width=1448&top_left_y=505&top_left_x=310" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_597", "problem": "靠近地面运行的近地卫星的加速度大小为 $a_{1}$, 地球同步轨道上的卫星的加速度大小为 $a_{2}$, 赤道上随地球一同运转 (相对地面静止) 的物体的加速度大小为 $a_{3}$, 则 ( )\nA: $a_{1}=a_{3}>a_{2}$\nB: $a_{1}>a_{2}>a_{3}$\nC: $a_{1}>a_{3}>a_{2}$\nD: $a_{3}>a_{2}>a_{1}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n靠近地面运行的近地卫星的加速度大小为 $a_{1}$, 地球同步轨道上的卫星的加速度大小为 $a_{2}$, 赤道上随地球一同运转 (相对地面静止) 的物体的加速度大小为 $a_{3}$, 则 ( )\n\nA: $a_{1}=a_{3}>a_{2}$\nB: $a_{1}>a_{2}>a_{3}$\nC: $a_{1}>a_{3}>a_{2}$\nD: $a_{3}>a_{2}>a_{1}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_354", "problem": "2020 年 6 月 23 日, 我国北斗卫星导航系统最后一颗组网卫星成功发射, 这是一颗同步卫星。发射此类卫星时, 通常先将卫星发送到一个椭圆轨道上, 其近地点 $M$ 距地面高 $h_{1}$, 远地点 $N$ 距地面高 $h_{2}$, 进入该轨道正常运行时, 其周期为 $T_{1}$, 机械能为 $E_{1}$,\n通过 $M 、 N$ 两点时的速率分别是 $v_{1} 、 v_{2}$, 加速度大小分别是 $a_{1} 、 a_{2}$ 。当某次飞船通过 $N$点时, 地面指挥部发出指令, 点燃飞船上的发动机, 使飞船在短时间内加速后进入离地面高 $h_{2}$ 的圆形轨道, 开始绕地球做匀速圆周运动, 这时飞船的运动周期为 $T_{2}$, 速率为 $v_{3}$,加速度大小为 $a_{3}$, 机械能为 $E_{2}$ 。下列结论正确的是 ( )\n(1) $v_{1}>v_{3}$\n(2) $E_{2}>E_{1}$\n(3) $a_{2}>a_{3}$\n(4) $T_{1}>T_{2}$\n\n[图1]\nA: (1)(2)(3)\nB: (2)(3)\nC: (1)(2)\nD: (3)(4)\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年 6 月 23 日, 我国北斗卫星导航系统最后一颗组网卫星成功发射, 这是一颗同步卫星。发射此类卫星时, 通常先将卫星发送到一个椭圆轨道上, 其近地点 $M$ 距地面高 $h_{1}$, 远地点 $N$ 距地面高 $h_{2}$, 进入该轨道正常运行时, 其周期为 $T_{1}$, 机械能为 $E_{1}$,\n通过 $M 、 N$ 两点时的速率分别是 $v_{1} 、 v_{2}$, 加速度大小分别是 $a_{1} 、 a_{2}$ 。当某次飞船通过 $N$点时, 地面指挥部发出指令, 点燃飞船上的发动机, 使飞船在短时间内加速后进入离地面高 $h_{2}$ 的圆形轨道, 开始绕地球做匀速圆周运动, 这时飞船的运动周期为 $T_{2}$, 速率为 $v_{3}$,加速度大小为 $a_{3}$, 机械能为 $E_{2}$ 。下列结论正确的是 ( )\n(1) $v_{1}>v_{3}$\n(2) $E_{2}>E_{1}$\n(3) $a_{2}>a_{3}$\n(4) $T_{1}>T_{2}$\n\n[图1]\n\nA: (1)(2)(3)\nB: (2)(3)\nC: (1)(2)\nD: (3)(4)\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-02.jpg?height=343&width=352&top_left_y=571&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_631", "problem": "中国新闻网宣布: 在摩洛哥坠落的陨石被证实来自火星。某同学想根据平时收集的部分火星资料 (如图所示) 计算出火星的密度, 再与这颗陨石的密度进行比较。下列计\n\n| 火星-Mars |\n| :--- |\n| 火星的小档案 |\n| 直径 $d=6794 \\mathrm{~km}$ |\n| 质量 $M=6.4219 \\times 10^{23} \\mathrm{~kg}$ |\n| 表面重力加速度 $g_{0}=3.7 \\mathrm{~m} / \\mathrm{s}^{2}$ |\n| 近地卫星周期 $T=3.4 \\mathrm{~h}$ |\nA: $\\rho=\\frac{3 g_{0}}{2 \\pi G d}$\nB: $\\rho=\\frac{g_{0} T^{2}}{3 \\pi d}$\nC: $\\rho=\\frac{3 \\pi}{G T^{2}}$\nD: $\\rho=\\frac{6 M}{\\pi d^{3}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n中国新闻网宣布: 在摩洛哥坠落的陨石被证实来自火星。某同学想根据平时收集的部分火星资料 (如图所示) 计算出火星的密度, 再与这颗陨石的密度进行比较。下列计\n\n| 火星-Mars |\n| :--- |\n| 火星的小档案 |\n| 直径 $d=6794 \\mathrm{~km}$ |\n| 质量 $M=6.4219 \\times 10^{23} \\mathrm{~kg}$ |\n| 表面重力加速度 $g_{0}=3.7 \\mathrm{~m} / \\mathrm{s}^{2}$ |\n| 近地卫星周期 $T=3.4 \\mathrm{~h}$ |\n\nA: $\\rho=\\frac{3 g_{0}}{2 \\pi G d}$\nB: $\\rho=\\frac{g_{0} T^{2}}{3 \\pi d}$\nC: $\\rho=\\frac{3 \\pi}{G T^{2}}$\nD: $\\rho=\\frac{6 M}{\\pi d^{3}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_672", "problem": "宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 。在星球 $\\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 P 轻放在弹簧上端, 如图 (a) 所示, P 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\\mathrm{B}$ 上用完全相同的弹簧和物体 $\\mathrm{P}$ 完成同样的过程, 其 $a-x$ 关系如图 (b) 中虚线所示（图中 $a_{0}$未知）。已知两星球密度相等。星球 $\\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。\n\n求星球 B 的质量 $M$;\n\n\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 。在星球 $\\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 P 轻放在弹簧上端, 如图 (a) 所示, P 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\\mathrm{B}$ 上用完全相同的弹簧和物体 $\\mathrm{P}$ 完成同样的过程, 其 $a-x$ 关系如图 (b) 中虚线所示（图中 $a_{0}$未知）。已知两星球密度相等。星球 $\\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。\n\n求星球 B 的质量 $M$;\n\n\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-108.jpg?height=206&width=254&top_left_y=1299&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-108.jpg?height=369&width=348&top_left_y=1186&top_left_x=611" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_774", "problem": "What does the astronomical term ecliptic describe?\nA: The path of the Sun in the sky throughout a year.\nB: The axial tilt of the Earth throughout a year.\nC: The movement of the stars due to Earth's rotation.\nD: The central line through the axis of rotation.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat does the astronomical term ecliptic describe?\n\nA: The path of the Sun in the sky throughout a year.\nB: The axial tilt of the Earth throughout a year.\nC: The movement of the stars due to Earth's rotation.\nD: The central line through the axis of rotation.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_909", "problem": "NASA's Mars 2020 mission involves a large rover called Perseverance, and an audacious experiment to see if motor driven flight is possible on the red planet. The helicopter drone is called Ingenuity, shown in Figure 2, and is the first attempt to fly on another world.\n\n[figure1]\n\nFigure 2: The Ingenuity drone with its helicopter rotors ready to take off, with the solar panel used to charge its battery on the very top. In the background you can see a part of the Perseverance rover. Credit: NASA/JPL-Caltech.\n\nAlthough gravity is weaker on Mars, the Martian atmosphere is only about $1 \\%$ the density of that on Earth, which makes it very hard to get lift. To overcome this, the probe has two counterrotating blades (directly above each other) with a tip-to-tip length of $1.2 \\mathrm{~m}$ that will spin at 2400 rpm (about 5 times faster than helicopters on Earth), and is incredibly light with a mass of $1.8 \\mathrm{~kg}$. A consequence of the weight restriction is that the battery is small (and charged by a solar panel on top of the drone, above the blades). Since flying in such a thin atmosphere is a very high power activity the maximum flight duration is therefore short, and given the time needed to recharge, they will be limited to only one flight per day.\n\nIf the active region of the solar panel has an area of $544 \\mathrm{~cm}^{2}$ and Mars takes 687 days to orbit the Sun, determine the minimum amount of time (in minutes) the battery needs to charge for between flights. Assume the panel is $30 \\%$ efficient and the Sun is directly overhead throughout.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nNASA's Mars 2020 mission involves a large rover called Perseverance, and an audacious experiment to see if motor driven flight is possible on the red planet. The helicopter drone is called Ingenuity, shown in Figure 2, and is the first attempt to fly on another world.\n\n[figure1]\n\nFigure 2: The Ingenuity drone with its helicopter rotors ready to take off, with the solar panel used to charge its battery on the very top. In the background you can see a part of the Perseverance rover. Credit: NASA/JPL-Caltech.\n\nAlthough gravity is weaker on Mars, the Martian atmosphere is only about $1 \\%$ the density of that on Earth, which makes it very hard to get lift. To overcome this, the probe has two counterrotating blades (directly above each other) with a tip-to-tip length of $1.2 \\mathrm{~m}$ that will spin at 2400 rpm (about 5 times faster than helicopters on Earth), and is incredibly light with a mass of $1.8 \\mathrm{~kg}$. A consequence of the weight restriction is that the battery is small (and charged by a solar panel on top of the drone, above the blades). Since flying in such a thin atmosphere is a very high power activity the maximum flight duration is therefore short, and given the time needed to recharge, they will be limited to only one flight per day.\n\nIf the active region of the solar panel has an area of $544 \\mathrm{~cm}^{2}$ and Mars takes 687 days to orbit the Sun, determine the minimum amount of time (in minutes) the battery needs to charge for between flights. Assume the panel is $30 \\%$ efficient and the Sun is directly overhead throughout.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of mins, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c3e3c992a9c51eb3e471g-07.jpg?height=731&width=1285&top_left_y=500&top_left_x=385" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "mins" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_775", "problem": "Neptune's diameter is similar to ...\nA: Venus\nB: Jupiter\nC: Saturn\nD: Uranus\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nNeptune's diameter is similar to ...\n\nA: Venus\nB: Jupiter\nC: Saturn\nD: Uranus\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_347", "problem": "2020 年 7 月 23 日, 我国的“天问一号”火星探测器, 搭乘着长征五号遥四运载火箭,成功从地球飞向了火星。如图所示为“天问一号”发射过程的示意图, 从地球上发射之后经过地火转移轨道被火星捕获, 进入环火星圆轨道, 经变轨调整后, 进入着陆准备轨道。已知“天问一号” 火星探测器在轨道半径为 $r$ 的环火星圆轨道上运动时, 周期为 $T_{1}$, 在半长轴为 $a$ 的着陆准备轨道上运动时, 周期为 $T_{2}$, 则下列判断正确的是 ( )\n\n地火转移轨道地球\n[图1]\nA: 火星的平均密度一定大于 $\\frac{3 \\pi}{G T_{1}^{2}}$\nB: “天问一号”从环火星圆轨道进入着陆准备轨道时需减速\nC: “天问一号”在环火星圆轨道和着陆准备轨道上运动时满足 $\\frac{r^{2}}{T_{1}^{3}}=\\frac{a^{2}}{T_{2}^{3}}$\nD: “天问一号”在环火星圆轨道上的机械能大于其在着陆准备轨道上的机械能\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2020 年 7 月 23 日, 我国的“天问一号”火星探测器, 搭乘着长征五号遥四运载火箭,成功从地球飞向了火星。如图所示为“天问一号”发射过程的示意图, 从地球上发射之后经过地火转移轨道被火星捕获, 进入环火星圆轨道, 经变轨调整后, 进入着陆准备轨道。已知“天问一号” 火星探测器在轨道半径为 $r$ 的环火星圆轨道上运动时, 周期为 $T_{1}$, 在半长轴为 $a$ 的着陆准备轨道上运动时, 周期为 $T_{2}$, 则下列判断正确的是 ( )\n\n地火转移轨道地球\n[图1]\n\nA: 火星的平均密度一定大于 $\\frac{3 \\pi}{G T_{1}^{2}}$\nB: “天问一号”从环火星圆轨道进入着陆准备轨道时需减速\nC: “天问一号”在环火星圆轨道和着陆准备轨道上运动时满足 $\\frac{r^{2}}{T_{1}^{3}}=\\frac{a^{2}}{T_{2}^{3}}$\nD: “天问一号”在环火星圆轨道上的机械能大于其在着陆准备轨道上的机械能\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-052.jpg?height=496&width=864&top_left_y=2022&top_left_x=320" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_301", "problem": "由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心在三角形所在的平面内做角速度相等的圆周运动, 如图所示。已知星体 $\\mathrm{A}$ 的质量为 $4 m$, 星体 $\\mathrm{B} 、 \\mathrm{C}$ 的质量均为 $m$, 三角形边长为 $d$ 。求:\n\n 星体 $\\mathrm{A}$ 所受的合力大小 $F_{\\mathrm{A}}$;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心在三角形所在的平面内做角速度相等的圆周运动, 如图所示。已知星体 $\\mathrm{A}$ 的质量为 $4 m$, 星体 $\\mathrm{B} 、 \\mathrm{C}$ 的质量均为 $m$, 三角形边长为 $d$ 。求:\n\n 星体 $\\mathrm{A}$ 所受的合力大小 $F_{\\mathrm{A}}$;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-094.jpg?height=417&width=445&top_left_y=2193&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_723", "problem": "如图所示, 有一质量为 $M$ 、半径为 $R$ 、密度均匀的球体, 在距离球心 $O$ 为 $2 R$ 的 $P$点有一质量为 $m$ 的质点, 从 $M$ 中挖去一个半径为 $\\frac{1}{2} R$ 的球体, 设大球剩余部分对 $m$ 的万有引力为 $F_{1}$ 。若把质点 $m$ 移放在空腔中心 $O^{\\prime}$ 点, 设大球的剩余部分对该质点的万有引力为 $F_{2}$ 。已知质量分布均匀的球壳对壳内物体的引力为 0 , 万有引力常量为 $G, O$ 、 $O^{\\prime} 、 P$ 三点共线。下列说法正确的是（）\n\n[图1]\nA: $F_{1}$ 的大小为 $\\frac{7 G M m}{36 R^{2}}$\nB: $F_{2}$ 的大小为 $\\frac{G M m}{4 R^{2}}$\nC: 若把质点 $m$ 移放在 $O$ 点右侧, 距 $O$ 点 $\\frac{3}{4} R$ 处, 大球的剩余部分对该质点的万有引力与 $F_{2}$ 相同\nD: 若把质点 $m$ 移放在 $O$ 点右侧, 距 $O$ 点 $\\frac{3}{4} R$ 处, 大球的剩余部分对该质点的万有引力与 $F_{2}$ 不同\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 有一质量为 $M$ 、半径为 $R$ 、密度均匀的球体, 在距离球心 $O$ 为 $2 R$ 的 $P$点有一质量为 $m$ 的质点, 从 $M$ 中挖去一个半径为 $\\frac{1}{2} R$ 的球体, 设大球剩余部分对 $m$ 的万有引力为 $F_{1}$ 。若把质点 $m$ 移放在空腔中心 $O^{\\prime}$ 点, 设大球的剩余部分对该质点的万有引力为 $F_{2}$ 。已知质量分布均匀的球壳对壳内物体的引力为 0 , 万有引力常量为 $G, O$ 、 $O^{\\prime} 、 P$ 三点共线。下列说法正确的是（）\n\n[图1]\n\nA: $F_{1}$ 的大小为 $\\frac{7 G M m}{36 R^{2}}$\nB: $F_{2}$ 的大小为 $\\frac{G M m}{4 R^{2}}$\nC: 若把质点 $m$ 移放在 $O$ 点右侧, 距 $O$ 点 $\\frac{3}{4} R$ 处, 大球的剩余部分对该质点的万有引力与 $F_{2}$ 相同\nD: 若把质点 $m$ 移放在 $O$ 点右侧, 距 $O$ 点 $\\frac{3}{4} R$ 处, 大球的剩余部分对该质点的万有引力与 $F_{2}$ 不同\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-065.jpg?height=274&width=500&top_left_y=911&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_99", "problem": "宇宙中有一星球, 其半径为 $R$, 自转周期为 $T$, 引力常量为 $G$, 该天体的质量为 $M$ 。若一宇航员来到位于赤道的一斜坡前, 将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示, 小球垂直击中了斜坡, 落点为 $P$ 点, 求\n\n该星球赤道地面处的重力加速度 $g_{1}$;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n宇宙中有一星球, 其半径为 $R$, 自转周期为 $T$, 引力常量为 $G$, 该天体的质量为 $M$ 。若一宇航员来到位于赤道的一斜坡前, 将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示, 小球垂直击中了斜坡, 落点为 $P$ 点, 求\n\n该星球赤道地面处的重力加速度 $g_{1}$;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-114.jpg?height=340&width=648&top_left_y=150&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_344", "problem": "行星 $\\mathrm{A}$ 和行星 $\\mathrm{B}$ 是两个均匀球体, 行星 $\\mathrm{A}$ 的卫星沿圆轨道运行的周期为 $T_{A}$, 行星 $B$ 的卫星沿圆轨道运行的周期为 $T_{B}$, 两卫星绕各自行星的近表面轨道运行, 已知 $T_{A}: T_{B}=1: 4$, 行星 $\\mathrm{A} 、 \\mathrm{~B}$ 的半径之比为 $R_{\\mathrm{A}}: R_{\\mathrm{B}}=1: 2$, 则 ()\nA: 这两颗行星的质量之比 $m_{\\mathrm{A}}: m_{\\mathrm{B}}=2: 1$\nB: 这两颗行星表面的重力加速度之比 $g_{A}: g_{B}=2: 1$\nC: 这两颗行星的密度之比 $\\rho_{\\mathrm{A}}: \\rho_{\\mathrm{B}}=16: 1$\nD: 这两颗行星的同步卫星周期之比 $T_{\\mathrm{A}}: T_{\\mathrm{B}}=1: \\sqrt{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n行星 $\\mathrm{A}$ 和行星 $\\mathrm{B}$ 是两个均匀球体, 行星 $\\mathrm{A}$ 的卫星沿圆轨道运行的周期为 $T_{A}$, 行星 $B$ 的卫星沿圆轨道运行的周期为 $T_{B}$, 两卫星绕各自行星的近表面轨道运行, 已知 $T_{A}: T_{B}=1: 4$, 行星 $\\mathrm{A} 、 \\mathrm{~B}$ 的半径之比为 $R_{\\mathrm{A}}: R_{\\mathrm{B}}=1: 2$, 则 ()\n\nA: 这两颗行星的质量之比 $m_{\\mathrm{A}}: m_{\\mathrm{B}}=2: 1$\nB: 这两颗行星表面的重力加速度之比 $g_{A}: g_{B}=2: 1$\nC: 这两颗行星的密度之比 $\\rho_{\\mathrm{A}}: \\rho_{\\mathrm{B}}=16: 1$\nD: 这两颗行星的同步卫星周期之比 $T_{\\mathrm{A}}: T_{\\mathrm{B}}=1: \\sqrt{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_858", "problem": "David is walking down MIT's infinite corridor (latitude $42^{\\circ} 21^{\\prime} 33^{\\prime \\prime}$ ) when he suddenly sees the sun aligning with the window at the end of the corridor. Being the observational master he is, David immediately pulls out his compass and measures the Sun to be at an azimuth of $245.81^{\\circ}$. Forgetting to bring his jacket, he is painfully reminded as he walks outside that it has been less than 6 months since the previous winter solstice. Which of the following choices is closest to the current date? Assume the corridor is parallel to the surface of the Earth.\nA: January 15\nB: January 30\nC: February 15\nD: March 20\nE: April 1\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nDavid is walking down MIT's infinite corridor (latitude $42^{\\circ} 21^{\\prime} 33^{\\prime \\prime}$ ) when he suddenly sees the sun aligning with the window at the end of the corridor. Being the observational master he is, David immediately pulls out his compass and measures the Sun to be at an azimuth of $245.81^{\\circ}$. Forgetting to bring his jacket, he is painfully reminded as he walks outside that it has been less than 6 months since the previous winter solstice. Which of the following choices is closest to the current date? Assume the corridor is parallel to the surface of the Earth.\n\nA: January 15\nB: January 30\nC: February 15\nD: March 20\nE: April 1\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1181", "problem": "On $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.c. By empirically fitting a sinusoidal function (which is assumed to be the same for each track, just with a fixed phase difference between them) and assuming all conjunctions are separated by the average synodic period, we can give rough estimations for the separations of any given great conjunction. Note: be careful as your calculations will be very sensitive to rounding errors.\n\nii. Without having to read anything else off the graph, write down the equations for Tracks B and $C$, given the same restrictions on $\\phi_{B}$ and $\\phi_{C}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nOn $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.\n\nproblem:\nc. By empirically fitting a sinusoidal function (which is assumed to be the same for each track, just with a fixed phase difference between them) and assuming all conjunctions are separated by the average synodic period, we can give rough estimations for the separations of any given great conjunction. Note: be careful as your calculations will be very sensitive to rounding errors.\n\nii. Without having to read anything else off the graph, write down the equations for Tracks B and $C$, given the same restrictions on $\\phi_{B}$ and $\\phi_{C}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-07.jpg?height=706&width=1564&top_left_y=834&top_left_x=244", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-08.jpg?height=578&width=1566&top_left_y=196&top_left_x=242", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-09.jpg?height=1072&width=1564&top_left_y=1191&top_left_x=246" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_988", "problem": "There are approximately 450 geostationary satellites currently in orbit. They have an orbital period of 1 day and orbit in the plane of the equator, so are therefore always directly above the same spot. Assuming circular orbits and an equidistant arrangement, what is the distance between two geostationary satellites?\nA: $560 \\mathrm{~km}$\nB: $570 \\mathrm{~km}$\nC: $580 \\mathrm{~km}$\nD: $590 \\mathrm{~km}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThere are approximately 450 geostationary satellites currently in orbit. They have an orbital period of 1 day and orbit in the plane of the equator, so are therefore always directly above the same spot. Assuming circular orbits and an equidistant arrangement, what is the distance between two geostationary satellites?\n\nA: $560 \\mathrm{~km}$\nB: $570 \\mathrm{~km}$\nC: $580 \\mathrm{~km}$\nD: $590 \\mathrm{~km}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1119", "problem": "In November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \\mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked.\n[figure1]\n\nFigure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC.\n\nRight: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \\theta$. Credit: OpenStax, College Physics.\n\nMercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres.\n\nThe radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \\mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \\mu \\mathrm{s}$ pulses at $430 \\mathrm{MHz}$ with a transmitted power of $2.0 \\mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1.\n\nThe signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \\mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2).\n\nPhotons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole).\n\n[figure2]\n\nFigure 2: Left: Snapshots of the reflections of a single $100 \\mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \\mathrm{c} / \\mathrm{s}=1 \\mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967).\n\nTop right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967).\n\nBottom right: The same as the snapshots, but this time summed over the first $500 \\mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\\sim 5 \\mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967).\n\nThe Doppler shift with light is given as\n\n$$\n\\frac{\\Delta f}{f}=\\frac{v}{c}\n$$\n\nwhere $\\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light.\n\nEver since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time.\n\n[figure3]\n\nFigure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966).b. Averaging over a series of pulses from August 1965, after correcting for the relative motion of the Earth and Mercury and the rotation rate of the Earth during the observations, the difference between the frequencies of photons from the extreme left and right parts of an annulus received $500 \\mu$ s after the initial echo was $\\Delta f_{\\text {total }}=4.27 \\mathrm{~Hz}$.\n\ni. Given that the pulse was Doppler shifted twice (once when reflected, once again when received back at Aricebo) show that the rotational period of Mercury is $\\approx 60$ days. Assume that the axis of rotation is normal to the plane of observations. [ 1 day $=24$ hours]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \\mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked.\n[figure1]\n\nFigure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC.\n\nRight: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \\theta$. Credit: OpenStax, College Physics.\n\nMercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres.\n\nThe radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \\mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \\mu \\mathrm{s}$ pulses at $430 \\mathrm{MHz}$ with a transmitted power of $2.0 \\mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1.\n\nThe signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \\mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2).\n\nPhotons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole).\n\n[figure2]\n\nFigure 2: Left: Snapshots of the reflections of a single $100 \\mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \\mathrm{c} / \\mathrm{s}=1 \\mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967).\n\nTop right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967).\n\nBottom right: The same as the snapshots, but this time summed over the first $500 \\mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\\sim 5 \\mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967).\n\nThe Doppler shift with light is given as\n\n$$\n\\frac{\\Delta f}{f}=\\frac{v}{c}\n$$\n\nwhere $\\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light.\n\nEver since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time.\n\n[figure3]\n\nFigure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966).\n\nproblem:\nb. Averaging over a series of pulses from August 1965, after correcting for the relative motion of the Earth and Mercury and the rotation rate of the Earth during the observations, the difference between the frequencies of photons from the extreme left and right parts of an annulus received $500 \\mu$ s after the initial echo was $\\Delta f_{\\text {total }}=4.27 \\mathrm{~Hz}$.\n\ni. Given that the pulse was Doppler shifted twice (once when reflected, once again when received back at Aricebo) show that the rotational period of Mercury is $\\approx 60$ days. Assume that the axis of rotation is normal to the plane of observations. [ 1 day $=24$ hours]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of days, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-04.jpg?height=512&width=1374&top_left_y=652&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-05.jpg?height=1094&width=1560&top_left_y=218&top_left_x=248", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-06.jpg?height=994&width=897&top_left_y=1359&top_left_x=585", "https://cdn.mathpix.com/cropped/2024_03_14_6cde567bccf58dc9a2d2g-02.jpg?height=471&width=683&top_left_y=547&top_left_x=435" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "days" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_521", "problem": "为研究太阳系内行星的运动, 需要知道太阳的质量, 已知地球半径为 $R$, 地球质量为 $m$ ，太阳与地球中心间距为 $r$, 地球表面的重力加速度为 $g$, 地球绕太阳公转的周期为 $T$. 则太阳的质量为 $(\\quad)$\nA: $\\frac{4 \\pi^{2} r^{3}}{T^{2} R^{2} g}$\nB: $\\frac{T^{2} R^{2} g}{4 \\pi^{2} m r^{3}}$\nC: $\\frac{4 \\pi^{2} m g r^{2}}{r^{3} T^{2}}$\nD: $\\frac{4 \\pi^{2} m r^{3}}{T^{2} R^{2} g}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n为研究太阳系内行星的运动, 需要知道太阳的质量, 已知地球半径为 $R$, 地球质量为 $m$ ，太阳与地球中心间距为 $r$, 地球表面的重力加速度为 $g$, 地球绕太阳公转的周期为 $T$. 则太阳的质量为 $(\\quad)$\n\nA: $\\frac{4 \\pi^{2} r^{3}}{T^{2} R^{2} g}$\nB: $\\frac{T^{2} R^{2} g}{4 \\pi^{2} m r^{3}}$\nC: $\\frac{4 \\pi^{2} m g r^{2}}{r^{3} T^{2}}$\nD: $\\frac{4 \\pi^{2} m r^{3}}{T^{2} R^{2} g}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_869", "problem": "Two planets A and B orbit a star with coplanar orbital paths that don't intersect. The major axes of the orbits are perfectly aligned, but the major axis of A is larger than that of B. A and $\\mathrm{B}$ are observed to have eccentricities 0.5 and 0.75 , respectively. What is the minimal possible ratio of semi-major axes of $\\mathrm{A}$ to $\\mathrm{B}$ ?\nA: 1\nB: $\\frac{7}{6}$\nC: $\\frac{8}{3}$\nD: $\\frac{7}{2}$\nE: 6\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo planets A and B orbit a star with coplanar orbital paths that don't intersect. The major axes of the orbits are perfectly aligned, but the major axis of A is larger than that of B. A and $\\mathrm{B}$ are observed to have eccentricities 0.5 and 0.75 , respectively. What is the minimal possible ratio of semi-major axes of $\\mathrm{A}$ to $\\mathrm{B}$ ?\n\nA: 1\nB: $\\frac{7}{6}$\nC: $\\frac{8}{3}$\nD: $\\frac{7}{2}$\nE: 6\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_623", "problem": "宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 。在星球 $\\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\\mathrm{P}$ 轻放在弹簧上端, 如图 (a) 所示, $\\mathrm{P}$ 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\\mathrm{B}$ 上用完全相同的弹簧和物体 P 完成同样的过程, 其 $a-x$ 关系如图 (b) 中虚线所示。已知两星球密度相等。星球 $\\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。\n\n若将星球 $\\mathrm{A}$ 看成是以星球 $\\mathrm{B}$ 为中心天体的一颗卫星, 求星球 $\\mathrm{A}$ 的运行周期 $T_{1}$;\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 。在星球 $\\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\\mathrm{P}$ 轻放在弹簧上端, 如图 (a) 所示, $\\mathrm{P}$ 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\\mathrm{B}$ 上用完全相同的弹簧和物体 P 完成同样的过程, 其 $a-x$ 关系如图 (b) 中虚线所示。已知两星球密度相等。星球 $\\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。\n\n若将星球 $\\mathrm{A}$ 看成是以星球 $\\mathrm{B}$ 为中心天体的一颗卫星, 求星球 $\\mathrm{A}$ 的运行周期 $T_{1}$;\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-145.jpg?height=183&width=256&top_left_y=2464&top_left_x=340", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-145.jpg?height=429&width=414&top_left_y=2264&top_left_x=684" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_130", "problem": "如图所示, 宇宙空间有一种由三颗星体 $A 、 B 、 C$ 组成的三星体系, 它们分别位于等边三角形 $A B C$ 的三个顶点上, 绕一个固定且共同的圆心 $O$ 做匀速圆周运动, 轨道如\n图中实线所示, 其轨道半径 $r_{\\mathrm{A}}0$, 假设两星球的半径远小于两球球心之间的距离。则下列说法正确的是（）\n\n[图1]\nA: 星球 $\\mathrm{P} 、 \\mathrm{Q}$ 的轨道半径之比为 $\\frac{L_{1}+L_{2}}{L_{1}-L_{2}}$\nB: 星球 $\\mathrm{P}$ 的质量大于星球 $\\mathrm{Q}$ 的质量\nC: 星球 $\\mathrm{P} 、 \\mathrm{Q}$ 的线速度之和与线速度之差的比值 $\\frac{L_{1}}{L_{2}}$\nD: 星球 $\\mathrm{P} 、 \\mathrm{Q}$ 的质量之和与 $\\mathrm{P} 、 \\mathrm{Q}$ 质量之差的比值 $\\frac{L_{1}}{L_{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 2022 年 7 月 15 日, 由清华大学天文系祝伟教授牵头的国际团队近日宣布在宇宙中发现两个罕见的恒星系统。该系统均是由两颗互相绕行的中央恒星组成, 被气体和尘埃盘包围, 且该盘与中央恒星的轨道成一定角度, 呈现出“雾绕双星”的奇幻效\n果。如图所示为该双星模型的简化图, 已知 $O_{1} O_{2}=L_{1}, \\quad O_{1} O-O O_{2}=L_{2}>0$, 假设两星球的半径远小于两球球心之间的距离。则下列说法正确的是（）\n\n[图1]\n\nA: 星球 $\\mathrm{P} 、 \\mathrm{Q}$ 的轨道半径之比为 $\\frac{L_{1}+L_{2}}{L_{1}-L_{2}}$\nB: 星球 $\\mathrm{P}$ 的质量大于星球 $\\mathrm{Q}$ 的质量\nC: 星球 $\\mathrm{P} 、 \\mathrm{Q}$ 的线速度之和与线速度之差的比值 $\\frac{L_{1}}{L_{2}}$\nD: 星球 $\\mathrm{P} 、 \\mathrm{Q}$ 的质量之和与 $\\mathrm{P} 、 \\mathrm{Q}$ 质量之差的比值 $\\frac{L_{1}}{L_{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-038.jpg?height=460&width=573&top_left_y=358&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1138", "problem": "Recent years have seen an explosion in the discovery of new exoplanets. About $85 \\%$ of transiting exoplanets discovered by the NASA Kepler telescope have radii less than Neptune ( $\\sim 4 R_{\\oplus}$ ), meaning we are improving our understanding of what the transition between rocky Earth-size planets and gaseous Neptune-size planets looks like.\n\nGiven how common these \"super-Earths\" and \"gas dwarfs\" seem to be, it was odd that we didn't have any in our own Solar System. However, Batygin \\& Brown (2016) suggested that a hypothetical ninth planet (called 'Planet Nine') could explain some of the unusual properties of the orbits of objects in the Kuiper Belt. This planet is inferred to have a mass of $10 M_{\\oplus}$, and so would be an example of a super-Earth.\n\n[figure1]\n\nFigure 5: A plot of planet density versus radius for 33 extrasolar planets (circles) and the planets in our solar system (diamonds).\n\nCredit: Marcy et al. (2014).\n\nAnalysing exoplanets discovered by Kepler, Marcy et al. (2014) used a piecewise function to describe their planetary density data such that:\n\n$$\n\\begin{aligned}\n\\text { For } R_{\\mathrm{P}} \\leq 1.5 R_{\\oplus} & \\rho & =2.32+3.18 \\frac{R_{\\mathrm{P}}}{R_{\\oplus}}\\left[\\mathrm{g} \\mathrm{cm}^{-3}\\right] \\\\\n\\text { For } 1.5 R_{\\oplus}\\beta_{12}, \\alpha_{6}=\\beta_{6}$\nC: $\\alpha_{12}<\\beta_{12}, \\alpha_{6}=\\beta_{6}$\nD: $\\alpha_{12}=\\beta_{12}, \\alpha_{6}>\\beta_{6}$\nE: $\\alpha_{12}=\\beta_{12}, \\alpha_{6}<\\beta_{6}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAt 6am on March 20th, as the Sun is rising, Leo, who is at $\\left(40^{\\circ} \\mathrm{N}, 75^{\\circ} \\mathrm{W}\\right)$, plants a stick vertically on the ground. At that moment, he marks out a (finite) line on the ground in the direction of the shadow of the stick at that moment, labeling it with the current time. Every hour afterwards, on the hour, he marks out a new line in the current direction of the shadow, until the sun sets at $6 \\mathrm{pm}$.\n\nThree months later, Leo returns to the same spot, where the vertical stick and lines remain. Again, every hour on the hour, he marks out a line in the current direction of the shadow, until the sun sets.\n\nLet $\\alpha_{12}$ and $\\alpha_{6}$ be the azimuths of the lines drawn in the spring at $12 \\mathrm{pm}$ and $6 \\mathrm{pm}$, and $\\beta_{12}$ and $\\beta_{6}$ be the azimuths of the lines drawn in the summer at $12 \\mathrm{pm}$ and $6 \\mathrm{pm}$. Which of the following statements is true? Ignore atmospheric effects and the equation of time.\n\nA: $\\alpha_{12}=\\beta_{12}, \\alpha_{6}=\\beta_{6}$\nB: $\\alpha_{12}>\\beta_{12}, \\alpha_{6}=\\beta_{6}$\nC: $\\alpha_{12}<\\beta_{12}, \\alpha_{6}=\\beta_{6}$\nD: $\\alpha_{12}=\\beta_{12}, \\alpha_{6}>\\beta_{6}$\nE: $\\alpha_{12}=\\beta_{12}, \\alpha_{6}<\\beta_{6}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_7205fccc557018644b5cg-13.jpg?height=1613&width=1445&top_left_y=240&top_left_x=367" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_574", "problem": "如图为高分一号北斗导航系统两颗卫星在空中某一面内运动的示意图. 导航卫星 $\\mathrm{G}_{1}$和 $\\mathrm{G}_{2}$ 以及高分一号均可认为统地心 $O$ 做匀速圆同运动. 卫星 $\\mathrm{G}_{1}$ 和 $\\mathrm{G}_{2}$ 的轨道半径为 $r$,某时刻两颗导航卫星分别位于轨道上的 $A$ 和 $B$ 两位置, 高分一号在 $C$ 位置. 若卫星均顺时针运行, $\\angle A O B=60^{\\circ}$, 地球表面处的重力加速度为 $g$, 地球半径为 $R$, 不计卫星间的相互作用力. 则下列说法正确的是（ ）\n\n[图1]\nA: 卫星 $\\mathrm{G}_{1}$ 和 $\\mathrm{G}_{2}$ 的加速度大小相等且为 $\\frac{R}{r} g$\nB: 卫星 $\\mathrm{G}_{1}$ 由位置 $A$ 运动到位置 $B$ 所需的时间为 $\\frac{\\pi r}{3 R} \\sqrt{\\frac{r}{g}}$\nC: 若高分一号与卫星 $\\mathrm{G}_{1}$ 的周期之比为 $1: k(k>1$, 且为整数), 某时刻两者相距最近,则从此时刻起, 在卫星 $\\mathrm{G}_{1}$ 运动一周的过程中二者距离最近的次数为 (k-1)\nD: 若高分一号与卫星 $\\mathrm{G}_{1}$ 质量相等,由于高分一号的绕行速度大,则发射所需的最小能量更多\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图为高分一号北斗导航系统两颗卫星在空中某一面内运动的示意图. 导航卫星 $\\mathrm{G}_{1}$和 $\\mathrm{G}_{2}$ 以及高分一号均可认为统地心 $O$ 做匀速圆同运动. 卫星 $\\mathrm{G}_{1}$ 和 $\\mathrm{G}_{2}$ 的轨道半径为 $r$,某时刻两颗导航卫星分别位于轨道上的 $A$ 和 $B$ 两位置, 高分一号在 $C$ 位置. 若卫星均顺时针运行, $\\angle A O B=60^{\\circ}$, 地球表面处的重力加速度为 $g$, 地球半径为 $R$, 不计卫星间的相互作用力. 则下列说法正确的是（ ）\n\n[图1]\n\nA: 卫星 $\\mathrm{G}_{1}$ 和 $\\mathrm{G}_{2}$ 的加速度大小相等且为 $\\frac{R}{r} g$\nB: 卫星 $\\mathrm{G}_{1}$ 由位置 $A$ 运动到位置 $B$ 所需的时间为 $\\frac{\\pi r}{3 R} \\sqrt{\\frac{r}{g}}$\nC: 若高分一号与卫星 $\\mathrm{G}_{1}$ 的周期之比为 $1: k(k>1$, 且为整数), 某时刻两者相距最近,则从此时刻起, 在卫星 $\\mathrm{G}_{1}$ 运动一周的过程中二者距离最近的次数为 (k-1)\nD: 若高分一号与卫星 $\\mathrm{G}_{1}$ 质量相等,由于高分一号的绕行速度大,则发射所需的最小能量更多\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-34.jpg?height=416&width=580&top_left_y=1294&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_41", "problem": "在太阳系外发现的某恒星 $\\mathrm{a}$ 的质量为太阳系质量的 0.3 倍, 该恒星的一颗行星 $\\mathrm{b}$ 的质量是地球的 4 倍, 直径是地球的 1.5 倍, 公转周期为 10 天. 设该行星与地球均为质量分布均匀的球体, 且分别绕其中心天体做匀速圆周运动, 则 ( )\nA: 行星 $\\mathrm{b}$ 的第一宇宙速度与地球相同\nB: 行星 $\\mathrm{b}$ 绕恒星 $\\mathrm{a}$ 运行的角速度大于地球绕太阳运行的角速度\nC: 如果将物体从地球搬到行星 $\\mathrm{b}$ 上, 其重力是在地球上重力的 $\\frac{16}{9}$\nD: 行星 $\\mathrm{b}$ 与恒星 $\\mathrm{a}$ 的距离是日地距离的 $\\sqrt{\\frac{2}{73}}$ 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n在太阳系外发现的某恒星 $\\mathrm{a}$ 的质量为太阳系质量的 0.3 倍, 该恒星的一颗行星 $\\mathrm{b}$ 的质量是地球的 4 倍, 直径是地球的 1.5 倍, 公转周期为 10 天. 设该行星与地球均为质量分布均匀的球体, 且分别绕其中心天体做匀速圆周运动, 则 ( )\n\nA: 行星 $\\mathrm{b}$ 的第一宇宙速度与地球相同\nB: 行星 $\\mathrm{b}$ 绕恒星 $\\mathrm{a}$ 运行的角速度大于地球绕太阳运行的角速度\nC: 如果将物体从地球搬到行星 $\\mathrm{b}$ 上, 其重力是在地球上重力的 $\\frac{16}{9}$\nD: 行星 $\\mathrm{b}$ 与恒星 $\\mathrm{a}$ 的距离是日地距离的 $\\sqrt{\\frac{2}{73}}$ 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_100", "problem": "甲、乙两颗卫星在不同轨道上绕地球运动, 甲卫星的轨道是圆, 半径为 $R$, 乙卫星的轨道是椭圆, 其中 $P$ 点为近地点, 到地心的距离为 $a, Q$ 为远地点, 到地心的距离为 $b$ 。已知 $ar_{2}$, 则下面的表述正确的是 ( )\n\n[图1]\nA: 它们运转的周期相同\nB: 它们的线速度大小相同\nC: $m_{2}>m_{1}$\nD: 它们的加速度大小相同\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n某双星系统由两颗恒星构成, 质量分别为 $m_{1}$ 和 $m_{2}$, 距中心的距离分别为 $r_{1}$ 和 $r_{2}$,且 $r_{1}>r_{2}$, 则下面的表述正确的是 ( )\n\n[图1]\n\nA: 它们运转的周期相同\nB: 它们的线速度大小相同\nC: $m_{2}>m_{1}$\nD: 它们的加速度大小相同\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-106.jpg?height=400&width=466&top_left_y=928&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1098", "problem": "The Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$e. Taking the mass of $\\mathrm{M}_{8} 7^{*}$ as $6.5 \\times 10^{9} \\mathrm{M}_{\\odot}$ :\n\nii. One of the bright patches in Fig 4 seemed to move a quarter of the way around the ring between April 5 and 10 (from the left hand side to the bottom). Could it be attributable to gas moving in the ISCO? If so, is the spin likely to be positive or negative?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$\n\nproblem:\ne. Taking the mass of $\\mathrm{M}_{8} 7^{*}$ as $6.5 \\times 10^{9} \\mathrm{M}_{\\odot}$ :\n\nii. One of the bright patches in Fig 4 seemed to move a quarter of the way around the ring between April 5 and 10 (from the left hand side to the bottom). Could it be attributable to gas moving in the ISCO? If so, is the spin likely to be positive or negative?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of days, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-07.jpg?height=704&width=1414&top_left_y=698&top_left_x=331", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-08.jpg?height=374&width=1562&top_left_y=1698&top_left_x=263", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-09.jpg?height=468&width=686&top_left_y=1388&top_left_x=705" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "days" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_532", "problem": "《流浪地球 2 》中太空电梯非常吸引观众眼球。在影片中太空电梯高算入云, 在地表与太空间高速穿梭。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转。随着物理学的发展, 我们越来越倾向于我们周围的环境。在本题中, 我们将跟随宇航员, 进入太空中探索。本题中, 若未特殊说明, 地球\n半径为 $R$, 地球质量为 $M$, 引力常量为 $G$, 卫星与地球间的引力势能表达式为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$ ( $r$ 是卫星到地心的距离)\n\n如图所示, 该系统的甲、乙两颗卫星的轨道分别是同一平面内的椭圆和圆, 甲的轨道近地点正好在地面附近, 已知远地点与地面的距离为 $3 R$, 甲轨道的远地点与乙轨道的最近距离为 $2 R$, 乙的质量为 $m_{0}$, 甲、乙的周期之比是多少?\n\n为\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n《流浪地球 2 》中太空电梯非常吸引观众眼球。在影片中太空电梯高算入云, 在地表与太空间高速穿梭。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转。随着物理学的发展, 我们越来越倾向于我们周围的环境。在本题中, 我们将跟随宇航员, 进入太空中探索。本题中, 若未特殊说明, 地球\n半径为 $R$, 地球质量为 $M$, 引力常量为 $G$, 卫星与地球间的引力势能表达式为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$ ( $r$ 是卫星到地心的距离)\n\n如图所示, 该系统的甲、乙两颗卫星的轨道分别是同一平面内的椭圆和圆, 甲的轨道近地点正好在地面附近, 已知远地点与地面的距离为 $3 R$, 甲轨道的远地点与乙轨道的最近距离为 $2 R$, 乙的质量为 $m_{0}$, 甲、乙的周期之比是多少?\n\n为\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-026.jpg?height=389&width=308&top_left_y=688&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1083", "problem": "Recently a group of researchers announced that they had discovered an Earth-sized exoplanet around our nearest star, Proxima Centauri. Its closeness raises an intriguing possibility about whether or not we might be able to image it directly using telescopes. The difficulty comes from the small angular scales that need to be resolved and the extreme differences in brightness between the reflected light from the planet and the light given out by the star.\n\n[figure1]\n\nFigure 6: Artist's impression of the view from the surface of Proxima Centauri b. Credit: ESO / M. Kornmesser\n\nData about the star and the planet are summarised below:\n\n| Proxima Centauri (star) | | Proxima Centauri b (planet) | |\n| :--- | :--- | :--- | :--- |\n| Distance | $1.295 \\mathrm{pc}$ | Orbital period | 11.186 days |\n| Mass | $0.123 \\mathrm{M}_{\\odot}$ | Mass $(\\mathrm{min})$ | $\\approx 1.27 \\mathrm{M}_{\\oplus}$ |\n| Radius | $0.141 \\mathrm{R}_{\\odot}$ | Radius $(\\mathrm{min})$ | $\\approx 1.1 \\mathrm{R}_{\\oplus}$ |\n| Surface temperature | $3042 \\mathrm{~K}$ | | |\n| Apparent magnitude | 11.13 | | |\n\nThe following formulae may also be helpful:\n\n$$\nm-\\mathcal{M}=5 \\log \\left(\\frac{d}{10}\\right) \\quad \\mathcal{M}-\\mathcal{M}_{\\odot}=-2.5 \\log \\left(\\frac{L}{\\mathrm{~L}_{\\odot}}\\right) \\quad \\Delta m=2.5 \\log C R\n$$\n\nwhere $m$ is the apparent magnitude, $\\mathcal{M}$ is the absolute magnitude, $d$ is the distance in parsecs, and the contrast ratio $(C R)$ is defined as the ratio of fluxes from the star and planet, $C R=\\frac{f_{\\text {star }}}{f_{\\text {planet }}}$.d. Calculate the exposure time needed for a Keck II image of the exoplanet to have an SNR of 3. Assume that the telescope has perfect $A O$, is observed at the longest wavelength for which the planet can still be resolved from the star, all the received flux from the planet consists of photons of that longest wavelength , $\\varepsilon=0.1$ and $b=10^{9}$ photons $s^{-1}$ (so $b>>$ ). Comment on your answer.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nRecently a group of researchers announced that they had discovered an Earth-sized exoplanet around our nearest star, Proxima Centauri. Its closeness raises an intriguing possibility about whether or not we might be able to image it directly using telescopes. The difficulty comes from the small angular scales that need to be resolved and the extreme differences in brightness between the reflected light from the planet and the light given out by the star.\n\n[figure1]\n\nFigure 6: Artist's impression of the view from the surface of Proxima Centauri b. Credit: ESO / M. Kornmesser\n\nData about the star and the planet are summarised below:\n\n| Proxima Centauri (star) | | Proxima Centauri b (planet) | |\n| :--- | :--- | :--- | :--- |\n| Distance | $1.295 \\mathrm{pc}$ | Orbital period | 11.186 days |\n| Mass | $0.123 \\mathrm{M}_{\\odot}$ | Mass $(\\mathrm{min})$ | $\\approx 1.27 \\mathrm{M}_{\\oplus}$ |\n| Radius | $0.141 \\mathrm{R}_{\\odot}$ | Radius $(\\mathrm{min})$ | $\\approx 1.1 \\mathrm{R}_{\\oplus}$ |\n| Surface temperature | $3042 \\mathrm{~K}$ | | |\n| Apparent magnitude | 11.13 | | |\n\nThe following formulae may also be helpful:\n\n$$\nm-\\mathcal{M}=5 \\log \\left(\\frac{d}{10}\\right) \\quad \\mathcal{M}-\\mathcal{M}_{\\odot}=-2.5 \\log \\left(\\frac{L}{\\mathrm{~L}_{\\odot}}\\right) \\quad \\Delta m=2.5 \\log C R\n$$\n\nwhere $m$ is the apparent magnitude, $\\mathcal{M}$ is the absolute magnitude, $d$ is the distance in parsecs, and the contrast ratio $(C R)$ is defined as the ratio of fluxes from the star and planet, $C R=\\frac{f_{\\text {star }}}{f_{\\text {planet }}}$.\n\nproblem:\nd. Calculate the exposure time needed for a Keck II image of the exoplanet to have an SNR of 3. Assume that the telescope has perfect $A O$, is observed at the longest wavelength for which the planet can still be resolved from the star, all the received flux from the planet consists of photons of that longest wavelength , $\\varepsilon=0.1$ and $b=10^{9}$ photons $s^{-1}$ (so $b>>$ ). Comment on your answer.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-10.jpg?height=708&width=1082&top_left_y=551&top_left_x=493" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1182", "problem": "Some of the very first exoplanets to be discovered in large surveys were dubbed 'hot Jupiters' as they were similar in mass to Jupiter (i.e. a gas giant) but were much closer to their star than Mercury is to the Sun (and hence are in a very hot environment). Planetary formation models suggest that they were unlikely to have formed there, but instead formed much further out from the star and migrated inwards, due to gravitational interactions with other planets in the system. Studies of 'hot Jupiters' show that there is an overabundance of them with periods of $\\sim 3-4$ days, and very few with periods shorter than that. Since large, close-in planets should be the easiest to detect in all of the main methods of finding exoplanets, this scarcity is likely to be a real effect and suggests that exoplanets which are that close to their star are in a relatively rapid (by astronomical standards) inspiral towards destruction by their star.\n[figure1]\n\nFigure 6: Left: The orbital radius of several 'hot Jupiters' scaled by the Roche radius of the system (where tidal forces would destroy the planet). There is an expected pile up close to radii double the Roche radius (dotted line), and very few with radii smaller than that - those that are will inevitably spiral into the star and be destroyed by the tidal forces when they get too close. Credit: Birkby et al. (2014).\n\nRight: As the planets inspiral we should see a shift in when their transits occur. This figure shows the predicted size of the shift after a period of 10 years if the tidal dissipation quality factor $Q_{\\star}^{\\prime}=10^{6}$, as well as the current detection limit of 5 seconds (dotted line). Therefore measuring if there is any shift in the transit times over the course of a decade of observations can put stringent limits on the value of $Q_{\\star}^{\\prime}$. Credit: Birkby et al. (2014).\n\nThe Roche radius, where a planet will be torn apart due to the tidal forces acting on it, is defined as\n\n$$\na_{\\text {Roche }} \\approx 2.16 R_{P}\\left(\\frac{M_{\\star}}{M_{P}}\\right)^{1 / 3}\n$$\n\nwhere $R_{P}$ is the radius of the planet, $M_{P}$ is the mass of the planet and $M_{\\star}$ is the mass of the star. If a gas giant is knocked into a highly elliptical orbit (i.e. $e \\approx 1$ ) that has a periapsis $r_{\\text {peri }}v_{b}>v_{a}>v_{d}$\nB: $\\omega_{a}>\\omega_{b}>\\omega_{c}>\\omega_{d}$\nC: $a_{a}>a_{b}>a_{c}>a_{d}$\nD: $T_{c}v_{b}>v_{a}>v_{d}$\nB: $\\omega_{a}>\\omega_{b}>\\omega_{c}>\\omega_{d}$\nC: $a_{a}>a_{b}>a_{c}>a_{d}$\nD: $T_{c}R)$, 太阳光光子被完全吸收时产生的光压 $p$ 是多大?", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n光电效应和康普顿效应深入地揭示了光的粒子性的一面。前者表明光子具有能量,后者表明光子除了具有能量之外还具有动量。由狭义相对论可知, 一定的质量 $m$ 与一\n定的能量 $E$ 相对应: $E=m c^{2}$, 其中 $c$ 为真空中光速。\n光照射到物体表面时, 光子被物体吸收或反射时, 光都会对物体产生压强, 这就是“光压”。已知太阳半径为 $R$, 单位时间辐射的总能量为 $P_{0}$, 光速为 $c$ 。求:距离太阳中心 $r$ 处 $(r>R)$, 太阳光光子被完全吸收时产生的光压 $p$ 是多大?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_38", "problem": "2020 年 7 月 23 日, 我国“天问一号”火星探测器成功发射, 2021 年 2 月 10 日, 顺利进入环火星大椭圆轨道, 并变轨到近火星圆轨道运动, 将于 2021 年 5 月至 6 月择机实施火星着陆, 最终实现“绕、着、巡”三大目标。已知火星质量约为地球的 $\\frac{1}{10}$, 半径约为地球的 $\\frac{1}{2}$, 地球表面的重力加速度为 $g$, 火星和地球均绕太阳做逆时针方向的匀速圆周运动, 火星的公转周期是地球公转周期的两倍。质量为 $m$ 的着陆器在着陆火星前,会在火星表面附近经历一个时长为 $t$ ，速度由 $v$ 减速到零的过程. 若该减速过程可以视为一个坚直向下的匀减速直线运动, 忽略火星大气阻力, 求:\n\n探测器分别围绕火星和地球做圆周运动一周的最短时间之比。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n2020 年 7 月 23 日, 我国“天问一号”火星探测器成功发射, 2021 年 2 月 10 日, 顺利进入环火星大椭圆轨道, 并变轨到近火星圆轨道运动, 将于 2021 年 5 月至 6 月择机实施火星着陆, 最终实现“绕、着、巡”三大目标。已知火星质量约为地球的 $\\frac{1}{10}$, 半径约为地球的 $\\frac{1}{2}$, 地球表面的重力加速度为 $g$, 火星和地球均绕太阳做逆时针方向的匀速圆周运动, 火星的公转周期是地球公转周期的两倍。质量为 $m$ 的着陆器在着陆火星前,会在火星表面附近经历一个时长为 $t$ ，速度由 $v$ 减速到零的过程. 若该减速过程可以视为一个坚直向下的匀减速直线运动, 忽略火星大气阻力, 求:\n\n探测器分别围绕火星和地球做圆周运动一周的最短时间之比。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_490", "problem": "人造地球卫星的成功发射一般经过三个阶段, 即卫星依次完成近地轨道、转移轨道、预定轨道。。其中近地轨道和预定轨道可认为是卫星做匀速圆周运动, 转移轨道是椭圆轨道。下图为同一颗人造卫星的变轨原理简图:I 为近地圆轨道, III 为预定圆轨道, II 为转移轨道即椭圆轨道, 且 I、II 轨道相切于 $P$ 点, II、III 轨道相切于 $Q$ 点。已知 I 轨道半径为 $R$ 、III 轨道半径为 $4 R$, 卫星在三个轨道上运行的周期分别是 $T_{1} 、 T_{2} 、 T_{3}$, 且已知 $T_{l}=2 \\mathrm{~h}$, 卫星在各轨道经过 $P 、 Q$ 两点处的速度大小分别是 $v_{1} P 、 v_{2} P 、 v_{2} Q 、 v_{3} Q$, 则下列关于卫星正确说法是（）\n\n[图1]\nA: $v_{2} P>v_{1} P>v_{3} Q>v_{2} Q$\nB: $T_{2} \\approx 8 \\mathrm{~h}, T_{3}=16 \\mathrm{~h}$\nC: 卫星在 II 轨道上从 $P$ 点到 $Q$ 点的运行过程中, 动能增大, 引力势能减小, 机械能大小不变\nD: 卫星在三个轨道中 I 轨道的机械能最大, III 轨道的机械能最小, II 轨道的机械能居二者之间\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n人造地球卫星的成功发射一般经过三个阶段, 即卫星依次完成近地轨道、转移轨道、预定轨道。。其中近地轨道和预定轨道可认为是卫星做匀速圆周运动, 转移轨道是椭圆轨道。下图为同一颗人造卫星的变轨原理简图:I 为近地圆轨道, III 为预定圆轨道, II 为转移轨道即椭圆轨道, 且 I、II 轨道相切于 $P$ 点, II、III 轨道相切于 $Q$ 点。已知 I 轨道半径为 $R$ 、III 轨道半径为 $4 R$, 卫星在三个轨道上运行的周期分别是 $T_{1} 、 T_{2} 、 T_{3}$, 且已知 $T_{l}=2 \\mathrm{~h}$, 卫星在各轨道经过 $P 、 Q$ 两点处的速度大小分别是 $v_{1} P 、 v_{2} P 、 v_{2} Q 、 v_{3} Q$, 则下列关于卫星正确说法是（）\n\n[图1]\n\nA: $v_{2} P>v_{1} P>v_{3} Q>v_{2} Q$\nB: $T_{2} \\approx 8 \\mathrm{~h}, T_{3}=16 \\mathrm{~h}$\nC: 卫星在 II 轨道上从 $P$ 点到 $Q$ 点的运行过程中, 动能增大, 引力势能减小, 机械能大小不变\nD: 卫星在三个轨道中 I 轨道的机械能最大, III 轨道的机械能最小, II 轨道的机械能居二者之间\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-005.jpg?height=331&width=386&top_left_y=1779&top_left_x=344" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_237", "problem": "18 世纪, 数学家莫佩尔蒂和哲学家伏尔泰, 曾设想“穿透”地球 假设能够沿着地球两极连线开凿一条沿着地轴的隧道贯穿地球, 一个人可以从北极入口由静止自由落入隧道中, 忽略一切阻力, 此人可以从南极出口飞出, 则以下说法正确的是(已知此人的质量 $\\mathrm{m}=50 \\mathrm{~kg}$; 地球表面处重力加速度 $\\mathrm{g}$ 取 $10 \\mathrm{~m} / \\mathrm{s}^{2}$; 地球半径 $\\mathrm{R}=6.4 \\times 10^{6} \\mathrm{~m}$; 假设地球可视为质量分布均匀的球体, 均匀球壳对壳内任一点处的质点合引力为零)( )\nA: 人与地球构成的系统, 由于重力发生变化, 故机械能不守恒\nB: 人在下落过程中, 受到的万有引力与到地心的距离成正比\nC: 人从北极开始下落, 直到经过地心的过程中, 万有引力对人做功 $\\mathrm{W}=1.6 \\times 10^{9} \\mathrm{~J}$\nD: 当人下落经过距地心 $\\mathrm{R} / 2$ 瞬间, 人的瞬时速度大小为 $4 \\times 10^{3} \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n18 世纪, 数学家莫佩尔蒂和哲学家伏尔泰, 曾设想“穿透”地球 假设能够沿着地球两极连线开凿一条沿着地轴的隧道贯穿地球, 一个人可以从北极入口由静止自由落入隧道中, 忽略一切阻力, 此人可以从南极出口飞出, 则以下说法正确的是(已知此人的质量 $\\mathrm{m}=50 \\mathrm{~kg}$; 地球表面处重力加速度 $\\mathrm{g}$ 取 $10 \\mathrm{~m} / \\mathrm{s}^{2}$; 地球半径 $\\mathrm{R}=6.4 \\times 10^{6} \\mathrm{~m}$; 假设地球可视为质量分布均匀的球体, 均匀球壳对壳内任一点处的质点合引力为零)( )\n\nA: 人与地球构成的系统, 由于重力发生变化, 故机械能不守恒\nB: 人在下落过程中, 受到的万有引力与到地心的距离成正比\nC: 人从北极开始下落, 直到经过地心的过程中, 万有引力对人做功 $\\mathrm{W}=1.6 \\times 10^{9} \\mathrm{~J}$\nD: 当人下落经过距地心 $\\mathrm{R} / 2$ 瞬间, 人的瞬时速度大小为 $4 \\times 10^{3} \\mathrm{~m} / \\mathrm{s}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1214", "problem": "Recent years have seen an explosion in the discovery of new exoplanets. About $85 \\%$ of transiting exoplanets discovered by the NASA Kepler telescope have radii less than Neptune ( $\\sim 4 R_{\\oplus}$ ), meaning we are improving our understanding of what the transition between rocky Earth-size planets and gaseous Neptune-size planets looks like.\n\nGiven how common these \"super-Earths\" and \"gas dwarfs\" seem to be, it was odd that we didn't have any in our own Solar System. However, Batygin \\& Brown (2016) suggested that a hypothetical ninth planet (called 'Planet Nine') could explain some of the unusual properties of the orbits of objects in the Kuiper Belt. This planet is inferred to have a mass of $10 M_{\\oplus}$, and so would be an example of a super-Earth.\n\n[figure1]\n\nFigure 5: A plot of planet density versus radius for 33 extrasolar planets (circles) and the planets in our solar system (diamonds).\n\nCredit: Marcy et al. (2014).\n\nAnalysing exoplanets discovered by Kepler, Marcy et al. (2014) used a piecewise function to describe their planetary density data such that:\n\n$$\n\\begin{aligned}\n\\text { For } R_{\\mathrm{P}} \\leq 1.5 R_{\\oplus} & \\rho & =2.32+3.18 \\frac{R_{\\mathrm{P}}}{R_{\\oplus}}\\left[\\mathrm{g} \\mathrm{cm}^{-3}\\right] \\\\\n\\text { For } 1.5 R_{\\oplus}r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$c. The angular diameter of M87* as determined from the images gained by the EHT (shown in Fig 4) is 42 microarcseconds, and the galaxy is $16.8 \\mathrm{Mpc}$ away from us. Determine the minimum and maximum possible masses of the $\\mathrm{SMBH}$ in units of $\\mathrm{M}_{\\odot}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$\n\nproblem:\nc. The angular diameter of M87* as determined from the images gained by the EHT (shown in Fig 4) is 42 microarcseconds, and the galaxy is $16.8 \\mathrm{Mpc}$ away from us. Determine the minimum and maximum possible masses of the $\\mathrm{SMBH}$ in units of $\\mathrm{M}_{\\odot}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-07.jpg?height=704&width=1414&top_left_y=698&top_left_x=331", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-08.jpg?height=374&width=1562&top_left_y=1698&top_left_x=263", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-09.jpg?height=468&width=686&top_left_y=1388&top_left_x=705" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_686", "problem": "如图所示, “火星”探测飞行器 $P$ 绕火星做匀速圆周运动, 若“火星”探测飞行器某时刻的轨道半径为 $r$, 探测飞行器 $\\mathrm{P}$ 观测火星的最大张角为 $\\beta$, 下列说法正确的是 ( )\n\n[图1]\nA: 探测飞行器 $\\mathrm{P}$ 的轨道半径 $r$ 越大, 其周期越小\nB: 探测飞行器 $\\mathrm{P}$ 的轨道半径 $r$ 越大, 其速度越大\nC: 若测得周期和张角, 可得到火星的平均密度\nD: 若测得周期和轨道半径, 可得到探测器 $P$ 的质量\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, “火星”探测飞行器 $P$ 绕火星做匀速圆周运动, 若“火星”探测飞行器某时刻的轨道半径为 $r$, 探测飞行器 $\\mathrm{P}$ 观测火星的最大张角为 $\\beta$, 下列说法正确的是 ( )\n\n[图1]\n\nA: 探测飞行器 $\\mathrm{P}$ 的轨道半径 $r$ 越大, 其周期越小\nB: 探测飞行器 $\\mathrm{P}$ 的轨道半径 $r$ 越大, 其速度越大\nC: 若测得周期和张角, 可得到火星的平均密度\nD: 若测得周期和轨道半径, 可得到探测器 $P$ 的质量\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-112.jpg?height=388&width=454&top_left_y=1451&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-113.jpg?height=351&width=414&top_left_y=710&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_11", "problem": "2019 年 1 月 3 日, 我国“嫦娥四号”探测器在月球背面成功着陆并发回大量月背影\n\n像. 如图所示为位于月球背面的“嫦娥四号”探测器 $A$ 通过“鹊桥”中继站 $B$ 向地球传输电磁波信息的示意图. 拉格朗日 $L_{2}$ 点位于地月连线延长线上, “鹊桥”的运动可看成如下两种运动的合运动: 一是在地球和月球引力共同作用下, “鹊桥”在 $L_{2}$ 点附近与月球以相同的周期 $T_{0}$ 一起绕地球做匀速圆周运动; 二是在与地月连线垂直的平面内绕 $L_{2}$ 点做匀速圆周运动. 已知地球的质量为月球质量的 $n$ 倍, 地球到 $L_{2}$ 点的距离为月球到 $L_{2}$ 点的距离的 $k$ 倍, 地球半径、月球半径以及“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的半径均远小于月球到 $L_{2}$ 点的距离 (提示: “鹊桥”绕 $L_{2}$ 点做匀速圆周运动的向心力由地球和月球对其引力在过 $L_{2}$ 点与地月连线垂直的平面内的分量提供).[图1]\n若月球到 $L_{2}$ 点的距离 $r=6.5 \\times 10^{7} \\mathrm{~m}, k=7$, “鹊桥”接收到“嫦娥四号”传来的信息后需经 $t_{0}=60.0 \\mathrm{~s}$ 处理才能发出, 试估算“嫦娥四号”从发出信息到传回地球的最短时间（保留三位有效数字);", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n2019 年 1 月 3 日, 我国“嫦娥四号”探测器在月球背面成功着陆并发回大量月背影\n\n像. 如图所示为位于月球背面的“嫦娥四号”探测器 $A$ 通过“鹊桥”中继站 $B$ 向地球传输电磁波信息的示意图. 拉格朗日 $L_{2}$ 点位于地月连线延长线上, “鹊桥”的运动可看成如下两种运动的合运动: 一是在地球和月球引力共同作用下, “鹊桥”在 $L_{2}$ 点附近与月球以相同的周期 $T_{0}$ 一起绕地球做匀速圆周运动; 二是在与地月连线垂直的平面内绕 $L_{2}$ 点做匀速圆周运动. 已知地球的质量为月球质量的 $n$ 倍, 地球到 $L_{2}$ 点的距离为月球到 $L_{2}$ 点的距离的 $k$ 倍, 地球半径、月球半径以及“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的半径均远小于月球到 $L_{2}$ 点的距离 (提示: “鹊桥”绕 $L_{2}$ 点做匀速圆周运动的向心力由地球和月球对其引力在过 $L_{2}$ 点与地月连线垂直的平面内的分量提供).[图1]\n若月球到 $L_{2}$ 点的距离 $r=6.5 \\times 10^{7} \\mathrm{~m}, k=7$, “鹊桥”接收到“嫦娥四号”传来的信息后需经 $t_{0}=60.0 \\mathrm{~s}$ 处理才能发出, 试估算“嫦娥四号”从发出信息到传回地球的最短时间（保留三位有效数字);\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以s为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-020.jpg?height=160&width=662&top_left_y=1408&top_left_x=363" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_671", "problem": "如图所示, 北斗卫星导航系统中的一颗卫星 $a$ 位于赤道上空, 其对地张角为 $60^{\\circ}$ 。已知地球的半径为 $R$, 自转周期为 $T_{0}$, 表面的重力加速度为 $g$, 万有引力常量为 $G$ 。根据题中条件，可求出（）\n\n[图1]\nA: 地球的平均密度为 $\\frac{3 \\pi}{G T_{0}^{2}}$\nB: 静止卫星的轨道半径为 $\\sqrt[3]{\\frac{g T_{0}^{2} R^{2}}{4 \\pi^{2}}}$\nC: 卫星 $a$ 的周期为 $2 \\sqrt{2} T_{0}$\nD: $a$ 与近地卫星运行方向相反时, 二者不能直接通讯的连续时间为 $\\frac{8 \\pi \\sqrt{2 g R}}{3(2 \\sqrt{2}+1) g}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 北斗卫星导航系统中的一颗卫星 $a$ 位于赤道上空, 其对地张角为 $60^{\\circ}$ 。已知地球的半径为 $R$, 自转周期为 $T_{0}$, 表面的重力加速度为 $g$, 万有引力常量为 $G$ 。根据题中条件，可求出（）\n\n[图1]\n\nA: 地球的平均密度为 $\\frac{3 \\pi}{G T_{0}^{2}}$\nB: 静止卫星的轨道半径为 $\\sqrt[3]{\\frac{g T_{0}^{2} R^{2}}{4 \\pi^{2}}}$\nC: 卫星 $a$ 的周期为 $2 \\sqrt{2} T_{0}$\nD: $a$ 与近地卫星运行方向相反时, 二者不能直接通讯的连续时间为 $\\frac{8 \\pi \\sqrt{2 g R}}{3(2 \\sqrt{2}+1) g}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-054.jpg?height=414&width=442&top_left_y=787&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_284", "problem": "如图为某双星系统 $\\mathrm{A} 、 \\mathrm{~B}$ 绕其连线上的 $O$ 点做匀速圆周运动的示意图, 若 $\\mathrm{A}$ 星的轨道半径大于 $\\mathrm{B}$ 星的轨道半径, 双星的总质量 $M$, 双星间的距离为 $L$, 其运动周期为 $T$, 则 ( )\n\n[图1]\nA: $\\mathrm{A}$ 的质量一定大于 $\\mathrm{B}$ 的质量\nB: $\\mathrm{A}$ 的加速度一定大于 $\\mathrm{B}$ 的加速度\nC: $L$ 一定时, $M$ 越小, $T$ 越大\nD: $L$ 一定时, $\\mathrm{A}$ 的质量减小 $\\Delta m$ 而 $\\mathrm{B}$ 的质量增加 $\\Delta m$, 它们的向心力减小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图为某双星系统 $\\mathrm{A} 、 \\mathrm{~B}$ 绕其连线上的 $O$ 点做匀速圆周运动的示意图, 若 $\\mathrm{A}$ 星的轨道半径大于 $\\mathrm{B}$ 星的轨道半径, 双星的总质量 $M$, 双星间的距离为 $L$, 其运动周期为 $T$, 则 ( )\n\n[图1]\n\nA: $\\mathrm{A}$ 的质量一定大于 $\\mathrm{B}$ 的质量\nB: $\\mathrm{A}$ 的加速度一定大于 $\\mathrm{B}$ 的加速度\nC: $L$ 一定时, $M$ 越小, $T$ 越大\nD: $L$ 一定时, $\\mathrm{A}$ 的质量减小 $\\Delta m$ 而 $\\mathrm{B}$ 的质量增加 $\\Delta m$, 它们的向心力减小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-121.jpg?height=388&width=411&top_left_y=417&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_743", "problem": "What is the name of the JWST component highlighted below?\n\n[figure1]\nA: Stabilization flap\nB: Spacecraft bus\nC: Antenna\nD: Star tracker\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the name of the JWST component highlighted below?\n\n[figure1]\n\nA: Stabilization flap\nB: Spacecraft bus\nC: Antenna\nD: Star tracker\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_620a57bf13ecc39e0534g-2.jpg?height=408&width=508&top_left_y=1098&top_left_x=794" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_495", "problem": "当某一地外行星 (火星、木星、土星、天王星、海王星) 于绕日公转过程中运行到试卷第 48 页，共 150 页\n与地球、太阳成一直线的状态, 且地球恰好位于太阳和外行星之间的这种天文现象叫“冲日”, 冲日前后是观测地外行星的好时机。如图所示是土星冲日示意图, 已知地球质量为 $M$, 半径为 $R$, 公转周期是 1 年, 公转半径为 $r$, 土星质量是地球的 95 倍, 土星半径是地球的 9.5 倍, 土星的公转半径是地球的 9.5 倍。求: $\\left(\\sqrt{9.5^{3}} \\approx 29\\right)$\n土星的第一宇宙速度是地球的几倍?\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n当某一地外行星 (火星、木星、土星、天王星、海王星) 于绕日公转过程中运行到试卷第 48 页，共 150 页\n与地球、太阳成一直线的状态, 且地球恰好位于太阳和外行星之间的这种天文现象叫“冲日”, 冲日前后是观测地外行星的好时机。如图所示是土星冲日示意图, 已知地球质量为 $M$, 半径为 $R$, 公转周期是 1 年, 公转半径为 $r$, 土星质量是地球的 95 倍, 土星半径是地球的 9.5 倍, 土星的公转半径是地球的 9.5 倍。求: $\\left(\\sqrt{9.5^{3}} \\approx 29\\right)$\n土星的第一宇宙速度是地球的几倍?\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-049.jpg?height=286&width=491&top_left_y=848&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_129", "problem": "地球同步卫星的发射方法是变轨发射, 如图所示, 先把卫星发射到近地圆形轨道I 上, 当卫星到达 $P$ 点时, 发动机点火。使卫星进入粗圆轨道II, 其远地点恰好在地球赤道上空约 $36000 \\mathrm{~km}$ 处, 当卫星到达远地点 $Q$ 时, 发动机再次点火。使之进入同步轨道 III, 已知地球赤道上的重力加速度为 $g$, 物体在赤道表面上随地球自转的向心加速度大小为 $a$, 下列说法正确的是如果地球自转的（）\n\n[图1]\nA: 角速度突然变为原来的 $\\frac{g+a}{a}$ 倍, 那么赤道上的物体将会飘起来\nB: 卫星与地心连线在轨道II上单位时间内扫过的面积小于在轨道III上单位时间内扫过的面积\nC: 卫星在轨道III上运行时的机械能小于在轨道I上运行时的机械能\nD: 卫星在远地点 $Q$ 时的速度可能大于第一宇宙速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球同步卫星的发射方法是变轨发射, 如图所示, 先把卫星发射到近地圆形轨道I 上, 当卫星到达 $P$ 点时, 发动机点火。使卫星进入粗圆轨道II, 其远地点恰好在地球赤道上空约 $36000 \\mathrm{~km}$ 处, 当卫星到达远地点 $Q$ 时, 发动机再次点火。使之进入同步轨道 III, 已知地球赤道上的重力加速度为 $g$, 物体在赤道表面上随地球自转的向心加速度大小为 $a$, 下列说法正确的是如果地球自转的（）\n\n[图1]\n\nA: 角速度突然变为原来的 $\\frac{g+a}{a}$ 倍, 那么赤道上的物体将会飘起来\nB: 卫星与地心连线在轨道II上单位时间内扫过的面积小于在轨道III上单位时间内扫过的面积\nC: 卫星在轨道III上运行时的机械能小于在轨道I上运行时的机械能\nD: 卫星在远地点 $Q$ 时的速度可能大于第一宇宙速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-12.jpg?height=440&width=471&top_left_y=151&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_892", "problem": "One possible theory for why the gas giants have ring systems is that a small moon got too close to the parent planet. When the gravitational tidal forces (due to the difference between the strength of the planet's pull on the near and far sides of the moon) became greater than the gravitational forces holding the moon together, it was ripped apart. This minimum distance is called the \"Roche limit\", named after the French astronomer Edouard Roche who first calculated it. It is defined as when the gravitational force generated by the moon at its surface is equal to the tidal forces it experiences at that distance.\n\n[figure1]\n\nConsider a spherical planet with mass $M$ and radius $R$, and a perfectly rigid spherical moon with mass $m$ and radius $r$, orbiting the planet in a circular orbit of radius $d$. For a small particle of mass $u$ on the surface of the moon, the gravitational and tidal forces it experiences will be\n\n$$\nF_{\\text {grav }}=\\frac{G m u}{r^{2}} \\quad F_{\\text {tidal }}=\\frac{2 G M u r}{d^{3}}\n$$\n\nBy making these two expressions equal, derive an expression for the Roche limit, $d_{R L}$, purely in terms of $R$ and the uniform densities of the planet and the moon ( $\\rho_{P}$ and $\\rho_{m}$ respectively)", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nOne possible theory for why the gas giants have ring systems is that a small moon got too close to the parent planet. When the gravitational tidal forces (due to the difference between the strength of the planet's pull on the near and far sides of the moon) became greater than the gravitational forces holding the moon together, it was ripped apart. This minimum distance is called the \"Roche limit\", named after the French astronomer Edouard Roche who first calculated it. It is defined as when the gravitational force generated by the moon at its surface is equal to the tidal forces it experiences at that distance.\n\n[figure1]\n\nConsider a spherical planet with mass $M$ and radius $R$, and a perfectly rigid spherical moon with mass $m$ and radius $r$, orbiting the planet in a circular orbit of radius $d$. For a small particle of mass $u$ on the surface of the moon, the gravitational and tidal forces it experiences will be\n\n$$\nF_{\\text {grav }}=\\frac{G m u}{r^{2}} \\quad F_{\\text {tidal }}=\\frac{2 G M u r}{d^{3}}\n$$\n\nBy making these two expressions equal, derive an expression for the Roche limit, $d_{R L}$, purely in terms of $R$ and the uniform densities of the planet and the moon ( $\\rho_{P}$ and $\\rho_{m}$ respectively)\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6d91a7785df4f4beaa9ag-08.jpg?height=711&width=942&top_left_y=1135&top_left_x=591" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_975", "problem": "Special Relativity (SR) tells us that two observers will disagree about the duration of a time interval measured by each one's clock if one is moving at speed $v$ relative to the other, a phenomenon called time dilation. General Relativity (GR) tells us that gravitational fields dilate time too. This has an impact on satellites, since they travel at high orbital speeds (slowing down their clocks relative to the surface) but due to their altitude they are in a weaker gravitational field (speeding up their clocks relative to the surface). Which effect is dominant varies with orbital radius. Global Positioning System (GPS) satellites must compensate for this effect, since the satellites rely on accurate measurements of the time between sending and receiving a radio signal.\n\n[figure1]\n\nFigure 4: A scale diagram of the positions of the orbits for the International Space Station (ISS), GPS satellites and geostationary satellites, along with their orbital periods\n\nIn $\\mathrm{SR}$, time dilation can be calculated with\n\n$$\nt^{\\prime}=\\gamma t_{0} \\quad \\text { where } \\quad \\gamma=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}} \\quad \\text { so } \\quad \\Delta t_{\\mathrm{SR}}=t_{0}-t^{\\prime}=(1-\\gamma) t_{0}\n$$\n\nwhere $t_{0}$ is the time measured by the moving clock, $t^{\\prime}$ is the time measured by the observer, $c$ is the speed of light and $v$ is the speed of the object. A negative $\\Delta t$ indicates that the clocks are passing time slower relative to the observer, whilst a positive indicates they are passing quicker.\n\nGPS satellites have a period of exactly half a day. Use this to determine their orbital speed and hence show that for them $\\Delta t_{\\mathrm{SR}} \\approx-7 \\mu$ when $t_{0}=1$ day.\n\nIn GR, the overall effect (taking into account both the orbital motion and changing gravitational field strength) can be calculated by considering the measurements of time passing on the surface of the Earth and on the satellite as taken by an observer infinitely far away from Earth (so outside the gravitational field). It can be shown that\n\n$$\n\\Delta t_{\\text {overall }}=\\left(\\Gamma_{\\mathrm{GPS}}-\\Gamma_{\\mathrm{E}}\\right) t_{0} \\quad \\text { where } \\quad \\Gamma_{\\mathrm{GPS}}=\\sqrt{1-\\frac{3 G M_{E}}{a_{\\mathrm{GPS}} c^{2}}} \\quad \\text { and } \\quad \\Gamma_{\\mathrm{E}}=\\sqrt{1-\\frac{2 G M_{E}}{R_{E} c^{2}}}\n$$", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nSpecial Relativity (SR) tells us that two observers will disagree about the duration of a time interval measured by each one's clock if one is moving at speed $v$ relative to the other, a phenomenon called time dilation. General Relativity (GR) tells us that gravitational fields dilate time too. This has an impact on satellites, since they travel at high orbital speeds (slowing down their clocks relative to the surface) but due to their altitude they are in a weaker gravitational field (speeding up their clocks relative to the surface). Which effect is dominant varies with orbital radius. Global Positioning System (GPS) satellites must compensate for this effect, since the satellites rely on accurate measurements of the time between sending and receiving a radio signal.\n\n[figure1]\n\nFigure 4: A scale diagram of the positions of the orbits for the International Space Station (ISS), GPS satellites and geostationary satellites, along with their orbital periods\n\nIn $\\mathrm{SR}$, time dilation can be calculated with\n\n$$\nt^{\\prime}=\\gamma t_{0} \\quad \\text { where } \\quad \\gamma=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}} \\quad \\text { so } \\quad \\Delta t_{\\mathrm{SR}}=t_{0}-t^{\\prime}=(1-\\gamma) t_{0}\n$$\n\nwhere $t_{0}$ is the time measured by the moving clock, $t^{\\prime}$ is the time measured by the observer, $c$ is the speed of light and $v$ is the speed of the object. A negative $\\Delta t$ indicates that the clocks are passing time slower relative to the observer, whilst a positive indicates they are passing quicker.\n\nGPS satellites have a period of exactly half a day. Use this to determine their orbital speed and hence show that for them $\\Delta t_{\\mathrm{SR}} \\approx-7 \\mu$ when $t_{0}=1$ day.\n\nIn GR, the overall effect (taking into account both the orbital motion and changing gravitational field strength) can be calculated by considering the measurements of time passing on the surface of the Earth and on the satellite as taken by an observer infinitely far away from Earth (so outside the gravitational field). It can be shown that\n\n$$\n\\Delta t_{\\text {overall }}=\\left(\\Gamma_{\\mathrm{GPS}}-\\Gamma_{\\mathrm{E}}\\right) t_{0} \\quad \\text { where } \\quad \\Gamma_{\\mathrm{GPS}}=\\sqrt{1-\\frac{3 G M_{E}}{a_{\\mathrm{GPS}} c^{2}}} \\quad \\text { and } \\quad \\Gamma_{\\mathrm{E}}=\\sqrt{1-\\frac{2 G M_{E}}{R_{E} c^{2}}}\n$$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-10.jpg?height=742&width=1236&top_left_y=791&top_left_x=410" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_64", "problem": "由多颗星体构成的系统, 叫做多星系统. 有这样一种简单的四星系统: 质量刚好都相同的四个星体 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C} 、 \\mathrm{D}, \\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 分别位于等边三角形的三个顶点上, $\\mathrm{D}$ 位于等边三角形的中心. 在四者相互之间的万有引力作用下, $\\mathrm{D}$ 静止不动, $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 绕共同的圆心 $\\mathrm{D}$ 在等边三角形所在的平面内做相同周期的圆周运动. 若四个星体的质量均为 $m$, 三角形的边长为 $a$, 引力常量为 $G$, 则下列说法正确的是\n\n[图1]\nA: $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三个星体做圆周运动的半径均为 $\\frac{\\sqrt{3}}{2} a$\nB: A、B 两个星体之间的万有引力大小为 $\\frac{G m^{2}}{a^{2}}$\nC: $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三个星体做圆周运动的向心加速度大小均为 $\\frac{(\\sqrt{3}+3) G m}{a^{2}}$\nD: A、B、C 三个星体做圆周运动的周期均为 $2 \\pi a \\sqrt{\\frac{a}{(3+\\sqrt{3}) G m}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n由多颗星体构成的系统, 叫做多星系统. 