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This primary reference image was chosen from the reference images used for the image subtraction on which SN 2002ic was originally discovered. | This primary reference image was chosen from the reference images used for the image subtraction on which SN 2002ic was originally discovered. |
The flux differences calculated relative to each reference were combined in a average for each image to yield an average flux for the image. | The flux differences calculated relative to each reference were combined in a noise-weighted average for each image to yield an average flux for the image. |
As the observations were taken within a span of less than one hour on each night, the results from the images of a given night were averaged to produce a single light curve point for that night. | As the observations were taken within a span of less than one hour on each night, the results from the images of a given night were averaged to produce a single light curve point for that night. |
The reference zeropoint calculated for the primary reference image from the above USNO A1.0 calibration was used to set the magnitudes for the restof the measured fluxes. | The reference zeropoint calculated for the primary reference image from the above USNO A1.0 calibration was used to set the magnitudes for the restof the measured fluxes. |
Table 1 reports these magnitudes and associated measurement uncertainties. | Table \ref{tab:2002ic_lightcurve} reports these magnitudes and associated measurement uncertainties. |
An overall systematic uncertainty in the zeropoint calibration is not included in the listed errors. | An overall systematic uncertainty in the zeropoint calibration is not included in the listed errors. |
The USNO A1.0 catalog suffers from systematic field-to-field errors of ~0.25 magnitudes in the northern hemisphere 1996).. | The USNO A1.0 catalog suffers from systematic field-to-field errors of $\sim0.25$ magnitudes in the northern hemisphere \citep{usnoa1}. |
The conversion ofPOSS-E magnitudes to V-band magnitudes for a SN Ia is relatively robust, as a SN Ia near maximum resembles a ~10,000 K blackbody quite similar to Vega in the wavelength range from 4, 500-10,000 A.. | The conversion of magnitudes to V-band magnitudes for a SN Ia is relatively robust, as a SN Ia near maximum resembles a $\sim10,000$ K blackbody quite similar to Vega in the wavelength range from $4,500$ $10,000$ . |
At late times, the observations of | At late times, the observations of |
focal plane mask. | focal plane mask. |
At larecr aneular separation. wavefrout errors can create speckles witlout xoduciug a signal in the CLOWEFS camera. aud other calibration approaches must be used to reconstruct the PSF: for example telemetry from a ueher order WES. differcutial spectral iniagiug. or he ADI/LOCT technique. | At larger angular separation, wavefront errors can create speckles without producing a signal in the CLOWFS camera, and other calibration approaches must be used to reconstruct the PSF: for example telemetry from a higher order WFS, differential spectral imaging, or the ADI/LOCI technique. |
We also propose to use a now clupirical inage-based. PSF reconstruction aleorithiu. where CLOWTFS images are matched O science calera images to reconstruct the PSF. as opposed to relying on a model of the adaptive optics system. | We also propose to use a new empirical image-based PSF reconstruction algorithm, where CLOWFS images are matched to science camera images to reconstruct the PSF, as opposed to relying on a model of the adaptive optics system. |
This empirical approach is more robust. simple to implement. aud is made possible in our case by the snall nuniber of modes measured by the CLOWES. | This empirical approach is more robust, simple to implement, and is made possible in our case by the small number of modes measured by the CLOWFS. |
A thorough description of the theory aud hardware implementation of CLOWFS was provided bv Caronoetal.(2009).. | A thorough description of the theory and hardware implementation of CLOWFS was provided by \citet{2009ApJ...693...75G}. |
