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What is the current status of Pluto? Pluto has been designated a planet in our solar system for years (ever since it was discovered in the last century), but in 2006 it was demoted.
What caused this decision? And is there a chance that it could be reversed?
Edit: well, http://www.dailygalaxy.com/my_weblog/2017/03/nasas-new-horizon-astronomers-declare-pluto-is-a-planet-so-is-jupiters-ocean-moon-europa.html is interesting; this is science, so anything could (potentially) change.
| If you are interested, there is an audio recording of the IAU General Assembly session on the definition of a planet http://www.jodcast.net/archive/200608IAU/
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Seeing cosmic activity now, really means it happens millions/billions of years ago? A Recent report about a cosmic burst 3.8 billion light years away. It is written as though it is happening now. However, my question is, if the event is 3.8 billion light years away, doesn't that mean we are continuously looking at history, or is it possible to detect activity in "realtime" despite the distance?
| From Wikipedia:
The finite speed of light is important in astronomy. Due to the vast distances involved, it can take a very long time for light to travel from its source to Earth. For example, it has taken 13 billion (13×109) years for light to travel to Earth from the faraway galaxies viewed in the Hubble Ultra Deep Field images. Those photographs, taken today, capture images of the galaxies as they appeared 13 billion years ago, when the universe was less than a billion years old. The fact that more distant objects appear to be younger, due to the finite speed of light, allows astronomers to infer the evolution of stars, of galaxies, and of the universe itself.
So this means that you actually see what happened a long time ago if it is very far away.
| {
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How does a spacecraft's orientation get determined What method are employed to determine a spacecraft orientation with respect to a known orientation or plane?
Are gyroscopes reliable enough to make this determination, for example in Voyager spacecraft.
| Most satellites, at least that I know about, use a series (usually three) of small star tracking telescopes. They are oriented in different directions and locate and lock on to known bright stars. Given the angles to the stars relative to the spacecraft body, the spacecraft knows its orientation in space.
| {
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Video of Earth spinning? If the Earth is spinning or rotating at a really fast speed, why haven't we seen any videos from space of it spinning when we get a lot of photos of it?
| You know I have been asking this same question, why aren't there videos of the earth's rotation from space? I've searched and only have found the opposite. Dish network had a station called "dish earth" that was up for about 3 years. Now if you select the 'dish earth' channel it will give you a different channel/station, NASA. What's really interesting is for a good 3 years this station showed a non-rotating earth in space and if you were to record it for a day, speed it up a bit, you will see the sun revolving around the earth and there's no playing around with it! Youtube have a few of these videos posted by people who had "dish earth". There's no mistake about it. In maybe 2 of those videos you will also see the moon passing by as well.
Even some of the videos from the ISS can support a non-rotating earth, look up sunrise or sunset in space on youtube.
Sometimes seeing is believing and all the math equations in the world can't alter reality. If there can be visual evidence of the earth's rotation, we would have had it long ago, there's no doubt in my mind.
| {
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Why does the homogeneity of the universe require inflation? They say inflation must have occured because the universe is very homogeneous. Otherwise, how could one part of the universe reach the same temperature as another when the distance between the parts is more than light could have traveled in the given time?
Why can't this problem be solved without inflation? If each part started with the same temperature to begin with, then they can have the same temperature irrespective of the distance between them. Am I missing something here?
| What Big Banged To Produce The Universe
From : http://universe-life.com/2011/12/10/eotoe-embarrassingly-obvious-theory-of-everything/
A commonsensible conjecture is that Universe Contraction is initiated following the Big-Bang event, as released moving gravitons (energy) start reconverting to mass (gravity) and eventually returning to black holes, steadily leading to the re-formation of The Universe Singularity, simultaneously with the inflation and expansion, i.e. that universal expansion and contraction are going on simultaneously.
Conjectured implications are that the Universe is a product of A Single Universal Black Hole with an extremely brief singularity of ALL the gravitons of the universe, which is feasible and possible and mandated because gravitation is a very weak force due to the small size of the gravitons, the primal mass-energy particles of the universe.
This implies also that when all the mass of the presently expanding universe is consumed by the present black holes, expansion will cease and be replaced with empansion back to THE Single Universal Black Hole.
| {
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How can Voyager 1 escape gravity of moons and planets? I think this one is pretty simple so excuse me for my ignorance. But since most planets in our solar system are very well tied to their orbit around the sun or orbit around their planet (for moons), I was wondering how can a really small spacecraft such as Voyager 1 avoid getting stuck into these orbits and avoid the gravitational force of these huge objects.
It's probably as simple as doing some math, but I imagine that it's because the spacecraft is so small compared to other objects that it makes it quite easy for it to escape gravity.
It's not easy for large objects (like moons) to escape their host planet gravity because they're much bigger, right?
| It is certainly possible to break the gravitational bind of a planets gravity. To do so you permanently you need to achieve escape velocity, which is an energetic statement that says a particle with enough kinetic energy will never 'come back down'. To calculate the associated velocity we equate the gravitational potential of a particle on the surface of a planet of mass $M$ and radius $r$ to the kinetic energy of the particle (mass $m$, velocity $v$):
$$\frac{GMm}{r^2}=\frac{1}{2}mv^2$$ ($G$ is Newton's gravitational constant) and noting we cancel $m$ and rearrange for $v$
$$v=\sqrt{\frac{2GM}{r}}.$$
Wikipedia gives a list of these velocities for various solar system bodies: linky.
Note we don't need to instantly achieve this velocity unless unpowered. With constant acceleration we just need to reach $v$ for the radius $r$ we are away from the body we're trying to escape.
| {
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Is dark matter around the Milky Way spread in a spiral shape (or, in a different shape)? Dark matter doesn't interact with electromagnetic radiation, but it, at least, participates in gravitational interactions as known from the discovery of dark matter. But does dark matter exist in a spiral shape around our galaxy?
| In current cosmological models, the Milky Way resides in a 'halo' of dark matter. Halo is a technical term - in this case, it means a spherically symmetric collection of dark matter. Since dark matter is not self-interacting and does not interact with other matter, it doesn't experience any sort of collisions or friction, and therefore never flattens out into a disk the way normal (baryonic) matter does. So, dark matter does not trace out a disk and does not follow spiral arms.
| {
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Are we going to be able to travel trough space deforming the space-time? I'm not talking about the speed of the spaceship. If we can deform space-time we needn't any type of propulsion. And how can the travel affect to it's pilots? Can they survive?
| The Alcubierre Warp Drive hasn't been proved theoretically impossible; although requiring an amount of energy equivalent to several solar masses appears to be one of the less difficult engineering challenges that would confront anyone attempting to build one.
| {
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If neutrinos travel faster than light, how much lead time would we have over detecting supernovas? In light of the recent story that neutrinos travel faster than photons, I realize the news about this is sensationalistic and many tests still remain, but let's ASSUME neutrinos are eventually proven to travel "60 ns faster than light". If so, how much lead time would they have over light from local supernovas (e.g. SN 1987A) and distant (e.g. SN 2011fe)?
What does the math look like to calculate this?
| I would think that if neutrinos travel faster than light the first thing one would need to know is their velocity.
Yesterday or today the Opera folks announced that they had found a loose cable connection and had calculated that the error it caused was the same as the discrepancy between the expected time of arrival and the time recorded by the experiment. It's all to be confirmed, of course.
Here's a link:
http://www.iol.co.za/scitech/science/news/was-einstein-s-theory-of-relativity-wrong-1.1240964
Use a search engine such as Google to find more.
| {
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Which direction before dawn to look for Comet Lovejoy (C/2011 W3)? Various websites today are reporting with photos and videos of Comet Lovejoy. However, I can't seem to find a definition of which direction to look for it tomorrow morning. I'm in Christchurch, New Zealand (roughly 43°S 173°E).
| According to Starry Night, Comet Lovejoy will rise this morning in the Southeast around 4 a.m. in Christchurch. Not a chance of seeing it here in Canada, alas!
| {
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How do we know the masses of single stars? I have recently read that we can only know the masses of stars in binary systems, because we use Kepler's third law to indirectly measure the mass. However, it is not hard to find measurements for the mass of stars not in binary systems. So how is the mass of these stars determined?
| The Hertzsprung–Russell diagram is the key to determining masses of individual stars. For stars on the main sequence, their properties are essentially determined by their mass. Age and metallicity are also interrelated factors, but of considerably less importance than mass. That is, if you tell me the mass of a star on the main sequence, I can tell you its temperature, luminosity, radius, etc., to reasonably good accuracy. This means that if you are able to measure the luminosity and temperature of a star, I can put it on a Hertzsprung–Russell diagram, and tell you how massive it is. Of course, calibrating this relationship in the first place required measuring the masses of stars directly using stars in binary systems, as you mention.
[Edit: I did not notice that the star you linked to specifically was Arcturus, for which this does not directly apply.] For a giant like Arcturus, masses are often determined in a bit more complicated manner. The Hertzsprung–Russell diagram still provides a guide, in developing models of stellar evolution that can produce the observed patterns of non-main-sequence stars on the HR diagram. As Arcturus is no longer on the main sequence, these stellar evolution models are used to find the mass that produces the combination of temperature, luminosity, and radius observed, from which a mass can be inferred.
| {
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How bright are auroras (aurorae)? Digital cameras are making the recent auroras look magnificent, but what are they like to the naked eye? Are they comparable in surface brightness to the Milky Way?
| Your eye can see them without having doubts about what it is. Is different than clouds and they are as magnificent as in the pictures. Usually the pictures are made with long exposures but not very long, between 2-30 seconds. The photo that I am posting is made with a 4 seconds exposure in a zone with a lot of light pollution. You could see this with the naked eye and it was actually more impressive as your eye has a larger field of view and higher sensibility. You can see the stars near the aurora, but usually aurora outshines them if is directly on top. The moon on the other hand is much brighter than aurora and even if it outshines it I saw full moon and aurora together (pic2).
With the full moon.
EXIF data.
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What is the simplest way to prove that Earth orbits the Sun? Assume you're talking to someone ignorant of the basic facts of astronomy.
How would you prove to them that Earth orbits the Sun? Similarly, how would you prove to them that the Moon orbits Earth?
| I originally had something about the constellations changing in the sky to show that the Earth orbits the sun, but that would still be the case if the Sun orbited the Earth instead. Now that I think about it, there is one thing that conclusively proves that the Earth orbits the sun: parallax. Over the course of one year many of the stars will move relative to each other. At the end of the year they will be back where they started. This is because the Earth moves around in a 2AU diameter circle, so that six months from your first observation, you'll be standing 2AU away from where you were then, and are viewing the stars from a (slightly, but observably) different angle.
To show that the moon orbits the earth you could observe its location at the same time every night, and see that it moves, and is always nearly the same distance from earth. It never goes into a retrograde motion. Assuming the earth is spherical, the only way this could be true is if the moon orbits the earth. You might also take the phases of the moon into account and model the Sun-Earth-Moon system to explain it.
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Do days and months on the Moon have names? On Earth we have various calendars, for example,
Days: Monday, Tuesday, Wednesday, etc., etc.
Months: January, February, March
Does the Moon have names for its "daily" rotations, etc.?
It sounds like a silly question, and I am not sure if I've asked it using the correct terminology. I suppose what I'm trying to ask is; from a viewpoint of someone living on the Moon - does it have "day" names?
| Proabably not, at least not that I've ever heard of. Since no one has ever lived there :), there has never been any sort of calendaring system needed. Even the longest Apollo missions were only there a few days.
I'm sure if there was ever a permanent base (or bases) there, some sort of time keeping system would be devised but it would also make sense to just use the Earth based UTC since that is the cycle the inhabitants would be on. The lunar day-night cycle would be a little too long to adapt to.
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Given a photo of the Moon, taken from Earth, is it possible to calculate the position of the photographer's site? Given a photo of the Moon, taken from Earth, is it possible to calculate the position (Earth longitude and latitude) of the photographer's site?
I am thinking about photos taken with a normal camera lens and not with a telescope, for example photos taken with a 300 mm lens like these one:
http://www.flickr.com/search/?q=100+300+moon&l=cc&ct=0&mt=all&adv=1
I assume the photo shows enough detailed features like recognizable craters and maria.
Is there any software capable of solving the problem?
Thank you.
Alessandro
Addendum:
A diagram of a simplified model of the problem:
R is the radius of the Earth.
D is the (average) distance from the center of the Moon to the center of the Earth.
P is the photographer site on the surface of the Earth.
| At the equator the dark side of the Moon grows down whereas by the poles it grows to either left or right. That's because, when you are at the opposite side of the planet, you are standing upside down. So the direction of the darkness can tell you how far north/south you are.
Regarding east and west.
At any given time, the Moon is supposed to be at a specific location.
If you know exactly how far east/west it is appearing and the exact time then you can deduce your east/west location.
The stars behind the Moon could of course yield that information.
So the answer is.. yes. The question is, to what level of accuracy can that be done with todays devices?
It would be kind of cool to have a device with which you can just take a photo of the moon and from that get your GPS coordinates. It might be useful in wartime when you might have to work without satellites, and when colonizing new planets.
One day, this ability will be here.
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Do nearby gamma ray busts/supernova damage more than just the ozone layer? So we know that many people are putting hard constraints on the galactic habitability zone based on the presence of nearby supernova/gamma ray bursts. But if they only affect the ozone layer, then I doubt that it's as hard of a constraint as many people think it is.
For one thing - there is practically no ozone layer around the planets of red dwarfs (and possibly even low-mass K-stars like Epsilon Eridani and Alpha Centauri B - IMHO, K-stars offer the best prospects for life on other planets.)