有这样一种简单的四星系统: 质量刚好都相同的四个星体 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C} 、 \\mathrm{D}, \\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 分别位于等边三角形的三个顶点上, $\\mathrm{D}$ 位于等边三角形的中心. 在四者相互之间的万有引力作用下, $\\mathrm{D}$ 静止不动, $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 绕共同的圆心 $\\mathrm{D}$ 在等边三角形所在的平面内做相同周期的圆周运动. 若四个星体的质量均为 $m$, 三角形的边长为 $a$, 引力常量为 $G$, 则下列说法正确的是\n\n[图1]\n\nA: $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三个星体做圆周运动的半径均为 $\\frac{\\sqrt{3}}{2} a$\nB: A、B 两个星体之间的万有引力大小为 $\\frac{G m^{2}}{a^{2}}$\nC: $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三个星体做圆周运动的向心加速度大小均为 $\\frac{(\\sqrt{3}+3) G m}{a^{2}}$\nD: A、B、C 三个星体做圆周运动的周期均为 $2 \\pi a \\sqrt{\\frac{a}{(3+\\sqrt{3}) G m}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-48.jpg?height=357&width=420&top_left_y=164&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_587", "problem": "天问一号探测器在 2 月 10 日成功被火星捕获, 进入环火星轨道。如图所示, 假设探测器绕火星先后在椭圆轨道 1 、近火圆轨道 2 上运行, $\\mathrm{A}$ 是两轨道的切点, $B$ 是粗圆轨道的远地点, 已知火星的质量与半径分别为 $M 、 R$, 火星的球心与 $B$ 点的距离为 $3 R$,探测器的质量为 $m$ 。若规定探测器距火星无限远时探测器的引力势能为 0 , 则探测器的引力势能的表达式为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $r$ 是探测器与火星的球心之间的距离, $G$ 为引力常量。下列说法正确的是 ( )\n\n[图1]\nA: 探测器在轨道 $1 、 2$ 上运行周期的比值为 $3 \\sqrt{3}$\nB: 探测器在轨道 2 上运行时, 动量大小为 $\\frac{m}{2} \\sqrt{\\frac{G M}{R}}$\nC: 探测器在轨道 1 上运行, 经过 $\\mathrm{A}$ 点时, 动能为 $\\frac{G M m}{2 R}$\nD: 探测器在轨道 1 上运行, 经过 $\\mathrm{A}$ 点时, 动能大于 $\\frac{G M m}{2 R}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n天问一号探测器在 2 月 10 日成功被火星捕获, 进入环火星轨道。如图所示, 假设探测器绕火星先后在椭圆轨道 1 、近火圆轨道 2 上运行, $\\mathrm{A}$ 是两轨道的切点, $B$ 是粗圆轨道的远地点, 已知火星的质量与半径分别为 $M 、 R$, 火星的球心与 $B$ 点的距离为 $3 R$,探测器的质量为 $m$ 。若规定探测器距火星无限远时探测器的引力势能为 0 , 则探测器的引力势能的表达式为 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $r$ 是探测器与火星的球心之间的距离, $G$ 为引力常量。下列说法正确的是 ( )\n\n[图1]\n\nA: 探测器在轨道 $1 、 2$ 上运行周期的比值为 $3 \\sqrt{3}$\nB: 探测器在轨道 2 上运行时, 动量大小为 $\\frac{m}{2} \\sqrt{\\frac{G M}{R}}$\nC: 探测器在轨道 1 上运行, 经过 $\\mathrm{A}$ 点时, 动能为 $\\frac{G M m}{2 R}$\nD: 探测器在轨道 1 上运行, 经过 $\\mathrm{A}$ 点时, 动能大于 $\\frac{G M m}{2 R}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-081.jpg?height=480&width=537&top_left_y=999&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_543", "problem": "如图, 地球和某行星在同一轨道平面内同向绕太阳做匀速圆周运动。地球的运转周期为 $T$ 。地球和太阳的连线与地球和行星的连线所夹的角叫地球对该行星的观察视角 (简称视角)。已知该行星的最大视角为 $\\theta$, 当行星处于最大视角处时, 是地球上天文爱好者观察该行星的最佳时期。\n若某时刻该行星正处于最佳观察期，则该行星下一次处于最佳观察期至少需经历\n多长时间？\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图, 地球和某行星在同一轨道平面内同向绕太阳做匀速圆周运动。地球的运转周期为 $T$ 。地球和太阳的连线与地球和行星的连线所夹的角叫地球对该行星的观察视角 (简称视角)。已知该行星的最大视角为 $\\theta$, 当行星处于最大视角处时, 是地球上天文爱好者观察该行星的最佳时期。\n若某时刻该行星正处于最佳观察期，则该行星下一次处于最佳观察期至少需经历\n多长时间？\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-029.jpg?height=359&width=334&top_left_y=246&top_left_x=336", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-030.jpg?height=443&width=440&top_left_y=150&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_732", "problem": "如图所示, 发射地球同步卫星时, 先将卫星发射至近地圆轨道 1 , 然后经点火, 使其沿椭圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3. 轨道 $1 、 2$ 相切于 $Q$ 点,轨道 2、3 相切于 $P$ 点, 假设在整个过程中卫星的质量保持不变, 则当卫星分别在 $1 、 2$ 、 3 轨道上正常运行时, 下列说法正确的是（）\n\n[图1]\nA: 卫星在轨道 3 上的动能小于在轨道 1 上的动能\nB: 卫星在轨道 3 上的机械能小于在轨道 1 上的机械能\nC: 卫星在轨道 1 上经过 $Q$ 点时的动能等于它在轨道 2 上经过 $Q$ 点时的动能\nD: 卫星在轨道 2 上经过 $P$ 点时的机械能小于它在轨道 3 上经过 $P$ 点时的机械能\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 发射地球同步卫星时, 先将卫星发射至近地圆轨道 1 , 然后经点火, 使其沿椭圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3. 轨道 $1 、 2$ 相切于 $Q$ 点,轨道 2、3 相切于 $P$ 点, 假设在整个过程中卫星的质量保持不变, 则当卫星分别在 $1 、 2$ 、 3 轨道上正常运行时, 下列说法正确的是（）\n\n[图1]\n\nA: 卫星在轨道 3 上的动能小于在轨道 1 上的动能\nB: 卫星在轨道 3 上的机械能小于在轨道 1 上的机械能\nC: 卫星在轨道 1 上经过 $Q$ 点时的动能等于它在轨道 2 上经过 $Q$ 点时的动能\nD: 卫星在轨道 2 上经过 $P$ 点时的机械能小于它在轨道 3 上经过 $P$ 点时的机械能\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-112.jpg?height=414&width=397&top_left_y=153&top_left_x=361" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1208", "problem": "The James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.b. To achieve suitable sampling, an image will be considered diffraction limited when it has $\\geq 2$ pixels per $\\theta_{\\text {FWHM. }}$. The diameter of the JWST primary mirror is $6.5 \\mathrm{~m}$, however since it is composed of hexagons and hexagonal in shape, it is not straightforward to work out the equivalent circular mirror diameter. To a good approximation it can be taken to be $6.0 \\mathrm{~m}$.\n\nii. Hence, determine which of the three imaging instruments is diffraction limited for the greatest fraction of its wavelength range.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.\n\nproblem:\nb. To achieve suitable sampling, an image will be considered diffraction limited when it has $\\geq 2$ pixels per $\\theta_{\\text {FWHM. }}$. The diameter of the JWST primary mirror is $6.5 \\mathrm{~m}$, however since it is composed of hexagons and hexagonal in shape, it is not straightforward to work out the equivalent circular mirror diameter. To a good approximation it can be taken to be $6.0 \\mathrm{~m}$.\n\nii. Hence, determine which of the three imaging instruments is diffraction limited for the greatest fraction of its wavelength range.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mu \\mathrm{m}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-09.jpg?height=618&width=1466&top_left_y=596&top_left_x=296", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-10.jpg?height=482&width=1536&top_left_y=1118&top_left_x=267" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mu \\mathrm{m}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_524", "problem": "地球的公转轨道接近圆, 但彗星的运动轨道则是一个非常扁的椭圆 (如图)。天文学家哈雷成功预言哈雷彗星的回归, 哈雷彗星最近出现的时间是 1986 年, 预测下次飞近地球将在 2061 年。设哈雷彗星在近日点与太阳中心的距离为 $r_{1}$, 在远日点与太阳中心的距离为 $r_{2}$ 。地球公转半径为 $R$ 。则 ( )\n\n[图1]\nA: $r_{l} \\approx 18 R$\nB: $r_{1}+r_{2} \\approx 36 R$\nC: 哈雷彗星在近日点和远日点的加速度大小之比为 $\\frac{r_{2}^{2}}{r_{1}^{2}}$\nD: 哈雷彗星在近日点和远日点的速度大小之比为 $\\sqrt{\\frac{r_{2}}{r_{1}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n地球的公转轨道接近圆, 但彗星的运动轨道则是一个非常扁的椭圆 (如图)。天文学家哈雷成功预言哈雷彗星的回归, 哈雷彗星最近出现的时间是 1986 年, 预测下次飞近地球将在 2061 年。设哈雷彗星在近日点与太阳中心的距离为 $r_{1}$, 在远日点与太阳中心的距离为 $r_{2}$ 。地球公转半径为 $R$ 。则 ( )\n\n[图1]\n\nA: $r_{l} \\approx 18 R$\nB: $r_{1}+r_{2} \\approx 36 R$\nC: 哈雷彗星在近日点和远日点的加速度大小之比为 $\\frac{r_{2}^{2}}{r_{1}^{2}}$\nD: 哈雷彗星在近日点和远日点的速度大小之比为 $\\sqrt{\\frac{r_{2}}{r_{1}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-042.jpg?height=271&width=868&top_left_y=150&top_left_x=320" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_856", "problem": "We observe that a quasar's brightness varies within less than a day. What is the best upper bound on the quasar's size that you can derive from this information?\nA: $8 \\mathrm{kpc}$\nB: $170 \\mathrm{AU}$\nC: $3 \\mathrm{AU}$\nD: 3 Sun Radii\nE: $1 \\mathrm{pc}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWe observe that a quasar's brightness varies within less than a day. What is the best upper bound on the quasar's size that you can derive from this information?\n\nA: $8 \\mathrm{kpc}$\nB: $170 \\mathrm{AU}$\nC: $3 \\mathrm{AU}$\nD: 3 Sun Radii\nE: $1 \\mathrm{pc}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_56d1b5239b3c83be7aceg-05.jpg?height=621&width=1610&top_left_y=1771&top_left_x=274" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_893", "problem": "Figure 4 below is a composite image which depicts a transit of the International Space Station (ISS) across the disc of the Sun. The image comprises 26 individual photographs which were taken at regular time intervals during the transit. The total duration of the transit was less than one second. In this question we will ignore any effects caused by the rotation of the Earth.\n\n[figure1]\n\nFigure 4: A composite of a selection of the frames taken with a high-speed camera of a transit of the ISS in front of the Sun, taken from Northamptonshire at 10:22 BST on $17^{\\text {th }}$ June 2022. Credit: Jamie Cooper Photography\n\nDuring the transit, the angular diameter of the Sun as viewed from the position of the camera was $31^{\\prime} 29^{\\prime \\prime}$. Use the image to calculate the angle $\\theta_{1}$ subtended by the ISS between the first and last photographs, as viewed from the position of the camera. Note that the centre of the solar disc is NOT in the field of view of the photograph. [You are given that $1^{\\circ}=60^{\\prime}=$ $3600^{\\prime \\prime}$.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nFigure 4 below is a composite image which depicts a transit of the International Space Station (ISS) across the disc of the Sun. The image comprises 26 individual photographs which were taken at regular time intervals during the transit. The total duration of the transit was less than one second. In this question we will ignore any effects caused by the rotation of the Earth.\n\n[figure1]\n\nFigure 4: A composite of a selection of the frames taken with a high-speed camera of a transit of the ISS in front of the Sun, taken from Northamptonshire at 10:22 BST on $17^{\\text {th }}$ June 2022. Credit: Jamie Cooper Photography\n\nDuring the transit, the angular diameter of the Sun as viewed from the position of the camera was $31^{\\prime} 29^{\\prime \\prime}$. Use the image to calculate the angle $\\theta_{1}$ subtended by the ISS between the first and last photographs, as viewed from the position of the camera. Note that the centre of the solar disc is NOT in the field of view of the photograph. [You are given that $1^{\\circ}=60^{\\prime}=$ $3600^{\\prime \\prime}$.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-08.jpg?height=831&width=1588&top_left_y=738&top_left_x=240", "https://i.postimg.cc/cJ1dS67T/Screenshot-2024-04-06-at-19-54-53.png" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_22", "problem": "$\\mathrm{A} 、 \\mathrm{~B}$ 两颗卫星在同一平面内沿同一方向绕地球做匀速圆周运动, 它们之间的距离 $\\Delta r$随时间变化的关系如图所示, 不考虑 $\\mathrm{A} 、 \\mathrm{~B}$ 之间的万有引力, 已知地球的半径为 $0.8 r$,万有引力常量为 $G$, 卫星 $\\mathrm{A}$ 的线速度大于卫星 $\\mathrm{B}$ 的线速度, 则以下说法正确的是 $(\\quad)$\n\n[图1]\nA: 卫星 A 的发射速度可能大于第二宇宙速度\nB: 地球的第一宇宙速度为 $\\frac{8 \\sqrt{5} \\pi r}{7 T}$\nC: 地球的密度为 $\\frac{192 \\pi}{49 G T^{2}}$\nD: 卫星 A 的加速度大小为 $\\frac{256 \\pi^{2} r}{49 T^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n$\\mathrm{A} 、 \\mathrm{~B}$ 两颗卫星在同一平面内沿同一方向绕地球做匀速圆周运动, 它们之间的距离 $\\Delta r$随时间变化的关系如图所示, 不考虑 $\\mathrm{A} 、 \\mathrm{~B}$ 之间的万有引力, 已知地球的半径为 $0.8 r$,万有引力常量为 $G$, 卫星 $\\mathrm{A}$ 的线速度大于卫星 $\\mathrm{B}$ 的线速度, 则以下说法正确的是 $(\\quad)$\n\n[图1]\n\nA: 卫星 A 的发射速度可能大于第二宇宙速度\nB: 地球的第一宇宙速度为 $\\frac{8 \\sqrt{5} \\pi r}{7 T}$\nC: 地球的密度为 $\\frac{192 \\pi}{49 G T^{2}}$\nD: 卫星 A 的加速度大小为 $\\frac{256 \\pi^{2} r}{49 T^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-094.jpg?height=488&width=877&top_left_y=830&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_965", "problem": "On $21^{\\text {st }}$ March the Sun is in the constellation of Pisces. In which of these constellations would it be possible to find Venus? You are given that Venus has a circular orbit of radius 0.723 au.\nA: Aries\nB: Leo\nC: Libra\nD: Gemini\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOn $21^{\\text {st }}$ March the Sun is in the constellation of Pisces. In which of these constellations would it be possible to find Venus? You are given that Venus has a circular orbit of radius 0.723 au.\n\nA: Aries\nB: Leo\nC: Libra\nD: Gemini\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://i.postimg.cc/13TYv90C/Screenshot-2024-04-06-at-22-08-18.png" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_555", "problem": "2023 年 10 月 26 日, 神舟十七号载人飞船与天和核心舱进行了对接, “太空之家”迎来汤洪波、唐胜杰、江新林 3 名中国航天史上最年轻的乘组入驻。如图为神舟十七号的发射与交会对接过程示意图, 图中(1)为飞船的近地圆轨道, 其轨道半径为 $R_{1}$, (2)为椭圆变轨轨道, (3)为天和核心舱所在的圆轨道, 其轨道半径为 $R_{2}, P 、 Q$ 分别为(2)轨道与(1)、(3)轨道的交会点。关于神舟十七号载人飞船与天和核心舱交会对接过程，下列说法正确的是 ( )\n\n[图1]\nA: 飞船在轨道 3 上运行的速度大于第一宇宙速度\nB: 飞船从(2)轨道到变轨到(3)轨道需要在 $Q$ 点点火加速\nC: 飞船在(1)轨道的动能一定大于天和核心舱在(3)轨道的动能\nD: 若核心舱在(3)轨道运行周期为 $T$, 则飞船在(2)轨道从 $P$ 到 $Q$ 的时间为 $$ \\frac{1}{2} \\sqrt{\\left(\\frac{R_{1}+R_{2}}{2 R_{2}}\\right)^{3} T} $$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2023 年 10 月 26 日, 神舟十七号载人飞船与天和核心舱进行了对接, “太空之家”迎来汤洪波、唐胜杰、江新林 3 名中国航天史上最年轻的乘组入驻。如图为神舟十七号的发射与交会对接过程示意图, 图中(1)为飞船的近地圆轨道, 其轨道半径为 $R_{1}$, (2)为椭圆变轨轨道, (3)为天和核心舱所在的圆轨道, 其轨道半径为 $R_{2}, P 、 Q$ 分别为(2)轨道与(1)、(3)轨道的交会点。关于神舟十七号载人飞船与天和核心舱交会对接过程，下列说法正确的是 ( )\n\n[图1]\n\nA: 飞船在轨道 3 上运行的速度大于第一宇宙速度\nB: 飞船从(2)轨道到变轨到(3)轨道需要在 $Q$ 点点火加速\nC: 飞船在(1)轨道的动能一定大于天和核心舱在(3)轨道的动能\nD: 若核心舱在(3)轨道运行周期为 $T$, 则飞船在(2)轨道从 $P$ 到 $Q$ 的时间为 $$ \\frac{1}{2} \\sqrt{\\left(\\frac{R_{1}+R_{2}}{2 R_{2}}\\right)^{3} T} $$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-035.jpg?height=394&width=466&top_left_y=1962&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_297", "problem": "2020 年 5 月 5 日 18 时, 长征五号 $B$ 运载火箭成功将新一代载人飞船试验船送入预定轨道, 拉开了我国载人航天工程“第三步”任务序幕, 下图是试验船返回地球的示意图。假设地球半径为 $R$, 地球表面的重力加速度为 $g_{0}$, 试验船初始时在距地球表面高度为 $3 R$的圆形轨道 $\\mathrm{I}$ 上运动, 然后在 $A$ 点点火变轨进入粗圆轨道 II, 到达近地点 $B$ (忽略距离地表的高度)再次点火进入近地轨道 III 绕地球位圆周运动。下列说法中正确的是 ( )\n\n[图1]\nA: 在 $A$ 点点火的目的是增大试验船的速度\nB: 在 $B$ 点点火的目的是增大试验船的速度\nC: 试验船在轨道 I 上运行时的加速度大于在轨道 II 上运行时的加速度\nD: 试验船在轨道 III 上绕地球运行一周所需的时间为 $2 \\pi \\sqrt{\\frac{R}{g_{0}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年 5 月 5 日 18 时, 长征五号 $B$ 运载火箭成功将新一代载人飞船试验船送入预定轨道, 拉开了我国载人航天工程“第三步”任务序幕, 下图是试验船返回地球的示意图。假设地球半径为 $R$, 地球表面的重力加速度为 $g_{0}$, 试验船初始时在距地球表面高度为 $3 R$的圆形轨道 $\\mathrm{I}$ 上运动, 然后在 $A$ 点点火变轨进入粗圆轨道 II, 到达近地点 $B$ (忽略距离地表的高度)再次点火进入近地轨道 III 绕地球位圆周运动。下列说法中正确的是 ( )\n\n[图1]\n\nA: 在 $A$ 点点火的目的是增大试验船的速度\nB: 在 $B$ 点点火的目的是增大试验船的速度\nC: 试验船在轨道 I 上运行时的加速度大于在轨道 II 上运行时的加速度\nD: 试验船在轨道 III 上绕地球运行一周所需的时间为 $2 \\pi \\sqrt{\\frac{R}{g_{0}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-046.jpg?height=420&width=457&top_left_y=2000&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_340", "problem": "在宇宙中有两颗星组成的孤立“双星系统”, “双星系统”离其他恒星较远，通常可忽略其他星体对“双星系统”的引力作用。星 $\\mathrm{A}$ 和星 $\\mathrm{B}$ 的质量分别为 $M_{1}$ 和 $M_{2}$, 它们都绕二者连线上的某点做周期为 $T$ 的匀速圆周运动。已知引力常量为 $G$, 求星 $\\mathrm{A}$ 和星 $\\mathrm{B}$间的距离 $L$ 。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n在宇宙中有两颗星组成的孤立“双星系统”, “双星系统”离其他恒星较远，通常可忽略其他星体对“双星系统”的引力作用。星 $\\mathrm{A}$ 和星 $\\mathrm{B}$ 的质量分别为 $M_{1}$ 和 $M_{2}$, 它们都绕二者连线上的某点做周期为 $T$ 的匀速圆周运动。已知引力常量为 $G$, 求星 $\\mathrm{A}$ 和星 $\\mathrm{B}$间的距离 $L$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_689", "problem": "2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为第一个首次探测火星就实现“绕、落、巡”任务的国家。为了简化问题，可认为地球和火星在同一平面上绕太阳做匀速圆周运动, 如图 1 所示。已知地球的公转周期为 $T_{1}$, 公转轨道半径为 $r_{1}$, 火星的公转周期为 $T_{2}$, 火星质量为 $M$ 。如图 2 所示, 以火星为参考系, 质量为 $m_{1}$ 的探测器沿 1 号轨道到达 $B$ 点时速度为 $v_{1}, B$ 点到火星球心的距离为 $r_{3}$, 此时启动发动机, 在极短时间内一次性喷出部分气体, 喷气后探测器质量变为 $m_{2}$ 、速度变为与 $v_{1}$ 垂直的 $v_{2}$,然后进入以 $B$ 点为远火点的椭圆轨道 2 。已知万有引力势能公式 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $M$为中心天体的质量, $m$ 为卫星的质量, $G$ 为引力常量, $r$ 为卫星到中心天体球心的距离。求探测器沿 2 号轨道运动至近火点的速度 $v_{3}$ 的大小。[图1]\n\n图1\n\n[图2]\n\n图2", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为第一个首次探测火星就实现“绕、落、巡”任务的国家。为了简化问题，可认为地球和火星在同一平面上绕太阳做匀速圆周运动, 如图 1 所示。已知地球的公转周期为 $T_{1}$, 公转轨道半径为 $r_{1}$, 火星的公转周期为 $T_{2}$, 火星质量为 $M$ 。如图 2 所示, 以火星为参考系, 质量为 $m_{1}$ 的探测器沿 1 号轨道到达 $B$ 点时速度为 $v_{1}, B$ 点到火星球心的距离为 $r_{3}$, 此时启动发动机, 在极短时间内一次性喷出部分气体, 喷气后探测器质量变为 $m_{2}$ 、速度变为与 $v_{1}$ 垂直的 $v_{2}$,然后进入以 $B$ 点为远火点的椭圆轨道 2 。已知万有引力势能公式 $E_{\\mathrm{p}}=-\\frac{G M m}{r}$, 其中 $M$为中心天体的质量, $m$ 为卫星的质量, $G$ 为引力常量, $r$ 为卫星到中心天体球心的距离。求探测器沿 2 号轨道运动至近火点的速度 $v_{3}$ 的大小。[图1]\n\n图1\n\n[图2]\n\n图2\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-006.jpg?height=452&width=534&top_left_y=1493&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-006.jpg?height=451&width=911&top_left_y=1488&top_left_x=881" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_406", "problem": "“双星系统”由相距较近的恒星组成, 每个恒星的半径远小于两个恒星之间的距离,而且双星系统一般远离其他天体, 它们在相互间的万有引力作用下, 绕某一点做匀速圆周运动, 如图所示为某一双星系统, $\\mathrm{A}$ 星球的质量为 $m_{1}, \\mathrm{~B}$ 星球的质量为 $m_{2}$, 它们中心之间的距离为 $L$, 引力常量为 $G$, 则下列说法正确的是 ( )\n\n[图1]\nA: A 星球的轨道半径为 $R=\\frac{m_{1}}{m_{1}+m_{2}} L$\nB: 双星运行的周期为 $T=2 \\pi L \\sqrt{\\frac{L}{G\\left(m_{1}+m_{2}\\right)}}$\nC: B 星球的轨道半径为 $r=\\frac{m_{2}}{m_{1}} L$\nD: 若近似认为 $\\mathrm{B}$ 星球绕 $\\mathrm{A}$ 星球中心做圆周运动, 则 $\\mathrm{B}$ 星球的运行周期为 $$ T=2 \\pi L \\sqrt{\\frac{L}{G m_{2}}} $$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n“双星系统”由相距较近的恒星组成, 每个恒星的半径远小于两个恒星之间的距离,而且双星系统一般远离其他天体, 它们在相互间的万有引力作用下, 绕某一点做匀速圆周运动, 如图所示为某一双星系统, $\\mathrm{A}$ 星球的质量为 $m_{1}, \\mathrm{~B}$ 星球的质量为 $m_{2}$, 它们中心之间的距离为 $L$, 引力常量为 $G$, 则下列说法正确的是 ( )\n\n[图1]\n\nA: A 星球的轨道半径为 $R=\\frac{m_{1}}{m_{1}+m_{2}} L$\nB: 双星运行的周期为 $T=2 \\pi L \\sqrt{\\frac{L}{G\\left(m_{1}+m_{2}\\right)}}$\nC: B 星球的轨道半径为 $r=\\frac{m_{2}}{m_{1}} L$\nD: 若近似认为 $\\mathrm{B}$ 星球绕 $\\mathrm{A}$ 星球中心做圆周运动, 则 $\\mathrm{B}$ 星球的运行周期为 $$ T=2 \\pi L \\sqrt{\\frac{L}{G m_{2}}} $$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-036.jpg?height=291&width=365&top_left_y=1011&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_419", "problem": "地球表面上两极的重力加速度约为 $9.83 \\mathrm{~m} / \\mathrm{s}^{2}$, 而赤道上的重力加速度约为 $9.78 \\mathrm{~m} / \\mathrm{s}^{2}$,即赤道上的重力加速度比两极的重力加速度小约 $\\frac{1}{200}$, 赤道上有一观察者, 日落后, 他用天文望远镜观察被太阳光照射的地球同步卫星, 他在一天的时间内看不到此卫星的时间为 $t$, 若将地球看成球体, 且地球的质量分布均匀, 半径约为 $6.4 \\times 10^{3} \\mathrm{~km}$, 取 $\\sqrt[3]{200} \\approx 6, \\sin 10^{\\circ}=\\frac{1}{6}$, 则通过以上数据估算可得（）\nA: 同步卫星的高度约为 $3.2 \\times 10^{3} \\mathrm{~km}$\nB: 同步卫星的高度约为 $3.2 \\times 10^{5} \\mathrm{~km}$\nC: 看不到同步卫星的时间与看得到卫星的时间之比约为 $1: 17$\nD: 看不到同步卫星的时间与看得到卫星的时间之比约为 $2: 17$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球表面上两极的重力加速度约为 $9.83 \\mathrm{~m} / \\mathrm{s}^{2}$, 而赤道上的重力加速度约为 $9.78 \\mathrm{~m} / \\mathrm{s}^{2}$,即赤道上的重力加速度比两极的重力加速度小约 $\\frac{1}{200}$, 赤道上有一观察者, 日落后, 他用天文望远镜观察被太阳光照射的地球同步卫星, 他在一天的时间内看不到此卫星的时间为 $t$, 若将地球看成球体, 且地球的质量分布均匀, 半径约为 $6.4 \\times 10^{3} \\mathrm{~km}$, 取 $\\sqrt[3]{200} \\approx 6, \\sin 10^{\\circ}=\\frac{1}{6}$, 则通过以上数据估算可得（）\n\nA: 同步卫星的高度约为 $3.2 \\times 10^{3} \\mathrm{~km}$\nB: 同步卫星的高度约为 $3.2 \\times 10^{5} \\mathrm{~km}$\nC: 看不到同步卫星的时间与看得到卫星的时间之比约为 $1: 17$\nD: 看不到同步卫星的时间与看得到卫星的时间之比约为 $2: 17$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-18.jpg?height=266&width=649&top_left_y=1923&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_931", "problem": "In the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year.\n\n[figure1]\n\nFigure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com\n\nCalculate (to the nearest day) how much earlier in the Gregorian calendar (on average) a given month in the Islamic calendar will start compared to the previous year.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year.\n\n[figure1]\n\nFigure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com\n\nCalculate (to the nearest day) how much earlier in the Gregorian calendar (on average) a given month in the Islamic calendar will start compared to the previous year.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_2c19fdb17927c588761dg-09.jpg?height=800&width=1110&top_left_y=862&top_left_x=473" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_82", "problem": "已知地球的质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。赤道上地球表面附近的重力加速度用 $g_{e}$ 表示, 北极处地球表面附近的重力加速度用 $g_{N}$ 表示, 将地球视为均匀球体。\n用 $g_{e} 、 g_{N}$ 和半径 $R$ 表示地球自转周期;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n已知地球的质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。赤道上地球表面附近的重力加速度用 $g_{e}$ 表示, 北极处地球表面附近的重力加速度用 $g_{N}$ 表示, 将地球视为均匀球体。\n用 $g_{e} 、 g_{N}$ 和半径 $R$ 表示地球自转周期;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-093.jpg?height=579&width=599&top_left_y=1735&top_left_x=360" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_979", "problem": "From the upper parts of the Sun's corona comes a stream of charged particles called the solar wind, meaning the Sun is slowly losing some of its mass (although the effect is negligible compared to the mass loss in nuclear fusion). The particles travel at supersonic speeds until the pressure from interstellar space causes their speed to drop to subsonic speeds instead - this transition is called the termination shock, and has been explored by the two Voyager probes as they leave the solar system.\n[figure1]\n\nFigure 3: Left: A demonstration of a termination shock formed with water flowing from a tap into a sink. Right: The Voyager spacecraft crossing the termination shock of the Solar System.\n\nAt the orbit of the Earth, the solar wind is measured to have a density of 7 protons $\\mathrm{cm}^{-3}$ and to be travelling at $500 \\mathrm{~km} \\mathrm{~s}^{-1}$. Calculate $\\Delta M / \\Delta t$, giving your answer in units of $\\mathrm{M}_{\\odot}$ year $^{-1}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nFrom the upper parts of the Sun's corona comes a stream of charged particles called the solar wind, meaning the Sun is slowly losing some of its mass (although the effect is negligible compared to the mass loss in nuclear fusion). The particles travel at supersonic speeds until the pressure from interstellar space causes their speed to drop to subsonic speeds instead - this transition is called the termination shock, and has been explored by the two Voyager probes as they leave the solar system.\n[figure1]\n\nFigure 3: Left: A demonstration of a termination shock formed with water flowing from a tap into a sink. Right: The Voyager spacecraft crossing the termination shock of the Solar System.\n\nAt the orbit of the Earth, the solar wind is measured to have a density of 7 protons $\\mathrm{cm}^{-3}$ and to be travelling at $500 \\mathrm{~km} \\mathrm{~s}^{-1}$. Calculate $\\Delta M / \\Delta t$, giving your answer in units of $\\mathrm{M}_{\\odot}$ year $^{-1}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c744602885fab54c0985g-8.jpg?height=454&width=1280&top_left_y=835&top_left_x=386" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_19", "problem": "将太阳系八大行星绕太阳的运转视作匀速圆周运动。设行星的轨道半径为 $R$, 环绕周期为 $T$, 角速度为 $\\omega$, 环绕速度为 $v$, 下列描述它们之间的关系图像中正确的是 ( )\nA: [图1]\nB: [图2]\nC: [图3]\nD: [图4]\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n将太阳系八大行星绕太阳的运转视作匀速圆周运动。设行星的轨道半径为 $R$, 环绕周期为 $T$, 角速度为 $\\omega$, 环绕速度为 $v$, 下列描述它们之间的关系图像中正确的是 ( )\n\nA: [图1]\nB: [图2]\nC: [图3]\nD: [图4]\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-099.jpg?height=282&width=308&top_left_y=1561&top_left_x=454", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-099.jpg?height=280&width=325&top_left_y=1559&top_left_x=1114", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-099.jpg?height=283&width=349&top_left_y=1829&top_left_x=745", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-099.jpg?height=285&width=326&top_left_y=2096&top_left_x=748" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_452", "problem": "我国航天技术水平在世界处于领先地位，对于人造卫星的发射，有人提出了利用“地球隧道”发射人造卫星的构想：沿地球的一条弦挖一通道，在通道的两个出口处分别将等质量的待发射卫星部件同时释放，部件将在通道中间位置“碰撞组装”成卫星并静止下来; 另在通道的出口处由静止释放一个大质量物体，大质量物体会在通道与待发射的卫星碰撞, 只要物体质量相比卫星质量足够大, 卫星获得足够速度就会从对向通道口射出。\n\n(以下计算中, 已知地球的质量为 $M_{0}$, 地球半径为 $R_{0}$, 引力常量为 $G$, 可忽略通道 $A B$的内径大小和地球自转影响。)\n\n如图丙所示, 如果质量为 $m$ 的待发射卫星已静止在通道中心 $O^{\\prime}$ 处, 由 $A$ 处静止释放另一质量为 $M$ 的物体, 物体到达 $O^{\\prime}$ 处与卫星发生弹性正碰, 设 $M$ 远大于 $m$, 计算时可取 $\\frac{m}{M} \\approx 0$ 。卫星从图丙示通道右侧 $B$ 处飞出, 为使飞出速度达到地球第一宇宙速度, $h$ 应为多大?\n\n[图1]\n\n丙", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n我国航天技术水平在世界处于领先地位，对于人造卫星的发射，有人提出了利用“地球隧道”发射人造卫星的构想：沿地球的一条弦挖一通道，在通道的两个出口处分别将等质量的待发射卫星部件同时释放，部件将在通道中间位置“碰撞组装”成卫星并静止下来; 另在通道的出口处由静止释放一个大质量物体，大质量物体会在通道与待发射的卫星碰撞, 只要物体质量相比卫星质量足够大, 卫星获得足够速度就会从对向通道口射出。\n\n(以下计算中, 已知地球的质量为 $M_{0}$, 地球半径为 $R_{0}$, 引力常量为 $G$, 可忽略通道 $A B$的内径大小和地球自转影响。)\n\n如图丙所示, 如果质量为 $m$ 的待发射卫星已静止在通道中心 $O^{\\prime}$ 处, 由 $A$ 处静止释放另一质量为 $M$ 的物体, 物体到达 $O^{\\prime}$ 处与卫星发生弹性正碰, 设 $M$ 远大于 $m$, 计算时可取 $\\frac{m}{M} \\approx 0$ 。卫星从图丙示通道右侧 $B$ 处飞出, 为使飞出速度达到地球第一宇宙速度, $h$ 应为多大?\n\n[图1]\n\n丙\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-067.jpg?height=408&width=462&top_left_y=687&top_left_x=1294" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1010", "problem": "On $24^{\\text {th }}$ August 2016, astronomers discovered a planet orbiting the closest star to the Sun, Proxima Centauri, situated 4.22 light years away, which fulfils a long-standing dream of science-fiction writers: a world that is close enough for humans to send their first interstellar spacecraft.\n\nAstronomers have noted how the motion of Proxima Centauri changed in the first months of 2016, with the star moving towards and away from the Earth, as seen in the figure below. Sometimes Proxima Centauri is approaching Earth at $5 \\mathrm{~km} \\mathrm{hour}^{-1}-$ normal human walking pace - and at times receding at the same speed. This regular pattern of changing radial velocities caused by an unseen planet, which they named Proxima Centauri B, repeats and results in tiny Doppler shifts in the star's light, making the light appear slightly redder, then bluer.\n\n[figure1]\n\nUsing a simple approximation, the equilibrium temperature of a planet can be calculated as\n\n$$\nT_{\\text {Planet }}=T_{\\text {Star }} \\sqrt{\\frac{R_{\\text {Star }}}{2 d}}\n$$\n\nwhere $d$ is the distance between the star and the planet. Given that the astronomers discovered that the orbit of the planet is in fact an ellipse with an eccentricity of 0.35 , and that the star has a surface temperature of $3000 \\mathrm{~K}$ and a radius of $0.14 R_{\\odot^{\\prime}}$ what are the minimum and maximum equilibrium temperatures of Proxima Centauri B?\n\nComment on whether or not the planet is in the habitable zone of Proxima Centauri.\n\n[The habitable zone is the band around a star where a planet can have water on its surface in liquid form, at normal pressure.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a range interval.\n\nproblem:\nOn $24^{\\text {th }}$ August 2016, astronomers discovered a planet orbiting the closest star to the Sun, Proxima Centauri, situated 4.22 light years away, which fulfils a long-standing dream of science-fiction writers: a world that is close enough for humans to send their first interstellar spacecraft.\n\nAstronomers have noted how the motion of Proxima Centauri changed in the first months of 2016, with the star moving towards and away from the Earth, as seen in the figure below. Sometimes Proxima Centauri is approaching Earth at $5 \\mathrm{~km} \\mathrm{hour}^{-1}-$ normal human walking pace - and at times receding at the same speed. This regular pattern of changing radial velocities caused by an unseen planet, which they named Proxima Centauri B, repeats and results in tiny Doppler shifts in the star's light, making the light appear slightly redder, then bluer.\n\n[figure1]\n\nUsing a simple approximation, the equilibrium temperature of a planet can be calculated as\n\n$$\nT_{\\text {Planet }}=T_{\\text {Star }} \\sqrt{\\frac{R_{\\text {Star }}}{2 d}}\n$$\n\nwhere $d$ is the distance between the star and the planet. Given that the astronomers discovered that the orbit of the planet is in fact an ellipse with an eccentricity of 0.35 , and that the star has a surface temperature of $3000 \\mathrm{~K}$ and a radius of $0.14 R_{\\odot^{\\prime}}$ what are the minimum and maximum equilibrium temperatures of Proxima Centauri B?\n\nComment on whether or not the planet is in the habitable zone of Proxima Centauri.\n\n[The habitable zone is the band around a star where a planet can have water on its surface in liquid form, at normal pressure.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an interval, e.g. ANSWER=(1,2] \\cup[7,+\\infty)", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6d91a7785df4f4beaa9ag-10.jpg?height=545&width=1602&top_left_y=1007&top_left_x=227" ], "answer": null, "solution": null, "answer_type": "IN", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_478", "problem": "电影中的太空电梯非常吸引人。现假设已经建成了如图所示的太空电梯, 其通过超级缆绳将地球赤道上的固定基地、同步空间站和配重空间站连接在一起，它们随地球同步旋转。图中配重空间站比同步空间站更高, $P$ 是缆绳上的一个平台。则下列说法正确的是 ( )\n\n[图1]\nA: 太空电梯上各点线速度的平方与该点离地球球心的距离成反比\nB: 宇航员在配重空间站时处于完全失重状态\nC: 若从 $P$ 平台向外自由释放一个小物块, 则小物块会一边朝 $P$ 点转动的方向向前运动一边落向地球\nD: 若两空间站之间缆绳断裂, 配重空间站将绕地球做椭圆运动, 且断裂处为椭圆的远地点\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n电影中的太空电梯非常吸引人。现假设已经建成了如图所示的太空电梯, 其通过超级缆绳将地球赤道上的固定基地、同步空间站和配重空间站连接在一起，它们随地球同步旋转。图中配重空间站比同步空间站更高, $P$ 是缆绳上的一个平台。则下列说法正确的是 ( )\n\n[图1]\n\nA: 太空电梯上各点线速度的平方与该点离地球球心的距离成反比\nB: 宇航员在配重空间站时处于完全失重状态\nC: 若从 $P$ 平台向外自由释放一个小物块, 则小物块会一边朝 $P$ 点转动的方向向前运动一边落向地球\nD: 若两空间站之间缆绳断裂, 配重空间站将绕地球做椭圆运动, 且断裂处为椭圆的远地点\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-043.jpg?height=406&width=874&top_left_y=865&top_left_x=357" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_150", "problem": "在寻找假想的第九大行星的过程中，天文学家发现了 2018VG18“外海王星天体”,外海王星天体的实际运行轨道与计算中的轨道数据有偏差, 而这通常被认为是受到了假想中第九行星（轨道处在太阳与外海王星天体之间）的引力扰动所致。如图所示, 在运行轨道不变的情况下, 当存在假想中的第九行星时, 跟没有假想中的第九行星相比, 2018VG18 ( )\n\n[图1]\nA: 公转周期更大\nB: 平均速率更小\nC: 自转周期更大\nD: 公转向心加速度更大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n在寻找假想的第九大行星的过程中，天文学家发现了 2018VG18“外海王星天体”,外海王星天体的实际运行轨道与计算中的轨道数据有偏差, 而这通常被认为是受到了假想中第九行星（轨道处在太阳与外海王星天体之间）的引力扰动所致。如图所示, 在运行轨道不变的情况下, 当存在假想中的第九行星时, 跟没有假想中的第九行星相比, 2018VG18 ( )\n\n[图1]\n\nA: 公转周期更大\nB: 平均速率更小\nC: 自转周期更大\nD: 公转向心加速度更大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-086.jpg?height=283&width=605&top_left_y=1503&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_860", "problem": "The fictional towns of Baia and Caia are located at $\\left(66.56^{\\circ} \\mathrm{N}, 67.55^{\\circ} \\mathrm{E}\\right)$ and $\\left(\\delta, 18.95^{\\circ} \\mathrm{E}\\right)$, respectively. It is known that the spherical triangle with vertices at Baia, Caia, and the North Pole covers $6.75 \\%$ of Earth's surface. Compute $\\delta$.\nA: $66.56^{\\circ} \\mathrm{N}$\nB: $55.25^{\\circ} \\mathrm{N}$\nC: $23.44^{\\circ} \\mathrm{N}$\nD: $55.25^{\\circ} \\mathrm{S}$\nE: $66.56^{\\circ} \\mathrm{S}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe fictional towns of Baia and Caia are located at $\\left(66.56^{\\circ} \\mathrm{N}, 67.55^{\\circ} \\mathrm{E}\\right)$ and $\\left(\\delta, 18.95^{\\circ} \\mathrm{E}\\right)$, respectively. It is known that the spherical triangle with vertices at Baia, Caia, and the North Pole covers $6.75 \\%$ of Earth's surface. Compute $\\delta$.\n\nA: $66.56^{\\circ} \\mathrm{N}$\nB: $55.25^{\\circ} \\mathrm{N}$\nC: $23.44^{\\circ} \\mathrm{N}$\nD: $55.25^{\\circ} \\mathrm{S}$\nE: $66.56^{\\circ} \\mathrm{S}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_101", "problem": "如图所示设地球的质量为 $M$ 且绕太阳做匀速圆周运动, 当地球运动到 $D$ 点时, 有一质量为 $m$ 的飞船由静止开始从 $D$ 点只在恒力 $F$ 的作用下沿 $D C$ 方向做匀加速直线运动, 再过两个月, 飞船在 $C$ 处再次掠过地球上空, 假设太阳与地球的万有引力作用不改变飞船所受恒力 $F$ 的大小和方向, 飞船到地球表面的距离远小于地球与太阳间的距离,则地球与太阳间的万有引力大小（）\n\n[图1]\nA: $\\frac{M F \\pi^{2}}{3 m}$\nB: $\\frac{M F \\pi^{2}}{6 m}$\nC: $\\frac{M F \\pi^{2}}{9 m}$\nD: $\\frac{M F \\pi^{2}}{18 m}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示设地球的质量为 $M$ 且绕太阳做匀速圆周运动, 当地球运动到 $D$ 点时, 有一质量为 $m$ 的飞船由静止开始从 $D$ 点只在恒力 $F$ 的作用下沿 $D C$ 方向做匀加速直线运动, 再过两个月, 飞船在 $C$ 处再次掠过地球上空, 假设太阳与地球的万有引力作用不改变飞船所受恒力 $F$ 的大小和方向, 飞船到地球表面的距离远小于地球与太阳间的距离,则地球与太阳间的万有引力大小（）\n\n[图1]\n\nA: $\\frac{M F \\pi^{2}}{3 m}$\nB: $\\frac{M F \\pi^{2}}{6 m}$\nC: $\\frac{M F \\pi^{2}}{9 m}$\nD: $\\frac{M F \\pi^{2}}{18 m}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://i.postimg.cc/X7JzxxVN/image.png" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_548", "problem": "将物体以一定初速度 $v_{0}=4 \\mathrm{~m} / \\mathrm{s}$ 在某行星表面坚直上抛, 从抛出开始计时, 落回抛出点前, 物体第 $1 \\mathrm{~s}$ 内和第 $4 \\mathrm{~s}$ 内通过的位移大小相等。已知该行星半径和地球半径相同的,地球表面的重力加速度为 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 不计空气阻力, 则下列说法正确的是 ( )\nA: 地球的平均密度和该行星的平均密度之比为 1:5\nB: 该行星和地球的第一宇宙速度之比为 $1: \\sqrt{5}$\nC: 将物体在地球表面以相同的初速度 $v_{0}$ 坚直抛出后上升的最大高度 $1.6 \\mathrm{~m}$\nD: 该行星表面的自由落体加速度大小为 $1 \\mathrm{~m} / \\mathrm{s}^{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n将物体以一定初速度 $v_{0}=4 \\mathrm{~m} / \\mathrm{s}$ 在某行星表面坚直上抛, 从抛出开始计时, 落回抛出点前, 物体第 $1 \\mathrm{~s}$ 内和第 $4 \\mathrm{~s}$ 内通过的位移大小相等。已知该行星半径和地球半径相同的,地球表面的重力加速度为 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 不计空气阻力, 则下列说法正确的是 ( )\n\nA: 地球的平均密度和该行星的平均密度之比为 1:5\nB: 该行星和地球的第一宇宙速度之比为 $1: \\sqrt{5}$\nC: 将物体在地球表面以相同的初速度 $v_{0}$ 坚直抛出后上升的最大高度 $1.6 \\mathrm{~m}$\nD: 该行星表面的自由落体加速度大小为 $1 \\mathrm{~m} / \\mathrm{s}^{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_91", "problem": "2020 年 7 月 31 日, 北斗闪耀, 泽沐八方。北斗三号全球卫星导航系统 (如图甲所示）建成暨开通仪式在北京举行。如图乙所示为 55 颗卫星绕地球在不同轨道上运动的 $\\lg T-\\lg r$ 图像, 其中 $\\mathrm{T}$ 为卫星的周期, $r$ 为卫星的轨道半径, 1 和 2 为其中的两颗卫星。\n已知引力常量为 $G$, 下列说法正确的是 ( )\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙\nA: 地球的半径为 $x_{0}$\nB: 地球质量为 $\\frac{4 \\pi^{2} 10^{b}}{G}$\nC: 卫星 1 和 2 运动的线速度大小之比为 $x_{1}: x_{2}$\nD: 卫星 1 和 2 向心加速度大小之比为 $10^{2 x_{2}}: 10^{2 x_{1}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年 7 月 31 日, 北斗闪耀, 泽沐八方。北斗三号全球卫星导航系统 (如图甲所示）建成暨开通仪式在北京举行。如图乙所示为 55 颗卫星绕地球在不同轨道上运动的 $\\lg T-\\lg r$ 图像, 其中 $\\mathrm{T}$ 为卫星的周期, $r$ 为卫星的轨道半径, 1 和 2 为其中的两颗卫星。\n已知引力常量为 $G$, 下列说法正确的是 ( )\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙\n\nA: 地球的半径为 $x_{0}$\nB: 地球质量为 $\\frac{4 \\pi^{2} 10^{b}}{G}$\nC: 卫星 1 和 2 运动的线速度大小之比为 $x_{1}: x_{2}$\nD: 卫星 1 和 2 向心加速度大小之比为 $10^{2 x_{2}}: 10^{2 x_{1}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-007.jpg?height=302&width=260&top_left_y=226&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-007.jpg?height=317&width=502&top_left_y=230&top_left_x=614" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1062", "problem": "On $21^{\\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history.\n\n## Total Solar Eclipse of 2017 Aug 21\n\n[figure1]\n\nFigure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC.\n\nThe path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse (\"GE\"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question:\n\n- The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\\prime} 48.7^{\\prime \\prime}$ and $16^{\\prime} 03.4^{\\prime \\prime}$, respectively, where the notation $x x^{\\prime} y y . y^{\\prime \\prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$\n- The latitude and longitude of the location of GE are $36^{\\circ} 58.0^{\\prime} \\mathrm{N}$ and $87^{\\circ} 40.3^{\\prime} \\mathrm{W}$, respectively\n- Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\\odot}=695700 \\mathrm{~km}, R_{\\oplus}=$ $6371 \\mathrm{~km}$ and $R_{\\text {Moon }}=1737 \\mathrm{~km}$, and a day to be 24 hours\n- Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \\mathrm{~km}$ and $384400 \\mathrm{~km}$, respectively\n- As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction\n\nFor an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as:\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nThe point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration (\"GD\") was at co-ordinates of $37^{\\circ} 35^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 07^{\\prime} \\mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \\mathrm{~s}$ longer than the value calculated in part $\\mathrm{c}$.\n\n[figure2]\n\nFigure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \\& Google Maps.d. The town of Carbondale, Illinois, is the closest big town to the point of GD, with co-ordinates $37^{\\circ} 44^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 13^{\\prime} \\mathrm{W}$ longitude. Assuming the path of maximum totality can be treated as linear as it passes through the region around GD and GE:\n\niii. Hence calculate (to the nearest $0.1 \\mathrm{~s}$ ) how much shorter totality was for residents in Carbondale compared with CP. Take the duration at CP to be the same as GD, the width of the path to be $0.5 \\mathrm{~km}$ less than for GE (so the Moon's shadow is elliptical), and the speed of the Moon's shadow to have only been affected by the change in latitude.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn $21^{\\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history.\n\n## Total Solar Eclipse of 2017 Aug 21\n\n[figure1]\n\nFigure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC.\n\nThe path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse (\"GE\"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question:\n\n- The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\\prime} 48.7^{\\prime \\prime}$ and $16^{\\prime} 03.4^{\\prime \\prime}$, respectively, where the notation $x x^{\\prime} y y . y^{\\prime \\prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$\n- The latitude and longitude of the location of GE are $36^{\\circ} 58.0^{\\prime} \\mathrm{N}$ and $87^{\\circ} 40.3^{\\prime} \\mathrm{W}$, respectively\n- Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\\odot}=695700 \\mathrm{~km}, R_{\\oplus}=$ $6371 \\mathrm{~km}$ and $R_{\\text {Moon }}=1737 \\mathrm{~km}$, and a day to be 24 hours\n- Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \\mathrm{~km}$ and $384400 \\mathrm{~km}$, respectively\n- As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction\n\nFor an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as:\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nThe point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration (\"GD\") was at co-ordinates of $37^{\\circ} 35^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 07^{\\prime} \\mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \\mathrm{~s}$ longer than the value calculated in part $\\mathrm{c}$.\n\n[figure2]\n\nFigure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \\& Google Maps.\n\nproblem:\nd. The town of Carbondale, Illinois, is the closest big town to the point of GD, with co-ordinates $37^{\\circ} 44^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 13^{\\prime} \\mathrm{W}$ longitude. Assuming the path of maximum totality can be treated as linear as it passes through the region around GD and GE:\n\niii. Hence calculate (to the nearest $0.1 \\mathrm{~s}$ ) how much shorter totality was for residents in Carbondale compared with CP. Take the duration at CP to be the same as GD, the width of the path to be $0.5 \\mathrm{~km}$ less than for GE (so the Moon's shadow is elliptical), and the speed of the Moon's shadow to have only been affected by the change in latitude.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-08.jpg?height=1011&width=1014&top_left_y=497&top_left_x=521", "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-09.jpg?height=859&width=1213&top_left_y=924&top_left_x=410", "https://cdn.mathpix.com/cropped/2024_03_14_062b7eb54e0b05b0a6dfg-08.jpg?height=486&width=604&top_left_y=959&top_left_x=429" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1177", "problem": "On $21^{\\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history.\n\n## Total Solar Eclipse of 2017 Aug 21\n\n[figure1]\n\nFigure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC.\n\nThe path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse (\"GE\"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question:\n\n- The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\\prime} 48.7^{\\prime \\prime}$ and $16^{\\prime} 03.4^{\\prime \\prime}$, respectively, where the notation $x x^{\\prime} y y . y^{\\prime \\prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$\n- The latitude and longitude of the location of GE are $36^{\\circ} 58.0^{\\prime} \\mathrm{N}$ and $87^{\\circ} 40.3^{\\prime} \\mathrm{W}$, respectively\n- Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\\odot}=695700 \\mathrm{~km}, R_{\\oplus}=$ $6371 \\mathrm{~km}$ and $R_{\\text {Moon }}=1737 \\mathrm{~km}$, and a day to be 24 hours\n- Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \\mathrm{~km}$ and $384400 \\mathrm{~km}$, respectively\n- As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction\n\nFor an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as:\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nThe point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration (\"GD\") was at co-ordinates of $37^{\\circ} 35^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 07^{\\prime} \\mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \\mathrm{~s}$ longer than the value calculated in part $\\mathrm{c}$.\n\n[figure2]\n\nFigure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \\& Google Maps.d. The town of Carbondale, Illinois, is the closest big town to the point of GD, with co-ordinates $37^{\\circ} 44^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 13^{\\prime} \\mathrm{W}$ longitude. Assuming the path of maximum totality can be treated as linear as it passes through the region around GD and GE:\n\ni. Calculate the co-ordinates of the closest point (\" $\\mathrm{CP}^{\\prime \\prime}$ ) to Carbondale on the path of maximum totality.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn $21^{\\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history.\n\n## Total Solar Eclipse of 2017 Aug 21\n\n[figure1]\n\nFigure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC.\n\nThe path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse (\"GE\"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question:\n\n- The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\\prime} 48.7^{\\prime \\prime}$ and $16^{\\prime} 03.4^{\\prime \\prime}$, respectively, where the notation $x x^{\\prime} y y . y^{\\prime \\prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$\n- The latitude and longitude of the location of GE are $36^{\\circ} 58.0^{\\prime} \\mathrm{N}$ and $87^{\\circ} 40.3^{\\prime} \\mathrm{W}$, respectively\n- Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\\odot}=695700 \\mathrm{~km}, R_{\\oplus}=$ $6371 \\mathrm{~km}$ and $R_{\\text {Moon }}=1737 \\mathrm{~km}$, and a day to be 24 hours\n- Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \\mathrm{~km}$ and $384400 \\mathrm{~km}$, respectively\n- As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction\n\nFor an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as:\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nThe point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration (\"GD\") was at co-ordinates of $37^{\\circ} 35^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 07^{\\prime} \\mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \\mathrm{~s}$ longer than the value calculated in part $\\mathrm{c}$.\n\n[figure2]\n\nFigure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \\& Google Maps.\n\nproblem:\nd. The town of Carbondale, Illinois, is the closest big town to the point of GD, with co-ordinates $37^{\\circ} 44^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 13^{\\prime} \\mathrm{W}$ longitude. Assuming the path of maximum totality can be treated as linear as it passes through the region around GD and GE:\n\ni. Calculate the co-ordinates of the closest point (\" $\\mathrm{CP}^{\\prime \\prime}$ ) to Carbondale on the path of maximum totality.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\circ, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-08.jpg?height=1011&width=1014&top_left_y=497&top_left_x=521", "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-09.jpg?height=859&width=1213&top_left_y=924&top_left_x=410" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\circ" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_229", "problem": "2021 年 6 月 17 日, 搭载神舟十二号载人飞船的长征二号 $\\mathrm{F}$ 遥十二运载火箭在酒泉卫星发射中心点成功发射升空, 神舟十二号载人飞船与天和核心舱及天舟二号组合体成功对接，将中国三名航天员送入“太空家园”，核心舱绕地球飞行的轨道可视为圆轨道，轨道离地面的高度约为地球半径的 $\\frac{1}{16}$, 运行周期约为 $90 \\mathrm{~min}$, 引力常量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$ 。下列说法中正确的是（）\n\n[图1]\nA: 核心舱在轨道上飞行的速度约为 $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: 仅根据题中数据即可估算出地球密度\nC: “太空家园”中静止状态的宇航员的加速度为 0 \nD: 理论上火箭从酒泉发射比从文昌发射更节省燃料\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2021 年 6 月 17 日, 搭载神舟十二号载人飞船的长征二号 $\\mathrm{F}$ 遥十二运载火箭在酒泉卫星发射中心点成功发射升空, 神舟十二号载人飞船与天和核心舱及天舟二号组合体成功对接，将中国三名航天员送入“太空家园”，核心舱绕地球飞行的轨道可视为圆轨道，轨道离地面的高度约为地球半径的 $\\frac{1}{16}$, 运行周期约为 $90 \\mathrm{~min}$, 引力常量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$ 。下列说法中正确的是（）\n\n[图1]\n\nA: 核心舱在轨道上飞行的速度约为 $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: 仅根据题中数据即可估算出地球密度\nC: “太空家园”中静止状态的宇航员的加速度为 0 \nD: 理论上火箭从酒泉发射比从文昌发射更节省燃料\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-014.jpg?height=371&width=622&top_left_y=157&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_384", "problem": "拉格朗日点指在两个大天体引力作用下, 能使小物体稳定的点 (小物体质量相对两\n大天体可忽略不计)。这些点的存在由法国数学家拉格朗日于 1772 年推导证明的, 1906 年首次发现运动于木星轨道上的小行星 (见脱罗央群小行星) 在木星和太阳的作用下处于拉格朗日点上。在每个由两大天体构成的系统中, 按推论有 5 个拉格朗日点, 其中连线上有三个拉格朗日点, 分别是 $L_{1} 、 L_{2} 、 L_{3}$, 如图所示。我国发射的“鹊桥”卫星就在地月系统平衡点 $L_{2}$ 点做周期运动, 通过定期轨控保持轨道的稳定性, 可实现对着陆器和巡视器的中继通信覆盖, 首次实现地月 $L_{2}$ 点周期轨道的长期稳定运行。设某两个天体系统的中心天体质量为 $M$, 环绕天体质量为 $m$, 两天体间距离为 $L$, 万有引力常量为 $G, L_{1}$ 点到中心天体的距离为 $R_{1}, L_{2}$ 点到中心天体的距离为 $R_{2}$ 。求:\n\n为了进一步的通信覆盖, 发射两颗质量均为 $m_{0}$ 的卫星, 分别处于 $L_{1} 、 L_{2}$ 点, $L_{1}$ 、 $L_{2}$ 到环绕天体的距离近似相等 (远小于 $L$ ), 两卫星与环绕天体同步绕中心天体做圆周运动, 忽略卫星间的引力, 求中心天体对环绕天体的引力与它对两卫星的引力之和的比值?\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n拉格朗日点指在两个大天体引力作用下, 能使小物体稳定的点 (小物体质量相对两\n大天体可忽略不计)。这些点的存在由法国数学家拉格朗日于 1772 年推导证明的, 1906 年首次发现运动于木星轨道上的小行星 (见脱罗央群小行星) 在木星和太阳的作用下处于拉格朗日点上。在每个由两大天体构成的系统中, 按推论有 5 个拉格朗日点, 其中连线上有三个拉格朗日点, 分别是 $L_{1} 、 L_{2} 、 L_{3}$, 如图所示。我国发射的“鹊桥”卫星就在地月系统平衡点 $L_{2}$ 点做周期运动, 通过定期轨控保持轨道的稳定性, 可实现对着陆器和巡视器的中继通信覆盖, 首次实现地月 $L_{2}$ 点周期轨道的长期稳定运行。设某两个天体系统的中心天体质量为 $M$, 环绕天体质量为 $m$, 两天体间距离为 $L$, 万有引力常量为 $G, L_{1}$ 点到中心天体的距离为 $R_{1}, L_{2}$ 点到中心天体的距离为 $R_{2}$ 。求:\n\n为了进一步的通信覆盖, 发射两颗质量均为 $m_{0}$ 的卫星, 分别处于 $L_{1} 、 L_{2}$ 点, $L_{1}$ 、 $L_{2}$ 到环绕天体的距离近似相等 (远小于 $L$ ), 两卫星与环绕天体同步绕中心天体做圆周运动, 忽略卫星间的引力, 求中心天体对环绕天体的引力与它对两卫星的引力之和的比值?\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-098.jpg?height=469&width=620&top_left_y=1299&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1074", "problem": "The James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.c. Computer models suggest the first galaxies formed around $z \\sim 10-20$. One of the best ways to look for high-redshift galaxies is to try and detect the emission from the Lyman alpha (Lya) emission line at $\\lambda_{\\text {emit }}=121.6 \\mathrm{~nm}$ as it is a relatively bright line. Some of the brightest galaxies in that initial era of galaxy formation would have an absolute magnitude of $\\mathcal{M} \\sim 20$. In this question, you are given that $\\Omega_{0, \\mathrm{~m}}=0.3, \\Omega_{0, \\Lambda}=0.7, \\Omega_{0, \\mathrm{r}}=0$ and $\\mathrm{H}_{0}=70 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\n\ni. Calculate the redshift at which the Lya line is detected in the centre of the F200W filter.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\\text {th }}$ December 2021 . Its mirror is approximately $6.5 \\mathrm{~m}$ in diameter, much larger than the $2.4 \\mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies.\n[figure1]\n\nFigure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA.\n\nThe resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\\lambda$. The resolution limit of a CCD is set by the size of the pixels.\n\nThree of the imaging cameras on JWST are tabulated with some properties below:\n\n| Instrument | Wavelength range $(\\mu \\mathrm{m})$ | CCD plate scale (arcseconds / pixel) |\n| :---: | :---: | :---: |\n| NIRCam (short wave) | $0.6-2.3$ | 0.031 |\n| NIRCam (long wave) | $2.4-5.0$ | 0.065 |\n| MIRI | $5.6-25.5$ | 0.11 |\n\nAn arcsecond is a measure of angle where $1^{\\circ}=3600$ arcseconds.\n\nThe familiar variation in intensity on a screen, $I_{\\text {slit }}$, due to diffraction through an infinitely tall single slit is given as\n\n$$\nI_{\\text {slit }}=I_{0}\\left(\\frac{\\sin (x)}{x}\\right)^{2}, \\text { where } \\quad x=\\frac{\\pi D \\theta}{\\lambda}\n$$\n\nand $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as\n\n$$\nI_{\\mathrm{circ}}=I_{0}\\left(\\frac{2 J_{1}(x)}{x}\\right)^{2} .\n$$\n\nHere $J_{1}(x)$ is the Bessel function of the first kind and is calculated as\n\n$$\nJ_{n}(x)=\\sum_{r=0}^{\\infty} \\frac{(-1)^{r}}{r !(n+r) !}\\left(\\frac{x}{2}\\right)^{n+2 r} \\quad \\text { so } \\quad J_{1}(x)=\\frac{x}{2}\\left(1-\\frac{x^{2}}{8}+\\frac{x^{4}}{192}-\\ldots\\right) .\n$$\n\nThe $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\\text {slit }}$ is at $x_{\\min }=\\pi$ meaning that $\\theta_{\\min , \\text { slit }}=\\lambda / D$, whilst for $I_{\\text {circ }}$ it is at $x_{\\min }=3.8317 \\ldots$ so $\\theta_{\\min , \\mathrm{circ}} \\approx 1.22 \\lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM).\n[figure2]\n\nFigure 6: Left: The $I_{\\text {slit }}$ (purple) and $I_{\\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different.\n\nRight: How $x_{\\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\\text {circ }}$.\n\nAs well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \\mu \\mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \\mathrm{nJy}\\left(1 \\mathrm{Jy}=10^{-26} \\mathrm{~W} \\mathrm{~m}^{-2} \\mathrm{~Hz}^{-1}\\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe.\n\nThe scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as\n\n$$\na=(1+z)^{-1} \\quad \\text { where } \\quad z \\equiv \\frac{\\lambda_{\\text {obs }}-\\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}\n$$\nwith $\\lambda_{\\text {obs }}$ the observed wavelength and $\\lambda_{\\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\\mathrm{H}_{0}}$, and current Hubble distance, $D_{\\mathrm{H}_{0}}$, as\n\n$$\nt_{\\mathrm{H}_{0}} \\equiv H_{0}^{-1} \\quad \\text { and } \\quad D_{\\mathrm{H}_{0}} \\equiv c t_{\\mathrm{H}_{0}} \\text {. }\n$$\n\nHere the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is\n\n$$\nE(z)=\\frac{H}{H_{0}} \\equiv\\left[\\Omega_{0, m}(1+z)^{3}+\\Omega_{0, \\Lambda}+\\Omega_{0, r}(1+z)^{4}\\right]^{1 / 2},\n$$\n\nwhere $\\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\\Lambda$ indicate the contribution to $\\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as\n\n$$\nt=t_{\\mathrm{H}_{0}} \\int_{0}^{(1+z)^{-1}} \\frac{a}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nIf $\\Omega_{0, r}=0$ and $\\Omega_{0, m}+\\Omega_{0, \\Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\\int\\left(b^{2}+x^{2}\\right)^{-1 / 2} \\mathrm{~d} x=\\ln \\left(x+\\sqrt{b^{2}+x^{2}}\\right)+C$ this integral can be evaluated analytically to give\n\n$$\nt=t_{\\mathrm{H}_{0}} \\frac{2}{3 \\Omega_{0, \\Lambda}^{1 / 2}} \\ln \\left[\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}}\\right)^{1 / 2}(1+z)^{-3 / 2}+\\left(\\frac{\\Omega_{0, \\Lambda}}{\\Omega_{0, m}(1+z)^{3}}+1\\right)^{1 / 2}\\right]\n$$\n\nFinally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \\equiv L / 4 \\pi D_{L}^{2}$ ) is given as\n\n$$\nD_{L}=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{0}^{z_{i}} \\frac{1}{E(z)} \\mathrm{d} z=\\left(1+z_{i}\\right) D_{\\mathrm{H}_{0}} \\int_{a_{i}}^{1} \\frac{1}{\\left(\\Omega_{0, m} a+\\Omega_{0, \\Lambda} a^{4}+\\Omega_{0, r}\\right)^{1 / 2}} \\mathrm{~d} a\n$$\n\nwhere $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.\n\nproblem:\nc. Computer models suggest the first galaxies formed around $z \\sim 10-20$. One of the best ways to look for high-redshift galaxies is to try and detect the emission from the Lyman alpha (Lya) emission line at $\\lambda_{\\text {emit }}=121.6 \\mathrm{~nm}$ as it is a relatively bright line. Some of the brightest galaxies in that initial era of galaxy formation would have an absolute magnitude of $\\mathcal{M} \\sim 20$. In this question, you are given that $\\Omega_{0, \\mathrm{~m}}=0.3, \\Omega_{0, \\Lambda}=0.7, \\Omega_{0, \\mathrm{r}}=0$ and $\\mathrm{H}_{0}=70 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$.\n\ni. Calculate the redshift at which the Lya line is detected in the centre of the F200W filter.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-09.jpg?height=618&width=1466&top_left_y=596&top_left_x=296", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-10.jpg?height=482&width=1536&top_left_y=1118&top_left_x=267" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_336", "problem": "设想地球没有自转, 坚直向下通过地心把地球钻通. 如果在这个通过地心的笔直的管道的一端无初速度地放下一物体，下列说法正确的是（）\nA: 物体在地心时, 它与地心间距离为零, 地球对物体的万有引力无穷大\nB: 物体在地心时, 地球对它的万有引力为零\nC: 物体在管道中将往返运动, 通过地心时加速度为零, 速率最大\nD: 物体运动到地心时由于万有引力为零, 它将静止在地心不动\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n设想地球没有自转, 坚直向下通过地心把地球钻通. 如果在这个通过地心的笔直的管道的一端无初速度地放下一物体，下列说法正确的是（）\n\nA: 物体在地心时, 它与地心间距离为零, 地球对物体的万有引力无穷大\nB: 物体在地心时, 地球对它的万有引力为零\nC: 物体在管道中将往返运动, 通过地心时加速度为零, 速率最大\nD: 物体运动到地心时由于万有引力为零, 它将静止在地心不动\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_74", "problem": "有人提出了一种不用火箭发射人造地球卫星的设想。其设想如下: 沿地球的一条弦挖一通道, 如图乙所示. 在通道的两个出口处 $A$ 和 $B$, 分别将质量为 $M$ 的物体和质量为 $m$ 的待发射卫星同时自由释放, 只要 $M$ 比 $m$ 足够大, 碰撞后, 质量为 $m$ 的物体, 即待发射的卫星就会从通道口 $B$ 冲出通道。(已知地球表面的重力加速度为 $g$, 地球半径为 $\\left.R_{0}\\right)$\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\n如图乙所示, 是在地球上距地心 $h$ 处沿一条弦挖了一条光滑的通道 $A B$, 从 $A$ 点处静止释放一个质量为 $m$ 的物体，物体下落到通道中点 $O^{\\prime}$ 处的速度多大?", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n有人提出了一种不用火箭发射人造地球卫星的设想。其设想如下: 沿地球的一条弦挖一通道, 如图乙所示. 在通道的两个出口处 $A$ 和 $B$, 分别将质量为 $M$ 的物体和质量为 $m$ 的待发射卫星同时自由释放, 只要 $M$ 比 $m$ 足够大, 碰撞后, 质量为 $m$ 的物体, 即待发射的卫星就会从通道口 $B$ 冲出通道。(已知地球表面的重力加速度为 $g$, 地球半径为 $\\left.R_{0}\\right)$\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\n如图乙所示, 是在地球上距地心 $h$ 处沿一条弦挖了一条光滑的通道 $A B$, 从 $A$ 点处静止释放一个质量为 $m$ 的物体，物体下落到通道中点 $O^{\\prime}$ 处的速度多大?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-024.jpg?height=229&width=232&top_left_y=682&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-024.jpg?height=237&width=368&top_left_y=681&top_left_x=570" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_263", "problem": "在宇宙中也存在由质量相等的四颗星组成的“四星系统”, “四星系统”离其他恒星较远，通常可忽略其他星体对“四星系统”的引力作用。已观测到稳定的“四星系统”存在两种基本的构成形式: 一种是四颗星稳定地分布在边长为 $a$ 的正方形的四个顶点上, 均围绕正方形对角线的交点做匀速圆周运动, 如下图 (1) 所示。另一种形式是有三颗星位于等边三角形的三个顶点上, 第四颗星刚好位于三角形的中心不动, 三颗星沿外接于等边三角形的半径为 $a$ 的圆形轨道运行, 如下图 (2) 所示。假设两种形式的“四星系统”中每个星的质量均为 $m$, 已知引力常量为 $G$, 求这两种形式下的周期 $T_{1}$ 和 $T_{2}$ 。\n\n[图1]\n\n(1)\n\n[图2]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n在宇宙中也存在由质量相等的四颗星组成的“四星系统”, “四星系统”离其他恒星较远，通常可忽略其他星体对“四星系统”的引力作用。已观测到稳定的“四星系统”存在两种基本的构成形式: 一种是四颗星稳定地分布在边长为 $a$ 的正方形的四个顶点上, 均围绕正方形对角线的交点做匀速圆周运动, 如下图 (1) 所示。另一种形式是有三颗星位于等边三角形的三个顶点上, 第四颗星刚好位于三角形的中心不动, 三颗星沿外接于等边三角形的半径为 $a$ 的圆形轨道运行, 如下图 (2) 所示。假设两种形式的“四星系统”中每个星的质量均为 $m$, 已知引力常量为 $G$, 求这两种形式下的周期 $T_{1}$ 和 $T_{2}$ 。\n\n[图1]\n\n(1)\n\n[图2]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[ $T_{1}$, $T_{2}$]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-072.jpg?height=325&width=371&top_left_y=2259&top_left_x=337", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-072.jpg?height=325&width=354&top_left_y=2259&top_left_x=745", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-073.jpg?height=388&width=371&top_left_y=1231&top_left_x=337", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-074.jpg?height=309&width=331&top_left_y=431&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ " $T_{1}$", " $T_{2}$" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_357", "problem": "如图所示, 发射同步卫星的一种程序是: 先让卫星进入一个近地的圆轨道, 然后在 $P$ 点点火加速, 进入粗圆形转移轨道 (该陏圆轨道的近地点为近地轨道上的 $P$, 远地点为同步轨道上的 $Q)$, 到达远地点时再次自动点火加速, 进入同步轨道. 设卫星在近地圆轨道上运行的速率为 $v_{1}$, 在 $P$ 点短时间加速后的速率为 $v_{2}$, 沿转移轨道刚到达远地点 $Q$ 时的速率为 $v_{3}$, 在 $Q$ 点短时间加速后进入同步轨道后的速率为 $v_{4}$, 则四个速率的大小排列正确的是（\n\n[图1]\nA: $v_{1}>v_{2}>v_{3}>v_{4}$\nB: $v_{2}>v_{1}>v_{3}>v_{4}$\nC: $v_{1}>v_{2}>v_{4}>v_{3}$\nD: $v_{2}>v_{1}>v_{4}>v_{3}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 发射同步卫星的一种程序是: 先让卫星进入一个近地的圆轨道, 然后在 $P$ 点点火加速, 进入粗圆形转移轨道 (该陏圆轨道的近地点为近地轨道上的 $P$, 远地点为同步轨道上的 $Q)$, 到达远地点时再次自动点火加速, 进入同步轨道. 设卫星在近地圆轨道上运行的速率为 $v_{1}$, 在 $P$ 点短时间加速后的速率为 $v_{2}$, 沿转移轨道刚到达远地点 $Q$ 时的速率为 $v_{3}$, 在 $Q$ 点短时间加速后进入同步轨道后的速率为 $v_{4}$, 则四个速率的大小排列正确的是（\n\n[图1]\n\nA: $v_{1}>v_{2}>v_{3}>v_{4}$\nB: $v_{2}>v_{1}>v_{3}>v_{4}$\nC: $v_{1}>v_{2}>v_{4}>v_{3}$\nD: $v_{2}>v_{1}>v_{4}>v_{3}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-87.jpg?height=409&width=483&top_left_y=795&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_265", "problem": "如图所示为科学家模拟水星探测器进入水星表面绕行轨道的过程示意图, 假设水星的半径为 $R$, 探测器在距离水星表面高度为 $3 R$ 的圆形轨道 $\\mathrm{I}$ 上做匀速圆周运动, 运行的周期为 $T$, 在到达轨道的 $P$ 点时变轨进入椭圆轨道 II, 到达轨道 II 的“近水星点” $Q$ 时,再次变轨进入近水星轨道III绕水星做匀速圆周运动, 从而实施对水星探测的任务, 则下列说法正确的是（）\n\n[图1]\nA: 水星探测器在 $P 、 Q$ 两点变轨的过程中速度均减小\nB: 水星探测器在轨道 II 上运行的周期小于 $T$\nC: 水星探测器在轨道 I 和轨道 II 上稳定运行经过 $P$ 时加速度大小不相等\nD: 若水星探测器在轨道 II 上经过 $P$ 点时的速度大小为 $\\mathrm{v}_{\\mathrm{P}}$, 在轨道III上做圆周运动的速度大小为 $v_{3}$, 则有 $v_{3}>\\mathrm{v}_{\\mathrm{P}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示为科学家模拟水星探测器进入水星表面绕行轨道的过程示意图, 假设水星的半径为 $R$, 探测器在距离水星表面高度为 $3 R$ 的圆形轨道 $\\mathrm{I}$ 上做匀速圆周运动, 运行的周期为 $T$, 在到达轨道的 $P$ 点时变轨进入椭圆轨道 II, 到达轨道 II 的“近水星点” $Q$ 时,再次变轨进入近水星轨道III绕水星做匀速圆周运动, 从而实施对水星探测的任务, 则下列说法正确的是（）\n\n[图1]\n\nA: 水星探测器在 $P 、 Q$ 两点变轨的过程中速度均减小\nB: 水星探测器在轨道 II 上运行的周期小于 $T$\nC: 水星探测器在轨道 I 和轨道 II 上稳定运行经过 $P$ 时加速度大小不相等\nD: 若水星探测器在轨道 II 上经过 $P$ 点时的速度大小为 $\\mathrm{v}_{\\mathrm{P}}$, 在轨道III上做圆周运动的速度大小为 $v_{3}$, 则有 $v_{3}>\\mathrm{v}_{\\mathrm{P}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-081.jpg?height=451&width=485&top_left_y=1773&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_682", "problem": "如图所示, 质量相等、周期均为 $T$ 的两颗人造地球卫星, 1 轨卫星道为圆、 2 轨道为椭圆。 $A 、 B$ 两点是椭圆长轴两端, $A$ 距离地心为 $r \\circ C$ 为椭圆短轴端点且是两轨道的交点, 到地心距离为 $2 r$, 卫星 1 的速率为 $v$, 下列说法正确的是 ( )\n\n[图1]\nA: $C$ 点到椭圆中心的距离为 $r$\nB: 卫星 2 在 $C$ 点的速率等于 $v$\nC: 卫星 2 在 $C$ 点的向心加速度等于 $\\frac{v^{2}}{2 r}$\nD: 卫星 2 由 $A$ 到 $C$ 的时间等于 $\\frac{T}{4}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 质量相等、周期均为 $T$ 的两颗人造地球卫星, 1 轨卫星道为圆、 2 轨道为椭圆。 $A 、 B$ 两点是椭圆长轴两端, $A$ 距离地心为 $r \\circ C$ 为椭圆短轴端点且是两轨道的交点, 到地心距离为 $2 r$, 卫星 1 的速率为 $v$, 下列说法正确的是 ( )\n\n[图1]\n\nA: $C$ 点到椭圆中心的距离为 $r$\nB: 卫星 2 在 $C$ 点的速率等于 $v$\nC: 卫星 2 在 $C$ 点的向心加速度等于 $\\frac{v^{2}}{2 r}$\nD: 卫星 2 由 $A$ 到 $C$ 的时间等于 $\\frac{T}{4}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-060.jpg?height=363&width=491&top_left_y=158&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_998", "problem": "Two light sources, $\\mathrm{A}$ and $\\mathrm{B}$, emitting their light isotropically (i.e. equally in all directions) are placed at distance $r$ and $2 r$ respectively from a detector, which shows they have the same apparent brightness (i.e. $b_{A} / b_{B}=1$ ). If $\\mathrm{A}$ is moved to $2 r$ and $\\mathrm{B}$ is moved to $3 r$, what is the new ratio of apparent brightness $b_{A}^{\\prime} / b_{B}^{\\prime}$ ?\nA: $2 / 3$\nB: $4 / 9$\nC: $3 / 4$\nD: $9 / 16$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo light sources, $\\mathrm{A}$ and $\\mathrm{B}$, emitting their light isotropically (i.e. equally in all directions) are placed at distance $r$ and $2 r$ respectively from a detector, which shows they have the same apparent brightness (i.e. $b_{A} / b_{B}=1$ ). If $\\mathrm{A}$ is moved to $2 r$ and $\\mathrm{B}$ is moved to $3 r$, what is the new ratio of apparent brightness $b_{A}^{\\prime} / b_{B}^{\\prime}$ ?\n\nA: $2 / 3$\nB: $4 / 9$\nC: $3 / 4$\nD: $9 / 16$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_296", "problem": "北斗卫星导航系统空间段计划由 35 颗卫星组成,包括 5 颗静止同步轨道卫星和 3\n颗倾斜同步轨道卫星, 以及 27 颗相同高度的中轨道卫星. 中轨道卫星运行在 3 个轨道面上, 轨道面之间相隔 $120^{\\circ}$ 均匀分布, 如图所示. 已知同步轨道、中轨道、倾斜同步轨道卫星距地面的高度分别约为 $6 R 、 4 R 、 6 R$ ( $R$ 为地球半径), 则\n\n[图1]\nA: 静止同步轨道卫星和倾斜同步轨道卫星的周期不同\nB: 3 个轨道面上的中轨道卫星角速度的值均相同\nC: 同步轨道卫星与中轨道卫星周期的比值约为 $\\frac{7}{5} \\sqrt{\\frac{7}{5}}$\nD: 倾斜同步轨道卫星与中轨道卫星角速度的比值约为 $\\frac{5}{7}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n北斗卫星导航系统空间段计划由 35 颗卫星组成,包括 5 颗静止同步轨道卫星和 3\n颗倾斜同步轨道卫星, 以及 27 颗相同高度的中轨道卫星. 中轨道卫星运行在 3 个轨道面上, 轨道面之间相隔 $120^{\\circ}$ 均匀分布, 如图所示. 已知同步轨道、中轨道、倾斜同步轨道卫星距地面的高度分别约为 $6 R 、 4 R 、 6 R$ ( $R$ 为地球半径), 则\n\n[图1]\n\nA: 静止同步轨道卫星和倾斜同步轨道卫星的周期不同\nB: 3 个轨道面上的中轨道卫星角速度的值均相同\nC: 同步轨道卫星与中轨道卫星周期的比值约为 $\\frac{7}{5} \\sqrt{\\frac{7}{5}}$\nD: 倾斜同步轨道卫星与中轨道卫星角速度的比值约为 $\\frac{5}{7}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-50.jpg?height=469&width=491&top_left_y=405&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_926", "problem": "Jupiter's apparent magnitude at opposition in 2022 of $m=-2.94$ is the brightest for $\\sim 70$ years due to it happening close to Jupiter's perihelion. What is the difference in apparent magnitude between Jupiter's brightest and faintest possible oppositions? Jupiter has a semi-major axis of 5.20 au and an eccentricity of 0.0489 . Assume the Earth's orbit is circular.\nA: 0.38\nB: 0.43\nC: 0.48\nD: 0.53\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nJupiter's apparent magnitude at opposition in 2022 of $m=-2.94$ is the brightest for $\\sim 70$ years due to it happening close to Jupiter's perihelion. What is the difference in apparent magnitude between Jupiter's brightest and faintest possible oppositions? Jupiter has a semi-major axis of 5.20 au and an eccentricity of 0.0489 . Assume the Earth's orbit is circular.\n\nA: 0.38\nB: 0.43\nC: 0.48\nD: 0.53\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1065", "problem": "In the heart of every star, nuclear fusion is taking place. For most stars that involves hydrogen being turned into helium, a process that starts by bringing two protons close enough that the strong nuclear force can act upon them. The smallest stars are the ones that have a core that is only just hot enough for fusion to occur, whilst in the biggest ones the radiation pressure of the photons given out by the fusion reaction pushing on the stellar material can overcome the gravitational forces holding it together.\n[figure1]\n\nFigure 5: Left: The lowest mass star we know of, EBLM J0555-57Ab, was found by von Boetticher et al. (2017) and is about the size of Saturn with a mass of $0.081 M_{\\odot}$. Credit: Amanda Smith, University of Cambridge. Right: The highest mass star we know of, R136a1, is in the centre of the clump of stars on the right of this HST image of the Tarantula Nebula. Schneider et al. (2014) suggest it has a mass of $315 M_{\\odot}$, which is above what stellar evolution models allow. Despite its large mass, other stars have far bigger radii. Credit: NASA \\& ESA.\n\nFor a spherical main sequence star made of a plasma (a fully ionized gas of electrons and nuclei) that is acting like an ideal gas, the temperature at the core can be approximately calculated as\n\n$$\nT_{\\mathrm{int}} \\simeq \\frac{G M \\bar{\\mu}}{k_{\\mathrm{B}} R} \\quad \\text { where } \\quad \\bar{\\mu}=\\frac{m_{\\mathrm{p}}}{2 X+3 Y / 4+Z / 2} .\n$$\n\nIn this equation, $M$ is the mass of the star, $R$ is its radius, $k_{\\mathrm{B}}$ is the Boltzmann constant, and $\\bar{\\mu}$ is the mean mass of the plasma particles (i.e nuclei and electrons) with $m_{\\mathrm{p}}$ the mass of a proton.\n\nClassically, the core of the Sun is not hot enough for fusion, and yet fusion is clearly happening. The key is that it is a fundamentally quantum process, and so protons are able to 'quantum tunnel' through the Coloumb barrier (see Figure 6), allowing fusion to occur at lower temperatures. In quantum mechanics, fusion will happen when $b=\\lambda$ where $\\lambda$ is the de Broglie wavelength of the proton, related to the momentum of the proton by $\\lambda=h / p$.\n\n[figure2]\n\nFigure 6: A diagram showing the way a particle can pass through a classically impenetrable potential barrier due to its wave-like properties on the quantum scale.\n\nCredit: Brooks/Cole - Thomson Learning.\n\nIn the smallest stars, electron degeneracy prevents them from compressing in radius and thus stops the core reaching $T_{\\text {int }} \\gtrsim T_{\\text {quantum }}$. At the limit of electron degeneracy, the number density of electrons $n_{\\mathrm{e}}=1 / \\lambda_{\\mathrm{e}}^{3}$ where $\\lambda_{\\mathrm{e}}$ is the de Broglie wavelength of the electrons.