Ilere. it suffices to remember that it operates thanks to an optimized dual zone focal plane mask. absorbing at its ceuter. and reflective in an annulus whose outer οσο defines the iuner working augle of the svstem (1.5 A/D for SCExAO). | Here, it suffices to remember that it operates thanks to an optimized dual zone focal plane mask, absorbing at its center, and reflective in an annulus whose outer edge defines the inner working angle of the system (1.5 $\lambda/D$ for SCExAO). |
Figure 1. shows the actual iuplementation of CLOWES on SCExAO. | Figure \ref{fig:bench} shows the actual implementation of CLOWFS on SCExAO. |
A lens re-hnages the reflective ring of the occulting mask. ou a detector deliberately placed out of focus. referred to as the CLOWES camera. | A lens re-images the reflective ring of the occulting mask on a detector deliberately placed out of focus, referred to as the CLOWFS camera. |
During the (long) exposure on the science camera. the CLOWES camera acquires a continuous stream of short (typically mullisecoud) exposures. | During the (long) exposure on the science camera, the CLOWFS camera acquires a continuous stream of short (typically millisecond) exposures. |
Au example of one such CLOWFS caicra image is provided in Fie. 2:: | An example of one such CLOWFS camera image is provided in Fig. \ref{fig:clowfs_img}: |
variations of the distribution of iuteusitv in this inagoe are used to identity drifts in pointing as well as cliauges in focus. | variations of the distribution of intensity in this image are used to identify drifts in pointing as well as changes in focus. |
Chuvonetal.(2009) have demonstrated that over a small range (~0.2A/ D) of poiutiug crrors. a linear model relates the changes in CLOWFS Muages o the actual poiuting error. and took advantage of this ina close-loop svsteun. stabilizing he pointing at the level of 10A/D over extcuded yeriods of time (~1 hr). | \citet{2009ApJ...693...75G} have demonstrated that over a small range $\sim 0.2 \lambda/D$ ) of pointing errors, a linear model relates the changes in CLOWFS images to the actual pointing error, and took advantage of this in a close-loop system, stabilizing the pointing at the level of $10^{-3} \lambda/D$ over extended periods of time $\sim 1$ hr). |
Additional calibration of the coronagraphic eaks cue to the residual tip-tilt error can be achieved im post-processing. | Additional calibration of the coronagraphic leaks due to the residual tip-tilt error can be achieved in post-processing. |
This calibration relies on the one-to-one correspondence that exists οποσα maages sinultaneouslv acquired on the scieuce aud the CLOWFS cameras. | This calibration relies on the one-to-one correspondence that exists between images simultaneously acquired on the science and the CLOWFS cameras. |
lu either close or open loop. recording at high temporal rate CLOWEFS images during a long exposure ou the science camera can help xediet the level of coronagraphlic leaks attributable Oo pointing errors. | In either close or open loop, recording at high temporal rate CLOWFS images during a long exposure on the science camera can help predict the level of coronagraphic leaks attributable to pointing errors. |
A synthetic pointing leak nuage cau be | A synthetic pointing leak image can be |
statistics. which measure the spatial structure of the turbulence. ancl in particular the correlation length? | statistics, which measure the spatial structure of the turbulence, and in particular the correlation length? |
The correlation length is a natural measure of the outer scale of the turbulence. aud should be resolved and independent of resolution in a converged simulation. | The correlation length is a natural measure of the outer scale of the turbulence, and should be resolved and independent of resolution in a converged simulation. |
We consider only the azimuthal correlation length. as this is most straightforward to compute. and is most often. under resolved in global simulations (IGN). | We consider only the azimuthal correlation length, as this is most straightforward to compute, and is most often under resolved in global simulations (HGK). |