With this information, I am wondering wondering about the outcome, particularly in regard to life: would a nearby supernova really do so much damage to planets around those stars?
For instance, would a supernova really cause more damage than, say, the K/T extinction event 65 million years ago? Also, given that much marine life is shielded from UV rays by layers or ocean water, is it really going to cause significant amounts of damage to such life in that environment?
As a side note, maybe this about surviving gamma ray bursts is relevant for complex life too (although this response might be imperfect for now.)
| It really depends on the range. A star going supernova is going to absolutely obliterate any planet that's closely orbiting it. For solely ozone/atmospheric damage, you're going to have to be several light years away. Probably anything within 10 light years is going to suffer a severe extinction event at the very least. 15-20 maybe a touch of atmospheric damage. I vaguely recall 25 ly being the point where you're pretty much safe.
| {
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What are the chances that a deadly asteroid will hit Earth in the next decade? What are the chances that an asteroid that will kill multiple people will hit Earth in the next decade?
| For statistical purposes, the difference between the odds of two people dying and the odds of one person dying are not signficant so, depending on who you reference...
1 in 200,000
1 in 700,000
1 in 40,000 << This seems pretty high, considering that this is about the rate of death by chicken pox. However this also includes comets and other things besides just "meteorites"
1 in 765,000,000 << risk due to major impact
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Why did the asteroid belt between Mars and Jupiter form as it did? I'm curious about why the asteroid belt wasn't pulled by Mars's or Jupiter's gravity or formed into either moons or planets. Why did it form into an asteroid belt instead?
| The original mass of the solar disk at that position in the Solar System is speculated to be about the same as Earth. Due to gravitational perturbations of Jupiter and Mars, the Asteroid Belt was too chaotic to allow a planet to fully form. Instead of relatively gentle collisions, allowing them to accrete, the impacts of planetesimals were highly energetic.
In fact, during the first few million years of formation about 99.9% of the original material was ejected, and we're left with the Asteroid Belt as it is today. Some of the asteroids were far enough out from the sun to accumulate ice, and it is thought that many bombarded the early Earth and that's how the oceans were formed.
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What will we see between the CMB and the current oldest object seen? The cosmic background radiation (CMB) is estimated to be from 13.7 billion years ago (BYA), and very shortly after the big bang compared to that time frame. The oldest coherent objects we've detected are around 500 million years after the big bang, making them 13.2 BYA.
If I understand these stories correctly, that means that we have not yet detected any radiation that can be clearly identified to be emitted from a source between these two times (otherwise NASA would announce an even older discovery). I don't think that that is implying that there's nothing there to see, although it's obviously the case that it is beyond our current technology to see in that window. The severe redshift and small solid angle are two complicating factors, another is that most galaxies were only beginning to for around 13.2 BYA.
Regarding astronomical objects of ages between 13.2 and 13.7 BYA:
*
*What is there to see? Obviously, the 13.7 BYA point has the problem of near homogeneity but what about the times in-between? Are there cosmic structures that we could see there but no where else? Things that predate proto-galaxies? After the CMB, did matter stop radiating? Are there other near homogenous emissions we could look for? Does that make any sense? What is an accurate characterization of that time frame? Does physics make good predictions for astronomers to chase? What discoveries in that time frame are next?
*What telescopes will see them? The record holding earliest protogalaxies I reference are highly red-shifted light and the CMB is $cm$ wavelength. Would new discoveries in this time frame mostly be in-between those two wavelengths? Are there promising telescopes that might open up new areas of study here?
| Well, we have to rely on theory here. The $\lambda$CDM model predicts that there should be lots of hot optically opaque hydrogen forming into the first stars and galaxies. Particular objects of interest will be the Population III stars which formed and died very quickly.
The best eyes on this scene in the near future future will be the JWST and the E-ELT. Both should have the power to see the first galaxy formations but not the wavelength range to reach much further back...
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Is there such a thing as "North" in outerspace? On Earth, North is determined by the magnetic poles of our planet. Is there such a thing as "North" in outerspace? To put it another way, is there any other way for astronauts to navigate besides starcharts? For instance, if an astronauts spaceship were to be placed somewhere (outside of our solar system) in the milkyway galaxy, would there be a way for them to orient themselves?
| On Earth the north and the south are defined by the south and north magnetic poles respectively of the Earth's Magnetic field. In space there is a thing called the Galactic Magnetic field which permeates galaxies, including the Milky Way (http://arxiv.org/abs/astro-ph/0207240). However, the strength of the Galactic magnetic field is much much lower than the Earth's magnetic field. Our compasses wont be able to detect this and be much more influenced by magnetic fields of nearby planets or stars, rather than the Galactic field. The Galactic magnetic field acts on much larger scales. In 1997 one scientist discovered that the supernova remnants (left overs of an exploded star) align them selves to Milky Way's magnetic field. Intergalactic magnetic fields have also been recently discovered but I believe little is known about their structure.
So in short, I think, theoretically there can be a "north" in space because there is a magnetic field present. Whether or not we can make use of it (due to its weak strength) is a different story. There are other coordinate systems that are used e.g the Galactic Coordinate system and Right ascension/declination.
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Can CMEs disturb a planet orbit? Yesterday (March 6), a massive X-class solar flare erupted from the Sun.
I was wondering if this kind of solar flares can affect in some way the orbit of a planet if aimed directly. Let's say, for example, Mercury.
Can this happen at all?
| CMEs can't disrupt planetary orbits. When you consider a collision you should compare the momenta of the colliding masses and while the speed of CMEs is up to two orders of magnitude larger than orbital speeds of the planets, its total mass is more than ten orders of magnitude smaller. Moreover, since CME's wave spreads over a huge area and since planetary radius is at least 4-5 orders of magnitude smaller than the radius of a planetary orbit, only a tiny fraction of a CME actually hits a planet. Consequently, the momentum change is too tiny to produce a noticeable disruption to a planetary orbit.
Here are some numbers: Mercury's mass: 3.3×1023 kg, CME's mass: 1.6×1012kg, Mercury's average orbital velocity: 47.87 km/s, the speed of fastest CMEs: 3200 km/s, approximate ratio of Mercury's cross-section to the area of a sphere enclosing its orbit's perihelion: 7×10-10.
Even assuming that the mass of a CME is concentrated within 1% of the full spherical angle and heads directly for Mercury, we have a collision between a CME of mass
\begin{equation}
100 \cdot 7×10^{-10} \cdot 1.6×10^{12}kg = 112511 kg
\end{equation}
and momentum 3.6×108 kg m/s with a planet whose momentum is 1.58×1028 kg m/s.
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Determining cloud cover from observer to near the horizon Does there exist a Clear Sky Chart with the following enhancements?:
1 - Actual Cloud Cover (Offered Visually and not just Colors with a Legend, Over Time/Past & Predictive)
2 - Simulate/Predict Cloud Cover taking into account the direction from Observer to Observed Object and Angle of view - Close to the horizon (May be helpful to know when you can reasonably start/end tracking something you want to catch that night that is close to the horizon)
The reason I'm curious is:
A) I wonder if it's just not feasible for any/many reasons.
B) It would be great help to know this information.
In general, does anyone know of other Earth Weather, Clear Sky Clocks and Charts or anything else that gives more information?...anything related will be helpful.
EDIT: I would love to find this lecture "You can do better than Clear Sky Chart" mentioned: http://stjornuskodun.blog.is/blog/stjornuskodun/entry/966941/
| It sounds as if you are not aware of the enormous amount of work which has been put into the Clear Sky Charts we have by two extremely generous and knowledgeable gentlemen: Alan Rahill of Environment Canada, who teased their supercomputer into generating the weather maps, and Attilla Danko, who persuaded Alan's maps to generate customized charts for hundreds of individual sites. Perhaps if you would write to them expressing your wishes politely, they might be able to accommodate you.
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Can the Hanbury-Brown and Twiss effect be used to measure the size of composite objects like galaxies? I know that the Hanbury-Brown and Twiss effect can be used to measure the size of stars. Can it also be used to measure the size of galaxies?
| In theory, yes, In practice its a lot trickier...
Within a simple interferometric measurement (1 baseline) You are only measuring the spatial frequencies parallel to that baseline, with a nice symmetric star you can assume a simple physical model of a circle and whatever angular diameter you recover that would be your 'stellar'diameter. With extended sources such as galaxies the physical model you assume for image reconstruction is fairly arbritrary and would therefore take a lot of processing time.
Also several orientations of the baseline (or simultaneous over $>1$ baselines) would be required to find the angular extent in different directions...
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Why don't we have a better telescope than the Hubble Space Telescope? The Hubble Space Telescope (HST) was launched in 1990, more than 20 years ago, but I know that it was supposed to be launched in 1986, 24 years ago. Since it only took 66 years from the fist plane to the first man on the Moon why don't we have a better telescope in space after 24 years?
| A big reason is that through subsidized servicing missions the Hubble has been substantially upgraded over the years. Very few of the original HST instruments remain, and that has resulted in a dramatic expansion of the science gathering capabilities of Hubble over time.
Also, there has been a Hubble successor in the works for quite some time. The JWST is far larger and far more capable than the Hubble in many ways (although it lacks some capabilities the HST has, such as UV and full visible light coverage) but it is a hugely expensive project that is substantially over budget and behind schedule. But once it gets into space people won't be asking this question any more.
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What does velocity dispersion (sigma) reveal about a galaxy? I'm getting hung up on this term. In studying SMBHs, I see that velocity dispersion strongly correlates with mass. Just what is the velocity dispersion? How can the velocity dispersion of the galaxy be expressed in one figure (sigma) if it has to be measured all over the galaxy?
I can imagine the velocity dispersion changes with radius, so "which" point is used? Why exactly is it that higher velocity dispersion is correlated with higher Mass? And is all this referring to the bulge only or to the entire galaxy? So to get velocity dispersion, is "one slit" at the center enough or do you scan across?
I hope someone can answer this without relying heavily on the maths. I am really just trying to understand the concept, because I think that velocity dispersion is real, not just a mathematical construct.
| The concept behind velocity dispersion comes from statistical mechanics, in which you're generally describing gases or fluids. In gases, you characterize the energy of the system by the temperature, which is a measure of the average thermal (random) motion of particles.
Astronomers do the same thing for galaxies. The "velocity dispersion" is one (or many) ways of measuring the effective average 'thermal' velocity of the system, as an indicator of the 'thermal' energy. I put 'thermal' in quotes, because its only analogous to the classical concept of temperature.
In any case, because of something called 'the virial theorem', the average kinetic energy (i.e. the average thermal motion) is directly related to the potential energy of the system. Knowing the velocity dispersion therefore tells you the potential energy of the system, which tells you roughly how massive the galaxy is. That's the primary insight given by the velocity dispersion.
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Vision vs. limiting magnitude Does anyone know how the acuity of your vision translates to a difference in limiting magnitude?
e.g., the kind of answer I'm looking for would be
"For each factor of 2 improvement in your vision (20/80 to 20/40, 20/40 to 20/20, 20/20 to 20/10, etc., your personal limiting magnitude for point sources increases by 1, independent of seeing or sky brightness."
(except I just made those numbers and conditions up out of thin air)
I also implicitly introduced the ansatz of a smooth logarithmic dependence. Yes, no, approximately so?
| There are two causes of limiting magnitude. First is pure sensitivity, where no matter how dark the sky is, one can't see stars fainter than about 6.5. If your vision loses acuity and blurs things out, it may not at first have much effect on limiting magnitude, especially if you use averted vision where things are not seen that sharply anyway. Of course, enough blurring spreads the signal over too many of your retina's rods and cones causing you to no longer be able to identify the star due to 'eyenoise'. If you have light pollution either natural (Moon, twilight) or artificial, things get bad much faster when you blur things and your eye can no longer see the signals because of low contrast.
I know it's not giving you a good answer because the situation can be complex. But you can easily do an experiment yourself. Get drugstore eyeglasses in various strengths (+1, +1.5 diopters, etc.) and look at a part of the sky where the magnitudes are well known, e.g. the bowl of the Little Dipper has stars of magnitude +2, +3, +4, and +5.
Translating to the values on an eyechart can be ambiguous because making out letters is a completely different process than seeing limited magnitudes. I have noticed several different conversion tables while searching. You might want to do an experiment with your own eyes by looking at standard eyecharts with the same glasses you used to estimate the limiting visual magnitudes.
| {
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Has a human ever perished in space? Apollo 13 returned safely. The Challenger was leaving when it exploded. The Columbia was coming back when it burned up, as was that Russian guy who was profiled on National Public Radio (NPR) and that recent book. Has any human ever died in space?
| There have only been 3 recorded deaths that occurred in space (that is, greater than 60 miles above the Earth). The crew of the Russian capsule Soyuz 11, died when their capsule depressurised during preparations for re-entry. It wasn't known they had died until the re-entry capsule was opened on Earth as communications had been lost with the capsule during re-entry. All three crew members died as a result of a loss of pressure in the capsule. These are the only recorded fatalities to have occurred outside of the Earths atmosphere and in space. You can read more about the series of events that lead to the accident here. There have been no deaths outside of the Earths gravitational pull (if this is what you are inferring) though since no humans have ever left it and as Stuart says, it technically extends to the edge of the universe.
| {
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Can extraterrestials detect our messages? We transmitted several messages to the space and listening to space for signs of intelligent life for years (SETI).
Assuming they have at least the same technology we have, could they detect these messages we sent out?