\n\nIn the largest stars, radiation pressure pushes on the outer layers of the star stronger than gravity pulls them in. The brightest luminosity for a star is known as the Eddington luminosity, $L_{\\text {Edd }}$. The acceleration due to radiation pressure can be calculated as\n\n$$\ng_{\\mathrm{rad}}=\\frac{\\kappa_{\\mathrm{e}} I}{c} \\quad \\text { where } \\quad \\kappa_{\\mathrm{e}}=\\frac{\\sigma_{\\mathrm{T}}}{2 m_{\\mathrm{p}}}(1+X)\n$$\n\nand $\\kappa_{\\mathrm{e}}$ is the electron opacity of the stellar material, $\\sigma_{\\mathrm{T}}$ is the Thomson scattering cross-section for electrons $\\left(=66.5 \\mathrm{fm}^{2}\\right.$ ), $X$ is the hydrogen fraction, and $I$ is the intensity of radiation (in $\\mathrm{W} \\mathrm{m}^{-2}$ ). Assuming main-sequence stars follow a mass-luminosity relation of $L \\propto M^{3}$, the maximum mass of a star can be found by considering one that is radiating at $L_{\\text {Edd }}$.d. Assuming the star to be of uniform density at this limit with $\\rho=m_{p} n_{e}$ and the electrons to be in thermal equilibrium with the plasma, show that the minimum mass of a star for which $T_{\\text {int }}=T_{\\text {quantum }}$ is $\\approx 0.1 \\mathrm{M}_{\\odot}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIn the heart of every star, nuclear fusion is taking place. For most stars that involves hydrogen being turned into helium, a process that starts by bringing two protons close enough that the strong nuclear force can act upon them. The smallest stars are the ones that have a core that is only just hot enough for fusion to occur, whilst in the biggest ones the radiation pressure of the photons given out by the fusion reaction pushing on the stellar material can overcome the gravitational forces holding it together.\n[figure1]\n\nFigure 5: Left: The lowest mass star we know of, EBLM J0555-57Ab, was found by von Boetticher et al. (2017) and is about the size of Saturn with a mass of $0.081 M_{\\odot}$. Credit: Amanda Smith, University of Cambridge. Right: The highest mass star we know of, R136a1, is in the centre of the clump of stars on the right of this HST image of the Tarantula Nebula. Schneider et al. (2014) suggest it has a mass of $315 M_{\\odot}$, which is above what stellar evolution models allow. Despite its large mass, other stars have far bigger radii. Credit: NASA \\& ESA.\n\nFor a spherical main sequence star made of a plasma (a fully ionized gas of electrons and nuclei) that is acting like an ideal gas, the temperature at the core can be approximately calculated as\n\n$$\nT_{\\mathrm{int}} \\simeq \\frac{G M \\bar{\\mu}}{k_{\\mathrm{B}} R} \\quad \\text { where } \\quad \\bar{\\mu}=\\frac{m_{\\mathrm{p}}}{2 X+3 Y / 4+Z / 2} .\n$$\n\nIn this equation, $M$ is the mass of the star, $R$ is its radius, $k_{\\mathrm{B}}$ is the Boltzmann constant, and $\\bar{\\mu}$ is the mean mass of the plasma particles (i.e nuclei and electrons) with $m_{\\mathrm{p}}$ the mass of a proton.\n\nClassically, the core of the Sun is not hot enough for fusion, and yet fusion is clearly happening. The key is that it is a fundamentally quantum process, and so protons are able to 'quantum tunnel' through the Coloumb barrier (see Figure 6), allowing fusion to occur at lower temperatures. In quantum mechanics, fusion will happen when $b=\\lambda$ where $\\lambda$ is the de Broglie wavelength of the proton, related to the momentum of the proton by $\\lambda=h / p$.\n\n[figure2]\n\nFigure 6: A diagram showing the way a particle can pass through a classically impenetrable potential barrier due to its wave-like properties on the quantum scale.\n\nCredit: Brooks/Cole - Thomson Learning.\n\nIn the smallest stars, electron degeneracy prevents them from compressing in radius and thus stops the core reaching $T_{\\text {int }} \\gtrsim T_{\\text {quantum }}$. At the limit of electron degeneracy, the number density of electrons $n_{\\mathrm{e}}=1 / \\lambda_{\\mathrm{e}}^{3}$ where $\\lambda_{\\mathrm{e}}$ is the de Broglie wavelength of the electrons.\n\nIn the largest stars, radiation pressure pushes on the outer layers of the star stronger than gravity pulls them in. The brightest luminosity for a star is known as the Eddington luminosity, $L_{\\text {Edd }}$. The acceleration due to radiation pressure can be calculated as\n\n$$\ng_{\\mathrm{rad}}=\\frac{\\kappa_{\\mathrm{e}} I}{c} \\quad \\text { where } \\quad \\kappa_{\\mathrm{e}}=\\frac{\\sigma_{\\mathrm{T}}}{2 m_{\\mathrm{p}}}(1+X)\n$$\n\nand $\\kappa_{\\mathrm{e}}$ is the electron opacity of the stellar material, $\\sigma_{\\mathrm{T}}$ is the Thomson scattering cross-section for electrons $\\left(=66.5 \\mathrm{fm}^{2}\\right.$ ), $X$ is the hydrogen fraction, and $I$ is the intensity of radiation (in $\\mathrm{W} \\mathrm{m}^{-2}$ ). Assuming main-sequence stars follow a mass-luminosity relation of $L \\propto M^{3}$, the maximum mass of a star can be found by considering one that is radiating at $L_{\\text {Edd }}$.\n\nproblem:\nd. Assuming the star to be of uniform density at this limit with $\\rho=m_{p} n_{e}$ and the electrons to be in thermal equilibrium with the plasma, show that the minimum mass of a star for which $T_{\\text {int }}=T_{\\text {quantum }}$ is $\\approx 0.1 \\mathrm{M}_{\\odot}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-08.jpg?height=712&width=1508&top_left_y=546&top_left_x=271", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-09.jpg?height=514&width=1010&top_left_y=186&top_left_x=523" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_977", "problem": "In the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year.\n\n[figure1]\n\nFigure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com\n\nIn other Islamic countries, the start of a new month is determined through direct telescopic observations of the Moon, looking for a very thin crescent. This is a very hard measurement to make and the human eye struggles to recognise the presence of a crescent until about $0.6 \\%$ of the lunar disc is illuminated. Calculate how many hours this is after the end of the astronomical New Moon. Assume the lunar orbit is circular.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year.\n\n[figure1]\n\nFigure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com\n\nIn other Islamic countries, the start of a new month is determined through direct telescopic observations of the Moon, looking for a very thin crescent. This is a very hard measurement to make and the human eye struggles to recognise the presence of a crescent until about $0.6 \\%$ of the lunar disc is illuminated. Calculate how many hours this is after the end of the astronomical New Moon. Assume the lunar orbit is circular.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of hours, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_2c19fdb17927c588761dg-09.jpg?height=800&width=1110&top_left_y=862&top_left_x=473", "https://i.postimg.cc/XqXhYpnb/Screenshot-2024-04-06-at-21-54-07.png" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "hours" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_752", "problem": "The moment of inertia of a solid cylinder with radius $R$ and mass $M$ is given by ..\nA: $M R^{2} / 2$\nB: $2 M R^{2} / 5$\nC: $3 M R^{2} / 10$\nD: $M R^{2} / 3$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe moment of inertia of a solid cylinder with radius $R$ and mass $M$ is given by ..\n\nA: $M R^{2} / 2$\nB: $2 M R^{2} / 5$\nC: $3 M R^{2} / 10$\nD: $M R^{2} / 3$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_227", "problem": "据报道, 天文学家近日发现了一颗距地球 40 光年的“超级地球”, 命名为\n\n“55Cancrie\", 该行星绕母星 (中心天体) 运行的周期约为地球绕太阳运行周期的 $\\frac{1}{480}$,母星的体积约为太阳的 60 倍. 假设母星与太阳密度相同, “ 55 Cancrie”与地球均做匀速圆周运动，则“55Cancrie”与地球的（）\nA: 轨道半径之比约为 $\\sqrt[3]{\\frac{60}{480}}$\nB: 轨道半径之比约为 $\\sqrt[3]{\\frac{60}{480^{2}}}$\nC: 向心加速度之比约为 $\\sqrt[3]{60 \\times 480^{2}}$\nD: 向心加速度之比约为 $\\sqrt[3]{60^{2} \\times 480}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n据报道, 天文学家近日发现了一颗距地球 40 光年的“超级地球”, 命名为\n\n“55Cancrie\", 该行星绕母星 (中心天体) 运行的周期约为地球绕太阳运行周期的 $\\frac{1}{480}$,母星的体积约为太阳的 60 倍. 假设母星与太阳密度相同, “ 55 Cancrie”与地球均做匀速圆周运动，则“55Cancrie”与地球的（）\n\nA: 轨道半径之比约为 $\\sqrt[3]{\\frac{60}{480}}$\nB: 轨道半径之比约为 $\\sqrt[3]{\\frac{60}{480^{2}}}$\nC: 向心加速度之比约为 $\\sqrt[3]{60 \\times 480^{2}}$\nD: 向心加速度之比约为 $\\sqrt[3]{60^{2} \\times 480}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_69", "problem": "地球赤道上有一物体随地球的自转而做圆周运动, 所受的向心力为 $F_{1}$, 向心加速度为 $a_{1}$, 线速度为 $v_{1}$, 角速度为 $\\omega_{1}$, 绕地球表面附近做圆周运动的人造卫星（高度忽略）所受的向心力为 $F_{2}$, 向心加速度为 $a_{2}$, 线速度为 $v_{2}$, 角速度为 $\\omega_{2}$; 地球同步卫星所受的向心力为 $F_{3}$, 向心加速度为 $a_{3}$, 线速度为 $v_{3}$, 角速度为 $\\omega_{3}$; 地球表面重力加速度为 $g$,第一宇宙速度为 $v$, 假设三者质量相等, 则 ( )\nA: $F_{2}>F_{1}>F_{3}$\nB: $\\omega_{1}=\\omega_{3}<\\omega_{2}$\nC: $v_{1}=v_{2}=v>v_{3}$\nD: $a_{1}>a_{2}=g>a_{3}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球赤道上有一物体随地球的自转而做圆周运动, 所受的向心力为 $F_{1}$, 向心加速度为 $a_{1}$, 线速度为 $v_{1}$, 角速度为 $\\omega_{1}$, 绕地球表面附近做圆周运动的人造卫星（高度忽略）所受的向心力为 $F_{2}$, 向心加速度为 $a_{2}$, 线速度为 $v_{2}$, 角速度为 $\\omega_{2}$; 地球同步卫星所受的向心力为 $F_{3}$, 向心加速度为 $a_{3}$, 线速度为 $v_{3}$, 角速度为 $\\omega_{3}$; 地球表面重力加速度为 $g$,第一宇宙速度为 $v$, 假设三者质量相等, 则 ( )\n\nA: $F_{2}>F_{1}>F_{3}$\nB: $\\omega_{1}=\\omega_{3}<\\omega_{2}$\nC: $v_{1}=v_{2}=v>v_{3}$\nD: $a_{1}>a_{2}=g>a_{3}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_783", "problem": "As a star collapses at the end of its life, the triple-alpha reaction takes place. Which one of these equations describes this reaction correctly?\nA: ${ }_{1}^{2} \\mathrm{H}+{ }_{1}^{2} \\mathrm{H}+{ }_{1}^{2} \\mathrm{H} \\rightarrow{ }_{3}^{6} \\mathrm{Li}+\\gamma$\nB: ${ }_{2}^{4} \\mathrm{He}+{ }_{2}^{4} \\mathrm{He}+{ }_{2}^{4} \\mathrm{He} \\rightarrow{ }_{6}^{12} \\mathrm{C}+\\gamma$\nC: ${ }_{1}^{2} \\mathrm{H}+{ }_{1}^{2} \\mathrm{H}+{ }_{2}^{4} \\mathrm{He}+{ }_{2}^{4} \\mathrm{He} \\rightarrow{ }_{6}^{12} \\mathrm{C}+\\gamma$\nD: ${ }_{1}^{2} \\mathrm{H}+{ }_{2}^{4} \\mathrm{He} \\rightarrow{ }_{3}^{6} \\mathrm{Li}+\\gamma$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAs a star collapses at the end of its life, the triple-alpha reaction takes place. Which one of these equations describes this reaction correctly?\n\nA: ${ }_{1}^{2} \\mathrm{H}+{ }_{1}^{2} \\mathrm{H}+{ }_{1}^{2} \\mathrm{H} \\rightarrow{ }_{3}^{6} \\mathrm{Li}+\\gamma$\nB: ${ }_{2}^{4} \\mathrm{He}+{ }_{2}^{4} \\mathrm{He}+{ }_{2}^{4} \\mathrm{He} \\rightarrow{ }_{6}^{12} \\mathrm{C}+\\gamma$\nC: ${ }_{1}^{2} \\mathrm{H}+{ }_{1}^{2} \\mathrm{H}+{ }_{2}^{4} \\mathrm{He}+{ }_{2}^{4} \\mathrm{He} \\rightarrow{ }_{6}^{12} \\mathrm{C}+\\gamma$\nD: ${ }_{1}^{2} \\mathrm{H}+{ }_{2}^{4} \\mathrm{He} \\rightarrow{ }_{3}^{6} \\mathrm{Li}+\\gamma$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_572", "problem": "人类对来知事物的好奇和科学家们的不懈努力, 使人类对宇宙的认识越来越丰富。\n\n开普勒坚信哥白尼的“日心说”, 在研究了导师第谷在 20 余年中坚持对天体进行系统观测得到的大量精确资料后, 得出了开普勒三定律。为人们解决行星运动问题提供了依据, 也为牛顿发现万有引力定律提供了基础。开普勒认为, 所有行星围绕太阳运动的轨道都是椭圆, 太阳处在所有粗圆的一个焦点上、行星轨道半长轴的三次方与其公转周期的二次方的比值是一个常量。实际上行星的轨道与圆十分接近, 在中学阶段的研究中我们按圆轨道处理, 请你以地球绕太阳公转为例, 若太阳的质量为 $M$, 引力常量为 $G$ 。根据万有引力定律和牛牛顿运动定律推导出此常量的表达式;", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n人类对来知事物的好奇和科学家们的不懈努力, 使人类对宇宙的认识越来越丰富。\n\n开普勒坚信哥白尼的“日心说”, 在研究了导师第谷在 20 余年中坚持对天体进行系统观测得到的大量精确资料后, 得出了开普勒三定律。为人们解决行星运动问题提供了依据, 也为牛顿发现万有引力定律提供了基础。开普勒认为, 所有行星围绕太阳运动的轨道都是椭圆, 太阳处在所有粗圆的一个焦点上、行星轨道半长轴的三次方与其公转周期的二次方的比值是一个常量。实际上行星的轨道与圆十分接近, 在中学阶段的研究中我们按圆轨道处理, 请你以地球绕太阳公转为例, 若太阳的质量为 $M$, 引力常量为 $G$ 。根据万有引力定律和牛牛顿运动定律推导出此常量的表达式;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_311", "problem": "预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\\left(r_{2}>r_{1}\\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求:\n\n在 $t=1.5 t_{0}$ 时线圈中产生感应电流的大小;\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\\left(r_{2}>r_{1}\\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求:\n\n在 $t=1.5 t_{0}$ 时线圈中产生感应电流的大小;\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-134.jpg?height=369&width=388&top_left_y=1780&top_left_x=340", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-134.jpg?height=294&width=505&top_left_y=1869&top_left_x=867" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1056", "problem": "Some of the very first exoplanets to be discovered in large surveys were dubbed 'hot Jupiters' as they were similar in mass to Jupiter (i.e. a gas giant) but were much closer to their star than Mercury is to the Sun (and hence are in a very hot environment). Planetary formation models suggest that they were unlikely to have formed there, but instead formed much further out from the star and migrated inwards, due to gravitational interactions with other planets in the system. Studies of 'hot Jupiters' show that there is an overabundance of them with periods of $\\sim 3-4$ days, and very few with periods shorter than that. Since large, close-in planets should be the easiest to detect in all of the main methods of finding exoplanets, this scarcity is likely to be a real effect and suggests that exoplanets which are that close to their star are in a relatively rapid (by astronomical standards) inspiral towards destruction by their star.\n[figure1]\n\nFigure 6: Left: The orbital radius of several 'hot Jupiters' scaled by the Roche radius of the system (where tidal forces would destroy the planet). There is an expected pile up close to radii double the Roche radius (dotted line), and very few with radii smaller than that - those that are will inevitably spiral into the star and be destroyed by the tidal forces when they get too close. Credit: Birkby et al. (2014).\n\nRight: As the planets inspiral we should see a shift in when their transits occur. This figure shows the predicted size of the shift after a period of 10 years if the tidal dissipation quality factor $Q_{\\star}^{\\prime}=10^{6}$, as well as the current detection limit of 5 seconds (dotted line). Therefore measuring if there is any shift in the transit times over the course of a decade of observations can put stringent limits on the value of $Q_{\\star}^{\\prime}$. Credit: Birkby et al. (2014).\n\nThe Roche radius, where a planet will be torn apart due to the tidal forces acting on it, is defined as\n\n$$\na_{\\text {Roche }} \\approx 2.16 R_{P}\\left(\\frac{M_{\\star}}{M_{P}}\\right)^{1 / 3}\n$$\n\nwhere $R_{P}$ is the radius of the planet, $M_{P}$ is the mass of the planet and $M_{\\star}$ is the mass of the star. If a gas giant is knocked into a highly elliptical orbit (i.e. $e \\approx 1$ ) that has a periapsis $r_{\\text {peri }}v_{2}$\nB: $v_{1}=v_{2}$\nC: $v_{1}v_{2}$\nB: $v_{1}=v_{2}$\nC: $v_{1}_{e}=407 \\mathrm{~L}_{\\odot} \\mathrm{pc}^{-2}\\right.$. Given that the scatter in the FP relation introduces an uncertainty in D of $\\pm 17 \\%$, calculate the distance to the galaxy (in Mpc) and its (absolute) uncertainty using the FP relation.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nGW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nThe rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as\n\n$$\n\\dot{f}_{\\mathrm{GW}}=\\frac{96}{5} \\pi^{8 / 3}\\left(\\frac{G \\mathcal{M}}{c^{3}}\\right)^{5 / 3} f_{\\mathrm{GW}}^{11 / 3}\n$$\n\nproblem:\nb. For NGC 4993 we measure $R_{e}=15.5$ arcseconds, $\\sigma=171 \\mathrm{~km} \\mathrm{~s}^{-1}$, and $\\left\\langle 1_{r}>_{e}=407 \\mathrm{~L}_{\\odot} \\mathrm{pc}^{-2}\\right.$. Given that the scatter in the FP relation introduces an uncertainty in D of $\\pm 17 \\%$, calculate the distance to the galaxy (in Mpc) and its (absolute) uncertainty using the FP relation.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{Mpc}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{Mpc}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_703", "problem": "天文学家通过观测两个黑洞并合的事件, 间接验证了引力波的存在。该事件中甲、乙两个黑洞的质量分别为太阳质量的 36 倍和 29 倍, 假设这两个黑洞绕它们连线上的某\n点做圆周运动, 且两个黑洞的间距缓慢减小。若该双星系统在运动过程中, 各自质量不变且不受其他星系的影响，则关于这两个黑洞的运动，下列说法正确的是（）\nA: 甲、乙两个黑洞运行的线速度大小之比为 $36: 29$\nB: 甲、乙两个黑洞运行的角速度大小始终相等\nC: 随着甲、乙两个黑洞的间距缓慢减小，它们运行的周期也在减小\nD: 甲、乙两个黑洞做圆周运动的向心加速度大小始终相等\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n天文学家通过观测两个黑洞并合的事件, 间接验证了引力波的存在。该事件中甲、乙两个黑洞的质量分别为太阳质量的 36 倍和 29 倍, 假设这两个黑洞绕它们连线上的某\n点做圆周运动, 且两个黑洞的间距缓慢减小。若该双星系统在运动过程中, 各自质量不变且不受其他星系的影响，则关于这两个黑洞的运动，下列说法正确的是（）\n\nA: 甲、乙两个黑洞运行的线速度大小之比为 $36: 29$\nB: 甲、乙两个黑洞运行的角速度大小始终相等\nC: 随着甲、乙两个黑洞的间距缓慢减小，它们运行的周期也在减小\nD: 甲、乙两个黑洞做圆周运动的向心加速度大小始终相等\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1029", "problem": "In the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are $\\propto \\log L$, this can be described by the empirical relation\n\n$$\n\\langle\\mathcal{M}\\rangle=-2.43(\\log P-1)-4.05\n$$\n\nwhere $P$ is the period measured in days and $\\langle\\mathcal{M}\\rangle$ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of $10 \\mathrm{pc}$. The apparent and absolute magnitudes are related as\n\n$$\nm-\\mathfrak{M}=5 \\log d-5\n$$\n\nwhere $d$ is the distance measured in pc.\n[figure1]\n\nFigure 1: Left: Colour composite view of the circumstellar nebula around RS Puppis assembled from HST images. Credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration.\n\nRight: The light curve of RS Puppis. Phase 0 corresponds to the when the star is at its brightest, and the data has been folded over a known period of 41.5 days. Credit: AAVSO / P. Kervella et al. (2017).\n\nIn the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about $100 \\mathrm{pc}$, however her method with Cepheid variables extended this range to more than $10^{6} \\mathrm{pc}$, enabling much better determination of distances across the Milky Way and into nearby galaxies.\n\nThe variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days.\n\nMeasure the mean apparent magnitude from Figure 1, and find the distance to the star, given that absorption of the light by interstellar dust means the star appears 1.42 magnitudes fainter.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are $\\propto \\log L$, this can be described by the empirical relation\n\n$$\n\\langle\\mathcal{M}\\rangle=-2.43(\\log P-1)-4.05\n$$\n\nwhere $P$ is the period measured in days and $\\langle\\mathcal{M}\\rangle$ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of $10 \\mathrm{pc}$. The apparent and absolute magnitudes are related as\n\n$$\nm-\\mathfrak{M}=5 \\log d-5\n$$\n\nwhere $d$ is the distance measured in pc.\n[figure1]\n\nFigure 1: Left: Colour composite view of the circumstellar nebula around RS Puppis assembled from HST images. Credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration.\n\nRight: The light curve of RS Puppis. Phase 0 corresponds to the when the star is at its brightest, and the data has been folded over a known period of 41.5 days. Credit: AAVSO / P. Kervella et al. (2017).\n\nIn the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about $100 \\mathrm{pc}$, however her method with Cepheid variables extended this range to more than $10^{6} \\mathrm{pc}$, enabling much better determination of distances across the Milky Way and into nearby galaxies.\n\nThe variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days.\n\nMeasure the mean apparent magnitude from Figure 1, and find the distance to the star, given that absorption of the light by interstellar dust means the star appears 1.42 magnitudes fainter.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of pc, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_2c19fdb17927c588761dg-06.jpg?height=606&width=1412&top_left_y=1102&top_left_x=320" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "pc" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1013", "problem": "The cubesat CAPSTONE will enter a near-rectilinear halo orbit (NRHO) of the Moon on $13^{\\text {th }}$ November 2022 as a direct test of the orbit planned for the Lunar Gateway - a space station due to be built in orbit around the Moon by the end of the decade. It is a polar orbit, going from $1500 \\mathrm{~km}$ above the North pole to $70000 \\mathrm{~km}$ above the South pole. Treating it as an ellipse, what is the eccentricity of the orbit? The Moon has a radius of $1740 \\mathrm{~km}$.\nA: 0.83\nB: 0.88\nC: 0.91\nD: 0.96\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe cubesat CAPSTONE will enter a near-rectilinear halo orbit (NRHO) of the Moon on $13^{\\text {th }}$ November 2022 as a direct test of the orbit planned for the Lunar Gateway - a space station due to be built in orbit around the Moon by the end of the decade. It is a polar orbit, going from $1500 \\mathrm{~km}$ above the North pole to $70000 \\mathrm{~km}$ above the South pole. Treating it as an ellipse, what is the eccentricity of the orbit? The Moon has a radius of $1740 \\mathrm{~km}$.\n\nA: 0.83\nB: 0.88\nC: 0.91\nD: 0.96\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://i.postimg.cc/SKTTw95q/Screenshot-2024-04-06-at-19-25-08.png" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_158", "problem": "两颗相距较远的行星 $\\mathrm{A} 、 \\mathrm{~B}$ 的半径分别为 $R_{A} 、 R_{B}$, 距 $\\mathrm{A} 、 \\mathrm{~B}$ 行星中心 $r$ 处, 各有一卫星分别围绕行星做匀速圆周运动, 线速度的平方 $v^{2}$ 随半径 $r$ 变化的关系如图甲所示,两图线左端的纵坐标相同; 卫星做匀速圆周运动的周期为 $T, \\lg T-\\lg r$ 的图像如图乙所示的两平行直线, 它们的截距分别为 $b_{A} 、 b_{B}$. 已知两图像数据均采用国际单位,\n\n$b_{\\mathrm{B}}-b_{\\mathrm{A}}=\\lg \\sqrt{3}$, 行星可看作质量分布均匀的球体, 忽略行星的自转和其他星球的影响,\n下列说法正确的是（ ）\n\n[图1]\n\n甲\n\n[图2]\nA: 图乙中两条直线的斜率均为 $\\frac{3}{2}$\nB: 行星 A、B 的质量之比为 $1: 3$\nC: 行星 A、B 的密度之比为 $1: 9$\nD: 行星 A、B 表面的重力加速度大小之比为 $3: 1$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n两颗相距较远的行星 $\\mathrm{A} 、 \\mathrm{~B}$ 的半径分别为 $R_{A} 、 R_{B}$, 距 $\\mathrm{A} 、 \\mathrm{~B}$ 行星中心 $r$ 处, 各有一卫星分别围绕行星做匀速圆周运动, 线速度的平方 $v^{2}$ 随半径 $r$ 变化的关系如图甲所示,两图线左端的纵坐标相同; 卫星做匀速圆周运动的周期为 $T, \\lg T-\\lg r$ 的图像如图乙所示的两平行直线, 它们的截距分别为 $b_{A} 、 b_{B}$. 已知两图像数据均采用国际单位,\n\n$b_{\\mathrm{B}}-b_{\\mathrm{A}}=\\lg \\sqrt{3}$, 行星可看作质量分布均匀的球体, 忽略行星的自转和其他星球的影响,\n下列说法正确的是（ ）\n\n[图1]\n\n甲\n\n[图2]\n\nA: 图乙中两条直线的斜率均为 $\\frac{3}{2}$\nB: 行星 A、B 的质量之比为 $1: 3$\nC: 行星 A、B 的密度之比为 $1: 9$\nD: 行星 A、B 表面的重力加速度大小之比为 $3: 1$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-056.jpg?height=260&width=534&top_left_y=247&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-056.jpg?height=314&width=398&top_left_y=237&top_left_x=906" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_108", "problem": "“天问一号”从地球发射后, 在如图甲所示的 $P$ 点沿地火转移轨道到 $Q$ 点, 再依次进入如图乙所示在近火点相切的调相轨道和停泊轨道, 则天问一号 ( )\n\n[图1]\n\n甲\n\n[图2]\n\n乙\nA: 从 $P$ 点转移到 $Q$ 点的时间小于 6 个月\nB: 在近火点实施制动可以实现从调相轨道转移到停泊轨道\nC: 在停泊轨道运行的周期比在调相轨道上运行的周期大\nD: 在相等时间内与火星球心的连线在停泊轨道上扫过的面积小于在调相轨道上扫过的面积\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n“天问一号”从地球发射后, 在如图甲所示的 $P$ 点沿地火转移轨道到 $Q$ 点, 再依次进入如图乙所示在近火点相切的调相轨道和停泊轨道, 则天问一号 ( )\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\nA: 从 $P$ 点转移到 $Q$ 点的时间小于 6 个月\nB: 在近火点实施制动可以实现从调相轨道转移到停泊轨道\nC: 在停泊轨道运行的周期比在调相轨道上运行的周期大\nD: 在相等时间内与火星球心的连线在停泊轨道上扫过的面积小于在调相轨道上扫过的面积\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-082.jpg?height=363&width=352&top_left_y=2217&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-082.jpg?height=323&width=571&top_left_y=2237&top_left_x=728" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_595", "problem": "如图为天文学家观测到的旋浴星系的旋转曲线, 该曲线在旋浴星系发光区之外并没有按天文学家预想的那样, 而是和预想曲线发生了较大偏差, 这引起了科学家们极大的兴趣, 我们知道, 根据人造卫星运行的速度和高度, 就可以测出地球的总质量, 根据地球绕太阳运行的速度和地球与太阳的距离, 就可以测出太阳的总质量, 同理, 根据某个星系内恒星或气团围绕该星系中心运行的速度和它们与“星系中心”的距离, 天文学家就可以估算出这个星系在该恒星或气团所处范围内物质的质量和分布, 经天文学家计算分析得出的结论是: 旋浴星系的总质量远大于星系中可见星体质量的总和, 请根据本题所引用的科普材料判断以下说法正确的是( )\n\n[图1]\nA: 根据牛顿定律和万有引力定律推导出的旋浴预想曲线应该是图中上面那条曲线\nB: 旋浴星系的观测曲线和预想曲线的较大差别说明万有引力定律错了, 需要创建新的定律\nC: 旋浴星系的旋转曲线中下降的曲线部分意味着星系中很可能包含了更多的不可见的物质\nD: 旋浴星系的旋转曲线中平坦的曲线部分意味着星系中很可能包含了更多的不可见的物质\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图为天文学家观测到的旋浴星系的旋转曲线, 该曲线在旋浴星系发光区之外并没有按天文学家预想的那样, 而是和预想曲线发生了较大偏差, 这引起了科学家们极大的兴趣, 我们知道, 根据人造卫星运行的速度和高度, 就可以测出地球的总质量, 根据地球绕太阳运行的速度和地球与太阳的距离, 就可以测出太阳的总质量, 同理, 根据某个星系内恒星或气团围绕该星系中心运行的速度和它们与“星系中心”的距离, 天文学家就可以估算出这个星系在该恒星或气团所处范围内物质的质量和分布, 经天文学家计算分析得出的结论是: 旋浴星系的总质量远大于星系中可见星体质量的总和, 请根据本题所引用的科普材料判断以下说法正确的是( )\n\n[图1]\n\nA: 根据牛顿定律和万有引力定律推导出的旋浴预想曲线应该是图中上面那条曲线\nB: 旋浴星系的观测曲线和预想曲线的较大差别说明万有引力定律错了, 需要创建新的定律\nC: 旋浴星系的旋转曲线中下降的曲线部分意味着星系中很可能包含了更多的不可见的物质\nD: 旋浴星系的旋转曲线中平坦的曲线部分意味着星系中很可能包含了更多的不可见的物质\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-89.jpg?height=663&width=874&top_left_y=1119&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_670", "problem": "如图所示, 甲、乙两颗卫星绕地球做圆周运动, 已知甲卫星的周期为 $N$ 小时, 每过 $9 N$ 小时, 乙卫星都要运动到与甲卫星同居于地球一侧且三者共线的位置上, 则甲、乙两颗卫星的线速度之比为 ( )\n\n[图1]\nA: $\\frac{\\sqrt[3]{9}}{2}$\nB: $\\frac{\\sqrt[3]{3}}{2}$\nC: $\\frac{2}{\\sqrt[3]{3}}$\nD: $\\frac{2}{\\sqrt[3]{9}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 甲、乙两颗卫星绕地球做圆周运动, 已知甲卫星的周期为 $N$ 小时, 每过 $9 N$ 小时, 乙卫星都要运动到与甲卫星同居于地球一侧且三者共线的位置上, 则甲、乙两颗卫星的线速度之比为 ( )\n\n[图1]\n\nA: $\\frac{\\sqrt[3]{9}}{2}$\nB: $\\frac{\\sqrt[3]{3}}{2}$\nC: $\\frac{2}{\\sqrt[3]{3}}$\nD: $\\frac{2}{\\sqrt[3]{9}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-096.jpg?height=286&width=325&top_left_y=1716&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_547", "problem": "如图所示, 由 $\\mathrm{A} 、 \\mathrm{~B}$ 组成的双星系统, 绕它们连线上的一点做匀速圆周运动, 其运行周期为 $T, \\mathrm{~A} 、 \\mathrm{~B}$ 间的距离为 $L$, 它们的线速度之比 $\\frac{v_{1}}{v_{2}}=2$, 则 ( )\n\n[图1]\nA: $\\mathrm{AB}$ 角速度比为: $\\frac{\\omega_{A}}{\\omega_{B}}=\\frac{2}{1}$\nB: $\\mathrm{AB}$ 质量比为: $\\frac{M_{A}}{M_{B}}=\\frac{2}{1}$\nC: A 星球质量为: $M_{A}=\\frac{4 \\pi^{2} L^{3}}{G T^{2}}$\nD: 两星球质量之和为: $M_{A}+M_{B}=\\frac{4 \\pi^{2} L^{3}}{G T^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 由 $\\mathrm{A} 、 \\mathrm{~B}$ 组成的双星系统, 绕它们连线上的一点做匀速圆周运动, 其运行周期为 $T, \\mathrm{~A} 、 \\mathrm{~B}$ 间的距离为 $L$, 它们的线速度之比 $\\frac{v_{1}}{v_{2}}=2$, 则 ( )\n\n[图1]\n\nA: $\\mathrm{AB}$ 角速度比为: $\\frac{\\omega_{A}}{\\omega_{B}}=\\frac{2}{1}$\nB: $\\mathrm{AB}$ 质量比为: $\\frac{M_{A}}{M_{B}}=\\frac{2}{1}$\nC: A 星球质量为: $M_{A}=\\frac{4 \\pi^{2} L^{3}}{G T^{2}}$\nD: 两星球质量之和为: $M_{A}+M_{B}=\\frac{4 \\pi^{2} L^{3}}{G T^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-73.jpg?height=397&width=400&top_left_y=1389&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_760", "problem": "If the Earth had the density of a neutron star, what would be the diameter of the Earth?\nA: between 1 - $100 \\mathrm{~m}$\nB: between $100-500 \\mathrm{~m}$\nC: between 500 - $1000 \\mathrm{~m}$\nD: between $1000-5000 \\mathrm{~m}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIf the Earth had the density of a neutron star, what would be the diameter of the Earth?\n\nA: between 1 - $100 \\mathrm{~m}$\nB: between $100-500 \\mathrm{~m}$\nC: between 500 - $1000 \\mathrm{~m}$\nD: between $1000-5000 \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1173", "problem": "The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.a. Let $r$ denote distance from the centre of a star. We define the variables $\\rho(r), p(r)$ and $T(r)$ to be the density, pressure and temperature at radius $r$ respectively, and $m(r)$ to be the mass enclosed within radius $r$. We will now try and derive an estimate for the pressure at the centre of the Sun.\nii. We can get a good estimate of the central pressure if we use $m$ as our independent variable rather than $r$. Derive an expression for $\\mathrm{dm} / \\mathrm{dr}$ in terms of $r$ and $\\rho$, and hence express $\\mathrm{dp} / \\mathrm{dm}$ in terms of $m$ and $r$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.\n\nproblem:\na. Let $r$ denote distance from the centre of a star. We define the variables $\\rho(r), p(r)$ and $T(r)$ to be the density, pressure and temperature at radius $r$ respectively, and $m(r)$ to be the mass enclosed within radius $r$. We will now try and derive an estimate for the pressure at the centre of the Sun.\nii. We can get a good estimate of the central pressure if we use $m$ as our independent variable rather than $r$. Derive an expression for $\\mathrm{dm} / \\mathrm{dr}$ in terms of $r$ and $\\rho$, and hence express $\\mathrm{dp} / \\mathrm{dm}$ in terms of $m$ and $r$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-06.jpg?height=536&width=1508&top_left_y=560&top_left_x=272", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-07.jpg?height=748&width=1185&top_left_y=1850&top_left_x=433" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_736", "problem": "英国《自然》杂志、美国太空网 2017 年 4 月 19 日共同发布消息称，一颗温度适中的岩态行星 LHS 1140b 在经过小型 LHS 1140 矮恒星时发生凌星现象。这颗新发现的超级地球”与恒星的距离、岩石构成以及存在液态水的可能性, 使其成为目前寻找外星生命的最佳选择。假设行星 LHS $1140 b$ 绕 LHS 1140 恒星和地球绕太阳的运动均看作匀速圆周运动, 下表是网上公布的相关数据, 则下列说法正确的是（）\n\n| 恒星 | 太阳 | 质量为 $M$ |\n| :--- | :--- | :--- |\n\n\n| | LHS 1140 | 质量为 $0.6 M$ |\n| :---: | :---: | :---: |\n| 行星 | 地球 | 质量为 $m$
轨道半径为 $r$ |\n| | LHS 1140b | 质量为 $6.6 m$
轨道半径为 $1.4 r$ |\nA: LHS $1140 b$ 与地球运行的速度大小之比为 $\\sqrt{\\frac{5}{7}}$\nB: LHS $1140 b$ 与地球运行的周期之比为 $\\frac{7 \\sqrt{21}}{15}$\nC: LHS $1140 b$ 的第一宇宙速度与地球的第一宇宙速度之比为 $\\sqrt{\\frac{3}{7}}$\nD: LHS $1140 b$ 的密度是地球密度的 $\\frac{6.6}{1.4^{3}}$ 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n英国《自然》杂志、美国太空网 2017 年 4 月 19 日共同发布消息称，一颗温度适中的岩态行星 LHS 1140b 在经过小型 LHS 1140 矮恒星时发生凌星现象。这颗新发现的超级地球”与恒星的距离、岩石构成以及存在液态水的可能性, 使其成为目前寻找外星生命的最佳选择。假设行星 LHS $1140 b$ 绕 LHS 1140 恒星和地球绕太阳的运动均看作匀速圆周运动, 下表是网上公布的相关数据, 则下列说法正确的是（）\n\n| 恒星 | 太阳 | 质量为 $M$ |\n| :--- | :--- | :--- |\n\n\n| | LHS 1140 | 质量为 $0.6 M$ |\n| :---: | :---: | :---: |\n| 行星 | 地球 | 质量为 $m$
轨道半径为 $r$ |\n| | LHS 1140b | 质量为 $6.6 m$
轨道半径为 $1.4 r$ |\n\nA: LHS $1140 b$ 与地球运行的速度大小之比为 $\\sqrt{\\frac{5}{7}}$\nB: LHS $1140 b$ 与地球运行的周期之比为 $\\frac{7 \\sqrt{21}}{15}$\nC: LHS $1140 b$ 的第一宇宙速度与地球的第一宇宙速度之比为 $\\sqrt{\\frac{3}{7}}$\nD: LHS $1140 b$ 的密度是地球密度的 $\\frac{6.6}{1.4^{3}}$ 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_500", "problem": "关于黑洞和暗物质(暗物质被称为“世纪之谜” . 它“霸占”了宇宙 $95 \\%$ 的地盘, 却摸不到看不着)的问题, 以下说法正确的是(黑洞临界半径公式取为 $c=\\sqrt{\\frac{2 G M}{r}}, c$ 为光速, $G$ 为万有引力常量, $M$ 为黑洞质量\nA: 如果地球成为黑洞的话, 那么它的临界半径为 $r=\\frac{v^{2}}{c^{2}} R(R$ 为地球的半径, $v$ 为第二宇宙速度)\nB: 如果太阳成为黑洞, 那么灿烂的阳光依然存在, 只是太阳光到地球的时间变得更长\nC: 有两颗星球(质量分别为 $M_{1}$ 和 $M_{2}$ )的距离为 $L$, 不考虑周围其他星球的影响, 由牛顿运动定律计算所得的周期为 $T$, 由于宇宙充满均匀的暗物质, 所以观察测量所得的周期比 $T$ 大\nD: 有两颗星球甲和乙(质量分别为 $M_{1}$ 和 $M_{2}$ ) 的距离为 $L$, 不考虑周围其他星球的影响, 它们运动的周期为 $T$, 如果其中甲的质量减小 $\\Delta m$ 而乙的质量增大 $\\Delta m$, 距离 $L$不变, 那么它们的周期依然为 $T$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n关于黑洞和暗物质(暗物质被称为“世纪之谜” . 它“霸占”了宇宙 $95 \\%$ 的地盘, 却摸不到看不着)的问题, 以下说法正确的是(黑洞临界半径公式取为 $c=\\sqrt{\\frac{2 G M}{r}}, c$ 为光速, $G$ 为万有引力常量, $M$ 为黑洞质量\n\nA: 如果地球成为黑洞的话, 那么它的临界半径为 $r=\\frac{v^{2}}{c^{2}} R(R$ 为地球的半径, $v$ 为第二宇宙速度)\nB: 如果太阳成为黑洞, 那么灿烂的阳光依然存在, 只是太阳光到地球的时间变得更长\nC: 有两颗星球(质量分别为 $M_{1}$ 和 $M_{2}$ )的距离为 $L$, 不考虑周围其他星球的影响, 由牛顿运动定律计算所得的周期为 $T$, 由于宇宙充满均匀的暗物质, 所以观察测量所得的周期比 $T$ 大\nD: 有两颗星球甲和乙(质量分别为 $M_{1}$ 和 $M_{2}$ ) 的距离为 $L$, 不考虑周围其他星球的影响, 它们运动的周期为 $T$, 如果其中甲的质量减小 $\\Delta m$ 而乙的质量增大 $\\Delta m$, 距离 $L$不变, 那么它们的周期依然为 $T$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1036", "problem": "In the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are $\\propto \\log L$, this can be described by the empirical relation\n\n$$\n\\langle\\mathcal{M}\\rangle=-2.43(\\log P-1)-4.05\n$$\n\nwhere $P$ is the period measured in days and $\\langle\\mathcal{M}\\rangle$ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of $10 \\mathrm{pc}$. The apparent and absolute magnitudes are related as\n\n$$\nm-\\mathfrak{M}=5 \\log d-5\n$$\n\nwhere $d$ is the distance measured in pc.\n[figure1]\n\nFigure 1: Left: Colour composite view of the circumstellar nebula around RS Puppis assembled from HST images. Credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration.\n\nRight: The light curve of RS Puppis. Phase 0 corresponds to the when the star is at its brightest, and the data has been folded over a known period of 41.5 days. Credit: AAVSO / P. Kervella et al. (2017).\n\nIn the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about $100 \\mathrm{pc}$, however her method with Cepheid variables extended this range to more than $10^{6} \\mathrm{pc}$, enabling much better determination of distances across the Milky Way and into nearby galaxies.\n\nThe variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days.\nIn 2018 the Gaia satellite measured a parallax angle of 0.5844 milliarcseconds (where there are 3600 arcseconds in a degree). What does this suggest is the distance to the star?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are $\\propto \\log L$, this can be described by the empirical relation\n\n$$\n\\langle\\mathcal{M}\\rangle=-2.43(\\log P-1)-4.05\n$$\n\nwhere $P$ is the period measured in days and $\\langle\\mathcal{M}\\rangle$ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of $10 \\mathrm{pc}$. The apparent and absolute magnitudes are related as\n\n$$\nm-\\mathfrak{M}=5 \\log d-5\n$$\n\nwhere $d$ is the distance measured in pc.\n[figure1]\n\nFigure 1: Left: Colour composite view of the circumstellar nebula around RS Puppis assembled from HST images. Credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration.\n\nRight: The light curve of RS Puppis. Phase 0 corresponds to the when the star is at its brightest, and the data has been folded over a known period of 41.5 days. Credit: AAVSO / P. Kervella et al. (2017).\n\nIn the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about $100 \\mathrm{pc}$, however her method with Cepheid variables extended this range to more than $10^{6} \\mathrm{pc}$, enabling much better determination of distances across the Milky Way and into nearby galaxies.\n\nThe variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days.\nIn 2018 the Gaia satellite measured a parallax angle of 0.5844 milliarcseconds (where there are 3600 arcseconds in a degree). What does this suggest is the distance to the star?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of pc, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_2c19fdb17927c588761dg-06.jpg?height=606&width=1412&top_left_y=1102&top_left_x=320", "https://i.postimg.cc/ydj3nNq2/Screenshot-2024-04-06-at-21-36-47.png" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "pc" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_161", "problem": "由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心 $O$ 在三角形所在平面内以相同角速度做匀速圆周运动。如图所示, 三颗星体的质量均为 $m$, 三角形的边长为 $a$, 引力常量为 $G$, 下列说法正确的是（）\n\n[图1]\nA: 每个星体受到引力大小均为 $3 \\frac{G m^{2}}{a^{2}}$\nB: 每个星体的角速度均为 $\\sqrt{\\frac{3 G m}{a^{3}}}$\nC: 若 $a$ 不变, $m$ 是原来的 2 倍, 则周期是原来的 $\\frac{1}{2}$\nD: 若 $m$ 不变, $a$ 是原来的 4 倍, 则线速度是原来的 $\\frac{1}{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心 $O$ 在三角形所在平面内以相同角速度做匀速圆周运动。如图所示, 三颗星体的质量均为 $m$, 三角形的边长为 $a$, 引力常量为 $G$, 下列说法正确的是（）\n\n[图1]\n\nA: 每个星体受到引力大小均为 $3 \\frac{G m^{2}}{a^{2}}$\nB: 每个星体的角速度均为 $\\sqrt{\\frac{3 G m}{a^{3}}}$\nC: 若 $a$ 不变, $m$ 是原来的 2 倍, 则周期是原来的 $\\frac{1}{2}$\nD: 若 $m$ 不变, $a$ 是原来的 4 倍, 则线速度是原来的 $\\frac{1}{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-062.jpg?height=568&width=646&top_left_y=213&top_left_x=385", "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-062.jpg?height=685&width=700&top_left_y=1485&top_left_x=538" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_339", "problem": "1772 年, 法籍意大利数学家拉格朗日在论文《三体问题》中指出: 两个质量相差悬殊的天体 (如太阳和地球) 所在同一平面上有 5 个特殊点, 如图中 $L_{1} 、 L_{2} 、 L_{3} 、 L_{4}$ 、 $L_{5}$ 所示, 人们称之为拉格朗日点。若飞行器位于这些点上, 会在太阳与地球共同引力作用下, 可以不消耗燃料而保持与地球同步绕太阳做圆周运动。已知太阳质量为地球质量的 33 万倍, 日地距离为太阳半径的 215 倍, $L_{4}$ 和 $L_{5}$ 对称, 且与太阳连线的夹角为 $120^{\\circ}$ ，则下列说法正确的是（）\n\n[图1]\nA: 人造卫星在 $L_{1}$ 和 $L_{2}$ 的向心加速度不同\nB: $L_{3}$ 距太阳中心的距离比 $L_{2}$ 距太阳中心的距离大\nC: $L_{1} 、 L_{2}$ 的公转线速度相同\nD: $L_{4}$ 的旋转中心点在太阳内部\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n1772 年, 法籍意大利数学家拉格朗日在论文《三体问题》中指出: 两个质量相差悬殊的天体 (如太阳和地球) 所在同一平面上有 5 个特殊点, 如图中 $L_{1} 、 L_{2} 、 L_{3} 、 L_{4}$ 、 $L_{5}$ 所示, 人们称之为拉格朗日点。若飞行器位于这些点上, 会在太阳与地球共同引力作用下, 可以不消耗燃料而保持与地球同步绕太阳做圆周运动。已知太阳质量为地球质量的 33 万倍, 日地距离为太阳半径的 215 倍, $L_{4}$ 和 $L_{5}$ 对称, 且与太阳连线的夹角为 $120^{\\circ}$ ，则下列说法正确的是（）\n\n[图1]\n\nA: 人造卫星在 $L_{1}$ 和 $L_{2}$ 的向心加速度不同\nB: $L_{3}$ 距太阳中心的距离比 $L_{2}$ 距太阳中心的距离大\nC: $L_{1} 、 L_{2}$ 的公转线速度相同\nD: $L_{4}$ 的旋转中心点在太阳内部\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-048.jpg?height=348&width=485&top_left_y=1411&top_left_x=337", "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-049.jpg?height=414&width=608&top_left_y=730&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_933", "problem": "On $19^{\\text {th }}$ July 2020 the first Arab interplanetary mission was launched on its way to Mars, consisting of the Hope probe designed by the United Arab Emirates Space Agency. When it reaches Mars in February 2021 it will do a short burn at a distance of $49400 \\mathrm{~km}$ away from the surface to slow it down and put it into an elliptical capture orbit which will bring it as close as only $1000 \\mathrm{~km}$ above the planet. Given the mass of Mars is $6.39 \\times 10^{23} \\mathrm{~kg}$ and its radius is $3390 \\mathrm{~km}$, what will be the period of this orbit?\nA: 37.3 hours\nB: 40.9 hours\nC: 105 hours\nD: 116 hours\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOn $19^{\\text {th }}$ July 2020 the first Arab interplanetary mission was launched on its way to Mars, consisting of the Hope probe designed by the United Arab Emirates Space Agency. When it reaches Mars in February 2021 it will do a short burn at a distance of $49400 \\mathrm{~km}$ away from the surface to slow it down and put it into an elliptical capture orbit which will bring it as close as only $1000 \\mathrm{~km}$ above the planet. Given the mass of Mars is $6.39 \\times 10^{23} \\mathrm{~kg}$ and its radius is $3390 \\mathrm{~km}$, what will be the period of this orbit?\n\nA: 37.3 hours\nB: 40.9 hours\nC: 105 hours\nD: 116 hours\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://i.postimg.cc/PJ98BF6j/Screenshot-2024-04-06-at-22-11-52.png" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_440", "problem": "我国的“嫦娥奔月”月球探测工程启动至今, 以“绕、落、回”为发展过程。中国国家航天局目前计划于 2020 年发射嫦娥工程第二阶段的月球车嫦娥四号。中国探月计划总工程师吴伟仁近期透露, 此台月球车很可能在离地球较远的月球背面着陆, 假设运载火箭先将 “嫦娥四号” 月球探测器成功送入太空, 由地月转移轨道进入半径为 $r_{l}=100$ 公里环月圆轨道I后成功变轨到近月点为 15 公里的椭圆轨道II, 在从 15 公里高度降至近月表面圆轨道III, 最后成功实现登月。若取两物体相距无穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\\mathrm{P}}=-G \\frac{M m}{r}$ （式中 $G$ 为引力常数）。已知月球质量 $M_{0}$, 月球半径为 $R$, 发射的“嫦娥四号”探测器质量\n为 $m_{0}$, 引力常量 $G$ 。 则关于“嫦娥四号” 登月过程的说法正确的是 ( )\n\n[图1]\nA: “嫦娥四号”探测器在轨道I上运行的动能大于在轨道III运行的动能\nB: “嫦娥四号”探测器从轨道I上变轨到轨道III上时, 势能减小了 $G M_{0} m_{0}\\left(\\frac{1}{r_{1}}-\\frac{1}{R}\\right)$\nC: “嫦娥四号”探测器在轨道III上运行时机械能等于在轨道I运行时的机械能\nD: 落月的“嫦娥四号” 探测器从轨道III回到轨道I, 所要提供的最小能量是 $$ \\frac{G M_{0} m}{2}\\left(\\frac{1}{R}-\\frac{1}{r_{1}}\\right) $$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n我国的“嫦娥奔月”月球探测工程启动至今, 以“绕、落、回”为发展过程。中国国家航天局目前计划于 2020 年发射嫦娥工程第二阶段的月球车嫦娥四号。中国探月计划总工程师吴伟仁近期透露, 此台月球车很可能在离地球较远的月球背面着陆, 假设运载火箭先将 “嫦娥四号” 月球探测器成功送入太空, 由地月转移轨道进入半径为 $r_{l}=100$ 公里环月圆轨道I后成功变轨到近月点为 15 公里的椭圆轨道II, 在从 15 公里高度降至近月表面圆轨道III, 最后成功实现登月。若取两物体相距无穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\\mathrm{P}}=-G \\frac{M m}{r}$ （式中 $G$ 为引力常数）。已知月球质量 $M_{0}$, 月球半径为 $R$, 发射的“嫦娥四号”探测器质量\n为 $m_{0}$, 引力常量 $G$ 。 则关于“嫦娥四号” 登月过程的说法正确的是 ( )\n\n[图1]\n\nA: “嫦娥四号”探测器在轨道I上运行的动能大于在轨道III运行的动能\nB: “嫦娥四号”探测器从轨道I上变轨到轨道III上时, 势能减小了 $G M_{0} m_{0}\\left(\\frac{1}{r_{1}}-\\frac{1}{R}\\right)$\nC: “嫦娥四号”探测器在轨道III上运行时机械能等于在轨道I运行时的机械能\nD: 落月的“嫦娥四号” 探测器从轨道III回到轨道I, 所要提供的最小能量是 $$ \\frac{G M_{0} m}{2}\\left(\\frac{1}{R}-\\frac{1}{r_{1}}\\right) $$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-069.jpg?height=357&width=808&top_left_y=250&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_923", "problem": "Looking up into the UK sky at $10 \\mathrm{pm}$ in late September, which of the following bright stars is NOT visible?\nA: Deneb (Right ascension $=20^{\\mathrm{h}} 41^{\\mathrm{m}}$, declination $\\left.=+45.28^{\\circ}\\right)$\nB: Vega (Right ascension $=18^{\\mathrm{h}} 37^{\\mathrm{m}}$, declination $\\left.=+38.78^{\\circ}\\right)$\nC: Capella (Right ascension $=05^{\\mathrm{h}} 17^{\\mathrm{m}}$, declination $=+46.00^{\\circ}$ )\nD: Sirius (Right ascension $=06^{\\mathrm{h}} 45^{\\mathrm{m}}$, declination $=-16.72^{\\circ}$ )\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nLooking up into the UK sky at $10 \\mathrm{pm}$ in late September, which of the following bright stars is NOT visible?\n\nA: Deneb (Right ascension $=20^{\\mathrm{h}} 41^{\\mathrm{m}}$, declination $\\left.=+45.28^{\\circ}\\right)$\nB: Vega (Right ascension $=18^{\\mathrm{h}} 37^{\\mathrm{m}}$, declination $\\left.=+38.78^{\\circ}\\right)$\nC: Capella (Right ascension $=05^{\\mathrm{h}} 17^{\\mathrm{m}}$, declination $=+46.00^{\\circ}$ )\nD: Sirius (Right ascension $=06^{\\mathrm{h}} 45^{\\mathrm{m}}$, declination $=-16.72^{\\circ}$ )\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_512", "problem": "2019 年 4 月 10 日晚 9 时许, 全球多地天文学家同步公布的首张黑洞照片（如图）。假设银河系中两个黑洞 $\\mathrm{A} 、 \\mathrm{~B}$, 它们以二者连线上的 $O$ 点为圆心做匀速圆周运动, 测得 $\\mathrm{A} 、 \\mathrm{~B}$ 到 $O$ 点的距离分别为 $r$ 和 $2 r$ 。黑洞 $\\mathrm{A} 、 \\mathrm{~B}$ 均可看成质量分布均匀的球体，不考虑其他星球对黑洞的引力, 两黑洞的半径均远小于它们之间的距离。下列说法正确的是\nA: 黑洞 $\\mathrm{A} 、 \\mathrm{~B}$ 的质量之比为 $1: 2$\nB: 黑洞 A、B 所受引力之比为 $1: 2$\nC: 黑洞 $\\mathrm{A} 、 \\mathrm{~B}$ 的角速度之比为 $1: 2$\nD: 黑洞 $\\mathrm{A} 、 \\mathrm{~B}$ 的线速度之比为 $1: 2$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2019 年 4 月 10 日晚 9 时许, 全球多地天文学家同步公布的首张黑洞照片（如图）。假设银河系中两个黑洞 $\\mathrm{A} 、 \\mathrm{~B}$, 它们以二者连线上的 $O$ 点为圆心做匀速圆周运动, 测得 $\\mathrm{A} 、 \\mathrm{~B}$ 到 $O$ 点的距离分别为 $r$ 和 $2 r$ 。黑洞 $\\mathrm{A} 、 \\mathrm{~B}$ 均可看成质量分布均匀的球体，不考虑其他星球对黑洞的引力, 两黑洞的半径均远小于它们之间的距离。下列说法正确的是\n\nA: 黑洞 $\\mathrm{A} 、 \\mathrm{~B}$ 的质量之比为 $1: 2$\nB: 黑洞 A、B 所受引力之比为 $1: 2$\nC: 黑洞 $\\mathrm{A} 、 \\mathrm{~B}$ 的角速度之比为 $1: 2$\nD: 黑洞 $\\mathrm{A} 、 \\mathrm{~B}$ 的线速度之比为 $1: 2$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_359", "problem": "最近中国宇航局公布了天眼射电望远镜最新发现的一个行星系统, 该系统拥有一颗由岩石和气体构成的行星围绕一颗的类太阳恒星运行。经观测, 行星与恒星之间的距离是地、日间距离的 $\\frac{1}{N}$, 恒星质量是太阳质量的 $k$ 倍，则下列叙述正确的是 ( )\nA: 行星公转周期和地球公转周期的比值是 $N^{-\\frac{3}{2}} k^{-\\frac{1}{2}}$\nB: 行星公转周期和地球公转周期的比值是 $N^{\\frac{3}{2}} k^{\\frac{1}{2}}$\nC: 行星公转线速度和地球公转线速度的比值是 $N^{\\frac{1}{2}} k^{\\frac{1}{2}}$\nD: 行星公转线速度和地球公转线速度的比值是 $N^{-\\frac{1}{2}} k^{-\\frac{1}{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n最近中国宇航局公布了天眼射电望远镜最新发现的一个行星系统, 该系统拥有一颗由岩石和气体构成的行星围绕一颗的类太阳恒星运行。经观测, 行星与恒星之间的距离是地、日间距离的 $\\frac{1}{N}$, 恒星质量是太阳质量的 $k$ 倍，则下列叙述正确的是 ( )\n\nA: 行星公转周期和地球公转周期的比值是 $N^{-\\frac{3}{2}} k^{-\\frac{1}{2}}$\nB: 行星公转周期和地球公转周期的比值是 $N^{\\frac{3}{2}} k^{\\frac{1}{2}}$\nC: 行星公转线速度和地球公转线速度的比值是 $N^{\\frac{1}{2}} k^{\\frac{1}{2}}$\nD: 行星公转线速度和地球公转线速度的比值是 $N^{-\\frac{1}{2}} k^{-\\frac{1}{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-067.jpg?height=265&width=830&top_left_y=153&top_left_x=610", "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-067.jpg?height=245&width=699&top_left_y=980&top_left_x=681" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_437", "problem": "中国科幻大片《流浪地球 2 》中描述的“太空电梯”让人印象深刻。科学家们在地球同步轨道上建造了一个空间站, 再用超级缆绳连接地球赤道上的固定基地, 通过超级缆绳承载太空电梯, 使轿厢沿绳索从地球基地直入太空, 而向空间站运送货物。原理如图甲所示, 图中的太空电梯正停在离地面高 $R$ 处的站点修整, 并利用太阳能给蓄电池充电。图乙中 $r$ 为货物到地心的距离, $R$ 为地球半径, 曲线 $A$ 为地球引力对货物产生的加速度大小与 $r$ 的关系; 直线 $B$ 为货物由于地球自转而产生的向心加速度大小与 $r$ 的关系。关于相对地面静止在不同高度的电梯中货物, 下列说法正确的有（）\n\n[图1]\n\n[图2]\n\n乙\nA: 货物的线速度随着 $r$ 的增大而减小\nB: 货物在 $r=R$ 处的线速度等于第一宇宙速度\nC: 地球自转的周期 $T=2 \\pi \\sqrt{\\frac{r_{0}}{a_{0}}}$\nD: 图甲中质量为 $m$ 的货物对电梯底板的压力大小为 $\\frac{m g_{0}}{4}-\\frac{2 m r_{0} a_{0}}{R}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n中国科幻大片《流浪地球 2 》中描述的“太空电梯”让人印象深刻。科学家们在地球同步轨道上建造了一个空间站, 再用超级缆绳连接地球赤道上的固定基地, 通过超级缆绳承载太空电梯, 使轿厢沿绳索从地球基地直入太空, 而向空间站运送货物。原理如图甲所示, 图中的太空电梯正停在离地面高 $R$ 处的站点修整, 并利用太阳能给蓄电池充电。图乙中 $r$ 为货物到地心的距离, $R$ 为地球半径, 曲线 $A$ 为地球引力对货物产生的加速度大小与 $r$ 的关系; 直线 $B$ 为货物由于地球自转而产生的向心加速度大小与 $r$ 的关系。关于相对地面静止在不同高度的电梯中货物, 下列说法正确的有（）\n\n[图1]\n\n[图2]\n\n乙\n\nA: 货物的线速度随着 $r$ 的增大而减小\nB: 货物在 $r=R$ 处的线速度等于第一宇宙速度\nC: 地球自转的周期 $T=2 \\pi \\sqrt{\\frac{r_{0}}{a_{0}}}$\nD: 图甲中质量为 $m$ 的货物对电梯底板的压力大小为 $\\frac{m g_{0}}{4}-\\frac{2 m r_{0} a_{0}}{R}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-027.jpg?height=334&width=985&top_left_y=1663&top_left_x=344", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-027.jpg?height=257&width=346&top_left_y=1688&top_left_x=1409" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_544", "problem": "2019 年 4 月 10 日，人类首张黑洞“照片”问世.黑洞是爱因斯坦广义相对论预言存在的一种天体, 它具有的超强引力使光也无法逃离它的势力范围, 即黑洞的逃逸速度大于光速. 理论分析表明.星球的逃逸速度是其第一宇宙速度的 $\\sqrt{2}$ 倍. 已知地球绕太阳公转的轨道半径约为 $1.5 \\times 10^{11} \\mathrm{~m}$, 公转周期约为 $3.15 \\times 10^{7} \\mathrm{~s}$, 假设太阳演变为黑洞, 它的半径最大为 $\\left(\\right.$ 太阳的质量不变, 光速 $\\left.c=3.0 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}\\right)$\nA: $1 \\mathrm{~km}$\nB: $3 \\mathrm{~km}$\nC: $100 \\mathrm{~km}$\nD: $300 \\mathrm{~km}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2019 年 4 月 10 日，人类首张黑洞“照片”问世.黑洞是爱因斯坦广义相对论预言存在的一种天体, 它具有的超强引力使光也无法逃离它的势力范围, 即黑洞的逃逸速度大于光速. 理论分析表明.星球的逃逸速度是其第一宇宙速度的 $\\sqrt{2}$ 倍. 已知地球绕太阳公转的轨道半径约为 $1.5 \\times 10^{11} \\mathrm{~m}$, 公转周期约为 $3.15 \\times 10^{7} \\mathrm{~s}$, 假设太阳演变为黑洞, 它的半径最大为 $\\left(\\right.$ 太阳的质量不变, 光速 $\\left.c=3.0 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}\\right)$\n\nA: $1 \\mathrm{~km}$\nB: $3 \\mathrm{~km}$\nC: $100 \\mathrm{~km}$\nD: $300 \\mathrm{~km}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_115", "problem": "2019 年诺贝尔物理学奖授予了三位天文学家, 以表彰他们对于人类对宇宙演化方面的了解所作的贡献。其中两位学者的贡献是首次发现地外行星, 其主要原理是恒星和其行星在引力作用下构成一个“双星系统”, 恒星在周期性运动时, 可通过观察其光谱的周期性变化知道其运动周期, 从而证实其附近存在行星。若观测到的某恒星运动周期为 $T$, 并测得该恒星与行星的距离为 $L$, 已知万有引力常量为 $G$, 则由这些物理量可以求得 ( )\nA: 行星的质量\nB: 恒星的质量\nC: 恒星与行星的质量之和\nD: 恒星与行星圆周运动的半径之比\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2019 年诺贝尔物理学奖授予了三位天文学家, 以表彰他们对于人类对宇宙演化方面的了解所作的贡献。其中两位学者的贡献是首次发现地外行星, 其主要原理是恒星和其行星在引力作用下构成一个“双星系统”, 恒星在周期性运动时, 可通过观察其光谱的周期性变化知道其运动周期, 从而证实其附近存在行星。若观测到的某恒星运动周期为 $T$, 并测得该恒星与行星的距离为 $L$, 已知万有引力常量为 $G$, 则由这些物理量可以求得 ( )\n\nA: 行星的质量\nB: 恒星的质量\nC: 恒星与行星的质量之和\nD: 恒星与行星圆周运动的半径之比\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_290", "problem": "2019 年 4 月 10 日, 天文学家召开全球新闻发布会, 宣布首次直接拍摄到黑洞的照片。黑洞是一种密度极大、引力极大的天体, 以至于光都无法逃逸（光速为 $c$ )。若黑洞的质量为 $M$, 半径为 $R$, 引力常量为 $G$, 其逃逸速度公式为 $v^{\\prime}=\\sqrt{\\frac{2 G M}{R}}$ 。如果天文学家观测到一天体以速度 $v$ 绕某黑洞做半径为 $r$ 的匀速圆周运动, 则下列说法正确的有\nA: $M=\\frac{v^{2} r}{G}$\nB: $M=G v^{2} r$\nC: 该黑洞的最大半径为 $\\frac{2 G M}{c^{2}}$\nD: 该黑洞表面的重力加速度为 $\\frac{G M}{R^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2019 年 4 月 10 日, 天文学家召开全球新闻发布会, 宣布首次直接拍摄到黑洞的照片。黑洞是一种密度极大、引力极大的天体, 以至于光都无法逃逸（光速为 $c$ )。若黑洞的质量为 $M$, 半径为 $R$, 引力常量为 $G$, 其逃逸速度公式为 $v^{\\prime}=\\sqrt{\\frac{2 G M}{R}}$ 。如果天文学家观测到一天体以速度 $v$ 绕某黑洞做半径为 $r$ 的匀速圆周运动, 则下列说法正确的有\n\nA: $M=\\frac{v^{2} r}{G}$\nB: $M=G v^{2} r$\nC: 该黑洞的最大半径为 $\\frac{2 G M}{c^{2}}$\nD: 该黑洞表面的重力加速度为 $\\frac{G M}{R^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_367", "problem": "利用金星凌日现象, 我们可以估算出地球与太阳之间的平均距离。日地平均距离也被定义为 1 个天文单位 (1A.U.), 是天文学中常用的距离单位。\n\n金星轨道在地球轨道内侧, 某些特殊时刻, 地球、金星、太阳恰在一条直线上, 这时从地球上可以看到金星就像一个小黑点一样在太阳表面缓慢移动, 如图甲所示, 天文学称之为“金星凌日”。在地球上的不同地点, 比如图乙中的 $A 、 B$ 两点, 它们在同一时刻观察到的金星在日面上的位置是不同的，我们分别记为 $A^{\\prime} 、 B^{\\prime}$ 。\n\n 在 $A 、 B$ 两地分别同时拍摄金星凌日的照片, 然后将其重合起来观察, 如图丙所示。发现太阳的直径是两列轨迹之间距离 $n$ 倍。若已知太阳直径对地面观察者的张角 (亦称视直径) 为 $\\alpha, \\alpha$ 为很小的角, 可认为: $\\sin \\alpha=\\tan \\alpha=\\alpha$, 请写出日地距离的表达式 (用 $l 、 k 、 n 、 \\alpha$ 表示)。\n\n[图1]\n\n甲\n\n[图2]\n\n丙\n\n[图3]\n\n乙", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n利用金星凌日现象, 我们可以估算出地球与太阳之间的平均距离。日地平均距离也被定义为 1 个天文单位 (1A.U.), 是天文学中常用的距离单位。\n\n金星轨道在地球轨道内侧, 某些特殊时刻, 地球、金星、太阳恰在一条直线上, 这时从地球上可以看到金星就像一个小黑点一样在太阳表面缓慢移动, 如图甲所示, 天文学称之为“金星凌日”。在地球上的不同地点, 比如图乙中的 $A 、 B$ 两点, 它们在同一时刻观察到的金星在日面上的位置是不同的，我们分别记为 $A^{\\prime} 、 B^{\\prime}$ 。\n\n 在 $A 、 B$ 两地分别同时拍摄金星凌日的照片, 然后将其重合起来观察, 如图丙所示。发现太阳的直径是两列轨迹之间距离 $n$ 倍。若已知太阳直径对地面观察者的张角 (亦称视直径) 为 $\\alpha, \\alpha$ 为很小的角, 可认为: $\\sin \\alpha=\\tan \\alpha=\\alpha$, 请写出日地距离的表达式 (用 $l 、 k 、 n 、 \\alpha$ 表示)。\n\n[图1]\n\n甲\n\n[图2]\n\n丙\n\n[图3]\n\n乙\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-038.jpg?height=214&width=240&top_left_y=201&top_left_x=474", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-038.jpg?height=249&width=280&top_left_y=178&top_left_x=751", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-038.jpg?height=205&width=853&top_left_y=480&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1093", "problem": "It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1).\n[figure1]\n\nFigure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\\sim 40^{\\circ}$. Credit: Daniel V. Schroeder / Weber State University.\n\nRight: The same idea but viewed from Iceland at a latitude of $65^{\\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland.\n\nDuring the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees.\n\nA simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\\delta$, is given as:\n\n$$\nh=-\\left(90^{\\circ}-\\phi\\right) \\cos (A)+\\delta\n$$\n\nThe solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\\delta=+23.44^{\\circ}$ at the June solstice (roughly $21^{\\text {st }}$ June) to a minimum of $\\delta=-23.44^{\\circ}$ on the December solstice (roughly $21^{\\text {st }}$ December).\n\nIt can be shown using spherical trigonometry that the precise model connecting $\\delta, h, \\phi$ and $A$ is:\n\n$$\n\\sin (\\delta)=\\sin (h) \\sin (\\phi)+\\cos (h) \\cos (\\phi) \\cos (A) .\n$$\n\nUsing the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2.\n\n[figure2]\n\nFigure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia.\n\nBy using further spherical trigonometry, we can derive a second helpful equation in the precise model:\n\n$$\n\\sin (h)=\\sin (\\phi) \\sin (\\delta)+\\cos (\\phi) \\cos (\\delta) \\cos (H)\n$$\n\nHere, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\\circ}=24^{\\mathrm{h}}$.d. This exam is being taken on $24^{\\text {th }}$ January and is 3 hours long.\n\niii. What is the latitude with the longest sunrise today? Give its duration in minutes and seconds.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIt is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1).\n[figure1]\n\nFigure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\\sim 40^{\\circ}$. Credit: Daniel V. Schroeder / Weber State University.\n\nRight: The same idea but viewed from Iceland at a latitude of $65^{\\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland.\n\nDuring the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees.\n\nA simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\\delta$, is given as:\n\n$$\nh=-\\left(90^{\\circ}-\\phi\\right) \\cos (A)+\\delta\n$$\n\nThe solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\\delta=+23.44^{\\circ}$ at the June solstice (roughly $21^{\\text {st }}$ June) to a minimum of $\\delta=-23.44^{\\circ}$ on the December solstice (roughly $21^{\\text {st }}$ December).\n\nIt can be shown using spherical trigonometry that the precise model connecting $\\delta, h, \\phi$ and $A$ is:\n\n$$\n\\sin (\\delta)=\\sin (h) \\sin (\\phi)+\\cos (h) \\cos (\\phi) \\cos (A) .\n$$\n\nUsing the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2.\n\n[figure2]\n\nFigure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia.\n\nBy using further spherical trigonometry, we can derive a second helpful equation in the precise model:\n\n$$\n\\sin (h)=\\sin (\\phi) \\sin (\\delta)+\\cos (\\phi) \\cos (\\delta) \\cos (H)\n$$\n\nHere, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\\circ}=24^{\\mathrm{h}}$.\n\nproblem:\nd. This exam is being taken on $24^{\\text {th }}$ January and is 3 hours long.\n\niii. What is the latitude with the longest sunrise today? Give its duration in minutes and seconds.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~min} 3 \\mathrm{~s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-04.jpg?height=668&width=1478&top_left_y=523&top_left_x=290", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-05.jpg?height=648&width=1234&top_left_y=738&top_left_x=385" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~min} 3 \\mathrm{~s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_487", "problem": "2018 年 12 月 8 日凌晨, 我国成功发射一枚火箭, 将“嫦娥四号”探测器送上了天空,历经 110 个小时的飞行后，在离月球仅 100 公里的距离完美“刹车”, 进入近月轨道运行; 12 月 30 日 8 时 55 分, “嫦娥四号”在环月轨道成功实施变轨控制, 顺利进入月球背面的预定着陆准备轨道; 2019 年 1 月 3 日 10 时 15 分北京航天飞行控制中心向“嫦娥四号”探测器发出着陆指令: 开启变推力发动机, 逐步将探测器的速度降到零, 并不断调整姿态,在距月面 100 米处悬停，选定相对平坦区域后缓慢垂直下降，实现了世界上首次在月球\n背面软着陆。探测器在着陆过程中沿坚直方向运动, 设悬停前减速阶段变推力发动机的平均作用力为 $F$, 经过时间 $t$ 将探测器的速度由 $v$ 减小到 0 。已知探测器质量为 $m$, 在近月轨道做匀速圆周运动的周期为 $T$, 引力常量为 $G$, 月球可视为质量分布均匀的球体,着陆过程中“嫦娥四号”探测器质量不变。则通过以上数据可求得（）\n\n[图1]\nA: 月球表面的重力加速度为 $g_{\\text {月 }}=\\frac{F}{m}$\nB: 月球的半径 $R=\\left(\\frac{(F t-m v) T^{2}}{4 m t \\pi^{2}}\\right)^{\\frac{1}{3}}$\nC: 月球的质量 $M=\\frac{(F t-m v)^{3} T^{4}}{16 \\pi^{4} G m^{3} t^{3}}$\nD: 月球的平均密度 $\\rho=\\frac{G T^{2}}{3 \\pi}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2018 年 12 月 8 日凌晨, 我国成功发射一枚火箭, 将“嫦娥四号”探测器送上了天空,历经 110 个小时的飞行后，在离月球仅 100 公里的距离完美“刹车”, 进入近月轨道运行; 12 月 30 日 8 时 55 分, “嫦娥四号”在环月轨道成功实施变轨控制, 顺利进入月球背面的预定着陆准备轨道; 2019 年 1 月 3 日 10 时 15 分北京航天飞行控制中心向“嫦娥四号”探测器发出着陆指令: 开启变推力发动机, 逐步将探测器的速度降到零, 并不断调整姿态,在距月面 100 米处悬停，选定相对平坦区域后缓慢垂直下降，实现了世界上首次在月球\n背面软着陆。探测器在着陆过程中沿坚直方向运动, 设悬停前减速阶段变推力发动机的平均作用力为 $F$, 经过时间 $t$ 将探测器的速度由 $v$ 减小到 0 。已知探测器质量为 $m$, 在近月轨道做匀速圆周运动的周期为 $T$, 引力常量为 $G$, 月球可视为质量分布均匀的球体,着陆过程中“嫦娥四号”探测器质量不变。则通过以上数据可求得（）\n\n[图1]\n\nA: 月球表面的重力加速度为 $g_{\\text {月 }}=\\frac{F}{m}$\nB: 月球的半径 $R=\\left(\\frac{(F t-m v) T^{2}}{4 m t \\pi^{2}}\\right)^{\\frac{1}{3}}$\nC: 月球的质量 $M=\\frac{(F t-m v)^{3} T^{4}}{16 \\pi^{4} G m^{3} t^{3}}$\nD: 月球的平均密度 $\\rho=\\frac{G T^{2}}{3 \\pi}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-010.jpg?height=351&width=480&top_left_y=493&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_207", "problem": "继 2004 年开始在四川西昌发射“嫦娥系列”卫星开展探月工程后, 2020 年 7 月 23\n\n日 12 时 41 分，我国首次火星探测任务“天问一号”在海南文昌航天发射场由长征 5 号运载火箭发射升空，直接送入地火转移轨道，开启了我国探测行星之旅。“天问一号”将飞行约 7 个月抵达火星, 并通过 2 至 3 个月的环绕飞行后着陆火星表面, 开展火星的表面形貌、土壤特性、物质成分、水冰、大气、电离层、磁场等科学探测。下列说法正确的是 ( )\nA: “天问一号”选择在海南文昌发射，是为了在同等发射条件下，提高同型火箭的运载能力\nB: 在地火转移轨道飞行时, “天问一号”是一颗人造行星, 与地球、火星共同绕太阳公转\nC: 若“嫦娥一号”绕月与“天问一号”绕火星都可看做是圆周运动, 则它们的运行周期的平方和轨道半径的 3 次方的比值 $\\frac{T^{2}}{R^{3}}$ 相等\nD: “天问一号”绕火星运动, 必须在火星赤道平面, 并且周期与火星自转周期相同\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n继 2004 年开始在四川西昌发射“嫦娥系列”卫星开展探月工程后, 2020 年 7 月 23\n\n日 12 时 41 分，我国首次火星探测任务“天问一号”在海南文昌航天发射场由长征 5 号运载火箭发射升空，直接送入地火转移轨道，开启了我国探测行星之旅。“天问一号”将飞行约 7 个月抵达火星, 并通过 2 至 3 个月的环绕飞行后着陆火星表面, 开展火星的表面形貌、土壤特性、物质成分、水冰、大气、电离层、磁场等科学探测。下列说法正确的是 ( )\n\nA: “天问一号”选择在海南文昌发射，是为了在同等发射条件下，提高同型火箭的运载能力\nB: 在地火转移轨道飞行时, “天问一号”是一颗人造行星, 与地球、火星共同绕太阳公转\nC: 若“嫦娥一号”绕月与“天问一号”绕火星都可看做是圆周运动, 则它们的运行周期的平方和轨道半径的 3 次方的比值 $\\frac{T^{2}}{R^{3}}$ 相等\nD: “天问一号”绕火星运动, 必须在火星赤道平面, 并且周期与火星自转周期相同\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-074.jpg?height=594&width=785&top_left_y=640&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_212", "problem": "2021 年 4 月 29 日, 长征五号 $B$ 遥二运载火箭搭载中国载人航天工程中第一个空间站核心舱“天和核心舱”，在海南文昌航天发射场发射升空。最终“天和核心舱”顺利进入离地约 $400 \\mathrm{~km}$ 高的预定圆轨道, 运行速率约为 $7.7 \\mathrm{~km} / \\mathrm{s}$, 宇航员 $24 \\mathrm{~h}$ 内可以看到 16 次日出日落。已知万有引力常量 $G$, 根据以上信息能估算出 ( )\nA: 地球的半径\nB: 地球的质量\nC: “天和核心舱”的质量\nD: “天和核心舱”受到的地球引力\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2021 年 4 月 29 日, 长征五号 $B$ 遥二运载火箭搭载中国载人航天工程中第一个空间站核心舱“天和核心舱”，在海南文昌航天发射场发射升空。最终“天和核心舱”顺利进入离地约 $400 \\mathrm{~km}$ 高的预定圆轨道, 运行速率约为 $7.7 \\mathrm{~km} / \\mathrm{s}$, 宇航员 $24 \\mathrm{~h}$ 内可以看到 16 次日出日落。已知万有引力常量 $G$, 根据以上信息能估算出 ( )\n\nA: 地球的半径\nB: 地球的质量\nC: “天和核心舱”的质量\nD: “天和核心舱”受到的地球引力\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1038", "problem": "The NEOWISE telescope discovered a new comet on $27^{\\text {th }}$ March 2020, later given the official designation C/2020 F3 NEOWISE. Although when first discovered it only had an apparent magnitude of 18.0, it would become sufficiently bright that it could be seen with the naked eye by observers throughout the northern hemisphere, and was one of the brightest comets since Hale-Bopp in 1997.\n\n[figure1]\n\nFigure 4: The comet C/2020 F3 NEOWISE as seen from the UK in late July. Credit: Alex Calverley.\n\nOn its discovery date the comet was 1.702 au from the Earth and 2.089 au from the Sun, and at perihelion (when it was closest to the Sun) on $3^{\\text {rd }}$ July 2020 it was only 0.294649 au from the Sun.\n\nGiven the comet's orbit has an eccentricity of 0.999188 , estimate the year of its next perihelion.\n\nFor a spherical object, the luminosity of the reflected light is a function of how much of the lit surface of the object we can see, known as the phase (for example crescent and gibbous phases of the moon).\n\nThis correction is known as the phase factor, $p(\\theta)$, such that the total power reflected is related to the total power incident as $P_{\\text {ref }}=P_{\\text {inc }} \\times p(\\theta)$. You are given that\n\n$$\np(\\theta)=B\\left[\\left(1-\\frac{\\theta}{\\pi}\\right) \\cos \\theta+\\frac{1}{\\pi} \\sin \\theta\\right]\n$$\n\nwhere $B$ is a constant and $\\theta$ is the phase angle, as defined in Fig 5, and is measured in radians (a measure of angle such that $2 \\pi$ radians $=360^{\\circ}$.\n\n[figure2]\n\nFigure 5: The phase angle is defined as the angle between the Earth and Sun as viewed from the object. Credit: Wikipedia.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe NEOWISE telescope discovered a new comet on $27^{\\text {th }}$ March 2020, later given the official designation C/2020 F3 NEOWISE. Although when first discovered it only had an apparent magnitude of 18.0, it would become sufficiently bright that it could be seen with the naked eye by observers throughout the northern hemisphere, and was one of the brightest comets since Hale-Bopp in 1997.\n\n[figure1]\n\nFigure 4: The comet C/2020 F3 NEOWISE as seen from the UK in late July. Credit: Alex Calverley.\n\nOn its discovery date the comet was 1.702 au from the Earth and 2.089 au from the Sun, and at perihelion (when it was closest to the Sun) on $3^{\\text {rd }}$ July 2020 it was only 0.294649 au from the Sun.\n\nGiven the comet's orbit has an eccentricity of 0.999188 , estimate the year of its next perihelion.\n\nFor a spherical object, the luminosity of the reflected light is a function of how much of the lit surface of the object we can see, known as the phase (for example crescent and gibbous phases of the moon).\n\nThis correction is known as the phase factor, $p(\\theta)$, such that the total power reflected is related to the total power incident as $P_{\\text {ref }}=P_{\\text {inc }} \\times p(\\theta)$. You are given that\n\n$$\np(\\theta)=B\\left[\\left(1-\\frac{\\theta}{\\pi}\\right) \\cos \\theta+\\frac{1}{\\pi} \\sin \\theta\\right]\n$$\n\nwhere $B$ is a constant and $\\theta$ is the phase angle, as defined in Fig 5, and is measured in radians (a measure of angle such that $2 \\pi$ radians $=360^{\\circ}$.\n\n[figure2]\n\nFigure 5: The phase angle is defined as the angle between the Earth and Sun as viewed from the object. Credit: Wikipedia.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c3e3c992a9c51eb3e471g-10.jpg?height=859&width=1285&top_left_y=593&top_left_x=385", "https://cdn.mathpix.com/cropped/2024_03_06_c3e3c992a9c51eb3e471g-11.jpg?height=639&width=868&top_left_y=203&top_left_x=594" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_199", "problem": "为“照亮”“嫦娥四号”“驾临”月球背面之路, 一颗承载地月中转通信任务的中继卫星将在“嫦娥四号”发射前半年进入到地月拉格朗日点 $L_{2}$, 如图。在该点, 地球、月球和中继卫星始终位于同一直线上, 且中继卫星绕地球做圆周运动的周期与月球绕地球做圆周运动的周期相同，则（）\n\n[图1]\nA: 中继卫星绕地球做圆周运动的周期为一年\nB: 中继卫星做圆周运动的向心力仅由地球提供\nC: 中继卫星的线速度小于月球运动的线速度\nD: 中继卫星的向心加速度大于月球运动的向心加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n为“照亮”“嫦娥四号”“驾临”月球背面之路, 一颗承载地月中转通信任务的中继卫星将在“嫦娥四号”发射前半年进入到地月拉格朗日点 $L_{2}$, 如图。在该点, 地球、月球和中继卫星始终位于同一直线上, 且中继卫星绕地球做圆周运动的周期与月球绕地球做圆周运动的周期相同，则（）\n\n[图1]\n\nA: 中继卫星绕地球做圆周运动的周期为一年\nB: 中继卫星做圆周运动的向心力仅由地球提供\nC: 中继卫星的线速度小于月球运动的线速度\nD: 中继卫星的向心加速度大于月球运动的向心加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-19.jpg?height=380&width=465&top_left_y=2237&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_695", "problem": "2019 年 3 月 10 日, 长征三号乙运载火箭将“中星 $6 \\mathrm{C}$ ”通信卫星（记为卫星I）送入地\n球同步轨道上, 主要为我国、东南亚、澳洲和南太平洋岛国等地区提供通信与广播业务。在同平面内的圆轨道上有一颗中轨道卫星II, 它运动的每个周期内都有一段时间 $t(t$ 未知）无法直接接收到卫星I发出的电磁波信号，因为其轨道上总有一段区域没有被卫星I 发出的电磁波信号覆盖到, 这段区域对应的圆心角为 $2 \\alpha$ 。已知卫星I对地球的张角为 $2 \\beta$,地球自转周期为 $T_{0}$, 万有引力常量为 $G$, 则根据题中条件, 可求出 ( )\n\n[图1]\nA: 地球的平均密度为 $\\frac{3 \\pi}{G T_{0}{ }^{2}}$\nB: 卫星I、II的角速度之比为 $\\frac{\\sin \\beta}{\\sin (\\alpha-\\beta)}$\nC: 卫星II的周期为 $\\sqrt{\\frac{\\sin ^{3} \\beta}{\\sin ^{3}(\\alpha-\\beta)}} \\cdot T_{0}$\nD: 题中时间 $t$ 为 $\\sqrt{\\frac{\\sin ^{3} \\beta}{\\sin ^{3}(\\alpha-\\beta)}} \\cdot \\frac{\\alpha}{\\pi} T_{0}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2019 年 3 月 10 日, 长征三号乙运载火箭将“中星 $6 \\mathrm{C}$ ”通信卫星（记为卫星I）送入地\n球同步轨道上, 主要为我国、东南亚、澳洲和南太平洋岛国等地区提供通信与广播业务。在同平面内的圆轨道上有一颗中轨道卫星II, 它运动的每个周期内都有一段时间 $t(t$ 未知）无法直接接收到卫星I发出的电磁波信号，因为其轨道上总有一段区域没有被卫星I 发出的电磁波信号覆盖到, 这段区域对应的圆心角为 $2 \\alpha$ 。已知卫星I对地球的张角为 $2 \\beta$,地球自转周期为 $T_{0}$, 万有引力常量为 $G$, 则根据题中条件, 可求出 ( )\n\n[图1]\n\nA: 地球的平均密度为 $\\frac{3 \\pi}{G T_{0}{ }^{2}}$\nB: 卫星I、II的角速度之比为 $\\frac{\\sin \\beta}{\\sin (\\alpha-\\beta)}$\nC: 卫星II的周期为 $\\sqrt{\\frac{\\sin ^{3} \\beta}{\\sin ^{3}(\\alpha-\\beta)}} \\cdot T_{0}$\nD: 题中时间 $t$ 为 $\\sqrt{\\frac{\\sin ^{3} \\beta}{\\sin ^{3}(\\alpha-\\beta)}} \\cdot \\frac{\\alpha}{\\pi} T_{0}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-002.jpg?height=448&width=573&top_left_y=610&top_left_x=336", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-002.jpg?height=400&width=506&top_left_y=2310&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_383", "problem": "2018 年 12 月 8 日, 嫦娥四号发射升空。将实现人类历史上首次月球背面登月。随着嫦娥奔月梦想的实现，我国不断刷新深空探测的中国高度。嫦娥卫星整个飞行过程可分为三个轨道段: 绕地飞行调相轨道段、地月转移轨道段、绕月飞行轨道段我们用如图所示的模型来简化描绘嫦娥卫星飞行过程, 假设调相轨道和绕月轨道的半长轴分别为 $a 、 b$, 公转周期分别为 $T_{1} 、 T_{2}$ 。关于嫦娥卫星的飞行过程, 下列说法正确的是（）\n\n[图1]\nA: $\\frac{a^{3}}{T_{1}^{2}}=\\frac{b^{3}}{T_{2}^{2}}$\nB: 嫦娥卫星在地月转移轨道上运行的速度应大于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nC: 从调相轨道切入到地月转移轨道时, 卫星在 $P$ 点必须减速\nD: 从地月转移轨道切入到绕月轨道时, 卫星在 $Q$ 点必须减速\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2018 年 12 月 8 日, 嫦娥四号发射升空。将实现人类历史上首次月球背面登月。随着嫦娥奔月梦想的实现，我国不断刷新深空探测的中国高度。嫦娥卫星整个飞行过程可分为三个轨道段: 绕地飞行调相轨道段、地月转移轨道段、绕月飞行轨道段我们用如图所示的模型来简化描绘嫦娥卫星飞行过程, 假设调相轨道和绕月轨道的半长轴分别为 $a 、 b$, 公转周期分别为 $T_{1} 、 T_{2}$ 。关于嫦娥卫星的飞行过程, 下列说法正确的是（）\n\n[图1]\n\nA: $\\frac{a^{3}}{T_{1}^{2}}=\\frac{b^{3}}{T_{2}^{2}}$\nB: 嫦娥卫星在地月转移轨道上运行的速度应大于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nC: 从调相轨道切入到地月转移轨道时, 卫星在 $P$ 点必须减速\nD: 从地月转移轨道切入到绕月轨道时, 卫星在 $Q$ 点必须减速\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-20.jpg?height=277&width=400&top_left_y=1615&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_883", "problem": "What is the perihelion of comet \"SMukherjee2017\"?\nA: 3.0 $\\mathrm{AU}$\nB: 3.5 AU\nC: 4.0 AU\nD: $4.5 \\mathrm{AU}$\nE: $5.0 \\mathrm{AU}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the perihelion of comet \"SMukherjee2017\"?\n\nA: 3.0 $\\mathrm{AU}$\nB: 3.5 AU\nC: 4.0 AU\nD: $4.5 \\mathrm{AU}$\nE: $5.0 \\mathrm{AU}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_531", "problem": "宇宙中两颗相距较近的天体称为“双星”, 它们以两者连线上的某一点为圆心做匀速圆周运动, 而不至于因万有引力的作用吸引到一起. 设两者的质量分别为 $m_{1}$ 和 $m_{2}$ 且 $m_{1}>m_{2}$则下列说法正确的是( )\nA: 两天体做圆周运动的周期相等\nB: 两天体做圆周运动的向心加速度大小相等\nC: $m_{1}$ 的轨道半径大于 $m_{2}$ 的轨道半径\nD: $m_{2}$ 的轨道半径大于 $m_{l}$ 的轨道半径\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇宙中两颗相距较近的天体称为“双星”, 它们以两者连线上的某一点为圆心做匀速圆周运动, 而不至于因万有引力的作用吸引到一起. 设两者的质量分别为 $m_{1}$ 和 $m_{2}$ 且 $m_{1}>m_{2}$则下列说法正确的是( )\n\nA: 两天体做圆周运动的周期相等\nB: 两天体做圆周运动的向心加速度大小相等\nC: $m_{1}$ 的轨道半径大于 $m_{2}$ 的轨道半径\nD: $m_{2}$ 的轨道半径大于 $m_{l}$ 的轨道半径\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_649", "problem": "假设在半径为 $R$ 的某天体上发射一颗该天体的卫星, 已知引力常量为 $G$, 忽略该天体自转。