The correlation function at radius r on the equatorial plane is where 9/ is deviation fom average value of variable f at r. In practice. we average J? in small area Γ.ΝM across the equatorial plane. normalize. ancl average in time: Note that the correlation Iunction for magnetic field is defined as where is defined in 82. | The correlation function at radius $r$ on the equatorial plane is where $\delta f$ is deviation from average value of variable $f$ at r. In practice, we average $R$ in small area $r\Delta r \Delta\theta$ across the equatorial plane, normalize, and average in time: Note that the correlation function for magnetic field is defined as where $b^{\mu}$ is defined in 2. |
Then is the correlation length at radius 7. | Then is the correlation length at radius $r$. |
Figure 13. shows the azimuthal correlation length for densitv p. internal energy. wv. magnetic field 6. and 8. for all runs. | Figure \ref{correlation} shows the azimuthal correlation length for density $\rho$, internal energy $u$, magnetic field $b$, and $\theta_e$ for all runs. |
Evidently the correlation lengths (angles) are nearly independent of r. except close to the outer boundary where the models are not in a steady state. | Evidently the correlation lengths (angles) are nearly independent of $r$, except close to the outer boundary where the models are not in a steady state. |
The correlation length varies between about 0.27 al the lowest resolution to O.la a the highest resolution for all variables except 6. | The correlation length varies between about $0.2\pi$ at the lowest resolution to $0.1\pi$ at the highest resolution for all variables except $b$. |
Since H/r.~0.3 [or all models over a wide range in radius (Figure 15)). (his corresponds (assuming flat space geometry) to 1 to 2 vertical scale heights. | Since $H/r \sim 0.3$ for all models over a wide range in radius (Figure \ref{scale_height}) ), this corresponds (assuming flat space geometry) to $1$ to $2$ vertical scale heights. |
TrMou.=1.4.5.8.95][3.5.0.24.0.09.0]. | $\tau_{\rm ff}[\nu_{\rm GHz}=1.4, 5, 8, 95]=[3.5, 0.24, 0.09, 0]$. |
For interesting values of M (z10SAL.vr.!) free-Iree optical depth is not important for the observations reported here. | For interesting values of $\dot M$ $\simlt 10^{-8}\,M_\odot\,{\rm yr}^{-1}$ ) free-free optical depth is not important for the observations reported here. |
Equation 4 provides (he starting point for the discussion. | Equation \ref{eq:Lthin3} provides the starting point for the discussion. |
This equation shows that the spectral luminosity can constrain M/ie provided that we have a good grasp of the blast wave dvnamics and microplvsies of particle acceleration and magnetic field generation. | This equation shows that the spectral luminosity can constrain $\dot M/w$ provided that we have a good grasp of the blast wave dynamics and microphysics of particle acceleration and magnetic field generation. |
We have argued above that pzz3. €,220.1 and c,&4x10 |. | We have argued above that $p\approx 3$, $\epsilon_e\approx 0.1$ and $v_s\approx 4\times 10^9\,$ $^{-1}$. |
We adopt these values and proceed with the discussion. | We adopt these values and proceed with the discussion. |
To start with we can see that the observations reported here (Table 1)) vield (he lowest limits on the radio Iuminositv of very voung Ia supernovae. L,zI0ergs.HIz+. | To start with we can see that the observations reported here (Table \ref{tab:RadioLog}) ) yield the lowest limits on the radio luminosity of very young Ia supernovae, $L_\nu \simlt 10^{24}{\rm erg}~{\rm s}^{-1}{\rm Hz}^{-1}$. |
This limit then constrains the following parameter. B=eMir, where e=v/eptc. | This limit then constrains the following parameter, $\mathcal{B}\equiv\epsilon\dot M/w$, where $\epsilon\equiv\sqrt{\epsilon_{\rm B}\epsilon_{\rm e}}$. |
We have deliberately not quoted the limits on AV from previous literature since radio measurements vield not M but B and this quantity depends on the unknown parameter. ej which (as summarized above) ean vary. by. orders of magnitude. | We have deliberately not quoted the limits on $\dot M$ from previous literature since radio measurements yield not $\dot M$ but $\mathcal{B}$ and this quantity depends on the unknown parameter, $\epsilon_B$ which (as summarized above) can vary by orders of magnitude. |
Despite this it is clear that the observations reported here present the most sensitive limits to M to date (Figure 2)). | Despite this it is clear that the observations reported here present the most sensitive limits to $\dot M$ to date (Figure \ref{fig:lc}) ). |