If not, how powerful transmitter is needed to make our word heard, let's say, from 1000 light years? Which wavelength we should use to transmit?
ps: Apparently not so many ET related questions on this site, maybe I asked on the wrong site?
| Here's a somewhat technical document talking about insterstellar beacons. A beacon would act as a "searchlight", sweeping across the sky, so that the time spent on each target star would be rather short. With a beam aperture able to illuminate 1% of the sky and working at 0.5 Hz they calculate that 6.9 GW power will be "visible" as far as 6000 light years. It would need an antenna array with about 5 km diameter, and the receiver dish would need to be about the same size. Some other systems are also discussed.
The message that was sent from the Arecibo telescope was beamed to a globular cluster (M13) that is 25000 ly away. Not only is the power level probably insufficient to get there, but in 25000 years M13 will no longer be in the beam anyway.
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Why can't dark matter be black holes? Since 90 % of matter is what we cannot see, why can't it be black-holes from early on? Is is possible to figure out that there are no black holes in the line of sight of various stars/galaxies we observe?
| Part of the reason is a matter of ratios.
We're not talking about a fraction of the total mass that can be observed. We're talking multiples. There's about 10x as much dark matter in any large galaxy than there is normal matter that can be observed. If this was all in black holes, then there would be 10x as many black holes as there are stars. That would be pretty obvious actually, as those black holes would interact with other matter in the galaxy (in say, nebulae, both cold and hot), and the effects would be quite obvious and catastrophic. For one, it would make life on earth impossible, as there should be around 10 (give or take a few) black holes hovering around our sun at less than interstellar distances (just by the law of averages), probably spewing x-rays and subatomic particles our way. Nevermind the major gravitational effects they would have on the formation of planets. We're pretty sure that's not happening.
Also, the velocity curves of galaxies where velocities of stars is graphed vs distance from the galactic center, demonstrate that the mass of the galaxy is quite evenly distributed. This would not be the case if the proposed black holes were either exceptionally massive (as with the black hole at the center of most or perhaps all galaxies) or even not-quite-so-massive-but-evenly-distributed.
Instead, dark matter seems to completely lack interaction with other matter except by gravity. This effect can be seen in colliding galaxies, where the galaxies' dark matter halos appear to keep moving while the normal, detectable matter gets left behind.
| {
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Limitations in using FLEX as a DMFT solver When using the fluctuating exchange approximation (FLEX) as a dynamical mean field theory (DMFT) solver, Kotliar, et al. (p. 898) suggest that it is only reliable for when the interaction strength, $U$, is less than half the bandwidth. How would one verify this? Also, is there a general technique for establishing this type of limit?
To clarify, DMFT is an approximation to the Anderson impurity model, and FLEX is a perturbative expansion in the interaction strength about the band, low interaction strength limit.
| The criterion you mention is roughly the threshold for the formation of the Coulomb gap in the Hubbard model or the local moment in the Anderson model. It is a common break-down region for many approaches starting from one of the limits (insulator/local moments versus conductor/mixed valence).
For perturbation theory in $U$, see the PRB 36, 675 (1986) by Horvatić et al. and references to and form that paper. A more comprehensive discussion can be found in the
monograph by Hewson. As far as I remember, perturbation in $U$ on the level of self-energy does not give the expected exponential dependence on $U$ for the Kondo temperature.
Unfortunately, I don't know specifics of FLEX method to help you in more detail.
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What is the Holevo-Schumacher-Westmoreland capacity of a Pauli channel? Suppose you are given an $n$-qubit quantum channel defined as $\mathcal{E}(\rho) = \sum_{i} p_i X_i \rho X_i^\dagger$, where $X_i$ denotes an $n$-fold tensor product of Pauli matrices and $\{p_i\}$ is a probability distribution. The Holevo-Schumacher-Westmoreland capacity of the channel is defined by
$$
\chi(\mathcal{E}) = \max_{\{q_j, \rho_j\}} \left[S\left(\sum_j q_j \rho_j\right) -\sum_j q_j S\left(\rho_j\right) \right],
$$
where $S$ denotes the von Neumann entropy of a density matrix (see, for example, http://theory.physics.helsinki.fi/~kvanttilaskenta/Lecture13.pdf). Is it known how to calculate this number as a function of $p_i$ and $n$?
| Finding the HSW capacity is an optimization problem which I believe is moderately tractable. There is an iterative numerical method outlined in this paper of mine ("Capacities of quantum channels and how to find them."). A different, although somewhat similar, method was detailed in the paper "Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures" by Masahito Hayashi, Hiroshi Imai, Keiji Matsumoto, Mary Beth Ruskai and Toshiyuki Shimono. If the number of qubits $n$ is not quite small, however, the high dimensionality of the space is going to keep these techniques from working.
| {
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Uniqueness of supersymmetric heterotic string theory Usually we say there are two types of heterotic strings, namely $E_8\times E_8$ and $Spin(32)/\mathbb{Z}_2$. (Let's forget about non-supersymmetric heterotic strings for now.)
The standard argument goes as follows.
*
*To have a supersymmetric heterotic string theory in 10d, you need to use a chiral CFT with central charge 16, such that its character $Z$ satisfies two conditions:
*
*$Z(-1/\tau)=Z(\tau)$
*$Z(\tau+1)=\exp(2\pi i/3) Z(\tau)$
*Such a chiral CFT, if we use the lattice construction, needs an even self-dual lattice of rank 16.
*There are only two such lattices, corresponding to the two already mentioned above.
We can replace the lattice construction with free fermion construction, and we still get the same result. But mathematically speaking, there might still be a chiral CFT of central charge 16, with the correct property, right? Is it studied anywhere?
| I think that the two solutions are the only modular-invariant chiral CFTs with the right central charge. They have the right transformation law under $\tau\to\tau+1$ and especially (and less trivially) $\tau\to-1/\tau$ where $\tau$ is the complex structure of the world sheet torus. That's needed for a consistent path integral interpretation of the histories and for unitarity when used as a portion of string theory.
| {
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Bogomol'nyi-Prasad-Sommerfield (BPS) states: Mathematical definition What is the proper mathematical definition of BPS states?
In string theory the BPS states correspond either to coherent sheaves or special Lagrangians of Calabi-Yau manifold depending upon the type of string theory considered. but in SUSY quantum field theories in 4d, there are no CYs as far I know (which is very little) and in gravity theories, these corresponds to some Black Holes. So what is the general mathematical definition of BPS states which is independent of the theory in consideration, say a general SUSY Quantum field theories, be it QFT, string theory, Gravity and in any dimension.
| The BPS bound was discovered independently of supersymmetry, but it was then better understood as general feature of the supersymmetry algebra. Look at the original paper by Witten and Olive. BPS states are states which saturate the BPS bound, forming "short" representations an extended supersymmetry algebra. Such representations have special properties, which often can be thought of as consequence of some fraction of the supersymmetry which remains in their presence. The examples you cite are special cases, in all those cases the special property of the objects you mention is a deduced from the requirement that the states involved form a short representation of the supersymmetry algebra.
| {
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Quantum mechanics as classical field theory Can we view the normal, non-relativistic quantum mechanics as a classical fields?
I know, that one can derive the Schrödinger equation from the Lagrangian density
$${\cal L} ~=~ \frac{i\hbar}{2} (\psi^* \dot\psi - \dot\psi^* \psi) - \frac{\hbar^2}{2m}\nabla\psi^* \cdot \nabla \psi - V({\rm r},t) \psi^*\psi$$
and the principle of least action. But I also heard, that this is not the true quantum mechanical picture as one has no probabilistic interpretation.
I hope the answer is not to obvious, but the topic is very hard to Google (as I get always results for QFT :)). So any pointers to the literature are welcome.
| I am no expert in classical fields, but I guess you have no entanglement there, that is, there is no difference with a single particle, but there is a big difference with more than one.
| {
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Constructing a CP map with some decaying property Given some observable $\mathcal O \in \mathcal H$ it is simple to construct a CP (completely positive) map $\Phi:\mathcal{H}\mapsto \mathcal{H}$ that conserves this quantity. All one has to observe is that
$$ \text{Tr}(\mathcal O \, \Phi[\rho]) = \text{Tr}(\Phi^*[\mathcal O] \rho).$$
Therefore, if we impose $\Phi^*[\mathcal O] = \mathcal O$, then $\text{Tr}(\mathcal O \, \Phi[\rho])=\text{Tr}(\mathcal O \rho), \; \forall \rho\in \mathcal H$. That amounts to impose that the Kraus operators of $\Phi^*$ should commute with $\mathcal O$.
I'd like, however, to construct a trace-preserving CP map for which the expectation value of $\mathcal O$ does not increase for any $\rho \in \mathcal H$. More explicitly, given $\mathcal O\in \mathcal H$, I want to construct $\Gamma:\mathcal H \mapsto \mathcal H$ such that
$$ \text{Tr}(\mathcal O\, \Gamma[\rho]) \le \text{Tr}(\mathcal O \rho), \; \forall \rho \in \mathcal H .$$
How would you go about that? Any ideas?
| The formal condition---the correspondent of $\Phi^*[\mathcal{O}]=\mathcal{O}$ in the other case---is $\Gamma^*[\mathcal{O}]\leq \mathcal{O}$. For $\mathcal{O}>0$ (all eigenvalues strictly positive), if one multiplies on the right and on the left by $\mathcal{O}^{-1/2}$, one can see that this condition is equivalent to the map with Kraus operators $\mathcal{O}^{1/2}K_i\mathcal{O}^{-1/2}$ being trace non-increasing.
| {
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Rigorous proof of Bohr-Sommerfeld quantization Bohr-Sommerfeld quantization provides an approximate recipe for recovering the spectrum of a quantum integrable system. Is there a mathematically rigorous explanation why this recipe works? In particular, I suppose it gives an exact description of the large quantum number asymptotics, which should be a theorem.
Also, is there a way to make the recipe more precise by adding corrections of some sort?
| Here's a very basic way to see this easily:
This is Action, search for the "Abbreviated Action", and it has the SI unit of Joule-second.
This is the equation that Planck (and later, Einstein) used:
$$E=nhf$$
for $n=1,2,3...$ and $f$ in frequency, in unit of 1/second).
This means that the Planck's constant has also an unit of Joule-second, therefore, you can interpret $nh$ as the Action of the system (since $n$ is dimensionless, it will maintain the unit, and it is only used to give the "correct" answer, since not every mechanical system has $h$ as the Action, $n$ should be used).
So
$$nh=\int p\,dq$$
which is the Sommerfield rule for quantization.
This is just an intuition of how it works.
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which letter to use for a CFT? In math, one says "let $G$ be a group", "let $A$ be an algebra", ...
For groups, the typical letters are $G$, $H$, $K$, ...
For algebras, the typical letters are $A$, $B$, ...
I want to say things such as "let xxx be a conformal field theory" and "let xxx $\subset$ xxx be a conformal inclusion".
Which letters should I use? What is the usual way people go about this?
Here, I'm mostly thinking about chiral CFTs, but the question is also relevant for full (modular invariant) CFTs.
| Ben-Zvi & Frenkel denote vertex algebras $V$,$W$,... They're using the labels specifically for the spaces of states, but one could also use them to refer the whole package.
Alternately, one sometimes sees all caps abbreviations: $YM_2$, $SYM_{4,G}$,...
There is not to my knowledge any conventional notation for morphisms of field theories.
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Convert state Vectors to Bloch Sphere angles I think this question is a bit low brow for the forum. I want to take a state vector $ \alpha |0\rangle + \beta |1\rangle $ to the two bloch angles. What's the best way? I tried to just factor out the phase from $\alpha$, but then ended up with a divide by zero when trying to compute $\phi$ from $\beta$.
| $\phi$ is the relative phase between $\alpha$ and $\beta$ (so the phase of $\alpha/\beta$). You will only get zero or divide-by-zero when $\alpha=0$ or $\beta=0$. But in that case, $\phi$ is arbitrary. And when $\alpha$ or $\beta$ are close to zero, you are near the poles of the Bloch sphere, and $\phi$ doesn't really matter that much.
| {
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How to write a paper in physics? I really like to do research in physics and like to calculate to see what happen. However, I really find it hard to write a paper, to explain the results I obtained and to put them in order. One of the reasons is the lack of my vocabulary.
*
*How do I write physics well? I think that writing physics is more dependent of an author's taste than writing mathematics is.
*Are there any good reference I can consult when writing?
*Or could you give me advice and tips on writing a paper?
*What do you take into account when you start writing a paper?
*What are your strategies on the process such as structuring the paper, writing a draft, polishing it, etc?
*In addition, it is helpful to give me examples of great writing with the reason why you think it is good.
*Do you have specific recommendations?
| In addition to the Joe's answer, a bunch of good advices is here:
*
*G. M. Whitesides, Whitesides' Group: Writing a Paper (Adv Mat 2004), doi:10.1002/adma.200400767
Its two main points are:
*
*Start writing a draft as soon as you have some results, not - when the research is complete (as the later may never come).
*Write in a way which is the most convenient to the reader, not - the writer.
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Electricity & Magnetism - Is an electric field infinite? The inverse square law for an electric field is:
$$
E = \frac{Q}{4\pi\varepsilon_{0}r^2}
$$
Here: $$\frac{Q}{\varepsilon_{0}}$$
is the source strength of the charge. It is the point charge divided by the vacuum permittivity or electric constant, I would like very much to know what is meant by source strength as I can't find it anywhere on the internet. Coming to the point an electric field is also described as:
$$Ed = \frac{Fd}{Q} = \Delta V$$
This would mean that an electric field can act only over a certain distance. But according to the Inverse Square Law, the denominator is the surface area of a sphere and we can extend this radius to infinity and still have a value for the electric field. Does this mean that any electric field extends to infinity but its intensity diminishes with increasing length? If that is so, then an electric field is capable of applying infinite energy on any charged particle since from the above mentioned equation, if the distance over which the electric field acts is infinite, then the work done on any charged particle by the field is infinite, therefore the energy supplied by an electric field is infinite. This clashes directly with energy-mass conservation laws. Maybe I don't understand this concept properly, I was hoping someone would help me understand this better.
| The Landau Pole is not a problem for QED because at scales much smaller than it (the Planck scale, which is smaller than the Landau pole by 260 orders of magnitude) the (negative) gravitational self-energy of the particle will more than cancel out its electromagnetic self-energy. So string theory is not necessary in this case, just gravity.