\n\n若卫星贴近该天体的表面做匀速圆周运动的周期为 $T_{2}$, 则该天体的密度是多少?", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n假设在半径为 $R$ 的某天体上发射一颗该天体的卫星, 已知引力常量为 $G$, 忽略该天体自转。\n\n若卫星贴近该天体的表面做匀速圆周运动的周期为 $T_{2}$, 则该天体的密度是多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_223", "problem": "地球赤道表面上的一物体质量为 $m_{1}$, 它相对地心的速度为 $v_{1}$, 地球同步卫星离地面的高度为 $h$, 它相对地心的速度为 $v_{2}$, 其质量为 $m_{2}$ 。已知地球的质量为 $M$, 半径为\n$R$, 自转角速度为 $\\omega$, 表面的重力加速度为 $g$, 地球的第一宇宙速度为 $v$, 万有引力常量为 $G$ 。下列各式成立的是 $(\\quad)$\nA: $v_{1}$ 小于 $v$\nB: $\\frac{v_{1}}{R}=\\frac{v_{2}}{R+h}$\nC: $m_{1} g=\\frac{m_{1} v_{1}^{2}}{R}$\nD: $\\frac{v}{v_{2}}=\\sqrt{\\frac{R+h}{R}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n地球赤道表面上的一物体质量为 $m_{1}$, 它相对地心的速度为 $v_{1}$, 地球同步卫星离地面的高度为 $h$, 它相对地心的速度为 $v_{2}$, 其质量为 $m_{2}$ 。已知地球的质量为 $M$, 半径为\n$R$, 自转角速度为 $\\omega$, 表面的重力加速度为 $g$, 地球的第一宇宙速度为 $v$, 万有引力常量为 $G$ 。下列各式成立的是 $(\\quad)$\n\nA: $v_{1}$ 小于 $v$\nB: $\\frac{v_{1}}{R}=\\frac{v_{2}}{R+h}$\nC: $m_{1} g=\\frac{m_{1} v_{1}^{2}}{R}$\nD: $\\frac{v}{v_{2}}=\\sqrt{\\frac{R+h}{R}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_581", "problem": "在星球 $\\mathrm{A}$ 上将一小物块 $P$ 坚直向上抛出, $P$ 的速度的二次方 $v^{2}$ 与位移 $x$ 间的关系如图中实线所示; 在另一星球 $B$ 上用另一小物块 $Q$ 完成同样的过程, $Q$ 的 $v^{2}-x$ 关系如图中虚线所示. 已知 $\\mathrm{A}$ 的半径是 $B$ 的半径的 $\\frac{1}{3}$, 若两星球均为质量均匀分布的球体 (球的体积公式为 $V=\\frac{4}{3} \\pi r^{3}, r$ 为球的半径), 两星球上均没有空气, 不考虑两星球的自转,则 ( )\n\n[图1]\nA: A 表面的重力加速度是 $B$ 表面的重力加速度的 9 倍\nB: $P$ 抛出后落回原处的时间是 $Q$ 抛出后落回原处的时间的 $\\frac{1}{9}$\nC: A 的密度是 $B$ 的密度的 9 倍\nD: A 的第一宇宙速度是 $B$ 的第一宇宙速度的 $\\sqrt{3}$ 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n在星球 $\\mathrm{A}$ 上将一小物块 $P$ 坚直向上抛出, $P$ 的速度的二次方 $v^{2}$ 与位移 $x$ 间的关系如图中实线所示; 在另一星球 $B$ 上用另一小物块 $Q$ 完成同样的过程, $Q$ 的 $v^{2}-x$ 关系如图中虚线所示. 已知 $\\mathrm{A}$ 的半径是 $B$ 的半径的 $\\frac{1}{3}$, 若两星球均为质量均匀分布的球体 (球的体积公式为 $V=\\frac{4}{3} \\pi r^{3}, r$ 为球的半径), 两星球上均没有空气, 不考虑两星球的自转,则 ( )\n\n[图1]\n\nA: A 表面的重力加速度是 $B$ 表面的重力加速度的 9 倍\nB: $P$ 抛出后落回原处的时间是 $Q$ 抛出后落回原处的时间的 $\\frac{1}{9}$\nC: A 的密度是 $B$ 的密度的 9 倍\nD: A 的第一宇宙速度是 $B$ 的第一宇宙速度的 $\\sqrt{3}$ 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-114.jpg?height=380&width=582&top_left_y=1101&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_638", "problem": "人造地球卫星是发射数量最多, 用途最广, 发展最快的航天器。已知引力常量为 $G$, 地球半径为 $R$, 地球表面的重力加速度为 $\\mathrm{g}$, 地球自转的周期为 $T$ 。\n求地球同步卫星离地高度 $h_{l}$", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n人造地球卫星是发射数量最多, 用途最广, 发展最快的航天器。已知引力常量为 $G$, 地球半径为 $R$, 地球表面的重力加速度为 $\\mathrm{g}$, 地球自转的周期为 $T$ 。\n求地球同步卫星离地高度 $h_{l}$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1066", "problem": "On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\\mathrm{x}, \\mathrm{y}, \\mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\\circ}$, and each orbital plane has 4 satellites.\n[figure1]\n\nFigure 1: The current set up of the GPS system used on Earth.\n\nCredits: Left: Peter H. Dana, University of Colorado;\n\nRight: GPS Standard Positioning Service Specification, $4^{\\text {th }}$ edition\n\nThe orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\\circ}$, and hence about $38 \\%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required.\n\nIn the future we hope to colonise Mars, and so for navigation purposes it is likely that a type of GPS system will eventually be established on Mars too. Mars has a mass of $6.42 \\times 10^{23} \\mathrm{~kg}$, a mean radius of $3390 \\mathrm{~km}$, a sidereal day of $24 \\mathrm{~h} 37 \\mathrm{mins}$, and two (low mass) moons with essentially circular orbits and semi-major axes of $9377 \\mathrm{~km}$ (Phobos) and $23460 \\mathrm{~km}$ (Deimos).c. Using suitable calculations, explore the viability of a 24-satellite GPS constellation similar to the one used on Earth, in a semi-synchronous Martian orbit, by considering:\n\ni. Would the moons prevent such an orbit?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\\mathrm{x}, \\mathrm{y}, \\mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\\circ}$, and each orbital plane has 4 satellites.\n[figure1]\n\nFigure 1: The current set up of the GPS system used on Earth.\n\nCredits: Left: Peter H. Dana, University of Colorado;\n\nRight: GPS Standard Positioning Service Specification, $4^{\\text {th }}$ edition\n\nThe orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\\circ}$, and hence about $38 \\%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required.\n\nIn the future we hope to colonise Mars, and so for navigation purposes it is likely that a type of GPS system will eventually be established on Mars too. Mars has a mass of $6.42 \\times 10^{23} \\mathrm{~kg}$, a mean radius of $3390 \\mathrm{~km}$, a sidereal day of $24 \\mathrm{~h} 37 \\mathrm{mins}$, and two (low mass) moons with essentially circular orbits and semi-major axes of $9377 \\mathrm{~km}$ (Phobos) and $23460 \\mathrm{~km}$ (Deimos).\n\nproblem:\nc. Using suitable calculations, explore the viability of a 24-satellite GPS constellation similar to the one used on Earth, in a semi-synchronous Martian orbit, by considering:\n\ni. Would the moons prevent such an orbit?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~m}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-04.jpg?height=512&width=1474&top_left_y=555&top_left_x=292" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ " \\mathrm{~m}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1194", "problem": "In the heart of every star, nuclear fusion is taking place. For most stars that involves hydrogen being turned into helium, a process that starts by bringing two protons close enough that the strong nuclear force can act upon them. The smallest stars are the ones that have a core that is only just hot enough for fusion to occur, whilst in the biggest ones the radiation pressure of the photons given out by the fusion reaction pushing on the stellar material can overcome the gravitational forces holding it together.\n[figure1]\n\nFigure 5: Left: The lowest mass star we know of, EBLM J0555-57Ab, was found by von Boetticher et al. (2017) and is about the size of Saturn with a mass of $0.081 M_{\\odot}$. Credit: Amanda Smith, University of Cambridge. Right: The highest mass star we know of, R136a1, is in the centre of the clump of stars on the right of this HST image of the Tarantula Nebula. Schneider et al. (2014) suggest it has a mass of $315 M_{\\odot}$, which is above what stellar evolution models allow. Despite its large mass, other stars have far bigger radii. Credit: NASA \\& ESA.\n\nFor a spherical main sequence star made of a plasma (a fully ionized gas of electrons and nuclei) that is acting like an ideal gas, the temperature at the core can be approximately calculated as\n\n$$\nT_{\\mathrm{int}} \\simeq \\frac{G M \\bar{\\mu}}{k_{\\mathrm{B}} R} \\quad \\text { where } \\quad \\bar{\\mu}=\\frac{m_{\\mathrm{p}}}{2 X+3 Y / 4+Z / 2} .\n$$\n\nIn this equation, $M$ is the mass of the star, $R$ is its radius, $k_{\\mathrm{B}}$ is the Boltzmann constant, and $\\bar{\\mu}$ is the mean mass of the plasma particles (i.e nuclei and electrons) with $m_{\\mathrm{p}}$ the mass of a proton.\n\nClassically, the core of the Sun is not hot enough for fusion, and yet fusion is clearly happening. The key is that it is a fundamentally quantum process, and so protons are able to 'quantum tunnel' through the Coloumb barrier (see Figure 6), allowing fusion to occur at lower temperatures. In quantum mechanics, fusion will happen when $b=\\lambda$ where $\\lambda$ is the de Broglie wavelength of the proton, related to the momentum of the proton by $\\lambda=h / p$.\n\n[figure2]\n\nFigure 6: A diagram showing the way a particle can pass through a classically impenetrable potential barrier due to its wave-like properties on the quantum scale.\n\nCredit: Brooks/Cole - Thomson Learning.\n\nIn the smallest stars, electron degeneracy prevents them from compressing in radius and thus stops the core reaching $T_{\\text {int }} \\gtrsim T_{\\text {quantum }}$. At the limit of electron degeneracy, the number density of electrons $n_{\\mathrm{e}}=1 / \\lambda_{\\mathrm{e}}^{3}$ where $\\lambda_{\\mathrm{e}}$ is the de Broglie wavelength of the electrons.\n\nIn the largest stars, radiation pressure pushes on the outer layers of the star stronger than gravity pulls them in. The brightest luminosity for a star is known as the Eddington luminosity, $L_{\\text {Edd }}$. The acceleration due to radiation pressure can be calculated as\n\n$$\ng_{\\mathrm{rad}}=\\frac{\\kappa_{\\mathrm{e}} I}{c} \\quad \\text { where } \\quad \\kappa_{\\mathrm{e}}=\\frac{\\sigma_{\\mathrm{T}}}{2 m_{\\mathrm{p}}}(1+X)\n$$\n\nand $\\kappa_{\\mathrm{e}}$ is the electron opacity of the stellar material, $\\sigma_{\\mathrm{T}}$ is the Thomson scattering cross-section for electrons $\\left(=66.5 \\mathrm{fm}^{2}\\right.$ ), $X$ is the hydrogen fraction, and $I$ is the intensity of radiation (in $\\mathrm{W} \\mathrm{m}^{-2}$ ). Assuming main-sequence stars follow a mass-luminosity relation of $L \\propto M^{3}$, the maximum mass of a star can be found by considering one that is radiating at $L_{\\text {Edd }}$.a. Given the Sun's composition has hydrogen fraction, $X=0.72$, helium fraction $Y=0.26$ and 'metals' (i.e. any element lithium and heavier) fraction $Z=0.02$, estimate the temperature at the centre of the Sun.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn the heart of every star, nuclear fusion is taking place. For most stars that involves hydrogen being turned into helium, a process that starts by bringing two protons close enough that the strong nuclear force can act upon them. The smallest stars are the ones that have a core that is only just hot enough for fusion to occur, whilst in the biggest ones the radiation pressure of the photons given out by the fusion reaction pushing on the stellar material can overcome the gravitational forces holding it together.\n[figure1]\n\nFigure 5: Left: The lowest mass star we know of, EBLM J0555-57Ab, was found by von Boetticher et al. (2017) and is about the size of Saturn with a mass of $0.081 M_{\\odot}$. Credit: Amanda Smith, University of Cambridge. Right: The highest mass star we know of, R136a1, is in the centre of the clump of stars on the right of this HST image of the Tarantula Nebula. Schneider et al. (2014) suggest it has a mass of $315 M_{\\odot}$, which is above what stellar evolution models allow. Despite its large mass, other stars have far bigger radii. Credit: NASA \\& ESA.\n\nFor a spherical main sequence star made of a plasma (a fully ionized gas of electrons and nuclei) that is acting like an ideal gas, the temperature at the core can be approximately calculated as\n\n$$\nT_{\\mathrm{int}} \\simeq \\frac{G M \\bar{\\mu}}{k_{\\mathrm{B}} R} \\quad \\text { where } \\quad \\bar{\\mu}=\\frac{m_{\\mathrm{p}}}{2 X+3 Y / 4+Z / 2} .\n$$\n\nIn this equation, $M$ is the mass of the star, $R$ is its radius, $k_{\\mathrm{B}}$ is the Boltzmann constant, and $\\bar{\\mu}$ is the mean mass of the plasma particles (i.e nuclei and electrons) with $m_{\\mathrm{p}}$ the mass of a proton.\n\nClassically, the core of the Sun is not hot enough for fusion, and yet fusion is clearly happening. The key is that it is a fundamentally quantum process, and so protons are able to 'quantum tunnel' through the Coloumb barrier (see Figure 6), allowing fusion to occur at lower temperatures. In quantum mechanics, fusion will happen when $b=\\lambda$ where $\\lambda$ is the de Broglie wavelength of the proton, related to the momentum of the proton by $\\lambda=h / p$.\n\n[figure2]\n\nFigure 6: A diagram showing the way a particle can pass through a classically impenetrable potential barrier due to its wave-like properties on the quantum scale.\n\nCredit: Brooks/Cole - Thomson Learning.\n\nIn the smallest stars, electron degeneracy prevents them from compressing in radius and thus stops the core reaching $T_{\\text {int }} \\gtrsim T_{\\text {quantum }}$. At the limit of electron degeneracy, the number density of electrons $n_{\\mathrm{e}}=1 / \\lambda_{\\mathrm{e}}^{3}$ where $\\lambda_{\\mathrm{e}}$ is the de Broglie wavelength of the electrons.\n\nIn the largest stars, radiation pressure pushes on the outer layers of the star stronger than gravity pulls them in. The brightest luminosity for a star is known as the Eddington luminosity, $L_{\\text {Edd }}$. The acceleration due to radiation pressure can be calculated as\n\n$$\ng_{\\mathrm{rad}}=\\frac{\\kappa_{\\mathrm{e}} I}{c} \\quad \\text { where } \\quad \\kappa_{\\mathrm{e}}=\\frac{\\sigma_{\\mathrm{T}}}{2 m_{\\mathrm{p}}}(1+X)\n$$\n\nand $\\kappa_{\\mathrm{e}}$ is the electron opacity of the stellar material, $\\sigma_{\\mathrm{T}}$ is the Thomson scattering cross-section for electrons $\\left(=66.5 \\mathrm{fm}^{2}\\right.$ ), $X$ is the hydrogen fraction, and $I$ is the intensity of radiation (in $\\mathrm{W} \\mathrm{m}^{-2}$ ). Assuming main-sequence stars follow a mass-luminosity relation of $L \\propto M^{3}$, the maximum mass of a star can be found by considering one that is radiating at $L_{\\text {Edd }}$.\n\nproblem:\na. Given the Sun's composition has hydrogen fraction, $X=0.72$, helium fraction $Y=0.26$ and 'metals' (i.e. any element lithium and heavier) fraction $Z=0.02$, estimate the temperature at the centre of the Sun.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~K}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-08.jpg?height=712&width=1508&top_left_y=546&top_left_x=271", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-09.jpg?height=514&width=1010&top_left_y=186&top_left_x=523" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~K}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_307", "problem": "如图, 赤道上空有 2 颗人造卫星 $\\mathrm{A} 、 \\mathrm{~B}$ 绕地球做同方向的匀速圆周运动, 地球半径为 $R$, 卫星 $\\mathrm{A} 、 \\mathrm{~B}$ 的轨道半径分别为 $\\frac{5}{4} R 、 \\frac{5}{3} R$, 卫星 $\\mathrm{B}$ 的运动周期为 $T$, 某时刻 2 颗卫星与地心在同一直线上, 两颗卫星之间保持用光信号直接通信, 则（）\n\n[图1]\nA: 卫星 $\\mathrm{A}$ 的加速度小于 $\\mathrm{B}$ 的加速度\nB: 卫星 $A 、 B$ 的周期之比为 $\\frac{3 \\sqrt{3}}{8}$\nC: 再经时间 $t=\\frac{3(8 \\sqrt{3}+9) T}{148}$, 两颗卫星之间的通信将中断\nD: 为了使赤道上任一点任一时刻均能接收到卫星 A 所在轨道的卫星的信号, 该轨道至少需要 3 颗卫星\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图, 赤道上空有 2 颗人造卫星 $\\mathrm{A} 、 \\mathrm{~B}$ 绕地球做同方向的匀速圆周运动, 地球半径为 $R$, 卫星 $\\mathrm{A} 、 \\mathrm{~B}$ 的轨道半径分别为 $\\frac{5}{4} R 、 \\frac{5}{3} R$, 卫星 $\\mathrm{B}$ 的运动周期为 $T$, 某时刻 2 颗卫星与地心在同一直线上, 两颗卫星之间保持用光信号直接通信, 则（）\n\n[图1]\n\nA: 卫星 $\\mathrm{A}$ 的加速度小于 $\\mathrm{B}$ 的加速度\nB: 卫星 $A 、 B$ 的周期之比为 $\\frac{3 \\sqrt{3}}{8}$\nC: 再经时间 $t=\\frac{3(8 \\sqrt{3}+9) T}{148}$, 两颗卫星之间的通信将中断\nD: 为了使赤道上任一点任一时刻均能接收到卫星 A 所在轨道的卫星的信号, 该轨道至少需要 3 颗卫星\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-64.jpg?height=277&width=302&top_left_y=2346&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_87", "problem": "设想从地球赤道平面内架设一垂直于地面延伸到太空的电梯，电梯的箱体可以将人从地面运送到地球同步轨道的空间站。已知地球表面两极处的重力加速度为 $g$, 地球自转周期为 $T$, 地球半径为 $R$, 万有引力常量为 $G$ 。求同步轨道空间站距地面的高度 $h$;", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n设想从地球赤道平面内架设一垂直于地面延伸到太空的电梯，电梯的箱体可以将人从地面运送到地球同步轨道的空间站。已知地球表面两极处的重力加速度为 $g$, 地球自转周期为 $T$, 地球半径为 $R$, 万有引力常量为 $G$ 。求同步轨道空间站距地面的高度 $h$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_412", "problem": "“重力探矿” 是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用,\n\n其原理可简述如下: 如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向，当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向 (即 $P O$ 方向 $)$ 上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金\n矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常数为 $G$ 。若在水平地面上以 $P$ 点为圆心、半径为 $L$ 的范围内发现: 重力加速度反常值在 $\\delta$与 $k \\delta(k>1)$ 之间变化, 且重力加速度反常的最大值出现在 $P$ 点, 如果这种反常是由于地下存在某一球形区域黄金矿石造成的, 试求此球形区域球心的深度和球形区域的体积。\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n“重力探矿” 是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用,\n\n其原理可简述如下: 如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向，当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向 (即 $P O$ 方向 $)$ 上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金\n矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常数为 $G$ 。若在水平地面上以 $P$ 点为圆心、半径为 $L$ 的范围内发现: 重力加速度反常值在 $\\delta$与 $k \\delta(k>1)$ 之间变化, 且重力加速度反常的最大值出现在 $P$ 点, 如果这种反常是由于地下存在某一球形区域黄金矿石造成的, 试求此球形区域球心的深度和球形区域的体积。\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[此球形区域球心的深度, 此球形区域球心的体积]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-014.jpg?height=417&width=560&top_left_y=728&top_left_x=357" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "此球形区域球心的深度", "此球形区域球心的体积" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_725", "problem": "2023 年春节期间, 中国科幻电影《流浪地球 2 》热播。假设地球逃离太阳系过程如图所示, 地球现在绕太阳在圆轨道I上运行, 运动到 $A$ 点加速变轨进入椭圆轨道II, 在粗圆轨道II上运动到远日点 $B$ 时再次加速变轨, 从而摆脱太阳的束缚, 则地球 $(\\quad)$\n\n[图1]\nA: 在轨道I通过 $A$ 点的速度小于在轨道II通过 $A$ 点的速度\nB: 沿轨道II运行时, 在 $A$ 点的加速度大于在 $B$ 点的加速度\nC: 沿轨道II运行时, 由 $A$ 点运动到 $B$ 点的过程中, 动能逐渐增大\nD: 沿轨道I和轨道II运行时, 相同时间内地球与太阳连线扫过的面积相等\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2023 年春节期间, 中国科幻电影《流浪地球 2 》热播。假设地球逃离太阳系过程如图所示, 地球现在绕太阳在圆轨道I上运行, 运动到 $A$ 点加速变轨进入椭圆轨道II, 在粗圆轨道II上运动到远日点 $B$ 时再次加速变轨, 从而摆脱太阳的束缚, 则地球 $(\\quad)$\n\n[图1]\n\nA: 在轨道I通过 $A$ 点的速度小于在轨道II通过 $A$ 点的速度\nB: 沿轨道II运行时, 在 $A$ 点的加速度大于在 $B$ 点的加速度\nC: 沿轨道II运行时, 由 $A$ 点运动到 $B$ 点的过程中, 动能逐渐增大\nD: 沿轨道I和轨道II运行时, 相同时间内地球与太阳连线扫过的面积相等\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-031.jpg?height=343&width=454&top_left_y=1365&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_628", "problem": "质量为 $m$ 的宇宙飞船, 在离月球地面高度 $h$ 处沿圆形轨道绕月球运行。为使飞船到达月球表面 $B$ 点, 喷气发动机在 $A$ 点做一次极短时间的喷气。从喷口射出的气流方向与圆周轨道相切且相对飞船的速度为 $u$, 月球半径为 $R, h=\\frac{R}{16}, A 、 B$ 两点与球心在一直线上, 其速度与飞船到球心的距离成反比。月球表面重力加速度为 $g$ 。若以无穷远处为飞船引力势能的零势能点, 飞船和球心距离为 $r$ 时, 引力势能的表达式为 $E_{\\mathrm{P}}=-\\frac{G M m}{r}$ 。 ( $M$ 是月球质量, 本题中未知)。求:\n\n 飞船在圆形轨道上运行的速度大小 $v_{A}$;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n质量为 $m$ 的宇宙飞船, 在离月球地面高度 $h$ 处沿圆形轨道绕月球运行。为使飞船到达月球表面 $B$ 点, 喷气发动机在 $A$ 点做一次极短时间的喷气。从喷口射出的气流方向与圆周轨道相切且相对飞船的速度为 $u$, 月球半径为 $R, h=\\frac{R}{16}, A 、 B$ 两点与球心在一直线上, 其速度与飞船到球心的距离成反比。月球表面重力加速度为 $g$ 。若以无穷远处为飞船引力势能的零势能点, 飞船和球心距离为 $r$ 时, 引力势能的表达式为 $E_{\\mathrm{P}}=-\\frac{G M m}{r}$ 。 ( $M$ 是月球质量, 本题中未知)。求:\n\n 飞船在圆形轨道上运行的速度大小 $v_{A}$;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-076.jpg?height=271&width=300&top_left_y=1235&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_203", "problem": "预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\\left(r_{2}>r_{1}\\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求:\n\n若已知空间站绕地球做匀速圆周运动的周期为 $T$, 地球的半径为 $R$ ，地球表面的重力加速度为 $g$, 则该空间站距离地球表面的高度 $h$ 为多少? (用所给物理量符号表示)\n[图1]\n\n图甲\n\n[图2]\n\n图乙", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\\left(r_{2}>r_{1}\\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求:\n\n若已知空间站绕地球做匀速圆周运动的周期为 $T$, 地球的半径为 $R$ ，地球表面的重力加速度为 $g$, 则该空间站距离地球表面的高度 $h$ 为多少? (用所给物理量符号表示)\n[图1]\n\n图甲\n\n[图2]\n\n图乙\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-134.jpg?height=369&width=388&top_left_y=1780&top_left_x=340", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-134.jpg?height=294&width=505&top_left_y=1869&top_left_x=867" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_639", "problem": "如图, 三个质点 $a 、 b 、 c$ 质量分别为 $m_{1} 、 m_{2} 、 M\\left(M \\gg m_{1}, M \\gg m_{2}\\right)$ 。在 $c$ 的万有引力作用下, $a 、 b$ 在同一平面内绕 $c$ 沿逆时针方向做匀速圆周运动, 它们的周期之比 $T_{a}$ : $T_{b}=1: 8, a 、 b$ 的轨道半径分别为 $r_{a}$ 和 $r_{b}$ 。从图示位置开始, 在 $b$ 运动一周的过程中,则 ( )\n\n[图1]\nA: $r_{a}: r_{b}=1: 4$\nB: B. $a 、 b$ 距离最近的次数为 8 次\nC: $a 、 b$ 距离最远的次数为 9 次\nD: $a 、 b 、 c$ 共线的次数为 16 次\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图, 三个质点 $a 、 b 、 c$ 质量分别为 $m_{1} 、 m_{2} 、 M\\left(M \\gg m_{1}, M \\gg m_{2}\\right)$ 。在 $c$ 的万有引力作用下, $a 、 b$ 在同一平面内绕 $c$ 沿逆时针方向做匀速圆周运动, 它们的周期之比 $T_{a}$ : $T_{b}=1: 8, a 、 b$ 的轨道半径分别为 $r_{a}$ 和 $r_{b}$ 。从图示位置开始, 在 $b$ 运动一周的过程中,则 ( )\n\n[图1]\n\nA: $r_{a}: r_{b}=1: 4$\nB: B. $a 、 b$ 距离最近的次数为 8 次\nC: $a 、 b$ 距离最远的次数为 9 次\nD: $a 、 b 、 c$ 共线的次数为 16 次\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-078.jpg?height=434&width=460&top_left_y=1505&top_left_x=358" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_910", "problem": "Given the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\\gtrsim 80 \\%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet.\n[figure1]\n\nFigure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\\sim 100 \\%$ iron) might look like.\n\nThe Roche limiting distance, $a_{\\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is\n\n$$\na_{\\min }=2.44 R_{\\star}\\left(\\frac{\\rho_{\\star}}{\\rho_{p}}\\right)^{1 / 3}\n$$\n\nwhere $R_{\\star}$ is the radius of the star, $\\rho_{\\star}$ is the density of the star and $\\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as\n\n$$\n\\log _{10}\\left(\\frac{R}{R_{E}}\\right)=0.295 \\log _{10}\\left(\\frac{M}{M_{E}}\\right)+\\alpha\n$$\n\nwhere $\\alpha=0.0286$ in the pure silicate case and $\\alpha=-0.1090$ in the pure iron case.\n\nDerive a formula for the period of an exoplanet orbiting at the Roche limit, and hence show it is dependent only on the variable $\\rho_{p}$ (i.e. independent of the properties of the parent star).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nGiven the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\\gtrsim 80 \\%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet.\n[figure1]\n\nFigure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\\sim 100 \\%$ iron) might look like.\n\nThe Roche limiting distance, $a_{\\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is\n\n$$\na_{\\min }=2.44 R_{\\star}\\left(\\frac{\\rho_{\\star}}{\\rho_{p}}\\right)^{1 / 3}\n$$\n\nwhere $R_{\\star}$ is the radius of the star, $\\rho_{\\star}$ is the density of the star and $\\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as\n\n$$\n\\log _{10}\\left(\\frac{R}{R_{E}}\\right)=0.295 \\log _{10}\\left(\\frac{M}{M_{E}}\\right)+\\alpha\n$$\n\nwhere $\\alpha=0.0286$ in the pure silicate case and $\\alpha=-0.1090$ in the pure iron case.\n\nDerive a formula for the period of an exoplanet orbiting at the Roche limit, and hence show it is dependent only on the variable $\\rho_{p}$ (i.e. independent of the properties of the parent star).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-09.jpg?height=620&width=1468&top_left_y=861&top_left_x=292" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1160", "problem": "In science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System.\n[figure1]\n\nFigure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right).\n\nThis question will look at the real distances between asteroids.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System.\n[figure1]\n\nFigure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right).\n\nThis question will look at the real distances between asteroids.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~W}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f87d81e0622ba23867ceg-4.jpg?height=618&width=1260&top_left_y=584&top_left_x=388" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ " \\mathrm{~W}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_501", "problem": "如图所示为人类历史上第一张黑洞照片。黑洞是一种密度极大、引力极大的天体,\n\n以至于光都无法逃逸, 科学家一般通过观测绕黑洞运行的天体的运动规律间接研究黑洞。已知某黑洞的逃逸速度为 $v=\\sqrt{\\frac{2 G M}{R}}$, 其中引力常量为 $G, M$ 是该黑洞的质量, $R$ 是该黑洞的半径。若天文学家观测到与该黑洞相距为 $r$ 的天体以周期 $T$ 绕该黑洞做匀速圆周运动，则下列关于该黑洞的说法正确的是（）\n\n[图1]\nA: 该黑洞的质量为 $\\frac{G T^{2}}{4 \\pi r^{3}}$\nB: 该黑洞的质量为 $\\frac{4 \\pi r^{3}}{G T^{2}}$\nC: 该黑洞的最大半径为 $\\frac{4 \\pi^{2} r^{3}}{c^{2}}$\nD: 该黑洞的最大半径为 $\\frac{8 \\pi^{2} r^{3}}{c^{2} T^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示为人类历史上第一张黑洞照片。黑洞是一种密度极大、引力极大的天体,\n\n以至于光都无法逃逸, 科学家一般通过观测绕黑洞运行的天体的运动规律间接研究黑洞。已知某黑洞的逃逸速度为 $v=\\sqrt{\\frac{2 G M}{R}}$, 其中引力常量为 $G, M$ 是该黑洞的质量, $R$ 是该黑洞的半径。若天文学家观测到与该黑洞相距为 $r$ 的天体以周期 $T$ 绕该黑洞做匀速圆周运动，则下列关于该黑洞的说法正确的是（）\n\n[图1]\n\nA: 该黑洞的质量为 $\\frac{G T^{2}}{4 \\pi r^{3}}$\nB: 该黑洞的质量为 $\\frac{4 \\pi r^{3}}{G T^{2}}$\nC: 该黑洞的最大半径为 $\\frac{4 \\pi^{2} r^{3}}{c^{2}}$\nD: 该黑洞的最大半径为 $\\frac{8 \\pi^{2} r^{3}}{c^{2} T^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-093.jpg?height=231&width=329&top_left_y=1689&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_305", "problem": "如图所示, 假设地球半径为 $R$, 地球表面的重力加速度为 $g$, 飞船在轨道半径为 $R$的近地圆轨道I上运动, 到达轨道上 $A$ 点时点火进入椭圆轨道II, 到达粗圆轨道II的远地点 $B$ 点时再次点火进入距地而高度为 $3 R$ 的圆轨道III绕地球做圆周运动, 不考虑飞船质量的变化，点火时间极短。下列分析正确的是（）\n\n[图1]\nA: 飞船在轨道 I 上绕地球运行的周期为 $2 \\pi \\sqrt{\\frac{R}{g}}$\nB: 飞船在 II、III 轨道上通过 $B$ 点的加速度相等\nC: 飞船在轨道 III 上运行速率为 $\\sqrt{\\frac{g R}{3}}$\nD: 飞船在轨道 I 上的机械能比在轨道 III 上的机械能大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 假设地球半径为 $R$, 地球表面的重力加速度为 $g$, 飞船在轨道半径为 $R$的近地圆轨道I上运动, 到达轨道上 $A$ 点时点火进入椭圆轨道II, 到达粗圆轨道II的远地点 $B$ 点时再次点火进入距地而高度为 $3 R$ 的圆轨道III绕地球做圆周运动, 不考虑飞船质量的变化，点火时间极短。下列分析正确的是（）\n\n[图1]\n\nA: 飞船在轨道 I 上绕地球运行的周期为 $2 \\pi \\sqrt{\\frac{R}{g}}$\nB: 飞船在 II、III 轨道上通过 $B$ 点的加速度相等\nC: 飞船在轨道 III 上运行速率为 $\\sqrt{\\frac{g R}{3}}$\nD: 飞船在轨道 I 上的机械能比在轨道 III 上的机械能大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-020.jpg?height=476&width=523&top_left_y=156&top_left_x=318" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_283", "problem": "如图所示, 某次发射远地圆轨道卫星时, 先让卫星进入一个近地的圆轨道I, 在此轨道运行的卫星的轨道半径为 $R_{1}$ 、周期为 $T_{1}$; 然后在 $P$ 点点火加速, 进入陏圆形转移轨道II, 在此轨道运行的卫星的周期为 $T_{2}$; 到达远地点 $\\mathrm{Q}$ 时再次点火加速, 进入远地圆轨道III, 在此轨道运行的卫星的轨道半径为 $R_{3}$ 、周期为 $T_{3}$ (轨道II的近地点为 $\\mathrm{I}$ 上的 $\\mathrm{P}$ 点,远地点为轨道III上的 $\\mathrm{Q}$ 点). 已知 $R_{3}=2 R_{1}$, 则下列关系正确的是 ( )\n\n[图1]\nA: $T_{3}=2 \\sqrt{2} T_{1}$\nB: $T_{2}=\\frac{3 \\sqrt{6}}{8} T_{1}$\nC: $T_{2}=\\frac{3 \\sqrt{3}}{8} T_{3}$\nD: $T_{3}=\\frac{3}{4} \\sqrt{6} T_{1}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 某次发射远地圆轨道卫星时, 先让卫星进入一个近地的圆轨道I, 在此轨道运行的卫星的轨道半径为 $R_{1}$ 、周期为 $T_{1}$; 然后在 $P$ 点点火加速, 进入陏圆形转移轨道II, 在此轨道运行的卫星的周期为 $T_{2}$; 到达远地点 $\\mathrm{Q}$ 时再次点火加速, 进入远地圆轨道III, 在此轨道运行的卫星的轨道半径为 $R_{3}$ 、周期为 $T_{3}$ (轨道II的近地点为 $\\mathrm{I}$ 上的 $\\mathrm{P}$ 点,远地点为轨道III上的 $\\mathrm{Q}$ 点). 已知 $R_{3}=2 R_{1}$, 则下列关系正确的是 ( )\n\n[图1]\n\nA: $T_{3}=2 \\sqrt{2} T_{1}$\nB: $T_{2}=\\frac{3 \\sqrt{6}}{8} T_{1}$\nC: $T_{2}=\\frac{3 \\sqrt{3}}{8} T_{3}$\nD: $T_{3}=\\frac{3}{4} \\sqrt{6} T_{1}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-26.jpg?height=345&width=348&top_left_y=176&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_538", "problem": "经国际小行星命名委员会命名的“神舟星”和“杨利伟星”的轨道均处在火星和木星轨道之间, 已知“神舟星”平均每天绕太阳运行 174 万公里, “杨利伟星”平均每天绕太阳运行 145 万公里, 假设两行星均绕太阳做匀速圆周运动, 则两星相比较 ( )\nA: “神舟星”的轨道半径大\nB: “神舟星”的公转周期大\nC: “神舟星”的加速度大\nD: “神舟星”受到的向心力大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n经国际小行星命名委员会命名的“神舟星”和“杨利伟星”的轨道均处在火星和木星轨道之间, 已知“神舟星”平均每天绕太阳运行 174 万公里, “杨利伟星”平均每天绕太阳运行 145 万公里, 假设两行星均绕太阳做匀速圆周运动, 则两星相比较 ( )\n\nA: “神舟星”的轨道半径大\nB: “神舟星”的公转周期大\nC: “神舟星”的加速度大\nD: “神舟星”受到的向心力大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_895", "problem": "When Mars has its opposition in 2022, an observer in the UK will see it in Taurus. In roughly which month will this opposition take place? [Mars' opposition corresponds to when it is closest in its orbit to the Earth.]\nA: March\nB: June\nC: September\nD: December\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhen Mars has its opposition in 2022, an observer in the UK will see it in Taurus. In roughly which month will this opposition take place? [Mars' opposition corresponds to when it is closest in its orbit to the Earth.]\n\nA: March\nB: June\nC: September\nD: December\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_568", "problem": "太阳主要是由 e、 ${ }_{1}^{1} \\mathrm{H}$ 和 ${ }_{2}^{4} \\mathrm{He}$ 等粒子组成的。维持太阳辐射的是其内部的核聚变反应,\n\n核反应方程是 $2 \\mathrm{e}+4{ }_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}$, 该核反应产生的核能最后转化为辐射能。根据目前关于\n恒星演化的理论, 若由于聚变反应而使太阳中的 ${ }_{1}^{1} \\mathrm{H}$ 核数目从现有数减少 $10 \\%$, 太阳将离开主序星阶段而转入红巨星的演化阶段。为了简化模型, 假定目前太阳全部由 $\\mathrm{e}$ 和 ${ }_{1}{ }^{1} \\mathrm{H}$核组成, 并据此回答下列问题。\n\n已知地球半径 $R=6.4 \\times 10^{6} \\mathrm{~m}$, 地球质量 $m=6.0 \\times 10^{24} \\mathrm{~kg}$, 日地中心的距离 $r=1.5 \\times 10^{11} \\mathrm{~m}$, 地球表面处的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}, 1$ 年约为 $3.2 \\times 10^{7} \\mathrm{~s}$, \n\n已知质子质量 $m_{\\mathrm{p}}=1.6726 \\times 10^{-27} \\mathrm{~kg},{ }_{2}^{4} \\mathrm{He}$ 质量 $m_{\\alpha}=6.6458 \\times 10^{-27} \\mathrm{~kg}$, 电子质量 $m_{\\mathrm{e}}=9.1 \\times 10^{-31} \\mathrm{~kg}$, 光速 $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$ 。求题中所述的核聚变反应所释放的核能。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n太阳主要是由 e、 ${ }_{1}^{1} \\mathrm{H}$ 和 ${ }_{2}^{4} \\mathrm{He}$ 等粒子组成的。维持太阳辐射的是其内部的核聚变反应,\n\n核反应方程是 $2 \\mathrm{e}+4{ }_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}$, 该核反应产生的核能最后转化为辐射能。根据目前关于\n恒星演化的理论, 若由于聚变反应而使太阳中的 ${ }_{1}^{1} \\mathrm{H}$ 核数目从现有数减少 $10 \\%$, 太阳将离开主序星阶段而转入红巨星的演化阶段。为了简化模型, 假定目前太阳全部由 $\\mathrm{e}$ 和 ${ }_{1}{ }^{1} \\mathrm{H}$核组成, 并据此回答下列问题。\n\n已知地球半径 $R=6.4 \\times 10^{6} \\mathrm{~m}$, 地球质量 $m=6.0 \\times 10^{24} \\mathrm{~kg}$, 日地中心的距离 $r=1.5 \\times 10^{11} \\mathrm{~m}$, 地球表面处的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}, 1$ 年约为 $3.2 \\times 10^{7} \\mathrm{~s}$, \n\n已知质子质量 $m_{\\mathrm{p}}=1.6726 \\times 10^{-27} \\mathrm{~kg},{ }_{2}^{4} \\mathrm{He}$ 质量 $m_{\\alpha}=6.6458 \\times 10^{-27} \\mathrm{~kg}$, 电子质量 $m_{\\mathrm{e}}=9.1 \\times 10^{-31} \\mathrm{~kg}$, 光速 $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$ 。求题中所述的核聚变反应所释放的核能。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以J为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "J" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_510", "problem": "在银河系中, 双星系统的数量非常多。研究双星, 不但对于了解恒星形成和演化过程的多样性有重要的意义, 而且对于了解银河系的形成与和演化, 也是一个不可缺少的方面。假设在宇宙中远离其他星体的空间中存在由两个质量分别为 $4 m 、 m$ 的天体 $\\mathrm{A} 、 \\mathrm{~B}$组成的双星系统, 二者中心间的距离为 $L \\circ a 、 b$ 两点为两天体所在直线与天体 $B$ 表面的交点, 天体 $\\mathrm{B}$ 的半径为 $\\frac{L}{5}$ 。已知引力常量为 $G$, 则 $\\mathrm{A} 、 \\mathrm{~B}$ 两天体运动的周期和 $a 、 b$ 两点处质量为 $m_{0}$ 的物体（视为质点）所受万有引力大小之差为（）\nA: $2 \\pi \\sqrt{\\frac{L^{3}}{5 G m}}, 0$\nB: $2 \\pi \\sqrt{\\frac{L^{3}}{5 G m}}, \\frac{325 G m m_{0}}{36 L^{2}}$\nC: $\\pi \\sqrt{\\frac{L^{3}}{5 G m}}, 0$\nD: $\\pi \\sqrt{\\frac{L^{3}}{5 G m}}, \\frac{325 G m m_{0}}{36 L^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n在银河系中, 双星系统的数量非常多。研究双星, 不但对于了解恒星形成和演化过程的多样性有重要的意义, 而且对于了解银河系的形成与和演化, 也是一个不可缺少的方面。假设在宇宙中远离其他星体的空间中存在由两个质量分别为 $4 m 、 m$ 的天体 $\\mathrm{A} 、 \\mathrm{~B}$组成的双星系统, 二者中心间的距离为 $L \\circ a 、 b$ 两点为两天体所在直线与天体 $B$ 表面的交点, 天体 $\\mathrm{B}$ 的半径为 $\\frac{L}{5}$ 。已知引力常量为 $G$, 则 $\\mathrm{A} 、 \\mathrm{~B}$ 两天体运动的周期和 $a 、 b$ 两点处质量为 $m_{0}$ 的物体（视为质点）所受万有引力大小之差为（）\n\nA: $2 \\pi \\sqrt{\\frac{L^{3}}{5 G m}}, 0$\nB: $2 \\pi \\sqrt{\\frac{L^{3}}{5 G m}}, \\frac{325 G m m_{0}}{36 L^{2}}$\nC: $\\pi \\sqrt{\\frac{L^{3}}{5 G m}}, 0$\nD: $\\pi \\sqrt{\\frac{L^{3}}{5 G m}}, \\frac{325 G m m_{0}}{36 L^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_658", "problem": "2011 年 11 月 3 日凌晨, “神舟八号”飞船与“天宫一号”空间站成功对接。对接后,空间站在离地面三百多公里的轨道上绕地球做匀速圆周运动。现已测出其绕地球球心做匀速圆周运动的周期为 $T$, 已知地球半径为 $R$ 、地球表面重力加速度 $g$ 、万有引力常量为 $G$ ，则根据以上数据能够计算的物理量是\nA: 地球的平均密度\nB: 空间站所在处的重力加速度大小\nC: 空间站绕行的线速度大小\nD: 空间站所受的万有引力大小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2011 年 11 月 3 日凌晨, “神舟八号”飞船与“天宫一号”空间站成功对接。对接后,空间站在离地面三百多公里的轨道上绕地球做匀速圆周运动。现已测出其绕地球球心做匀速圆周运动的周期为 $T$, 已知地球半径为 $R$ 、地球表面重力加速度 $g$ 、万有引力常量为 $G$ ，则根据以上数据能够计算的物理量是\n\nA: 地球的平均密度\nB: 空间站所在处的重力加速度大小\nC: 空间站绕行的线速度大小\nD: 空间站所受的万有引力大小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1045", "problem": "The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.c. Considering the evaluated equations for $\\tau, R$, and $q$ we can use this with the measured luminosity of the Sun to get a new estimate for the central temperature.\n\ni. Considering the simplified equation for $q$ and assuming that the core has a mass of $0.35 \\mathrm{M}_{\\odot}$, throughout which T and $\\rho$ are constant, and that the Sun's luminosity is equal to the power produced by the $p-p$ chain fusion processes occurring within its core, estimate the central temperature. [You are given that $u=3 p_{c} / 2 p_{c}$.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.\n\nproblem:\nc. Considering the evaluated equations for $\\tau, R$, and $q$ we can use this with the measured luminosity of the Sun to get a new estimate for the central temperature.\n\ni. Considering the simplified equation for $q$ and assuming that the core has a mass of $0.35 \\mathrm{M}_{\\odot}$, throughout which T and $\\rho$ are constant, and that the Sun's luminosity is equal to the power produced by the $p-p$ chain fusion processes occurring within its core, estimate the central temperature. [You are given that $u=3 p_{c} / 2 p_{c}$.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~K}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-06.jpg?height=536&width=1508&top_left_y=560&top_left_x=272", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-07.jpg?height=748&width=1185&top_left_y=1850&top_left_x=433", "https://cdn.mathpix.com/cropped/2024_03_14_e9aa0a135004f2f4a278g-07.jpg?height=418&width=1446&top_left_y=190&top_left_x=430" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~K}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_952", "problem": "Mercury is the innermost of the Solar System's planets and so is most influenced by gravitational interactions with the Sun, tying its rotational period to its orbital period in a similar way to the tidal locking between the Moon and the Earth. It orbits the Sun in a 3:2 resonance, meaning that it rotates on its axis three times for every two orbits of the Sun.\n\n[figure1]\n\nFigure 1: True colour image of Mercury taken by the probe MESSENGER after its closest approach in 2008. Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie.\n\nMercury has an orbital period of 88 Earth days and a radius of $2440 \\mathrm{~km}$, and spins in the same direction as it orbits (both are anti-clockwise as viewed from high above the Sun).\n\nThinking carefully about the geometry of the situation, calculate the length of a solar day as observed on Mercury (i.e. the length of time from one noon to the next). Hint: you should find it is longer than a Mercurian year.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nMercury is the innermost of the Solar System's planets and so is most influenced by gravitational interactions with the Sun, tying its rotational period to its orbital period in a similar way to the tidal locking between the Moon and the Earth. It orbits the Sun in a 3:2 resonance, meaning that it rotates on its axis three times for every two orbits of the Sun.\n\n[figure1]\n\nFigure 1: True colour image of Mercury taken by the probe MESSENGER after its closest approach in 2008. Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie.\n\nMercury has an orbital period of 88 Earth days and a radius of $2440 \\mathrm{~km}$, and spins in the same direction as it orbits (both are anti-clockwise as viewed from high above the Sun).\n\nThinking carefully about the geometry of the situation, calculate the length of a solar day as observed on Mercury (i.e. the length of time from one noon to the next). Hint: you should find it is longer than a Mercurian year.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_13148c5721a741e30941g-06.jpg?height=1005&width=1330&top_left_y=734&top_left_x=363", "https://i.postimg.cc/9MKLp26P/Screenshot-2024-04-06-at-22-50-51.png" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_439", "problem": "我国航天技术走在世界的前列，探月工程“绕、落、回”三步走的最后一步即将完成,即月球投测器实现采样返回. 如图所示为该过程简化后的示意图, 探测器从圆轨道 1 上的 $\\mathrm{A}$ 点减速后变轨到椭圆轨道 2 , 之后又在轨道 2 上的 $\\mathrm{B}$ 点变轨到近月圆轨道 3 . 已知探测器在轨道 1 上的运行周期为 $T_{1}, O$ 为月球球心, $\\mathrm{C}$ 为轨道 3 上的一点, $\\mathrm{AC}$ 与 $\\mathrm{AO}$之间的最大夹角为 $\\theta$. 下列说法正确的是 ( )\n\n[图1]\nA: 探测器在轨道 2 运行时的机械能大于在轨道 1 运行时的机械能\nB: 探测器在轨道 $1 、 2 、 3$ 运行时的周期大小关系为 $\\mathrm{T}_{1}<\\mathrm{T}_{2}<\\mathrm{T}_{3}$\nC: 探测器在轨道 2 上运行和在圆轨道 1 上运行, 加速度大小相等的位置有两个\nD: 探测器在轨道 3 上运行时的周期为 $\\sqrt{\\sin ^{3} \\theta} \\mathrm{T}_{1}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n我国航天技术走在世界的前列，探月工程“绕、落、回”三步走的最后一步即将完成,即月球投测器实现采样返回. 如图所示为该过程简化后的示意图, 探测器从圆轨道 1 上的 $\\mathrm{A}$ 点减速后变轨到椭圆轨道 2 , 之后又在轨道 2 上的 $\\mathrm{B}$ 点变轨到近月圆轨道 3 . 已知探测器在轨道 1 上的运行周期为 $T_{1}, O$ 为月球球心, $\\mathrm{C}$ 为轨道 3 上的一点, $\\mathrm{AC}$ 与 $\\mathrm{AO}$之间的最大夹角为 $\\theta$. 下列说法正确的是 ( )\n\n[图1]\n\nA: 探测器在轨道 2 运行时的机械能大于在轨道 1 运行时的机械能\nB: 探测器在轨道 $1 、 2 、 3$ 运行时的周期大小关系为 $\\mathrm{T}_{1}<\\mathrm{T}_{2}<\\mathrm{T}_{3}$\nC: 探测器在轨道 2 上运行和在圆轨道 1 上运行, 加速度大小相等的位置有两个\nD: 探测器在轨道 3 上运行时的周期为 $\\sqrt{\\sin ^{3} \\theta} \\mathrm{T}_{1}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-65.jpg?height=848&width=900&top_left_y=855&top_left_x=384" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1081", "problem": "The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.a. Let $r$ denote distance from the centre of a star. We define the variables $\\rho(r), p(r)$ and $T(r)$ to be the density, pressure and temperature at radius $r$ respectively, and $m(r)$ to be the mass enclosed within radius $r$. We will now try and derive an estimate for the pressure at the centre of the Sun.\niii. Assuming that the pressure at the surface, $\\mathrm{p}_{\\mathrm{s}}$, is negligible compared to the pressure at the centre of the Sun, $p_{c}$, the edge of the core is at $r=0.20 R_{\\odot}$ and encloses a mass of $m=0.35$ $\\mathrm{M}_{\\odot}$, and that $\\mathrm{dp} / \\mathrm{dm}$ is constant throughout the star and equal to the value at the edge of the core, calculate a value for $\\mathrm{p}_{\\mathrm{c}}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.\n\nproblem:\na. Let $r$ denote distance from the centre of a star. We define the variables $\\rho(r), p(r)$ and $T(r)$ to be the density, pressure and temperature at radius $r$ respectively, and $m(r)$ to be the mass enclosed within radius $r$. We will now try and derive an estimate for the pressure at the centre of the Sun.\niii. Assuming that the pressure at the surface, $\\mathrm{p}_{\\mathrm{s}}$, is negligible compared to the pressure at the centre of the Sun, $p_{c}$, the edge of the core is at $r=0.20 R_{\\odot}$ and encloses a mass of $m=0.35$ $\\mathrm{M}_{\\odot}$, and that $\\mathrm{dp} / \\mathrm{dm}$ is constant throughout the star and equal to the value at the edge of the core, calculate a value for $\\mathrm{p}_{\\mathrm{c}}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~Pa}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-06.jpg?height=536&width=1508&top_left_y=560&top_left_x=272", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-07.jpg?height=748&width=1185&top_left_y=1850&top_left_x=433", "https://cdn.mathpix.com/cropped/2024_03_14_e9aa0a135004f2f4a278g-05.jpg?height=554&width=1051&top_left_y=1071&top_left_x=431" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ " \\mathrm{~Pa}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1155", "problem": "In November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \\mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked.\n[figure1]\n\nFigure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC.\n\nRight: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \\theta$. Credit: OpenStax, College Physics.\n\nMercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres.\n\nThe radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \\mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \\mu \\mathrm{s}$ pulses at $430 \\mathrm{MHz}$ with a transmitted power of $2.0 \\mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1.\n\nThe signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \\mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2).\n\nPhotons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole).\n\n[figure2]\n\nFigure 2: Left: Snapshots of the reflections of a single $100 \\mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \\mathrm{c} / \\mathrm{s}=1 \\mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967).\n\nTop right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967).\n\nBottom right: The same as the snapshots, but this time summed over the first $500 \\mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\\sim 5 \\mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967).\n\nThe Doppler shift with light is given as\n\n$$\n\\frac{\\Delta f}{f}=\\frac{v}{c}\n$$\n\nwhere $\\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light.\n\nEver since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time.\n\n[figure3]\n\nFigure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966).a. Calculate the power of each echo received by the Aricebo telescope and hence determine the total number of photons in each echo, given the echo was detected $579.3 \\mathrm{~s}$ after being transmitted and Mercury's surface only reflects $6.5 \\%$ of the incident radio photons. Assume $\\theta=0.16^{\\circ}$ and the reflected photons from Mercury are scattered randomly within only the hemisphere facing Earth.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIn November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \\mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked.\n[figure1]\n\nFigure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC.\n\nRight: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \\theta$. Credit: OpenStax, College Physics.\n\nMercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres.\n\nThe radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \\mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \\mu \\mathrm{s}$ pulses at $430 \\mathrm{MHz}$ with a transmitted power of $2.0 \\mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1.\n\nThe signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \\mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2).\n\nPhotons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole).\n\n[figure2]\n\nFigure 2: Left: Snapshots of the reflections of a single $100 \\mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \\mathrm{c} / \\mathrm{s}=1 \\mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967).\n\nTop right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967).\n\nBottom right: The same as the snapshots, but this time summed over the first $500 \\mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\\sim 5 \\mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967).\n\nThe Doppler shift with light is given as\n\n$$\n\\frac{\\Delta f}{f}=\\frac{v}{c}\n$$\n\nwhere $\\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light.\n\nEver since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time.\n\n[figure3]\n\nFigure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966).\n\nproblem:\na. Calculate the power of each echo received by the Aricebo telescope and hence determine the total number of photons in each echo, given the echo was detected $579.3 \\mathrm{~s}$ after being transmitted and Mercury's surface only reflects $6.5 \\%$ of the incident radio photons. Assume $\\theta=0.16^{\\circ}$ and the reflected photons from Mercury are scattered randomly within only the hemisphere facing Earth.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-04.jpg?height=512&width=1374&top_left_y=652&top_left_x=338", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-05.jpg?height=1094&width=1560&top_left_y=218&top_left_x=248", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-06.jpg?height=994&width=897&top_left_y=1359&top_left_x=585" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1142", "problem": "On $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.a. In ideal observing conditions the two planets are far enough apart that they should be (just about) distinguishable to the naked eye, however to some observers in imperfect conditions they would appear as a single bright dot, brighter than either planet on its own.\n\nii. Although they appeared close in angle, there was a very considerable distance between the two planets. At conjunction, Jupiter was 5.926 au from Earth whilst Saturn was 10.827 au (see Fig 5). If they were actually next to each other in space such that they could be treated as a single object, how far from the Earth (in au) would they need to be to have the same apparent magnitude as calculated in the previous part? For simplicity, assume that both planets can be modelled as (very low luminosity) stars so that the change in brightness is only due to changing the distance from the Earth (i.e. ignore the complications from the changing distance from the Sun affecting the number of reflected photons and the changing geometry affecting the illuminated fraction of the planet's surface).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.\n\nproblem:\na. In ideal observing conditions the two planets are far enough apart that they should be (just about) distinguishable to the naked eye, however to some observers in imperfect conditions they would appear as a single bright dot, brighter than either planet on its own.\n\nii. Although they appeared close in angle, there was a very considerable distance between the two planets. At conjunction, Jupiter was 5.926 au from Earth whilst Saturn was 10.827 au (see Fig 5). If they were actually next to each other in space such that they could be treated as a single object, how far from the Earth (in au) would they need to be to have the same apparent magnitude as calculated in the previous part? For simplicity, assume that both planets can be modelled as (very low luminosity) stars so that the change in brightness is only due to changing the distance from the Earth (i.e. ignore the complications from the changing distance from the Sun affecting the number of reflected photons and the changing geometry affecting the illuminated fraction of the planet's surface).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of au, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-07.jpg?height=706&width=1564&top_left_y=834&top_left_x=244", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-08.jpg?height=578&width=1566&top_left_y=196&top_left_x=242", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-09.jpg?height=1072&width=1564&top_left_y=1191&top_left_x=246" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "au" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_48", "problem": "如图所示, 横截面积为 $A$ 、质量为 $m$ 的柱状飞行器沿半径为 $R$ 的圆形轨道在高空绕地球做无动力运行。将地球看作质量为 $M$ 的均匀球体。万有引力常量为 $G$ 。\n\n求飞行器在轨道半径为 $R$ 的高空绕地球做圆周运动的线速度;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 横截面积为 $A$ 、质量为 $m$ 的柱状飞行器沿半径为 $R$ 的圆形轨道在高空绕地球做无动力运行。将地球看作质量为 $M$ 的均匀球体。万有引力常量为 $G$ 。\n\n求飞行器在轨道半径为 $R$ 的高空绕地球做圆周运动的线速度;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-058.jpg?height=445&width=314&top_left_y=1391&top_left_x=354" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1179", "problem": "Why is the Moon heavily cratered, but not the Earth?\nA: The Moon has stronger gravity, so it attracts more space debris\nB: The Moon formed earlier than the Earth, so it had more time to be bombarded by asteroids\nC: The craters on Earth were eroded by the oceans and atmosphere over a long period of time\nD: The Moon orbits around the Earth in addition to orbiting around the Sun, so it collects more space debris\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhy is the Moon heavily cratered, but not the Earth?\n\nA: The Moon has stronger gravity, so it attracts more space debris\nB: The Moon formed earlier than the Earth, so it had more time to be bombarded by asteroids\nC: The craters on Earth were eroded by the oceans and atmosphere over a long period of time\nD: The Moon orbits around the Earth in addition to orbiting around the Sun, so it collects more space debris\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_182", "problem": "质量为 $100 \\mathrm{~kg}$ 的“勇气”号火星车于 2004 年成功登陆在火星表面。若“勇气”号在离火星表面 $12 \\mathrm{~m}$ 时与降落伞自动脱离, 被气囊包裹的“勇气”号下落到地面后又弹跳到 $18 \\mathrm{~m}$ 高处, 这样上下碰撞了若干次后, 才静止在火星表面上。已知火星的半径为地球半径的 0.5 倍, 质量为地球质量的 0.1 倍。若“勇气”号第一次碰撞火星地面时, 气囊和地面的接触时间为 $0.7 \\mathrm{~s}$, 其损失的机械能为它与降落伞自动脱离处 (即离火星地面 $12 \\mathrm{~m}$ 时) 动能的 70\\%, (地球表面的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 不考虑火星表面空气阻力) 求:\n\n “勇气”号在它与降落伞自动脱离处（即离火星地面 $12 \\mathrm{~m}$ 时）的速度;", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n质量为 $100 \\mathrm{~kg}$ 的“勇气”号火星车于 2004 年成功登陆在火星表面。若“勇气”号在离火星表面 $12 \\mathrm{~m}$ 时与降落伞自动脱离, 被气囊包裹的“勇气”号下落到地面后又弹跳到 $18 \\mathrm{~m}$ 高处, 这样上下碰撞了若干次后, 才静止在火星表面上。已知火星的半径为地球半径的 0.5 倍, 质量为地球质量的 0.1 倍。若“勇气”号第一次碰撞火星地面时, 气囊和地面的接触时间为 $0.7 \\mathrm{~s}$, 其损失的机械能为它与降落伞自动脱离处 (即离火星地面 $12 \\mathrm{~m}$ 时) 动能的 70\\%, (地球表面的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 不考虑火星表面空气阻力) 求:\n\n “勇气”号在它与降落伞自动脱离处（即离火星地面 $12 \\mathrm{~m}$ 时）的速度;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以m/s为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m/s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_425", "problem": "'重力探矿是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用, 其原理可简述如下：如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向, 当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向（即 $P O$ 方向）上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常量为 $G$ 。\n设球形区域体积为 $V$, 球心深度为 $d$ ( $d$ 远小于地球半径), $\\overline{P Q}=x$, 求:\n球形区域内黄金矿石在 $Q$ 点产生的加速度大小;\n\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n'重力探矿是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用, 其原理可简述如下：如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向, 当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向（即 $P O$ 方向）上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常量为 $G$ 。\n设球形区域体积为 $V$, 球心深度为 $d$ ( $d$ 远小于地球半径), $\\overline{P Q}=x$, 求:\n球形区域内黄金矿石在 $Q$ 点产生的加速度大小;\n\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-047.jpg?height=434&width=551&top_left_y=2319&top_left_x=364" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_274", "problem": "如图所示, 有 $a 、 b 、 c 、 d$ 四颗卫星, $a$ 未发射在地球赤道上随地球一起转动, $b$ 为近地轨道卫星, $c$ 为地球同步卫星, $d$ 为高空探测卫星, 所有卫星的运动均视为匀速圆周运动，重力加速度为 $g$, 则下列关于四颗卫星的说法正确的是（）\n\n[图1]\nA: $a$ 卫星的向心加速度等于重力加速度 $g$\nB: $b$ 卫星与地心连线在单位时间扫过的面积等于 $c$ 卫星与地心连线在单位时间扫过的面积\nC: $b 、 c$ 卫星轨道半径的三次方与周期平方之比相等\nD: $a$ 卫星的运行周期大于 $d$ 卫星的运行周期\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 有 $a 、 b 、 c 、 d$ 四颗卫星, $a$ 未发射在地球赤道上随地球一起转动, $b$ 为近地轨道卫星, $c$ 为地球同步卫星, $d$ 为高空探测卫星, 所有卫星的运动均视为匀速圆周运动，重力加速度为 $g$, 则下列关于四颗卫星的说法正确的是（）\n\n[图1]\n\nA: $a$ 卫星的向心加速度等于重力加速度 $g$\nB: $b$ 卫星与地心连线在单位时间扫过的面积等于 $c$ 卫星与地心连线在单位时间扫过的面积\nC: $b 、 c$ 卫星轨道半径的三次方与周期平方之比相等\nD: $a$ 卫星的运行周期大于 $d$ 卫星的运行周期\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-032.jpg?height=348&width=1036&top_left_y=163&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_661", "problem": "宇航员飞到一个被稠密气体包围的某行星上进行科学探索。他站在该行星表面, 从静止释放一个质量为 $m$ 的物体, 由于气体阻力的作用, 其加速度 $a$ 随下落位移 $x$ 变化的关系图像如图所示。已知该星球半径为 $R$, 万有引力常量为 $G$ 。下列说法正确的是 $(\\quad)$\n\n[图1]\nA: 该行星的平均密度为 $\\frac{3 a_{0}}{4 \\pi G R}$\nB: 该行星的第一宇宙速度为 $\\sqrt{a_{0} R}$\nC: 卫星在距该行星表面高 $h$ 处的圆轨道上运行的周期为 $\\frac{4 \\pi}{R} \\sqrt{\\frac{(R+h)^{3}}{a_{0}}}$\nD: 从释放到速度刚达到最大的过程中, 物体克服阻力做功 $\\frac{m a_{0} x_{0}}{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇航员飞到一个被稠密气体包围的某行星上进行科学探索。他站在该行星表面, 从静止释放一个质量为 $m$ 的物体, 由于气体阻力的作用, 其加速度 $a$ 随下落位移 $x$ 变化的关系图像如图所示。已知该星球半径为 $R$, 万有引力常量为 $G$ 。下列说法正确的是 $(\\quad)$\n\n[图1]\n\nA: 该行星的平均密度为 $\\frac{3 a_{0}}{4 \\pi G R}$\nB: 该行星的第一宇宙速度为 $\\sqrt{a_{0} R}$\nC: 卫星在距该行星表面高 $h$ 处的圆轨道上运行的周期为 $\\frac{4 \\pi}{R} \\sqrt{\\frac{(R+h)^{3}}{a_{0}}}$\nD: 从释放到速度刚达到最大的过程中, 物体克服阻力做功 $\\frac{m a_{0} x_{0}}{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-115.jpg?height=354&width=391&top_left_y=571&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_189", "problem": "宇宙中有一孤立星系, 中心天体周围有三颗行星, 如图所示。中心天体质量远大于行星质量, 不考虑行星之间的万有引力, 三颗行星的运动轨道中, 有两个为圆轨道, 半径分别为 $r_{1} 、 r_{3}$, 一个为粗圆轨道, 半长轴为 $a, a=r_{3}$ 。在 $\\Delta t$ 时间内, 行星II、行星III 与中心天体连线扫过的面积分别为 $S_{2} 、 S_{3}$ 。行星 I 的速率为 $v_{1}$, 行星 II 在 $B$ 点的速率为 $v_{2 \\mathrm{~B}}$, 行星 II在 $E$ 点的速率为 $v_{2 \\mathrm{E}}$, 行星III的速率为 $v_{3}$, 下列说法正确的是 ( )\n\n[图1]\nA: $S_{2}r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$f. A particle close to M87* moves directly from risco $^{\\text {to }} r_{\\text {ph }}$ (and subsequently into the black hole). What is the extra distance travelled by it due to the curvature of spacetime, as described in Fig 5 ? Give your answer in au, and assume M87* is non-spinning.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$\n\nproblem:\nf. A particle close to M87* moves directly from risco $^{\\text {to }} r_{\\text {ph }}$ (and subsequently into the black hole). What is the extra distance travelled by it due to the curvature of spacetime, as described in Fig 5 ? Give your answer in au, and assume M87* is non-spinning.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-07.jpg?height=704&width=1414&top_left_y=698&top_left_x=331", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-08.jpg?height=374&width=1562&top_left_y=1698&top_left_x=263", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-09.jpg?height=468&width=686&top_left_y=1388&top_left_x=705" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_53", "problem": "某宇宙飞船在赤道所在平面内绕地球做匀速圆周运动, 假设地球赤道平面与其公转平面共面, 地球半径为 R. 日落后 3 小时时, 站在地球赤道上的小明, 刚好观察到头顶正上方的宇宙飞船正要进入地球阴影区，则\nA: 宇宙飞船距地面高度为 $\\sqrt{2} \\mathrm{R}$\nB: 在宇宙飞船中的宇航员观测地球, 其张角为 $90^{\\circ}$\nC: 宇航员绕地球一周经历的“夜晚”时间为 6 小时\nD: 若宇宙飞船的周期为 $T$ ，则宇航员绕地球一周经历的“夜晚”时间为 $T / 4$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n某宇宙飞船在赤道所在平面内绕地球做匀速圆周运动, 假设地球赤道平面与其公转平面共面, 地球半径为 R. 日落后 3 小时时, 站在地球赤道上的小明, 刚好观察到头顶正上方的宇宙飞船正要进入地球阴影区，则\n\nA: 宇宙飞船距地面高度为 $\\sqrt{2} \\mathrm{R}$\nB: 在宇宙飞船中的宇航员观测地球, 其张角为 $90^{\\circ}$\nC: 宇航员绕地球一周经历的“夜晚”时间为 6 小时\nD: 若宇宙飞船的周期为 $T$ ，则宇航员绕地球一周经历的“夜晚”时间为 $T / 4$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-25.jpg?height=312&width=648&top_left_y=1169&top_left_x=344" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_968", "problem": "In the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year.\n\n[figure1]\n\nFigure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com\n\nGiven that the Islamic year $1429 \\mathrm{AH}$ fell completely within the Gregorian year $2008 \\mathrm{CE}$, calculate the Gregorian year in which the Islamic calendar started, and predict the Gregorian year when (at least in part of it) they will both have the same numerical value of year.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year.\n\n[figure1]\n\nFigure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com\n\nGiven that the Islamic year $1429 \\mathrm{AH}$ fell completely within the Gregorian year $2008 \\mathrm{CE}$, calculate the Gregorian year in which the Islamic calendar started, and predict the Gregorian year when (at least in part of it) they will both have the same numerical value of year.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_2c19fdb17927c588761dg-09.jpg?height=800&width=1110&top_left_y=862&top_left_x=473" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_378", "problem": "长征五号遥四运载火箭直接将我国首次执行火星探测任务的“天问一号””探测器送入地火转移轨道, 自此“天问一号”开启了奔向火星的旅程。如图所示为“天问一号”的运动轨迹图, 下列说法正确的是 ( )\n\n[图1]\nA: 发射阶段的末速度已经超过了第二宇宙速度\nB: 探测器沿不同轨道经过图中的 $A$ 点时的速度都相同\nC: 探测器沿不同轨道经过图中的 $A$ 点时的加速度都相同\nD: “天问一号”在火星着陆时, 发动机要向运动的反方向喷气\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n长征五号遥四运载火箭直接将我国首次执行火星探测任务的“天问一号””探测器送入地火转移轨道, 自此“天问一号”开启了奔向火星的旅程。如图所示为“天问一号”的运动轨迹图, 下列说法正确的是 ( )\n\n[图1]\n\nA: 发射阶段的末速度已经超过了第二宇宙速度\nB: 探测器沿不同轨道经过图中的 $A$ 点时的速度都相同\nC: 探测器沿不同轨道经过图中的 $A$ 点时的加速度都相同\nD: “天问一号”在火星着陆时, 发动机要向运动的反方向喷气\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-019.jpg?height=425&width=802&top_left_y=484&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1048", "problem": "GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\n[figure1]\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.c. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\\odot}$, a radius of $1.41 R_{\\odot}$ and a surface temperature of $112000 \\mathrm{~K}$, with a measured $v_{\\infty}$ of $1545 \\mathrm{~km} \\mathrm{~s}^{-1}$.\niii. If the expansion is purely driven by the direct impact of the stellar winds, then the radius at time $t, R(t)$, can be related to $\\dot{M}$ with the given formula. If $n_{0}=16 \\mathrm{~cm}^{-3}$ and $m_{H}=1.67 \\times 10^{-27}$ $\\mathrm{kg}$, calculate the observed mass loss rate based upon the properties of the nebula. Compare it with the predicted one from earlier and comment on your answer.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nGW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\n[figure1]\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nproblem:\nc. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\\odot}$, a radius of $1.41 R_{\\odot}$ and a surface temperature of $112000 \\mathrm{~K}$, with a measured $v_{\\infty}$ of $1545 \\mathrm{~km} \\mathrm{~s}^{-1}$.\niii. If the expansion is purely driven by the direct impact of the stellar winds, then the radius at time $t, R(t)$, can be related to $\\dot{M}$ with the given formula. If $n_{0}=16 \\mathrm{~cm}^{-3}$ and $m_{H}=1.67 \\times 10^{-27}$ $\\mathrm{kg}$, calculate the observed mass loss rate based upon the properties of the nebula. Compare it with the predicted one from earlier and comment on your answer.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~J}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-06.jpg?height=802&width=1308&top_left_y=1709&top_left_x=383" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~J}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_716", "problem": "在暑期科幻电影《独行月球》中, 有大量的与航天知识有关的情景。已知月球的质量约为地球质量的 $\\frac{1}{81}$, 月球半径约为地球半径的 $\\frac{1}{4}$, 月球绕地球公转的周期约为 27 天。则下列说法正确的是\nA: 从地球向月球发射的探月飞船的发射速度应大于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nB: 用同一速度坚直上抛一物体, 在月球表面的最大上升高度是地球表面的 $\\frac{81}{16}$ 倍\nC: 月球到地心的距离大约是地球同步卫星轨道半径的 9 倍\nD: 月球的第一宇宙速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n在暑期科幻电影《独行月球》中, 有大量的与航天知识有关的情景。已知月球的质量约为地球质量的 $\\frac{1}{81}$, 月球半径约为地球半径的 $\\frac{1}{4}$, 月球绕地球公转的周期约为 27 天。则下列说法正确的是\n\nA: 从地球向月球发射的探月飞船的发射速度应大于 $11.2 \\mathrm{~km} / \\mathrm{s}$\nB: 用同一速度坚直上抛一物体, 在月球表面的最大上升高度是地球表面的 $\\frac{81}{16}$ 倍\nC: 月球到地心的距离大约是地球同步卫星轨道半径的 9 倍\nD: 月球的第一宇宙速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_228", "problem": "如图所示, 质量分别为 $M$ 和 $m$ 的两个星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ (均视为质点) 在它们之间的引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 之间的距离为 $L$ 。已知星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 和 $O$三点始终共线, $\\mathrm{A}$ 和 $\\mathrm{B}$ 分别在 $O$ 点的两侧。引力常量为 $G$ 。\n\n求星球 $\\mathrm{B}$ 的线速度大小 $v$;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 质量分别为 $M$ 和 $m$ 的两个星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ (均视为质点) 在它们之间的引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 之间的距离为 $L$ 。已知星球 $\\mathrm{A}$ 和 $\\mathrm{B}$ 和 $O$三点始终共线, $\\mathrm{A}$ 和 $\\mathrm{B}$ 分别在 $O$ 点的两侧。引力常量为 $G$ 。\n\n求星球 $\\mathrm{B}$ 的线速度大小 $v$;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-031.jpg?height=429&width=488&top_left_y=154&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_270", "problem": "如图 (i) 所示, 真空中两正点电荷 $\\mathrm{A} 、 \\mathrm{~B}$ 固定在 $x$ 轴上, 其中 $\\mathrm{A}$ 位于坐标原点。\n\n一质量为 $m$ 、电量为 $q$ (电量远小于 $\\mathrm{A} 、 \\mathrm{~B}$ ) 的带正电小球 $\\mathrm{a}$ 仅在电场力作用下, 以大小为 $v_{0}$ 的初速度从 $x=x_{1}$ 处沿 $x$ 轴正方向运动。取无穷远处势能为零, $\\mathrm{a}$ 在 $\\mathrm{A} 、 \\mathrm{~B}$ 间由于受 $\\mathrm{A} 、 \\mathrm{~B}$ 的电场力作用而具有的电势能 $E_{p}$ 随位置 $x$ 变化关系如图 (ii) 所示, 图中 $E_{N_{1}}$, $E_{2}$ 均为已知, 且 $\\mathrm{a}$ 在 $x=x_{2}$ 处受到的电场力为零。\n\n比较 $\\mathrm{A} 、 \\mathrm{~B}$ 两电荷电量 $Q_{A} 、 Q_{B}$ 的大小关系;\n\n\n[图1]\n\n图(i)\n\n[图2]\n\n图(ii)\n\n[图3]\n\n图(iii)\n\n[图4]\n\n图(iv)", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图 (i) 所示, 真空中两正点电荷 $\\mathrm{A} 、 \\mathrm{~B}$ 固定在 $x$ 轴上, 其中 $\\mathrm{A}$ 位于坐标原点。\n\n一质量为 $m$ 、电量为 $q$ (电量远小于 $\\mathrm{A} 、 \\mathrm{~B}$ ) 的带正电小球 $\\mathrm{a}$ 仅在电场力作用下, 以大小为 $v_{0}$ 的初速度从 $x=x_{1}$ 处沿 $x$ 轴正方向运动。取无穷远处势能为零, $\\mathrm{a}$ 在 $\\mathrm{A} 、 \\mathrm{~B}$ 间由于受 $\\mathrm{A} 、 \\mathrm{~B}$ 的电场力作用而具有的电势能 $E_{p}$ 随位置 $x$ 变化关系如图 (ii) 所示, 图中 $E_{N_{1}}$, $E_{2}$ 均为已知, 且 $\\mathrm{a}$ 在 $x=x_{2}$ 处受到的电场力为零。\n\n比较 $\\mathrm{A} 、 \\mathrm{~B}$ 两电荷电量 $Q_{A} 、 Q_{B}$ 的大小关系;\n\n\n[图1]\n\n图(i)\n\n[图2]\n\n图(ii)\n\n[图3]\n\n图(iii)\n\n[图4]\n\n图(iv)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-065.jpg?height=119&width=379&top_left_y=1982&top_left_x=336", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-065.jpg?height=294&width=331&top_left_y=1846&top_left_x=768", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-065.jpg?height=157&width=368&top_left_y=1940&top_left_x=1135", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-065.jpg?height=197&width=274&top_left_y=1935&top_left_x=1528" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_694", "problem": "北京时间 2019 年 4 月 10 日, 人类首次利用虚拟射电望远镜, 在紧邻巨椭圆星系 M87 的中心成功捕获世界首张黑洞图像。科学研究表明, 当天体的逃逸速度（即第二宇宙速度, 为第一宇宙速度的 $\\sqrt{2}$ 倍）超过光速时, 该天体就是黑洞。已知某天体质量为 $M$,万有引力常量为 $G$, 光速为 $c$, 则要使该天体成为黑洞, 其半径应小于 ( )\nA: $\\frac{2 G M}{c^{2}}$\nB: $\\frac{2 c^{2}}{G M}$\nC: $\\frac{\\sqrt{2} G M}{c^{2}}$\nD: $\\frac{G M}{c^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n北京时间 2019 年 4 月 10 日, 人类首次利用虚拟射电望远镜, 在紧邻巨椭圆星系 M87 的中心成功捕获世界首张黑洞图像。科学研究表明, 当天体的逃逸速度（即第二宇宙速度, 为第一宇宙速度的 $\\sqrt{2}$ 倍）超过光速时, 该天体就是黑洞。已知某天体质量为 $M$,万有引力常量为 $G$, 光速为 $c$, 则要使该天体成为黑洞, 其半径应小于 ( )\n\nA: $\\frac{2 G M}{c^{2}}$\nB: $\\frac{2 c^{2}}{G M}$\nC: $\\frac{\\sqrt{2} G M}{c^{2}}$\nD: $\\frac{G M}{c^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1042", "problem": "The surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.a. When it reached first perihelion, radio signals from the probe took $446.58 \\mathrm{~s}$ to reach Earth.\n\ni. Show that the spacecraft's perihelion is $\\approx 0.5$ au, giving your answer to 4 s.f., and hence estimate the launch date, assuming the Earth's orbit is circular. Note that 2020 is a leap year and take 1 year $=365.25$ days. [Hint: You may wish to use a numerical method.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.\n\nproblem:\na. When it reached first perihelion, radio signals from the probe took $446.58 \\mathrm{~s}$ to reach Earth.\n\ni. Show that the spacecraft's perihelion is $\\approx 0.5$ au, giving your answer to 4 s.f., and hence estimate the launch date, assuming the Earth's orbit is circular. Note that 2020 is a leap year and take 1 year $=365.25$ days. [Hint: You may wish to use a numerical method.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-10.jpg?height=792&width=1572&top_left_y=598&top_left_x=241", "https://cdn.mathpix.com/cropped/2024_03_14_6cde567bccf58dc9a2d2g-09.jpg?height=571&width=597&top_left_y=588&top_left_x=455" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_6", "problem": "2022 年 11 月 29 日 23 时 08 分, 搭载着神舟十五号载人飞船的长征二号 $F$ 遥十五运载火箭在酒泉卫星发射中心升空, 11 月 30 日 5 时 42 分, 神舟十五号载人飞船与天和核心舱成功完成自主交会对接。如图为神舟十五号的发射与交会对接过程示意图，图中 I 为近地圆轨道, 其轨道半径可认为等于地球半径 $R$, II 为椭圆变轨轨道, III 为天和核心舱所在轨道, 其轨道半径为 $r_{0}, P 、 Q$ 分别为轨道 II 与 I、III 轨道的交会点, 已知神舟十五号的质量为 $m_{0}$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 若取两物体相距无\n穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\\mathrm{P}}=-\\frac{G M m}{r}$ (式中 $G$ 为引力常量)。求:\n要使神舟十五号从轨道 I 迁移到轨道 III, 所要提供的最小能量\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2022 年 11 月 29 日 23 时 08 分, 搭载着神舟十五号载人飞船的长征二号 $F$ 遥十五运载火箭在酒泉卫星发射中心升空, 11 月 30 日 5 时 42 分, 神舟十五号载人飞船与天和核心舱成功完成自主交会对接。如图为神舟十五号的发射与交会对接过程示意图，图中 I 为近地圆轨道, 其轨道半径可认为等于地球半径 $R$, II 为椭圆变轨轨道, III 为天和核心舱所在轨道, 其轨道半径为 $r_{0}, P 、 Q$ 分别为轨道 II 与 I、III 轨道的交会点, 已知神舟十五号的质量为 $m_{0}$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 若取两物体相距无\n穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\\mathrm{P}}=-\\frac{G M m}{r}$ (式中 $G$ 为引力常量)。求:\n要使神舟十五号从轨道 I 迁移到轨道 III, 所要提供的最小能量\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-039.jpg?height=631&width=560&top_left_y=510&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_780", "problem": "What is a synodic month?