From Figure 2. we note that 6 can be constrained as follows: the optically thin regime offers a lower bound whereas the optically thick regime (SSA and free-Iree) an upper bound. | From Figure \ref{fig:lc} we note that $\mathcal{B}$ can be constrained as follows: the optically thin regime offers a lower bound whereas the optically thick regime (SSA and free-free) an upper bound. |
The upper bound is not interesting since the simplest explanation lor the absence of radio emission is that the explosion takes place in a vacuum (or very low density. circumstellar medium). | The upper bound is not interesting since the simplest explanation for the absence of radio emission is that the explosion takes place in a vacuum (or very low density circumstellar medium). |
Besides. fom past studies. there is no indication of M. in the range of LO°AL.vr.| that is indicated [rom the upper bounds. | Besides, from past studies, there is no indication of $\dot M$ in the range of $10^{-6}\,M_\odot\,{\rm yr}^{-1}$ that is indicated from the upper bounds. |
The constraint deduced for each observation are summarized in Table 1.. | The constraint deduced for each observation are summarized in Table \ref{tab:RadioLog}. |
Combining all the constraints we find Mzlx10"ue0.1/e)M.vr.|l. | Combining all the constraints we find $\dot M\simlt 1\times
10^{-8} w_7(0.1/\epsilon)M_{\odot} {\rm yr}^{-1}$. |
Past X-ray. observations. (wpically undertaken no earlier (han a week past the explosion. have resulted in upper limits at the level of Ly<10?ergs!|.. | Past X-ray observations, typically undertaken no earlier than a week past the explosion, have resulted in upper limits at the level of $L_X\simlt 10^{39}\,{\rm erg\,s}^{-1}$. |
A claim of detection of emission from 22005ke (Ly~2xI0Pergs!: ?)) has been disputed by 2).. | A claim of detection of emission from 2005ke $L_X\sim 2\times
10^{38}{\rm erg\,s}^{-1}$; \citealt{immler+06}) ) has been disputed by \citet{hughes+07}. |
2) reported a detection in a 2-month stack of NRT data of 2201108. | \citet{immler11} reported a detection in a 2-month stack of XRT data of 2011by. |
However. subsequent high angular resolution Chandra observations by ο) strongly suggest that the emission arose [rom a steady source | However, subsequent high angular resolution Chandra observations by \citet{pooley11} strongly suggest that the emission arose from a steady source |
Combining. the APAICAT. LRIS2 imagine and Εςο datasets resulted. in a sample of 09 spectroscopicallv identified quasars. | Combining the APMCAT, IRIS2 imaging and FCSS datasets resulted in a sample of 69 spectroscopically identified quasars. |
In Section 3.1 we examine the b;—ἐν colours of the quasars to determine which quasars are red. | In Section \ref{detection} we examine the $b_J - K$ colours of the quasars to determine which quasars are red. |
We investigate the uncertainties in the data to ascertain if the uncertainties caused blue quasars to appear red. | We investigate the uncertainties in the data to ascertain if the uncertainties caused blue quasars to appear red. |
Section compares the colours of the constructed quasar sample to the colours of a sub-set of LBQS quasars to characterise the creddness! of the quasar sample. | Section \ref{comparison} compares the colours of the constructed quasar sample to the colours of a sub-set of LBQS quasars to characterise the `reddness' of the quasar sample. |
Then in Section 3.4. we examine selection effects. ancl estimate the red. quasar fraction. | Then in Section \ref{selection}, we examine selection effects and estimate the red quasar fraction. |
Finally. Section 3.3. examines the relative 6,& colours of our quasar sample. ancl compares them to the relative colours of the LBOQS sub-set. | Finally, Section \ref{relativebjMk} examines the relative $b_J - k$ colours of our quasar sample, and compares them to the relative colours of the LBQS sub-set. |
We show the 5;dv colours of our quasar sample in Figure l.. ll quasars are red. because they satisfy 5;fyο3.5. corresponding to 20.16 of the 69 quasars with 5; and Ix magnituces. | We show the $b_J - K$ colours of our quasar sample in Figure \ref{colourvsMag}, , 11 quasars are red because they satisfy $b_J - K \geq 3.5$, corresponding to $\approx 0.16$ of the 69 quasars with $b_J$ and K magnitudes. |