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Fermi statistics and Berry phase When the positions of two fermions are exchanged adiabatically in three-dimensional space, we know that the wave function gains a factor of $-1$. Is this related to Berry's phase?
| The answer is yes.
The angular momentum of a (scalar) particle moving in the background of a Dirac magnetic monopole of an odd magnetic charge becomes half integral. (The magnetic charge must be an integer by the Dirac quantization condition.)
Under these conditions, the phase acquired by the wavefunction through a 360 degrees rotation is $-1$. By the spin statistics theorem, the particle must be quantized as a fermion, even though its dynamics is described by bosonic coordinates.
This phase can be obtained as a Berry phase. This principle was used by Witten in his seminal paper "E. Witten, Current algebra, baryons, and quark confinement, Nucl. Phys. B 223 (1983) 433-444", to demonstrate that the Skyrmions are fermions.
Witten obtained the change of sign of the Skyrmion wave function through a Berry phase computation. Here, the magnetic monopole field stems from the Wess-Zumino-Witten term in the Skyrmion Lagrangian.
A clear exposition of Witten's work with the emphasis on the analogy with the motion in the background of a Dirac monopole is given in I.J.R. Atchinson's paper : "Berry phases, magnetic monopoles and Wess Zumino terms, or how the Skyrmion got its spin"
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Did anyone claim that quantum theory meant lasers would never work I've been reading 'How the Hippies saved Physics', which describes a design for a superluminal communication device, of which the crucial part was a laser which duplicated an incoming photon many times. The reason this won't work is what is now known as the no-cloning theorem - a quantum state can't be duplicated in this way. It may appear that a laser can do this, but it can't.
The thing is that I have vague memories of reading that when the laser was first talked about, it was claimed that quantum theory forbade such a device. What I'd like to know is whether there were such claims, and if so were they based on the idea that a laser would duplicate a quantum state.
| Yeah, Neils Bohr and John Von Neumann were skeptics:
Many prominent physicists thought it could not even work, based on their knowledge of physical principles. In quantum mechanics, the uncertainty principle developed by Einstein, says that the energy (and therefore the frequency, by E=hv) of a photon can't be known to great precision in a short time. In masers, photons last for a very short time. Therefore, no less than Neils Bohr and John von Neumann thought it couldn't work, even after it had been created. The solution to this apparent paradox is that, though the photons all have the same frequency and direction, which atoms do the emitting and when remains unknown. The emitting atoms maintain an anonymity that avoids uncertainty >violation.
Read more: Why Was the Laser Light Invented? | eHow.com http://www.ehow.com/about_5250763_laser-light-invented_.html#ixzz1uIC0yArE
| {
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'Applications' of surface tension
What are some common applications, uses, exploitations of the properties of surface tension?
Here is what I mean. A water strider can walk on water, that is a consequence of surface tension. This is a consequence, but it is not human made.
On the other hand, I heard that in the construction of some tents, the upper cover of the tent is the rain protector. It is not really impermeable, but if water is placed on it then the water surface tension does not let the water pass through the fine, small pores of the tent cover. However, if you touch the cover while water is on it, you break the surface tension and water passes through.
I would say that the above fact is a clever use of the effect of surface tension. Are there any other known applications, or interesting experiments regarding the surface or interfacial tension?
| The waterproofing of tents (or of Gore-Tex that maybe more of us will be familar with) is really to do with wetting of the fabric rather than surface tension. Though having said that, the wetting i.e. contact angle, is a play-off between the water-air, fabric-water and fabric-air interfacial tensions, so it is related to surface tension.
Anyhow, back to surface tension. How about making nano-sized motors?
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Angular momentum operator and expectation values I was reading some notes and it says that $\langle L_z^2\rangle=\langle L^2\rangle$ IFF the system is radially symmetric. I can see that in order that the LHS of the statement implies that $\langle L_x^2\rangle=0=\langle L_y^2\rangle$. But I am not sure how to rigorously prove to myself that in order to make those 2 vanish we must have an isotropic system. Could someone please kindly explain this? Many thanks.
| I am assuming you're talking about the expectation values of these operators on hydrogenlike wave functions?
Consider the eigenvalues of $\hat{L_z}$ and $\hat{L^2}$ operating on some eigenstate $|\psi>$:
$L_z|\psi>=\hbar m|\psi>$ and $L^2|\psi>=l(l+1)\hbar^2|\psi>$.
So applying $\hat{L_z}$ twice to get $\hat{L_z^2}$ gives $\hbar^2m^2|\psi>$.
Therefore, $<L_z^2>=<\psi|L_z^2|\psi>=\hbar^2m^2<\psi|\psi>=\hbar^2m^2$.
Similarly, $<L^2>=<\psi|L^2|\psi>=\hbar^2l(l+1)<\psi|\psi>=\hbar^2l(l+1)$
If we equate the two, so $<L_z^2>=<L^2>$, we get $\hbar^2m^2 = \hbar^2l(l+1)$.
Since the allowed values of $m$ are limited by $\pm l$, the only valid solution is the solution where $m=0$ and $l=0$.
This solution, yields the spherical harmonic $Y_0^0(\theta,\phi) = \sqrt{\frac{1}{4\pi}}$ which is spherically symmetric because there is no $\theta$ or $\phi$ dependence.
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A certain regularization and renormalization scheme In a certain lecture of Witten's about some QFT in $1+1$ dimensions, I came across these two statements of regularization and renormalization, which I could not prove,
(1) $\int ^\Lambda \frac{d^2 k}{(2\pi)^2}\frac{1}{k^2 + q_i ^2 \vert \sigma \vert ^2} = - \frac{1}{2\pi} ln \vert q _ i \vert - \frac{1}{2\pi}ln \frac{\vert \sigma\vert}{\mu}$
(..there was an overall $\sum _i q_i$ in the above but I don't think that is germane to the point..)
(2) $\int ^\Lambda \frac{d^2 k}{(2\pi)^2}\frac{1}{k^2 + \vert \sigma \vert ^2} = \frac{1}{2\pi} (ln \frac{\Lambda}{\mu} - ln \frac{\vert \sigma \vert }{\mu} )$
I tried doing dimensional regularization and Pauli-Villar's (motivated by seeing that $\mu$ which looks like an IR cut-off) but nothing helped me reproduce the above equations.
I would glad if someone can help prove these above two equations.
| Let's just look at the integral
$$\int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2+\alpha^2}.$$
The other integrals should follow from this one.
Introduce the Pauli-Villars regulator,
$$\begin{eqnarray*}
\int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2+\alpha^2}
&\rightarrow& \int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2+\alpha^2} - \int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2+\Lambda^2} \\
&=& (\Lambda^2-\alpha^2)\int \frac{d^2k}{(2\pi)^2} \frac{1}{(k^2+\alpha^2)(k^2+\Lambda^2)} \\
&=& (\Lambda^2-\alpha^2)\int_0^1 dx\, \int\frac{d^2k}{(2\pi)^2} \frac{1}{(k^2 + \beta^2)^2} \\
&=& (\Lambda^2-\alpha^2)\int_0^1 dx\, \frac{1}{2} \frac{2\pi}{(2\pi)^2}
\int_0^\infty dk^2\,\frac{1}{(k^2 + \beta^2)^2} \\
&=& (\Lambda^2-\alpha^2) \frac{1}{4\pi}
\int_0^1 dx\, \frac{1}{\beta^2} \\
&=& (\Lambda^2-\alpha^2) \frac{1}{4\pi}
\int_0^1 dx\, \frac{1}{\Lambda^2 - x(\Lambda^2-\alpha^2)} \\
&=& -\frac{1}{2\pi} \ln \frac{|\alpha|}{\Lambda}
\end{eqnarray*}$$
Where we have combined denominators with the Feynman parameter $x$, with the intermediate variable $\beta^2 = \Lambda^2 - x(\Lambda^2-\alpha^2)$.
Of course, this could also be approached with dimensional regularization with the same result.
Addendum: After regularization we must renormalize.
Using the minimal subtraction prescription we find
$$\int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2+\alpha^2}
\rightarrow -\frac{1}{2\pi} \ln \frac{|\alpha|}{\mu},$$
as required.
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Is the environment around an asteroid harsher than in interplanetary space? In the wikipedia article about NEAR Shoemaker it is mentioned that the craft stopped operating under these conditions:
At 7 p.m. EST on February 28, 2001 the last data signals were received from NEAR Shoemaker before it was shut down. A final attempt to communicate with the spacecraft on December 10, 2002 was unsuccessful. This was likely due to the extreme -279 °F (-173 °C, 100 K) conditions the probe experienced while on Eros.[6]
I could understand the lack of sunlight during the Eros night being a contributing factor, but that is not mentioned specifically. Are the conditions near an asteroid much harsher for spacecraft than interplanetary space?
| A spacecraft in space should only lose heat by radiation to space, since it's not in direct contact with anything.
NEAR Shoemaker actually soft-landed on the surface of Eros. After the landing, its contact with the surface meant that it would lose heat by direct conduction, making it more difficult to maintain the temperature necessary for operation.
Conditions near an asteroid probably wouldn't be a problem, except for the lack of sunlight that you mentioned.
Disclaimer: I don't know how much of the spacecraft was in contact with the surface, or how much heat it would have lost through the contact area. This is largely speculation.
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What does the Fermi Energy really signify in a Semiconductor? In understanding the behavior of semiconductors, I'm coming across a description of the Fermi Energy here and at Wikipedia's page (Fermi Energy, Fermi Level). If I understand correctly, the Fermi Level refers to the energy state at which there's a 50% chance of finding an electron. This varies with temperature. The Fermi Energy is the highest occupied energy state of fermions at absolute zero.
I'm a little confused as to the relation of the two terms. Additionally, in a semiconductor, the Fermi Energy falls between the valence band and the conduction band. However, my understanding is that electrons cannot exist between the two bands -- so why isn't the Fermi Energy the top of the valence band?
| The reason for this apparent contradiction is that you have two "separate" quantum effects.
*
*Fermi-Dirac distribution describes the energies of single particles in a system comprising many identical particles that obey the Pauli exclusion principle. Distribution is calculated for potential-free space and is temperature dependant.
*You put electrons into the material, and in the material they feel potential of atomic cores. This potential restrict possible energetic states that are available for electrons, that is it makes bands, where electrons can behave almost freely (according to Fermi-Dirac distribution), but makes energetic states between the bands forbidden.
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The speed of sound is proportional to the square root of absolute temperature. What happens at extremely high temperatures? The speed cannot increase unboundedly of course, so what happens?
| It's a bit misleading to simply say the speed of sound is proportional to $\sqrt{T}$ because life is a bit more complicated than that. You've probably seen http://en.wikipedia.org/wiki/Speed_of_sound and this does indeed say in the introduction that the speed of sound is a function of the square root of the absolute temperature. However if you read on you'll find:
$$c = \sqrt{\frac{P}{\rho}}$$
but for an ideal gas the ratio P/$\rho$ is roughly proportional to temperature hence you get $c \propto \sqrt{T}$.
Imagine doing an experiment in the lab at atmospheric pressure where you raise the temperature and see what effect it has on the speed of sound. In this experiment the pressure, $P$, is constant at one atmosphere, so as you increase the temperature the density falls and the speed of sound does indeed increase. But because the density is falling the mean free path of the air molecules increases, and at some temperature the mean free path becomes comparable with the wavelength of sound. When this happens the air will no longer conduct sound so the speed of sound ceases to be physically meaningful.
You could do a different experiment where you put a known amount of gas into a container of constant volume and then increase the temperature. In this experiment the density is constant, so as you increase the temperature the pressure increases and the speed of sound increases. In this experiment the mean free path is roughly constant so you don't run into the problem with the first experiment. However as the temperature rises the gas molecules will dissociate and then ionise to form a plasma. The speed of sound is then given by http://en.wikipedia.org/wiki/Speed_of_sound#Speed_in_plasma. You can keep increasing the temperature and the speed will indeed carry on increasing until it runs into a relativistic limit.
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Impervious nature of solid matter due to quantum degeneracy pressure On Wikipedia the following statement is made without reference:
Freeman Dyson showed that the imperviousness of solid matter is due to
quantum degeneracy pressure rather than electrostatic repulsion as had
been previously assumed.
Can anyone find the appropriate reference(s)?
| Wikipedia to the rescue!