\nA: The time between two full moons.\nB: The time between two Venus appearances.\nC: The time for $1 / 12$ th Earth orbit around the Sun.\nD: The time for one Earth rotation in respect to the Sun.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is a synodic month?\n\nA: The time between two full moons.\nB: The time between two Venus appearances.\nC: The time for $1 / 12$ th Earth orbit around the Sun.\nD: The time for one Earth rotation in respect to the Sun.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_351", "problem": "天问一号火星探测器的发射标志着我国的航天事业迈进了新时代, 设地球绕太阳的公转周期为 $T$, 环绕太阳公转的轨道半径为 $r_{1}$, 火星环绕太阳公转的轨道半径为 $r_{2}$, 火星的半径为 $R$, 万有引力常量为 $G$, 下列说法正确的是 ( )\nA: 太阳的质量为 $\\frac{4 \\pi^{2} r_{2}^{3}}{G T^{2}}$\nB: 火星绕太阳公转的角速度大小为 $\\frac{2 \\pi}{T}\\left(\\frac{r_{2}}{r_{1}}\\right)^{\\frac{3}{2}}$\nC: 火星表面的重力加速度大小为 $\\frac{4 \\pi^{2} r_{1}^{3}}{R^{2} T^{2}}$\nD: 从火星与地球相距最远到地球与火星相距最近的最短时间为 $\\frac{r_{2}^{\\frac{3}{2}} T}{2\\left(r_{2}^{\\frac{3}{2}}-r_{1}^{\\frac{3}{2}}\\right)}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n天问一号火星探测器的发射标志着我国的航天事业迈进了新时代, 设地球绕太阳的公转周期为 $T$, 环绕太阳公转的轨道半径为 $r_{1}$, 火星环绕太阳公转的轨道半径为 $r_{2}$, 火星的半径为 $R$, 万有引力常量为 $G$, 下列说法正确的是 ( )\n\nA: 太阳的质量为 $\\frac{4 \\pi^{2} r_{2}^{3}}{G T^{2}}$\nB: 火星绕太阳公转的角速度大小为 $\\frac{2 \\pi}{T}\\left(\\frac{r_{2}}{r_{1}}\\right)^{\\frac{3}{2}}$\nC: 火星表面的重力加速度大小为 $\\frac{4 \\pi^{2} r_{1}^{3}}{R^{2} T^{2}}$\nD: 从火星与地球相距最远到地球与火星相距最近的最短时间为 $\\frac{r_{2}^{\\frac{3}{2}} T}{2\\left(r_{2}^{\\frac{3}{2}}-r_{1}^{\\frac{3}{2}}\\right)}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_430", "problem": "如图所示, 甲、乙两卫星在某行星的球心的同一平面内做圆周运动, 某时刻恰好处于行星上 $A$ 点的正上方, 从该时刻算起, 在同一段时间内, 甲卫星恰好又有 5 次经过 $A$点的正上方, 乙卫星恰好又有 3 次经过 $A$ 点的正上方, 不计行星自转的影响, 下列关于这两颗卫星的说法正确的是\n\n[图1]\nA: 甲、乙两卫量的周期之比为 $2: 3$\nB: 甲、乙两卫星的角速度之比为 $3: 5$\nC: 甲、乙两卫星的轨道半径之比为 $\\sqrt[3]{\\frac{9}{25}}$\nD: 若甲、乙两卫星质量相同, 则甲的机械能大于乙的机械能\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 甲、乙两卫星在某行星的球心的同一平面内做圆周运动, 某时刻恰好处于行星上 $A$ 点的正上方, 从该时刻算起, 在同一段时间内, 甲卫星恰好又有 5 次经过 $A$点的正上方, 乙卫星恰好又有 3 次经过 $A$ 点的正上方, 不计行星自转的影响, 下列关于这两颗卫星的说法正确的是\n\n[图1]\n\nA: 甲、乙两卫量的周期之比为 $2: 3$\nB: 甲、乙两卫星的角速度之比为 $3: 5$\nC: 甲、乙两卫星的轨道半径之比为 $\\sqrt[3]{\\frac{9}{25}}$\nD: 若甲、乙两卫星质量相同, 则甲的机械能大于乙的机械能\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-001.jpg?height=511&width=562&top_left_y=818&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_567", "problem": "地球质量大约是月球质量的 81 倍, 地月距离约为 38 万千米, 两者中心连线上有一个被称作“拉格朗日点” 的位置, 一飞行器处于该点, 在几乎不消耗燃料的情况下与月球同步绕地球做圆周运动, 则这个点到地球的距离约为 ( )\nA: 3.8 万千米\nB: 5.8 万千米\nC: 32 万千米\nD: 34 万千米\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球质量大约是月球质量的 81 倍, 地月距离约为 38 万千米, 两者中心连线上有一个被称作“拉格朗日点” 的位置, 一飞行器处于该点, 在几乎不消耗燃料的情况下与月球同步绕地球做圆周运动, 则这个点到地球的距离约为 ( )\n\nA: 3.8 万千米\nB: 5.8 万千米\nC: 32 万千米\nD: 34 万千米\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_525", "problem": "2018 年 3 月 30 日, 我国在西昌卫星发射中心用长征三号乙运载火箭, 以“一箭双星”方式成功发射第三十、三十一颗北斗导航卫星. 已知“北斗第三十颗导航卫星”做匀速圆周运动的轨道半径小于地球同步卫星轨道半径, 运行速度为 $v$, 向心加速度为 $a$;地球表面的重力加速度为 $g$, 引力常量为 $G$. 下列判断正确的是 ( )\nA: 该导航卫星的运行周期大于 24 小时\nB: 地球质量为 $\\frac{v^{4}}{G a}$\nC: 该导航卫星的轨道半径与地球半径之比为 $\\sqrt{g}: \\sqrt{a}$\nD: 该导航卫星的运行速度与地球第一宇宙速度之比 $\\sqrt{a}: \\sqrt{g}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2018 年 3 月 30 日, 我国在西昌卫星发射中心用长征三号乙运载火箭, 以“一箭双星”方式成功发射第三十、三十一颗北斗导航卫星. 已知“北斗第三十颗导航卫星”做匀速圆周运动的轨道半径小于地球同步卫星轨道半径, 运行速度为 $v$, 向心加速度为 $a$;地球表面的重力加速度为 $g$, 引力常量为 $G$. 下列判断正确的是 ( )\n\nA: 该导航卫星的运行周期大于 24 小时\nB: 地球质量为 $\\frac{v^{4}}{G a}$\nC: 该导航卫星的轨道半径与地球半径之比为 $\\sqrt{g}: \\sqrt{a}$\nD: 该导航卫星的运行速度与地球第一宇宙速度之比 $\\sqrt{a}: \\sqrt{g}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_138", "problem": "我国的“天链一号”地球同步轨道卫星, 可为载人航天器及中、低轨道卫星提供数据通讯服务。如图为“天链一号” $\\mathrm{a}$ 、赤道平面内的低轨道卫星 b、地球三者的位置关系示意图, $O$ 为地心, 地球相对卫星 $\\mathrm{a} 、 \\mathrm{~b}$ 的张角分别为 $\\theta_{1}$ 和 $\\theta_{2}$ ( $\\theta_{2}$ 图中未标出), 卫星 $\\mathrm{a}$ 的轨道半径是 $\\mathrm{b}$ 的 4 倍, 已知卫星 $\\mathrm{a} 、 \\mathrm{~b}$ 绕地球同向运行, 卫星 $\\mathrm{a}$ 周期为 $T$, 在运行过程中由于地球的遮挡, 卫星 $\\mathrm{b}$ 会进入与卫星 $\\mathrm{a}$ 通讯的盲区。卫星间的通讯信号视为沿直线传播，忽略信号传输时间，下列分析正确的是（ ）\n\n[图1]\nA: 张角 $\\theta_{1}$ 和 $\\theta_{2}$ 满足 $\\sin \\theta_{2}=4 \\sin \\theta_{1}$\nB: 卫星 $\\mathrm{b}$ 的周期为 $\\frac{T}{8}$\nC: 卫星 $\\mathrm{b}$ 每次在盲区运行的时间 $\\frac{\\theta_{1}+\\theta_{2}}{14 \\pi} T$\nD: 卫星 $\\mathrm{b}$ 每次在盲区运行的时间为 $\\frac{\\theta_{1}+\\theta_{2}}{16 \\pi} T$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n我国的“天链一号”地球同步轨道卫星, 可为载人航天器及中、低轨道卫星提供数据通讯服务。如图为“天链一号” $\\mathrm{a}$ 、赤道平面内的低轨道卫星 b、地球三者的位置关系示意图, $O$ 为地心, 地球相对卫星 $\\mathrm{a} 、 \\mathrm{~b}$ 的张角分别为 $\\theta_{1}$ 和 $\\theta_{2}$ ( $\\theta_{2}$ 图中未标出), 卫星 $\\mathrm{a}$ 的轨道半径是 $\\mathrm{b}$ 的 4 倍, 已知卫星 $\\mathrm{a} 、 \\mathrm{~b}$ 绕地球同向运行, 卫星 $\\mathrm{a}$ 周期为 $T$, 在运行过程中由于地球的遮挡, 卫星 $\\mathrm{b}$ 会进入与卫星 $\\mathrm{a}$ 通讯的盲区。卫星间的通讯信号视为沿直线传播，忽略信号传输时间，下列分析正确的是（ ）\n\n[图1]\n\nA: 张角 $\\theta_{1}$ 和 $\\theta_{2}$ 满足 $\\sin \\theta_{2}=4 \\sin \\theta_{1}$\nB: 卫星 $\\mathrm{b}$ 的周期为 $\\frac{T}{8}$\nC: 卫星 $\\mathrm{b}$ 每次在盲区运行的时间 $\\frac{\\theta_{1}+\\theta_{2}}{14 \\pi} T$\nD: 卫星 $\\mathrm{b}$ 每次在盲区运行的时间为 $\\frac{\\theta_{1}+\\theta_{2}}{16 \\pi} T$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-111.jpg?height=400&width=440&top_left_y=685&top_left_x=340", "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-112.jpg?height=391&width=440&top_left_y=798&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_599", "problem": "宇宙中存在一些离其他恒星较远的、由质量相等的三颗星组成的三星系统, 可忽略其他星体对三星系统的影响。稳定的三星系统存在两种基本形式: 一种是三颗星位于同一直线上, 两颗星围绕中央星在同一半径为 $R$ 的轨道上运行, 如图甲所示, 周期为 $T_{1}$;另一种是三颗星位于边长为 $R$ 的等边三角形的三个顶点上, 并沿等边三角形的外接圆运行, 如图乙所示, 周期为 $T_{2}$ 。则 $T_{1}: T_{2}$ 为 ( )\n\n[图1]\n\n甲\n\n[图2]\n\n乙\nA: $\\sqrt{\\frac{3}{5}}$\nB: $2 \\sqrt{\\frac{3}{5}}$\nC: $3 \\sqrt{\\frac{3}{5}}$\nD: $4 \\sqrt{\\frac{3}{5}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n宇宙中存在一些离其他恒星较远的、由质量相等的三颗星组成的三星系统, 可忽略其他星体对三星系统的影响。稳定的三星系统存在两种基本形式: 一种是三颗星位于同一直线上, 两颗星围绕中央星在同一半径为 $R$ 的轨道上运行, 如图甲所示, 周期为 $T_{1}$;另一种是三颗星位于边长为 $R$ 的等边三角形的三个顶点上, 并沿等边三角形的外接圆运行, 如图乙所示, 周期为 $T_{2}$ 。则 $T_{1}: T_{2}$ 为 ( )\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\nA: $\\sqrt{\\frac{3}{5}}$\nB: $2 \\sqrt{\\frac{3}{5}}$\nC: $3 \\sqrt{\\frac{3}{5}}$\nD: $4 \\sqrt{\\frac{3}{5}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-048.jpg?height=359&width=440&top_left_y=1648&top_left_x=337", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-048.jpg?height=277&width=285&top_left_y=1683&top_left_x=840" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_747", "problem": "An object's spectral energy distribution (SED) is formally given by ...\nA: $d E / d \\lambda$\nB: $d E / d t$\nC: $d z / d \\lambda$\nD: $d z / d t$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn object's spectral energy distribution (SED) is formally given by ...\n\nA: $d E / d \\lambda$\nB: $d E / d t$\nC: $d z / d \\lambda$\nD: $d z / d t$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_414", "problem": "由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心 $\\mathrm{O}$ 在三角形所在的平面内做相同角速度的圆周运动 (图示为 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三颗星体质量不相同时的一般情况). 若 $\\mathrm{A}$ 星体质量为 $2 \\mathrm{~m}, \\mathrm{~B} 、 \\mathrm{C}$ 两星体的质量均为 $\\mathrm{m}$, 三角形边长为 $\\mathrm{a}$. 则 ( )\n\n[图1]\nA: B 星体所受的合力与 $\\mathrm{A}$ 星体所受合力之比为 $1: 2$\nB: 圆心 $\\mathrm{O}$ 与 $\\mathrm{B}$ 的连线与 $\\mathrm{BC}$ 夹角 $\\theta$ 的正切值为 $\\frac{\\sqrt{3}}{2}$\nC: A、B 星体做圆周运动的线速度大小之比为 $\\sqrt{3}: \\sqrt{7}$\nD: 此三星体做圆周运动的角速度大小为 $2 \\sqrt{\\frac{G m^{2}}{a^{3}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心 $\\mathrm{O}$ 在三角形所在的平面内做相同角速度的圆周运动 (图示为 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三颗星体质量不相同时的一般情况). 若 $\\mathrm{A}$ 星体质量为 $2 \\mathrm{~m}, \\mathrm{~B} 、 \\mathrm{C}$ 两星体的质量均为 $\\mathrm{m}$, 三角形边长为 $\\mathrm{a}$. 则 ( )\n\n[图1]\n\nA: B 星体所受的合力与 $\\mathrm{A}$ 星体所受合力之比为 $1: 2$\nB: 圆心 $\\mathrm{O}$ 与 $\\mathrm{B}$ 的连线与 $\\mathrm{BC}$ 夹角 $\\theta$ 的正切值为 $\\frac{\\sqrt{3}}{2}$\nC: A、B 星体做圆周运动的线速度大小之比为 $\\sqrt{3}: \\sqrt{7}$\nD: 此三星体做圆周运动的角速度大小为 $2 \\sqrt{\\frac{G m^{2}}{a^{3}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-42.jpg?height=314&width=331&top_left_y=1405&top_left_x=337", "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-43.jpg?height=499&width=554&top_left_y=156&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_92", "problem": "“天问一号”探测器需要通过霍曼转移轨道从地球发送到火星, 地球轨道和火星轨道看成圆形轨道, 此时霍曼转移轨道是一个近日点 $M$ 和远日点 $P$ 都与地球轨道、火星轨道相切的椭圆轨道 (如图所示)。在近日点短暂点火后“天问一号”进入霍曼转移轨道，接着“天问一号”沿着这个轨道直至抵达远日点, 然后再次点火进入火星轨道。已知万有引力常量为 $G$, 太阳质量为 $m$, 地球轨道和火星轨道半径分别为 $r$ 和 $R$, 地球、火星、 “天问一号”运行方向都为逆时针方向。下列说法正确的是（）\n\n[图1]\nA: 两次点火时喷气方向都与运动方向相同\nB: 两次点火之间的时间为 $\\frac{\\pi}{2 \\sqrt{2}} \\sqrt{\\frac{(r+R)^{3}}{G m}}$\nC: “天问一号”与太阳连线单位时间在地球轨道上扫过的面积等于在火星轨道上扫过的面积\nD: “天问一号”在转移轨道上近日点的速度大小 $v_{1}$ 比远日点的速度大小 $v_{2}$ 大, 且满足 $\\frac{v_{1}}{v_{2}}=\\sqrt{\\frac{R}{r}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n“天问一号”探测器需要通过霍曼转移轨道从地球发送到火星, 地球轨道和火星轨道看成圆形轨道, 此时霍曼转移轨道是一个近日点 $M$ 和远日点 $P$ 都与地球轨道、火星轨道相切的椭圆轨道 (如图所示)。在近日点短暂点火后“天问一号”进入霍曼转移轨道，接着“天问一号”沿着这个轨道直至抵达远日点, 然后再次点火进入火星轨道。已知万有引力常量为 $G$, 太阳质量为 $m$, 地球轨道和火星轨道半径分别为 $r$ 和 $R$, 地球、火星、 “天问一号”运行方向都为逆时针方向。下列说法正确的是（）\n\n[图1]\n\nA: 两次点火时喷气方向都与运动方向相同\nB: 两次点火之间的时间为 $\\frac{\\pi}{2 \\sqrt{2}} \\sqrt{\\frac{(r+R)^{3}}{G m}}$\nC: “天问一号”与太阳连线单位时间在地球轨道上扫过的面积等于在火星轨道上扫过的面积\nD: “天问一号”在转移轨道上近日点的速度大小 $v_{1}$ 比远日点的速度大小 $v_{2}$ 大, 且满足 $\\frac{v_{1}}{v_{2}}=\\sqrt{\\frac{R}{r}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-041.jpg?height=454&width=420&top_left_y=167&top_left_x=361" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_936", "problem": "In the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year.\n\n[figure1]\n\nFigure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com\n\nIn some Islamic countries, odd-numbered months have 30 days and even-numbered months have 29 days. How often will leap years be in this system (where the twelfth month is also 30 days)? [This is analogous to the Gregorian system where there is a leap year roughly once every 4 years.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year.\n\n[figure1]\n\nFigure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com\n\nIn some Islamic countries, odd-numbered months have 30 days and even-numbered months have 29 days. How often will leap years be in this system (where the twelfth month is also 30 days)? [This is analogous to the Gregorian system where there is a leap year roughly once every 4 years.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of lunar years, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_2c19fdb17927c588761dg-09.jpg?height=800&width=1110&top_left_y=862&top_left_x=473" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "lunar years" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_37", "problem": "2020 年 7 月 23 日, 我国火星探测器 “天问一号” 发射成功, 2021 年 1 月 28 日, “天问一号” 飞行里程突破四亿千米, 图甲是火星探测器的运行路线图。假设探测器经过多次变轨后登陆火星的轨迹变化可抽象为如图乙所示, 探测器先在轨道 I 上运动, 经过 $P$点启动变轨发动机切换到圆轨道 II 上运动, 经过一段时间后, 再次经过 $P$ 点时启动变轨发动机切换到椭圆轨道III上运动。轨道上的 $P 、 Q 、 \\mathrm{~S}$ 三点与火星中心位于同一直线上, $P 、 Q$ 两点分别是粗圆轨道的远火星点和近火星点, 且 $P Q=2 Q S=2 l$ 。除了变轨瞬间,探测器在轨道上运行时均处于无动力航行状态。探测器在轨道 I、II、III 上经过 $P$ 点的速度分别为 $v_{1} 、 v_{2} 、 v_{3}$, 下列说法正确的是 ( )\n\n[图1]\n\n甲\n\n[图2]\n\n乙\nA: $v_{1}\\mathrm{a}_{\\mathrm{c}}>\\mathrm{a}_{\\mathrm{a}}$\nB: $a 、 b 、 c$ 的角速度大小关系为 $\\omega_{\\mathrm{a}}>\\omega_{\\mathrm{b}}>\\omega_{\\mathrm{c}}$\nC: $a 、 b 、 c$ 的线速度大小关系为 $\\mathrm{v}_{\\mathrm{a}}<\\mathrm{v}_{\\mathrm{b}}<\\mathrm{v}_{\\mathrm{c}}$\nD: $a 、 b 、 c$ 的周期大小关系为 $\\mathrm{T}_{\\mathrm{a}}<\\mathrm{T}_{\\mathrm{b}}<\\mathrm{T}_{\\mathrm{c}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, $a$ 为放在赤道上相对地球静止的物体随地球自转做匀速圆周运动, $b$ 为在地球表面附近做匀速圆周运动的人造卫星 (轨道半径近似等于地球半径), $c$ 为地球的同步卫星, 则以下说法中正确的是（ ）[图1]\n\nA: $a 、 b 、 c$ 的向心加速度大小关系为 $\\mathrm{a}_{\\mathrm{b}}>\\mathrm{a}_{\\mathrm{c}}>\\mathrm{a}_{\\mathrm{a}}$\nB: $a 、 b 、 c$ 的角速度大小关系为 $\\omega_{\\mathrm{a}}>\\omega_{\\mathrm{b}}>\\omega_{\\mathrm{c}}$\nC: $a 、 b 、 c$ 的线速度大小关系为 $\\mathrm{v}_{\\mathrm{a}}<\\mathrm{v}_{\\mathrm{b}}<\\mathrm{v}_{\\mathrm{c}}$\nD: $a 、 b 、 c$ 的周期大小关系为 $\\mathrm{T}_{\\mathrm{a}}<\\mathrm{T}_{\\mathrm{b}}<\\mathrm{T}_{\\mathrm{c}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-089.jpg?height=389&width=554&top_left_y=1122&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_453", "problem": "2018 年 6 月 14 日 11 时 06 分，探月工程嫦娥四号任务“鹊桥”中继星成为世界首颗成功进入地月拉格朗日 $L_{2}$ 点的 Halo 使命轨道的卫星, 为地月信息联通搭建“天桥”。如图所示, 该 $L_{2}$ 点位于地球与月球连线的延长线上, “鹊桥”位于该点, 在几乎不消耗燃料的情况下与月球同步绕地球做圆周运动。已知地球、月球和“鹊桥”的质量分别为 $M_{e}$ 、 $M_{m} 、 m$ ，地球和月球之间的平均距离为 $R, L_{2}$ 点离月球的距离为 $x$, 则\n\n[图1]\nA: “鹊桥”的线速度大于月球的线速度\nB: “鹊桥”的向心加速度小于月球的向心加速度\nC: $x$ 满足 $\\frac{M_{e}}{(R+x)^{2}}+\\frac{M_{m}}{x^{2}}=\\frac{M_{e}}{R^{3}}(R+x)$\nD: $x$ 满足 $\\frac{M_{e}}{(R+x)^{2}}+\\frac{M_{e}}{x^{2}}=\\frac{m}{R^{3}}(R+x)$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2018 年 6 月 14 日 11 时 06 分，探月工程嫦娥四号任务“鹊桥”中继星成为世界首颗成功进入地月拉格朗日 $L_{2}$ 点的 Halo 使命轨道的卫星, 为地月信息联通搭建“天桥”。如图所示, 该 $L_{2}$ 点位于地球与月球连线的延长线上, “鹊桥”位于该点, 在几乎不消耗燃料的情况下与月球同步绕地球做圆周运动。已知地球、月球和“鹊桥”的质量分别为 $M_{e}$ 、 $M_{m} 、 m$ ，地球和月球之间的平均距离为 $R, L_{2}$ 点离月球的距离为 $x$, 则\n\n[图1]\n\nA: “鹊桥”的线速度大于月球的线速度\nB: “鹊桥”的向心加速度小于月球的向心加速度\nC: $x$ 满足 $\\frac{M_{e}}{(R+x)^{2}}+\\frac{M_{m}}{x^{2}}=\\frac{M_{e}}{R^{3}}(R+x)$\nD: $x$ 满足 $\\frac{M_{e}}{(R+x)^{2}}+\\frac{M_{e}}{x^{2}}=\\frac{m}{R^{3}}(R+x)$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-06.jpg?height=446&width=571&top_left_y=1096&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1001", "problem": "Although Saturn is famous for its rings, all of the gas giants in the Solar System have ring systems. The outer ring is known as the Adams ring and is very thin. Normally such a thin structure would widen over time so there needs to be a process keeping it constrained. One hypothesis is that the Neptunian moon Galatea, with an orbit just slightly smaller than the ring, acts as a 'shepherd moon' by having a 42 : 43 orbital resonance with particles in the ring, in terms of the period of their orbits. The ring and the moon are shown in Figure 1.\n[figure1]\n\nFigure 1: Left: Neptune as seen by the Voyager 2 mission in August 1989, a few days before its flyby. Credit: NASA / JPL / Voyager-ISS / Justin Cowart.\n\nRight: Neptune and its ring system as imaged in the infrared by the NIRCam instrument on the James Webb Space Telescope in July 2022. Multiple moons and rings are visible, with Galatea and the Adams ring labelled. Credit: NASA / ESA / CSA / STScI / Joseph DePasquale.\n\nThe semi-major axis of Galatea is $61953 \\mathrm{~km}$. Assume the moon and the ring particles travel in circular orbits.\n\nMeasurements of the amplitude of ripples in the ring caused by gravitational attraction towards the moon indicate the ring particles closest to the moon have a radial acceleration of $0.15 \\mathrm{~mm} \\mathrm{~s}^{-1}$. Estimate the mass of the moon.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nAlthough Saturn is famous for its rings, all of the gas giants in the Solar System have ring systems. The outer ring is known as the Adams ring and is very thin. Normally such a thin structure would widen over time so there needs to be a process keeping it constrained. One hypothesis is that the Neptunian moon Galatea, with an orbit just slightly smaller than the ring, acts as a 'shepherd moon' by having a 42 : 43 orbital resonance with particles in the ring, in terms of the period of their orbits. The ring and the moon are shown in Figure 1.\n[figure1]\n\nFigure 1: Left: Neptune as seen by the Voyager 2 mission in August 1989, a few days before its flyby. Credit: NASA / JPL / Voyager-ISS / Justin Cowart.\n\nRight: Neptune and its ring system as imaged in the infrared by the NIRCam instrument on the James Webb Space Telescope in July 2022. Multiple moons and rings are visible, with Galatea and the Adams ring labelled. Credit: NASA / ESA / CSA / STScI / Joseph DePasquale.\n\nThe semi-major axis of Galatea is $61953 \\mathrm{~km}$. Assume the moon and the ring particles travel in circular orbits.\n\nMeasurements of the amplitude of ripples in the ring caused by gravitational attraction towards the moon indicate the ring particles closest to the moon have a radial acceleration of $0.15 \\mathrm{~mm} \\mathrm{~s}^{-1}$. Estimate the mass of the moon.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of kg, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-05.jpg?height=574&width=1420&top_left_y=838&top_left_x=318" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "kg" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_663", "problem": "运城市位于山西省南端黄河金三角地区，与陕西、河南两省隔黄河而相望，近年来随着经济的增长, 环境却越来越糟糕: 冬天的雾霔极为严重, 某些企业对水和空气的污染让周边的居民苦不堪言...... 为对运城市及相邻区域内的环境进行有效的监测, 现计划发射一颗环境监测卫星, 要求该卫星每天同一时刻以相同的方向通过运城市的正上空,已知运城市的地理坐标在东经 $110^{\\circ} 30^{\\prime}$ 和北纬 $35^{\\circ} 30^{\\prime}$, 下列四选项为康杰中学的同学们对该卫星的运行参数的讨论稿，其中说法正确的是（ ）\nA: 该卫星轨道可以是周期为 120 分钟的近极地轨道\nB: 该卫星的轨道必须为地球同步卫星轨道\nC: 该卫星的轨道平面必过地心且与赤道平面呈 $35^{\\circ} 30^{\\prime}$ 的夹角\nD: 该卫星的转动角速度约为 $7.3 \\times 10^{-5} \\mathrm{rad} / \\mathrm{s}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n运城市位于山西省南端黄河金三角地区，与陕西、河南两省隔黄河而相望，近年来随着经济的增长, 环境却越来越糟糕: 冬天的雾霔极为严重, 某些企业对水和空气的污染让周边的居民苦不堪言...... 为对运城市及相邻区域内的环境进行有效的监测, 现计划发射一颗环境监测卫星, 要求该卫星每天同一时刻以相同的方向通过运城市的正上空,已知运城市的地理坐标在东经 $110^{\\circ} 30^{\\prime}$ 和北纬 $35^{\\circ} 30^{\\prime}$, 下列四选项为康杰中学的同学们对该卫星的运行参数的讨论稿，其中说法正确的是（ ）\n\nA: 该卫星轨道可以是周期为 120 分钟的近极地轨道\nB: 该卫星的轨道必须为地球同步卫星轨道\nC: 该卫星的轨道平面必过地心且与赤道平面呈 $35^{\\circ} 30^{\\prime}$ 的夹角\nD: 该卫星的转动角速度约为 $7.3 \\times 10^{-5} \\mathrm{rad} / \\mathrm{s}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1223", "problem": "GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\n[figure1]\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.c. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\\odot}$, a radius of $1.41 R_{\\odot}$ and a surface temperature of $112000 \\mathrm{~K}$, with a measured $v_{\\infty}$ of $1545 \\mathrm{~km} \\mathrm{~s}^{-1}$.\nii. The nebula is 5 arcmins in diameter $\\left(1^{\\circ}=60 \\mathrm{arcmin}\\right)$ and $4.8 \\mathrm{kpc}$ away, and at its edge is a bright thin shell of swept up material expanding at a rate of $30 \\mathrm{~km} \\mathrm{~s}^{-1}$. The age of such a nebula, $t$, is related to the current values of radius, $R$, and expansion speed, $v$, by $t=0.55 R / v$. Using this model, determine the age of the nebula.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nGW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\n[figure1]\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nproblem:\nc. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\\odot}$, a radius of $1.41 R_{\\odot}$ and a surface temperature of $112000 \\mathrm{~K}$, with a measured $v_{\\infty}$ of $1545 \\mathrm{~km} \\mathrm{~s}^{-1}$.\nii. The nebula is 5 arcmins in diameter $\\left(1^{\\circ}=60 \\mathrm{arcmin}\\right)$ and $4.8 \\mathrm{kpc}$ away, and at its edge is a bright thin shell of swept up material expanding at a rate of $30 \\mathrm{~km} \\mathrm{~s}^{-1}$. The age of such a nebula, $t$, is related to the current values of radius, $R$, and expansion speed, $v$, by $t=0.55 R / v$. Using this model, determine the age of the nebula.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-06.jpg?height=802&width=1308&top_left_y=1709&top_left_x=383" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_872", "problem": "Two protons $A$ and $B$ lie in the solar interior. In the rest frame of proton $A$, the proton $B$ approaches it radially from a large distance with speed $0.9 c$. In the rest frame of proton A,\nidentify the radius of the \"classically forbidden\" region for proton B (i.e. the region in which proton B cannot enter).\nA: $6.6 \\times 10^{-12} \\mathrm{~m}$\nB: $4.1 \\times 10^{-15} \\mathrm{~m}$\nC: $2.3 \\times 10^{-15} \\mathrm{~m}$\nD: $3.8 \\times 10^{-18} \\mathrm{~m}$\nE: $1.2 \\times 10^{-18} \\mathrm{~m}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo protons $A$ and $B$ lie in the solar interior. In the rest frame of proton $A$, the proton $B$ approaches it radially from a large distance with speed $0.9 c$. In the rest frame of proton A,\nidentify the radius of the \"classically forbidden\" region for proton B (i.e. the region in which proton B cannot enter).\n\nA: $6.6 \\times 10^{-12} \\mathrm{~m}$\nB: $4.1 \\times 10^{-15} \\mathrm{~m}$\nC: $2.3 \\times 10^{-15} \\mathrm{~m}$\nD: $3.8 \\times 10^{-18} \\mathrm{~m}$\nE: $1.2 \\times 10^{-18} \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_805", "problem": "For the five maneuvers described above, rank the resulting apogees from lowest to highest. Assume the change in velocity is small relative to orbital velocity, but not negligible.\nA: $2<3=4=5<1$\nB: $2=3<5<4=1$\nC: $2<3=4<5<1$\nD: $2<5<3=4<1$\nE: $2<3<5<4<1$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nFor the five maneuvers described above, rank the resulting apogees from lowest to highest. Assume the change in velocity is small relative to orbital velocity, but not negligible.\n\nA: $2<3=4=5<1$\nB: $2=3<5<4=1$\nC: $2<3=4<5<1$\nD: $2<5<3=4<1$\nE: $2<3<5<4<1$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_758", "problem": "The Jovian planets are ...\nA: Mercury, Venus, Earth, Mars\nB: Mercury, Venus, Earth, Mars, Jupiter, Saturn\nC: Jupiter, Saturn, Uranus, Neptune\nD: Uranus, Neptune\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe Jovian planets are ...\n\nA: Mercury, Venus, Earth, Mars\nB: Mercury, Venus, Earth, Mars, Jupiter, Saturn\nC: Jupiter, Saturn, Uranus, Neptune\nD: Uranus, Neptune\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_877", "problem": "A comet's orbit has the following characteristics: eccentricity $\\mathrm{e}=0.995$; aphelion distance $r_{a}=5 \\cdot 10^{4} \\mathrm{AU}$. Assume we know the mass of the Sun $M_{S}=1.98 \\cdot 10^{30} \\mathrm{~kg}$, and gravitational constant $G=6.67 \\cdot 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$. Determine the velocity of the comet at its aphelion.\nA: $34.76 \\mathrm{~m} / \\mathrm{s}$\nB: $20.57 \\mathrm{~m} / \\mathrm{s}$\nC: $187.91 \\mathrm{~m} / \\mathrm{s}$\nD: $63.38 \\mathrm{~m} / \\mathrm{s}$\nE: $9.19 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA comet's orbit has the following characteristics: eccentricity $\\mathrm{e}=0.995$; aphelion distance $r_{a}=5 \\cdot 10^{4} \\mathrm{AU}$. Assume we know the mass of the Sun $M_{S}=1.98 \\cdot 10^{30} \\mathrm{~kg}$, and gravitational constant $G=6.67 \\cdot 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$. Determine the velocity of the comet at its aphelion.\n\nA: $34.76 \\mathrm{~m} / \\mathrm{s}$\nB: $20.57 \\mathrm{~m} / \\mathrm{s}$\nC: $187.91 \\mathrm{~m} / \\mathrm{s}$\nD: $63.38 \\mathrm{~m} / \\mathrm{s}$\nE: $9.19 \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1184", "problem": "On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\\mathrm{x}, \\mathrm{y}, \\mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\\circ}$, and each orbital plane has 4 satellites.\n[figure1]\n\nFigure 1: The current set up of the GPS system used on Earth.\n\nCredits: Left: Peter H. Dana, University of Colorado;\n\nRight: GPS Standard Positioning Service Specification, $4^{\\text {th }}$ edition\n\nThe orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\\circ}$, and hence about $38 \\%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required.c. Using suitable calculations, explore the viability of a 24-satellite GPS constellation similar to the one used on Earth, in a semi-synchronous Martian orbit, by considering:\n\ni. Would the moons prevent such an orbit?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\\mathrm{x}, \\mathrm{y}, \\mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\\circ}$, and each orbital plane has 4 satellites.\n[figure1]\n\nFigure 1: The current set up of the GPS system used on Earth.\n\nCredits: Left: Peter H. Dana, University of Colorado;\n\nRight: GPS Standard Positioning Service Specification, $4^{\\text {th }}$ edition\n\nThe orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\\circ}$, and hence about $38 \\%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required.\n\nproblem:\nc. Using suitable calculations, explore the viability of a 24-satellite GPS constellation similar to the one used on Earth, in a semi-synchronous Martian orbit, by considering:\n\ni. Would the moons prevent such an orbit?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~km}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-04.jpg?height=512&width=1474&top_left_y=555&top_left_x=292" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ " \\mathrm{~km}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1129", "problem": "The Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\\text {peri }}=9.86 R_{\\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft.\n[figure1]\n\nFigure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman.\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nGiven that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.\n\nClose to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2.\n\n[figure2]\n\nFigure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL.\n\nWhen considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\\angle y c z$ in the figure, such that the shaded areas are the same).\n\n[figure3]\n\nFigure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia.\n\n\nThe eccentric anomaly can be related to the mean anomaly through Kepler's Equation,\n\n$$\nM=E-e \\sin E \\text {. }\n$$a. When the probe is at its closest perihelion:\n\ni. Calculate the apparent magnitude of the Sun, given that from Earth $m_{\\odot}=-26.74$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\\text {peri }}=9.86 R_{\\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft.\n[figure1]\n\nFigure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman.\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nGiven that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.\n\nClose to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2.\n\n[figure2]\n\nFigure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL.\n\nWhen considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\\angle y c z$ in the figure, such that the shaded areas are the same).\n\n[figure3]\n\nFigure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia.\n\n\nThe eccentric anomaly can be related to the mean anomaly through Kepler's Equation,\n\n$$\nM=E-e \\sin E \\text {. }\n$$\n\nproblem:\na. When the probe is at its closest perihelion:\n\ni. Calculate the apparent magnitude of the Sun, given that from Earth $m_{\\odot}=-26.74$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-04.jpg?height=708&width=1438&top_left_y=694&top_left_x=318", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-05.jpg?height=411&width=1539&top_left_y=383&top_left_x=264", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-05.jpg?height=603&width=714&top_left_y=1429&top_left_x=677" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1193", "problem": "On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\\mathrm{x}, \\mathrm{y}, \\mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\\circ}$, and each orbital plane has 4 satellites.\n[figure1]\n\nFigure 1: The current set up of the GPS system used on Earth.\n\nCredits: Left: Peter H. Dana, University of Colorado;\n\nRight: GPS Standard Positioning Service Specification, $4^{\\text {th }}$ edition\n\nThe orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\\circ}$, and hence about $38 \\%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required.ii. How would the GPS positional accuracy compare to Earth?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\\mathrm{x}, \\mathrm{y}, \\mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\\circ}$, and each orbital plane has 4 satellites.\n[figure1]\n\nFigure 1: The current set up of the GPS system used on Earth.\n\nCredits: Left: Peter H. Dana, University of Colorado;\n\nRight: GPS Standard Positioning Service Specification, $4^{\\text {th }}$ edition\n\nThe orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\\circ}$, and hence about $38 \\%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required.\n\nproblem:\nii. How would the GPS positional accuracy compare to Earth?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~m}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-04.jpg?height=512&width=1474&top_left_y=555&top_left_x=292" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ " \\mathrm{~m}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_303", "problem": "新时代的中国北斗导航系统是世界一流的。空间段由若干地球静止轨道卫星、倾斜地球同步轨道卫星和中圆地球轨道卫星组成。已知地球表面两极处的重力加速度为 $g_{0}$,赤道处的重力加速度为 $g_{1}$, 万有引力常量为 $G$ 。若把地球看成密度均匀、半径为 $R$ 的球\n体，下列说法正确的是（）\nA: 北斗地球同步卫星距离地球表面的高度 $h=\\left(\\sqrt[3]{\\frac{g_{1}}{g_{0}-g_{1}}}-1\\right) R$\nB: 北斗地球同步卫星距离地球表面的高度 $h=\\left(\\sqrt[3]{\\frac{g_{0}}{g_{0}-g_{1}}}-1\\right) R$\nC: 地球的平均密度 $\\rho=\\frac{3 g_{1}}{4 \\pi G R}$\nD: 地球的近地卫星的周期 $T_{0}=2 \\pi \\sqrt{\\frac{R}{g_{1}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n新时代的中国北斗导航系统是世界一流的。空间段由若干地球静止轨道卫星、倾斜地球同步轨道卫星和中圆地球轨道卫星组成。已知地球表面两极处的重力加速度为 $g_{0}$,赤道处的重力加速度为 $g_{1}$, 万有引力常量为 $G$ 。若把地球看成密度均匀、半径为 $R$ 的球\n体，下列说法正确的是（）\n\nA: 北斗地球同步卫星距离地球表面的高度 $h=\\left(\\sqrt[3]{\\frac{g_{1}}{g_{0}-g_{1}}}-1\\right) R$\nB: 北斗地球同步卫星距离地球表面的高度 $h=\\left(\\sqrt[3]{\\frac{g_{0}}{g_{0}-g_{1}}}-1\\right) R$\nC: 地球的平均密度 $\\rho=\\frac{3 g_{1}}{4 \\pi G R}$\nD: 地球的近地卫星的周期 $T_{0}=2 \\pi \\sqrt{\\frac{R}{g_{1}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_966", "problem": "\"Manhattanhenge\" is the name given to when, just before sunset or just after sunrise 4 times a year (twice for setting, and twice for rising), the Sun aligns with the east-west streets of the New York grid system. One of the setting dates this year was on 11th July. Which of these is another date you are likely to see the \"Manhattanhenge\" sunset?\nA: $30^{\\text {th }}$ May\nB: $30^{\\text {th }}$ June\nC: $11^{\\text {th }}$ December\nD: $11^{\\text {th }}$ January\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\n\"Manhattanhenge\" is the name given to when, just before sunset or just after sunrise 4 times a year (twice for setting, and twice for rising), the Sun aligns with the east-west streets of the New York grid system. One of the setting dates this year was on 11th July. Which of these is another date you are likely to see the \"Manhattanhenge\" sunset?\n\nA: $30^{\\text {th }}$ May\nB: $30^{\\text {th }}$ June\nC: $11^{\\text {th }}$ December\nD: $11^{\\text {th }}$ January\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_247", "problem": "2022 年 2 月 27 日上午, 长征八号遥二运载火箭在文昌航天发射场点火升空, 成功将 22 颗卫星送入预定轨道，这次发射，创造了我国一箭多星发射的新纪录。已知其中一颗卫星绕地球运行近似为匀速圆周运动, 到地面距离为 $h$, 地球半径为 $R$, 地球表面的重力加速度为 $g$, 下列说法正确的是 ( )\nA: 该卫星的向心加速度小于 $g$\nB: 该卫星的运行速度有可能等于第一宇宙速度\nC: 由题干条件无法求出地球的质量\nD: 由于稀薄大气的阻力影响, 该卫星运行的轨道半径会变小, 速度也变小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2022 年 2 月 27 日上午, 长征八号遥二运载火箭在文昌航天发射场点火升空, 成功将 22 颗卫星送入预定轨道，这次发射，创造了我国一箭多星发射的新纪录。已知其中一颗卫星绕地球运行近似为匀速圆周运动, 到地面距离为 $h$, 地球半径为 $R$, 地球表面的重力加速度为 $g$, 下列说法正确的是 ( )\n\nA: 该卫星的向心加速度小于 $g$\nB: 该卫星的运行速度有可能等于第一宇宙速度\nC: 由题干条件无法求出地球的质量\nD: 由于稀薄大气的阻力影响, 该卫星运行的轨道半径会变小, 速度也变小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_721", "problem": "地球赤道上有一物体随地球的自转而做圆周运动,所受的向心力为 $F_{l}$, 向心加速度为 $a_{1}$, 线速度为 $v_{1}$, 角速度为 $\\omega_{1}$; 绕地球表面附近做圆周运动的人造卫星受的向心力为 $F_{2}$, 向心加速度为 $a_{2}$, 线速度为 $v_{2}$, 角速度为 $\\omega_{2}$; 地球同步卫星所受的向心力为 $F_{3}$, 向心加速度为 $a_{3}$, 线速度为 $v_{3}$, 角速度为 $\\omega_{3}$. 地球表面重力加速度为 $g$,第一宇宙速度为 $\\mathrm{v}$,假设三者质量相等,下列结论中错误的是( )\nA: $F_{2}>F_{3}>F_{1}$\nB: $a_{2}=g>a_{3}>a_{1}$\nC: $v_{1}=v_{2}=v>v_{3}$\nD: $\\omega_{1}=\\omega_{3}<\\omega_{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球赤道上有一物体随地球的自转而做圆周运动,所受的向心力为 $F_{l}$, 向心加速度为 $a_{1}$, 线速度为 $v_{1}$, 角速度为 $\\omega_{1}$; 绕地球表面附近做圆周运动的人造卫星受的向心力为 $F_{2}$, 向心加速度为 $a_{2}$, 线速度为 $v_{2}$, 角速度为 $\\omega_{2}$; 地球同步卫星所受的向心力为 $F_{3}$, 向心加速度为 $a_{3}$, 线速度为 $v_{3}$, 角速度为 $\\omega_{3}$. 地球表面重力加速度为 $g$,第一宇宙速度为 $\\mathrm{v}$,假设三者质量相等,下列结论中错误的是( )\n\nA: $F_{2}>F_{3}>F_{1}$\nB: $a_{2}=g>a_{3}>a_{1}$\nC: $v_{1}=v_{2}=v>v_{3}$\nD: $\\omega_{1}=\\omega_{3}<\\omega_{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_308", "problem": "某天体科学家在太阳系外发现一颗类地球行星, 这颗类地行星绕中心恒星做圆周运动, 公转的周期为 146 天, 体积是地球体积的 8 倍, 行星表面的重力加速度是地球表面重力加速度的 2 倍, 它与中心恒星间的距离跟地球和太阳的距离相近。地球公转周期为 365 天，类地行星和地球均看作密度均匀的球体。求:\n\n类地行星的中心恒星质量与太阳的质量之比 $\\frac{M_{\\text {恒 }}}{M_{\\text {日 }}}$ 为多少?", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n某天体科学家在太阳系外发现一颗类地球行星, 这颗类地行星绕中心恒星做圆周运动, 公转的周期为 146 天, 体积是地球体积的 8 倍, 行星表面的重力加速度是地球表面重力加速度的 2 倍, 它与中心恒星间的距离跟地球和太阳的距离相近。地球公转周期为 365 天，类地行星和地球均看作密度均匀的球体。求:\n\n类地行星的中心恒星质量与太阳的质量之比 $\\frac{M_{\\text {恒 }}}{M_{\\text {日 }}}$ 为多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_460", "problem": "质量相等的甲、乙两颗卫星分别贴近某星球表面和地球表面围绕其做匀速圆周运动,已知该星球和地球的密度相同, 半径分别为 $R$ 和 $r$, 则 ( )\nA: 甲、乙两颗卫星的加速度之比等于 $R: r$\nB: 甲、乙两颗卫星所受的向心力之比等于 $1: 1$\nC: 甲、乙两颗卫星的线速度之比等于 $1: 1$\nD: 甲、乙两颗卫星的周期之比等于 $1: 1$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n质量相等的甲、乙两颗卫星分别贴近某星球表面和地球表面围绕其做匀速圆周运动,已知该星球和地球的密度相同, 半径分别为 $R$ 和 $r$, 则 ( )\n\nA: 甲、乙两颗卫星的加速度之比等于 $R: r$\nB: 甲、乙两颗卫星所受的向心力之比等于 $1: 1$\nC: 甲、乙两颗卫星的线速度之比等于 $1: 1$\nD: 甲、乙两颗卫星的周期之比等于 $1: 1$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_757", "problem": "A comet's tail points in the following direction:\nA: away from the Sun\nB: towards the Sun\nC: in the direction of movement\nD: against the direction of movement\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA comet's tail points in the following direction:\n\nA: away from the Sun\nB: towards the Sun\nC: in the direction of movement\nD: against the direction of movement\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_999", "problem": "What is the approximate average thickness of Saturn's rings?\nA: $10 \\mu \\mathrm{m}$\nB: $10 \\mathrm{~mm}$\nC: $10 \\mathrm{~m}$\nD: $10 \\mathrm{~km}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the approximate average thickness of Saturn's rings?\n\nA: $10 \\mu \\mathrm{m}$\nB: $10 \\mathrm{~mm}$\nC: $10 \\mathrm{~m}$\nD: $10 \\mathrm{~km}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_331", "problem": "地球到太阳的距离为水星到太阳距离的 2.6 倍, 那么地球和水星绕太阳运行的线速度之比为 (设地球和水星绕太阳运行的轨道为圆)（）\nA: $\\frac{1}{2.6}$\nB: $\\frac{2.6}{1}$\nC: $\\frac{1}{\\sqrt{2.6}}$\nD: $\\frac{\\sqrt{2.6}}{1}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n地球到太阳的距离为水星到太阳距离的 2.6 倍, 那么地球和水星绕太阳运行的线速度之比为 (设地球和水星绕太阳运行的轨道为圆)（）\n\nA: $\\frac{1}{2.6}$\nB: $\\frac{2.6}{1}$\nC: $\\frac{1}{\\sqrt{2.6}}$\nD: $\\frac{\\sqrt{2.6}}{1}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_771", "problem": "Galileo discovered with his telescope that Venus ...\nA: has a diameter similar to the Earth\nB: has mountains on the surface.\nC: has phases like the Moon.\nD: has clouds in the atmosphere.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nGalileo discovered with his telescope that Venus ...\n\nA: has a diameter similar to the Earth\nB: has mountains on the surface.\nC: has phases like the Moon.\nD: has clouds in the atmosphere.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_201", "problem": "太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间, 且三者几乎排成一条直线时, 天文学称这种现象为“行星冲日”。已知 2020 年 7 月 21 日土星冲日, 土星绕太阳运动的轨道半径约为地球绕太阳运动的轨道半径的 9.5 倍, 则下一次土星冲日的时间约为 ( )\nA: 2021 年 8 月\nB: 2022 年 7 月\nC: 2023 年 8 月\nD: 2024 年 7 月\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间, 且三者几乎排成一条直线时, 天文学称这种现象为“行星冲日”。已知 2020 年 7 月 21 日土星冲日, 土星绕太阳运动的轨道半径约为地球绕太阳运动的轨道半径的 9.5 倍, 则下一次土星冲日的时间约为 ( )\n\nA: 2021 年 8 月\nB: 2022 年 7 月\nC: 2023 年 8 月\nD: 2024 年 7 月\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_518", "problem": "2018 年 12 月 8 日中国在西昌卫星发射中心成功发射了嫦娥四号探测器, 经过地月转移飞行进入环月椭圆轨道, 然后实施近月制动, 顺利完成“太空刹车”, 被月球捕获,进入距月球表面约 $100 \\mathrm{~km}$ 的环月圆形轨道, 准备登陆月球背面。如图所示, 关于嫦娥四号在环月椭圆轨道和环月圆形轨道的运行, 下列说法正确的是 ( )\n\n[图1]\nA: 在环月粗圆轨道运行时, $A$ 点速度小于 $B$ 点速度\nB: 由环月椭圆轨道进入环月圆形轨道, 嫦娥四号的机械能减小\nC: 在环月制圆轨道上通过 $A$ 点的加速度比在环月圆形轨道通过 $A$ 点的加速度大\nD: 在环月椭圆轨道运行的周期比环月圆形轨道的周期小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2018 年 12 月 8 日中国在西昌卫星发射中心成功发射了嫦娥四号探测器, 经过地月转移飞行进入环月椭圆轨道, 然后实施近月制动, 顺利完成“太空刹车”, 被月球捕获,进入距月球表面约 $100 \\mathrm{~km}$ 的环月圆形轨道, 准备登陆月球背面。如图所示, 关于嫦娥四号在环月椭圆轨道和环月圆形轨道的运行, 下列说法正确的是 ( )\n\n[图1]\n\nA: 在环月粗圆轨道运行时, $A$ 点速度小于 $B$ 点速度\nB: 由环月椭圆轨道进入环月圆形轨道, 嫦娥四号的机械能减小\nC: 在环月制圆轨道上通过 $A$ 点的加速度比在环月圆形轨道通过 $A$ 点的加速度大\nD: 在环月椭圆轨道运行的周期比环月圆形轨道的周期小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-31.jpg?height=283&width=483&top_left_y=2451&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_56", "problem": "2021 年 5 月 15 日 7 时 18 分, 我国火星探测器“天问一号”的着陆巡视器 (其中巡视器就是“祝融号”火星车) 成功着陆于火星乌托邦平原南部预选着陆区, 这标志着我国是继美国、前苏联之后第三个成功进行火星探测的国家, 展示了中国在航天技术领域的强大实力, 为中国乃至国际航天事业迈出了历史性的一大步。“天问一号”的着陆巡视器从进入火星大气层到成功着陆经历了气动减速段、伞系减速段、动力减速段、悬停避障与缓速下降段, 其过程大致如图所示。已知火星质量为 $6.42 \\times 10^{23} \\mathrm{~kg}$ (约为地球质量的 0.11 倍)、半径为 $3395 \\mathrm{~km}$ (约为地球半径的 0.53 倍), “天问一号”的着陆巡视器质量为 $1.3 \\mathrm{t}$, 地球表面重力加速度为 $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$ 。（计算）设着陆巡视器在伞系减速段做的是坚直方向的匀减速直线运动, 试求火星大气对着陆巡视器的平均阻力。(结果保留 3 位有效数字)\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n2021 年 5 月 15 日 7 时 18 分, 我国火星探测器“天问一号”的着陆巡视器 (其中巡视器就是“祝融号”火星车) 成功着陆于火星乌托邦平原南部预选着陆区, 这标志着我国是继美国、前苏联之后第三个成功进行火星探测的国家, 展示了中国在航天技术领域的强大实力, 为中国乃至国际航天事业迈出了历史性的一大步。“天问一号”的着陆巡视器从进入火星大气层到成功着陆经历了气动减速段、伞系减速段、动力减速段、悬停避障与缓速下降段, 其过程大致如图所示。已知火星质量为 $6.42 \\times 10^{23} \\mathrm{~kg}$ (约为地球质量的 0.11 倍)、半径为 $3395 \\mathrm{~km}$ (约为地球半径的 0.53 倍), “天问一号”的着陆巡视器质量为 $1.3 \\mathrm{t}$, 地球表面重力加速度为 $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$ 。（计算）设着陆巡视器在伞系减速段做的是坚直方向的匀减速直线运动, 试求火星大气对着陆巡视器的平均阻力。(结果保留 3 位有效数字)\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以N为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-027.jpg?height=734&width=531&top_left_y=161&top_left_x=357" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "N" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_392", "problem": "影片《流浪地球》中地球脱离太阳系流浪的最终目标是进入离太阳系最近的比邻星系的合适轨道, 成为这颗恒星的行星。现实中欧洲南方天文台曾宣布在离地球最近的比邻星发现宜居行星“比邻星 $b$ ”, 该行星质量约为地球的 1.3 倍, 直径约为地球的 22 倍,绕比邻星公转周期 11.2 天, 与比邻星距离约为日地距离的 5\\%, 若不考虑星球的自转效应, 则 ( )\n\n[图1]\nA: 比邻星的质量大于太阳的质量\nB: 比邻星的密度小于太阳的密度\nC: “比邻星 $b$ ”的公转线速度大小小于地球的公转线速度大小\nD: “比邻星 $b$ ”表面的重力加速度小于地球表面的重力加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n影片《流浪地球》中地球脱离太阳系流浪的最终目标是进入离太阳系最近的比邻星系的合适轨道, 成为这颗恒星的行星。现实中欧洲南方天文台曾宣布在离地球最近的比邻星发现宜居行星“比邻星 $b$ ”, 该行星质量约为地球的 1.3 倍, 直径约为地球的 22 倍,绕比邻星公转周期 11.2 天, 与比邻星距离约为日地距离的 5\\%, 若不考虑星球的自转效应, 则 ( )\n\n[图1]\n\nA: 比邻星的质量大于太阳的质量\nB: 比邻星的密度小于太阳的密度\nC: “比邻星 $b$ ”的公转线速度大小小于地球的公转线速度大小\nD: “比邻星 $b$ ”表面的重力加速度小于地球表面的重力加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-036.jpg?height=203&width=417&top_left_y=841&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_859", "problem": "With the technology currently available, it would take hundreds of millennia to send a humanmade object to other stars. A possible solution to this problem is to use relativistic light sails, which consist of very small probes propelled by radiation pressure. It is estimated that on the reference frame of an Earth observer, these sails would take 20.0 years to reach Alpha Centauri, which is 4.37 light-years away from the Solar System. The velocity of a light sail can be assumed to be constant throughout the entire trip. How long would this trip be on the reference frame of the light sail?\nA: 18.5 years\nB: 19.0 years\nC: 19.5 years\nD: 20.0 years\nE: 20.5 years\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWith the technology currently available, it would take hundreds of millennia to send a humanmade object to other stars. A possible solution to this problem is to use relativistic light sails, which consist of very small probes propelled by radiation pressure. It is estimated that on the reference frame of an Earth observer, these sails would take 20.0 years to reach Alpha Centauri, which is 4.37 light-years away from the Solar System. The velocity of a light sail can be assumed to be constant throughout the entire trip. How long would this trip be on the reference frame of the light sail?\n\nA: 18.5 years\nB: 19.0 years\nC: 19.5 years\nD: 20.0 years\nE: 20.5 years\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_854", "problem": "The spectrum of a blackbody peaks at a wavelength inversely proportional to its temperature. This is known as Wein's law and is used to estimate stellar temperatures. The sun can be approximated as a blackbody with its peak wavelength in the visible portion of the spectrum and a surface temperature of $6000 \\mathrm{~K}$. Given this data, estimate the peak wavelength of a human being, assuming it to be a black body.\nA: $10 \\mathrm{~nm}$\nB: $10 \\mu \\mathrm{m}$\nC: $10 \\mathrm{~mm}$\nD: $10 \\mathrm{~m}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe spectrum of a blackbody peaks at a wavelength inversely proportional to its temperature. This is known as Wein's law and is used to estimate stellar temperatures. The sun can be approximated as a blackbody with its peak wavelength in the visible portion of the spectrum and a surface temperature of $6000 \\mathrm{~K}$. Given this data, estimate the peak wavelength of a human being, assuming it to be a black body.\n\nA: $10 \\mathrm{~nm}$\nB: $10 \\mu \\mathrm{m}$\nC: $10 \\mathrm{~mm}$\nD: $10 \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_759", "problem": "What is expected to happen at the end of the Sun's lifetime?\nA: Become a white dwarf\nB: Become a brown dwarf\nC: Explode as a supernova\nD: Become a neutron star or black hole\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is expected to happen at the end of the Sun's lifetime?\n\nA: Become a white dwarf\nB: Become a brown dwarf\nC: Explode as a supernova\nD: Become a neutron star or black hole\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_442", "problem": "宇宙中两个相距较近的星球可以看成双星, 它们只在相互间的万有引力作用下, 绕二球心连线上的某一固定点做周期相同的匀速圆周运动. 根据宇宙大爆炸理论. 双星间的距离在不断缓慢增加, 设双星仍做匀速圆周运动, 则下列说法正确的是(\nA: 双星相互间的万有引力减小\nB: 双星做圆周运动的角速度增大\nC: 双星做圆周运动的周期减小\nD: 双星做圆周运动的半径增大\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n宇宙中两个相距较近的星球可以看成双星, 它们只在相互间的万有引力作用下, 绕二球心连线上的某一固定点做周期相同的匀速圆周运动. 根据宇宙大爆炸理论. 双星间的距离在不断缓慢增加, 设双星仍做匀速圆周运动, 则下列说法正确的是(\n\nA: 双星相互间的万有引力减小\nB: 双星做圆周运动的角速度增大\nC: 双星做圆周运动的周期减小\nD: 双星做圆周运动的半径增大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_846", "problem": "The spectral line $H_{\\alpha}$ in the spectrum of a star is recorded as having displacement of $\\Delta \\lambda=$ $0.043 \\times 10^{-10} \\mathrm{~m}$. At rest, the spectral line has a wavelength of $\\lambda_{0}=6.563 \\times 10^{-7} \\mathrm{~m}$. Calculate the period of rotation for this star, if it is observed from its equatorial plane. We also know: $R_{\\text {star }}=8 \\times 10^{5} \\mathrm{~km}$.\nA: 29.59 days\nB: 14.63 days\nC: 21.15 days\nD: 34.39 days\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe spectral line $H_{\\alpha}$ in the spectrum of a star is recorded as having displacement of $\\Delta \\lambda=$ $0.043 \\times 10^{-10} \\mathrm{~m}$. At rest, the spectral line has a wavelength of $\\lambda_{0}=6.563 \\times 10^{-7} \\mathrm{~m}$. Calculate the period of rotation for this star, if it is observed from its equatorial plane. We also know: $R_{\\text {star }}=8 \\times 10^{5} \\mathrm{~km}$.\n\nA: 29.59 days\nB: 14.63 days\nC: 21.15 days\nD: 34.39 days\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1110", "problem": "The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.c. Considering the evaluated equations for $\\tau, R$, and $q$ we can use this with the measured luminosity of the Sun to get a new estimate for the central temperature.\n\niii. Since $q \\propto \\tau^{2} e^{-\\tau}$ and $\\tau \\propto T^{-1 / 3}$, it can be approximated at a given temperature as $q \\propto T^{\\alpha}$, quantifying the sensitivity of the fusion reaction to temperature. By considering $d(\\ln q) / d(\\ln$ T) give an expression for $\\alpha$ as a function of $\\tau$ and calculate it at your central temperature.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.\n\nproblem:\nc. Considering the evaluated equations for $\\tau, R$, and $q$ we can use this with the measured luminosity of the Sun to get a new estimate for the central temperature.\n\niii. Since $q \\propto \\tau^{2} e^{-\\tau}$ and $\\tau \\propto T^{-1 / 3}$, it can be approximated at a given temperature as $q \\propto T^{\\alpha}$, quantifying the sensitivity of the fusion reaction to temperature. By considering $d(\\ln q) / d(\\ln$ T) give an expression for $\\alpha$ as a function of $\\tau$ and calculate it at your central temperature.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-06.jpg?height=536&width=1508&top_left_y=560&top_left_x=272", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-07.jpg?height=748&width=1185&top_left_y=1850&top_left_x=433" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_260", "problem": "如图所示, $1 、 3$ 轨道均是卫星绕地球做圆周运动的轨道示意图, 1 轨道半径为 $R$, 2 轨道是一颗卫星绕地球做椭圆运动的轨道示意图, 3 轨道与 2 轨道相切于 $B$ 点, $O$ 点为地球球心, $A B$ 为椭圆的长轴, 两轨道和地心都在同一平面内。已知在 $1 、 2$ 两轨道上运动的卫星的周期相等, 万有引力常量为 $G$, 地球质量为 $M$, 三颗卫星的质量相等, 下列说法正确的是 ( )\n\n[图1]\nA: 卫星在 3 轨道上的机械能小于在 2 轨道上的机械能\nB: 若卫星在 I 轨道的速率为 $v_{l}$, 卫星在 2 轨道 $A$ 点的速率为 $\\mathrm{v}_{\\mathrm{A}}$, 则 $v_{l}<\\mathrm{v}_{\\mathrm{A}}$\nC: 若卫星在 $1 、 3$ 轨道的加速度大小分别为 $a_{1} 、 a_{3}$, 卫星在 2 轨道 $B$ 点加速度大小为 $a_{B}$, 则 $a_{B}=a_{3}\\sqrt{\\frac{5 G M}{8 R}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, $1 、 3$ 轨道均是卫星绕地球做圆周运动的轨道示意图, 1 轨道半径为 $R$, 2 轨道是一颗卫星绕地球做椭圆运动的轨道示意图, 3 轨道与 2 轨道相切于 $B$ 点, $O$ 点为地球球心, $A B$ 为椭圆的长轴, 两轨道和地心都在同一平面内。已知在 $1 、 2$ 两轨道上运动的卫星的周期相等, 万有引力常量为 $G$, 地球质量为 $M$, 三颗卫星的质量相等, 下列说法正确的是 ( )\n\n[图1]\n\nA: 卫星在 3 轨道上的机械能小于在 2 轨道上的机械能\nB: 若卫星在 I 轨道的速率为 $v_{l}$, 卫星在 2 轨道 $A$ 点的速率为 $\\mathrm{v}_{\\mathrm{A}}$, 则 $v_{l}<\\mathrm{v}_{\\mathrm{A}}$\nC: 若卫星在 $1 、 3$ 轨道的加速度大小分别为 $a_{1} 、 a_{3}$, 卫星在 2 轨道 $B$ 点加速度大小为 $a_{B}$, 则 $a_{B}=a_{3}\\sqrt{\\frac{5 G M}{8 R}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-116.jpg?height=468&width=523&top_left_y=1953&top_left_x=344" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1025", "problem": "Mercury is the innermost of the Solar System's planets and so is most influenced by gravitational interactions with the Sun, tying its rotational period to its orbital period in a similar way to the tidal locking between the Moon and the Earth. It orbits the Sun in a 3:2 resonance, meaning that it rotates on its axis three times for every two orbits of the Sun.\n\n[figure1]\n\nFigure 1: True colour image of Mercury taken by the probe MESSENGER after its closest approach in 2008. Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie.\n\nMercury has an orbital period of 88 Earth days and a radius of $2440 \\mathrm{~km}$, and spins in the same direction as it orbits (both are anti-clockwise as viewed from high above the Sun).\n
Hence calculate the speed an astronaut would need to move at whilst travelling along its equator in order to keep the Sun in the same position in the sky. Assume Mercury has no axial tilt, and give your answer in $\\mathrm{km} \\mathrm{h}^{-1}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nMercury is the innermost of the Solar System's planets and so is most influenced by gravitational interactions with the Sun, tying its rotational period to its orbital period in a similar way to the tidal locking between the Moon and the Earth. It orbits the Sun in a 3:2 resonance, meaning that it rotates on its axis three times for every two orbits of the Sun.\n\n[figure1]\n\nFigure 1: True colour image of Mercury taken by the probe MESSENGER after its closest approach in 2008. Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie.\n\nMercury has an orbital period of 88 Earth days and a radius of $2440 \\mathrm{~km}$, and spins in the same direction as it orbits (both are anti-clockwise as viewed from high above the Sun).\n
Hence calculate the speed an astronaut would need to move at whilst travelling along its equator in order to keep the Sun in the same position in the sky. Assume Mercury has no axial tilt, and give your answer in $\\mathrm{km} \\mathrm{h}^{-1}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of km/h, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_13148c5721a741e30941g-06.jpg?height=1005&width=1330&top_left_y=734&top_left_x=363" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "km/h" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_75", "problem": "在星球 $M$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $P$ 轻放在弹簧上端, $P$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $N$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $M$ 的半径是星球 $N$ 的 3 倍，则（）\n\n[图1]\nA: $M$ 的质量是 $N$ 的 6 倍\nB: $Q$ 的质量是 $P$ 的 6 倍\nC: $M$ 与 $N$ 的密度相等\nD: 贴近 $M$ 表面运行的卫星的周期是贴近 $N$ 表面运行的卫星的周期的 6 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n在星球 $M$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $P$ 轻放在弹簧上端, $P$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $N$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $M$ 的半径是星球 $N$ 的 3 倍，则（）\n\n[图1]\n\nA: $M$ 的质量是 $N$ 的 6 倍\nB: $Q$ 的质量是 $P$ 的 6 倍\nC: $M$ 与 $N$ 的密度相等\nD: 贴近 $M$ 表面运行的卫星的周期是贴近 $N$ 表面运行的卫星的周期的 6 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-098.jpg?height=357&width=516&top_left_y=1649&top_left_x=356" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_234", "problem": "2021 年 7 月 22 日上午 8 时我国首次公开了中国空间站核心舱组合体的运动轨道参数。现将其运行轨道简化为如图所示的绕地球 $O$ 运动的椭圆轨道, 地球位于制圆的一个焦点上, 其中 $A$ 为近地点, $B$ 为远地点。假设每隔 $\\Delta t$ 时间记录一次核心舱的位置, 记录点如图所示, 已知 $E$ 为椭圆轨道的中心, $C 、 D 、 E$ 在同一条直线上且 $C D \\perp A B, A B$的距离为 $2 a, C D$ 的距离为 $2 b$, 椭圆的面积公式为 $S=\\pi a b$, 则核心舱从 $C$ 运动到 $B$ 所需的最短时间为 ( )\n\n[图1]\nA: $\\frac{7 \\Delta t}{2}$\nB: $\\frac{7 \\Delta t}{2}+\\frac{7 \\sqrt{a^{2}-b^{2}} \\Delta t}{\\pi a}$\nC: $\\frac{7 \\Delta t}{2}+\\frac{7 \\sqrt{a^{2}-b^{2}} \\Delta t}{\\pi b}$\nD: $12 \\Delta t$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2021 年 7 月 22 日上午 8 时我国首次公开了中国空间站核心舱组合体的运动轨道参数。现将其运行轨道简化为如图所示的绕地球 $O$ 运动的椭圆轨道, 地球位于制圆的一个焦点上, 其中 $A$ 为近地点, $B$ 为远地点。假设每隔 $\\Delta t$ 时间记录一次核心舱的位置, 记录点如图所示, 已知 $E$ 为椭圆轨道的中心, $C 、 D 、 E$ 在同一条直线上且 $C D \\perp A B, A B$的距离为 $2 a, C D$ 的距离为 $2 b$, 椭圆的面积公式为 $S=\\pi a b$, 则核心舱从 $C$ 运动到 $B$ 所需的最短时间为 ( )\n\n[图1]\n\nA: $\\frac{7 \\Delta t}{2}$\nB: $\\frac{7 \\Delta t}{2}+\\frac{7 \\sqrt{a^{2}-b^{2}} \\Delta t}{\\pi a}$\nC: $\\frac{7 \\Delta t}{2}+\\frac{7 \\sqrt{a^{2}-b^{2}} \\Delta t}{\\pi b}$\nD: $12 \\Delta t$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-103.jpg?height=546&width=691&top_left_y=184&top_left_x=334", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-104.jpg?height=605&width=760&top_left_y=183&top_left_x=405" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_624", "problem": "2019 年 10 月 28 日发生了天王星冲日现象, 即太阳、地球、天王星处于同一直线,此时是观察天王星的最佳时间。已知日地距离为 $R_{0}$, 天王星和地球的公转周期分别为 $T$和 $T_{0}$, 则下列说法正确的是 ( )\nA: 天王星与太阳的距离为 $\\sqrt[3]{\\frac{T^{2}}{T_{0}^{2}}} R_{0}$\nB: 天王星与太阳的距离为 $\\sqrt{\\frac{T^{3}}{T_{0}^{3}}} R_{0}$\nC: 至少再经过 $t=\\frac{T T_{0}}{T-T_{0}}$ 时间, 天王星再次冲日\nD: 至少再经过 $t=\\frac{T T_{0}}{T+T_{0}}$ 时间，天王星再次冲日\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2019 年 10 月 28 日发生了天王星冲日现象, 即太阳、地球、天王星处于同一直线,此时是观察天王星的最佳时间。已知日地距离为 $R_{0}$, 天王星和地球的公转周期分别为 $T$和 $T_{0}$, 则下列说法正确的是 ( )\n\nA: 天王星与太阳的距离为 $\\sqrt[3]{\\frac{T^{2}}{T_{0}^{2}}} R_{0}$\nB: 天王星与太阳的距离为 $\\sqrt{\\frac{T^{3}}{T_{0}^{3}}} R_{0}$\nC: 至少再经过 $t=\\frac{T T_{0}}{T-T_{0}}$ 时间, 天王星再次冲日\nD: 至少再经过 $t=\\frac{T T_{0}}{T+T_{0}}$ 时间，天王星再次冲日\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_225", "problem": "科学家通过研究双中子星合并的引力波, 发现: 两颗中子星在合并前相距为 $L$ 时,两者绕连线上的某点每秒转 $n$ 圈; 经过缓慢演化一段时间后, 两者的距离变为 $k L$, 每秒转 $p n$ 圈, 则演化前后 ( )\nA: 两中子星运动周期为之前 $k p$ 倍\nB: 两中子星运动的角速度为之前 $\\frac{k}{p}$ 倍\nC: 两中子星质量之和为之前 $k^{3} p^{2}$ 倍\nD: 两中子星运动的线速度平方之和为之前 $\\frac{1}{k}$ 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n科学家通过研究双中子星合并的引力波, 发现: 两颗中子星在合并前相距为 $L$ 时,两者绕连线上的某点每秒转 $n$ 圈; 经过缓慢演化一段时间后, 两者的距离变为 $k L$, 每秒转 $p n$ 圈, 则演化前后 ( )\n\nA: 两中子星运动周期为之前 $k p$ 倍\nB: 两中子星运动的角速度为之前 $\\frac{k}{p}$ 倍\nC: 两中子星质量之和为之前 $k^{3} p^{2}$ 倍\nD: 两中子星运动的线速度平方之和为之前 $\\frac{1}{k}$ 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-073.jpg?height=185&width=391&top_left_y=173&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_900", "problem": "Star formation begins when dense clumps within giant molecular clouds, called diffuse nebulae, begin to collapse as their gravity exceeds the pressure from the temperature of the gas. For much of the collapse the clump remains transparent and so can radiate away the energy of the collapse. Consequently the clump stays at a very constant temperature (until a protostar begins to form at the core, when the temperature does rise rapidly as the gas becomes opaque), and so this part of the collapse happens essentially in freefall. This means it can happen rather fast (by astronomical timescales!).\n[figure1]\n\nFigure 2: Left: Orion in visible light.\n\nRight: With radio detections of giant molecular clouds superimposed.\n\nConsider one of the very dense clumps within a nebula. Assume it is spherical with an initial radius of $r_{0}$, and begins to collapse at time $t=0$, until it has shrunk to radius $r$ by some time $t$ later. If the initial density of this clump, $\\rho_{0}$, is uniform everywhere in the sphere (called a homologous collapse) then we can describe its collapse with the following formulae:\n\n$$\n\\theta+\\frac{1}{2} \\sin 2 \\theta=\\left(\\frac{8 \\pi G \\rho_{0}}{3}\\right)^{1 / 2} t \\quad \\text { where } \\quad \\frac{r}{r_{0}}=\\cos ^{2} \\theta\n$$\n\nThe duration of the freefall is $t_{\\mathrm{ff}}$, and by the end of that part of the collapse $r \\ll r_{0}$. Using the above equations derive an expression for $t_{\\mathrm{ff}}$ in its simplest form. What do you notice about your formula? [Note: $\\theta$ is in radians]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nStar formation begins when dense clumps within giant molecular clouds, called diffuse nebulae, begin to collapse as their gravity exceeds the pressure from the temperature of the gas. For much of the collapse the clump remains transparent and so can radiate away the energy of the collapse. Consequently the clump stays at a very constant temperature (until a protostar begins to form at the core, when the temperature does rise rapidly as the gas becomes opaque), and so this part of the collapse happens essentially in freefall. This means it can happen rather fast (by astronomical timescales!).\n[figure1]\n\nFigure 2: Left: Orion in visible light.\n\nRight: With radio detections of giant molecular clouds superimposed.\n\nConsider one of the very dense clumps within a nebula. Assume it is spherical with an initial radius of $r_{0}$, and begins to collapse at time $t=0$, until it has shrunk to radius $r$ by some time $t$ later. If the initial density of this clump, $\\rho_{0}$, is uniform everywhere in the sphere (called a homologous collapse) then we can describe its collapse with the following formulae:\n\n$$\n\\theta+\\frac{1}{2} \\sin 2 \\theta=\\left(\\frac{8 \\pi G \\rho_{0}}{3}\\right)^{1 / 2} t \\quad \\text { where } \\quad \\frac{r}{r_{0}}=\\cos ^{2} \\theta\n$$\n\nThe duration of the freefall is $t_{\\mathrm{ff}}$, and by the end of that part of the collapse $r \\ll r_{0}$. Using the above equations derive an expression for $t_{\\mathrm{ff}}$ in its simplest form. What do you notice about your formula? [Note: $\\theta$ is in radians]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c744602885fab54c0985g-7.jpg?height=702&width=1092&top_left_y=691&top_left_x=480" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1220", "problem": "Young, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\\star}=q \\ll h^{3}$.\n[figure1]\n\nFigure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF).\n\nRight: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset.\n\nSince the planet is assumed small $\\left(q \\ll h^{3}\\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \\approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via:\n\n$$\n\\frac{\\mathrm{d} L}{\\mathrm{~d} t}=\\Gamma\n$$\n\nIt is convenient to write the torque in terms of the reference value\n\n$$\n\\Gamma_{0}=\\left(\\frac{q}{h}\\right)^{2} \\Sigma_{p} r_{p}^{4} \\Omega_{p}^{2}\n$$\nc. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\\Gamma_{C}$. For a disc of uniform entropy ( $\\left.\\mathrm{d} s=0\\right)$, and with surface density profile $\\Sigma \\propto r^{-\\alpha}$, and pressure profile $P \\propto r^{-\\delta}$, Tanaka et al. (2002) and Paardekooper \\& Papaloizou (2009) find these torques are given by:\n\n$$\n\\begin{gathered}\n\\Gamma_{L}=(-3.20+0.86 \\alpha-2.33 \\delta) \\Gamma_{0} \\\\\n\\Gamma_{C}=5.97(1.5-\\alpha) \\Gamma_{0}\n\\end{gathered}\n$$\n\nWe assume the gas in the disc obeys the ideal gas law, so that:\n\n$$\n\\frac{P}{\\Sigma T}=\\text { constant }, \\quad \\mathrm{d} s=\\text { constant } \\times\\left(\\frac{1}{\\gamma-1} \\frac{\\mathrm{d} T}{T}-\\frac{\\mathrm{d} \\Sigma}{\\Sigma}\\right),\n$$\n\nwhere $T$ is the absolute temperature and $\\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy,\n\n$$\n\\Gamma=\\Gamma_{L}+\\Gamma_{C}=(5.76-(5.11+2.33 \\gamma) \\alpha) \\Gamma_{0}\n$$\n\n[Hint: if $\\frac{\\mathrm{d} y}{y}=\\lambda \\frac{\\mathrm{d} x}{x}$, then $y \\propto x^{\\lambda}$.]\n\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\Omega$ is $L=I \\Omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.e. If the disc has mass $M_{\\text {disc }}=0.01 \\mathrm{M} *$ and aspect ratio $h=0.05$, and if $q=5 \\times 10^{-6}$ and $\\tau_{0}=10$ years, find the elapsed time in years for the migration described in part $d$. to occur. Is this a feasible mechanism for planetary migration given the lifetime of the disc is roughly 10 Myr?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nYoung, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\\star}=q \\ll h^{3}$.\n[figure1]\n\nFigure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF).\n\nRight: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset.\n\nSince the planet is assumed small $\\left(q \\ll h^{3}\\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \\approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via:\n\n$$\n\\frac{\\mathrm{d} L}{\\mathrm{~d} t}=\\Gamma\n$$\n\nIt is convenient to write the torque in terms of the reference value\n\n$$\n\\Gamma_{0}=\\left(\\frac{q}{h}\\right)^{2} \\Sigma_{p} r_{p}^{4} \\Omega_{p}^{2}\n$$\nc. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\\Gamma_{C}$. For a disc of uniform entropy ( $\\left.\\mathrm{d} s=0\\right)$, and with surface density profile $\\Sigma \\propto r^{-\\alpha}$, and pressure profile $P \\propto r^{-\\delta}$, Tanaka et al. (2002) and Paardekooper \\& Papaloizou (2009) find these torques are given by:\n\n$$\n\\begin{gathered}\n\\Gamma_{L}=(-3.20+0.86 \\alpha-2.33 \\delta) \\Gamma_{0} \\\\\n\\Gamma_{C}=5.97(1.5-\\alpha) \\Gamma_{0}\n\\end{gathered}\n$$\n\nWe assume the gas in the disc obeys the ideal gas law, so that:\n\n$$\n\\frac{P}{\\Sigma T}=\\text { constant }, \\quad \\mathrm{d} s=\\text { constant } \\times\\left(\\frac{1}{\\gamma-1} \\frac{\\mathrm{d} T}{T}-\\frac{\\mathrm{d} \\Sigma}{\\Sigma}\\right),\n$$\n\nwhere $T$ is the absolute temperature and $\\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy,\n\n$$\n\\Gamma=\\Gamma_{L}+\\Gamma_{C}=(5.76-(5.11+2.33 \\gamma) \\alpha) \\Gamma_{0}\n$$\n\n[Hint: if $\\frac{\\mathrm{d} y}{y}=\\lambda \\frac{\\mathrm{d} x}{x}$, then $y \\propto x^{\\lambda}$.]\n\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\Omega$ is $L=I \\Omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.\n\nproblem:\ne. If the disc has mass $M_{\\text {disc }}=0.01 \\mathrm{M} *$ and aspect ratio $h=0.05$, and if $q=5 \\times 10^{-6}$ and $\\tau_{0}=10$ years, find the elapsed time in years for the migration described in part $d$. to occur. Is this a feasible mechanism for planetary migration given the lifetime of the disc is roughly 10 Myr?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{yrs}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_9bba4f2e5c10ed29bb97g-10.jpg?height=702&width=1416&top_left_y=654&top_left_x=317" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ " \\mathrm{yrs}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1039", "problem": "In July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.b. In reality, the effects of air resistance and the weight of the rocket are substantial. Once in the parking orbit it is travelling at $7.79 \\mathrm{~km} \\mathrm{~s}^{-1}$.\n\nii. The Apollo 11 spacecraft was in the parking orbit for 2 hours 32 mins 27 secs. How many revolutions of the Earth did it do?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIn July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.\n\nproblem:\nb. In reality, the effects of air resistance and the weight of the rocket are substantial. Once in the parking orbit it is travelling at $7.79 \\mathrm{~km} \\mathrm{~s}^{-1}$.\n\nii. The Apollo 11 spacecraft was in the parking orbit for 2 hours 32 mins 27 secs. How many revolutions of the Earth did it do?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-04.jpg?height=1010&width=1508&top_left_y=543&top_left_x=271", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-06.jpg?height=800&width=1586&top_left_y=518&top_left_x=240" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_385", "problem": "2007 年 4 月 24 日, 欧洲科学家宣布在太阳系之外发现了一颗可能适合人类居住的类地行星 Gliest581c.这颗围绕红矮星 Gliese 581 运行的星球有类似地球的温度，表面可能有液态水存在, 距离地球约为 20 光年, 直径约为地球的 1.5 倍, 质量约为地球的 5 倍, 绕红矮星 Gliese 581 运行的周期约为 13 天. 假设有一艘宇宙飞船飞临该星球表面附近轨道, 下列说法正确的是 ( )\nA: 飞船在 Gliest $581 \\mathrm{c}$ 表面附近运行的周期约为 13 天\nB: 飞船在 Gliest $581 \\mathrm{c}$ 表面附近运行时的速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 人在 Gliese $581 \\mathrm{c}$ 上所受重力比在地球上所受重力大\nD: Gliest $581 \\mathrm{c}$ 的平均密度比地球平均密度小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2007 年 4 月 24 日, 欧洲科学家宣布在太阳系之外发现了一颗可能适合人类居住的类地行星 Gliest581c.这颗围绕红矮星 Gliese 581 运行的星球有类似地球的温度，表面可能有液态水存在, 距离地球约为 20 光年, 直径约为地球的 1.5 倍, 质量约为地球的 5 倍, 绕红矮星 Gliese 581 运行的周期约为 13 天. 假设有一艘宇宙飞船飞临该星球表面附近轨道, 下列说法正确的是 ( )\n\nA: 飞船在 Gliest $581 \\mathrm{c}$ 表面附近运行的周期约为 13 天\nB: 飞船在 Gliest $581 \\mathrm{c}$ 表面附近运行时的速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nC: 人在 Gliese $581 \\mathrm{c}$ 上所受重力比在地球上所受重力大\nD: Gliest $581 \\mathrm{c}$ 的平均密度比地球平均密度小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_504", "problem": "2018 年 5 月 21 日, 我国发射人类首颗月球中继卫星“鹊桥”, 6 月 14 日进入使命轨道----地月拉格朗日 $L_{2}$ 轨道, 为在月球背面着陆的嫦娥四号与地球站之间提供通信链路。\n\n12 月 8 日, 我国成功发射嫦娥四号探测器, 并于 2019 年 1 月 3 日成功着陆于与月球背面, 通过中继卫星“鹊桥\"传回了月被影像图, 解开了古老月背的神秘面纱。如图所示, “鹊桥”中继星处于 $L_{2}$ 点上时, 会和月、地两个大天体保持相对静止的状态。设地球的质量为月球的 $k$ 倍，地月间距为 $L$, 拉格朗日 $L_{2}$ 点与月球间距为 $d$, 地球、月球和“鹊桥”均视为质点, 忽略太阳对“鹊桥”中继星的引力。则“鹊桥”中继星处于 $L_{2}$ 点上时, 下列选项正确的是 ( )\n\n[图1]\nA: “鹊桥”与月球的线速度之比为 $v_{\\text {穜 }}: v_{\\text {月 }}=\\sqrt{L}: \\sqrt{L+d}$\nB: “鹊桥”与月球的向心加速度之比为 $a_{\\text {鹊 }}: a_{\\text {月 }}=L:(L+d)$\nC: $k, L, d$ 之间在关系为 $\\frac{1}{(L+d)^{2}}+\\frac{1}{k d^{2}}=\\frac{L+d}{L^{3}}$\nD: $k, L, d$ 之间在关系为 $\\frac{1}{k(L+d)^{2}}+\\frac{1}{d^{2}}=\\frac{L+d}{L^{3}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2018 年 5 月 21 日, 我国发射人类首颗月球中继卫星“鹊桥”, 6 月 14 日进入使命轨道----地月拉格朗日 $L_{2}$ 轨道, 为在月球背面着陆的嫦娥四号与地球站之间提供通信链路。\n\n12 月 8 日, 我国成功发射嫦娥四号探测器, 并于 2019 年 1 月 3 日成功着陆于与月球背面, 通过中继卫星“鹊桥\"传回了月被影像图, 解开了古老月背的神秘面纱。如图所示, “鹊桥”中继星处于 $L_{2}$ 点上时, 会和月、地两个大天体保持相对静止的状态。设地球的质量为月球的 $k$ 倍，地月间距为 $L$, 拉格朗日 $L_{2}$ 点与月球间距为 $d$, 地球、月球和“鹊桥”均视为质点, 忽略太阳对“鹊桥”中继星的引力。