These red quasars have colours. 3.5 xbyIN: 5.6. which occupy the colour space between CGlikmanctal.(2004) red quasars and. LBOS. SDSS red quasars. | These red quasars have colours, 3.5 $\leq b_J - K \leq$ 5.6, which occupy the colour space between \citet{2004ApJ...607...60G} red quasars and LBQS, SDSS red quasars. |
Using the All-Skv. Optical Catalogue of Itadio/NX-Itay Sources in Fleseh&Llardeastle(2004)... we verified the existence (or lack thereof) of a raclio-counterpart for all the red. quasars discovered here. to a παν limit of 1niJy. | Using the All-Sky Optical Catalogue of Radio/X-Ray Sources in \citet{2004A&A...427..387F}, we verified the existence (or lack thereof) of a radio-counterpart for all the red quasars discovered here, to a flux limit of 1mJy. |
We found that 5 of these quasars had. no radio-counterpart and the remaining 6 were not in the catalogue. indicating they too are raclio-quiet. | We found that 5 of these quasars had no radio-counterpart and the remaining 6 were not in the catalogue, indicating they too are radio-quiet. |
We concluded that all of the red quasars in Figure 1 are raclio-quict quasars. | We concluded that all of the red quasars in Figure \ref{colourvsMag} are radio-quiet quasars. |
In Maddox&Llewett(2006) it was demonstrated. that the host galaxy. contribution to quasar lux is maximal in the Ix-band. for resolved. low-Iuminosity quasars at z« | In \citet{2006MNRAS.367..717M} it was demonstrated that the host galaxy contribution to quasar flux is maximal in the K-band for resolved, low-luminosity quasars at $z<1$. |
lob,dv is plotted against redshift in Figure 2: and a dotted. line tracks the number of quasars as a function of redshift. peaking at z & 1.5. | $b_J - K$ is plotted against redshift in Figure \ref{bjMKvsz}; and a dotted line tracks the number of quasars as a function of redshift, peaking at z $\approx 1.5$ . |
The dashed. line in Figure 2. traces the median quasar colour with redshift. demonstrating a dependence of the observed b;Ay colour on redshift. | The dashed line in Figure \ref{bjMKvsz} traces the median quasar colour with redshift, demonstrating a dependence of the observed $b_J - K$ colour on redshift. |
Section 3.3. examines the dependence of the observed b;/—A colour on redshift. | Section \ref{relativebjMk} examines the dependence of the observed $b_J - K$ colour on redshift. |
As the red quasars founc in Figure 1 are all luminous unresolved APMCAAT sources. and the majority are luminous quasars with z21 in Figure 2: anv host galaxy contribution to the 6;ἐν colours of the red quasars will be minimal. | As the red quasars found in Figure \ref{colourvsMag} are all luminous unresolved APMCAT sources, and the majority are luminous quasars with $z > 1$ in Figure \ref{bjMKvsz}; ; any host galaxy contribution to the $b_J - K$ colours of the red quasars will be minimal. |
La significant host galaxy contribution was present in the red quasars. identified. in Figure 1.. then the b;—A colours of the red. quasars woutle become increasingly redder as both Luminosity aid. redshift decreased. | If a significant host galaxy contribution was present in the red quasars identified in Figure \ref{colourvsMag}, then the $b_J - K$ colours of the red quasars would become increasingly redder as both luminosity and redshift decreased. |
In Figures 1 and 2.. a clear trend of increasingly red b,48 colours with decreasing luminosity anc recdshift is not apparent. | In Figures \ref{colourvsMag} and \ref{bjMKvsz}, a clear trend of increasingly red $b_J - K$ colours with decreasing luminosity and redshift is not apparent. |
The 11 red racio-quict quasars detected here are most likely the result. of reddening by dust. located. at the quasar recdshift. | The 11 red radio-quiet quasars detected here are most likely the result of reddening by dust located at the quasar redshift. |
We investigated the uncertainties in the observed by,A colours. to determine if the uncertainties caused bluc quasars to appear red. | We investigated the uncertainties in the observed $b_J - K$ colours, to determine if the uncertainties caused blue quasars to appear red. |
Uncertainty in the ὃν colours in Figure l has two sources. photometric uncertainties in the 5; and I|x magnitudesancl the effect. of quasar variability on multi-epoch observations. | Uncertainty in the $b_J - K$ colours in Figure \ref{colourvsMag} has two sources, photometric uncertainties in the $b_J$ and K magnitudesand the effect of quasar variability on multi-epoch observations. |