FJ Dyson and A Lenard: Stability of Matter, Parts I and II (J. Math. Phys., 8, 423-434
(1967); J. Math. Phys., 9, 698-711 (1968) ); FJ Dyson: Ground-State Energy of a Finite
System of Charged Particles (J.Math.Phys. 8, 1538-1545 (1967) )
I found the reference in ref 6 of http://en.wikipedia.org/wiki/Pauli_exclusion_principle
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Coulomb potential in 2D I know that the Coulomb potential is logarithmic is two dimensions, and that (see for instance this paper: http://pil.phys.uniroma1.it/~satlongrange/abstracts/samaj.pdf) a length scale naturally arises:
$$ V(\mathbf{x}) = - \ln \left( \frac{\left| \mathbf{x} \right|}{L} \right) $$
I can't see what's the physical meaning of this length scale, and, most of all, I can't see how this length scale can come up while deriving the 2D Coulomb potential by means of a Fourier transform:
$$ V(\mathbf{x}) = \int \frac{\mathrm{d^2 k}}{\left( 2 \pi \right)^2} \frac{e^{\mathrm{i} \mathbf{k} \cdot \mathbf{x}}}{\left| \mathbf{k} \right|^2} $$
I would appreciate some references where the two-dimensional Fourier transform is carried out explicitly and some insight about the physical meaning of L and how can it arise from the aforementioned integral.
| I don't know if you have solved it or not, but my suggestion is to add a convergence factor $$e^{-a\,r}$$ and get the integral, and then let $a \to 0$ (the same trick with yukawa potential)
You can do this integral by the mathematica.
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How do we recognize hardware used in accelerator physics When I see a new accelerator in real life or on a picture, I always find it interesting to see how many thing I can recognize. In that way, I can also get a small first idea of how the accelerator is working.
Here is a picture, I have taken of LEIR at CERN
Help me to be able to recognize even more stuff, than I can now(I will post a few answers myself)
Suggested answer form:
*
*Title
*Images
*One line description
*Link
| Sextupole magnet
Sextupole magnets are mainly used to correct for chromaticity.
http://en.wikipedia.org/wiki/Sextupole_magnet
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On constancy of cometary orbits how are the comets able to keep to a nearly fixed orbital period, though they lose a certain amount of mass during their perihelion?
| As seen from Kepler's laws the orbit is not dependent of the orbiting object's mass. Also, if you do derivation for the orbit (usually in polar coordinates), coordinates $\theta$ and $r$, are orbiting object's mass independent:
$$ \ddot{r} - r \dot{\theta}^2 = -\frac{GM}{r^2}$$
$$ r \ddot{\theta} + 2 \dot{r} \dot{\theta}$$
If you are looking for more intuitive answer, the conclusions above are true due to the fact that gravitational force is dependent on the mass of the object $F_g \propto m$. Since according to second Newton's law force is proportional to the mass of the object and its acceleration $F = m a$, when you want to find acceleration, the mass cancels out. So as long as you have object's initial velocity, you can predict the orbit.
This is true, as dmckee pointed out, if the other object's (i.e. sun's) position is fixed. In the case $m \ll M$, this is practically satisfied.
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If blue light has a higher energy than red light, why does it scatter more? As $E=hf=\frac{hc}{\lambda}$, blue light - with a smaller wavelength - should have a higher energy. However, it is the case that blue light scatters the most. Why is it that higher energy rays scatter more?
| In general, the scattering of light from some object depends on the how close the wavelength of light is to the size of the object.
To make an analogy, if a tidal wave with a wavelength of several kilometers hits a telegraph pole with a radius of 15 cm it isn't going to scatter very much. On the other hand, waves with a wavelength of a few cm, e.g. generated by you throwing a stone into the water, are going to be strongly scattered.
As you've said in your question, blue light has a smaller wavelength than red light. Assuming you are talking about the sky, the scattering is from particles much smaller than the wavelength of light. That means you'd expect light with the smaller wavelength to be scattered more strongly because it's nearer to the size of the objects doing the scattering.
The formula you quote is for the energy of a photon, but this is not relevant for Rayleigh scattering.
To expand the discussion a bit, when the particle size approaches or exceeds the wavelength of light the difference in the wavelengths disappears. If you look at scattering from e.g. a colloidal suspension with a one micron particle size the scattering is (mostly) wavelength independant.
If "scattering" can be extended to include diffraction you find the wavelength dependance is inverted. Red light is diffracted more strongly than blue light.
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The Planck constant $\hbar$, the angular momentum, and the action Is there anything interesting to say about the fact that the Planck constant $\hbar$, the angular momentum, and the action have the same units or is it a pure coincidence?
| The dimensions of
*
*the Planck constant $\hbar$,
*the action $S$, and
*the angular momentum,
are constrained by the following important facts:
*
*A conjugate pair of two observables is quantum mechanically related to the Planck constant $\hbar$ via a Heisenberg uncertainty relation.
*A conjugate pair of two variables is classically related to the action $S$ via Noether's Theorem, cf. e.g. this Phys.SE post. Listen e.g. to Richard Feynman approximately 50 minutes into this YouTube video.
*The conjugate variable to an angular momentum is an angle (angular position), which is usually treated as dimensionless.
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How can a Photon have a "frequency"? I picture light ray as a composition of photons with an energy equal to the frequency of the light ray according to $E=hf$. Is this the good way to picture this? Although I can solve elementary problems with the formulas, I've never really been comfortable with the idea of an object having or being related to a "frequency". Do I need to learn quantum field theory to really understand this?
| It is the quantum wave field $\psi$ (probability amplitude) for a particle which may have oscillations. So for a wave pulse advancing smoothly in some direction the oscillations inside the pulse correspond to the photon frequency and wavelength meanwhile the general movement of the pulse would be associated with the smooth movement of the particle. quite different. So the photon itself doesn't wiggle with the photon frequency, just the underlying abstract quantum field used to describe where we can find it.
Feynman said wave particle duality and quantum mechanics are very difficult to fully comprehend but we have to somehow get to grips with the dilemmas as the quantum mechanics results are outstandingly correct and accurate.
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Has any permanent magnet motor been proven to run? I have read lots of articles about permanent magnet motors, some of which claim the possibility and other which refute it. Is it possible to have a permanent magnet motor that runs on the magnetic force of permanent magnets?
| Why, here you go, an example of permanent magnet motor - a Curie engine.
Motors that combine permanent magnets and electromagnets are common too.
Thing is magnetic field as a means of storage of energy is subject of the same thermodynamics laws as the rest of the universe. If you want the motor to move, you must change the field. To change the field you must expend energy one way or another. Something like a motor that uses just permanent magnets with no external energy sources is impossible. Now ones that use permanent magnets and little more - like in the example, a heat source - these are common.
The linked engine uses the Curie effect - metals above certain temperature cease to be ferromagnetic - attracted by magnets. And while you could theoretically build such a motor in a way that restores spent heat energy, it will still need to spend more heat to remove ferromagnetic effect from a magnetized piece of metal than from unmagnetized one.
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Should any theory of physics respect the principle of conservation of angular momentum or linear momentum? Is it possible that a theory that can describe the universe at the planck scale can violate things that we now consider fundamental in nature?For example can it violate rotational and translational invariance and subsequently momentum wouldn't be conserved ?Should one consider these invariance principles to be fundamental that we must choose the lagrangian to respect them ?
| In once sense, the quantities you mention get violated all the time in quantum mechanics. Furthermore, it happens at levels that are far less ferocious, inaccessible, and hypothetical (if you can't get there it's hypothetical, by definition) than the Planck scale. Such violations rely on quantum uncertainty and result in various forms and combinations of virtual particles, that is, particles that "sort of" exist if you look at the world using sufficiently short time scales. Virtual particles have real, measurable impacts on everyday physics, e.g. screening or "blurring" of nuclear charge in atoms, which impacts chemistry. But on the average, such violations cannot endure in the face of the absolute conservation laws of physics.
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Is there a point in universe that is observable at present? We know that we can see distant galaxies only billions years before now. We can observe the nearest stars just several years before the present. Something on the Moon can be observed only some seconds in the past.
Continuing this scale, is there an object in the universe that can be observed just now, at present, or at least closer to the present than any other object?
I suppose such object should be located in the brain of the observer, but where in the brain exactly, given that brain has finite dementions.
The question can be formulated differently: where exactly is located the center of the sphere of the cosmological event horizon for a given observer?
| An answer in more or less the same spirit as Adam's:
You need to consider the time involved for the neural impulses to get to your brain and for your brain to make sense of the information. Neural impulses travel considerably slower than light, and you'd expect a number of neural impulse "bounces" to take place for any sort of "making sense" to take place.
Putting some rough numbers in, if the impulses go at $100\textrm{ m}/\textrm{s }$and there are (say) ten $10\textrm{ cm}$ bounces, then it takes your brain $0.01\textrm{ s}$ to "see". You might try and push these numbers, but it's unlikely you'll get anything significantly smaller than $100\textrm{ $\mu$s}$ for the response time.
My answer to your question is then, you see "now" anything close enough to your eyeballs that light reaches it within your brain's response time. With the numbers above it's anything inside a sphere of radius $30$ to $3000\textrm{ km}$, i.e. city-size to continent-size.
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Can every particle be regarded as being a combination of Black holes and White holes? Can the statement be regarded as true? That every particle, or element in the universe can be regarded as a combination of black hole and white hole in variable proportion.
| Over the years there have been suggestions that elementary particles may be black holes. However no-one has ever been able to make this quantitative and I doubt anyone believes it these days. There was some discussion of this in what is the difference between a blackhole and a point particle, and Googling will find you lots of hits on this subject.
I've never heard of the idea that particles may be a combination of a black hole and a white hole. It's hard to see how such an arrangement could be possible.
The term "white hole" is widely used in science fiction, but SF has the wrong idea about them. The equations of general relativity are time symmetric, and if you reverse the direction of time the black hole solution turns into a white hole. However there is no evidence that the time reversed solution has any physical significance, and as the comments above suggest, no-one (in the mainstream physics community) believes that white holes exist.
The nearest we come go to a white hole existing are suggestions that the Big Bang was a white hole. Have a look at John Baez's article on this for more info.
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Gamma Ray Bursts What is the maximum frequency of the Gamma Rays produced during supernovae? And how are these detected by telescopes without getting some serious damage done?
| There is the ground based observatory ( nice picture) Veritas.
VERITAS (Very Energetic Radiation Imaging Telescope Array System) is a major ground-based gamma-ray observatory located at the basecamp of the Fred Lawrence Whipple Observatory in southern Arizona, designed to observe and study very-high-energy (VHE) gamma-rays (energies above ~100 GeV).
Because it is very difficult to produce gamma-rays, the objects that emit them are very interesting to astrophysicists. High-energy gamma rays are associated with exploding stars (supernovae), pulsars , quasars , and black holes rather than with ordinary stars or galaxies.
There is the space based Fermi Large Area Telescope
The LAT is an imaging high-energy gamma-ray telescope covering the energy range from about 20 MeV to more than 300 GeV. Such gamma rays are emitted only in the most extreme conditions, by particles moving very nearly at the speed of light. The LAT's field of view covers about 20% of the sky at any time, and it scans continuously, covering the whole sky every three hours.
Currently the LAT scientific collaboration includes more than 400 scientists and students at more than 90 universities and laboratories in 12 countries. The collaboration has published papers on pulsars, active galactic nuclei, globular clusters, cosmic-ray electrons, gamma-ray bursts, binary stars, supernova remnants, diffuse gamma-ray sources and other subjects.
A recent analysis of 130GeV gammas mentioned in a blog drew my attention to it.
So the energies detected go fairly high and there are means of measuring the energetic photons, developed for the earth bound accelerators.
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Simple projectile motion problem A person wants to throw an object from the top of a tower $9,0m$ high towards a target which is $3,5m$ far from the place where the person is launching the object. Suppose that this object is thrown horizontally.
What I want to find out is:
*
*What initial speed does the object need to have in order to hit the target?
*What is the acceleration of the object one moment before it touches the ground?
I suppose that the answer to the second question is simply $9,81m/s^2$, but I am not sure about it because it looks so simple :D
As far as the first question is concerned, I was thinking about using the formula $y_{MAX} = \frac{v_0^2 + sin^2\theta}{2g}$, with $y_{MAX}$ being $3,5m$ and $\theta$ being $45°$, but the result in this case would be extremely unrealistic ^^
Am I going in the right direction (in both questions)?
| You are right about the second question.
for the first one. Ask yourself how long (time) does it have before it hits the ground (and what is this dependent on)? Some thought will reveal that the time it's in the air has to do only with its motion in the $y$ direction. So, if we only throw the ball horizontally, then the equation for how long with will be in the air comes from
$$y=\frac{1}{2}gt^2\rightarrow t=\sqrt{\frac{2y}{g}}$$
That is how long the object has before it comes in contact with the ground. Now, we also know how far we want the ball to get in the horizontal direction, 3.5 m. What equation has no acceleration in the $x$, and relates velocity, distance, and time?
$$x=v_xt\rightarrow v_x=\frac{x}{t}=x\sqrt{\frac{g}{2y}}=\sqrt{\frac{x^2 g}{2y}}$$
| {
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Lenses (refractor) or mirrors (reflector) telescope? What differentiates, in terms of practical quality, not technical implementation, a refractor from a reflector telescope?
Why would one prefer a refractor over a reflector, when reflectors come with such large diameters at a smaller price?
| First of all, Carson is correct in that refractor lenses climb exponentially in price as the size of the lens grows.
Reflecting telescopes are very useful for astronomy, and it's easy to spend a small amount of money and end up with a hefty tube that can gather a ton of light with very few drawbacks.
On the other hand, a high-quality small to midsize (and thus more transportable) refractor can more-easily perform many other useful duties such as daytime photography, easier viewing of land-based targets, and serving as an all-around travel (or spontaneous use) scope.
Apochromats, which feature some of the best lenses among refractors, have fallen in price considerably in recent years, so it's now possible to obtain a very versatile, top-quality refractor in the 60 to 110mm range. Beyond that size, many of these tasks become much more difficult and require increasingly sophisticated mounting setups -- this is especially true of photography.