则“鹊桥”中继星处于 $L_{2}$ 点上时, 下列选项正确的是 ( )\n\n[图1]\n\nA: “鹊桥”与月球的线速度之比为 $v_{\\text {穜 }}: v_{\\text {月 }}=\\sqrt{L}: \\sqrt{L+d}$\nB: “鹊桥”与月球的向心加速度之比为 $a_{\\text {鹊 }}: a_{\\text {月 }}=L:(L+d)$\nC: $k, L, d$ 之间在关系为 $\\frac{1}{(L+d)^{2}}+\\frac{1}{k d^{2}}=\\frac{L+d}{L^{3}}$\nD: $k, L, d$ 之间在关系为 $\\frac{1}{k(L+d)^{2}}+\\frac{1}{d^{2}}=\\frac{L+d}{L^{3}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-55.jpg?height=417&width=505&top_left_y=1733&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_111", "problem": "卫星 $\\mathrm{M}$ 在轨道I上做匀速圆周运动, 一段时间后在 $A$ 点变速进入轨道II, 运行一段时间后, 在 $B$ 点变速进入轨道半径为轨道I轨道半径 5 倍的轨道III, 最后在轨道III做匀速圆周运动, 在轨道III上的速率为 $v$, 则卫星在轨道II上的 $B$ 点速率可能是 ( )\n\n[图1]\nA: $\\frac{1}{5} v$\nB: $\\frac{\\sqrt{5}}{6} v$\nC: $\\frac{\\sqrt{3}}{3} v$\nD: $\\frac{\\sqrt{6}}{2} v$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n卫星 $\\mathrm{M}$ 在轨道I上做匀速圆周运动, 一段时间后在 $A$ 点变速进入轨道II, 运行一段时间后, 在 $B$ 点变速进入轨道半径为轨道I轨道半径 5 倍的轨道III, 最后在轨道III做匀速圆周运动, 在轨道III上的速率为 $v$, 则卫星在轨道II上的 $B$ 点速率可能是 ( )\n\n[图1]\n\nA: $\\frac{1}{5} v$\nB: $\\frac{\\sqrt{5}}{6} v$\nC: $\\frac{\\sqrt{3}}{3} v$\nD: $\\frac{\\sqrt{6}}{2} v$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-052.jpg?height=409&width=437&top_left_y=2331&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1122", "problem": "GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nThe rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as\n\n$$\n\\dot{f}_{\\mathrm{GW}}=\\frac{96}{5} \\pi^{8 / 3}\\left(\\frac{G \\mathcal{M}}{c^{3}}\\right)^{5 / 3} f_{\\mathrm{GW}}^{11 / 3}\n$$d. Combine your result from $c$. with the above equation to cancel out $\\mathcal{M}$ and so express the distance to the gravitational wave source, $r$, as a function of fundamental constants and the measurables $h, f_{G W}$, and $\\dot{f}_{G W}$ only.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nGW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nThe rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as\n\n$$\n\\dot{f}_{\\mathrm{GW}}=\\frac{96}{5} \\pi^{8 / 3}\\left(\\frac{G \\mathcal{M}}{c^{3}}\\right)^{5 / 3} f_{\\mathrm{GW}}^{11 / 3}\n$$\n\nproblem:\nd. Combine your result from $c$. with the above equation to cancel out $\\mathcal{M}$ and so express the distance to the gravitational wave source, $r$, as a function of fundamental constants and the measurables $h, f_{G W}$, and $\\dot{f}_{G W}$ only.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_660", "problem": "如图所示, 地球绕太阳做匀速圆周运动, 地球处在运动轨道 $b$ 位置时, 地球和太阳连线上的 $a$ 与 $e$ 位置、 $c$ 与 $d$ 位置均关于太阳对称. 当一无动力的探测器处在 $a$ 或 $c$ 位置时, 它仅在太阳和地球引力的共同作用下, 与地球一起以相同的角速度绕太阳做圆周运动, 下列说法正确的是\n\n[图1]\nA: 若地球和该探测器分别在 $b 、 d$ 位置, 它们也能以相同的角速度绕太阳运动\nB: 若地球和该探测器分别在 $b 、 e$ 位置, 它们也能以相同的角速度绕太阳运动\nC: 该探测器在 $a$ 位置受到太阳、地球引力的合力等于在 $c$ 位置受到太阳, 地球引力的合力\nD: 该探测器在 $a$ 位置受到太阳、地球引力的合力大于在 $c$ 位置受到太阳、地球引力的合力\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 地球绕太阳做匀速圆周运动, 地球处在运动轨道 $b$ 位置时, 地球和太阳连线上的 $a$ 与 $e$ 位置、 $c$ 与 $d$ 位置均关于太阳对称. 当一无动力的探测器处在 $a$ 或 $c$ 位置时, 它仅在太阳和地球引力的共同作用下, 与地球一起以相同的角速度绕太阳做圆周运动, 下列说法正确的是\n\n[图1]\n\nA: 若地球和该探测器分别在 $b 、 d$ 位置, 它们也能以相同的角速度绕太阳运动\nB: 若地球和该探测器分别在 $b 、 e$ 位置, 它们也能以相同的角速度绕太阳运动\nC: 该探测器在 $a$ 位置受到太阳、地球引力的合力等于在 $c$ 位置受到太阳, 地球引力的合力\nD: 该探测器在 $a$ 位置受到太阳、地球引力的合力大于在 $c$ 位置受到太阳、地球引力的合力\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-74.jpg?height=442&width=768&top_left_y=1041&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_248", "problem": "万有引力作用下的物体具有引力势能, 取无穷远处引力势能为零, 物体距星球球心\n距离为 $r$ 时的引力势能为 $E_{p}=-G \\frac{M m}{r}$ ( $G$ 为引力常量, $M 、 m$ 分别为星球和物体的质量), 在一半径为 $R$ 的星球上, 一物体从星球表面某高度处自由下落(不计空气阻力),自开始下落计时, 得到物体在星球表面下落高度 $H$ 随时间 $t$ 变化的图象如图所示, 则\n\n[图1]\nA: 在该星球表面上以 $\\frac{1}{t_{0}} \\sqrt{2 h R}$ 的初速度水平抛出一物体, 物体将不再落回星球表面\nB: 在该星球表面上以 $\\frac{2}{t_{0}} \\sqrt{h R}$ 的初速度水平抛出一物体, 物体将不再落回星球表面\nC: 在该星球表面上以 $\\frac{1}{t_{0}} \\sqrt{2 h R}$ 的初速度坚直上抛一物体, 物体将不再落回星球表面\nD: 在该星球表面上以 $\\frac{2}{t_{0}} \\sqrt{h R}$ 的初速度坚直上抛一物体, 物体将不再落回星球表面\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n万有引力作用下的物体具有引力势能, 取无穷远处引力势能为零, 物体距星球球心\n距离为 $r$ 时的引力势能为 $E_{p}=-G \\frac{M m}{r}$ ( $G$ 为引力常量, $M 、 m$ 分别为星球和物体的质量), 在一半径为 $R$ 的星球上, 一物体从星球表面某高度处自由下落(不计空气阻力),自开始下落计时, 得到物体在星球表面下落高度 $H$ 随时间 $t$ 变化的图象如图所示, 则\n\n[图1]\n\nA: 在该星球表面上以 $\\frac{1}{t_{0}} \\sqrt{2 h R}$ 的初速度水平抛出一物体, 物体将不再落回星球表面\nB: 在该星球表面上以 $\\frac{2}{t_{0}} \\sqrt{h R}$ 的初速度水平抛出一物体, 物体将不再落回星球表面\nC: 在该星球表面上以 $\\frac{1}{t_{0}} \\sqrt{2 h R}$ 的初速度坚直上抛一物体, 物体将不再落回星球表面\nD: 在该星球表面上以 $\\frac{2}{t_{0}} \\sqrt{h R}$ 的初速度坚直上抛一物体, 物体将不再落回星球表面\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-62.jpg?height=443&width=457&top_left_y=441&top_left_x=323" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_701", "problem": "已知地球自转周期为 $T_{0}$, 有一颗与同步卫星在同一轨道平面的低轨道卫星, 自西向东绕地球运行, 其运行半径为同步轨道半径的四分之一, 该卫星两次在同一城市的正上方出现的时间间隔可能是\nA: $\\frac{T_{0}}{4}$\nB: $\\frac{3 T_{0}}{7}$\nC: $\\frac{3 T_{0}}{4}$\nD: $\\frac{T_{0}}{7}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n已知地球自转周期为 $T_{0}$, 有一颗与同步卫星在同一轨道平面的低轨道卫星, 自西向东绕地球运行, 其运行半径为同步轨道半径的四分之一, 该卫星两次在同一城市的正上方出现的时间间隔可能是\n\nA: $\\frac{T_{0}}{4}$\nB: $\\frac{3 T_{0}}{7}$\nC: $\\frac{3 T_{0}}{4}$\nD: $\\frac{T_{0}}{7}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_106", "problem": "4 月 29 日 11 时 23 分，在中国文昌航天发射场，长征五号 B 运载火箭将中国空间站工程首个航天器“天和”核心舱顺利送入太空, 任务取得圆满成功。未来两年, “天和”\n将在距离地面约 $400 \\mathrm{~km}$ 的圆轨道上，静候“天舟”“神舟”“问天”“梦天”等航天器的陆续来访, 共同完成空间站组装建造和关键技术在轨验证等建“宫”大业。假设地球的半径 $R=6400 \\mathrm{~km}$ 。下列说法正确的是（ ）\nA: “天和”核心舱的发射速度小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: “天和”核心舱的运行周期约等于 24 小时\nC: “天和”核心舱的加速度约等于 $9 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: “天和”核心舱在圆轨道上运行速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n4 月 29 日 11 时 23 分，在中国文昌航天发射场，长征五号 B 运载火箭将中国空间站工程首个航天器“天和”核心舱顺利送入太空, 任务取得圆满成功。未来两年, “天和”\n将在距离地面约 $400 \\mathrm{~km}$ 的圆轨道上，静候“天舟”“神舟”“问天”“梦天”等航天器的陆续来访, 共同完成空间站组装建造和关键技术在轨验证等建“宫”大业。假设地球的半径 $R=6400 \\mathrm{~km}$ 。下列说法正确的是（ ）\n\nA: “天和”核心舱的发射速度小于 $7.9 \\mathrm{~km} / \\mathrm{s}$\nB: “天和”核心舱的运行周期约等于 24 小时\nC: “天和”核心舱的加速度约等于 $9 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: “天和”核心舱在圆轨道上运行速度大于 $7.9 \\mathrm{~km} / \\mathrm{s}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_789", "problem": "The Large Magellanic Cloud is ..\nA: a dwarf galaxy orbiting the Milky Way.\nB: the closest planetary nebula to the Earth.\nC: a bright star cluster discovered by Magellan.\nD: the outer arm of the Milky Way named after Magellan.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe Large Magellanic Cloud is ..\n\nA: a dwarf galaxy orbiting the Milky Way.\nB: the closest planetary nebula to the Earth.\nC: a bright star cluster discovered by Magellan.\nD: the outer arm of the Milky Way named after Magellan.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_282", "problem": "我国发射的“天问一号”火星探测器到达火用后开展了一系列复杂的变轨操作: 2021 年 2 月 10 日, 探测器第一次到达近火点时被火星捕获, 成功实现火星环绕, 进入周期为 10 天的大椭圆轨道; 2 月 15 日, 探测器第一次到达远火点时进行变轨, 调整轨道平面与近火点高度, 环火轨道变为经过火星南北两极的极轨; 2 月 20 日, 探测器第二次到达近火点时进行轨道调整, 进入周期为 4 天的调相轨道; 2 月 24 日, 探测器第三次运行至近火点时顺利实施第三次近火制动, 成功进入停泊轨道。极轨、调相轨道、停泊轨道在同一平面内。探测器在这四次变轨过程中\n\n[图1]\nA: 沿大粗圆轨道经过远火点与变轨后在极轨上经过远火点的加速度方向垂直\nB: 沿极轨到达近火点变轨时制动减速才能进入调相轨道\nC: 沿极轨、调相轨道经过近火点时的加速度都相等\nD: 大椭圆轨道半长轴 $r_{1}$ 与调相轨道半长轴 $r_{2}$ 的比值为 $\\frac{\\sqrt[3]{400}}{4}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n我国发射的“天问一号”火星探测器到达火用后开展了一系列复杂的变轨操作: 2021 年 2 月 10 日, 探测器第一次到达近火点时被火星捕获, 成功实现火星环绕, 进入周期为 10 天的大椭圆轨道; 2 月 15 日, 探测器第一次到达远火点时进行变轨, 调整轨道平面与近火点高度, 环火轨道变为经过火星南北两极的极轨; 2 月 20 日, 探测器第二次到达近火点时进行轨道调整, 进入周期为 4 天的调相轨道; 2 月 24 日, 探测器第三次运行至近火点时顺利实施第三次近火制动, 成功进入停泊轨道。极轨、调相轨道、停泊轨道在同一平面内。探测器在这四次变轨过程中\n\n[图1]\n\nA: 沿大粗圆轨道经过远火点与变轨后在极轨上经过远火点的加速度方向垂直\nB: 沿极轨到达近火点变轨时制动减速才能进入调相轨道\nC: 沿极轨、调相轨道经过近火点时的加速度都相等\nD: 大椭圆轨道半长轴 $r_{1}$ 与调相轨道半长轴 $r_{2}$ 的比值为 $\\frac{\\sqrt[3]{400}}{4}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-039.jpg?height=731&width=1087&top_left_y=1465&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_46", "problem": "开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。\n[图1]根据开普勒第三定律, 求卫星在II轨道运动时的周期大小;", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。\n[图1]根据开普勒第三定律, 求卫星在II轨道运动时的周期大小;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-022.jpg?height=436&width=1398&top_left_y=230&top_left_x=342" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_79", "problem": "2017 年, 人类第一次直接探测到来自双中子星合并的引力波。根据科学家们复原的过程, 在两颗中子星合并前约 $100 \\mathrm{~s}$ 时, 它们相距约 $400 \\mathrm{~km}$, 绕二者连线上的某点每秒转动 12 圈, 将两颗中子星都看作是质量均匀分布的球体, 由这些数据、万有引力常量并利用牛顿力学知识, 可以估算出这一时刻两颗中子星（）\nA: 质量之积\nB: 质量之和\nC: 速率之和\nD: 各自的自转角速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2017 年, 人类第一次直接探测到来自双中子星合并的引力波。根据科学家们复原的过程, 在两颗中子星合并前约 $100 \\mathrm{~s}$ 时, 它们相距约 $400 \\mathrm{~km}$, 绕二者连线上的某点每秒转动 12 圈, 将两颗中子星都看作是质量均匀分布的球体, 由这些数据、万有引力常量并利用牛顿力学知识, 可以估算出这一时刻两颗中子星（）\n\nA: 质量之积\nB: 质量之和\nC: 速率之和\nD: 各自的自转角速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_637", "problem": "预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\\left(r_{2}>r_{1}\\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求:\n\n在 $0 \\sim 3 t_{0}$ 时间内线圈中产生的焦耳热;\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\\left(r_{2}>r_{1}\\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求:\n\n在 $0 \\sim 3 t_{0}$ 时间内线圈中产生的焦耳热;\n\n[图1]\n\n图甲\n\n[图2]\n\n图乙\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-134.jpg?height=369&width=388&top_left_y=1780&top_left_x=340", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-134.jpg?height=294&width=505&top_left_y=1869&top_left_x=867" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_142", "problem": "我国“天宫一号”圆满完成相关科学实验, 于 2018 年“受控”坠落. 若某航天器变轨后仍绕地球做匀速圆周运动, 但动能增大为原来的 4 倍, 不考虑航天器质量的变化, 则变轨后, 下列说法正确的是 ( )\nA: 航天器的轨道半径变为原来的 $1 / 4$\nB: 航天器的向心加速度变为原来的 4 倍\nC: 航天器的周期变为原来的 $1 / 4$\nD: 航天器的角速度变为原来的 4 倍\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n我国“天宫一号”圆满完成相关科学实验, 于 2018 年“受控”坠落. 若某航天器变轨后仍绕地球做匀速圆周运动, 但动能增大为原来的 4 倍, 不考虑航天器质量的变化, 则变轨后, 下列说法正确的是 ( )\n\nA: 航天器的轨道半径变为原来的 $1 / 4$\nB: 航天器的向心加速度变为原来的 4 倍\nC: 航天器的周期变为原来的 $1 / 4$\nD: 航天器的角速度变为原来的 4 倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_793", "problem": "Let $T_{\\odot, C}$ and $T_{\\odot, S}$ be the temperatures at the core and the surface of the sun, respectively. Similarly, let $T_{A, C}$ and $T_{A, S}$ be the temperatures at the core and surface of the red giant Arcturus, and let $T_{S, C}$ and $T_{S, S}$ be the temperatures at the core and surface of the white dwarf Sirius B. Which of the following inequalities is true?\nA: $\\frac{T_{\\odot, C}}{T_{\\odot, S}}<\\frac{T_{A, C}}{T_{A, S}}<\\frac{T_{S, C}}{T_{S, S}}$\nB: $\\frac{T_{\\odot, C}}{T_{\\odot, S}}<\\frac{T_{S, C}}{T_{S, S}}<\\frac{T_{A, C}}{T_{A, S}}$\nC: $\\frac{T_{A, C}}{T_{A, S}}<\\frac{T_{\\odot, C}}{T_{\\odot, S}}<\\frac{T_{S, C}}{T_{S, S}}$\nD: $\\frac{T_{S, C}}{T_{S, S}}<\\frac{T_{\\odot, C}}{T_{\\odot, S}}<\\frac{T_{A, C}}{T_{A, S}}$\nE: $\\frac{T_{S, C}}{T_{S, S}}<\\frac{T_{A, C}}{T_{A, S}}<\\frac{T_{\\odot, C}}{T_{\\odot, S}}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nLet $T_{\\odot, C}$ and $T_{\\odot, S}$ be the temperatures at the core and the surface of the sun, respectively. Similarly, let $T_{A, C}$ and $T_{A, S}$ be the temperatures at the core and surface of the red giant Arcturus, and let $T_{S, C}$ and $T_{S, S}$ be the temperatures at the core and surface of the white dwarf Sirius B. Which of the following inequalities is true?\n\nA: $\\frac{T_{\\odot, C}}{T_{\\odot, S}}<\\frac{T_{A, C}}{T_{A, S}}<\\frac{T_{S, C}}{T_{S, S}}$\nB: $\\frac{T_{\\odot, C}}{T_{\\odot, S}}<\\frac{T_{S, C}}{T_{S, S}}<\\frac{T_{A, C}}{T_{A, S}}$\nC: $\\frac{T_{A, C}}{T_{A, S}}<\\frac{T_{\\odot, C}}{T_{\\odot, S}}<\\frac{T_{S, C}}{T_{S, S}}$\nD: $\\frac{T_{S, C}}{T_{S, S}}<\\frac{T_{\\odot, C}}{T_{\\odot, S}}<\\frac{T_{A, C}}{T_{A, S}}$\nE: $\\frac{T_{S, C}}{T_{S, S}}<\\frac{T_{A, C}}{T_{A, S}}<\\frac{T_{\\odot, C}}{T_{\\odot, S}}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1004", "problem": "A comet orbits the Sun with a period of 172 years and eccentricity 0.94 . It is currently at a distance of 60 au away from the Sun. After which of these times will the comet be moving the fastest?\nA: 43 years\nB: 86 years\nC: 129 years\nD: 172 years\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA comet orbits the Sun with a period of 172 years and eccentricity 0.94 . It is currently at a distance of 60 au away from the Sun. After which of these times will the comet be moving the fastest?\n\nA: 43 years\nB: 86 years\nC: 129 years\nD: 172 years\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_728", "problem": "宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 。在星球 $\\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 P 轻放在弹簧上端, 如图 (a) 所示, P 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\\mathrm{B}$ 上用完全相同的弹簧和物体 $\\mathrm{P}$ 完成同样的过程, 其 $a-x$ 关系如图 (b) 中虚线所示（图中 $a_{0}$未知）。已知两星球密度相等。星球 $\\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。\n\n求星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 的表面重力加速度的比值;\n\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 。在星球 $\\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 P 轻放在弹簧上端, 如图 (a) 所示, P 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\\mathrm{B}$ 上用完全相同的弹簧和物体 $\\mathrm{P}$ 完成同样的过程, 其 $a-x$ 关系如图 (b) 中虚线所示（图中 $a_{0}$未知）。已知两星球密度相等。星球 $\\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。\n\n求星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 的表面重力加速度的比值;\n\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-108.jpg?height=206&width=254&top_left_y=1299&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-108.jpg?height=369&width=348&top_left_y=1186&top_left_x=611", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-108.jpg?height=266&width=306&top_left_y=1860&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_21", "problem": "已成为我国首个人造太阳系小行星的“嫦娥二号”，2014 年 2 月再次刷新我国深空探测最远距离纪录, 超过 7000 万公里, “嫦娥二号”是我国探月工程二期的先导星, 它先在距月球表面高度为 $h$ 的轨道上做匀速圆周运动, 运行周期为 $T$; 然后从月球轨道出发飞赴日地拉格朗日 $L_{2}$ 点（物体在该点受日、地引力平衡）进行科学探测。若以 $R$ 表示月球的半径, 引力常量为 $G$, 则 ( )\nA: “嫦娥二号”卫星绕月运行时的线速度为 $\\frac{2 \\pi R}{T}$\nB: 月球的质量为 $\\frac{4 \\pi^{2}(R+h)^{3}}{G T^{2}}$\nC: 物体在月球表面自由下落的加速度为 $\\frac{4 \\pi^{2} R}{T^{2}}$\nD: 嫦娥二号卫星在月球轨道需经过加速才能飞赴日地拉格朗日 $L_{2}$ 点\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n已成为我国首个人造太阳系小行星的“嫦娥二号”，2014 年 2 月再次刷新我国深空探测最远距离纪录, 超过 7000 万公里, “嫦娥二号”是我国探月工程二期的先导星, 它先在距月球表面高度为 $h$ 的轨道上做匀速圆周运动, 运行周期为 $T$; 然后从月球轨道出发飞赴日地拉格朗日 $L_{2}$ 点（物体在该点受日、地引力平衡）进行科学探测。若以 $R$ 表示月球的半径, 引力常量为 $G$, 则 ( )\n\nA: “嫦娥二号”卫星绕月运行时的线速度为 $\\frac{2 \\pi R}{T}$\nB: 月球的质量为 $\\frac{4 \\pi^{2}(R+h)^{3}}{G T^{2}}$\nC: 物体在月球表面自由下落的加速度为 $\\frac{4 \\pi^{2} R}{T^{2}}$\nD: 嫦娥二号卫星在月球轨道需经过加速才能飞赴日地拉格朗日 $L_{2}$ 点\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_128", "problem": "开普勒第三定律指出: 所有行星轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等, 即 $\\frac{a^{3}}{T^{2}}=k$, 其中 $a$ 表示椭圆轨道半长轴, $T$ 表示公转周期, 比值 $k$ 是一个对所有行星都相同的常量。同时, 开普勒第三定律对于轨迹为圆形和直线的运动依然适用: 圆形轨迹可以认为中心天体在圆心处, 半长轴为轨迹半径; 直线轨迹可以看成无限扁的椭圆轨迹, 长轴为物体与星球之间的距离。已知: 星球质量为 $M$, 在距离星球的距离为 $r$ 处有一物体, 该物体仅在星球引力的作用下运动。星球可视为质点且认为保持静止, 引力常量为 $G$, 则下列说法正确的是 ( )\nA: 该星球和物体的引力系统中常量 $k=\\frac{4 \\pi^{2}}{G M}$\nB: 要使物体绕星球做匀速圆周运动, 则物体的速度为 $v=\\sqrt{\\frac{2 G M}{r}}$\nC: 若物体绕星球沿粗圆轨道运动, 在靠近星球的过程中动能在减少\nD: 若物体由静止开始释放, 则该物体到达星球所经历的时间为 $t=\\pi r \\sqrt{\\frac{r}{8 G M}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n开普勒第三定律指出: 所有行星轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等, 即 $\\frac{a^{3}}{T^{2}}=k$, 其中 $a$ 表示椭圆轨道半长轴, $T$ 表示公转周期, 比值 $k$ 是一个对所有行星都相同的常量。同时, 开普勒第三定律对于轨迹为圆形和直线的运动依然适用: 圆形轨迹可以认为中心天体在圆心处, 半长轴为轨迹半径; 直线轨迹可以看成无限扁的椭圆轨迹, 长轴为物体与星球之间的距离。已知: 星球质量为 $M$, 在距离星球的距离为 $r$ 处有一物体, 该物体仅在星球引力的作用下运动。星球可视为质点且认为保持静止, 引力常量为 $G$, 则下列说法正确的是 ( )\n\nA: 该星球和物体的引力系统中常量 $k=\\frac{4 \\pi^{2}}{G M}$\nB: 要使物体绕星球做匀速圆周运动, 则物体的速度为 $v=\\sqrt{\\frac{2 G M}{r}}$\nC: 若物体绕星球沿粗圆轨道运动, 在靠近星球的过程中动能在减少\nD: 若物体由静止开始释放, 则该物体到达星球所经历的时间为 $t=\\pi r \\sqrt{\\frac{r}{8 G M}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_374", "problem": "下面是地球、火星的有关情况比较。根据以上信息, 关于地球及火星（行星的运动\n\n| 星球 | 地球 | 火星 |\n| :---: | :---: | :---: |\n| 公转半径 | $1.5 \\times 10^{8} \\mathrm{~km}$ | $2.25 \\times$
$10^{8} \\mathrm{~km}$ |\n| 自转周期 | 23 时 56 分 | 24 时 37 分 |\n| 表面温度 | $15^{\\circ} \\mathrm{C}$ | $-100^{\\circ} \\mathrm{C} \\sim 0^{\\circ} \\mathrm{C}$ |\n| 大气主要成分 | $78 \\%$ 的 $\\mathrm{N}_{2}, 21$
$\\%$ 的 $\\mathrm{O}_{2}$ | 约 $95 \\%$ 的
$\\mathrm{CO}_{2}$ |\n\n可看做圆周运动 ), 下列推测正确的是（）\nA: 地球公转的线速度小于火星公转的线速度\nB: 地球公转的向心加速度大于火星公转的向心加速度\nC: 地球的自转角速度小于火星的自转角速度\nD: 地球表面的重力加速度大于火星表面的重力加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n下面是地球、火星的有关情况比较。根据以上信息, 关于地球及火星（行星的运动\n\n| 星球 | 地球 | 火星 |\n| :---: | :---: | :---: |\n| 公转半径 | $1.5 \\times 10^{8} \\mathrm{~km}$ | $2.25 \\times$
$10^{8} \\mathrm{~km}$ |\n| 自转周期 | 23 时 56 分 | 24 时 37 分 |\n| 表面温度 | $15^{\\circ} \\mathrm{C}$ | $-100^{\\circ} \\mathrm{C} \\sim 0^{\\circ} \\mathrm{C}$ |\n| 大气主要成分 | $78 \\%$ 的 $\\mathrm{N}_{2}, 21$
$\\%$ 的 $\\mathrm{O}_{2}$ | 约 $95 \\%$ 的
$\\mathrm{CO}_{2}$ |\n\n可看做圆周运动 ), 下列推测正确的是（）\n\nA: 地球公转的线速度小于火星公转的线速度\nB: 地球公转的向心加速度大于火星公转的向心加速度\nC: 地球的自转角速度小于火星的自转角速度\nD: 地球表面的重力加速度大于火星表面的重力加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_71", "problem": "2022 年 11 月 29 日 23 时 08 分, 搭载着神舟十五号载人飞船的长征二号 $F$ 遥十五运载火箭在酒泉卫星发射中心升空, 11 月 30 日 5 时 42 分, 神舟十五号载人飞船与天和核心舱成功完成自主交会对接。如图为神舟十五号的发射与交会对接过程示意图，图中 I 为近地圆轨道, 其轨道半径可认为等于地球半径 $R$, II 为椭圆变轨轨道, III 为天和核心舱所在轨道, 其轨道半径为 $r_{0}, P 、 Q$ 分别为轨道 II 与 I、III 轨道的交会点, 已知神舟十五号的质量为 $m_{0}$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 若取两物体相距无\n穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\\mathrm{P}}=-\\frac{G M m}{r}$ (式中 $G$ 为引力常量)。求:\n\n神舟十五号在轨道 II 运动时从 $P$ 点运动到 $Q$ 点的最短时间;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2022 年 11 月 29 日 23 时 08 分, 搭载着神舟十五号载人飞船的长征二号 $F$ 遥十五运载火箭在酒泉卫星发射中心升空, 11 月 30 日 5 时 42 分, 神舟十五号载人飞船与天和核心舱成功完成自主交会对接。如图为神舟十五号的发射与交会对接过程示意图，图中 I 为近地圆轨道, 其轨道半径可认为等于地球半径 $R$, II 为椭圆变轨轨道, III 为天和核心舱所在轨道, 其轨道半径为 $r_{0}, P 、 Q$ 分别为轨道 II 与 I、III 轨道的交会点, 已知神舟十五号的质量为 $m_{0}$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 若取两物体相距无\n穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\\mathrm{P}}=-\\frac{G M m}{r}$ (式中 $G$ 为引力常量)。求:\n\n神舟十五号在轨道 II 运动时从 $P$ 点运动到 $Q$ 点的最短时间;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-039.jpg?height=631&width=560&top_left_y=510&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_820", "problem": "(CANCELED) When binary systems are really close together, they can execute an accretion process, in which one star (called the primary star) \"eats\" the mass of the other (called the secondary star), whose mass spirals down into the primary star, creating an accretion disk!\n\nFor an accretion disk with the outer edge $3 R$ from the center of the primary star (radius $R$ and mass $M$ ), calculate the energy lost by a test mass (mass $m$ ) where it touches the primary star from where it first enters the accretion disk.\n\nConsider the orbits to be Keplerian.\nA: $\\frac{G M m}{R}$\nB: $\\frac{1}{2} \\frac{G M m}{R}$\nC: $\\frac{5}{2} \\frac{G M m}{R}$\nD: $\\frac{1}{3} \\frac{G M m}{R}$\nE: $\\frac{3}{4} \\frac{G M m}{R}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\n(CANCELED) When binary systems are really close together, they can execute an accretion process, in which one star (called the primary star) \"eats\" the mass of the other (called the secondary star), whose mass spirals down into the primary star, creating an accretion disk!\n\nFor an accretion disk with the outer edge $3 R$ from the center of the primary star (radius $R$ and mass $M$ ), calculate the energy lost by a test mass (mass $m$ ) where it touches the primary star from where it first enters the accretion disk.\n\nConsider the orbits to be Keplerian.\n\nA: $\\frac{G M m}{R}$\nB: $\\frac{1}{2} \\frac{G M m}{R}$\nC: $\\frac{5}{2} \\frac{G M m}{R}$\nD: $\\frac{1}{3} \\frac{G M m}{R}$\nE: $\\frac{3}{4} \\frac{G M m}{R}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_497", "problem": "'重力探矿是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用, 其原理可简述如下：如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向, 当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向（即 $P O$ 方向）上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常量为 $G$ 。\n设球形区域体积为 $V$, 球心深度为 $d$ ( $d$ 远小于地球半径), $\\overline{P Q}=x$, 求:\n $Q$ 点处的重力加速度反常值;\n\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n'重力探矿是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用, 其原理可简述如下：如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向, 当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向（即 $P O$ 方向）上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常量为 $G$ 。\n设球形区域体积为 $V$, 球心深度为 $d$ ( $d$ 远小于地球半径), $\\overline{P Q}=x$, 求:\n $Q$ 点处的重力加速度反常值;\n\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-047.jpg?height=434&width=551&top_left_y=2319&top_left_x=364" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_729", "problem": "随着对宇宙的研究逐步开展, 科学家已多次探测到引力波。这证实了爱因斯坦 100 年前的预测，弥补了爱因斯坦广义相对论中最后一块缺失的“拼图”。双星的运动是产生引力波的来源之一，假设宇宙中有一由 $a 、 b$ 两颗星组成的双星系统，这两颗星在万有引力的作用下, 绕它们连线的某一点做匀速圆周运动, $a$ 星的运行周期为 $T, a 、 b$ 两颗星的距离为 $L, a 、 b$ 两颗星的轨道半径之差为 $\\Delta r$ 。已知 $a$ 星的轨道半径大于 $b$ 星的轨道半径，则（）\nA: $b$ 星的周期为 $\\frac{L-\\Delta r}{L+\\Delta r} T$\nB: $b$ 星的线速度大小为 $\\frac{\\pi(L-\\Delta r)}{T}$\nC: $a 、 b$ 两颗星的半径之比为 $\\frac{(L+\\Delta r)^{2}}{(L-\\Delta r)^{2}}$\nD: $a 、 b$ 两颗星的质量之比为 $\\frac{L-\\Delta r}{L+\\Delta r}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n随着对宇宙的研究逐步开展, 科学家已多次探测到引力波。这证实了爱因斯坦 100 年前的预测，弥补了爱因斯坦广义相对论中最后一块缺失的“拼图”。双星的运动是产生引力波的来源之一，假设宇宙中有一由 $a 、 b$ 两颗星组成的双星系统，这两颗星在万有引力的作用下, 绕它们连线的某一点做匀速圆周运动, $a$ 星的运行周期为 $T, a 、 b$ 两颗星的距离为 $L, a 、 b$ 两颗星的轨道半径之差为 $\\Delta r$ 。已知 $a$ 星的轨道半径大于 $b$ 星的轨道半径，则（）\n\nA: $b$ 星的周期为 $\\frac{L-\\Delta r}{L+\\Delta r} T$\nB: $b$ 星的线速度大小为 $\\frac{\\pi(L-\\Delta r)}{T}$\nC: $a 、 b$ 两颗星的半径之比为 $\\frac{(L+\\Delta r)^{2}}{(L-\\Delta r)^{2}}$\nD: $a 、 b$ 两颗星的质量之比为 $\\frac{L-\\Delta r}{L+\\Delta r}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_222", "problem": "据报道, 已经发射成功的“嫦娥四号”月球探测器在月球背面实现了软着陆, 并展开探测工作, 它将通过早先发射的“鹊桥”中继卫星与地球实现信号传输及控制。在地月连线上存在一点“拉格朗日 $L_{2}$ ”, “鹊桥”在随月球绕地球同步公转的同时, 沿“Halo 轨道” (轨道平面与地月连线垂直) 绕 $L_{2}$ 转动, 如图所示。已知“鹊桥”卫星位于“Halo 轨道”时, 在地月引力共同作用下具有跟月球绕地球公转相同的周期。根据图中有关数据结合有关物理知识，可估算出\n\n[图1]\n\n地-月距离: $384400 \\mathrm{~km}$ 鹊桥绕 $L_{2}$ 运转周期: $14 \\mathrm{~d}$\n\n月- $L_{2}$ 距离: $64500 \\mathrm{~km}$\n\nHalo轨道半径: $3500 \\mathrm{~km}$\n\n引力常量: $6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$\nA: “鹊桥”质量\nB: 月球质量\nC: 月球的公转周期\nD: “鹊桥”相对于 $L_{2}$ 的线速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n据报道, 已经发射成功的“嫦娥四号”月球探测器在月球背面实现了软着陆, 并展开探测工作, 它将通过早先发射的“鹊桥”中继卫星与地球实现信号传输及控制。在地月连线上存在一点“拉格朗日 $L_{2}$ ”, “鹊桥”在随月球绕地球同步公转的同时, 沿“Halo 轨道” (轨道平面与地月连线垂直) 绕 $L_{2}$ 转动, 如图所示。已知“鹊桥”卫星位于“Halo 轨道”时, 在地月引力共同作用下具有跟月球绕地球公转相同的周期。根据图中有关数据结合有关物理知识，可估算出\n\n[图1]\n\n地-月距离: $384400 \\mathrm{~km}$ 鹊桥绕 $L_{2}$ 运转周期: $14 \\mathrm{~d}$\n\n月- $L_{2}$ 距离: $64500 \\mathrm{~km}$\n\nHalo轨道半径: $3500 \\mathrm{~km}$\n\n引力常量: $6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$\n\nA: “鹊桥”质量\nB: 月球质量\nC: 月球的公转周期\nD: “鹊桥”相对于 $L_{2}$ 的线速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-056.jpg?height=503&width=622&top_left_y=154&top_left_x=494", "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-056.jpg?height=60&width=1380&top_left_y=1880&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_535", "problem": "2022 年 4 月 16 日神舟十三号载人飞船返回舱成功着陆, 三位航天员在空间站出差半年, 完成了两次太空行走和 20 多项科学实验, 并开展了两次“天宫课堂”活动, 刷新了中国航天新纪录。已知地球半径为 $R$, 地球表面重力加速度为 $g$ 。总质量为 $m_{0}$ 的空间站绕地球的运动可近似为匀速圆周运动, 距地球表面高度为 $h$, 阻力忽略不计。\n维持空间站的运行与舱内航天员的生活需要耗费大量电能, 某同学为其设计了太阳能电池板。太阳辐射的总功率为 $P_{0}$, 太阳与空间站的平均距离为 $r$, 且该太阳能电池板正对太阳的面积始终为 $S$, 假设该太阳能电池板的能量转化效率为 $\\eta$ 。求单位时间空间站通过太阳能电池板获得的电能。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2022 年 4 月 16 日神舟十三号载人飞船返回舱成功着陆, 三位航天员在空间站出差半年, 完成了两次太空行走和 20 多项科学实验, 并开展了两次“天宫课堂”活动, 刷新了中国航天新纪录。已知地球半径为 $R$, 地球表面重力加速度为 $g$ 。总质量为 $m_{0}$ 的空间站绕地球的运动可近似为匀速圆周运动, 距地球表面高度为 $h$, 阻力忽略不计。\n维持空间站的运行与舱内航天员的生活需要耗费大量电能, 某同学为其设计了太阳能电池板。太阳辐射的总功率为 $P_{0}$, 太阳与空间站的平均距离为 $r$, 且该太阳能电池板正对太阳的面积始终为 $S$, 假设该太阳能电池板的能量转化效率为 $\\eta$ 。求单位时间空间站通过太阳能电池板获得的电能。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1164", "problem": "Some of the very first exoplanets to be discovered in large surveys were dubbed 'hot Jupiters' as they were similar in mass to Jupiter (i.e. a gas giant) but were much closer to their star than Mercury is to the Sun (and hence are in a very hot environment). Planetary formation models suggest that they were unlikely to have formed there, but instead formed much further out from the star and migrated inwards, due to gravitational interactions with other planets in the system. Studies of 'hot Jupiters' show that there is an overabundance of them with periods of $\\sim 3-4$ days, and very few with periods shorter than that. Since large, close-in planets should be the easiest to detect in all of the main methods of finding exoplanets, this scarcity is likely to be a real effect and suggests that exoplanets which are that close to their star are in a relatively rapid (by astronomical standards) inspiral towards destruction by their star.\n[figure1]\n\nFigure 6: Left: The orbital radius of several 'hot Jupiters' scaled by the Roche radius of the system (where tidal forces would destroy the planet). There is an expected pile up close to radii double the Roche radius (dotted line), and very few with radii smaller than that - those that are will inevitably spiral into the star and be destroyed by the tidal forces when they get too close. Credit: Birkby et al. (2014).\n\nRight: As the planets inspiral we should see a shift in when their transits occur. This figure shows the predicted size of the shift after a period of 10 years if the tidal dissipation quality factor $Q_{\\star}^{\\prime}=10^{6}$, as well as the current detection limit of 5 seconds (dotted line). Therefore measuring if there is any shift in the transit times over the course of a decade of observations can put stringent limits on the value of $Q_{\\star}^{\\prime}$. Credit: Birkby et al. (2014).\n\nThe Roche radius, where a planet will be torn apart due to the tidal forces acting on it, is defined as\n\n$$\na_{\\text {Roche }} \\approx 2.16 R_{P}\\left(\\frac{M_{\\star}}{M_{P}}\\right)^{1 / 3}\n$$\n\nwhere $R_{P}$ is the radius of the planet, $M_{P}$ is the mass of the planet and $M_{\\star}$ is the mass of the star. If a gas giant is knocked into a highly elliptical orbit (i.e. $e \\approx 1$ ) that has a periapsis $r_{\\text {peri }}V_{1}>V_{4}>V_{3}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 某次发射远地圆轨道卫星时, 先让卫星进入一个近地的圆轨道I, 在此轨道正常运行时, 卫星的轨道半径为 $\\mathrm{R}_{1}$ 、周期为 $\\mathrm{T}_{1}$ 、经过 $\\mathrm{p}$ 点的速度大小为 $\\mathrm{V}_{1}$ 、加速度大小为 $a_{1}$; 然后在 $\\mathrm{P}$ 点点火加速, 进入粗圆形转移轨道II, 在此轨道正常运行时, 卫星的周期为 $\\mathrm{T}_{2}$, 经过 $\\mathrm{p}$ 点的速度大小为 $\\mathrm{V}_{2}$ 、加速度大小为 $a_{2}$, 经过 $\\mathrm{Q}$ 点速度大小为 $\\mathrm{V}_{3}$; 稳定运行数圈后达远地点 $\\mathrm{Q}$ 时再次点火加速, 进入远地圆轨道III在此轨道正常运行时, 卫星的轨道半径为 $\\mathrm{R}_{3}$ 、周期为 $\\mathrm{T}_{3}$ 、经过 $\\mathrm{Q}$ 点速度大小为 $\\mathrm{V}_{4}$ (轨道II的近地点和远地点分别为轨道 $I$ 上的 $P$ 点、轨道III上的 $Q$ 点). 已知 $R_{3}=2 R_{1}$, 则下列关系正确的是 $(\\quad)$\n\n[图1]\n\nA: $\\mathrm{T}_{2}=3 \\sqrt{3} \\mathrm{~T}_{1}$\nB: $\\mathrm{T}_{2}=\\frac{3 \\sqrt{3}}{8} \\mathrm{~T}_{3}$\nC: $a_{1}=a_{2}$\nD: $V_{2}>V_{1}>V_{4}>V_{3}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-12.jpg?height=309&width=306&top_left_y=2384&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_667", "problem": "“信使号”水星探测器按计划将在今年陨落在水星表面。工程师通过向后释放推进系统中的高压氦气来提升轨道, 使其寿命再延长一个月, 如图所示, 释放氦气前, 探测器在贴近水星表面的圆形轨道I上做匀速圆周运动, 释放氦气后探测器进入粗圆轨道II, 忽略探测器在粗圆轨道上所受阻力, 则下列说法正确的是（）\n\n[图1]\nA: 探测器在轨道I上 $E$ 点速率大于在轨道II上 $E$ 点速率\nB: 探测器在轨道II上任意位置的速率都大于在轨道I上速率\nC: 探测器在轨道I和轨道II上的 $E$ 点处加速度不相同\nD: 探测器在轨道II上远离水星过程中, 动能减少但势能增加\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n“信使号”水星探测器按计划将在今年陨落在水星表面。工程师通过向后释放推进系统中的高压氦气来提升轨道, 使其寿命再延长一个月, 如图所示, 释放氦气前, 探测器在贴近水星表面的圆形轨道I上做匀速圆周运动, 释放氦气后探测器进入粗圆轨道II, 忽略探测器在粗圆轨道上所受阻力, 则下列说法正确的是（）\n\n[图1]\n\nA: 探测器在轨道I上 $E$ 点速率大于在轨道II上 $E$ 点速率\nB: 探测器在轨道II上任意位置的速率都大于在轨道I上速率\nC: 探测器在轨道I和轨道II上的 $E$ 点处加速度不相同\nD: 探测器在轨道II上远离水星过程中, 动能减少但势能增加\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-105.jpg?height=254&width=394&top_left_y=1758&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1060", "problem": "GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nThe rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as\n\n$$\n\\dot{f}_{\\mathrm{GW}}=\\frac{96}{5} \\pi^{8 / 3}\\left(\\frac{G \\mathcal{M}}{c^{3}}\\right)^{5 / 3} f_{\\mathrm{GW}}^{11 / 3}\n$$a. The galaxy NGC 4993 is measured to have a redshift of $z=0.00980 \\pm 0.00079$. Assuming it follows Hubble's Law, $v=H_{0} d$, where $H_{0}=73.24 \\pm 1.74 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$, as determined by Hubble Space Telescope (HST) measurements of Cepheid variables, calculate the distance to the galaxy (in Mpc) and its (absolute) uncertainty. Give your distance to an appropriate number of significant figures.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nGW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \\& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$.\n\nAnother way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation\n\n$$\n\\log \\left(\\frac{D}{(1+z)^{2}}\\right)=-\\log R_{e}+\\alpha \\log \\sigma-\\beta \\log \\left\\langle I_{r}\\right\\rangle_{e}+\\gamma\n$$\n\nwhere $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\\sigma$ is the velocity dispersion in $\\mathrm{km} \\mathrm{s}^{-1},\\left\\langle I_{r}\\right\\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\\odot} \\mathrm{pc}^{-2}$, and $\\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\\alpha=1.24, \\beta=0.82$, and $\\gamma=2.194$.\n\nFigure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017).\n\nBy measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\\omega$, then the dimensionless strain parameter $h$ is\n\n$$\nh \\simeq \\frac{G}{c^{4}} \\frac{1}{r} \\mu a^{2} \\omega^{2}\n$$\n\nwhere $r$ is the luminosity distance, $c$ is the speed of light, $\\mu=m_{1} m_{2} / M_{\\text {tot }}$ is the reduced mass and $M_{\\text {tot }}=m_{1}+m_{2}$ is the total mass.\n\nThe rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as\n\n$$\n\\dot{f}_{\\mathrm{GW}}=\\frac{96}{5} \\pi^{8 / 3}\\left(\\frac{G \\mathcal{M}}{c^{3}}\\right)^{5 / 3} f_{\\mathrm{GW}}^{11 / 3}\n$$\n\nproblem:\na. The galaxy NGC 4993 is measured to have a redshift of $z=0.00980 \\pm 0.00079$. Assuming it follows Hubble's Law, $v=H_{0} d$, where $H_{0}=73.24 \\pm 1.74 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$, as determined by Hubble Space Telescope (HST) measurements of Cepheid variables, calculate the distance to the galaxy (in Mpc) and its (absolute) uncertainty. Give your distance to an appropriate number of significant figures.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{Mpc}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{Mpc}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_706", "problem": "太空中存在一些离其它恒星较远的、由质量相等的三颗星组成的三星系统, 通常可忽略其它星体对它们的引力作用. 已观测到稳定的三星系统存在两种基本的构成形式:一种是三颗星位于同一直线上, 两颗星围绕中央星在同一半径为 $\\mathrm{R}$ 的圆轨道上运行; 另一种形式是三颗星位于等边三角形的三个顶点上, 并沿外接于等边三角形的圆形轨道运行. 设这三个星体的质量均为 $\\mathrm{M}$, 并设两种系统的运动周期相同，则 （ )\n\n[图1]\nA: 直线三星系统运动的线速度大小为 $v=\\sqrt{\\frac{G M}{R}}$\nB: 三星系统的运动周期为 $T=4 \\pi R \\sqrt{\\frac{R}{5 G M}}$\nC: 三角形三星系统中星体间的距离为 $L=\\sqrt[3]{\\frac{12}{5}} R$\nD: 三角形三星系统的线速度大小为 $v=\\frac{1}{2} \\sqrt{\\frac{5 G M}{R}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n太空中存在一些离其它恒星较远的、由质量相等的三颗星组成的三星系统, 通常可忽略其它星体对它们的引力作用. 已观测到稳定的三星系统存在两种基本的构成形式:一种是三颗星位于同一直线上, 两颗星围绕中央星在同一半径为 $\\mathrm{R}$ 的圆轨道上运行; 另一种形式是三颗星位于等边三角形的三个顶点上, 并沿外接于等边三角形的圆形轨道运行. 设这三个星体的质量均为 $\\mathrm{M}$, 并设两种系统的运动周期相同，则 （ )\n\n[图1]\n\nA: 直线三星系统运动的线速度大小为 $v=\\sqrt{\\frac{G M}{R}}$\nB: 三星系统的运动周期为 $T=4 \\pi R \\sqrt{\\frac{R}{5 G M}}$\nC: 三角形三星系统中星体间的距离为 $L=\\sqrt[3]{\\frac{12}{5}} R$\nD: 三角形三星系统的线速度大小为 $v=\\frac{1}{2} \\sqrt{\\frac{5 G M}{R}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-26.jpg?height=320&width=391&top_left_y=2184&top_left_x=336", "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-27.jpg?height=360&width=354&top_left_y=808&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1148", "problem": "The speed of light is considered to be the speed limit of the Universe, however knots of plasma in the jets from active galactic nuclei (AGN) have been observed to be moving with apparent transverse speeds in excess of this, called superluminal speeds. Some of the more extreme examples can be appearing to move at up to 6 times the speed of light (see Figure 7).\n[figure1]\n\nFigure 7: Left: The jet coming from the elliptical galaxy M87 as viewed by the Hubble Space Telescope (HST). Right: Sequence of HST images showing motion at six times the speed of light. The slanting lines track the moving features, and the speeds are given in units of the velocity of light, $c$. Credit: NASA / Space Telescope Science Institute / John Biretta.\n\nThis can be explained by understanding that the jet is offset by an angle $\\theta$ from the sightline to Earth, and that the real speed of the plasma knot, $v$, is less than $c$, and from it we can define the scaled speed $\\beta \\equiv v / c$.\n\nSuperluminal jets are not limited just to AGN, as they have also been observed from systems within our own galaxy. A particularly famous one is the 'microquasar' GRS $1915+105$, which is a low mass X-ray binary consisting of a small star orbiting a black hole. A symmetrical jet with components approaching and receding from us is observed (as expected for jets coming from the poles of the black hole), and the apparent transverse motion of material in those jets has been measured using very high resolution radio imaging. Fender et. al (1999) measure these motions to be $\\mu_{a}=23.6$ mas day $^{-1}$ and $\\mu_{r}=$ 10.0 mas day $^{-1}$ for the approaching and receding jet respectively ( 1 mas $=1$ milliarcsecond, a unit of angle, and there are 3600 arcseconds in a degree) and the distance to the system as $11 \\mathrm{kpc}$.\n\nIn practice, for a given $\\beta_{\\text {app }}$ the values of $\\beta$ and $\\theta$ are degenerate and it is unlikely that the orientation of the jet is such that $\\beta_{\\text {app }}$ has been maximised, so the value in part $\\mathrm{c}$. is just a lower limit. However, since there are two jets then if we assume that they are from the same event (and so equal in speed but opposite in direction) we can break this degeneracy.\n\nSince it is a binary system, we can gain information about the masses of the objects by looking at their period and radial velocity. Formally, the relationship is\n\n$$\n\\frac{\\left(M_{\\mathrm{BH}} \\sin i\\right)^{3}}{\\left(M_{\\mathrm{BH}}+M_{\\star}\\right)^{2}}=\\frac{P_{\\mathrm{orb}} K_{d}^{3}}{2 \\pi G}\n$$\n\nwhere $M_{\\mathrm{BH}}$ is the mass of the black hole, $M_{\\star}$ is the mass of the orbiting star, $i$ is the inclination of the orbit, $P_{\\text {orb }}$ is the orbital period, and $K_{d}$ is the amplitude of the radial velocity curve. Normally the inclination can't be measured, however if we assume that the orbit is perpendicular to the jets then $i=\\theta$ and we can measure the mass of the black hole.e. Show that $\\beta \\cos \\theta$ can be expressed purely as a function of $\\mu_{a}$ and $\\mu_{r}$, and hence use your value of $\\theta$ to calculate the value of $\\beta$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe speed of light is considered to be the speed limit of the Universe, however knots of plasma in the jets from active galactic nuclei (AGN) have been observed to be moving with apparent transverse speeds in excess of this, called superluminal speeds. Some of the more extreme examples can be appearing to move at up to 6 times the speed of light (see Figure 7).\n[figure1]\n\nFigure 7: Left: The jet coming from the elliptical galaxy M87 as viewed by the Hubble Space Telescope (HST). Right: Sequence of HST images showing motion at six times the speed of light. The slanting lines track the moving features, and the speeds are given in units of the velocity of light, $c$. Credit: NASA / Space Telescope Science Institute / John Biretta.\n\nThis can be explained by understanding that the jet is offset by an angle $\\theta$ from the sightline to Earth, and that the real speed of the plasma knot, $v$, is less than $c$, and from it we can define the scaled speed $\\beta \\equiv v / c$.\n\nSuperluminal jets are not limited just to AGN, as they have also been observed from systems within our own galaxy. A particularly famous one is the 'microquasar' GRS $1915+105$, which is a low mass X-ray binary consisting of a small star orbiting a black hole. A symmetrical jet with components approaching and receding from us is observed (as expected for jets coming from the poles of the black hole), and the apparent transverse motion of material in those jets has been measured using very high resolution radio imaging. Fender et. al (1999) measure these motions to be $\\mu_{a}=23.6$ mas day $^{-1}$ and $\\mu_{r}=$ 10.0 mas day $^{-1}$ for the approaching and receding jet respectively ( 1 mas $=1$ milliarcsecond, a unit of angle, and there are 3600 arcseconds in a degree) and the distance to the system as $11 \\mathrm{kpc}$.\n\nIn practice, for a given $\\beta_{\\text {app }}$ the values of $\\beta$ and $\\theta$ are degenerate and it is unlikely that the orientation of the jet is such that $\\beta_{\\text {app }}$ has been maximised, so the value in part $\\mathrm{c}$. is just a lower limit. However, since there are two jets then if we assume that they are from the same event (and so equal in speed but opposite in direction) we can break this degeneracy.\n\nSince it is a binary system, we can gain information about the masses of the objects by looking at their period and radial velocity. Formally, the relationship is\n\n$$\n\\frac{\\left(M_{\\mathrm{BH}} \\sin i\\right)^{3}}{\\left(M_{\\mathrm{BH}}+M_{\\star}\\right)^{2}}=\\frac{P_{\\mathrm{orb}} K_{d}^{3}}{2 \\pi G}\n$$\n\nwhere $M_{\\mathrm{BH}}$ is the mass of the black hole, $M_{\\star}$ is the mass of the orbiting star, $i$ is the inclination of the orbit, $P_{\\text {orb }}$ is the orbital period, and $K_{d}$ is the amplitude of the radial velocity curve. Normally the inclination can't be measured, however if we assume that the orbit is perpendicular to the jets then $i=\\theta$ and we can measure the mass of the black hole.\n\nproblem:\ne. Show that $\\beta \\cos \\theta$ can be expressed purely as a function of $\\mu_{a}$ and $\\mu_{r}$, and hence use your value of $\\theta$ to calculate the value of $\\beta$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-10.jpg?height=812&width=1458&top_left_y=504&top_left_x=296" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_770", "problem": "In astronomy, the concept of black bodies is very important to better calculate the radiation of stars. Which one is the correct definition of a black body?\nA: An idealized physical object that reflects all electromagnetic radiation.\nB: An idealized physical object that absorbs all electromagnetic radiation.\nC: An idealized physical object that reflects all polarized radiation.\nD: An idealized physical object that absorbs all polarized radiation.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn astronomy, the concept of black bodies is very important to better calculate the radiation of stars. Which one is the correct definition of a black body?\n\nA: An idealized physical object that reflects all electromagnetic radiation.\nB: An idealized physical object that absorbs all electromagnetic radiation.\nC: An idealized physical object that reflects all polarized radiation.\nD: An idealized physical object that absorbs all polarized radiation.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_52", "problem": "2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为世界上第一个首次探测火星就实现“绕、落、巡”三项任务的国家。\n\n 为了简化问题, 可以认为地球和火星在同一平面上绕太阳做匀速圆周运动, 如图 1 所示。已知地球的公转周期为 $T_{1}$, 火星的公转周期为 $T_{2}$ 。\n\n考虑到飞行时间和节省燃料, 地球和火星处于图 1 中相对位置时是在地球上发射火星探测器的最佳时机, 推导在地球上相邻两次发射火星探测器最佳时机的时间间隔 $\\Delta t$ 。\n\n\n[图1]\n\n图1\n\n[图2]\n\n图2", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为世界上第一个首次探测火星就实现“绕、落、巡”三项任务的国家。\n\n 为了简化问题, 可以认为地球和火星在同一平面上绕太阳做匀速圆周运动, 如图 1 所示。已知地球的公转周期为 $T_{1}$, 火星的公转周期为 $T_{2}$ 。\n\n考虑到飞行时间和节省燃料, 地球和火星处于图 1 中相对位置时是在地球上发射火星探测器的最佳时机, 推导在地球上相邻两次发射火星探测器最佳时机的时间间隔 $\\Delta t$ 。\n\n\n[图1]\n\n图1\n\n[图2]\n\n图2\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-035.jpg?height=432&width=508&top_left_y=1937&top_left_x=337", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-035.jpg?height=417&width=871&top_left_y=1939&top_left_x=844" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_195", "problem": "开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。\n[图1]求卫星在 I 轨道运动时的速度大小;", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。\n[图1]求卫星在 I 轨道运动时的速度大小;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-022.jpg?height=436&width=1398&top_left_y=230&top_left_x=342" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_935", "problem": "What is the period of the comet from the previous question?\nA: 2.1 years\nB: 2.9 years\nC: 5.2 years\nD: 11.2 years\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the period of the comet from the previous question?\n\nA: 2.1 years\nB: 2.9 years\nC: 5.2 years\nD: 11.2 years\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_494", "problem": "我国已掌握“半弹道跳跃式高速再入返回技术”, 为实现“嫦娥”飞船月地返回任务奠定基础。如图所示, 假设与地球同球心的虚线球面为地球大气层边界, 虚线球面外侧没有空气, 返回舱从 $a$ 点无动力滑入大气层, 然后经 $b$ 点从 $c$ 点“跳出”, 再经 $d$ 点从 $e$ 点 “跃入”实现多次减速, 可避免损坏返回舱。 $d$ 点为轨迹最高点, 离地面高 $h$, 已知地球质量为 $M$ ，半径为 $R$ ，引力常量为 $G$ 。则返回舱（）\n\n[图1]\nA: 在 $d$ 点加速度小于 $\\frac{G M}{(R+h)^{2}}$\nB: 在 $d$ 点速度小于 $\\sqrt{\\frac{G M}{R+h}}$\nC: 虚线球面上 $c 、 e$ 两点离地面高度相等, 所以 $\\mathrm{v}_{\\mathrm{c}}$ 和 $\\mathrm{v}_{\\mathrm{e}}$ 大小相等\nD: 虚线球面上 $a 、 c$ 两点离地面高度相等, 所以 $\\mathrm{v}_{\\mathrm{a}}$ 和 $\\mathrm{v}_{\\mathrm{c}}$ 大小相等\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n我国已掌握“半弹道跳跃式高速再入返回技术”, 为实现“嫦娥”飞船月地返回任务奠定基础。如图所示, 假设与地球同球心的虚线球面为地球大气层边界, 虚线球面外侧没有空气, 返回舱从 $a$ 点无动力滑入大气层, 然后经 $b$ 点从 $c$ 点“跳出”, 再经 $d$ 点从 $e$ 点 “跃入”实现多次减速, 可避免损坏返回舱。 $d$ 点为轨迹最高点, 离地面高 $h$, 已知地球质量为 $M$ ，半径为 $R$ ，引力常量为 $G$ 。则返回舱（）\n\n[图1]\n\nA: 在 $d$ 点加速度小于 $\\frac{G M}{(R+h)^{2}}$\nB: 在 $d$ 点速度小于 $\\sqrt{\\frac{G M}{R+h}}$\nC: 虚线球面上 $c 、 e$ 两点离地面高度相等, 所以 $\\mathrm{v}_{\\mathrm{c}}$ 和 $\\mathrm{v}_{\\mathrm{e}}$ 大小相等\nD: 虚线球面上 $a 、 c$ 两点离地面高度相等, 所以 $\\mathrm{v}_{\\mathrm{a}}$ 和 $\\mathrm{v}_{\\mathrm{c}}$ 大小相等\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-117.jpg?height=363&width=348&top_left_y=2377&top_left_x=357" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_250", "problem": "已知地球自转周期为 $\\mathrm{T}$, 地球半径为 $\\mathrm{R}$, 地球极地表面的重力加速度为 $\\mathrm{g}$,关于在地球同步轨道运行的卫星, 下列说法正确的是( )\nA: 角速度为 $\\frac{2 \\pi}{T}$\nB: 距地高度为 $\\sqrt[3]{\\frac{g R^{2} T^{2}}{4 \\pi^{2}}}$\nC: 线速度为 $\\sqrt[3]{\\frac{2 \\pi g R^{2}}{T}}$\nD: 加速度为 $\\sqrt[3]{\\frac{16 g R^{2} \\pi^{4}}{T^{4}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n已知地球自转周期为 $\\mathrm{T}$, 地球半径为 $\\mathrm{R}$, 地球极地表面的重力加速度为 $\\mathrm{g}$,关于在地球同步轨道运行的卫星, 下列说法正确的是( )\n\nA: 角速度为 $\\frac{2 \\pi}{T}$\nB: 距地高度为 $\\sqrt[3]{\\frac{g R^{2} T^{2}}{4 \\pi^{2}}}$\nC: 线速度为 $\\sqrt[3]{\\frac{2 \\pi g R^{2}}{T}}$\nD: 加速度为 $\\sqrt[3]{\\frac{16 g R^{2} \\pi^{4}}{T^{4}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1217", "problem": "In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.a. Assume the Sun was initially made of pure hydrogen, carries out nuclear fusion at a constant rate and will continue to do so until the hydrogen in its core is used up. If the mass of the core is $10 \\%$ of the star, and $0.7 \\%$ of the mass in each fusion reaction is converted into energy, show that the Sun's lifespan on the main sequence is approximately 10 billion years.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.\n\nproblem:\na. Assume the Sun was initially made of pure hydrogen, carries out nuclear fusion at a constant rate and will continue to do so until the hydrogen in its core is used up. If the mass of the core is $10 \\%$ of the star, and $0.7 \\%$ of the mass in each fusion reaction is converted into energy, show that the Sun's lifespan on the main sequence is approximately 10 billion years.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of years, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-06.jpg?height=611&width=1448&top_left_y=505&top_left_x=310" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "years" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_916", "problem": "Given the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\\gtrsim 80 \\%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet.\n[figure1]\n\nFigure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\\sim 100 \\%$ iron) might look like.\n\nThe Roche limiting distance, $a_{\\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is\n\n$$\na_{\\min }=2.44 R_{\\star}\\left(\\frac{\\rho_{\\star}}{\\rho_{p}}\\right)^{1 / 3}\n$$\n\nwhere $R_{\\star}$ is the radius of the star, $\\rho_{\\star}$ is the density of the star and $\\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as\n\n$$\n\\log _{10}\\left(\\frac{R}{R_{E}}\\right)=0.295 \\log _{10}\\left(\\frac{M}{M_{E}}\\right)+\\alpha\n$$\n\nwhere $\\alpha=0.0286$ in the pure silicate case and $\\alpha=-0.1090$ in the pure iron case.\n\nThe measured period is 0.1768913 days. If it is only made of iron and silicate, estimate the minimum percentage of iron.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nGiven the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\\gtrsim 80 \\%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet.\n[figure1]\n\nFigure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\\sim 100 \\%$ iron) might look like.\n\nThe Roche limiting distance, $a_{\\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is\n\n$$\na_{\\min }=2.44 R_{\\star}\\left(\\frac{\\rho_{\\star}}{\\rho_{p}}\\right)^{1 / 3}\n$$\n\nwhere $R_{\\star}$ is the radius of the star, $\\rho_{\\star}$ is the density of the star and $\\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as\n\n$$\n\\log _{10}\\left(\\frac{R}{R_{E}}\\right)=0.295 \\log _{10}\\left(\\frac{M}{M_{E}}\\right)+\\alpha\n$$\n\nwhere $\\alpha=0.0286$ in the pure silicate case and $\\alpha=-0.1090$ in the pure iron case.\n\nThe measured period is 0.1768913 days. If it is only made of iron and silicate, estimate the minimum percentage of iron.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of percentage, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_3776e2d93eca0bbf48b9g-09.jpg?height=620&width=1468&top_left_y=861&top_left_x=292" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "percentage" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_68", "problem": "2021 年 5 月 15 日中国首次火星探测任务“天问一号”探测器的着陆巡视器在火星乌托邦平原南部预选着陆区成功着陆, 在火星上首次留下中国印迹, 迈出了中国星际探测征程的重要一步。“天问一号”探测器需要通过霍曼转移轨道从地球发射到火星, 地球轨道和火星轨道近似看成圆形轨道, 霍曼转移轨道是一个在近日点 $M$ 和远日点 $P$ 分别与地球轨道、火星轨道相切的椭圆轨道 (如图所示), 在近日点短暂点火后“天向一号”进入霍曼转移轨道, 接着“天问一号”沿着这个轨道运行直至抵达远日点, 然后再次点火进入火星轨道。已知引力常量为 $G$, 太阳质量为 $m$, 地球轨道和火星轨道半径分别为 $r$ 和 $R$ ，地球、火星、“天向一号”运行方向都为逆时针方向。若只考虑太阳对“天问一号”的作用力, 下列说法正确的是（）\n\n[图1]\nA: 两次点火喷射方向都与速度方向相反\nB: “天问-号”运行中, 在霍曼转移轨道上 $P$ 点的加速度比在火星轨道上 $P$ 点的加速度小\nC: 两次点火之间的时间间隔为 $\\frac{\\pi}{2 \\sqrt{2}} \\sqrt{\\frac{(R+r)^{3}}{G m}}$\nD: “天问一号”在地球轨道上的角速度小于在火星轨道上的角速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2021 年 5 月 15 日中国首次火星探测任务“天问一号”探测器的着陆巡视器在火星乌托邦平原南部预选着陆区成功着陆, 在火星上首次留下中国印迹, 迈出了中国星际探测征程的重要一步。“天问一号”探测器需要通过霍曼转移轨道从地球发射到火星, 地球轨道和火星轨道近似看成圆形轨道, 霍曼转移轨道是一个在近日点 $M$ 和远日点 $P$ 分别与地球轨道、火星轨道相切的椭圆轨道 (如图所示), 在近日点短暂点火后“天向一号”进入霍曼转移轨道, 接着“天问一号”沿着这个轨道运行直至抵达远日点, 然后再次点火进入火星轨道。已知引力常量为 $G$, 太阳质量为 $m$, 地球轨道和火星轨道半径分别为 $r$ 和 $R$ ，地球、火星、“天向一号”运行方向都为逆时针方向。若只考虑太阳对“天问一号”的作用力, 下列说法正确的是（）\n\n[图1]\n\nA: 两次点火喷射方向都与速度方向相反\nB: “天问-号”运行中, 在霍曼转移轨道上 $P$ 点的加速度比在火星轨道上 $P$ 点的加速度小\nC: 两次点火之间的时间间隔为 $\\frac{\\pi}{2 \\sqrt{2}} \\sqrt{\\frac{(R+r)^{3}}{G m}}$\nD: “天问一号”在地球轨道上的角速度小于在火星轨道上的角速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-110.jpg?height=585&width=554&top_left_y=1569&top_left_x=363" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_657", "problem": "开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。\n[图1]在牛顿力学体系中, 当两个质量分别为 $m_{1} 、 m_{2}$ 的质点相距为 $r$ 时具有的势能, 称为引力势能, 其大小为 $\\mathrm{E}_{\\mathrm{P}}=-\\frac{G m_{1} m_{2}}{r}$ (规定无穷远处势能为零)卫星在 I 轨道的 $\\mathrm{P}$ 点点火加速, 变轨到II轨道,卫星在 I 轨道的 P 点, 变轨到II轨道, 求则至少需对卫星做多少功(不考虑卫星质量的变化和所受的阻力).", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。\n[图1]在牛顿力学体系中, 当两个质量分别为 $m_{1} 、 m_{2}$ 的质点相距为 $r$ 时具有的势能, 称为引力势能, 其大小为 $\\mathrm{E}_{\\mathrm{P}}=-\\frac{G m_{1} m_{2}}{r}$ (规定无穷远处势能为零)卫星在 I 轨道的 $\\mathrm{P}$ 点点火加速, 变轨到II轨道,卫星在 I 轨道的 P 点, 变轨到II轨道, 求则至少需对卫星做多少功(不考虑卫星质量的变化和所受的阻力).\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-022.jpg?height=436&width=1398&top_left_y=230&top_left_x=342" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_509", "problem": "春分时，当太阳光直射地球赤道时，某天文爱好者在地球表面上某处用天文望远镜恰好观测到其正上方有一颗被太阳光照射的地球同步卫星。下列关于在日落后的 12 小时内，该观察者看不见此卫星的时间的判断正确的是（已知地球半径为 $R$, 地球表面处的重力加速度为 $g$, 地球自转角速度为 $\\omega$, 不考虑大气对光的折射）( )\nA: $t=\\frac{1}{\\omega} \\arcsin \\sqrt[3]{\\frac{R \\omega^{2}}{g}}$\nB: $t=\\frac{2}{\\omega} \\arcsin \\sqrt[3]{\\frac{g}{R \\omega^{2}}}$\nC: $t=\\frac{2}{\\omega} \\arcsin \\sqrt[3]{\\frac{R \\omega^{2}}{g}}$\nD: $t=\\frac{1}{\\omega} \\arcsin \\sqrt[3]{\\frac{g}{R \\omega^{2}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n春分时，当太阳光直射地球赤道时，某天文爱好者在地球表面上某处用天文望远镜恰好观测到其正上方有一颗被太阳光照射的地球同步卫星。下列关于在日落后的 12 小时内，该观察者看不见此卫星的时间的判断正确的是（已知地球半径为 $R$, 地球表面处的重力加速度为 $g$, 地球自转角速度为 $\\omega$, 不考虑大气对光的折射）( )\n\nA: $t=\\frac{1}{\\omega} \\arcsin \\sqrt[3]{\\frac{R \\omega^{2}}{g}}$\nB: $t=\\frac{2}{\\omega} \\arcsin \\sqrt[3]{\\frac{g}{R \\omega^{2}}}$\nC: $t=\\frac{2}{\\omega} \\arcsin \\sqrt[3]{\\frac{R \\omega^{2}}{g}}$\nD: $t=\\frac{1}{\\omega} \\arcsin \\sqrt[3]{\\frac{g}{R \\omega^{2}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-068.jpg?height=434&width=525&top_left_y=154&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_401", "problem": "鸟神星是太阳系内已知的第三大矮行星, 已知其质量为 $m$, 绕太阳做匀速圆周运动(近似认为)的周期为 $T_{1}$, 鸟神星的自转周期为 $T_{2}$, 表面的重力加速度为 $g$, 引力常量为 $G$,根据这些已知量可得\nA: 鸟神星的半径为 $\\frac{G m}{g}$\nB: 鸟神星到太阳的距离为 $\\sqrt[3]{\\frac{G m T_{1}}{4 \\pi}}$\nC: 鸟神星的同步卫星的轨道半径为 $\\sqrt[3]{\\frac{G m T_{2}^{2}}{4 \\pi^{2}}}$\nD: 鸟神星的第一宇宙速度为 $\\sqrt{m g G}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n鸟神星是太阳系内已知的第三大矮行星, 已知其质量为 $m$, 绕太阳做匀速圆周运动(近似认为)的周期为 $T_{1}$, 鸟神星的自转周期为 $T_{2}$, 表面的重力加速度为 $g$, 引力常量为 $G$,根据这些已知量可得\n\nA: 鸟神星的半径为 $\\frac{G m}{g}$\nB: 鸟神星到太阳的距离为 $\\sqrt[3]{\\frac{G m T_{1}}{4 \\pi}}$\nC: 鸟神星的同步卫星的轨道半径为 $\\sqrt[3]{\\frac{G m T_{2}^{2}}{4 \\pi^{2}}}$\nD: 鸟神星的第一宇宙速度为 $\\sqrt{m g G}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_781", "problem": "Why are Cepheid stars relevant for astronomers?\nA: To measure interstellar mass.\nB: To measure galactic distances.\nC: To measure galactic energy-density.\nD: To measure interstellar density.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhy are Cepheid stars relevant for astronomers?\n\nA: To measure interstellar mass.\nB: To measure galactic distances.\nC: To measure galactic energy-density.\nD: To measure interstellar density.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_413", "problem": "2018 年 6 月 14 日。承担嫦娥四号中继通信任务的“鹊桥”中继星抵达绕地月第二拉\n格朗日点的轨道, 第二拉格朗日点是地月连线延长线上的一点, 处于该位置上的卫星与月球同步绕地球公转, 则该卫星的（）\n\n[图1]\nA: 向心力仅来自于地球引力\nB: 线速度大于月球的线速度\nC: 角速度大于月球的角速度\nD: 向心加速度大于月球的向心加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2018 年 6 月 14 日。承担嫦娥四号中继通信任务的“鹊桥”中继星抵达绕地月第二拉\n格朗日点的轨道, 第二拉格朗日点是地月连线延长线上的一点, 处于该位置上的卫星与月球同步绕地球公转, 则该卫星的（）\n\n[图1]\n\nA: 向心力仅来自于地球引力\nB: 线速度大于月球的线速度\nC: 角速度大于月球的角速度\nD: 向心加速度大于月球的向心加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-19.jpg?height=397&width=437&top_left_y=333&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_884", "problem": "A student group from California Institute of Technology is planning a massive prank for this year. This year, they are planning on launching a rocket to Massachusetts Institute of Technology and making it explode above the so called 'Great Dome' of MIT. The rocket will contain a giant parachute printed with a logo of Caltech, so that the Great Dome will be covered with Caltech logo. How much distance does this rocket need to fly? The longitude and latitude information of two locations are given in the table below. (Assume that the Earth is a perfect sphere with radius of $6371 \\mathrm{~km}$.)\n\n| | Latitude | Longtitude |\n| :--- | :--- | :--- |\n| The Great Dome of
MIT | $42.3601^{\\circ} \\mathrm{N}$ | $71.0942^{\\circ} \\mathrm{W}$ |\n| Caltech | $34.1377^{\\circ} \\mathrm{N}$ | $118.1253^{\\circ} \\mathrm{W}\nA: $3890 \\mathrm{~km}$\nB: $4160 \\mathrm{~km}$\nC: $4780 \\mathrm{~km}$\nD: $4910 \\mathrm{~km}$\nE: $5290 \\mathrm{~km}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA student group from California Institute of Technology is planning a massive prank for this year. This year, they are planning on launching a rocket to Massachusetts Institute of Technology and making it explode above the so called 'Great Dome' of MIT. The rocket will contain a giant parachute printed with a logo of Caltech, so that the Great Dome will be covered with Caltech logo. How much distance does this rocket need to fly? The longitude and latitude information of two locations are given in the table below. (Assume that the Earth is a perfect sphere with radius of $6371 \\mathrm{~km}$.)\n\n| | Latitude | Longtitude |\n| :--- | :--- | :--- |\n| The Great Dome of
MIT | $42.3601^{\\circ} \\mathrm{N}$ | $71.0942^{\\circ} \\mathrm{W}$ |\n| Caltech | $34.1377^{\\circ} \\mathrm{N}$ | $118.1253^{\\circ} \\mathrm{W}\n\nA: $3890 \\mathrm{~km}$\nB: $4160 \\mathrm{~km}$\nC: $4780 \\mathrm{~km}$\nD: $4910 \\mathrm{~km}$\nE: $5290 \\mathrm{~km}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_148", "problem": "如图所示, 从地面上 $A$ 点发射一枚远程弹道导弹, 假设导弹仅在地球引力作用下沿 $A C B$ 粗圆轨道飞行并击中地面目标 $B, C$ 为轨道的远地点, 距地面高度为 $h$. 已知地球半径为 $R$, 地球质量为 $M$, 引力常量为 $G$. 则下列结论正确的是 ( )\n\n[图1]\nA: 导弹在 $C$ 点的速度大于 $\\sqrt{\\frac{G M}{R+h}}$\nB: 导弹在 $C$ 点的速度等于 $\\sqrt[3]{\\frac{G M}{R+h}}$\nC: 导弹在 $c$ 点的加速度等于 $\\frac{G M}{(R+h)^{2}}$\nD: 导弹在 $C$ 点的加速度大于 $\\frac{G M}{(R-h)^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 从地面上 $A$ 点发射一枚远程弹道导弹, 假设导弹仅在地球引力作用下沿 $A C B$ 粗圆轨道飞行并击中地面目标 $B, C$ 为轨道的远地点, 距地面高度为 $h$. 已知地球半径为 $R$, 地球质量为 $M$, 引力常量为 $G$. 则下列结论正确的是 ( )\n\n[图1]\n\nA: 导弹在 $C$ 点的速度大于 $\\sqrt{\\frac{G M}{R+h}}$\nB: 导弹在 $C$ 点的速度等于 $\\sqrt[3]{\\frac{G M}{R+h}}$\nC: 导弹在 $c$ 点的加速度等于 $\\frac{G M}{(R+h)^{2}}$\nD: 导弹在 $C$ 点的加速度大于 $\\frac{G M}{(R-h)^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-56.jpg?height=411&width=297&top_left_y=1274&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_474", "problem": "质量为 $100 \\mathrm{~kg}$ 的“勇气”号火星车于 2004 年成功登陆在火星表面。若“勇气”号在离火星表面 $12 \\mathrm{~m}$ 时与降落伞自动脱离, 被气囊包裹的“勇气”号下落到地面后又弹跳到 $18 \\mathrm{~m}$ 高处, 这样上下碰撞了若干次后, 才静止在火星表面上。已知火星的半径为地球半径的 0.5 倍, 质量为地球质量的 0.1 倍。若“勇气”号第一次碰撞火星地面时, 气囊和地面的接触时间为 $0.7 \\mathrm{~s}$, 其损失的机械能为它与降落伞自动脱离处 (即离火星地面 $12 \\mathrm{~m}$ 时) 动能的 70\\%, (地球表面的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 不考虑火星表面空气阻力) 求:\n\n“勇气”号和气囊第一次与火星碰撞时所受到的平均冲力。", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n质量为 $100 \\mathrm{~kg}$ 的“勇气”号火星车于 2004 年成功登陆在火星表面。若“勇气”号在离火星表面 $12 \\mathrm{~m}$ 时与降落伞自动脱离, 被气囊包裹的“勇气”号下落到地面后又弹跳到 $18 \\mathrm{~m}$ 高处, 这样上下碰撞了若干次后, 才静止在火星表面上。已知火星的半径为地球半径的 0.5 倍, 质量为地球质量的 0.1 倍。若“勇气”号第一次碰撞火星地面时, 气囊和地面的接触时间为 $0.7 \\mathrm{~s}$, 其损失的机械能为它与降落伞自动脱离处 (即离火星地面 $12 \\mathrm{~m}$ 时) 动能的 70\\%, (地球表面的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 不考虑火星表面空气阻力) 求:\n\n“勇气”号和气囊第一次与火星碰撞时所受到的平均冲力。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以N为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "N" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_861", "problem": "An interesting phenomena that happens in the Solar System is the capture of comets in the interstellar medium. Assume that a comet with a mass of $7.15 * 10^{16} \\mathrm{~kg}$ is captured by the solar system. The perihelion of this comet's orbit after it is captured is equal to $4.64 \\mathrm{AU}$, and its velocity with respect to the Sun before being captured by the Solar System was very small. Calculate the velocity of the comet at the perihelion.\nA: $87.1 \\mathrm{~km} / \\mathrm{s}$\nB: $45.9 \\mathrm{~km} / \\mathrm{s}$\nC: $5.67 \\mathrm{~km} / \\mathrm{s}$\nD: $105.4 \\mathrm{~km} / \\mathrm{s}$\nE: $19.6 \\mathrm{~km} / \\mathrm{s}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn interesting phenomena that happens in the Solar System is the capture of comets in the interstellar medium. Assume that a comet with a mass of $7.15 * 10^{16} \\mathrm{~kg}$ is captured by the solar system. The perihelion of this comet's orbit after it is captured is equal to $4.64 \\mathrm{AU}$, and its velocity with respect to the Sun before being captured by the Solar System was very small. Calculate the velocity of the comet at the perihelion.\n\nA: $87.1 \\mathrm{~km} / \\mathrm{s}$\nB: $45.9 \\mathrm{~km} / \\mathrm{s}$\nC: $5.67 \\mathrm{~km} / \\mathrm{s}$\nD: $105.4 \\mathrm{~km} / \\mathrm{s}$\nE: $19.6 \\mathrm{~km} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_410", "problem": "如图所示, 在某行星表面上有一倾斜的圆盘, 面与水平面的夹角为 $30^{\\circ}$, 盘面上离转轴距离 $L$ 处有小物体与圆盘保持相对静止, 绕垂直于盘面的固定对称轴以恒定角速度转动, 角速度为 $\\omega$ 时, 小物块刚要滑动, 物体与盘面间的动摩擦因数为 $\\frac{2 \\sqrt{3}}{3}$ (设最大静摩擦力等于滑动摩擦力), 星球的半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( )\n\n[图1]\nA: 这个行星的质量 $\\frac{2 \\omega^{2} R^{2} L}{G}$\nB: 这个行星的第一宇宙速度 $v=2 \\omega \\sqrt{L R}$\nC: 这个行星的密度是 $\\rho=\\frac{3 \\omega^{2} L}{\\pi G R}$\nD: 离行星表面距离为 $R$ 的地方的重力加速度为 $\\frac{1}{2} \\omega^{2} L$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 在某行星表面上有一倾斜的圆盘, 面与水平面的夹角为 $30^{\\circ}$, 盘面上离转轴距离 $L$ 处有小物体与圆盘保持相对静止, 绕垂直于盘面的固定对称轴以恒定角速度转动, 角速度为 $\\omega$ 时, 小物块刚要滑动, 物体与盘面间的动摩擦因数为 $\\frac{2 \\sqrt{3}}{3}$ (设最大静摩擦力等于滑动摩擦力), 星球的半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( )\n\n[图1]\n\nA: 这个行星的质量 $\\frac{2 \\omega^{2} R^{2} L}{G}$\nB: 这个行星的第一宇宙速度 $v=2 \\omega \\sqrt{L R}$\nC: 这个行星的密度是 $\\rho=\\frac{3 \\omega^{2} L}{\\pi G R}$\nD: 离行星表面距离为 $R$ 的地方的重力加速度为 $\\frac{1}{2} \\omega^{2} L$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-101.jpg?height=274&width=448&top_left_y=1913&top_left_x=356" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_343", "problem": "13. 一同学准备设计一个绕地球纬度圈飞行的卫星, 绕行方向与地球自转方向相同, 且要求其在一天绕地球 3 周,则该卫星与地球静止同步卫星相比,下列说法正确的是 ( )\nA: 该卫星与地球静止同步卫星可能不在同一轨道平面内\nB: 该卫星离地高度与地球静止同步卫星的离地高度之比为 $\\left(\\frac{1}{9}\\right)^{\\frac{1}{3}}$\nC: 该卫星线速度与地球静止同步卫星的线速度之比为 $4^{\\frac{1}{6}}$\nD: 该卫星与地球静止同步卫星的向心加速度之比为 $3^{\\frac{4}{3}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n13. 一同学准备设计一个绕地球纬度圈飞行的卫星, 绕行方向与地球自转方向相同, 且要求其在一天绕地球 3 周,则该卫星与地球静止同步卫星相比,下列说法正确的是 ( )\n\nA: 该卫星与地球静止同步卫星可能不在同一轨道平面内\nB: 该卫星离地高度与地球静止同步卫星的离地高度之比为 $\\left(\\frac{1}{9}\\right)^{\\frac{1}{3}}$\nC: 该卫星线速度与地球静止同步卫星的线速度之比为 $4^{\\frac{1}{6}}$\nD: 该卫星与地球静止同步卫星的向心加速度之比为 $3^{\\frac{4}{3}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_122", "problem": "有一轨道平面与赤道平面重合的侦察卫星, 轨道高度为 $R$, 飞行方向与地球自转方向相同。设地球自转周期为 $T_{0}$, 半径为 $R$, 地球赤道处的重力加速度为 $g$ 。位于赤道的某一地面基站在某时刻恰好与该卫星建立起通信链路, 则该地面基站能不间断的从侦察卫星上下载侦察数据的时间为（）\nA: A. $\\frac{2 \\pi}{3\\left(\\sqrt{\\frac{g}{R}}-\\frac{2 \\pi}{T_{0}}\\right)}$\nB: [图1]\nC: [图2]\nD: [图3]\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n有一轨道平面与赤道平面重合的侦察卫星, 轨道高度为 $R$, 飞行方向与地球自转方向相同。设地球自转周期为 $T_{0}$, 半径为 $R$, 地球赤道处的重力加速度为 $g$ 。位于赤道的某一地面基站在某时刻恰好与该卫星建立起通信链路, 则该地面基站能不间断的从侦察卫星上下载侦察数据的时间为（）\n\nA: A. $\\frac{2 \\pi}{3\\left(\\sqrt{\\frac{g}{R}}-\\frac{2 \\pi}{T_{0}}\\right)}$\nB: [图1]\nC: [图2]\nD: [图3]\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-085.jpg?height=177&width=434&top_left_y=1416&top_left_x=1051", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-085.jpg?height=169&width=423&top_left_y=1640&top_left_x=405", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-085.jpg?height=166&width=434&top_left_y=1642&top_left_x=1051", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-086.jpg?height=460&width=466&top_left_y=170&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1205", "problem": "The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.b. The Sun is composed predominantly of ionized hydrogen and helium, with approximate mass fractions $X=0.70$ and $Y=0.30$ respectively (taken to be constant throughout the Sun).\nii. Using the Virial Theorem, and given $E_{G} \\approx G M_{\\odot}^{2} / R_{\\odot}$, estimate the Sun's mean temperature.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.\n\nproblem:\nb. The Sun is composed predominantly of ionized hydrogen and helium, with approximate mass fractions $X=0.70$ and $Y=0.30$ respectively (taken to be constant throughout the Sun).\nii. Using the Virial Theorem, and given $E_{G} \\approx G M_{\\odot}^{2} / R_{\\odot}$, estimate the Sun's mean temperature.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~K}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-06.jpg?height=536&width=1508&top_left_y=560&top_left_x=272", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-07.jpg?height=748&width=1185&top_left_y=1850&top_left_x=433" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~K}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_157", "problem": "如图所示, 探月卫星的发射过程可简化如下：首先进入绕地球运行的“停泊轨道”,在该轨道的 $P$ 处通过变速再进入“地月转移轨道”, 在快要到达月球时, 对卫星再次变速,卫星被月球引力“俘获”后, 成为环月卫星, 最终在环绕月球的“工作轨道”绕月飞行（视为圆周运动), 对月球进行探测. “工作轨道”周期为 $T$ 、距月球表面的高度为 $h$, 月球半径为 $R$, 引力常量为 $G$, 忽略其他天体对探月卫星在“工作轨道”上环绕运动的影响。\n\n求探月卫星在“工作轨道”上环绕的线速度大小;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 探月卫星的发射过程可简化如下：首先进入绕地球运行的“停泊轨道”,在该轨道的 $P$ 处通过变速再进入“地月转移轨道”, 在快要到达月球时, 对卫星再次变速,卫星被月球引力“俘获”后, 成为环月卫星, 最终在环绕月球的“工作轨道”绕月飞行（视为圆周运动), 对月球进行探测. “工作轨道”周期为 $T$ 、距月球表面的高度为 $h$, 月球半径为 $R$, 引力常量为 $G$, 忽略其他天体对探月卫星在“工作轨道”上环绕运动的影响。\n\n求探月卫星在“工作轨道”上环绕的线速度大小;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-140.jpg?height=314&width=727&top_left_y=2210&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1126", "problem": "The Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$e. Taking the mass of $\\mathrm{M}_{8} 7^{*}$ as $6.5 \\times 10^{9} \\mathrm{M}_{\\odot}$ :\niii. Determine the minimum and maximum ISCO periods for Sgr A* and hence suggest a possible reason why $\\mathrm{M} 87$ has been imaged first, even though Sgr A* has a larger angular diameter, given that each 'exposure' with the EHT was 7 mins long (with multiple exposures from each observing run added together for the final image from each night).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*).\n[figure1]\n\nFigure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration.\n\nRight: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa.\n\nSome data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system.\n\n| Facility | Location | $X(\\mathrm{~m})$ | $Y(\\mathrm{~m})$ | $Z(\\mathrm{~m})$ |\n| :--- | :--- | :---: | :---: | :---: |\n| ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 |\n| APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 |\n| JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 |\n| LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 |\n| PV | Spain | 5088967.8 | -301681.2 | 3825012.2 |\n| SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 |\n| SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 |\n| SPT | Antarctica | 809.8 | -816.9 | -6359568.7 |\n\nThe minimum angle, $\\theta_{\\min }$ (in radians) that can be resolved by a VLBI array is given by the equation\n\n$$\n\\theta_{\\min }=\\frac{\\lambda_{\\mathrm{obs}}}{d_{\\max }},\n$$\n\nwhere $\\lambda_{\\text {obs }}$ is the observing wavelength and $d_{\\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation.\n\nAn important length scale when discussing black holes is the gravitational radius, $r_{g}=\\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \\equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \\sqrt{3+2 \\sqrt{2}}) r_{g}$ and $(3 \\sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for.\n\n[figure2]\n\nFigure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration.\n\nThe EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by\n\n$$\nE=m c^{2}\\left(\\frac{1-\\frac{2 r_{g}}{r}}{\\sqrt{1-\\frac{3 r_{g}}{r}}}\\right)\n$$\n\nand the radius of the ISCO, $r_{\\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised.\n\nWe expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \\equiv J / J_{\\max }$ where $J$ is the angular momentum of the black hole and $J_{\\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \\leq a \\leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by\n\n$$\n\\omega^{2}=\\frac{G M}{\\left(r_{\\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\\right)^{2}}\n$$\n\n[figure3]\n\nFigure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972).\n\nThe spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by\n\n$$\n\\Delta l=\\int_{r_{2}}^{r_{1}}\\left(1-\\frac{2 r_{g}}{r}\\right)^{-1 / 2} \\mathrm{~d} r\n$$\n\nproblem:\ne. Taking the mass of $\\mathrm{M}_{8} 7^{*}$ as $6.5 \\times 10^{9} \\mathrm{M}_{\\odot}$ :\niii. Determine the minimum and maximum ISCO periods for Sgr A* and hence suggest a possible reason why $\\mathrm{M} 87$ has been imaged first, even though Sgr A* has a larger angular diameter, given that each 'exposure' with the EHT was 7 mins long (with multiple exposures from each observing run added together for the final image from each night).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-07.jpg?height=704&width=1414&top_left_y=698&top_left_x=331", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-08.jpg?height=374&width=1562&top_left_y=1698&top_left_x=263", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-09.jpg?height=468&width=686&top_left_y=1388&top_left_x=705" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_446", "problem": "在星球 $\\mathrm{M}$ 上一轻弹簧坚直固定于水平桌面, 物体 $P$ 轻放在弹簧上由静止释放, 其加速度 $a$ 与弹簧压缩量 $x$ 的关系如图 $P$ 线所示。另一星球 $\\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样过程, 其加速度 $a$ 与弹簧压缩量 $x$ 的关系如 $Q$ 线所示, 下列说法正确的是 ( )\n\n[图1]\nA: 同一物体在 $M$ 星球表面与在 $N$ 星球表面重力大小之比为 3: 1\nB: 物体 $P 、 Q$ 的质量之比是 $6: 1$\nC: $\\mathrm{M}$ 星球上物体 $\\mathrm{R}$ 由静止开始做加速度为 $3 a_{0}$ 的匀加速直线运动, 通过位移 $x_{0}$ 时的速度为 $\\sqrt{3 a_{0} x_{0}}$\nD: 图中 $P 、 Q$ 下落的最大速度之比为 $\\frac{\\sqrt{6}}{2}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n在星球 $\\mathrm{M}$ 上一轻弹簧坚直固定于水平桌面, 物体 $P$ 轻放在弹簧上由静止释放, 其加速度 $a$ 与弹簧压缩量 $x$ 的关系如图 $P$ 线所示。另一星球 $\\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样过程, 其加速度 $a$ 与弹簧压缩量 $x$ 的关系如 $Q$ 线所示, 下列说法正确的是 ( )\n\n[图1]\n\nA: 同一物体在 $M$ 星球表面与在 $N$ 星球表面重力大小之比为 3: 1\nB: 物体 $P 、 Q$ 的质量之比是 $6: 1$\nC: $\\mathrm{M}$ 星球上物体 $\\mathrm{R}$ 由静止开始做加速度为 $3 a_{0}$ 的匀加速直线运动, 通过位移 $x_{0}$ 时的速度为 $\\sqrt{3 a_{0} x_{0}}$\nD: 图中 $P 、 Q$ 下落的最大速度之比为 $\\frac{\\sqrt{6}}{2}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-101.jpg?height=554&width=811&top_left_y=1825&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_908", "problem": "On the evening of 27th July 2018, there was a total lunar eclipse (where the Moon passes into the Earth's shadow). As viewed from Oxford, totality began whilst the Moon was below the horizon. Interestingly, moonrise was at $8: 55 \\mathrm{pm}$, yet sunset was only at $9: 01 \\mathrm{pm}$, so for 6 minutes both the fully eclipsed Moon and setting Sun were visible above the horizon. This very rare event is known as a selenelion. What is the explanation behind this seemingly impossible observation?\nA: The Moon is in an orbit with a non-zero eccentricity\nB: Prominences on the Sun at the time of the eclipse\nC: The non-zero light travel time to cover the large distance between the Moon and the Earth\nD: The effect of atmospheric refraction meaning the Sun and Moon only appear to be above the horizon\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOn the evening of 27th July 2018, there was a total lunar eclipse (where the Moon passes into the Earth's shadow). As viewed from Oxford, totality began whilst the Moon was below the horizon. Interestingly, moonrise was at $8: 55 \\mathrm{pm}$, yet sunset was only at $9: 01 \\mathrm{pm}$, so for 6 minutes both the fully eclipsed Moon and setting Sun were visible above the horizon. This very rare event is known as a selenelion. What is the explanation behind this seemingly impossible observation?\n\nA: The Moon is in an orbit with a non-zero eccentricity\nB: Prominences on the Sun at the time of the eclipse\nC: The non-zero light travel time to cover the large distance between the Moon and the Earth\nD: The effect of atmospheric refraction meaning the Sun and Moon only appear to be above the horizon\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_90", "problem": "宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 。在星球 $\\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\\mathrm{P}$ 轻放在弹簧上端, 如图 (a) 所示, $\\mathrm{P}$ 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\\mathrm{B}$ 上用完全相同的弹簧和物体 P 完成同样的过程, 其 $a-x$ 关系如图 (b) 中虚线所示。已知两星球密度相等。星球 $\\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。\n\n若将星球 $\\mathrm{A}$ 看成是以星球 $\\mathrm{B}$ 为中心天体的一颗卫星, 星球 $\\mathrm{A}$ 的运行周期为 $T_{1}$\n\n若将星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 看成是远离其他星球的双星模型, 这样算得的两星球做匀速圆周运动的周期为 $T_{2}$ 。求此情形中的周期 $T_{2}$ 与上述周期 $T_{1}$ 的比值。\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 。在星球 $\\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\\mathrm{P}$ 轻放在弹簧上端, 如图 (a) 所示, $\\mathrm{P}$ 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\\mathrm{B}$ 上用完全相同的弹簧和物体 P 完成同样的过程, 其 $a-x$ 关系如图 (b) 中虚线所示。已知两星球密度相等。星球 $\\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。\n\n若将星球 $\\mathrm{A}$ 看成是以星球 $\\mathrm{B}$ 为中心天体的一颗卫星, 星球 $\\mathrm{A}$ 的运行周期为 $T_{1}$\n\n若将星球 $\\mathrm{A}$ 和星球 $\\mathrm{B}$ 看成是远离其他星球的双星模型, 这样算得的两星球做匀速圆周运动的周期为 $T_{2}$ 。求此情形中的周期 $T_{2}$ 与上述周期 $T_{1}$ 的比值。\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-145.jpg?height=183&width=256&top_left_y=2464&top_left_x=340", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-145.jpg?height=429&width=414&top_left_y=2264&top_left_x=684" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_673", "problem": "太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间, 且三者几乎排成一条直线的现象, 天文学称为“行星冲日”。已知地球及各地外行星绕太阳运动的轨道半径如表所示（）\n\n| | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| 轨道半径