The photometric uncertainty in the 5; magnitudes. was obtained. from. Drinkwater where it was foundto be 0.1mag. | The photometric uncertainty in the $b_J$ magnitudes was obtained from \citet{2000A&A...355..900D} where it was foundto be 0.1mag. |
For the Ix-band magnitudes the uncertainty was calculated by. SExtractor | For the K-band magnitudes the uncertainty was calculated by SExtractor |
matter how far away it is frou the ceuter of its rost galaxy. | matter how far away it is from the center of its host galaxy. |
They also have the advantage that much of their leh in οuitted in a sinele line. so obtaimine an accurate radial velocity Or a aueary 1bebula can be straightforward. | They also have the advantage that much of their light is emitted in a single line, so obtaining an accurate radial velocity for a planetary nebula can be straightforward. |
However. the surface density of panetary nenmlae does not have the large spatial extent of the οobular clusteY svstenis. aud they have not vet provided as muuch information about tie fornatio1 istexw of galaxies at large radii as elobulus. | However, the surface density of planetary nebulae does not have the large spatial extent of the globular cluster systems, and they have not yet provided as much information about the formation history of galaxies at large radii as globulars. |
A final approach. reviewed at this necting bv David Buote. is to use the properties of the hot eas enuttiis in A-ravs foitnd around lines earlv-tvpe galaxies. | A final approach, reviewed at this meeting by David Buote, is to use the properties of the hot gas emitting in X-rays found around luminous early-type galaxies. |
With sufficienty accurate N-rav müaeiue spectroscopy. it is feasible to determine the dark matter distribution and metal abuudauces in hot gas. as well as testing for asvuuuetrics that mich indicate objects for which the assumption of νεrostatic equilibrium iu the eas at large radii is questionable. | With sufficiently accurate X-ray imaging spectroscopy, it is feasible to determine the dark matter distribution and metal abundances in hot gas, as well as testing for asymmetries that might indicate objects for which the assumption of hydrostatic equilibrium in the gas at large radii is questionable. |
An obvious goal is to combine as many approaches as possible. as they cach lave ciffereut ses of assuniptions aud posside systematic errors. which night be revealed through careful intercomparison. | An obvious goal is to combine as many approaches as possible, as they each have different sets of assumptions and possible systematic errors, which might be revealed through careful intercomparison. |
The two dunensioial distribution of light iu galaxies has been fairly well characterized within about ΤΠ... | The two dimensional distribution of light in galaxies has been fairly well characterized within about $1 R_e$. |
Oue of the uses of these data ds to trv to coustrain the threenuensiona shapes of galaxies. | One of the uses of these data is to try to constrain the three-dimensional shapes of galaxies. |
An unconstrained inverso- of two-dimensionalcata to the intrinsic threc-dimensioial shape is problematic (Rvbicki 1987). but either through coustraimed mversioi techniques RRv«Cll 1992. Liiubas. Madox. Loveday 1992) or throieh t1e addition of kinematic data FEraux. Thueworth. de Zeeww 1991. Bak Staler 2000). some progress can be iade. | An unconstrained inversion of two-dimensional data to the intrinsic three-dimensional shape is problematic (Rybicki 1987), but either through constrained inversion techniques Ryden 1992, Lambas, Maddox, Loveday 1992) or through the addition of kinematic data Franx, Illingworth, de Zeeuw 1991, Bak Statler 2000), some progress can be made. |
Overall. the evidence sugeess that at cast a small amorut of triaxialitv is common. with most ealaxies |eine nearly oblate aud a niolest yaction ucarly prolate. | Overall, the evidence suggests that at least a small amount of triaxiality is common, with most galaxies being nearly oblate and a modest fraction nearly prolate. |