Most amateur astronomers I've talked to recommend switching to a reflector at that point, because the primary use of larger scopes at that point becomes astronomy, and it is here that reflectors have a great bang-for-buck.
| {
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Electrostatic Pressure Concept There was a Question bothering me.
I tried solving it But couldn't So I finally went up to my teacher asked him for help . He told me that there was a formula for Electrostatic pressure $\rightarrow$
$$\mbox{Pressure}= \frac{\sigma^2}{2\epsilon_0}$$
And we had just to multiply it to the projected area = $\pi r^2$
When i asked him about the pressure thing he never replied.
So what is it actually.Can someone Derive it/Explain it please.
|
Take the case of 3-D conductor and derive pressure exerted on a plate slightly separated from that conductor.
First try to derive it on your own if not comfortable take help from uploaded picture.
| {
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Rotationally invariant body and principal axis Suppose a rigid body is invariant under a rotation around an axis $\mathsf{A}$ by a given angle $0 \leq \alpha_0 < 2\pi$ (and also every multiple of $\alpha_0$).
Is it true that in this case the axis $\mathsf{A}$ is a principal axis of the rigid body?
If so, how to prove it? Do you have any references for a proof?
| At least as phrased ("by a given angle $\alpha_0$"), axis $\mathsf{A}$ can be trivially shown not to necessarily be a principal axis. If $\alpha_0$ is a multiple of $2\pi$, every rigid body is invariant under a rotation about any axis.
| {
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How does universal inflation fit with the Planck length? If the universe is undergoing inflation, and there is a minimum scale that things can exist at (the Planck length), does that mean that new Planck-sized domains have to be continuously popping into existence? If not, does that mean that the Planck length is constantly changing?
| This Wikipedia article on the Metric expansion of space might provide a better answer, but here goes my understanding, good or bad.
I don't think it makes sense to think that new Plank domains come into existence. After all, the Plank distance is just a unit of measure. The distance between objects in the Universe does grow, so expressing the distance between them will require an ever-increasing number of Plank distances, but nothing new has been created. The Plank length, as a unit of measure, is the same as it was. There was an early time when everything fit in a Plank-distance-sized Universe.
Easier: you measure the diameter of a baloon with a long ruler. As you inflate it, it will take more and more centimeters to express its diameter, but no new space is created.
| {
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Calculating the Uncertainty for an Average Value How would I calculate the uncertainty for the average of this set?
$32.5 \pm 0.1$
$32.0 \pm 0.1$
$32.3 \pm 0.1$
| You can callculate the standart deviation as show in this link http://en.wikipedia.org/wiki/Standard_deviation#Generalizing_from_two_numbers
The standart deviation $\sigma=\sqrt{\frac{\sum_{i=1}^n a_i^2}{n}-\left(\frac{\sum_{i=1}^n a_i}{n}\right)^2}$, where $a_i$ is the $i$-th number in your set and $n$ is the number of numbers you have in your set (in your example $a_1=32.5$, $a_2=32.0$, $a_3=32.3$ so $n=3$)
Using the numbers from your question I got $\sigma \approx 0.20548$
| {
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Why/how does an electron emit a photon when decelerating? I've had two special relativity courses so far but none really gave me a clear description of the process.
| Charged particles are permanently coupled to the electromagnetic field, it's an experimental fact and the very essential feature of charges. As any coupled (compound) system, the system (electron + EMF) has its center of mass variables and the "relative motion" (or "internal") variables. Generally, when you act on one of a constituents of a compound system, you transfer energy to its center of mass and to internal variables, both energies being additive. Just like hitting a ball - you push it as a whole and you make it vibrate "internally" (you excite the "shape dynamics", too).
I think when you act on the electron, the previous state of "relative motion" of the compound system (electron + EMF) gets perturbed and one observes relaxation of this perturbation as electromagnetic waves.
| {
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Gravity on the International Space Station We created a table in my physics class which contained the strength of gravity on different planet and objects in space. At altitude 0 (Earth), the gravitational strength is 100%. On the Moon at altitude 240,000 miles, it's 0.028%. And on the International Space Station at 4,250 miles, the gravitational strength compared to the surface of the earth is 89%.
Here's my question:
Why is the strength of gravity compared to the surface of the Earth 89% even though it appears like the ISS has no gravity since we see astronauts just "floating" around?
| The astronauts are just floating around because they are each in orbit. Because they are at the same altitude as the ISS they are in the same orbit and moving at the same speed as it and so appear weightless.
| {
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Why do magnetic field lines go from North to South? Why magnetic lines comes from north to south out side of the magnet
is any magnetic lines comes from south to north if so in which direction
What is the reason of magnetic lineS
| Making magnetic field lines go from north to south is just a convention. An equally valid convention could have been magnetic field lines going from south to north. The magnetic field lines are a conceptual tool to visualize the magnetic field.
An interesting point about magnetic fields is that they are divergenceless (i.e. $\nabla\cdot\vec{B}=0$). What this means is that the elementary unit that gives a static magnetic field is a dipole (e.g. a bar magnet).
| {
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Is there any way to survive solarwinter like in Sunshine - movie? Is there any way to survive solarwinter like in Sunshine - movie?
Solar winter is where for some reason sun looses its capasity to produce radiation( heat etc.). It doesn't loose everything but some of its radiation energy( say 50 %) That causes earth to cool down causing next "ice age"
| Food could be grown using UV lights, powered by nuclear fission.
We could probably do it.
But it would be the spece equivalent of a human being on a life support machine - all our time and energy would be consumed just with survival, so while humans as a species might survive, our society, culture and science would probably slow down to a crawl, or disappear completely. Most other species would die off, so it would be a pretty dismal future.
| {
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Ideal gas with two kinds of particles, Grand canonical partition function Consider an ideal gas contained in a volume V at temperature T. If all particles are identical the Grand canonical partition function can be calculated using
$$Z_g(V,T,z) := \sum_{N=0}^\infty z^N Z_c(N,V,T)$$
where $z$ is the fugacity, and
$$Z_c(N,V,T):= \frac{1}{N!h^{3N}} \int_\Gamma e^{- \beta H(X)} dX$$
is the canonical partition function.
For identical particles I can compute the Grand canonical partition function. Now I am asked about two different kind of particles with masses $m_2= 2m_1$ and energies of
$$h_1(p) = \frac{p^2}{2m_1}$$
and
$$h_2(p) = \frac{p^2}{2m_2} + \Delta$$
where $\Delta > 0$ is a constant. The fugacitys are given as $z_{1,2}=e^{\beta \mu_{1,2}}$, where $\mu_{1,2}$ are the corresponding chemical potentials. The task is to find the partition function.
I am sorry to admit, but I have absolutly no clue where to start. Since no potential is present the Hamiltionian should be
$$H(X) = \sum_{i=1}^{N_1} \frac{p_i^2}{2m_1} + \sum_{i=1}^{N_2} \frac{p_j^2}{2m_2} + N_2 \Delta$$
where $N_1+N_2=N$ are the numbers of particles of the differen kind. First I don't understand where that $\Delta$ comes from? And I have no idea how to continue. Simply inserting $H$ in $Z_c$ does not give any usefull results.
| Here's my shot at it and my whole thought process so we are checking each other.
If we put in the Hamiltonian into the exponent and talk it through, we get the following:
$$
\int e^{-\beta( \sum_i p^{2}_i/2m_1 +\sum_j p^{2}_j/2m_2)}
$$
but what is the measure of integration? well they are non interacting, so we have to look at each individual particle, of each type, and then sum over all the possible momenta they could have
$$
\int e^{-\beta( \sum_i p^{2}_i/2m_1 +\sum_j p^{2}_j/2m_2)}\Pi_i d^3p_i \Pi_j d^3p_j
$$
where I have ignored anything to do with space recklessly. now my guess for the density of states part is
$$
\frac{d^3p d^3 x}{\hbar^3}=\frac{4\pi V p^2 dp}{\hbar^3}
$$
(there is no spatial dependence, so I just integrated over the whole volume) so really I had
$$
\int e^{-\beta H}\frac{d^{3n}p d^{3n} x}{\hbar^{3n}}
$$
and I need to put that in
$$
\frac{(4\pi)^{N_1 +N_2}V^{N_1+N_2}}{\hbar^{3(N_1+N_2)}} \int_{0}^{\infty} e^{-\beta( \sum_i p^{2}_i/2m_1 +\sum_j p^{2}_j/2m_2)}(\Pi_i p^{2}_i dp_i)( \Pi_j p^{2}_j dp_j)
$$
This is a ton of do-able individual integrals, all multiplied, because the sum in the exponents are multiplied down below. I have been really sloppy about numerical factors (like $e^{-\beta N_2 \Delta}$) and constants, sorry. I hope this helps/is correct.
| {
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Does this relation about direction of particles make sense? Maybe I've just stared at this statement too long and I've missed something obvious. Nevertheless, here's the problem: Landau-Lifshitz vol. 1§16, problem 1.
Consider (classical) collision of two particles in center of mass coordinates. Before the collision, the particles are just traveling towards each other and in CM coordinates then the direction of the velocities of two particles should be opposite to each other, i.e.
$$
\theta_1 = \theta_2 + \pi ,
$$
where $\theta_i$ is the angle between the velocity of particle $i$ and the coordinate axis.
However, in the solution in Landau-Lifshitz, they state that
"In the C system, the corresponding angles are related by $\theta_1 = \pi - \theta_2$."
Is there a mistake in L-L? If not, can you explain me the relation above?
| The difference comes from the picture--- the $\theta_1$ and $\theta_2$ in the original statement are both relative to the positive x-axis, while in the solution $\theta_2$ is the final angle relative to the initial velocity of the corresponding particle, so if the velocity is along the x-axis, $\theta_1$ is the angle relative to the x-axis, and $\theta_2$ is relative to the minus x axis.
| {
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How do we prove the existence of a multiverse? How do we prove that a multiverse exists?
Scientists are talking about our universe not being the only universe, but even if that is true, how can we prove the existence of multiverse? We are being 'confined' in this universe and there is no way we can know what is happening outside, right?
| You don't. That's because Multiverse is not a scientific hypothesis. It speculates that there are numerous universes out there. Those are not observable. They would have each their own set of laws of physics much different than our own. As conceptualized, the Multiverse theory can’t be tested. For this reason, the majority of physics theorists state that Multiverse does not belong to the body of sciences.
One leading physics theorist, Nima Arkani-Hamed, sees it differently. And, he even came up with a Higgs boson mass threshold (140 GeV) that, in his view, would support Multiverse. However, other physicists have rebutted his logic by arguing that given the speculative nature of Multiverse there are no such relevant threshold applicable. For a threshold to be relevant you would have to experimentally prove that a Higgs boson of 140 GeV does confirm the existence of Multiverse. That’s not going to happen.
| {
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Collision of a black hole & a white hole A black hole and white hole experience a direct collision.
What happens? What shall be the result of such a collision?
| In an entirely time symmetric situation, and assuming the universe does not contain any other fields besides the two B&W holes, the only way to determine what is the preferred time direction is by looking which of the two holes have the bigger entropy.
Each hole has an entropy proportional to the square of its mass. if $M_{bh} \gg M_{wh}$, then we can consider the white hole a "temporary fluctuation" in a background of forward-oriented thermodynamic increase (using the convention that black-holes increase their entropy "forward" in time), in the case where $M_{bh} \ll M_{wh}$ is the other way around: The preferred thermodynamic arrow is backward and the black hole is a temporary fluctuation going forward.
In the case where $M_{bh} \approx M_{wh}$ there is no preferred thermodynamic time direction. If thermodynamic time is considered a flow, this would be a fixed point.
| {
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What are electromagnetic fields made of? I am trying to understand electromagnetic fields so I have two question related to them.
*
*What is a electromagnetic field made of? Is it made of photons / virtual photons?
*How about a static electric or magnetic field?
| The most fundamental thing in physics, is the way that we conduct phyiscs. And physics will be as good as we are conducting it.
We conduct physics by using 1) logic and 2) the scientific method.
In that context - a scientific theory is just us Humans, trying to give an accurate description of our unerstanding of our reality and our experiments - by creating mathematical models that best fit that understanding.
If at any point our 1) description, 2) understanding, 3) experiments are flawed, so will our physics.
So, the best model that we have today, is that of a Universe where fields are fundumental and they are not "made of" anything simpler.
That doesn't mean that the Universe is that way, we are just describing, that way.
If the real Universe turns out to agree with our theoretical one then that's good and the more it does the more accurate our predictions about this world would be... And ofcourse if the theory turns out to not describe our reality at all, then our theory is not so usefull.
So fields in this model are not made of anything. But maybe in the future, a model that has fields made of something rather than nothing will prove itself even more accurate at describing our understanding of our reality...
So we don't know what fields are made of, so far in our models they are not made of anything.
| {
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Path traced out by a point While studying uniform circular motion at school, one of my friends asked a question:
"How do I prove that the path traced out by a particle such that an applied force of constant magnitude acts on it perpendicular to its velocity is a circle?" Our physics teacher said it was not exactly a very simple thing to prove.
I really wish to know how one can prove it.Thank you!
| If you don't want or know how to solve a pair of simultaneous differential equations, try this more elementary approach using complex numbers and ordinary time derivatives.
Consider the arbitrary path, with parameter t, in the complex plane:
$r(t)e^{i\theta(t)}$
The "velocity" is the time derivative:
$[\frac{dr}{dt} + ir(t)\frac{d\theta}{dt}] e^{i\theta(t)}$
The "acceleration" is the 2nd time derivative:
$\{[\frac{d^2r}{dt^2} - r(t)(\frac{d\theta}{dt})^2] + i[2(\frac{dr}{dt}\frac{d\theta}{dt}) + r(t)\frac{d^2\theta}{dt^2}]\}e^{i\theta(t)}$
We require that the "acceleration" be orthogonal to the "velocity". In polar representation, this just means that the ratio of the two complex numbers is an imaginary number (multiplication by $i$ is a rotation of 90 degrees in the complex plane).