R/AU | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 |\nA: 土星相邻两次冲日的时间间隔是 450 天左右\nB: 土星相邻两次冲日的时间间隔是 378 天左右\nC: 火星相邻两次冲日的时间间隔最长\nD: 海王星相邻两次冲日的时间间隔最长\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间, 且三者几乎排成一条直线的现象, 天文学称为“行星冲日”。已知地球及各地外行星绕太阳运动的轨道半径如表所示（）\n\n| | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| 轨道半径
R/AU | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 |\n\nA: 土星相邻两次冲日的时间间隔是 450 天左右\nB: 土星相邻两次冲日的时间间隔是 378 天左右\nC: 火星相邻两次冲日的时间间隔最长\nD: 海王星相邻两次冲日的时间间隔最长\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-090.jpg?height=131&width=564&top_left_y=414&top_left_x=746" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_180", "problem": "开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。\n[图1]在牛顿力学体系中, 当两个质量分别为 $m_{1} 、 m_{2}$ 的质点相距为 $r$ 时具有的势能, 称为引力势能, 其大小为 $\\mathrm{E}_{\\mathrm{P}}=-\\frac{G m_{1} m_{2}}{r}$ (规定无穷远处势能为零)卫星在 I 轨道的 $\\mathrm{P}$ 点点火加速, 变轨到II轨道,根据开普勒第二定律, 求卫星在椭圆轨道II运动时, 在近地点 $\\mathrm{P}$ 与在远地点 $\\mathrm{Q}$ 的速率之比", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。\n[图1]在牛顿力学体系中, 当两个质量分别为 $m_{1} 、 m_{2}$ 的质点相距为 $r$ 时具有的势能, 称为引力势能, 其大小为 $\\mathrm{E}_{\\mathrm{P}}=-\\frac{G m_{1} m_{2}}{r}$ (规定无穷远处势能为零)卫星在 I 轨道的 $\\mathrm{P}$ 点点火加速, 变轨到II轨道,根据开普勒第二定律, 求卫星在椭圆轨道II运动时, 在近地点 $\\mathrm{P}$ 与在远地点 $\\mathrm{Q}$ 的速率之比\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-022.jpg?height=436&width=1398&top_left_y=230&top_left_x=342" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_533", "problem": "如图所示, 探月卫星的发射过程可简化如下：首先进入绕地球运行的“停泊轨道”,在该轨道的 $P$ 处通过变速再进入“地月转移轨道”, 在快要到达月球时, 对卫星再次变速,卫星被月球引力“俘获”后, 成为环月卫星, 最终在环绕月球的“工作轨道”绕月飞行（视为圆周运动), 对月球进行探测. “工作轨道”周期为 $T$ 、距月球表面的高度为 $h$, 月球半径为 $R$, 引力常量为 $G$, 忽略其他天体对探月卫星在“工作轨道”上环绕运动的影响。\n\n求月球的第一宇宙速度。\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 探月卫星的发射过程可简化如下：首先进入绕地球运行的“停泊轨道”,在该轨道的 $P$ 处通过变速再进入“地月转移轨道”, 在快要到达月球时, 对卫星再次变速,卫星被月球引力“俘获”后, 成为环月卫星, 最终在环绕月球的“工作轨道”绕月飞行（视为圆周运动), 对月球进行探测. “工作轨道”周期为 $T$ 、距月球表面的高度为 $h$, 月球半径为 $R$, 引力常量为 $G$, 忽略其他天体对探月卫星在“工作轨道”上环绕运动的影响。\n\n求月球的第一宇宙速度。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-140.jpg?height=314&width=727&top_left_y=2210&top_left_x=333" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_404", "problem": "2021 年 1 月 20 日, 我国在西昌卫星发射中心用长征三号乙运载火箭, 成功将天通一号 03 星发射升空, 它将与天通一号 01 星、 02 星组网运行。若 03 星绕地球做圆周运动的轨道半径为 02 星的 $a$ 倍, 02 星做圆周运动的向心加速度为 01 星的 $b$ 倍, 已知 01 星的运行周期为 $T$ ，则 03 星的运行周期为\nA: $a^{\\frac{3}{2}} b^{\\frac{2}{3}} T$\nB: $a^{\\frac{3}{2}} b^{-\\frac{1}{2}} T$\nC: $a^{\\frac{3}{2}} b^{-\\frac{3}{4}} T$\nD: $a^{\\frac{3}{2}} b^{\\frac{3}{4}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2021 年 1 月 20 日, 我国在西昌卫星发射中心用长征三号乙运载火箭, 成功将天通一号 03 星发射升空, 它将与天通一号 01 星、 02 星组网运行。若 03 星绕地球做圆周运动的轨道半径为 02 星的 $a$ 倍, 02 星做圆周运动的向心加速度为 01 星的 $b$ 倍, 已知 01 星的运行周期为 $T$ ，则 03 星的运行周期为\n\nA: $a^{\\frac{3}{2}} b^{\\frac{2}{3}} T$\nB: $a^{\\frac{3}{2}} b^{-\\frac{1}{2}} T$\nC: $a^{\\frac{3}{2}} b^{-\\frac{3}{4}} T$\nD: $a^{\\frac{3}{2}} b^{\\frac{3}{4}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_744", "problem": "The famous Drake equation attempts to answer the following question:\nA: Will the Sun become a black hole?\nB: Is the universe infinitely large?\nC: How old is the visible universe?\nD: Are we alone in the universe?\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe famous Drake equation attempts to answer the following question:\n\nA: Will the Sun become a black hole?\nB: Is the universe infinitely large?\nC: How old is the visible universe?\nD: Are we alone in the universe?\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_904", "problem": "A comet is orbiting the Sun in an orbit with a semi-major axis of 5.0 au and an eccentricity, $e=0.80$. Calculate its semi-minor axis, $b$.\nA: $1.0 \\mathrm{au}$\nB: $2.0 \\mathrm{au}$\nC: $3.0 \\mathrm{au}$\nD: $4.0 \\mathrm{au}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA comet is orbiting the Sun in an orbit with a semi-major axis of 5.0 au and an eccentricity, $e=0.80$. Calculate its semi-minor axis, $b$.\n\nA: $1.0 \\mathrm{au}$\nB: $2.0 \\mathrm{au}$\nC: $3.0 \\mathrm{au}$\nD: $4.0 \\mathrm{au}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_788", "problem": "Which one of the following expressions is correct for the Lorentz factor $\\gamma$ and $\\beta=$ $v / c$ ?\nA: $\\frac{1}{\\gamma^{2}}=1+\\beta^{2}$\nB: $\\frac{1}{\\gamma^{2}}=\\beta \\gamma^{2}$\nC: $\\frac{1}{\\gamma^{2}}=1-\\beta^{2}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhich one of the following expressions is correct for the Lorentz factor $\\gamma$ and $\\beta=$ $v / c$ ?\n\nA: $\\frac{1}{\\gamma^{2}}=1+\\beta^{2}$\nB: $\\frac{1}{\\gamma^{2}}=\\beta \\gamma^{2}$\nC: $\\frac{1}{\\gamma^{2}}=1-\\beta^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_794", "problem": "At which of the following phases of the Moon's orbit is the tidal bulge of Earth largest?\nA: Full\nB: First Quarter\nC: Waxing Gibbous\nD: Waning Gibbous\nE: Waxing Crescent\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAt which of the following phases of the Moon's orbit is the tidal bulge of Earth largest?\n\nA: Full\nB: First Quarter\nC: Waxing Gibbous\nD: Waning Gibbous\nE: Waxing Crescent\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_119", "problem": "如图所示, I为北斗卫星导航系统中的静止轨道卫星, 其对地张角为 $2 \\theta$; II为地球试卷第 75 页，共 122 页\n的近地卫星。已知地球的自转周期为 $T_{0}$, 万有引力常量为 $G$, 根据题中条件, 可求出\n\n[图1]\nA: 地球的平均密度为 $\\frac{3 \\pi}{G T_{0}^{2} \\sin ^{3} \\theta}$\nB: 卫星I和卫星II的加速度之比为 $\\sin ^{2} 2 \\theta$\nC: 卫星II的周期为 $\\frac{T_{0}}{\\sqrt{\\sin ^{3} \\theta}}$\nD: 卫星II运动的周期内无法直接接收到卫星发出电磁波信号的时间为 $\\frac{(\\pi+2 \\theta) T_{0}}{2 \\pi} \\sqrt{\\sin ^{3} \\theta}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, I为北斗卫星导航系统中的静止轨道卫星, 其对地张角为 $2 \\theta$; II为地球试卷第 75 页，共 122 页\n的近地卫星。已知地球的自转周期为 $T_{0}$, 万有引力常量为 $G$, 根据题中条件, 可求出\n\n[图1]\n\nA: 地球的平均密度为 $\\frac{3 \\pi}{G T_{0}^{2} \\sin ^{3} \\theta}$\nB: 卫星I和卫星II的加速度之比为 $\\sin ^{2} 2 \\theta$\nC: 卫星II的周期为 $\\frac{T_{0}}{\\sqrt{\\sin ^{3} \\theta}}$\nD: 卫星II运动的周期内无法直接接收到卫星发出电磁波信号的时间为 $\\frac{(\\pi+2 \\theta) T_{0}}{2 \\pi} \\sqrt{\\sin ^{3} \\theta}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-076.jpg?height=525&width=554&top_left_y=346&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_745", "problem": "Which one of these wavelengths is considered infrared radiation?\nA: 150 meters\nB: 150 millimeters\nC: 150 micrometers\nD: 150 nanometers\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhich one of these wavelengths is considered infrared radiation?\n\nA: 150 meters\nB: 150 millimeters\nC: 150 micrometers\nD: 150 nanometers\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_830", "problem": "Assuming that the radius of the smaller star is 2 solar radii, what is the distance to the system?\nA: 75 parsecs\nB: 85 parsecs\nC: 100 parsecs\nD: 115 parsecs\nE: 150 parsecs\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAssuming that the radius of the smaller star is 2 solar radii, what is the distance to the system?\n\nA: 75 parsecs\nB: 85 parsecs\nC: 100 parsecs\nD: 115 parsecs\nE: 150 parsecs\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_147", "problem": "2020 年 11 月 24 日凌晨, 搭载嫦娥五号探测器的长征五号遥五运载火箭从文昌航天发射场顺利升空，12 月 17 日“嫦娥五号”返回器携带月球样品在预定区域安全着陆,在落地之前, 它在大气层打个“水漂”。现简化过程如图所示, 以地心为中心、半径为 $r_{1}$的圆周为大气层的边界, 忽略大气层外空气阻力。已知地球半径为 $R$, 返回器从 $o$ 点进入大气层，经 $a 、 b 、 c 、 d$ 回到地面，其中 $o 、 b 、 d$ 为轨道和大气层外边界的交点， $c$ 点到地心的距离为 $r_{2}$, 地球表面重力加速度为 $g$, 以下结论正确的是 ( )\n\n[图1]\nA: 返回器在 $o$ 点动能大于在 $b$ 点的动能\nB: 返回器在 $b$ 点动能大于在 $d$ 点的动能\nC: 返回器在 $c$ 点的加速度大于 $\\frac{g R^{2}}{r_{1}^{2}}$\nD: 返回器通过 $a$ 点处于失重状态\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2020 年 11 月 24 日凌晨, 搭载嫦娥五号探测器的长征五号遥五运载火箭从文昌航天发射场顺利升空，12 月 17 日“嫦娥五号”返回器携带月球样品在预定区域安全着陆,在落地之前, 它在大气层打个“水漂”。现简化过程如图所示, 以地心为中心、半径为 $r_{1}$的圆周为大气层的边界, 忽略大气层外空气阻力。已知地球半径为 $R$, 返回器从 $o$ 点进入大气层，经 $a 、 b 、 c 、 d$ 回到地面，其中 $o 、 b 、 d$ 为轨道和大气层外边界的交点， $c$ 点到地心的距离为 $r_{2}$, 地球表面重力加速度为 $g$, 以下结论正确的是 ( )\n\n[图1]\n\nA: 返回器在 $o$ 点动能大于在 $b$ 点的动能\nB: 返回器在 $b$ 点动能大于在 $d$ 点的动能\nC: 返回器在 $c$ 点的加速度大于 $\\frac{g R^{2}}{r_{1}^{2}}$\nD: 返回器通过 $a$ 点处于失重状态\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-080.jpg?height=729&width=1122&top_left_y=198&top_left_x=361" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_13", "problem": "人造地球卫星绕地球做匀速圆周运动, 例如卫星的周期增大到原来的 8 倍, 卫星仍做匀速圆周运动，则下列说法中正确的是（ ）\nA: 卫星的向心加速度增大到原来的 4 倍\nB: 卫星的角速度减小到原来的 $1 / 4$\nC: 卫星的动能将增大到原来的 4 倍\nD: 卫星的线速度减小到原来的 $1 / 2$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n人造地球卫星绕地球做匀速圆周运动, 例如卫星的周期增大到原来的 8 倍, 卫星仍做匀速圆周运动，则下列说法中正确的是（ ）\n\nA: 卫星的向心加速度增大到原来的 4 倍\nB: 卫星的角速度减小到原来的 $1 / 4$\nC: 卫星的动能将增大到原来的 4 倍\nD: 卫星的线速度减小到原来的 $1 / 2$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_616", "problem": "已知地球半径为 $R$, 地球表面的重力加速度为 $g$ 。质量为 $m$ 的宇宙飞船在半径为 $2 R$的轨道 1 上绕地球中心 $O$ 做圆两运动。现飞船在轨道 1 的 $A$ 点加速到陏圆轨道 2 上,再在远地点 $B$ 点加速, 从而使飞船转移到半径为 $4 R$ 的轨道 3 上, 如图所示。若相距 $r$的两物体间引力势能为 $E_{\\mathrm{p}}=-G \\frac{M m}{r}$, 求:\n\n飞船在轨道 2 上从 $A$ 点到 $B$ 点飞行的时间。\n\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n已知地球半径为 $R$, 地球表面的重力加速度为 $g$ 。质量为 $m$ 的宇宙飞船在半径为 $2 R$的轨道 1 上绕地球中心 $O$ 做圆两运动。现飞船在轨道 1 的 $A$ 点加速到陏圆轨道 2 上,再在远地点 $B$ 点加速, 从而使飞船转移到半径为 $4 R$ 的轨道 3 上, 如图所示。若相距 $r$的两物体间引力势能为 $E_{\\mathrm{p}}=-G \\frac{M m}{r}$, 求:\n\n飞船在轨道 2 上从 $A$ 点到 $B$ 点飞行的时间。\n\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-084.jpg?height=425&width=423&top_left_y=153&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_457", "problem": "假设地球半径为 $R$, 地球表面的重力加速度为 $g_{0}$. “神舟九号”飞船沿距地球表面高度为 $3 R$ 的圆形轨道I运动, 到达轨道的 $A$ 点, 点火变轨进入粗圆轨道II, 到达轨道II的近地点 $B$ 再次点火进入近地轨道III绕地球做圆周运动. 下列判断正确的是 ( )\n\n[图1]\nA: 飞船在轨道III跟轨道I的线速度大小之比为 $1: 2$\nB: 飞船在轨道III跟轨道I的线速度大小之比为 $2: 1$\nC: 飞船在轨道I绕地球运动一周所需的时间为 $2 \\pi \\sqrt{\\frac{27 R}{g_{0}}}$\nD: 飞船在轨道I绕地球运动一周所需的时间为 $16 \\pi \\sqrt{\\frac{R}{g_{0}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n假设地球半径为 $R$, 地球表面的重力加速度为 $g_{0}$. “神舟九号”飞船沿距地球表面高度为 $3 R$ 的圆形轨道I运动, 到达轨道的 $A$ 点, 点火变轨进入粗圆轨道II, 到达轨道II的近地点 $B$ 再次点火进入近地轨道III绕地球做圆周运动. 下列判断正确的是 ( )\n\n[图1]\n\nA: 飞船在轨道III跟轨道I的线速度大小之比为 $1: 2$\nB: 飞船在轨道III跟轨道I的线速度大小之比为 $2: 1$\nC: 飞船在轨道I绕地球运动一周所需的时间为 $2 \\pi \\sqrt{\\frac{27 R}{g_{0}}}$\nD: 飞船在轨道I绕地球运动一周所需的时间为 $16 \\pi \\sqrt{\\frac{R}{g_{0}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-36.jpg?height=377&width=417&top_left_y=1865&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_688", "problem": "我国航天技术水平在世界处于领先地位，对于人造卫星的发射，有人提出了利用“地球隧道”发射人造卫星的构想：沿地球的一条弦挖一通道，在通道的两个出口处分别将等质量的待发射卫星部件同时释放，部件将在通道中间位置“碰撞组装”成卫星并静止下来; 另在通道的出口处由静止释放一个大质量物体，大质量物体会在通道与待发射的卫星碰撞, 只要物体质量相比卫星质量足够大, 卫星获得足够速度就会从对向通道口射出。\n\n(以下计算中, 已知地球的质量为 $M_{0}$, 地球半径为 $R_{0}$, 引力常量为 $G$, 可忽略通道 $A B$的内径大小和地球自转影响。)\n\n如图甲所示, 将一个质量为 $m_{0}$ 的质点置于质量分布均匀的球形天体内, 质点离球心 $O$ 的距离为 $r$ 。已知天体内部半径在 $r \\sim R$ 之间的“球壳”部分（如甲示阴影部分）对质点的万有引力为零, 求质点所受万有引力的大小 $F_{\\mathrm{r}}$ 。\n\n[图1]\n\n甲", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n我国航天技术水平在世界处于领先地位，对于人造卫星的发射，有人提出了利用“地球隧道”发射人造卫星的构想：沿地球的一条弦挖一通道，在通道的两个出口处分别将等质量的待发射卫星部件同时释放，部件将在通道中间位置“碰撞组装”成卫星并静止下来; 另在通道的出口处由静止释放一个大质量物体，大质量物体会在通道与待发射的卫星碰撞, 只要物体质量相比卫星质量足够大, 卫星获得足够速度就会从对向通道口射出。\n\n(以下计算中, 已知地球的质量为 $M_{0}$, 地球半径为 $R_{0}$, 引力常量为 $G$, 可忽略通道 $A B$的内径大小和地球自转影响。)\n\n如图甲所示, 将一个质量为 $m_{0}$ 的质点置于质量分布均匀的球形天体内, 质点离球心 $O$ 的距离为 $r$ 。已知天体内部半径在 $r \\sim R$ 之间的“球壳”部分（如甲示阴影部分）对质点的万有引力为零, 求质点所受万有引力的大小 $F_{\\mathrm{r}}$ 。\n\n[图1]\n\n甲\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-067.jpg?height=411&width=417&top_left_y=688&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1201", "problem": "On $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.a. In ideal observing conditions the two planets are far enough apart that they should be (just about) distinguishable to the naked eye, however to some observers in imperfect conditions they would appear as a single bright dot, brighter than either planet on its own.\n\ni. During the conjunction, the apparent magnitudes of Jupiter and Saturn were $m_{J}=-1.97$ and $m_{S}=0.63$, respectively (ignoring dimming by the atmosphere). What would be the apparent magnitude of the two planets if they appeared to an observer as a single point? [Hint: it is not simply $-1.97-0.63=-2.60$ ]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.\n\nproblem:\na. In ideal observing conditions the two planets are far enough apart that they should be (just about) distinguishable to the naked eye, however to some observers in imperfect conditions they would appear as a single bright dot, brighter than either planet on its own.\n\ni. During the conjunction, the apparent magnitudes of Jupiter and Saturn were $m_{J}=-1.97$ and $m_{S}=0.63$, respectively (ignoring dimming by the atmosphere). What would be the apparent magnitude of the two planets if they appeared to an observer as a single point? [Hint: it is not simply $-1.97-0.63=-2.60$ ]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-07.jpg?height=706&width=1564&top_left_y=834&top_left_x=244", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-08.jpg?height=578&width=1566&top_left_y=196&top_left_x=242", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-09.jpg?height=1072&width=1564&top_left_y=1191&top_left_x=246" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_699", "problem": "2021 年 9 月 17 日, “神舟十二号”返回舱在东风着陆场安全降落。返回舱从工作轨道I返回地面的运动轨迹如图, 椭圆轨道II与圆轨道I、III分别相切于 $P 、 Q$ 两点, 返回舱从轨道III上适当位置减速后进入大气层, 最后在东风着陆场着陆。下列说法正确的是\n\n[图1]\nA: 返回舱在I轨道上 $P$ 需要向运动方向的反方向喷气进入II轨道\nB: 返回舱在II轨道上运动的周期小于返回舱在III轨道上运动的周期\nC: 返回舱在III轨道上 $Q$ 点的速度的大小大于II轨道上 $P$ 点速度的大小\nD: 返回舱在I轨道上经过 $P$ 点时的加速度等于在II轨道上经过 $P$ 点时的加速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2021 年 9 月 17 日, “神舟十二号”返回舱在东风着陆场安全降落。返回舱从工作轨道I返回地面的运动轨迹如图, 椭圆轨道II与圆轨道I、III分别相切于 $P 、 Q$ 两点, 返回舱从轨道III上适当位置减速后进入大气层, 最后在东风着陆场着陆。下列说法正确的是\n\n[图1]\n\nA: 返回舱在I轨道上 $P$ 需要向运动方向的反方向喷气进入II轨道\nB: 返回舱在II轨道上运动的周期小于返回舱在III轨道上运动的周期\nC: 返回舱在III轨道上 $Q$ 点的速度的大小大于II轨道上 $P$ 点速度的大小\nD: 返回舱在I轨道上经过 $P$ 点时的加速度等于在II轨道上经过 $P$ 点时的加速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-093.jpg?height=397&width=440&top_left_y=176&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_169", "problem": "利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \\times 10^{3} \\mathrm{~W} / \\mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 水的密度 $\\rho=1.0 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。\n\n若利用太阳能发电, 需要发射一颗卫星到地球同步轨道上, 然后通过微波持续不断地将电能输送到地面, 这样就建成了宇宙太阳能发电站。已知地球同步轨道半径约为地球半径的 $2 \\sqrt{11}$ 倍。求卫星在地球同步轨道上向心加速度的大小;", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \\times 10^{3} \\mathrm{~W} / \\mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 水的密度 $\\rho=1.0 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。\n\n若利用太阳能发电, 需要发射一颗卫星到地球同步轨道上, 然后通过微波持续不断地将电能输送到地面, 这样就建成了宇宙太阳能发电站。已知地球同步轨道半径约为地球半径的 $2 \\sqrt{11}$ 倍。求卫星在地球同步轨道上向心加速度的大小;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~m} / \\mathrm{s}^{2}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m} / \\mathrm{s}^{2}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_153", "problem": "理论上已经证明, 质量分布均匀的球壳对壳内物体的万有引力为零。现假设地球是一半径为 $R$ 、质量分布均匀的实心球体, 将一个铁球分别放在地面以下 $\\frac{R}{3}$ 深处和放在地面上方 $\\frac{R}{3}$ 高度处, 则物体在两处的重力加速度之比为（）\nA: $32: 27$\nB: $9: 8$\nC: $81: 64$\nD: $4: 3$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n理论上已经证明, 质量分布均匀的球壳对壳内物体的万有引力为零。现假设地球是一半径为 $R$ 、质量分布均匀的实心球体, 将一个铁球分别放在地面以下 $\\frac{R}{3}$ 深处和放在地面上方 $\\frac{R}{3}$ 高度处, 则物体在两处的重力加速度之比为（）\n\nA: $32: 27$\nB: $9: 8$\nC: $81: 64$\nD: $4: 3$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1218", "problem": "In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.e. The Starkiller Base wants to destroy all the planets in a stellar system on the far side of the galaxy and so drains $0.10 M_{\\odot}$ from the Sun to charge its weapon. Assuming that the $U$ per unit volume of the Sun stays approximately constant during this process, calculate:\nii. The new radius of the Sun.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel.\n\nFor an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \\ll M$ ) the orbital velocity, $v_{\\text {orb }}$, is given by the formula $v_{\\text {orb }}=\\sqrt{\\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth.\n\n[figure1]\n\nFigure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm.\n\nFor this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by\n\n$$\nU=\\frac{3 G M^{2}}{5 R}\n$$\n\nand that the mass-luminosity relation of low-mass main sequence stars is given by $L \\propto M^{4}$.{r}}$.\n\nproblem:\ne. The Starkiller Base wants to destroy all the planets in a stellar system on the far side of the galaxy and so drains $0.10 M_{\\odot}$ from the Sun to charge its weapon. Assuming that the $U$ per unit volume of the Sun stays approximately constant during this process, calculate:\nii. The new radius of the Sun.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of m, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_204b2e236273ea30e8d2g-06.jpg?height=611&width=1448&top_left_y=505&top_left_x=310" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_991", "problem": "Currently, Polaris is very close to the north celestial pole (the projection of the Earth's rotational axis on the sky) and so all other stars appear to rotate around it. However, this axis is drawing out a large circle in the sky with an angular radius of $23.44^{\\circ}$ so Polaris will only temporarily be the pole star (see Figure 2). This precession of the rotational axis is mainly driven by the gravitational pull of the Moon and the Sun.\n[figure1]\n\nFigure 2: Top left: The Earth's rotational axis itself rotates slowly (white circle), in what is known as axial precession. Credit: David Battisti / University of Washington.\n\nTop right: Due to precession, the pole star has changed over time. About 5000 years ago, the star Thuban in the constellation of Draco was the pole star. Credit: Richard W. Pogge / Ohio State University.\n\nBottom: The position of the Sun at the spring equinox (where the celestial equator meets the ecliptic) has also changed over the same period, moving from Aries to Pisces. Credit: Guy Ottewell / Universal Workshop.\n\nAnother consequence is that the position of the Sun at the equinoxes varies slightly, moving slowly westwards. This gives rise to two definitions of a year:\n\n- a sidereal year (the time taken for the Earth to orbit the Sun once with respect to the background stars) $=365.256363$ days\n- a tropical year (the time taken for the Sun to return to the same position in the cycle of the seasons) $=365.242190$ days\n\nThe Gregorian calendar is a 400-year cycle with a system of leap years. The rule is: \"every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100 , but these centurial years are leap years if they are exactly divisible by 400. .\"\n\nAssuming the rate of axial precession remains constant, work out the time (in sidereal years) to complete a whole precessional cycle.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nCurrently, Polaris is very close to the north celestial pole (the projection of the Earth's rotational axis on the sky) and so all other stars appear to rotate around it. However, this axis is drawing out a large circle in the sky with an angular radius of $23.44^{\\circ}$ so Polaris will only temporarily be the pole star (see Figure 2). This precession of the rotational axis is mainly driven by the gravitational pull of the Moon and the Sun.\n[figure1]\n\nFigure 2: Top left: The Earth's rotational axis itself rotates slowly (white circle), in what is known as axial precession. Credit: David Battisti / University of Washington.\n\nTop right: Due to precession, the pole star has changed over time. About 5000 years ago, the star Thuban in the constellation of Draco was the pole star. Credit: Richard W. Pogge / Ohio State University.\n\nBottom: The position of the Sun at the spring equinox (where the celestial equator meets the ecliptic) has also changed over the same period, moving from Aries to Pisces. Credit: Guy Ottewell / Universal Workshop.\n\nAnother consequence is that the position of the Sun at the equinoxes varies slightly, moving slowly westwards. This gives rise to two definitions of a year:\n\n- a sidereal year (the time taken for the Earth to orbit the Sun once with respect to the background stars) $=365.256363$ days\n- a tropical year (the time taken for the Sun to return to the same position in the cycle of the seasons) $=365.242190$ days\n\nThe Gregorian calendar is a 400-year cycle with a system of leap years. The rule is: \"every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100 , but these centurial years are leap years if they are exactly divisible by 400. .\"\n\nAssuming the rate of axial precession remains constant, work out the time (in sidereal years) to complete a whole precessional cycle.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_2c19fdb17927c588761dg-07.jpg?height=890&width=1144&top_left_y=574&top_left_x=456" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1095", "problem": "Young, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\\star}=q \\ll h^{3}$.\n[figure1]\n\nFigure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF).\n\nRight: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset.\n\nSince the planet is assumed small $\\left(q \\ll h^{3}\\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \\approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via:\n\n$$\n\\frac{\\mathrm{d} L}{\\mathrm{~d} t}=\\Gamma\n$$\n\nIt is convenient to write the torque in terms of the reference value\n\n$$\n\\Gamma_{0}=\\left(\\frac{q}{h}\\right)^{2} \\Sigma_{p} r_{p}^{4} \\Omega_{p}^{2}\n$$\nc. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\\Gamma_{C}$. For a disc of uniform entropy ( $\\left.\\mathrm{d} s=0\\right)$, and with surface density profile $\\Sigma \\propto r^{-\\alpha}$, and pressure profile $P \\propto r^{-\\delta}$, Tanaka et al. (2002) and Paardekooper \\& Papaloizou (2009) find these torques are given by:\n\n$$\n\\begin{gathered}\n\\Gamma_{L}=(-3.20+0.86 \\alpha-2.33 \\delta) \\Gamma_{0} \\\\\n\\Gamma_{C}=5.97(1.5-\\alpha) \\Gamma_{0}\n\\end{gathered}\n$$\n\nWe assume the gas in the disc obeys the ideal gas law, so that:\n\n$$\n\\frac{P}{\\Sigma T}=\\text { constant }, \\quad \\mathrm{d} s=\\text { constant } \\times\\left(\\frac{1}{\\gamma-1} \\frac{\\mathrm{d} T}{T}-\\frac{\\mathrm{d} \\Sigma}{\\Sigma}\\right),\n$$\n\nwhere $T$ is the absolute temperature and $\\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy,\n\n$$\n\\Gamma=\\Gamma_{L}+\\Gamma_{C}=(5.76-(5.11+2.33 \\gamma) \\alpha) \\Gamma_{0}\n$$\n\n[Hint: if $\\frac{\\mathrm{d} y}{y}=\\lambda \\frac{\\mathrm{d} x}{x}$, then $y \\propto x^{\\lambda}$.]\n\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\Omega$ is $L=I \\Omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.c. Show that for a disc of uniform entropy, $\\Gamma_{=} \\Gamma_{L}+\\Gamma_{\\mathrm{C}}=(5.76-(5.11+2.33 \\gamma) \\alpha) \\Gamma_{0}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nYoung, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\\star}=q \\ll h^{3}$.\n[figure1]\n\nFigure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF).\n\nRight: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset.\n\nSince the planet is assumed small $\\left(q \\ll h^{3}\\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \\approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via:\n\n$$\n\\frac{\\mathrm{d} L}{\\mathrm{~d} t}=\\Gamma\n$$\n\nIt is convenient to write the torque in terms of the reference value\n\n$$\n\\Gamma_{0}=\\left(\\frac{q}{h}\\right)^{2} \\Sigma_{p} r_{p}^{4} \\Omega_{p}^{2}\n$$\nc. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\\Gamma_{C}$. For a disc of uniform entropy ( $\\left.\\mathrm{d} s=0\\right)$, and with surface density profile $\\Sigma \\propto r^{-\\alpha}$, and pressure profile $P \\propto r^{-\\delta}$, Tanaka et al. (2002) and Paardekooper \\& Papaloizou (2009) find these torques are given by:\n\n$$\n\\begin{gathered}\n\\Gamma_{L}=(-3.20+0.86 \\alpha-2.33 \\delta) \\Gamma_{0} \\\\\n\\Gamma_{C}=5.97(1.5-\\alpha) \\Gamma_{0}\n\\end{gathered}\n$$\n\nWe assume the gas in the disc obeys the ideal gas law, so that:\n\n$$\n\\frac{P}{\\Sigma T}=\\text { constant }, \\quad \\mathrm{d} s=\\text { constant } \\times\\left(\\frac{1}{\\gamma-1} \\frac{\\mathrm{d} T}{T}-\\frac{\\mathrm{d} \\Sigma}{\\Sigma}\\right),\n$$\n\nwhere $T$ is the absolute temperature and $\\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy,\n\n$$\n\\Gamma=\\Gamma_{L}+\\Gamma_{C}=(5.76-(5.11+2.33 \\gamma) \\alpha) \\Gamma_{0}\n$$\n\n[Hint: if $\\frac{\\mathrm{d} y}{y}=\\lambda \\frac{\\mathrm{d} x}{x}$, then $y \\propto x^{\\lambda}$.]\n\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\Omega$ is $L=I \\Omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.\n\nproblem:\nc. Show that for a disc of uniform entropy, $\\Gamma_{=} \\Gamma_{L}+\\Gamma_{\\mathrm{C}}=(5.76-(5.11+2.33 \\gamma) \\alpha) \\Gamma_{0}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_9bba4f2e5c10ed29bb97g-10.jpg?height=702&width=1416&top_left_y=654&top_left_x=317" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_890", "problem": "An astronomer takes a spectrum of a galaxy and observes that the hydrogen-alpha emission line is at a wavelength of 721.9 nanometers. In a laboratory on Earth, this same emission line is observed at a wavelength of 656.3 nanometers. Approximately what is the (proper) distance to this galaxy?\nA: $66 \\mathrm{Mpc}$\nB: $430 \\mathrm{Mpc}$\nC: $480 \\mathrm{Mpc}$\nD: $3900 \\mathrm{Mpc}$\nE: $4700 \\mathrm{Mpc}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn astronomer takes a spectrum of a galaxy and observes that the hydrogen-alpha emission line is at a wavelength of 721.9 nanometers. In a laboratory on Earth, this same emission line is observed at a wavelength of 656.3 nanometers. Approximately what is the (proper) distance to this galaxy?\n\nA: $66 \\mathrm{Mpc}$\nB: $430 \\mathrm{Mpc}$\nC: $480 \\mathrm{Mpc}$\nD: $3900 \\mathrm{Mpc}$\nE: $4700 \\mathrm{Mpc}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_317", "problem": "已知地球的质量是月球质量的 81 倍, 地球半径大约是月球半径的 4 倍, 不考虑地球、月球自转的影响, 以上数据可推算出 [ ]\nA: 地球表面的重力加速度与月球表面重力加速度之比为 9: 16\nB: 地球的平均密度与月球的平均密度之比为 $9: 8$\nC: 靠近地球表面沿圆轨道运动的航天器的周期与靠近月球表面沿圆轨道运行的航天器的周期之比约为 $8: 9$\nD: 靠近地球表面沿圆轨道运行的航天器的线速度与靠近月球表面沿圆轨道运行的航天器的线速度之比约为 $81: 4$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n已知地球的质量是月球质量的 81 倍, 地球半径大约是月球半径的 4 倍, 不考虑地球、月球自转的影响, 以上数据可推算出 [ ]\n\nA: 地球表面的重力加速度与月球表面重力加速度之比为 9: 16\nB: 地球的平均密度与月球的平均密度之比为 $9: 8$\nC: 靠近地球表面沿圆轨道运动的航天器的周期与靠近月球表面沿圆轨道运行的航天器的周期之比约为 $8: 9$\nD: 靠近地球表面沿圆轨道运行的航天器的线速度与靠近月球表面沿圆轨道运行的航天器的线速度之比约为 $81: 4$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_837", "problem": "Assume that the smaller star in the above binary star system is brighter than the larger star. What is the ratio of the radius of the smaller star to the radius of the larger star?\nA: 0.21\nB: 0.76\nC: 0.82\nD: 0.95\nE: 0.98\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAssume that the smaller star in the above binary star system is brighter than the larger star. What is the ratio of the radius of the smaller star to the radius of the larger star?\n\nA: 0.21\nB: 0.76\nC: 0.82\nD: 0.95\nE: 0.98\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_121", "problem": "如图所示, 有 $\\mathrm{A} 、 \\mathrm{~B}$ 两颗行星绕同一恒星 $\\mathrm{O}$ 做圆周运动, 运行方向相同。 $\\mathrm{A}$ 行星的周期为 $T_{1}, \\mathrm{~B}$ 行星的周期为 $T_{2}$, 在某一时刻两行星相距最近, 则 ( )\n\n[图1]\nA: 经过时间 $t=\\frac{T_{1} T_{2}}{T_{2}-T_{1}}$, 两行星将再次相距最近\nB: 经过时间 $t=T_{1}+T_{2}$, 两行星将再次相距最近\nC: 经过时间 $t=\\frac{n T_{1} T_{2}}{2\\left(T_{2}-T_{1}\\right)}(n=1,3,5, \\ldots)$, 两行星相距最远\nD: 经过时间 $t=\\frac{n\\left(T_{1}+T_{2}\\right)}{2}(n=1,3,5, \\ldots)$, 两行星相距最远\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 有 $\\mathrm{A} 、 \\mathrm{~B}$ 两颗行星绕同一恒星 $\\mathrm{O}$ 做圆周运动, 运行方向相同。 $\\mathrm{A}$ 行星的周期为 $T_{1}, \\mathrm{~B}$ 行星的周期为 $T_{2}$, 在某一时刻两行星相距最近, 则 ( )\n\n[图1]\n\nA: 经过时间 $t=\\frac{T_{1} T_{2}}{T_{2}-T_{1}}$, 两行星将再次相距最近\nB: 经过时间 $t=T_{1}+T_{2}$, 两行星将再次相距最近\nC: 经过时间 $t=\\frac{n T_{1} T_{2}}{2\\left(T_{2}-T_{1}\\right)}(n=1,3,5, \\ldots)$, 两行星相距最远\nD: 经过时间 $t=\\frac{n\\left(T_{1}+T_{2}\\right)}{2}(n=1,3,5, \\ldots)$, 两行星相距最远\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-083.jpg?height=363&width=411&top_left_y=2383&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_709", "problem": "2018 年 12 月 27 日, 我国北斗卫星导航系统开始提供全球服务, 标志着北斗系统正式迈入全球时代。覆盖全球的北斗卫星导航系统是由静止轨道卫星 (即地球同步卫星)和非静止轨道卫星共 35 颗组成的。卫星绕地球近似做匀速圆周运动。已知其中一颗地球同步卫星距离地球表面的高度为 $h$, 地球质量为 $M$, 地球半径为 $R$, 引力常量为 $G$ 。\n如图所示, $O$ 点为地球的球心, $P$ 点处有一颗地球同步卫星, $P$ 点所在的虚线圆轨道为同步卫星绕地球运动的轨道。已知 $h=5.6 R$ 。忽略大气等一切影响因素, 请论证说明要使卫星通讯覆盖全球, 至少需要几颗地球同步卫星。 $\\left(\\cos 81^{\\circ} \\approx 0.15\\right.$, $\\sin 81^{\\circ} \\approx 0.99$ )\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n2018 年 12 月 27 日, 我国北斗卫星导航系统开始提供全球服务, 标志着北斗系统正式迈入全球时代。覆盖全球的北斗卫星导航系统是由静止轨道卫星 (即地球同步卫星)和非静止轨道卫星共 35 颗组成的。卫星绕地球近似做匀速圆周运动。已知其中一颗地球同步卫星距离地球表面的高度为 $h$, 地球质量为 $M$, 地球半径为 $R$, 引力常量为 $G$ 。\n如图所示, $O$ 点为地球的球心, $P$ 点处有一颗地球同步卫星, $P$ 点所在的虚线圆轨道为同步卫星绕地球运动的轨道。已知 $h=5.6 R$ 。忽略大气等一切影响因素, 请论证说明要使卫星通讯覆盖全球, 至少需要几颗地球同步卫星。 $\\left(\\cos 81^{\\circ} \\approx 0.15\\right.$, $\\sin 81^{\\circ} \\approx 0.99$ )\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以颗为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-133.jpg?height=223&width=1162&top_left_y=1259&top_left_x=321", "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-133.jpg?height=262&width=643&top_left_y=2159&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "颗" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_366", "problem": "如图所示, 地球卫星开始在圆形低轨道 1 运行, 在 $P$ 处点火后, 地球卫星沿椭圆轨道 2 运行, 在远地点 $Q$ 再次点火, 将地球卫星送入更高的圆形轨道 3. 若地球卫星在 $1 、 3$ 轨道上运行的速率分别为 $v_{1} 、 v_{3}$, 在 2 轨道上经过 $P 、 Q$ 处的速率分别为 $v_{2 P} 、 v_{2 Q}$,则\n\n[图1]\nA: $v_{3}v_{2 O}$\nD: $v_{1}>v_{2 P}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 地球卫星开始在圆形低轨道 1 运行, 在 $P$ 处点火后, 地球卫星沿椭圆轨道 2 运行, 在远地点 $Q$ 再次点火, 将地球卫星送入更高的圆形轨道 3. 若地球卫星在 $1 、 3$ 轨道上运行的速率分别为 $v_{1} 、 v_{3}$, 在 2 轨道上经过 $P 、 Q$ 处的速率分别为 $v_{2 P} 、 v_{2 Q}$,则\n\n[图1]\n\nA: $v_{3}v_{2 O}$\nD: $v_{1}>v_{2 P}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_9938578583ce82f2e878g-01.jpg?height=397&width=402&top_left_y=818&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_196", "problem": "火星被认为是太阳系中最有可能存在地外生命的行星, 对人类来说充满着神奇!\n\n2021 年 2 月, 包括我国“天问一号”在内的国际三个火星探测器全部抵达火星。若火星的密度为 $\\rho$, 火星的一颗天然卫星绕火星做匀速圆周运动, 其线速度为 $v$, 运行周期为 $T$ 。已知引力常量为 $G$, 则可以求得 ( )\nA: 火星的半径\nB: 火星的质量\nC: 天然卫星的质量\nD: 天然卫星距离火星表面的高度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n火星被认为是太阳系中最有可能存在地外生命的行星, 对人类来说充满着神奇!\n\n2021 年 2 月, 包括我国“天问一号”在内的国际三个火星探测器全部抵达火星。若火星的密度为 $\\rho$, 火星的一颗天然卫星绕火星做匀速圆周运动, 其线速度为 $v$, 运行周期为 $T$ 。已知引力常量为 $G$, 则可以求得 ( )\n\nA: 火星的半径\nB: 火星的质量\nC: 天然卫星的质量\nD: 天然卫星距离火星表面的高度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1166", "problem": "In July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.d. For the patched conics approach (solid lines):\n\ni. Find the distance from the centre of the Earth to point $C$, and hence the semi-major axes of both ellipses.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nIn July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made.\n[figure1]\n\nFigure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA.\n\nRight: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica.\n\n| Stage | Initial Mass $(\\mathrm{t})$ | Final mass $(\\mathrm{t})$ | $I_{\\mathrm{sp}}(\\mathrm{s})$ | Burn duration $(\\mathrm{s})$ |\n| :---: | :---: | :---: | :---: | :---: |\n| S-IC | 2283.9 | 135.6 | 263 | 168 |\n| S-II | 483.7 | 39.9 | 421 | 384 |\n| S-IV (Burn 1) | 121.0 | - | 421 | 147 |\n| S-IV (Burn 2) | - | 13.2 | 421 | 347 |\n| Apollo Spacecraft | 49.7 | - | - | - |\n\nTable 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \\mathrm{t}=1000 \\mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB.\n\nThe Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \\mathrm{t}(1$ tonne, $\\mathrm{t}=1000 \\mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \\mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was\nthe heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1.\n\nThe thrust of the rocket is given as\n\n$$\nF=-I_{\\mathrm{sp}} g_{0} \\dot{m}\n$$\n\nwhere the specific impulse, $I_{\\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \\mathrm{~m} \\mathrm{~s}^{-2}$ ) and $\\dot{m} \\equiv \\mathrm{d} m / \\mathrm{d} t$ is the rate of change of mass of the rocket with time.\n\nThe thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket).\n\nBy the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2.\n\nThe first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\\mathrm{C}$ where the gravitational force on the spacecraft is equal\nfrom both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\\mathrm{A}$ to $\\mathrm{B}$ via $\\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast.\n[figure2]\n\nFigure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA.\n\nBottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal.\n\nFor the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \\mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \\mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \\times 10^{8} \\mathrm{~m}$. Take the radius of the Earth to be $6370 \\mathrm{~km}$, the radius of the Moon to be $1740 \\mathrm{~km}$, and the mass of the Moon to be $7.35 \\times 10^{22} \\mathrm{~kg}$.\n\nproblem:\nd. For the patched conics approach (solid lines):\n\ni. Find the distance from the centre of the Earth to point $C$, and hence the semi-major axes of both ellipses.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~m}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-04.jpg?height=1010&width=1508&top_left_y=543&top_left_x=271", "https://cdn.mathpix.com/cropped/2024_03_14_0117b7b4f76996307b50g-06.jpg?height=800&width=1586&top_left_y=518&top_left_x=240" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~m}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_28", "problem": "有 $a 、 b 、 c 、 d$ 四颗地球卫星: $a$ 还未发射, 在地球赤道上随地球表面一起转动, $b$在地球的近地圆轨道上正常运行, $c$ 是地球同步卫星, $d$ 是高空探测卫星, 各卫星排列位置如图, 则下列说法正确的是（）\n\n[图1]\nA: 向心加速度大小关系是: $a_{b}>a_{c}>a_{d}>a_{a}$, 速度大小关系是: $v_{a}>v_{b}>v_{c}>v_{d}$\nB: 在相同时间内 $b$ 转过的弧长最长, $a 、 c$ 转过的弧长对应的角度相等\nC: $c$ 在 4 小时内转过的圆心角是 $\\frac{\\pi}{2}, a$ 在 2 小时内转过的圆心角是 $\\frac{\\pi}{6}$\nD: $b$ 的周期一定小于 $d$ 的周期， $d$ 的周期一定大于 24 小时\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n有 $a 、 b 、 c 、 d$ 四颗地球卫星: $a$ 还未发射, 在地球赤道上随地球表面一起转动, $b$在地球的近地圆轨道上正常运行, $c$ 是地球同步卫星, $d$ 是高空探测卫星, 各卫星排列位置如图, 则下列说法正确的是（）\n\n[图1]\n\nA: 向心加速度大小关系是: $a_{b}>a_{c}>a_{d}>a_{a}$, 速度大小关系是: $v_{a}>v_{b}>v_{c}>v_{d}$\nB: 在相同时间内 $b$ 转过的弧长最长, $a 、 c$ 转过的弧长对应的角度相等\nC: $c$ 在 4 小时内转过的圆心角是 $\\frac{\\pi}{2}, a$ 在 2 小时内转过的圆心角是 $\\frac{\\pi}{6}$\nD: $b$ 的周期一定小于 $d$ 的周期， $d$ 的周期一定大于 24 小时\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-072.jpg?height=228&width=799&top_left_y=1071&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1190", "problem": "A \"supermoon\" is a new or full moon that occurs with the Moon at or near its closest approach to Earth in a given orbit (perigee). The media commonly associates supermoons with extreme brightness and size, sometimes implying that the Moon itself will become larger and have an impact on human behaviour, but just how different is a supermoon compared to the 'normal' Moon we see each month?\n\nLunar Data:\n\nSynodic Period Anomalistic Period Semi-major axis Orbit eccentricity\n\n$$\n\\begin{aligned}\n& =29.530589 \\text { days (time between same phases e.g. full moon to full moon) } \\\\\n& =27.554550 \\text { days (time between perigees i.e. perigee to perigee) } \\\\\n& =3.844 \\times 10^{5} \\mathrm{~km} \\\\\n& =0.0549 \\\\\n& =1738.1 \\mathrm{~km}\n\\end{aligned}\n$$\n\n$$\n\\begin{array}{ll}\n\\text { Radius of the Moon } & =1738.1 \\mathrm{~km} \\\\\n\\text { Mass of the Moon } & =7.342 \\times 10^{22} \\mathrm{~kg}\n\\end{array}\n$$\n\nIn this question, we will only consider a full moon that is at perigee to be a supermoon.calculate the mean difference in the distance between the apogee and perigee (The data given in this question allows the mean orbital parameters to be calculated. Note that in reality, perturbations in the lunar orbit mean that the perigee and apogee continually change over the course of the year).", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nA \"supermoon\" is a new or full moon that occurs with the Moon at or near its closest approach to Earth in a given orbit (perigee). The media commonly associates supermoons with extreme brightness and size, sometimes implying that the Moon itself will become larger and have an impact on human behaviour, but just how different is a supermoon compared to the 'normal' Moon we see each month?\n\nLunar Data:\n\nSynodic Period Anomalistic Period Semi-major axis Orbit eccentricity\n\n$$\n\\begin{aligned}\n& =29.530589 \\text { days (time between same phases e.g. full moon to full moon) } \\\\\n& =27.554550 \\text { days (time between perigees i.e. perigee to perigee) } \\\\\n& =3.844 \\times 10^{5} \\mathrm{~km} \\\\\n& =0.0549 \\\\\n& =1738.1 \\mathrm{~km}\n\\end{aligned}\n$$\n\n$$\n\\begin{array}{ll}\n\\text { Radius of the Moon } & =1738.1 \\mathrm{~km} \\\\\n\\text { Mass of the Moon } & =7.342 \\times 10^{22} \\mathrm{~kg}\n\\end{array}\n$$\n\nIn this question, we will only consider a full moon that is at perigee to be a supermoon.\n\nproblem:\ncalculate the mean difference in the distance between the apogee and perigee (The data given in this question allows the mean orbital parameters to be calculated. Note that in reality, perturbations in the lunar orbit mean that the perigee and apogee continually change over the course of the year).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of km, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "km" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1175", "problem": "A day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4).\n\n[figure1]\n\nFigure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun.\n\nCredit: Wikipedia.\n\nThe length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time.\n\n[Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \\times 10^{-11} \\mathrm{~m}^{3} \\mathrm{~kg}^{-1} \\mathrm{~s}^{-2}, R_{\\oplus}=6371 \\mathrm{~km}, M_{\\oplus}=5.972 \\times 10^{24} \\mathrm{~kg}, M_{\\odot}=$ $1.989 \\times 10^{30} \\mathrm{~kg}$ and $1 \\mathrm{au}=1.496 \\times 10^{11} \\mathrm{~m}$.]\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\\frac{2}{5} M R^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\omega$ is $L=I \\omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.\n\nThe speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$a. Given the Sun's composition has hydrogen fraction, $X=0.72$, helium fraction $Y=0.26$ and 'metals' (i.e. any element lithium and heavier) fraction $Z=0.02$, estimate the temperature at the centre of the Sun.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nA day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4).\n\n[figure1]\n\nFigure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun.\n\nCredit: Wikipedia.\n\nThe length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time.\n\n[Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \\times 10^{-11} \\mathrm{~m}^{3} \\mathrm{~kg}^{-1} \\mathrm{~s}^{-2}, R_{\\oplus}=6371 \\mathrm{~km}, M_{\\oplus}=5.972 \\times 10^{24} \\mathrm{~kg}, M_{\\odot}=$ $1.989 \\times 10^{30} \\mathrm{~kg}$ and $1 \\mathrm{au}=1.496 \\times 10^{11} \\mathrm{~m}$.]\n\n## Helpful equations:\n\nThe moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\\frac{2}{5} M R^{2}$.\n\nThe angular momentum, $L$, of a spinning object with an angular velocity of $\\omega$ is $L=I \\omega=r \\times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.\n\nThe speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nproblem:\na. Given the Sun's composition has hydrogen fraction, $X=0.72$, helium fraction $Y=0.26$ and 'metals' (i.e. any element lithium and heavier) fraction $Z=0.02$, estimate the temperature at the centre of the Sun.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of sidereal days, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-06.jpg?height=1276&width=782&top_left_y=567&top_left_x=657" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "sidereal days" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_411", "problem": "2013 年 12 月 2 日 1 时 30 分, “嫦娥三号”探测器由长征三号乙运载火箭从西昌卫星发射中心发射, 首次实现月球软着陆和月面巡视勘察. 嫦娥三号的飞行轨道示意图如图所示. 假设“嫦娥三号”在环月段圆轨道和椭圆轨道上运动时，只受到月球的万有引力. 则 ( )\n\n[图1]\nA: 若已知嫦娥三号环月段圆轨道的半径、运动周期和引力常量, 则可以计算出月球的密度\nB: 嫦娥三号由环月段圆轨道变轨进入环月段椭圆轨道时, 应让发动机点火使其加速\nC: 嫦娥三号在环月段椭圆轨道上 $\\mathrm{P}$ 点的动能大于 $\\mathrm{Q}$ 点的动能\nD: 嫦娥三号在动力下降阶段, 其引力势能减小\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2013 年 12 月 2 日 1 时 30 分, “嫦娥三号”探测器由长征三号乙运载火箭从西昌卫星发射中心发射, 首次实现月球软着陆和月面巡视勘察. 嫦娥三号的飞行轨道示意图如图所示. 假设“嫦娥三号”在环月段圆轨道和椭圆轨道上运动时，只受到月球的万有引力. 则 ( )\n\n[图1]\n\nA: 若已知嫦娥三号环月段圆轨道的半径、运动周期和引力常量, 则可以计算出月球的密度\nB: 嫦娥三号由环月段圆轨道变轨进入环月段椭圆轨道时, 应让发动机点火使其加速\nC: 嫦娥三号在环月段椭圆轨道上 $\\mathrm{P}$ 点的动能大于 $\\mathrm{Q}$ 点的动能\nD: 嫦娥三号在动力下降阶段, 其引力势能减小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-88.jpg?height=400&width=763&top_left_y=160&top_left_x=361" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1032", "problem": "Figure 4 below is a composite image which depicts a transit of the International Space Station (ISS) across the disc of the Sun. The image comprises 26 individual photographs which were taken at regular time intervals during the transit. The total duration of the transit was less than one second. In this question we will ignore any effects caused by the rotation of the Earth.\n\n[figure1]\n\nFigure 4: A composite of a selection of the frames taken with a high-speed camera of a transit of the ISS in front of the Sun, taken from Northamptonshire at 10:22 BST on $17^{\\text {th }}$ June 2022. Credit: Jamie Cooper Photography\nThe orbital period of the ISS is approximately $93 \\mathrm{~min}$. Estimate the frame rate of the camera used to photograph the transit.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nFigure 4 below is a composite image which depicts a transit of the International Space Station (ISS) across the disc of the Sun. The image comprises 26 individual photographs which were taken at regular time intervals during the transit. The total duration of the transit was less than one second. In this question we will ignore any effects caused by the rotation of the Earth.\n\n[figure1]\n\nFigure 4: A composite of a selection of the frames taken with a high-speed camera of a transit of the ISS in front of the Sun, taken from Northamptonshire at 10:22 BST on $17^{\\text {th }}$ June 2022. Credit: Jamie Cooper Photography\nThe orbital period of the ISS is approximately $93 \\mathrm{~min}$. Estimate the frame rate of the camera used to photograph the transit.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of fps, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-08.jpg?height=831&width=1588&top_left_y=738&top_left_x=240" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "fps" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_733", "problem": "如图所示, 有两颗卫星绕某星球做椭圆轨道运动, 两颗卫星的近地点均与星球表面很近 (可视为相切), 卫星 1 和卫星 2 的轨道远地点到星球表面的最近距离分别为\n\n$h_{1} 、 h_{2}$, 卫星 1 和卫星 2 的环绕周期之比为 $k$ 。忽略星球自转的影响, 已知引力常量为\n\n$G$, 星球表面的重力加速度为 $g_{c}$ 。则星球的平均密度为 ( )\n\n[图1]\nA: $\\frac{3 g_{c}\\left(1-k^{\\frac{2}{3}}\\right)}{2 \\pi G\\left(h_{2} k^{\\frac{2}{3}}-h_{1}\\right)}$\nB: $\\frac{3 g_{c}\\left(1-k^{\\frac{3}{2}}\\right)}{2 \\pi G\\left(h_{2} k^{\\frac{3}{2}}-h_{1}\\right)}$\nC: $\\frac{3 g_{c}\\left(1-k^{\\frac{3}{2}}\\right)}{4 \\pi G\\left(h_{2} k^{\\frac{3}{2}}-h_{1}\\right)}$\nD: $\\frac{3 g_{c}\\left(1-k^{\\frac{2}{3}}\\right)}{4 \\pi G\\left(h_{2} k^{\\frac{2}{3}}-h_{1}\\right)}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 有两颗卫星绕某星球做椭圆轨道运动, 两颗卫星的近地点均与星球表面很近 (可视为相切), 卫星 1 和卫星 2 的轨道远地点到星球表面的最近距离分别为\n\n$h_{1} 、 h_{2}$, 卫星 1 和卫星 2 的环绕周期之比为 $k$ 。忽略星球自转的影响, 已知引力常量为\n\n$G$, 星球表面的重力加速度为 $g_{c}$ 。则星球的平均密度为 ( )\n\n[图1]\n\nA: $\\frac{3 g_{c}\\left(1-k^{\\frac{2}{3}}\\right)}{2 \\pi G\\left(h_{2} k^{\\frac{2}{3}}-h_{1}\\right)}$\nB: $\\frac{3 g_{c}\\left(1-k^{\\frac{3}{2}}\\right)}{2 \\pi G\\left(h_{2} k^{\\frac{3}{2}}-h_{1}\\right)}$\nC: $\\frac{3 g_{c}\\left(1-k^{\\frac{3}{2}}\\right)}{4 \\pi G\\left(h_{2} k^{\\frac{3}{2}}-h_{1}\\right)}$\nD: $\\frac{3 g_{c}\\left(1-k^{\\frac{2}{3}}\\right)}{4 \\pi G\\left(h_{2} k^{\\frac{2}{3}}-h_{1}\\right)}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-019.jpg?height=257&width=482&top_left_y=1628&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_889", "problem": "Estimate the mass of a globular cluster with a radial velocity dispersion $\\sigma_{r}=16.2 \\mathrm{~km} / \\mathrm{s}$. The cluster has an angular diameter of $\\theta=3.56^{\\prime}$ and is a distance $d=9630$ pc away from us.\nA: $6.05 \\times 10^{35} \\mathrm{~kg}$\nB: $9.71 \\times 10^{35} \\mathrm{~kg}$\nC: $1.01 \\times 10^{36} \\mathrm{~kg}$\nD: $3.03 \\times 10^{36} \\mathrm{~kg}$\nE: $5.96 \\times 10^{36} \\mathrm{~kg}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nEstimate the mass of a globular cluster with a radial velocity dispersion $\\sigma_{r}=16.2 \\mathrm{~km} / \\mathrm{s}$. The cluster has an angular diameter of $\\theta=3.56^{\\prime}$ and is a distance $d=9630$ pc away from us.\n\nA: $6.05 \\times 10^{35} \\mathrm{~kg}$\nB: $9.71 \\times 10^{35} \\mathrm{~kg}$\nC: $1.01 \\times 10^{36} \\mathrm{~kg}$\nD: $3.03 \\times 10^{36} \\mathrm{~kg}$\nE: $5.96 \\times 10^{36} \\mathrm{~kg}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_200", "problem": "如图所示, 有一质量为 $M$, 半径为 $R$, 密度均匀的球体, 在距离球心 $O$ 为 $2 R$ 的地方有一质量为 $m$ 的质点, 现从 $M$ 中挖去一半径为 $\\frac{R}{2}$ 的球体, 试求:\n若在挖空部分填满另外一种密度为原来 2 倍的物质, 求填充后的实心体对质点 $m$的引力大小。\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 有一质量为 $M$, 半径为 $R$, 密度均匀的球体, 在距离球心 $O$ 为 $2 R$ 的地方有一质量为 $m$ 的质点, 现从 $M$ 中挖去一半径为 $\\frac{R}{2}$ 的球体, 试求:\n若在挖空部分填满另外一种密度为原来 2 倍的物质, 求填充后的实心体对质点 $m$的引力大小。\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-062.jpg?height=286&width=443&top_left_y=174&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_912", "problem": "For which of these lines of latitude will a vertical stick in the ground have no shadow at local midday on $21^{\\text {st }}$ December 2021 ?\nA: Tropic of Cancer\nB: Equator\nC: Tropic of Capricorn\nD: Antarctic Circle\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nFor which of these lines of latitude will a vertical stick in the ground have no shadow at local midday on $21^{\\text {st }}$ December 2021 ?\n\nA: Tropic of Cancer\nB: Equator\nC: Tropic of Capricorn\nD: Antarctic Circle\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://i.postimg.cc/hPkWyb0R/Screenshot-2024-04-06-at-21-28-46.png" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_994", "problem": "On $5^{\\text {th }}$ September 2016, the Rosetta mission has finally found the Philae lander on Comet 67P/Churyumov-Gerasimenko. Considering that Philae ( $1 \\times 1 \\times 1 \\mathrm{~m})$ appeared in an image from the high-resolution camera (with $2048 \\times 2048$ pixels and field of view $2.2^{\\circ} \\times 2.2^{\\circ}$ ) as $25 \\times 25$ pixels, from what distance did Rosetta manage to image Philae?\nA: $1.1 \\mathrm{~km}$\nB: $2.1 \\mathrm{~km}$\nC: $12.2 \\mathrm{~km}$\nD: $26.8 \\mathrm{~km}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOn $5^{\\text {th }}$ September 2016, the Rosetta mission has finally found the Philae lander on Comet 67P/Churyumov-Gerasimenko. Considering that Philae ( $1 \\times 1 \\times 1 \\mathrm{~m})$ appeared in an image from the high-resolution camera (with $2048 \\times 2048$ pixels and field of view $2.2^{\\circ} \\times 2.2^{\\circ}$ ) as $25 \\times 25$ pixels, from what distance did Rosetta manage to image Philae?\n\nA: $1.1 \\mathrm{~km}$\nB: $2.1 \\mathrm{~km}$\nC: $12.2 \\mathrm{~km}$\nD: $26.8 \\mathrm{~km}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_190", "problem": "2019 年 1 月 3 日, 我国“嫦娥四号”探测器在月球背面成功着陆并发回大量月背影\n\n像. 如图所示为位于月球背面的“嫦娥四号”探测器 $A$ 通过“鹊桥”中继站 $B$ 向地球传输电磁波信息的示意图. 拉格朗日 $L_{2}$ 点位于地月连线延长线上, “鹊桥”的运动可看成如下两种运动的合运动: 一是在地球和月球引力共同作用下, “鹊桥”在 $L_{2}$ 点附近与月球以相同的周期 $T_{0}$ 一起绕地球做匀速圆周运动; 二是在与地月连线垂直的平面内绕 $L_{2}$ 点做匀速圆周运动. 已知地球的质量为月球质量的 $n$ 倍, 地球到 $L_{2}$ 点的距离为月球到 $L_{2}$ 点的距离的 $k$ 倍, 地球半径、月球半径以及“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的半径均远小于月球到 $L_{2}$ 点的距离 (提示: “鹊桥”绕 $L_{2}$ 点做匀速圆周运动的向心力由地球和月球对其引力在过 $L_{2}$ 点与地月连线垂直的平面内的分量提供).[图1]\n试推导“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的周期 $T$ 的（近似）表达式. 若 $k=7, n=81$, $T_{0}=27.3$ 天, 求出 $T$ 的天数（取整数）.", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n2019 年 1 月 3 日, 我国“嫦娥四号”探测器在月球背面成功着陆并发回大量月背影\n\n像. 如图所示为位于月球背面的“嫦娥四号”探测器 $A$ 通过“鹊桥”中继站 $B$ 向地球传输电磁波信息的示意图. 拉格朗日 $L_{2}$ 点位于地月连线延长线上, “鹊桥”的运动可看成如下两种运动的合运动: 一是在地球和月球引力共同作用下, “鹊桥”在 $L_{2}$ 点附近与月球以相同的周期 $T_{0}$ 一起绕地球做匀速圆周运动; 二是在与地月连线垂直的平面内绕 $L_{2}$ 点做匀速圆周运动. 已知地球的质量为月球质量的 $n$ 倍, 地球到 $L_{2}$ 点的距离为月球到 $L_{2}$ 点的距离的 $k$ 倍, 地球半径、月球半径以及“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的半径均远小于月球到 $L_{2}$ 点的距离 (提示: “鹊桥”绕 $L_{2}$ 点做匀速圆周运动的向心力由地球和月球对其引力在过 $L_{2}$ 点与地月连线垂直的平面内的分量提供).[图1]\n试推导“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的周期 $T$ 的（近似）表达式. 若 $k=7, n=81$, $T_{0}=27.3$ 天, 求出 $T$ 的天数（取整数）.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以天为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-020.jpg?height=160&width=662&top_left_y=1408&top_left_x=363" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "天" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_136", "problem": "北斗卫星导航系统是中国自主研发、独立运行的全球卫星导航系统, 北斗卫星导航系统由空间段、地面段和用户段三部分组成。空间段包括 5 颗静止轨道卫星和 30 颗非静止轨道卫星。假设一颗非静止轨道卫星 $\\mathrm{a}$ 在轨道上绕行 $n$ 圈所用时间为 $t$ 。如图所示。已知地球的半径为 $R$, 地球表面处的重力加速度为 $g$, 万有引力常量为 $G$, 求:\n\n卫星 a 离地面的高度 $h$ 。\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n北斗卫星导航系统是中国自主研发、独立运行的全球卫星导航系统, 北斗卫星导航系统由空间段、地面段和用户段三部分组成。空间段包括 5 颗静止轨道卫星和 30 颗非静止轨道卫星。假设一颗非静止轨道卫星 $\\mathrm{a}$ 在轨道上绕行 $n$ 圈所用时间为 $t$ 。如图所示。已知地球的半径为 $R$, 地球表面处的重力加速度为 $g$, 万有引力常量为 $G$, 求:\n\n卫星 a 离地面的高度 $h$ 。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-148.jpg?height=343&width=457&top_left_y=931&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1213", "problem": "The surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.c. The Rayleigh criterion and speed of sound in a plasma are given.\n\niii. Later on in its mission, Solar Orbiter will have a perihelion of 0.284 au. Calculate the physical size on the Sun (in $\\mathrm{km}$ ) of each picture element in an image taken with $\\mathrm{HRI}_{\\text {EUv }}$ as well as the FOV in units of $R_{\\odot}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe surface of the Sun has a temperature of $\\sim 5700 \\mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \\mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras).\n[figure1]\n\nFigure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \\& NASA\n\nRight: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\\mathrm{HRI}_{\\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \\& NASA.\n\nLaunched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail.\n\nThe highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\\mathrm{Fe}^{9+}$ ) though is called $\\mathrm{Fe} \\mathrm{X} \\mathrm{('ten')} \\mathrm{by} \\mathrm{astronomers} \\mathrm{(as} \\mathrm{Fe} \\mathrm{I} \\mathrm{is} \\mathrm{the} \\mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument.\n\nThe photons detected by $\\mathrm{HRI}_{\\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\\mathrm{Fe} \\mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \\mathrm{eV}$ (where $1 \\mathrm{eV}=1.60 \\times 10^{-19} \\mathrm{~J}$ ). The HRI $\\mathrm{HUV}_{\\mathrm{EUV}}$ telescope has a $1000^{\\prime \\prime}$ by $1000^{\\prime \\prime}$ field of view (FOV, where $1^{\\circ}=3600^{\\prime \\prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \\mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \\mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \\mu \\mathrm{m}$.\n\nAlthough we are viewing the emissions of $\\mathrm{Fe} \\mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.\n\nproblem:\nc. The Rayleigh criterion and speed of sound in a plasma are given.\n\niii. Later on in its mission, Solar Orbiter will have a perihelion of 0.284 au. Calculate the physical size on the Sun (in $\\mathrm{km}$ ) of each picture element in an image taken with $\\mathrm{HRI}_{\\text {EUv }}$ as well as the FOV in units of $R_{\\odot}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-10.jpg?height=792&width=1572&top_left_y=598&top_left_x=241", "https://cdn.mathpix.com/cropped/2024_03_14_6cde567bccf58dc9a2d2g-13.jpg?height=166&width=665&top_left_y=1376&top_left_x=430" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_231", "problem": "卫星 1 和卫星 2 分别沿圆轨道和椭圆轨道环绕地球运行, 两轨道在同一平面内相交于 $A 、 B$ 两点, 卫星 2 在近地点离地心的距离是卫星 1 轨道半径的 $\\frac{1}{k}$ 倍, 如图所示, 某时刻两卫星与地心在同一直线上, $D$ 点为远地点, 当卫星 2 运行到 $A$ 点时速度方向与 $C D$连线平行。已知近地点的曲率半径为 $\\rho=\\frac{b^{2}}{a}$, 式中 $a 、 b$ 分别是椭圆的半长轴和半短轴,下列说法中正确的是（）\n\n[图1]\nA: 两卫星在 $A$ 点的加速度相同\nB: 卫星 2 在近地点的加速度大小是卫星 1 加速度大小的 $k$ 倍\nC: 卫星 2 在近地点的速率是卫星 1 速率的 $k$ 倍\nD: 卫星 2 运行到近地点时, 卫星 1 和卫星 2 的连线过地心\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n卫星 1 和卫星 2 分别沿圆轨道和椭圆轨道环绕地球运行, 两轨道在同一平面内相交于 $A 、 B$ 两点, 卫星 2 在近地点离地心的距离是卫星 1 轨道半径的 $\\frac{1}{k}$ 倍, 如图所示, 某时刻两卫星与地心在同一直线上, $D$ 点为远地点, 当卫星 2 运行到 $A$ 点时速度方向与 $C D$连线平行。已知近地点的曲率半径为 $\\rho=\\frac{b^{2}}{a}$, 式中 $a 、 b$ 分别是椭圆的半长轴和半短轴,下列说法中正确的是（）\n\n[图1]\n\nA: 两卫星在 $A$ 点的加速度相同\nB: 卫星 2 在近地点的加速度大小是卫星 1 加速度大小的 $k$ 倍\nC: 卫星 2 在近地点的速率是卫星 1 速率的 $k$ 倍\nD: 卫星 2 运行到近地点时, 卫星 1 和卫星 2 的连线过地心\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-104.jpg?height=314&width=642&top_left_y=1733&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_983", "problem": "The very first image released by the James Webb Space Telescope (JWST) was of a galaxy cluster called SMACS 0723. The image is considered to be Webb's first deep field, since a long exposure time of 12.5 hours was used to allow the light from very faint and distant galaxies to be seen. The spectrum of one such galaxy is shown in Figure 2.\n\n[figure1]\n\nFigure 2: Highly redshifted emission lines in the spectrum of a galaxy that is 13.1 billion years old, captured using the JWST's near-infrared spectrometer (NIRSpec). Credit: NASA, ESA, CSA, STScI.\n\nThe spectrum shows four bright hydrogen lines, which are part of the Balmer series (some of which are normally seen in the visible). The rest frame wavelengths of the longest four lines in the series are $410 \\mathrm{~nm}, 434 \\mathrm{~nm}, 486 \\mathrm{~nm}$ and $656 \\mathrm{~nm}$ (not all of which are visible in the spectrum).\n\nOnce a redshift is known, its recessional velocity can be calculated. At very high redshifts, such as these, General Relativity must be used. A conversion from redshift to recessional velocity is shown in Figure 3.\n\n[figure2]\n\nFigure 3: Conversion from redshift to recessional velocity for a linear approximation $(v=z c)$, using Special Relativity $\\left(v=c \\frac{(1+z)^{2}-1}{(1+z)^{2}+1}\\right)$, and using General Relativity $\\left(v=\\dot{a}(z) \\int_{0}^{z} \\frac{c d z^{\\prime}}{H\\left(z^{\\prime}\\right)}\\right)$. The grey area corresponds to a variety of values for cosmological parameters. The solid line corresponds to values approximately the same as the current measured cosmological parameters. Credit: Davis \\& Lineweaver (2001).\n\nTaking the value of the Hubble constant to be $H_{0}=70 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$, what is the distance to the galaxy? Give your answer in Mpc.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe very first image released by the James Webb Space Telescope (JWST) was of a galaxy cluster called SMACS 0723. The image is considered to be Webb's first deep field, since a long exposure time of 12.5 hours was used to allow the light from very faint and distant galaxies to be seen. The spectrum of one such galaxy is shown in Figure 2.\n\n[figure1]\n\nFigure 2: Highly redshifted emission lines in the spectrum of a galaxy that is 13.1 billion years old, captured using the JWST's near-infrared spectrometer (NIRSpec). Credit: NASA, ESA, CSA, STScI.\n\nThe spectrum shows four bright hydrogen lines, which are part of the Balmer series (some of which are normally seen in the visible). The rest frame wavelengths of the longest four lines in the series are $410 \\mathrm{~nm}, 434 \\mathrm{~nm}, 486 \\mathrm{~nm}$ and $656 \\mathrm{~nm}$ (not all of which are visible in the spectrum).\n\nOnce a redshift is known, its recessional velocity can be calculated. At very high redshifts, such as these, General Relativity must be used. A conversion from redshift to recessional velocity is shown in Figure 3.