If you stare at the two derivatives a bit, you see that this condition only holds if $r$ and $\frac{d\theta}{dt}$ are constants:
$r(t) = R$
$\frac{d\theta}{dt} = \omega$
Then the "velocity" is just:
$iR\omega e^{i\omega t}$
and the "acceleration" is just:
$-R\omega^2e^{i\omega t}$
So, the "velocity" and "acceleration" are indeed orthogonal. The path then is just:
$Re^{i\omega t}$
This is just a circle in the complex plane.
Of course, you could have done this with polar coordinates in the x-y plane but polar complex numbers make the derivatives so much easier!
| {
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Does $p=mc$ hold for photons? Known that $E=hf$, $p=hf/c=h/\lambda$, then if $p=mc$, where $m$ is the (relativistic) mass, then $E=mc^2$ follows directly as an algebraic fact. Is this the case?
| According to Special Relativity the relativistic energy for a particle is: $E^2= m^2c^4+p^2c^2$
The invariant quantity under relativistic transformations is the rest mass $m$ of the particle.
For a photon $m=0$
Using some simple algebra it is found $E=pc$ for a photon.
You will see this preserves the frequency and energy relationship.
The error in the question is that momentum $p$ is always related to mass and velocity ($p=mv$ where $c$ is placed in as $v$ for the photon), whereas for a massless particle this does not apply.
| {
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Why isn't it allowed to use a flash when taking pictures in a certain place? When I go to, for example, a museum I try to take some pictures.
Sometimes the museum staffs forbid me to use a flash. Do you know the reason? I don't think it is related to photo-electric effect, right?
| As user9886 explained, the main reason is probably not physical. There are indeed cases where strong flashes can damage pigments that are fairly stable in daylight. I know that some modern documents use rhodopsin based ink that makes it impossible to use an ordinary photocopier to copy them without destroying them. I'm not sure if there are a lot of historically used pigments that are similarly senitive.
The main other reasons are simply:
*
*It might annoy people. Especially in museums with a lot of visitors, you would have flashing all the time. Also, in some places (churches etc.) it might be considered rude or, I don't know, not doing the holiness of the place justice or something.
*Intellectual property. Museums often ban the use of cameras altogether because they don't want anyone to take photos of their pieces. In cases where pictures are allowed, they often forbid flashes and/or tripods, so that you can't take "professional" pictures, but you can still take private pictures for yourself.
*It might also be a bit of "we don't know why, but we've been allways doing it like this". Museums could tend to err on the side of caution when forbidding flashes.
| {
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What's the difference between Fermi Energy and Fermi Level? I'm a bit confused about the difference between these two concepts. According to Wikipedia the Fermi energy and Fermi level are closely related concepts. From my understanding, the Fermi energy is the highest occupied energy level of a system in absolute zero? Is that correct? Then what's the difference between Fermi energy and Fermi level?
| The Fermi energy is as you describe: it is the highest occupied level at absolute zero. The Fermi level is the chemical potential. It is the energy level with 50% chance of being occupied at finite temperature T. The Fermi energy does not depend on temperature; the Fermi level does depend on temperature.
| {
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Non linear QM and wave function collapse I heard that there have been some propositions about describing the collapse of the wave-function by adding non-linear terms, but I couldn't anything in any any textbooks or even articles (probably those propositions never reached a good level of consistency). However, I'd like to read about it. Could someone send me a reference?
| As far as I know, nonlinearities aren't compatible with Lorentz invariance. The overall probability renormalization factor also needs to be rescaled globally, although that might not be a problem if rejecting a probabilistic ontology.
| {
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Practical matter of the Higgs-Mechanism My maybe very naive question is, of what practical importance will the discovery of the Higgs-Mechanism be for our technological advance in the near future?
| I don't think that's a naive question at all, as there's quite a lot to it.
The immediate answer is "no difference". If you, like many of us, read science fiction as an impressionable youngster you're probably vaguely disappointed that warp engines and interdimensional drives haven't been invented yet, and you probably harbour hopes that some new discovery could still make them possible.
The problem is that the Standard Model is an extremely accurate description of the world at low energies. There are undoubtably new discoveries to come, but by definition they'll be at high energies that we don't see in the world around us. This probably means the technologies we'll invent have to be based on Standard Model physics. Any new discoverties aren't likely to have a big impact on technology. I say "probably" because I'm keenly aware of Arthur C. Clarke's first law and it would be a brave man to claim discoveries like the Higgs will never affect everyday technology.
However, discovering the Higgs has required enormous advances in areas like computing, and this is likely to have a big effect on all of us in the near future, especially if cloud computing takes off. After all, remember that the World Wide Web was invented at Cern.
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How long was a day at the creation of Earth? Since the earth is slowing its rotation, and as far as I know, each day is 1 second longer every about 1.5 years, how long was an earth day near the formation of earth (4.5 billion years ago)?
I wouldn't assume to just do 4.5b/1.5 and subtract, because you would think the rate of change is changing itself, as seen here from wikimedia. It is a graphical representation of data from INTERNATIONAL EARTH ROTATION AND REFERENCE SYSTEMS SERVICE. They decide when its time for a leap second (the last one being on Jun 30, 2012) The data can be found here.
|
... each day is 1 second longer every about 1.5 years
That figure is way off.
According to this Scientific American article, the Earth's rotation rate just after the collision that formed the Moon was about once every 6 hours. At that time, the Moon would have been about 25,000 kilometers away. The tidal effect of the Moon is the major reason the day has been lengthening, and the Moon's orbit has been widening.
The collision is believed to have taken place about 4.5 billion years ago, not long after the formation of the proto-Earth.
There are still some open questions about the impact hypothesis (see the linked Wikipedia article), so this is uncertain.
I strongly suspect that the impact would have erased any information about the Earth's rotation rate before the impact. (It might be possible to estimate the pre-collision rotation rate by modelling the initial formation of the Earth; I don't know whether there's been any research in this area.)
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Can a huge gravitational force cause visible distortions on an object In space, would it be possible to have an object generating such a huge gravitational force so it would be possible for an observer (not affected directly by gravitational force and the space time distortion) to see some visual distortions (bending) on another small object placed near it ?
(eg : a building on a very huge planet would have his lower base having a different size than the roof).
We assume that object would not collapse on himself because of the important gravitational force.
| This answer may not fully qualify because it is not seen from afar, but when we observe tides we are basically seeing a huge gravitational force (the moon and sun) causing visible distortions on an object (the ocean).
| {
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Friction at zero temperature? By the fluctuation-dissipation theorem (detailed-balance for Langevin equation), $$\sigma^2 = 2 \gamma k_B T$$ where $\sigma$ is the variance of noise, $\gamma$ is a friction coefficient, $k_B$ is Boltzmann's constant, and $T$ is temperature. So in principle, one can have $\gamma\neq 0$ while $T=0$ and $\sigma=0$.
Is it indeed possible to experimentally achieve a system whose temperature and noise approach zero, but whose friction coefficient $\gamma$ does not approach zero?
*
*If yes, what would be an example of such a system? What is the physical source of friction for such a system?
*If not, why not? Is there some sort of "quantum" correction to the fluctuation-dissipation theorem that rules out such zero-noise, non-zero friction systems?
| A simple approximation of your question, from the semiclassical point of view, could be this.
Imagine a ball running over a surface. This surface is made up of other little balls: these balls are considered little respect to the one running over them so they create a flat surface. Imagine that these little balls start to moving up an down: the surface won't be flat and we call what is experienced by the big ball in this way "friction".
So let the little balls be independent quantum harmonic oscillators. You find out that the energy of one of these is $E=h\nu (n+1/2)$. For the ground state, $T=0K$ and $n=0$ and $E=h\nu/2$. So it's not zero.
So at $T=0K$ there is still a remaining movement of the atoms of the lattice that constitute the surface and so they can cause friction in the sense said before.
| {
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Magnetic field inside a charged stream Outside a narrow charged stream (say, a beam of ions or electrons) is the same as observing a current through a conducting wire - there is a circular magnetic field around it.
What would happen inside a charged stream (for example, inside a conducting wire or inside a solar flare)? I have a feeling that symmetry will rule that there is no magnetic field, but I am not sure.
| The field is linearly proportional to r inside the stream (if the current density is uniform inside the beam) and falls off as 1/r outside. This is a simple application of Ampere's law. This leads to an attractive force compressing the beam, but always less than the electrostatic repulsive force pushing the beam out. The two balance when the beam is moving at the speed of light.
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If a superconductor has zero resistance, does it have infinite amperage? If amps = volts / ohms, and ohms is 0, then what is x volts / 0 ohms?
| the amperage is current flowing through the superconductor. All superconductor have a critical magnetic field they can counter before the superconducting phase breaks. This critical magnetic field also implies a critical electrical current, because all current will generate an associated trasverse magnetic field.
What happens to the omhic law in a superconductor is that a non-zero (but below critical) current can be sustained indefinitely even in zero voltage, so the ohmic law relationship becomes an indefinite $\frac{0}{0}$ expression, which can be understood as a breakdown of the dependence between the quantities
| {
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Sound frequency of dropping bomb Everyone has seen cartoons of bombs being dropped, accompanied by a whistling sound as they drop. This sound gets lower in frequency as the bomb nears the ground.
I've been lucky enough to not be near falling bombs, but I assume this sound is based on reality.
Why does the frequency drop? Or does it only drop when it is fallling at an oblique angle away from you, and is produced by doppler shift?
I would have thought that most bombs would fall pretty much straight down (after decelerating horizontally), and therefore they would always be coming slightly closer to me (if I'm on the ground), and thus the frequency should increase..
| In my opinion I think the whistling sounds come from a manufacturing defect or an engineering defect in the shape or dynamic balance of any rapidly moving object through the air. Eddy current caused behind the moving object can set up as standing waves creating a whistling noise, such eddies are used in the form of flow meters to measure the flow of a moving fluid through a pipe. If you notice the design of a simple bullet it has a spritzer nose for supersonic speed and when it slows the boat tail ass means better laminar flow. The creation of any noise at all means loss of energy. There also exists the possibility of psychological impact but since destruction is the primary role the sounds must usually be unwanted artifacts related to construction or design. Anything moving quickly through air will generate turbulence over 30 miles per hour and there seems no way to stop it. But such sounds are usually noise like hissing and not pure tones. Doppler or spin rate may cause frequency changes.
| {
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Will adding heat to a material increase or decrease entropy? Does adding heat to a material, thereby increasing electrical resistance in the material increase or decrease entropy?
Follow up questions:
Is there a situation were Heat flux ie. thermal flux, will change entropy?
Does increasing resistance to em transfer prevent work from being done?
| The resistivity of some materials increases with temperature and with others it decreases. The way we manufacture zero temperature coefficient resistors (stable resistors) is by balancing these effects. But ignoring that, your question is also about entropy, does it always increase with temperature?
An assumption of statistical mechanics (see Callen, page 28) is that "The entropy is continuous and differentiable and is a monotonically increasing function of the energy." Thus if rising energy always causes a rise in temperature, your statement will be true. Such a material would have a negative heat capacity. Surprisingly, there are a lot of hits for "negative heat capacity" in google, and so I suppose one of these examples will be a contradiction to the assumption that entropy always increases with temperature.
The easiest one to explain has to do with black holes. With black holes, the entropy is proportional to the surface area as was famously discovered by Hawking and others. On the other hand, the temperature decreases as the black hole becomes larger. Consequently, raising the temperature of the black hole (which is done by making it smaller), causes a decrease in the system's entropy (as the area of the black hole gets smaller).
| {
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Non-Newtonian Fluid Stop a Bullet? I just saw a YouTube video about Non-Newtonian fluids where people could actually walk on the surface of the fluid but if they stood still, they'd sink. Cool stuff.
Now, I'm wondering: Could a pool of Non-Newtonian fluid stop a bullet? Why or why not?
If so, if you put this stuff inside of a vest, it would make an effective bullet-proof vest, wouldn't it?
| Actually non-neutonian fluids are a bit heavy and massive with regard to their bullet stopping efficiency. They could work with other approaches.
This does not mean their tension surface is infinite. They are penetrable and a bullet applies huge surface tension
| {
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What's an efficient way to produce graphite on TEM-Grids? I am trying to produce graphene with few layers(<10) on a TEM-Grid. Until now I've been trying this with the scotch-tape-method with slight modifications. Unfortunately it requires a lot of time und there are often TEM-Grids without any flakes of the required thinness.
Is there a more efficient way to place graphene flakes having a size above $$ 150 \mu m * 150 \mu m $$
on a TEM-Grids? Is there a better quality of graphite blocks on the market than the SPI-1 quality?
| There are, in fact, a wide variety of techniques for producing graphene other than the scotch-tape method. A very good review of these techniques can be found in this recent review article:
http://onlinelibrary.wiley.com/doi/10.1002/adma.201202321/abstract
It is extremely difficult to obtain the dimensions you require using the scotch-tape method. In my former group, even the student who was the best (in our group obviously) at using the scotch-tape method could go as far as 20-30 $\mu m$ long flakes. I am not aware of the world record for graphene flake sizes using this method. Our group, very soon, switched to the CVD method to produce large area graphene; we were successfully able to produce graphene on TEM grids using this method.