\n\n[figure2]\n\nFigure 3: Conversion from redshift to recessional velocity for a linear approximation $(v=z c)$, using Special Relativity $\\left(v=c \\frac{(1+z)^{2}-1}{(1+z)^{2}+1}\\right)$, and using General Relativity $\\left(v=\\dot{a}(z) \\int_{0}^{z} \\frac{c d z^{\\prime}}{H\\left(z^{\\prime}\\right)}\\right)$. The grey area corresponds to a variety of values for cosmological parameters. The solid line corresponds to values approximately the same as the current measured cosmological parameters. Credit: Davis \\& Lineweaver (2001).\n\nTaking the value of the Hubble constant to be $H_{0}=70 \\mathrm{~km} \\mathrm{~s}^{-1} \\mathrm{Mpc}^{-1}$, what is the distance to the galaxy? Give your answer in Mpc.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of Mpc, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-06.jpg?height=985&width=1588&top_left_y=547&top_left_x=240", "https://cdn.mathpix.com/cropped/2024_03_06_116c30b1e79c82f9c667g-07.jpg?height=997&width=1334&top_left_y=187&top_left_x=361" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "Mpc" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_1015", "problem": "The image below was released in April 2019 by the Event Horizon Telescope collaboration and is considered to be one of the most significant astronomical images ever made. What is it of?\n\n[figure1]\nA: A supernova\nB: A planetary nebula\nC: A black hole\nD: A quasar\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe image below was released in April 2019 by the Event Horizon Telescope collaboration and is considered to be one of the most significant astronomical images ever made. What is it of?\n\n[figure1]\n\nA: A supernova\nB: A planetary nebula\nC: A black hole\nD: A quasar\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_13148c5721a741e30941g-04.jpg?height=614&width=1048&top_left_y=595&top_left_x=504" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_349", "problem": "在星球 $P$ 和星球 $Q$ 的表面, 以相同的初速度 $v_{0}$ 坚直上抛一小球, 小球在空中运动时的 $v-t$ 图像分别如图所示。假设两星球均为质量均匀分布的球体, 星球 $P$ 的半径是星球 $Q$ 半径的 3 倍, 下列说法正确的是（）\n\n[图1]\nA: 星球 $P$ 和星球 $Q$ 的质量之比为 $3: 1$\nB: 星球 $P$ 和星球 $Q$ 的密度之比为 $1: 1$\nC: 星球 $P$ 和星球 $Q$ 的第一宇宙速度之比为 3:1\nD: 星球 $P$ 和星球 $Q$ 的近地卫星周期之比为 $1: 3$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n在星球 $P$ 和星球 $Q$ 的表面, 以相同的初速度 $v_{0}$ 坚直上抛一小球, 小球在空中运动时的 $v-t$ 图像分别如图所示。假设两星球均为质量均匀分布的球体, 星球 $P$ 的半径是星球 $Q$ 半径的 3 倍, 下列说法正确的是（）\n\n[图1]\n\nA: 星球 $P$ 和星球 $Q$ 的质量之比为 $3: 1$\nB: 星球 $P$ 和星球 $Q$ 的密度之比为 $1: 1$\nC: 星球 $P$ 和星球 $Q$ 的第一宇宙速度之比为 3:1\nD: 星球 $P$ 和星球 $Q$ 的近地卫星周期之比为 $1: 3$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-035.jpg?height=368&width=494&top_left_y=1912&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_822", "problem": "The apparent magnitude of a star of radius $0.41 R_{\\odot}$ as observed from Earth appears to fluctuate by 0.037 . That is, the difference between the maximum and minimum apparent magnitudes is 0.037 . This fluctuation is caused by an exoplanet that orbits the star. Determine the radius of the exoplanet.\nA: $0.075 R_{\\odot}$\nB: $0.079 R_{\\odot}$\nC: $0.085 R_{\\odot}$\nD: $0.098 R_{\\odot}$\nE: $0.12 R_{\\odot}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe apparent magnitude of a star of radius $0.41 R_{\\odot}$ as observed from Earth appears to fluctuate by 0.037 . That is, the difference between the maximum and minimum apparent magnitudes is 0.037 . This fluctuation is caused by an exoplanet that orbits the star. Determine the radius of the exoplanet.\n\nA: $0.075 R_{\\odot}$\nB: $0.079 R_{\\odot}$\nC: $0.085 R_{\\odot}$\nD: $0.098 R_{\\odot}$\nE: $0.12 R_{\\odot}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_924", "problem": "In which constellation would you find the centre of the Milky Way?\nA: Ophiucus\nB: Coma Berenices\nC: Sagittarius\nD: Scorpius\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn which constellation would you find the centre of the Milky Way?\n\nA: Ophiucus\nB: Coma Berenices\nC: Sagittarius\nD: Scorpius\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_230", "problem": "$\\mathrm{A} 、 \\mathrm{~B}$ 两颗卫星在同一平面内沿同一方向绕地球做匀速圆周运动, 如图甲所示。两卫星之间的距离 $\\Delta r$ 随时间周期性变化, 如图乙所示。仅考虑地球对卫星的引力, 下列说法正确的是（ ）\n\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)\nA: A、B 的轨道半径之比为 $1: 3$\nB: A、B 的线速度之比为 $1: 2$\nC: $\\mathrm{A}$ 的运动周期大于 $\\mathrm{B}$ 的运动周期\nD: $A 、 B$ 的向心加速度之比为 4: 1\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n$\\mathrm{A} 、 \\mathrm{~B}$ 两颗卫星在同一平面内沿同一方向绕地球做匀速圆周运动, 如图甲所示。两卫星之间的距离 $\\Delta r$ 随时间周期性变化, 如图乙所示。仅考虑地球对卫星的引力, 下列说法正确的是（ ）\n\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)\n\nA: A、B 的轨道半径之比为 $1: 3$\nB: A、B 的线速度之比为 $1: 2$\nC: $\\mathrm{A}$ 的运动周期大于 $\\mathrm{B}$ 的运动周期\nD: $A 、 B$ 的向心加速度之比为 4: 1\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-035.jpg?height=311&width=277&top_left_y=644&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-035.jpg?height=297&width=445&top_left_y=657&top_left_x=674" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_208", "problem": "宇宙空间由一种由三颗星体 $A 、 B 、 C$ 组成的三星体系, 它们分别位于等边三角形 $A B C$的三个顶点上, 绕一个固定且共同的圆心 $O$ 做匀速圆周运动, 轨道如图中实线所示,其轨道半径 $r_{A}\\mathrm{v}_{\\mathrm{B}}$\nC: 卫星 $P$ 在 I 轨道的加速度大小为 $a_{0}$, 卫星 $Q$ 在II轨道 $A$ 点加速度大小为 $\\mathrm{a}_{\\mathrm{A}}$, 则 $a_{0}<\\mathrm{a}_{\\mathrm{A}}$\nD: 卫星 $P$ 在 I 轨道上受到的地球引力与卫星 $Q$ 在II轨道上经过两轨道交点时受到的地球引力大小相等\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, 曲线 $\\mathrm{I}$ 是一颗绕地球做圆周运动的卫星 $P$ 轨道的示意图, 其半径为 $R$;曲线II是一颗绕地球做椭圆运动的卫星 $Q$ 轨道的示意图, $O$ 点为地球球心, $A B$ 为椭圆的长轴, 两轨道和地心都在同一平面内, 已知在两轨道上运动的卫星的周期相等, 万有引力常量为 $G$ ，地球质量为 $M$ ，下列说法错误的是（ ）\n\n[图1]\n\nA: 椭圆轨道的长轴长度为 $2 R$\nB: 卫星 $P$ 在 I 轨道的速率为 $v_{0}$, 卫星 $Q$ 在II轨道 $B$ 点的速率为 $\\mathrm{v}_{\\mathrm{B}}$, 则 $v_{0}>\\mathrm{v}_{\\mathrm{B}}$\nC: 卫星 $P$ 在 I 轨道的加速度大小为 $a_{0}$, 卫星 $Q$ 在II轨道 $A$ 点加速度大小为 $\\mathrm{a}_{\\mathrm{A}}$, 则 $a_{0}<\\mathrm{a}_{\\mathrm{A}}$\nD: 卫星 $P$ 在 I 轨道上受到的地球引力与卫星 $Q$ 在II轨道上经过两轨道交点时受到的地球引力大小相等\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-52.jpg?height=299&width=391&top_left_y=1935&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1104", "problem": "The Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\\text {peri }}=9.86 R_{\\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft.\n[figure1]\n\nFigure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman.\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nGiven that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.\n\nClose to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2.\n\n[figure2]\n\nFigure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL.\n\nWhen considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\\angle y c z$ in the figure, such that the shaded areas are the same).\n\n[figure3]\n\nFigure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia.\n\n\nThe eccentric anomaly can be related to the mean anomaly through Kepler's Equation,\n\n$$\nM=E-e \\sin E \\text {. }\n$$d. Derive a formula for the distance from the focus for an elliptical orbit, $r$ (SP in the figure) in terms of the semi-major axis $a$, the eccentricity $e$, and the eccentric anomaly $E$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an equation.\nHere is some context information for this question, which might assist you in solving it:\nThe Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\\text {peri }}=9.86 R_{\\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft.\n[figure1]\n\nFigure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman.\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nGiven that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\\mathrm{km} \\mathrm{s}^{-1}$.\n\nClose to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2.\n\n[figure2]\n\nFigure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL.\n\nWhen considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\\angle y c z$ in the figure, such that the shaded areas are the same).\n\n[figure3]\n\nFigure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia.\n\n\nThe eccentric anomaly can be related to the mean anomaly through Kepler's Equation,\n\n$$\nM=E-e \\sin E \\text {. }\n$$\n\nproblem:\nd. Derive a formula for the distance from the focus for an elliptical orbit, $r$ (SP in the figure) in terms of the semi-major axis $a$, the eccentricity $e$, and the eccentric anomaly $E$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-04.jpg?height=708&width=1438&top_left_y=694&top_left_x=318", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-05.jpg?height=411&width=1539&top_left_y=383&top_left_x=264", "https://cdn.mathpix.com/cropped/2024_03_14_ffe0ae050771e0e3decbg-05.jpg?height=603&width=714&top_left_y=1429&top_left_x=677", "https://cdn.mathpix.com/cropped/2024_03_14_bf2d6c3a07c7dc22bd04g-2.jpg?height=469&width=554&top_left_y=1690&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_838", "problem": "An exoplanet was observed during its transit across the surface of a bright star. Estimate the variation of the apparent magnitude $(\\Delta \\mathrm{m})$ of the star caused by exoplanet's transit. During the transit, assume an Earth-based astronomer observes that the area covered by the exoplanet on the projected surface of the star represents $\\eta=2 \\%$ of the star's projected surface.\nA: -4.247\nB: 0.003\nC: 0.022\nD: 0.679\nE: -0.003\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn exoplanet was observed during its transit across the surface of a bright star. Estimate the variation of the apparent magnitude $(\\Delta \\mathrm{m})$ of the star caused by exoplanet's transit. During the transit, assume an Earth-based astronomer observes that the area covered by the exoplanet on the projected surface of the star represents $\\eta=2 \\%$ of the star's projected surface.\n\nA: -4.247\nB: 0.003\nC: 0.022\nD: 0.679\nE: -0.003\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_396", "problem": "有一颗绕地球做匀速圆周运动的卫星, 其运行的线速度是地球近地卫星的 $\\frac{\\sqrt{2}}{2}$, 卫星圆形轨道平面与地球赤道平面重合, 卫星上有太阳能收集板可以把光能转化为电能,提供卫星工作所必须的能量。已知地球表面重力加速度为 $g$, 地球半径为 $R$, 忽略地球公转, 此时太阳处于赤道平面上, 近似认为太阳光是平行光, 则下列说法正确的是 ( )\nA: 卫星的轨道半径为 $3 R$\nB: 卫星轨道所在位置的重力加速度为 $\\frac{1}{3} g$\nC: 卫星运动的周期为 $4 \\pi \\sqrt{\\frac{2 R}{g}}$\nD: 卫星绕地球一周, 太阳能收集板的工作时间为 $\\frac{10 \\pi}{3} \\sqrt{\\frac{2 R}{g}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n有一颗绕地球做匀速圆周运动的卫星, 其运行的线速度是地球近地卫星的 $\\frac{\\sqrt{2}}{2}$, 卫星圆形轨道平面与地球赤道平面重合, 卫星上有太阳能收集板可以把光能转化为电能,提供卫星工作所必须的能量。已知地球表面重力加速度为 $g$, 地球半径为 $R$, 忽略地球公转, 此时太阳处于赤道平面上, 近似认为太阳光是平行光, 则下列说法正确的是 ( )\n\nA: 卫星的轨道半径为 $3 R$\nB: 卫星轨道所在位置的重力加速度为 $\\frac{1}{3} g$\nC: 卫星运动的周期为 $4 \\pi \\sqrt{\\frac{2 R}{g}}$\nD: 卫星绕地球一周, 太阳能收集板的工作时间为 $\\frac{10 \\pi}{3} \\sqrt{\\frac{2 R}{g}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-047.jpg?height=366&width=514&top_left_y=1950&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_519", "problem": "已知某卫星在赤道上空的圆形轨道运行, 轨道半径为 $r_{1}$, 运行周期为 $T$, 卫星运动方向与地球自转方向相同, 不计空气阻力, 万有引力常量为 $G \\circ$ 。求:\n\n地球质量 $M$ 的大小;\n\n[图1]", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n已知某卫星在赤道上空的圆形轨道运行, 轨道半径为 $r_{1}$, 运行周期为 $T$, 卫星运动方向与地球自转方向相同, 不计空气阻力, 万有引力常量为 $G \\circ$ 。求:\n\n地球质量 $M$ 的大小;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_29925d26250e50e92016g-136.jpg?height=514&width=531&top_left_y=654&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_137", "problem": "有人设想: 可以在飞船从运行轨道进入返回地球程序时, 借飞船需要减速的机会,发射一个小型太空探测器, 从而达到节能的目的。如图所示, 飞船在圆轨道I上绕地球飞行, 其轨道半径为地球半径的 $k$ 倍 $(k>1)$ 。当飞船通过轨道I的 $A$ 点时, 飞船上的发射装置短暂工作, 将探测器沿飞船原运动方向射出, 并使探测器恰能完全脱离地球的引力范围, 即到达距地球无限远时的速度恰好为零, 而飞船在发射探测器后沿椭圆轨道II向前运动, 其近地点 $B$ 到地心的距离近似为地球半径 $R$ 。已知取无穷远处引力势能为零,物体距星球球心距离为 $r$ 时的引力势能 $E_{\\mathrm{p}}=-G \\frac{M m}{r}$ 。在飞船沿轨道I和轨道II以及探测器被射出后的运动过程中, 其动能和引力势能之和均保持不变。以上过程中飞船和探测器的质量均可视为不变, 已知地球表面的重力加速度为 $g$ 。则下列说法正确的是 $(\\quad)$\n\n[图1]\nA: 飞船在轨道 $\\mathrm{I}$ 运动的速度大小为 $\\sqrt{(k+1) g R}$\nB: 飞船在轨道I上的运行周期是在轨道II上运行周期的 $\\frac{2 k}{k+1}$ 倍\nC: 探测器刚离开飞船时的速度大小为 $\\sqrt{\\frac{2 g R}{k}}$\nD: 若飞船沿轨道II运动过程中, 通过 $A$ 点与 $B$ 点的速度大小与这两点到地心的距离成反比, 实现上述飞船和探测器的运动过程, 飞船与探测器的质量之比应满足 $\\frac{\\sqrt{2}}{1-\\sqrt{\\frac{2}{k+1}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n有人设想: 可以在飞船从运行轨道进入返回地球程序时, 借飞船需要减速的机会,发射一个小型太空探测器, 从而达到节能的目的。如图所示, 飞船在圆轨道I上绕地球飞行, 其轨道半径为地球半径的 $k$ 倍 $(k>1)$ 。当飞船通过轨道I的 $A$ 点时, 飞船上的发射装置短暂工作, 将探测器沿飞船原运动方向射出, 并使探测器恰能完全脱离地球的引力范围, 即到达距地球无限远时的速度恰好为零, 而飞船在发射探测器后沿椭圆轨道II向前运动, 其近地点 $B$ 到地心的距离近似为地球半径 $R$ 。已知取无穷远处引力势能为零,物体距星球球心距离为 $r$ 时的引力势能 $E_{\\mathrm{p}}=-G \\frac{M m}{r}$ 。在飞船沿轨道I和轨道II以及探测器被射出后的运动过程中, 其动能和引力势能之和均保持不变。以上过程中飞船和探测器的质量均可视为不变, 已知地球表面的重力加速度为 $g$ 。则下列说法正确的是 $(\\quad)$\n\n[图1]\n\nA: 飞船在轨道 $\\mathrm{I}$ 运动的速度大小为 $\\sqrt{(k+1) g R}$\nB: 飞船在轨道I上的运行周期是在轨道II上运行周期的 $\\frac{2 k}{k+1}$ 倍\nC: 探测器刚离开飞船时的速度大小为 $\\sqrt{\\frac{2 g R}{k}}$\nD: 若飞船沿轨道II运动过程中, 通过 $A$ 点与 $B$ 点的速度大小与这两点到地心的距离成反比, 实现上述飞船和探测器的运动过程, 飞船与探测器的质量之比应满足 $\\frac{\\sqrt{2}}{1-\\sqrt{\\frac{2}{k+1}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-011.jpg?height=348&width=374&top_left_y=1602&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1211", "problem": "On $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.c. By empirically fitting a sinusoidal function (which is assumed to be the same for each track, just with a fixed phase difference between them) and assuming all conjunctions are separated by the average synodic period, we can give rough estimations for the separations of any given great conjunction. Note: be careful as your calculations will be very sensitive to rounding errors.\n\ni. By reading off the graph, give an equation for Track $A$ of the form $\\theta=\\left|D \\sin \\left(\\frac{2 \\pi t}{\\lambda}+\\phi_{A}\\right)\\right|$, where $t$ is the (decimalised) date in years, and $D, \\lambda$, and $-\\pi / 2<\\phi_{A} \\leq \\pi / 2$ are values that need to be determined. [Hint: ensure your function passes through the 2020 data point, and the function is decreasing as it does.]", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn $21^{\\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4).\n\nWhen two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5).\n[figure1]\n\nFigure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 \" telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole.\n\nRight: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery\n\nTelescope. Credit: Levine / Elbert / Bosh / Lowell Observatory.\n\n[figure2]\n\nFigure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\\left(1 / 60^{\\text {th }}\\right.$ of a degree). Credit: Pete Lawrence.\n\nRight: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit:\n\ntimeanddate.com.\n\nThe time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth.\n\nFor circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\\theta=0^{\\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other.\n\nFig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function.\n\n[figure3]\n\nFigure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles.\n\nBottom: The same idea but extended over a much larger date range, up to $10000 \\mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \\& Telescope.\n\nproblem:\nc. By empirically fitting a sinusoidal function (which is assumed to be the same for each track, just with a fixed phase difference between them) and assuming all conjunctions are separated by the average synodic period, we can give rough estimations for the separations of any given great conjunction. Note: be careful as your calculations will be very sensitive to rounding errors.\n\ni. By reading off the graph, give an equation for Track $A$ of the form $\\theta=\\left|D \\sin \\left(\\frac{2 \\pi t}{\\lambda}+\\phi_{A}\\right)\\right|$, where $t$ is the (decimalised) date in years, and $D, \\lambda$, and $-\\pi / 2<\\phi_{A} \\leq \\pi / 2$ are values that need to be determined. [Hint: ensure your function passes through the 2020 data point, and the function is decreasing as it does.]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of ^{\\circ}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-07.jpg?height=706&width=1564&top_left_y=834&top_left_x=244", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-08.jpg?height=578&width=1566&top_left_y=196&top_left_x=242", "https://cdn.mathpix.com/cropped/2024_03_14_f4dc8cb2d9258a843a19g-09.jpg?height=1072&width=1564&top_left_y=1191&top_left_x=246" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "^{\\circ}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_592", "problem": "假设地球可视为质量均匀分布的球体.已知地球表面的重力加速度在两极的大小为 $g_{0}$ ，在赤道的大小为 $g$; 地球半径为 $R$, 引力常数为 $G$, 则\nA: 地球同步卫星距地表的高度为 $\\left(\\sqrt[3]{\\frac{g_{0}}{g_{0}-g}}-1\\right) R$\nB: 地球的质量为 $\\frac{g R^{2}}{G}$\nC: 地球的第一宇宙速度为 $\\sqrt{g_{0} R}$\nD: 地球密度为 $\\frac{3 g_{0}}{4 \\pi R G}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n假设地球可视为质量均匀分布的球体.已知地球表面的重力加速度在两极的大小为 $g_{0}$ ，在赤道的大小为 $g$; 地球半径为 $R$, 引力常数为 $G$, 则\n\nA: 地球同步卫星距地表的高度为 $\\left(\\sqrt[3]{\\frac{g_{0}}{g_{0}-g}}-1\\right) R$\nB: 地球的质量为 $\\frac{g R^{2}}{G}$\nC: 地球的第一宇宙速度为 $\\sqrt{g_{0} R}$\nD: 地球密度为 $\\frac{3 g_{0}}{4 \\pi R G}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_773", "problem": "Kepler's Laws state that the cube of a planet's semi-major axis is proportional to the ...\nA: square of the average distance.\nB: covered area by the planet.\nC: cube of the average distance.\nD: square of the orbital period.\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nKepler's Laws state that the cube of a planet's semi-major axis is proportional to the ...\n\nA: square of the average distance.\nB: covered area by the planet.\nC: cube of the average distance.\nD: square of the orbital period.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_505", "problem": "万有引力定律能够很好地将天体运行规律与地球上物体运动规律具有的内在一致性统一起来. 用弹簧秤称量一个相对于地球静止的小物体的重量, 随称量位置的变化可能会有不同的结果. 已知地球质量为 $M$, 万有引力常量为 $G$. 将地球视为半径为 $\\mathrm{R}$ 质量均匀分布的球体. 下列选项中说法正确的是\nA: 在赤道地面称量时, 弹簧秤读数为 $F_{1}=G \\frac{M m}{R^{2}}$\nB: 在北极地面称量时, 弹簧科读数为 $F_{0}=G \\frac{M m}{R^{2}}$\nC: 在北极上空高出地面 $h$ 处称量时, 弹簧秤读数为 $F_{2}=G \\frac{M m}{(R+h)^{2}}$\nD: 在赤道上空高出地面 $h$ 处称量时, 弹簧科读数为 $F_{3}=G \\frac{M m}{(R+h)^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n万有引力定律能够很好地将天体运行规律与地球上物体运动规律具有的内在一致性统一起来. 用弹簧秤称量一个相对于地球静止的小物体的重量, 随称量位置的变化可能会有不同的结果. 已知地球质量为 $M$, 万有引力常量为 $G$. 将地球视为半径为 $\\mathrm{R}$ 质量均匀分布的球体. 下列选项中说法正确的是\n\nA: 在赤道地面称量时, 弹簧秤读数为 $F_{1}=G \\frac{M m}{R^{2}}$\nB: 在北极地面称量时, 弹簧科读数为 $F_{0}=G \\frac{M m}{R^{2}}$\nC: 在北极上空高出地面 $h$ 处称量时, 弹簧秤读数为 $F_{2}=G \\frac{M m}{(R+h)^{2}}$\nD: 在赤道上空高出地面 $h$ 处称量时, 弹簧科读数为 $F_{3}=G \\frac{M m}{(R+h)^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1113", "problem": "The Sun is seen setting from London $\\left(\\varphi=51^{\\circ} 30^{\\prime} \\mathrm{N}, L=0^{\\circ} 8^{\\prime} \\mathrm{W}\\right)$ at 21:00 UT. At what time UT will it be seen setting in Cardiff ( $\\varphi=51^{\\circ} 30^{\\prime} \\mathrm{N}, L=3^{\\circ} 11^{\\prime} \\mathrm{W}$ ) on the same day?\nA: $21: 12$\nB: $21: 00$\nC: $20: 48$\nD: $20: 58$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe Sun is seen setting from London $\\left(\\varphi=51^{\\circ} 30^{\\prime} \\mathrm{N}, L=0^{\\circ} 8^{\\prime} \\mathrm{W}\\right)$ at 21:00 UT. At what time UT will it be seen setting in Cardiff ( $\\varphi=51^{\\circ} 30^{\\prime} \\mathrm{N}, L=3^{\\circ} 11^{\\prime} \\mathrm{W}$ ) on the same day?\n\nA: $21: 12$\nB: $21: 00$\nC: $20: 48$\nD: $20: 58$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_710", "problem": "现有质量相等的 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三个物体, 物体 $\\mathrm{A}$ 置于地球表面赤道上随地球一起自转, $\\mathrm{B}$ 是一颗近地轨道卫星, 在赤道正上方绕地球做匀速圆周运动（忽略其距离地面的高度), $\\mathrm{C}$ 是一颗地球同步卫星。 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三个物体所受地球的万有引力大小分别为 $F A$ 、 $F B$ 和 $F C$; 它们绕地心做圆周运动的线速度大小分别为 $v A 、 v B$ 和 $v C$; 角速度大小分别为 $\\omega A 、 \\omega B$ 和 $\\omega C$; 向心加速度大小分别为 $a A 、 a B$ 和 $a C$ 。则以下分析中正确的是 ( )\nA: $F B>F C>F A$\nB: $v B>v C>v A$\nC: $\\omega B>\\omega C>\\omega A$\nD: $a B>a C>a A$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n现有质量相等的 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三个物体, 物体 $\\mathrm{A}$ 置于地球表面赤道上随地球一起自转, $\\mathrm{B}$ 是一颗近地轨道卫星, 在赤道正上方绕地球做匀速圆周运动（忽略其距离地面的高度), $\\mathrm{C}$ 是一颗地球同步卫星。 $\\mathrm{A} 、 \\mathrm{~B} 、 \\mathrm{C}$ 三个物体所受地球的万有引力大小分别为 $F A$ 、 $F B$ 和 $F C$; 它们绕地心做圆周运动的线速度大小分别为 $v A 、 v B$ 和 $v C$; 角速度大小分别为 $\\omega A 、 \\omega B$ 和 $\\omega C$; 向心加速度大小分别为 $a A 、 a B$ 和 $a C$ 。则以下分析中正确的是 ( )\n\nA: $F B>F C>F A$\nB: $v B>v C>v A$\nC: $\\omega B>\\omega C>\\omega A$\nD: $a B>a C>a A$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_1198", "problem": "Plotting the position of the Sun in the sky at the same time every day, you get an interesting figure-ofeight shape known as an analemma (see Figure 1). For observers in the Northern hemisphere, you might expect to always see the Sun due South at midday, however on some days the Sun has already passed through that bearing and on others it needs a few more minutes before it gets there. This is due to two effects: the axial tilt of the Earth, and the fact the Earth's orbit is not perfectly circular\n\n[figure1]\n\nFigure 1: The analemma above was composed from images taken every few days at noon near the village of Callanish in the Outer Hebrides in Scotland. In the foreground are the Callanish Stones and the main photo was taken on the winter solstice (when the maximum angle the Sun reaches above the horizon is the lowest of the year, so is at the bottom of the analemma). Credit: Giuseppe Petricca.\n\nThe vertical co-ordinate of a point in the analemma is entirely determined by the Earth's axial tilt. This is known as the solar declination, $\\delta$, and varies sinusoidally throughout the year. The horizontal coordinate of a point in the analemma is determined by a combination of the Earth's axial tilt and the eccentricity of the Earth's orbit. Both of these individually vary sinusoidally, but the superposition of the two is no longer sinusoidal.\n\nWe will define $\\alpha$ as the angle between due South and the Sun at local midday as seen from Oxford, where a positive value means the Sun has already passed through due South (so is on the right of the figure above) whilst a negative value means the Sun has yet to pass through due South. If $\\alpha_{\\text {tilt }}$ is the contribution due to the axial tilt and $\\alpha_{\\text {ecc }}$ is the contribution due to the Earth's orbital eccentricity, then\n\n$$\n\\alpha=\\alpha_{\\text {tilt }}+\\alpha_{\\text {ecc }}\n$$\n\nIf the angle of the axial tilt is $\\varepsilon$ and the eccentricity of the Earth's orbit is $e$, and we assume that both are small enough that the sinusoidal approximation of $\\delta, \\alpha_{\\text {tilt }}$, and $\\alpha_{\\text {ecc }}$ apply, then we find the following boundary conditions:\n\n- $\\delta$ has a period of 1 year, an amplitude of $\\varepsilon$, is maximum at the summer solstice (21 $21^{\\text {st }}$ June) and minimum at the winter solstice $\\left(21^{\\text {st }}\\right.$ December $)$\n- $\\alpha_{\\text {tilt }}$ has a period of 0.5 years, an amplitude (in radians) of $\\tan ^{2}(\\varepsilon / 2)$, is zero at the solstices and the equinoxes (vernal equinox $=21^{\\text {st }}$ March, autumnal equinox $=21^{\\text {st }}$ September), and (using our sign convention) positive just after the vernal equinox\n- $\\alpha_{\\text {ecc }}$ has a period of 1 year, an amplitude (in radians) of $2 e$, is zero at the perihelion (4 $4^{\\text {th }}$ January) and the aphelion ( $6^{\\text {th }}$ July), and (using our sign convention) negative just after the perihelion\n\nGiven the $n^{\\text {th }}$ day of the year, a value can be calculated for $\\delta$ and $\\alpha$, and these are the co-ordinates for the analemma (it is drawn by these parametric equations). For the Earth, $\\varepsilon=23.44^{\\circ}$ and $e=0.0167$.\n\nConsider an alternative version of Earth, known as Earth 2.0. On this planet, the year is unchanged and the perihelion and aphelion are at the same time, but it has a different axial tilt, a different orbital eccentricity, and a different month for the vernal equinox (although it is still on the $21^{\\text {st }}$ day of that month). The analemma as viewed from Earth 2.0 is show in Figure 2 below.\n\n[figure2]\n\nFigure 2: The analemma of the Sun at midday as seen by an observer on Earth 2.0. In this situation, $\\alpha$ ranges from -26 mins 47 secs to 18 mins 56 secs. The circled letters correspond to the same (unknown) day of each month (for example $5^{\\text {th }}$ Jan, $5^{\\text {th }}$ Feb, $5^{\\text {th }}$ March etc.). Credit: Bob Urschel.a. Although $\\alpha$ is really an angle in radians (where $2 \\pi$ radians $=360^{\\circ}$ ), it is normally more useful to convert it into time units (essentially the time since the Sun was due South, or the time until the Sun reaches due South). Taking the mean solar day to be exactly 24 hours:\n\nii. Determine equations for $\\delta$ (in degrees) and tilt and ecc (both in minutes) as a function of the day of the year, $n$. Take $n=1$ to be 1 st January and $n=365$ to be 31 st December.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nPlotting the position of the Sun in the sky at the same time every day, you get an interesting figure-ofeight shape known as an analemma (see Figure 1). For observers in the Northern hemisphere, you might expect to always see the Sun due South at midday, however on some days the Sun has already passed through that bearing and on others it needs a few more minutes before it gets there. This is due to two effects: the axial tilt of the Earth, and the fact the Earth's orbit is not perfectly circular\n\n[figure1]\n\nFigure 1: The analemma above was composed from images taken every few days at noon near the village of Callanish in the Outer Hebrides in Scotland. In the foreground are the Callanish Stones and the main photo was taken on the winter solstice (when the maximum angle the Sun reaches above the horizon is the lowest of the year, so is at the bottom of the analemma). Credit: Giuseppe Petricca.\n\nThe vertical co-ordinate of a point in the analemma is entirely determined by the Earth's axial tilt. This is known as the solar declination, $\\delta$, and varies sinusoidally throughout the year. The horizontal coordinate of a point in the analemma is determined by a combination of the Earth's axial tilt and the eccentricity of the Earth's orbit. Both of these individually vary sinusoidally, but the superposition of the two is no longer sinusoidal.\n\nWe will define $\\alpha$ as the angle between due South and the Sun at local midday as seen from Oxford, where a positive value means the Sun has already passed through due South (so is on the right of the figure above) whilst a negative value means the Sun has yet to pass through due South. If $\\alpha_{\\text {tilt }}$ is the contribution due to the axial tilt and $\\alpha_{\\text {ecc }}$ is the contribution due to the Earth's orbital eccentricity, then\n\n$$\n\\alpha=\\alpha_{\\text {tilt }}+\\alpha_{\\text {ecc }}\n$$\n\nIf the angle of the axial tilt is $\\varepsilon$ and the eccentricity of the Earth's orbit is $e$, and we assume that both are small enough that the sinusoidal approximation of $\\delta, \\alpha_{\\text {tilt }}$, and $\\alpha_{\\text {ecc }}$ apply, then we find the following boundary conditions:\n\n- $\\delta$ has a period of 1 year, an amplitude of $\\varepsilon$, is maximum at the summer solstice (21 $21^{\\text {st }}$ June) and minimum at the winter solstice $\\left(21^{\\text {st }}\\right.$ December $)$\n- $\\alpha_{\\text {tilt }}$ has a period of 0.5 years, an amplitude (in radians) of $\\tan ^{2}(\\varepsilon / 2)$, is zero at the solstices and the equinoxes (vernal equinox $=21^{\\text {st }}$ March, autumnal equinox $=21^{\\text {st }}$ September), and (using our sign convention) positive just after the vernal equinox\n- $\\alpha_{\\text {ecc }}$ has a period of 1 year, an amplitude (in radians) of $2 e$, is zero at the perihelion (4 $4^{\\text {th }}$ January) and the aphelion ( $6^{\\text {th }}$ July), and (using our sign convention) negative just after the perihelion\n\nGiven the $n^{\\text {th }}$ day of the year, a value can be calculated for $\\delta$ and $\\alpha$, and these are the co-ordinates for the analemma (it is drawn by these parametric equations). For the Earth, $\\varepsilon=23.44^{\\circ}$ and $e=0.0167$.\n\nConsider an alternative version of Earth, known as Earth 2.0. On this planet, the year is unchanged and the perihelion and aphelion are at the same time, but it has a different axial tilt, a different orbital eccentricity, and a different month for the vernal equinox (although it is still on the $21^{\\text {st }}$ day of that month). The analemma as viewed from Earth 2.0 is show in Figure 2 below.\n\n[figure2]\n\nFigure 2: The analemma of the Sun at midday as seen by an observer on Earth 2.0. In this situation, $\\alpha$ ranges from -26 mins 47 secs to 18 mins 56 secs. The circled letters correspond to the same (unknown) day of each month (for example $5^{\\text {th }}$ Jan, $5^{\\text {th }}$ Feb, $5^{\\text {th }}$ March etc.). Credit: Bob Urschel.\n\nproblem:\na. Although $\\alpha$ is really an angle in radians (where $2 \\pi$ radians $=360^{\\circ}$ ), it is normally more useful to convert it into time units (essentially the time since the Sun was due South, or the time until the Sun reaches due South). Taking the mean solar day to be exactly 24 hours:\n\nii. Determine equations for $\\delta$ (in degrees) and tilt and ecc (both in minutes) as a function of the day of the year, $n$. Take $n=1$ to be 1 st January and $n=365$ to be 31 st December.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_9bba4f2e5c10ed29bb97g-04.jpg?height=1693&width=1470&top_left_y=550&top_left_x=293", "https://cdn.mathpix.com/cropped/2024_03_14_9bba4f2e5c10ed29bb97g-06.jpg?height=1207&width=1388&top_left_y=413&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_987", "problem": "Which of the following cities will experience the longest day during June?\nA: Edinburgh (longitude $=3.2^{\\circ} \\mathrm{W}$, latitude $=56.0^{\\circ} \\mathrm{N}$ )\nB: Rome (longitude $=12.5^{\\circ} \\mathrm{E}$, latitude $=41.9^{\\circ} \\mathrm{N}$ )\nC: Nairobi (longitude $=36.8^{\\circ} \\mathrm{E}$, latitude $=1.3^{\\circ} \\mathrm{S}$ )\nD: Sydney (longitude $=58.4^{\\circ} \\mathrm{W}$, latitude $=34.6^{\\circ} \\mathrm{S}$ )\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhich of the following cities will experience the longest day during June?\n\nA: Edinburgh (longitude $=3.2^{\\circ} \\mathrm{W}$, latitude $=56.0^{\\circ} \\mathrm{N}$ )\nB: Rome (longitude $=12.5^{\\circ} \\mathrm{E}$, latitude $=41.9^{\\circ} \\mathrm{N}$ )\nC: Nairobi (longitude $=36.8^{\\circ} \\mathrm{E}$, latitude $=1.3^{\\circ} \\mathrm{S}$ )\nD: Sydney (longitude $=58.4^{\\circ} \\mathrm{W}$, latitude $=34.6^{\\circ} \\mathrm{S}$ )\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_1168", "problem": "The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.a. Let $r$ denote distance from the centre of a star. We define the variables $\\rho(r), p(r)$ and $T(r)$ to be the density, pressure and temperature at radius $r$ respectively, and $m(r)$ to be the mass enclosed within radius $r$. We will now try and derive an estimate for the pressure at the centre of the Sun.\n\ni. By considering forces on a box of height $d r$ at radius $r$, show that $d p / d r=-\\rho G m / r^{2}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3).\n[figure1]\n\nFigure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta.\n\nRight: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\\odot}$ and a mass of $0.35 M_{\\odot}$. Credit: Kevin France / University of Colorado.\n\nEstimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun.\n\nThe primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4.\n\n[figure2]\n\nFigure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia.\n\nThe net reaction of the p-p chain is\n\n$$\n4_{1}^{1} \\mathrm{H} \\rightarrow{ }_{2}^{4} \\mathrm{He}+2 e^{+}+2 \\nu_{e}+2 \\gamma .\n$$\n\nThe rate-limiting step is the first one $\\left({ }_{1}^{1} \\mathrm{H}+{ }_{1}^{1} \\mathrm{H}\\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force.\n\nNuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening,\n\n$$\nR=\\frac{4}{3^{2.5} \\pi^{2}} \\frac{h}{\\mu_{r} m_{p}} \\frac{4 \\pi \\varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\\left(E_{0}\\right) \\tau^{2} e^{-\\tau}, \\quad \\text { where } \\quad \\mu_{r}=\\frac{A_{i} A_{j}}{A_{i}+A_{j}}\n$$\n\nand $S\\left(E_{0}\\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\\tau$ is a characteristic width of the Gamow peak,\n\n$$\n\\tau=\\frac{3 E_{0}}{k_{B} T} \\quad \\text { where } \\quad E_{0}=\\left(\\frac{b k_{B} T}{2}\\right)^{2 / 3} \\quad \\text { given } \\quad b=\\sqrt{\\frac{\\mu_{r} m_{p}}{2}} \\frac{\\pi Z_{i} Z_{j} e^{2}}{h \\varepsilon_{0}}\n$$\n\nHere $h$ is Planck's constant, $\\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is\n\n$$\nq=\\frac{\\rho}{m_{p}^{2}}\\left(\\frac{1}{1+\\delta_{i j}}\\right) \\frac{X_{i} X_{j}}{A_{i} A_{j}} R Q\n$$\n\nwhere $\\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise.\n\nEvaluating the fundamental constants and defining $T_{6} \\equiv \\frac{T}{10^{6} \\mathrm{~K}}$ gives\n\n$$\n\\tau=42.59\\left[Z_{i}^{2} Z_{j}^{2} \\mu_{r} T_{6}^{-1}\\right]^{1 / 3},\n$$\n\nwhilst for the proton-proton interaction $Q=13.366 \\mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\\delta_{i j}=1$, and $S\\left(E_{0}\\right)$ is $4.01 \\times 10^{-50} \\mathrm{keV} \\mathrm{m}^{2}$ (Adelberger et al. 2011), so\n\n$R=6.55 \\times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~m}^{3} \\mathrm{~s}^{-1}$ and $q=0.251 \\rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \\mathrm{~W} \\mathrm{~kg}^{-1}$.\n\nproblem:\na. Let $r$ denote distance from the centre of a star. We define the variables $\\rho(r), p(r)$ and $T(r)$ to be the density, pressure and temperature at radius $r$ respectively, and $m(r)$ to be the mass enclosed within radius $r$. We will now try and derive an estimate for the pressure at the centre of the Sun.\n\ni. By considering forces on a box of height $d r$ at radius $r$, show that $d p / d r=-\\rho G m / r^{2}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-06.jpg?height=536&width=1508&top_left_y=560&top_left_x=272", "https://cdn.mathpix.com/cropped/2024_03_14_92511eeeff6a809f304ag-07.jpg?height=748&width=1185&top_left_y=1850&top_left_x=433", "https://cdn.mathpix.com/cropped/2024_03_14_e9aa0a135004f2f4a278g-04.jpg?height=583&width=1011&top_left_y=605&top_left_x=454" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_172", "problem": "如图所示, $\\mathrm{A}$ 和 $\\mathrm{B}$ 两行星绕同一恒星 $\\mathrm{C}$ 做圆周运动, 旋转方向相同, $\\mathrm{A}$ 行星的周期为 $T_{1}, \\mathrm{~B}$ 行星的周期为 $T_{2}$, 某一时刻两行星相距最近, 则 $(\\quad)$\n\n[图1]\nA: 经过 $T_{1}+T_{2}$ 两行星再次相距最近\nB: 经过 $\\frac{T_{1} T_{2}}{T_{2}-T_{1}}$ 两行星再次相距最近\nC: 经过 $\\frac{T_{1}+T_{2}}{2}$ 两行星相距最远\nD: 经过 $\\frac{T_{1} T_{2}}{T_{2}-T_{1}}$ 两行星相距最远\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图所示, $\\mathrm{A}$ 和 $\\mathrm{B}$ 两行星绕同一恒星 $\\mathrm{C}$ 做圆周运动, 旋转方向相同, $\\mathrm{A}$ 行星的周期为 $T_{1}, \\mathrm{~B}$ 行星的周期为 $T_{2}$, 某一时刻两行星相距最近, 则 $(\\quad)$\n\n[图1]\n\nA: 经过 $T_{1}+T_{2}$ 两行星再次相距最近\nB: 经过 $\\frac{T_{1} T_{2}}{T_{2}-T_{1}}$ 两行星再次相距最近\nC: 经过 $\\frac{T_{1}+T_{2}}{2}$ 两行星相距最远\nD: 经过 $\\frac{T_{1} T_{2}}{T_{2}-T_{1}}$ 两行星相距最远\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-022.jpg?height=277&width=311&top_left_y=932&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_345", "problem": "2021 年 5 月 15 日, “天问一号”着陆器成功着陆于火星乌托邦平原南部预选着陆区,我国首次火星探测任务着陆火星取得圆满成功。如图为“天问一号”的地火转移轨道，为了节省燃料, 我们在火星与地球之间相对合适位置时发射“天问一号”。将火星与地球绕太阳的运动简化为在同一平面、沿同一方向的匀速圆周运动。下列说法正确的是 ( )\n\n[图1]\nA: 火星的公转周期大于地球的公转周期\nB: 火星公转的向心加速度大于地球公转的向心加速度\nC: “天问一号”在地火转移轨道上运动的周期小于地球绕太阳运动的周期\nD: “天问一号”从 $A$ 点运动到 $C$ 点的过程中处于加速状态\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2021 年 5 月 15 日, “天问一号”着陆器成功着陆于火星乌托邦平原南部预选着陆区,我国首次火星探测任务着陆火星取得圆满成功。如图为“天问一号”的地火转移轨道，为了节省燃料, 我们在火星与地球之间相对合适位置时发射“天问一号”。将火星与地球绕太阳的运动简化为在同一平面、沿同一方向的匀速圆周运动。下列说法正确的是 ( )\n\n[图1]\n\nA: 火星的公转周期大于地球的公转周期\nB: 火星公转的向心加速度大于地球公转的向心加速度\nC: “天问一号”在地火转移轨道上运动的周期小于地球绕太阳运动的周期\nD: “天问一号”从 $A$ 点运动到 $C$ 点的过程中处于加速状态\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-030.jpg?height=509&width=419&top_left_y=171&top_left_x=356" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_612", "problem": "如图所示, 飞船在地面指挥控制中心的控制下, 由近地点圆形轨道 $A$, 经粗圆轨道 $B$ 转变到远地点的圆轨道 $C$. 轨道 $A$ 与轨道 $B$ 相切于 $P$ 点, 轨道 $B$ 与轨道 $C$ 相切于 $Q$ 点,以下说法正确的是 ( )\n\n[图1]\nA: 卫星在轨道 $B$ 上由 $P$ 向 $Q$ 运动的过程中速率越来越小\nB: 卫星在轨道 $C$ 上经过 $Q$ 点的速率大于在轨道 $A$ 上经过 $P$ 点的速率\nC: 卫星在轨道 $B$ 上经过 $P$ 点的加速度与在轨道 $A$ 上经过 $P$ 点的加速度是相等的\nD: 卫星在轨道 $B$ 上经过 $Q$ 点的速度小于卫星在轨道 $\\mathrm{C}$ 上经过 $\\mathrm{Q}$ 点速度\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 飞船在地面指挥控制中心的控制下, 由近地点圆形轨道 $A$, 经粗圆轨道 $B$ 转变到远地点的圆轨道 $C$. 轨道 $A$ 与轨道 $B$ 相切于 $P$ 点, 轨道 $B$ 与轨道 $C$ 相切于 $Q$ 点,以下说法正确的是 ( )\n\n[图1]\n\nA: 卫星在轨道 $B$ 上由 $P$ 向 $Q$ 运动的过程中速率越来越小\nB: 卫星在轨道 $C$ 上经过 $Q$ 点的速率大于在轨道 $A$ 上经过 $P$ 点的速率\nC: 卫星在轨道 $B$ 上经过 $P$ 点的加速度与在轨道 $A$ 上经过 $P$ 点的加速度是相等的\nD: 卫星在轨道 $B$ 上经过 $Q$ 点的速度小于卫星在轨道 $\\mathrm{C}$ 上经过 $\\mathrm{Q}$ 点速度\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-119.jpg?height=423&width=465&top_left_y=160&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1003", "problem": "An asteroid in a circular orbit around the Sun is at its closest to Earth every 300 days. What is its orbital speed? Assume the Earth's orbit is also circular and that both orbit in the same direction.\nA: $21 \\mathrm{~km} \\mathrm{~s}^{-1}$\nB: $28 \\mathrm{~km} \\mathrm{~s}^{-1}$\nC: $32 \\mathrm{~km} \\mathrm{~s}^{-1}$\nD: $39 \\mathrm{~km} \\mathrm{~s}^{-1}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn asteroid in a circular orbit around the Sun is at its closest to Earth every 300 days. What is its orbital speed? Assume the Earth's orbit is also circular and that both orbit in the same direction.\n\nA: $21 \\mathrm{~km} \\mathrm{~s}^{-1}$\nB: $28 \\mathrm{~km} \\mathrm{~s}^{-1}$\nC: $32 \\mathrm{~km} \\mathrm{~s}^{-1}$\nD: $39 \\mathrm{~km} \\mathrm{~s}^{-1}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_922", "problem": "In Autumn 2022, NASA plans to launch a practice mission in preparation for sending humans back to the Moon. What is the name of the programme this is part of?\nA: Artemis\nB: Athena\nC: Diana\nD: Orion\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn Autumn 2022, NASA plans to launch a practice mission in preparation for sending humans back to the Moon. What is the name of the programme this is part of?\n\nA: Artemis\nB: Athena\nC: Diana\nD: Orion\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_264", "problem": "一球状行星的自转与地球自转的运动情况相似, 此行星的一昼夜为 $a$ 秒, 在星球上的不同位置用弹簧科测量同一物体的重力, 在此星球赤道上称得的重力是在北极处的 $b$倍 ( $b$ 小于 1$)$, 万有引力常量为 $\\mathrm{G}$, 则此行星的平均密度为 $(\\quad)$\nA: $\\frac{3 \\pi}{G a^{2}(1-b)}$\nB: $\\frac{3 \\pi}{G a^{2} b}$\nC: $\\frac{30 \\pi}{G a^{2}(1-b)}$\nD: $\\frac{30 \\pi}{G a^{2} b}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n一球状行星的自转与地球自转的运动情况相似, 此行星的一昼夜为 $a$ 秒, 在星球上的不同位置用弹簧科测量同一物体的重力, 在此星球赤道上称得的重力是在北极处的 $b$倍 ( $b$ 小于 1$)$, 万有引力常量为 $\\mathrm{G}$, 则此行星的平均密度为 $(\\quad)$\n\nA: $\\frac{3 \\pi}{G a^{2}(1-b)}$\nB: $\\frac{3 \\pi}{G a^{2} b}$\nC: $\\frac{30 \\pi}{G a^{2}(1-b)}$\nD: $\\frac{30 \\pi}{G a^{2} b}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_387", "problem": "已知某星球的近地卫星和同步卫星的周期分别为 $T$ 和 $8 T$, 星球半径为 $R$, 引力常量为 $G$, 星球赤道上有一静止的质量为 $m$ 的物体, 若把星球视为一个质量均匀的球体,则下列说法不正确的是（）\nA: 该星球的质量为 $\\frac{4 \\pi^{2} R^{3}}{G T^{2}}$\nB: 该星球的密度为 $\\frac{3 \\pi}{G T^{2}}$\nC: 该星球同步卫星的轨道半径为 $4 R$\nD: 赤道对该物体的支持力大小为 $\\frac{63 \\pi^{2} m R}{64 T^{2}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n已知某星球的近地卫星和同步卫星的周期分别为 $T$ 和 $8 T$, 星球半径为 $R$, 引力常量为 $G$, 星球赤道上有一静止的质量为 $m$ 的物体, 若把星球视为一个质量均匀的球体,则下列说法不正确的是（）\n\nA: 该星球的质量为 $\\frac{4 \\pi^{2} R^{3}}{G T^{2}}$\nB: 该星球的密度为 $\\frac{3 \\pi}{G T^{2}}$\nC: 该星球同步卫星的轨道半径为 $4 R$\nD: 赤道对该物体的支持力大小为 $\\frac{63 \\pi^{2} m R}{64 T^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_394", "problem": "某同学设计了一个滑梯游戏装置, 如图所示, 一光滑轨道 $A O$ 固定在水平桌面上,\n$O$ 点在桌面右侧边缘上。以 $O$ 点为圆心的 $\\frac{1}{4}$ 光滑圆弧轨道 $B D$ 坚直固定在桌子的右侧, $C$ 点为圆弧轨道 $B D$ 的中点。若宇航员利用该游戏装置分别在地球表面和火星表面进行模拟实验, 将小球放在光滑轨道 $A O$ 上某点由静止下滑, 小球越过 $O$ 点后飞出, 落在光滑圆弧轨道 $B D$ 上。忽略空气阻力, 已知地球表面的重力加速度大小为 $\\mathrm{g}$, 火星的质量约为地球质量的 $\\frac{1}{9}$, 火星的半径约为地球半径的 $\\frac{1}{2}$ 。在地球表面或在火星表面上, 下列说法正确的是（ ）\n\n[图1]\nA: 若小球恰能打到 $C$ 点, 则击中 $C$ 点时的速度方向与圆弧面垂直\nB: 小球释放点越低, 小球落到圆弧上时动能就越小\nC: 根据题目的条件可以得出火星表面的重力加速度大小 $g_{\\text {火 }}=\\frac{9}{4} g$\nD: 在地球和火星进行模拟实验时, 若都从光滑轨道上同一位置释放小球, 则小球将落在圆弧上的同一点\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n某同学设计了一个滑梯游戏装置, 如图所示, 一光滑轨道 $A O$ 固定在水平桌面上,\n$O$ 点在桌面右侧边缘上。以 $O$ 点为圆心的 $\\frac{1}{4}$ 光滑圆弧轨道 $B D$ 坚直固定在桌子的右侧, $C$ 点为圆弧轨道 $B D$ 的中点。若宇航员利用该游戏装置分别在地球表面和火星表面进行模拟实验, 将小球放在光滑轨道 $A O$ 上某点由静止下滑, 小球越过 $O$ 点后飞出, 落在光滑圆弧轨道 $B D$ 上。忽略空气阻力, 已知地球表面的重力加速度大小为 $\\mathrm{g}$, 火星的质量约为地球质量的 $\\frac{1}{9}$, 火星的半径约为地球半径的 $\\frac{1}{2}$ 。在地球表面或在火星表面上, 下列说法正确的是（ ）\n\n[图1]\n\nA: 若小球恰能打到 $C$ 点, 则击中 $C$ 点时的速度方向与圆弧面垂直\nB: 小球释放点越低, 小球落到圆弧上时动能就越小\nC: 根据题目的条件可以得出火星表面的重力加速度大小 $g_{\\text {火 }}=\\frac{9}{4} g$\nD: 在地球和火星进行模拟实验时, 若都从光滑轨道上同一位置释放小球, 则小球将落在圆弧上的同一点\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-092.jpg?height=457&width=489&top_left_y=708&top_left_x=338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_299", "problem": "2013 年 5 月 2 日凌晨 0 时 06 分, 我国“中星 11 号”通信卫星发射成功, “中星 11 号”是一颗地球同步卫星, 它主要用于为亚太地区等区域用户提供商业通信服务, 图 2 为发射过程的示意图, 先将卫星发射至近地圆轨道 1 , 速度为 $v_{1}$, 然后经点火, 使其沿粗圆轨道 2 运行, 在椭圆轨道上 $P 、 Q$ 点的速度分别为 $v_{2 \\mathrm{P}}$ 和 $v_{2 \\mathrm{Q}}$, 最后再一次点火, 将卫星送入同步圆轨道 3 , 速度为 $v_{3}$, 轨道 $1 、 2$ 相切于 $Q$ 点, 轨道 $2 、 3$ 相切于 $P$ 点, 则当卫星分别在 1、2、3 轨道上正常运行时, 以下说法正确的是（ ）\n\n[图1]\nA: 四个速率的大小顺序为: $v_{2 \\mathrm{Q}}>v_{2 \\mathrm{P}}v_{2 \\mathrm{P}}\\sqrt{\\frac{2 G M}{3 R}}$\nD: 如果卫星 1 的加速度为 $a$, 卫星 2 在 $P$ 点的加速度为 $a_{p}$, 则 $a\\sqrt{\\frac{2 G M}{3 R}}$\nD: 如果卫星 1 的加速度为 $a$, 卫星 2 在 $P$ 点的加速度为 $a_{p}$, 则 $a1)$, 地球绕太阳运动的周期为 $T_{0}$ 。如图为某时刻火星与地球距离最近时的示意图, 则到火星与地球再次距离最近所需的最短时间为 ( )\n\n[图1]\nA: $\\frac{k^{\\frac{3}{2}}}{k^{\\frac{3}{2}}-1} T_{0}$\nB: $\\frac{k^{\\frac{2}{3}}}{k^{\\frac{2}{3}}-1} T_{0}$\nC: $\\frac{k^{\\frac{3}{2}}+1}{k^{\\frac{3}{2}}} T_{0}$\nD: $\\frac{k^{\\frac{2}{3}}+1}{k^{\\frac{2}{3}}} T_{0}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n中国对火星探测不解追求, 火星与地球距离最近的时刻最适合登陆火星和在地面对火星进行观测。设定火星、地球绕太阳做匀速圆周运动的轨道在同一平面内, 火星绕太\n阳运动的轨道半径是地球绕太阳运动的轨道半径的 $k$ 倍 $(k>1)$, 地球绕太阳运动的周期为 $T_{0}$ 。如图为某时刻火星与地球距离最近时的示意图, 则到火星与地球再次距离最近所需的最短时间为 ( )\n\n[图1]\n\nA: $\\frac{k^{\\frac{3}{2}}}{k^{\\frac{3}{2}}-1} T_{0}$\nB: $\\frac{k^{\\frac{2}{3}}}{k^{\\frac{2}{3}}-1} T_{0}$\nC: $\\frac{k^{\\frac{3}{2}}+1}{k^{\\frac{3}{2}}} T_{0}$\nD: $\\frac{k^{\\frac{2}{3}}+1}{k^{\\frac{2}{3}}} T_{0}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-071.jpg?height=377&width=482&top_left_y=425&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_784", "problem": "What is the correct numerical value and unit of the Boltzmann constant?\nA: $1.38 \\times 10^{-21} \\mathrm{~m}^{3} \\cdot \\mathrm{kg} \\cdot \\mathrm{s}^{-2} \\cdot \\mathrm{K}^{-1}$\nB: $1.38 \\times 10^{-22} \\mathrm{~m}^{2} \\cdot \\mathrm{kg} \\cdot \\mathrm{s}^{-3} \\cdot \\mathrm{K}^{-1}$\nC: $1.38 \\times 10^{-23} \\mathrm{~m}^{2} \\cdot \\mathrm{kg} \\cdot \\mathrm{s}^{-2} \\cdot \\mathrm{K}^{-1}$\nD: $1.38 \\times 10^{-24} \\mathrm{~m}^{2} \\cdot \\mathrm{kg} \\cdot \\mathrm{s}^{-2} \\cdot \\mathrm{K}^{-2}$\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the correct numerical value and unit of the Boltzmann constant?\n\nA: $1.38 \\times 10^{-21} \\mathrm{~m}^{3} \\cdot \\mathrm{kg} \\cdot \\mathrm{s}^{-2} \\cdot \\mathrm{K}^{-1}$\nB: $1.38 \\times 10^{-22} \\mathrm{~m}^{2} \\cdot \\mathrm{kg} \\cdot \\mathrm{s}^{-3} \\cdot \\mathrm{K}^{-1}$\nC: $1.38 \\times 10^{-23} \\mathrm{~m}^{2} \\cdot \\mathrm{kg} \\cdot \\mathrm{s}^{-2} \\cdot \\mathrm{K}^{-1}$\nD: $1.38 \\times 10^{-24} \\mathrm{~m}^{2} \\cdot \\mathrm{kg} \\cdot \\mathrm{s}^{-2} \\cdot \\mathrm{K}^{-2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_219", "problem": "我国天文学家通过 FAST, 在武仙座球状星团 M13 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{l}$, 它们的轨道半径分别为 $\\mathrm{R}_{\\mathrm{A}} 、 \\mathrm{R}_{\\mathrm{B}}, \\mathrm{R}_{\\mathrm{A}}<\\mathrm{R}_{\\mathrm{B}}, C$ 为 $B$ 的卫星, 绕 $B$ 做逆时针匀速圆周运动, 周期为 $T_{2}$, 忽略 $A$ 与 $C$ 之间的引力, 且 $A$ 与 $B$ 之间的引力远大于 $C$ 与 $B$ 之间的引力。万有引力常量为 $G$, 则以下说法正确的是 ( )\n\n[图1]\nA: 若知道 $C$ 的轨道半径, 则可求出 $C$ 的质量\nB: 恒星 $B$ 的质量为 $M_{B}=\\frac{4 \\pi^{2} R_{B}\\left(R_{A}+R_{B}\\right)^{2}}{G T_{1}^{2}}$\nC: 若 $A$ 也有一颗运动周期为 $T_{2}$ 的卫星, 则其轨道半径一定小于 $C$ 的轨道半径\nD: 设 $A 、 B 、 C$ 三星由图示位置到再次共线的时间为 $t$, 则 $t=\\frac{T_{1} T_{2}}{\\left.4 T_{1}+T_{2}\\right)}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n我国天文学家通过 FAST, 在武仙座球状星团 M13 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{l}$, 它们的轨道半径分别为 $\\mathrm{R}_{\\mathrm{A}} 、 \\mathrm{R}_{\\mathrm{B}}, \\mathrm{R}_{\\mathrm{A}}<\\mathrm{R}_{\\mathrm{B}}, C$ 为 $B$ 的卫星, 绕 $B$ 做逆时针匀速圆周运动, 周期为 $T_{2}$, 忽略 $A$ 与 $C$ 之间的引力, 且 $A$ 与 $B$ 之间的引力远大于 $C$ 与 $B$ 之间的引力。万有引力常量为 $G$, 则以下说法正确的是 ( )\n\n[图1]\n\nA: 若知道 $C$ 的轨道半径, 则可求出 $C$ 的质量\nB: 恒星 $B$ 的质量为 $M_{B}=\\frac{4 \\pi^{2} R_{B}\\left(R_{A}+R_{B}\\right)^{2}}{G T_{1}^{2}}$\nC: 若 $A$ 也有一颗运动周期为 $T_{2}$ 的卫星, 则其轨道半径一定小于 $C$ 的轨道半径\nD: 设 $A 、 B 、 C$ 三星由图示位置到再次共线的时间为 $t$, 则 $t=\\frac{T_{1} T_{2}}{\\left.4 T_{1}+T_{2}\\right)}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-011.jpg?height=339&width=394&top_left_y=630&top_left_x=340", "https://cdn.mathpix.com/cropped/2024_04_01_ef01104c57d69d8b0f5ag-012.jpg?height=596&width=597&top_left_y=1558&top_left_x=341" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1128", "problem": "Wolf-Rayet (WR) stars are some of the hottest stars known, with very strong stellar winds causing considerable mass to the be lost to the interstellar medium (ISM). In binary systems between a WR star and a very large $\\mathrm{O}$ or $\\mathrm{B}$ spectral class star, where their strong stellar winds collide can create the conditions for the formation of dust which goes on to enrich the ISM.\n[figure1]\n\nFigure 3: Left: A view of the WR140 binary system taken with the James Webb Space Telescope (JWST) in July 2022, showing clearly at least 17 nested dust shells. Credit: NASA/ESA/CSA/STScI/JPL-Caltech.\n\nRight: A radial plot along the image in the three mid infrared JWST filters used corresponding to 7.7, 15 and\n\n$21 \\mu \\mathrm{m}$, as well as the model of the dust production. The peaks correspond to each shell. Shells 2 and 17 are indicated on the model and the median shell separation is shown by the grey vertical lines. The projected distance is given in arcseconds. Credit: Lau et al. (2022).\n\nThe WR140 system consists of a WR and an O star which produce dust very regularly when the two stars are close together, around periastron. They are in a highly elliptical orbit $(e=0.8993)$ with a period of 2895 days. Once far from the stars, these dust shells move through space at a remarkably constant speed as indicated by the regularity of the shells in the recent image taken with the James Webb Space Telescope (JWST), shown above in Figure 3. An artist's impression of the two stars in the system and the orbit (in the reference frame of the WR star) is shown in Figure 4 below.\n[figure2]\n\nFigure 4: Left: The relative size of the Sun, upper left, compared to the two stars in the system WR140. The\n\nO-type star is $\\sim 30 M_{\\odot}$, while its companion is $\\sim 10 M_{\\odot}$. Credit: NASA/JPL-Caltech.\n\nRight: The projected orbital configuration of WR 140 in the reference frame of the WR star. The red solid region around the periastron passage is where the O star is when dust is being formed. Credit: Lau et al. (2022).a. Take the distance to the system to be $1.64 \\mathrm{kpc}$.\n\nii. Shell 1 was observed by JWST on 27 th July 2022 to be 1.63 \" away from the central stars. It was formed during the last periastron passage of the $\\mathrm{O}$ star, which (as viewed from Earth) took place in December 2016. Taking light travel time into account, in what year was the periastron passage responsible for shell 17?", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nWolf-Rayet (WR) stars are some of the hottest stars known, with very strong stellar winds causing considerable mass to the be lost to the interstellar medium (ISM). In binary systems between a WR star and a very large $\\mathrm{O}$ or $\\mathrm{B}$ spectral class star, where their strong stellar winds collide can create the conditions for the formation of dust which goes on to enrich the ISM.\n[figure1]\n\nFigure 3: Left: A view of the WR140 binary system taken with the James Webb Space Telescope (JWST) in July 2022, showing clearly at least 17 nested dust shells. Credit: NASA/ESA/CSA/STScI/JPL-Caltech.\n\nRight: A radial plot along the image in the three mid infrared JWST filters used corresponding to 7.7, 15 and\n\n$21 \\mu \\mathrm{m}$, as well as the model of the dust production. The peaks correspond to each shell. Shells 2 and 17 are indicated on the model and the median shell separation is shown by the grey vertical lines. The projected distance is given in arcseconds. Credit: Lau et al. (2022).\n\nThe WR140 system consists of a WR and an O star which produce dust very regularly when the two stars are close together, around periastron. They are in a highly elliptical orbit $(e=0.8993)$ with a period of 2895 days. Once far from the stars, these dust shells move through space at a remarkably constant speed as indicated by the regularity of the shells in the recent image taken with the James Webb Space Telescope (JWST), shown above in Figure 3. An artist's impression of the two stars in the system and the orbit (in the reference frame of the WR star) is shown in Figure 4 below.\n[figure2]\n\nFigure 4: Left: The relative size of the Sun, upper left, compared to the two stars in the system WR140. The\n\nO-type star is $\\sim 30 M_{\\odot}$, while its companion is $\\sim 10 M_{\\odot}$. Credit: NASA/JPL-Caltech.\n\nRight: The projected orbital configuration of WR 140 in the reference frame of the WR star. The red solid region around the periastron passage is where the O star is when dust is being formed. Credit: Lau et al. (2022).\n\nproblem:\na. Take the distance to the system to be $1.64 \\mathrm{kpc}$.\n\nii. Shell 1 was observed by JWST on 27 th July 2022 to be 1.63 \" away from the central stars. It was formed during the last periastron passage of the $\\mathrm{O}$ star, which (as viewed from Earth) took place in December 2016. Taking light travel time into account, in what year was the periastron passage responsible for shell 17?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_9bba4f2e5c10ed29bb97g-07.jpg?height=830&width=1508&top_left_y=500&top_left_x=270", "https://cdn.mathpix.com/cropped/2024_03_14_9bba4f2e5c10ed29bb97g-07.jpg?height=530&width=1448&top_left_y=1962&top_left_x=294" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_373", "problem": "北斗卫星导航系统由地球同步静止轨道卫星 $a$ 、与地球自转周期相同的倾斜地球同步轨道卫星 $b$, 以及比它们轨道低一些的轨道卫星 $c$ 组成, 它们均为圆轨道卫星。若轨道卫星 $c$ 与地球同步静止轨道卫星 $a$ 在同一平面内同向旋转, 已知卫星 $c$ 的轨道半径为 $r$, 同步卫星轨道半径为 $4 r$, 地球自转周期为 $T$, 万有引力常量为 $G$, 下列说法正确的是 ( )\n\n[图1]\nA: 卫星 $a$ 的发射速度大于地球第一宇宙速度, 轨道运行速度小于地球第一宇宙速度\nB: 卫星 $a$ 与卫星 $b$ 具有相同的机械能\nC: 地球的质量为 $\\frac{4 \\pi^{2} r^{3}}{G T^{2}}$\nD: 卫星 $a$ 与卫星 $c$ 周期之比为 $8: 1$, 某时刻两者相距最近, 则经过 $\\frac{T}{8}$ 后, 两者再次相距最近\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n北斗卫星导航系统由地球同步静止轨道卫星 $a$ 、与地球自转周期相同的倾斜地球同步轨道卫星 $b$, 以及比它们轨道低一些的轨道卫星 $c$ 组成, 它们均为圆轨道卫星。若轨道卫星 $c$ 与地球同步静止轨道卫星 $a$ 在同一平面内同向旋转, 已知卫星 $c$ 的轨道半径为 $r$, 同步卫星轨道半径为 $4 r$, 地球自转周期为 $T$, 万有引力常量为 $G$, 下列说法正确的是 ( )\n\n[图1]\n\nA: 卫星 $a$ 的发射速度大于地球第一宇宙速度, 轨道运行速度小于地球第一宇宙速度\nB: 卫星 $a$ 与卫星 $b$ 具有相同的机械能\nC: 地球的质量为 $\\frac{4 \\pi^{2} r^{3}}{G T^{2}}$\nD: 卫星 $a$ 与卫星 $c$ 周期之比为 $8: 1$, 某时刻两者相距最近, 则经过 $\\frac{T}{8}$ 后, 两者再次相距最近\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_a9b05ce8eea7b0e40e5eg-065.jpg?height=426&width=748&top_left_y=2294&top_left_x=334" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_479", "problem": "如果你想利用通过同步卫星转发信号的无线电话与对方通话, 你讲完电话后, 至少要等多长时间才能听到对方通话？（已知地球质量 $M=6.0 \\times 10^{24} \\mathrm{~kg}$, 地球半径 $R=6.4 \\times 10^{6} \\mathrm{~m}$, 引力常量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$ )", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n如果你想利用通过同步卫星转发信号的无线电话与对方通话, 你讲完电话后, 至少要等多长时间才能听到对方通话？（已知地球质量 $M=6.0 \\times 10^{24} \\mathrm{~kg}$, 地球半径 $R=6.4 \\times 10^{6} \\mathrm{~m}$, 引力常量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{kg}^{2}$ )\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以s为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_762", "problem": "The picture below shows a very close and well-known galaxy:\n\n[figure1]\n\nWhat is the name of this galaxy?\nA: Pinwheel Galaxy\nB: Whirlpool Galaxy\nC: Bode Galaxy\nD: Triangulum Galaxy\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe picture below shows a very close and well-known galaxy:\n\n[figure1]\n\nWhat is the name of this galaxy?\n\nA: Pinwheel Galaxy\nB: Whirlpool Galaxy\nC: Bode Galaxy\nD: Triangulum Galaxy\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_43eda1938c0e7e74a4cbg-5.jpg?height=549&width=462&top_left_y=1096&top_left_x=817" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_868", "problem": "Ben Chen is an alien living on a system identical to earth, except his planet's obliquity is $0^{\\circ}$. Located at latitude $42.20^{\\circ}$, he wants to observe M52. Due to the open cluster being so dim, Ben needs perfect conditions to observe M52. Due to atmospheric effects, M52 can only be observed above an altitude of $30^{\\circ}$. Additionally, it must be during astronomical twilight (when the Sun is more than $18^{\\circ}$ below the horizon). Of the following dates, which is the earliest after the vernal equinox that Ben can observe the cluster? The coordinates of M52 are approximately $\\alpha=0 \\mathrm{~h}$ and $\\delta=60^{\\circ}$.\nA: April 21st\nB: June 21st\nC: August 21st\nD: October 21st\nE: December 21st\n", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nBen Chen is an alien living on a system identical to earth, except his planet's obliquity is $0^{\\circ}$. Located at latitude $42.20^{\\circ}$, he wants to observe M52. Due to the open cluster being so dim, Ben needs perfect conditions to observe M52. Due to atmospheric effects, M52 can only be observed above an altitude of $30^{\\circ}$. Additionally, it must be during astronomical twilight (when the Sun is more than $18^{\\circ}$ below the horizon). Of the following dates, which is the earliest after the vernal equinox that Ben can observe the cluster? The coordinates of M52 are approximately $\\alpha=0 \\mathrm{~h}$ and $\\delta=60^{\\circ}$.\n\nA: April 21st\nB: June 21st\nC: August 21st\nD: October 21st\nE: December 21st\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "text-only" }, { "id": "Astronomy_696", "problem": "2019 年 11 月 5 日, 第 49 颗北斗导航卫星成功发射, 为 2020 年完成北斗全球组网打下坚实基础。北斗卫星导航系统由不同轨道卫星组成, 其中北斗-IGSO3 卫星的运行轨道为倾斜地球同步轨道, 倾角为 $55.9^{\\circ}$, 高度约为 3.59 万千米; 北斗-M3 卫星运行轨道为中地球轨道, 倾角为 $55.3^{\\circ}$, 高度约为 2.16 万千米。已知地球半径约为 6400 千米,两颗卫星的运行轨道均可视为圆轨道，则下列说法中正确的是（）\n\n[图1]\nA: 北斗-IGSO3 卫星的线速度大于北斗-M3 卫星的线速度\nB: 北斗-IGSO3 卫星的周期大于北斗-M3 卫星的周期\nC: 北斗-IGSO3 卫星连续经过地球非赤道上某处正上方的时间间隔约为 $24 \\mathrm{~h}$\nD: 北斗-IGSO3 卫星与地面上的北京市的距离恒定\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n2019 年 11 月 5 日, 第 49 颗北斗导航卫星成功发射, 为 2020 年完成北斗全球组网打下坚实基础。北斗卫星导航系统由不同轨道卫星组成, 其中北斗-IGSO3 卫星的运行轨道为倾斜地球同步轨道, 倾角为 $55.9^{\\circ}$, 高度约为 3.59 万千米; 北斗-M3 卫星运行轨道为中地球轨道, 倾角为 $55.3^{\\circ}$, 高度约为 2.16 万千米。已知地球半径约为 6400 千米,两颗卫星的运行轨道均可视为圆轨道，则下列说法中正确的是（）\n\n[图1]\n\nA: 北斗-IGSO3 卫星的线速度大于北斗-M3 卫星的线速度\nB: 北斗-IGSO3 卫星的周期大于北斗-M3 卫星的周期\nC: 北斗-IGSO3 卫星连续经过地球非赤道上某处正上方的时间间隔约为 $24 \\mathrm{~h}$\nD: 北斗-IGSO3 卫星与地面上的北京市的距离恒定\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_cbd0a496f6e2fb8d7781g-022.jpg?height=477&width=594&top_left_y=1780&top_left_x=343" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_644", "problem": "如图所示, 某航天器围绕一颗半径为 $R$ 的行星做匀速圆周运动, 其环绕周期为 $T$,经过轨道上 $A$ 点时发出了一束激光, 与行星表面相切于 $B$ 点, 若测得激光束 $A B$ 与轨道半径 $A O$ 夹角为 $\\theta$, 引力常量为 $G$, 不考虑行星的自转, 下列说法正确的是 ( )\n\n[图1]\nA: 行星的质量为 $\\frac{4 \\pi^{2} R^{3}}{G T^{2} \\sin ^{3} \\theta}$\nB: 行星的平均密度为 $\\frac{3 \\pi}{G T^{2} \\sin ^{3} \\theta}$\nC: 行星表面的重力加速度为 $\\frac{4 \\pi^{2} R}{T^{2} \\sin ^{3} \\theta}$\nD: 行星赤道表面随行星自转做匀速圆周运动的线速度为 $\\frac{2 \\pi R}{T \\sin \\theta}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 某航天器围绕一颗半径为 $R$ 的行星做匀速圆周运动, 其环绕周期为 $T$,经过轨道上 $A$ 点时发出了一束激光, 与行星表面相切于 $B$ 点, 若测得激光束 $A B$ 与轨道半径 $A O$ 夹角为 $\\theta$, 引力常量为 $G$, 不考虑行星的自转, 下列说法正确的是 ( )\n\n[图1]\n\nA: 行星的质量为 $\\frac{4 \\pi^{2} R^{3}}{G T^{2} \\sin ^{3} \\theta}$\nB: 行星的平均密度为 $\\frac{3 \\pi}{G T^{2} \\sin ^{3} \\theta}$\nC: 行星表面的重力加速度为 $\\frac{4 \\pi^{2} R}{T^{2} \\sin ^{3} \\theta}$\nD: 行星赤道表面随行星自转做匀速圆周运动的线速度为 $\\frac{2 \\pi R}{T \\sin \\theta}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_86694e5d1e9acbe7af1ag-042.jpg?height=385&width=417&top_left_y=390&top_left_x=337" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" }, { "id": "Astronomy_1089", "problem": "On $21^{\\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history.\n\n## Total Solar Eclipse of 2017 Aug 21\n\n[figure1]\n\nFigure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC.\n\nThe path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse (\"GE\"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question:\n\n- The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\\prime} 48.7^{\\prime \\prime}$ and $16^{\\prime} 03.4^{\\prime \\prime}$, respectively, where the notation $x x^{\\prime} y y . y^{\\prime \\prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$\n- The latitude and longitude of the location of GE are $36^{\\circ} 58.0^{\\prime} \\mathrm{N}$ and $87^{\\circ} 40.3^{\\prime} \\mathrm{W}$, respectively\n- Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\\odot}=695700 \\mathrm{~km}, R_{\\oplus}=$ $6371 \\mathrm{~km}$ and $R_{\\text {Moon }}=1737 \\mathrm{~km}$, and a day to be 24 hours\n- Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \\mathrm{~km}$ and $384400 \\mathrm{~km}$, respectively\n- As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction\n\nFor an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as:\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nThe point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration (\"GD\") was at co-ordinates of $37^{\\circ} 35^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 07^{\\prime} \\mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \\mathrm{~s}$ longer than the value calculated in part $\\mathrm{c}$.\n\n[figure2]\n\nFigure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \\& Google Maps.c. Hence calculate the duration of the eclipse. Give your answer to the nearest $0.1 \\mathrm{~s}$.", "prompt": "You are participating in an international Astronomy competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nOn $21^{\\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history.\n\n## Total Solar Eclipse of 2017 Aug 21\n\n[figure1]\n\nFigure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC.\n\nThe path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse (\"GE\"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question:\n\n- The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\\prime} 48.7^{\\prime \\prime}$ and $16^{\\prime} 03.4^{\\prime \\prime}$, respectively, where the notation $x x^{\\prime} y y . y^{\\prime \\prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$\n- The latitude and longitude of the location of GE are $36^{\\circ} 58.0^{\\prime} \\mathrm{N}$ and $87^{\\circ} 40.3^{\\prime} \\mathrm{W}$, respectively\n- Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\\odot}=695700 \\mathrm{~km}, R_{\\oplus}=$ $6371 \\mathrm{~km}$ and $R_{\\text {Moon }}=1737 \\mathrm{~km}$, and a day to be 24 hours\n- Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \\mathrm{~km}$ and $384400 \\mathrm{~km}$, respectively\n- As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction\n\nFor an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as:\n\n$$\nv^{2}=G M\\left(\\frac{2}{r}-\\frac{1}{a}\\right)\n$$\n\nThe point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration (\"GD\") was at co-ordinates of $37^{\\circ} 35^{\\prime} \\mathrm{N}$ latitude and $89^{\\circ} 07^{\\prime} \\mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \\mathrm{~s}$ longer than the value calculated in part $\\mathrm{c}$.\n\n[figure2]\n\nFigure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \\& Google Maps.\n\nproblem:\nc. Hence calculate the duration of the eclipse. Give your answer to the nearest $0.1 \\mathrm{~s}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-08.jpg?height=1011&width=1014&top_left_y=497&top_left_x=521", "https://cdn.mathpix.com/cropped/2024_03_14_2827c35b7a4e24cd73bcg-09.jpg?height=859&width=1213&top_left_y=924&top_left_x=410" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "EN", "modality": "multi-modal" }, { "id": "Astronomy_346", "problem": "“开普勒 -47 ”, 该系统位于天鹅座内, 距离地球大约 5000 光年, 这一新的系统有一对互相围绕彼此运行的恒星，运行周期为 $T$, 其中一颗大恒星的质量为 $M$ ，另一颗小恒星质量只有大恒星质量的三分之一, 已知引力常量为 $G$, 则下列判断正确的是 ( )\nA: 大恒星与小恒星的角速度之比为 $1: 1\nB: 大恒星与小恒星的向心加速度之比为 1\nC: 大恒星与小恒星相距 $\\sqrt[3]{\\frac{G M T^{2}}{3 \\pi^{2}}}$\nD: 小恒星的轨道半径为 $\\frac{3}{4} \\sqrt[3]{\\frac{G M T^{2}}{3 \\pi^{2}}}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n“开普勒 -47 ”, 该系统位于天鹅座内, 距离地球大约 5000 光年, 这一新的系统有一对互相围绕彼此运行的恒星，运行周期为 $T$, 其中一颗大恒星的质量为 $M$ ，另一颗小恒星质量只有大恒星质量的三分之一, 已知引力常量为 $G$, 则下列判断正确的是 ( )\n\nA: 大恒星与小恒星的角速度之比为 $1: 1\nB: 大恒星与小恒星的向心加速度之比为 1\nC: 大恒星与小恒星相距 $\\sqrt[3]{\\frac{G M T^{2}}{3 \\pi^{2}}}$\nD: 小恒星的轨道半径为 $\\frac{3}{4} \\sqrt[3]{\\frac{G M T^{2}}{3 \\pi^{2}}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_105", "problem": "有一颗中高轨道卫星在赤道上空自西向东绕地球做圆周运动, 其轨道半径为地球同步卫星轨道半径的四分之一。某时刻该卫星正好经过赤道上某建筑物, 已知同步卫星的周期为 $\\mathrm{T}_{0}$ ，则下列说法正确的是（）\nA: 该卫星的周期为 $\\frac{T_{0}}{4}$\nB: 该卫星的向心力为同步卫星的 $\\frac{1}{16}$\nC: 再经 $\\frac{T_{0}}{8}$ 的时间该卫星将再次经过该建筑物\nD: 再经 $\\frac{T_{0}}{7}$ 的时间该卫星将再次经过该建筑物\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n有一颗中高轨道卫星在赤道上空自西向东绕地球做圆周运动, 其轨道半径为地球同步卫星轨道半径的四分之一。某时刻该卫星正好经过赤道上某建筑物, 已知同步卫星的周期为 $\\mathrm{T}_{0}$ ，则下列说法正确的是（）\n\nA: 该卫星的周期为 $\\frac{T_{0}}{4}$\nB: 该卫星的向心力为同步卫星的 $\\frac{1}{16}$\nC: 再经 $\\frac{T_{0}}{8}$ 的时间该卫星将再次经过该建筑物\nD: 再经 $\\frac{T_{0}}{7}$ 的时间该卫星将再次经过该建筑物\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "text-only" }, { "id": "Astronomy_634", "problem": "我国对火星的首次探测任务将于 2020 年正式开始实施, 要实现对火星的形貌、土\n\n壤、环境、大气等的探测, 研究火星上的水冰分布、物理场和内部结构等。现假想为研究火星上有关重力的实验, 在火星的表面附近做一个上拖实验, 将一个小球以某一初速度坚直向上抛出, 测得小球相邻两次经过抛出点上方 $h$ 处的时间间隔为 $t$, 作出 $t^{2}-h$ 图像如图所示, 已知火星的半径为 $R$, 引力常量为 $G$, 则火星的质量为 ( )\n\n[图1]\nA: $\\frac{4 a R^{2}}{G b}$\nB: $\\frac{4 b R^{2}}{G a}$\nC: $\\frac{8 a R^{2}}{G b}$\nD: $\\frac{8 b R^{2}}{G a}$\n", "prompt": "你正在参加一个国际天文竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n我国对火星的首次探测任务将于 2020 年正式开始实施, 要实现对火星的形貌、土\n\n壤、环境、大气等的探测, 研究火星上的水冰分布、物理场和内部结构等。现假想为研究火星上有关重力的实验, 在火星的表面附近做一个上拖实验, 将一个小球以某一初速度坚直向上抛出, 测得小球相邻两次经过抛出点上方 $h$ 处的时间间隔为 $t$, 作出 $t^{2}-h$ 图像如图所示, 已知火星的半径为 $R$, 引力常量为 $G$, 则火星的质量为 ( )\n\n[图1]\n\nA: $\\frac{4 a R^{2}}{G b}$\nB: $\\frac{4 b R^{2}}{G a}$\nC: $\\frac{8 a R^{2}}{G b}$\nD: $\\frac{8 b R^{2}}{G a}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_04_01_6842b9ceb844a90b34c3g-38.jpg?height=391&width=445&top_left_y=176&top_left_x=340" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Astronomy", "language": "ZH", "modality": "multi-modal" } ]