Since I don't know what your resources are, you may want to go through the above review article to find the method that is most convenient for you. My personal favorite is the layer-by-layer removal of graphene described in section 6.1 of the paper. I consider this method to be the chemical analogue of the scotch-tape method; except here you can precisely control the number of graphene layers!
| {
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Atomic structure and corresponding superpartner behavior If all quantum particles have a superpartner, what happens, if this has been able to be speculated based on theory, to the superpartners when the corresponding partners start forming atoms?
*
*Is there an S-hydrogen, S-helium, etc.?
*Or are superpartners free particles that cannot form structure?
*Are there spatial interactions; folds/intersections in supersymmetric space that form S-matter?
(edit-clarification)
Alright, maybe the idea that when matter forms, s-matter forms is fundamentally incorrect. Perhaps I put that out there first, even though I figured it to be false, to be falsified first.
However, that aside, these particles do exist (given assumption) and there must be some kind of interaction, lack of interaction, as in, these particles must be doing something. They can't just provide mass for these equations to work...even though that's why they were theorized in the first place, is that correct?
There must be physical ramifications of these particles existing. Have we had any insights into the behavior of these particles?
Do they always exist, do they decay, what is their state in their hidden dimension of space?
Does supersymmetry breaking have behavioral ramifications other than just having unequal mass?
Can we at all determine the composition of this hidden spatial dimension?
| Firstly the correspondence happens at the level of fundamental particles. Quarks imply squarks, leptons imply sleptons and so on. Further, the partners of fermions are bosons and vice versa, so their interaction don't mirror their partners and you should not be thinking about s-nuclons or s-Hydrogen.
Further, it's not that this electron has a particular selectron matched with it. It is that the category "electrons" is matched with corresponding category "selectons". So what happens when the electrons combine has no bearing on the fate of the sleptons. They have to make their own way.
| {
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Smooth trajectory on a smooth manifold Physicists talk about a smooth trajectory of a particle on a smooth manifold and they label it as $q(t)$ where $q_1(t)....q_n(t)$ are component functions coming from the homeomorphism.
I don't see how we can meaningfully talk about the whole trajectory this way as it might happen that the particle moves from one point to another point on the manifold and it might happen that the coordinate system at this point is different from that of previous point.
How do we ensure that these two patch up and we can meaningfully talk about $q(t)$ independent of the location of particle?
| Is there some specific problem you have which makes this a physics question and not just a mathematics question?
You essentially ask how we can state a parametrized curve, a one-dimensional submanifold, a smooth function from $t$ to $\mathcal M$. Clearly, if you use an atlas without global coordinates, but a set of coordinates on their respective patches, then you have to state the curve in all the coordinates. If the expressions $q_1(t),\dots,q_n(t)$ denote specific functional dependencies, then they are of course only valid in one coordinate system. In the spaces where the patches intersect, you know how to translate from one to the other by the definition of a manifold. If that system is not covering the whole space, then this is not the full information, but the full information should be available by solving the equations for the mechanical system. If the curve is given by a integral curve of some vector field and you have to compute it by say the Hamiltonian equations of motion, then the tangent vector is determined by the equations at every point and so in every patch. Say you want to compute where the trajectory has to go step by step. Then if you go along and the patch comes to an end, then you'll have to transform to the next patch, transform the equations with you and follow the same procedure againt.
| {
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Does gravity slow the expansion of the universe? Does gravity slow the expansion of the universe?
I read through the thread http://www.physicsforums.com/showthread.php?t=322633 and I have the same question. I know that the universe is not being stopped by gravity, but is the force of gravity slowing it down in any way? Without the force of gravity, would space expand faster?
Help me formulate this question better if you know what I am asking.
| The Friedmann equations for the expansion of space are (assuming flat space for simplicity):
$(1)\ (\frac{\dot a}{a})^2 = \frac{8 \pi G \rho + \Lambda}{3}$
$(2)\ \frac{\ddot a}{a}= -\frac{4 \pi G}{3}(\rho + 3P) + \frac{\Lambda}{3}$
where $a$ is the scale factor (roughly, how "expanded" space is), $\dot a$ is the rate of expansion and $\ddot a$ is the acceleration of the expansion.
If, "without the force of gravity", you mean "with $G = 0$", then we have:
$(3)\ (\frac{\dot a}{a})^2 = \frac{\Lambda}{3} \rightarrow a(t) = a(0)e^{\pm t \sqrt{\frac{\Lambda}{3}}}$
$(4)\ \frac{\ddot a}{a}= \frac{\Lambda}{3}$
So, "without gravity" in this particular sense, with $G = 0$, space is either expanding or contracting exponentially with time (for the special case of $\Lambda = 0$ , $\dot a = \ddot a = 0$)
Now, in the context of your question about an expanding universe, by inspection of equation (2), see that introducing "gravity" via giving $G$ a positive value (and, of course, assuming there is a non-zero mass density), this term "opposes" the cosmological constant term and can even reverse the acceleration of the expansion of space by making $\ddot a$ negative thus slowing the expansion.
| {
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Where do the terms microcanonical, canonical and grand canonical (ensemble) come from? Where do the terms microcanonical, canonical and grand canonical (ensemble) come from?
When were they coined and by whom? Is there any reason for the names or are they historical accidents?
| I do not know where they come from, but in french a canonical form is an expression which appears "naturally". For instance the canonical basis for the linear space spanned by the second degree polynomial is made of: $\{1,X,X^2\}$
| {
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Introduction to differential forms in thermodynamics I've studied differential geometry just enough to be confident with differential forms. Now I want to see application of this formalism in thermodynamics.
I'm looking for a small reference, to learn familiar concepts of (equilibrium?) thermodynamics formulated through differential forms.
Once again, it shouldn't be a complete book, a chapter at max, or an article.
UPD Although I've accepted David's answer, have a look at the Nick's one and my comment on it.
| There are two articles by S.G. Rajeev: Quantization of Contact Manifolds and Thermodynamics and A Hamilton-Jacobi Formalism for Thermodynamics in which he reviews the formulation of thermodynamics in terms of contact geometry and explains a number of examples such as van der Waals gases and the thermodynamics of black holes in this picture.
Contact geometry is intended primarily to applications of mechanical systems with time varying Hamiltonians by adding time to the phase space coordinates. The dimension of contact manifolds is thus odd. Contact geometry is formulated in terms of a basic one form, the contact one form:
$$ \alpha = dq^0 -p_i dq^i$$
($q^0$ is the time coordinate). The key observation in Rajeev's formulation is that one can identify the contact structure with the first law:
$$ \alpha = dU -TdS + PdV$$
| {
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How do mirrors work? My physics professor explained to me that electromagnetic waves are consisted of two components - electric and magnetic - which cause each other.
*
*Which part of the mirror actually reflects the wave?
*Which of those two wave components? Both?
*How come the wave doesn't get heavily distorted in the process?
I guess the actual electrons of atoms of silver play a role, but why isn't every material reflective, then? Because is isn't "perfectly" flat? If I lined up atoms of a non-metal element in a perfect plane (maybe several rows), would it reflect light just as mirrors do?
| The reflection could be viewed as a two step process. The incident wave causes the electrons in the silver to vibrate like in an antenna. Though by vibrating they also emit the same light. So it's the electrons at the surface of the silver that reflect the incoming wave. As you mentioned the wave is part electric and part magnetic, but these cannot be taken apart since they are each others cause and effect: without one the other wouldn't be there either, and therefore it must reflect both parts.
That silver (and all metals) don't distort is due to the fact that they are also very good conductors. This prevents the electromagnetic waves from entering the object. The boundary conditions which must hold (from being an conductor) result in the perfect reflection and that the resulting angle is equal to the incident angle.
Similar boundary conditions are there for non-conducting materials like plastic and glass. These similar conditions result in reflection of glass and the shine/reflection on other smooth surfaces (though there can be other causes too). Also Snell's law would follow from these boundary conditions.
In contrast to conducting materials it is possible for electromagnetic waves to enter non-conducting objects. As a consequence part of the incoming wave is transmitted into the material. The propagation or dampening of the wave through the material is largely dependent on the properties of the material. Some materials like glass hardly dampen the wave and you can see through them, while others like most plastics dampen them and thus are opaque.
| {
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Which universe had a beginning? The universe or the observable universe? When we say the universe had a beginning, do we mean the entire universe or the observable universe? Or did both of them have a beginning?
| You should never distinguish between the universe and the observable universe. This is a fallacy of assuming that the word "exist" means something more than "something we can measure and interact with". There is no accepted or concievable way to explore other universe, so one must be very careful when talking about their existence. It is usually just a figure of speech.
Physics operates based on a philosophy called positivism, which is operational--- in order to ask a question you should give a prescription for how it is to be answered experimentally. If you can't, it isn't part of physics, at least not yet. In this case, it doesn't seem likely that there is any operational definition to this question.
The model of the universe that one uses should be bounded by the cosmological horizon, and this horizon came from an inflating small-size deSitter horizon (this is the inflation theory). The start of inflation is shrouded in mystery for now, but any attempt to extend the concept of the universe outside the cosmological horizon is at best unobservable, and most likely incompatible with the quantum gravitational holographic principle, which asks that the spacetime have a description along the horizon boundary.
| {
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Hamiltonian and the space-time structure I'm reading Arnold's "Mathematical Methods of Classical Mechanics" but I failed to find rigorous development for the allowed forms of Hamiltonian.
Space-time structure dictates the form of Hamiltonian. Indeed, we know how the free particle should move in inertial frame of references (straight line) so Hamiltonian should respect this.
I know how the form of the free particle Lagrangian can be derived from Galileo transform (see Landau's mechanics).
I'm looking for a text that presents a rigorous incorporation of space-time structure into Hamiltonian mechanics. I'm not interested in Lagrangian or Newtonian approach, only Hamiltonian. The level of the abstraction should correspond to the one in Arnolds' book (symplectic manifolds, etc).
Basically, I want to be able to answer the following question: "Given certain metrics, find the form of kinetic energy".
| I tried the following: assuming that the metric only depends on the coordinates, $g(q)$, and the energy only depends on the the coordinates and the spatial momenta, $p_0 (q,p_i)$. I then considered variations like
$$
\delta S=\int \delta \theta=\int \delta(g_{ab}p^a dq^b)=\int \delta(g_{00}p^0dq^0+g_{0i}(p^idq^0+p^0dq^i)+g_{ij}p^idq^j)
$$
Then just remember for variations of metric components
$$
\delta g_{ab}=\frac{\partial g_{ab}}{\partial q^c}\delta q^c=\frac{\partial g_{ab}}{\partial q^0}\delta q^0 + \frac{\partial g_{ab}}{\partial q^i}\delta q^i
$$
and
$$
\delta p^0=\frac{\partial p^0}{\partial q^c}\delta q^c+\frac{\partial p^0}{\partial p^k}\delta p^k=\frac{\partial p^0}{\partial q^0}\delta q^0 + \frac{\partial p^0}{\partial q^i}\delta q^i +\frac{\partial p^0}{\partial p^k}\delta p^k
$$
You will want to isolate the coefficients of $\delta q^0$ and $\delta q^i$ and $\delta p^i$. These are Hamilton's Equations. You will need to integrate by parts when you have derivative's of variations remembering the the variations are zero at the end points. For example:
$$
g_{00}p^0 d(\delta q^0)=d(g_{00}p^0q^0)-d(g_{00}p^0)\delta q^0
$$
I think this will work.
| {
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What is the physical meaning of diffusion coefficient? In Fick's first law, the diffusion coefficient is velocity, but I do not understand the two-dimensional concept of this velocity. Imagine that solutes are diffusing from one side of a tube to another (this would be the same as persons running from one side of a street) to unify the concentration across the tube.
Here we have a one-dimensional flow in x direction. The diffusion coefficient should define the velocity of solutes or persons across the tube or street direction. How the two-dimensional velocity does this? I wish to understand the concept to imagine the actual meaning of the diffusion coefficient.
| Let stick to your example of the people running around in the street.
this would be the same as persons running from one side of a street
If all people are running in the same direction, then this would be convection, and not diffusion.
Suppose you have a large room, with 100 guys running around in random directions. If you give them a random initial position somewhere in the room, not a lot will change. Except that each man will be at a different location the next time you look.
Now suppose you let all the guys starting at the left side of the room. You're standing with your back to the wall, exactly in the middle. Initially, you will, every once in a while, see a guy passing your line of sight from left to right. No man will cross from right to left, because there is no one at the right hand side. At some point, there will be that many guys on the right hand side, that you will also see man running from right to left, until, at some point, you will not see any difference between left and right.
This is basically the physical concept behind diffusion. If you replace the man by molecules, you get a gas or liquid. These molecules also have some random velocity and direction.
The diffusion coefficient in this context, is measure for how fast the left and right side 'mix', and thus also determines how long you have to wait before you don't note any difference. A larger diffusion coefficient does mean faster mixing.
| {
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Can light exist in $2+1$ or $1+1$ spacetime dimensions? Spacetime of special relativity is frequently illustrated with its spatial part reduced to one or two spatial dimension (with light sector or cone, respectively). Taken literally, is it possible for $2+1$ or $1+1$ (flat) spacetime dimensions to accommodate Maxwell's equations and their particular solution - electromagnetic radiation (light)?
| Light may exist in a 2 dimensional space, but it wouldn't appear the same as ours. With 1 less dimension it would become a point. Add an additional dimension to the point and ,as all dimensions should be perpendicular to each other, you would end up with an electromagnetic wave in 3 dimensional space. This wave would intuitively appear strange due to not all of its components existing in our Time/Space.
| {
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