]t would be of cear mterest to extend these sucdies to the outer regious of eaOs:axies which iight be less influenced by evolulon and more closely reflect he conditions whei they formed. | It would be of clear interest to extend these studies to the outer regions of galaxies which might be less influenced by evolution and more closely reflect the conditions when they formed. |
CCD ALosaicὪὰ covering lareer areas are celine to make his feasible. altrough the laree surveys of galaxies used in the statistical struclies given above are a long way away. | CCD Mosaics covering larger areas are beginning to make this feasible, although the large surveys of galaxies used in the statistical studies given above are a long way away. |
Both the integrated ight aux the elonar clusters cau 0 studied this way. with the integrated ight offeriug πιο1 niore signal. aud he clusters potentially reaching to larger radi. bu being Linited by defining the two-dimensional shape with a modest iuuber of poiuts. | Both the integrated light and the globular clusters can be studied this way, with the integrated light offering much more signal, and the clusters potentially reaching to larger radii, but being limited by defining the two-dimensional shape with a modest number of points. |
Asa1 exaniple of how this work might develop iu the future. we present in Figure 2 our new results for the position angle aud ellipticity of the iuteerated | As an example of how this work might develop in the future, we present in Figure 2 our new results for the position angle and ellipticity of the integrated |
aud test whether its value remains equal to £=—1 along each geodesic. | and test whether its value remains equal to $\xi=-1$ along each geodesic. |
Starting from a fine raster of polnts on the image plane of an observer at infinity. we follow the geodesies backwards to the surface of the compact object or to differeut regious in the accretiou flow where the photons originate. | Starting from a fine raster of points on the image plane of an observer at infinity, we follow the geodesics backwards to the surface of the compact object or to different regions in the accretion flow where the photons originate. |
Following Johanuseu Psaltis (2010b). we consider an observer viewing the central object [rom a large distauce d aud at au inclination angle 0, [rou its rotation axis (see Fig. 1)). | Following Johannsen Psaltis (2010b), we consider an observer viewing the central object from a large distance $d$ and at an inclination angle $\theta_o$ from its rotation axis (see Fig. \ref{fig:geometry}) ). |
We set up a virtual image plane that is perpeudicular to the line of sight aud centered at o=0 of the spacetime. | We set up a virtual image plane that is perpendicular to the line of sight and centered at $\phi=0$ of the spacetime. |
We define the set of Cartesian coordinates (ay.20) ou the image plane such that the %y-axis is along the same fiducial plane aud the ay-anxis is perpendicular to it. | We define the set of Cartesian coordinates $(\alpha_0,\beta_0)$ on the image plane such that the $\beta_0$ -axis is along the same fiducial plane and the $\alpha_0$ -axis is perpendicular to it. |
We then convert the coordinates (o.20) ob a photon that reaches the nage plane to the coordinates (rj.0;.0;) in the spherical-polar system used for the metric (1)) with the relatious (see Joliauusen Psaltis 2010b) The photons that contribute to the image of the compact object are those with 3-momeuta that are perpendicular to the image plaue. | We then convert the coordinates $(\alpha_0,\beta_0)$ of a photon that reaches the image plane to the coordinates $(r_i,\theta_i, \phi_i)$ in the spherical-polar system used for the metric \ref{eq:GB}) ) with the relations (see Johannsen Psaltis 2010b) The photons that contribute to the image of the compact object are those with 3-momenta that are perpendicular to the image plane. |
This orthogonality coudition uniquely specifies the momenttun vector of a photon with the above coordinates. accordiug to the relatious (Johansen | This orthogonality condition uniquely specifies the momentum vector of a photon with the above coordinates, according to the relations (Johannsen |
proportion of as reported by ? for the LGRD«-SCRDs:. | proportion of as reported by \citet{zhme2004} for the LGRBs-SGRBs. |
7? and 7 have studied the distribution of openiug aueles. which is reproduced in Fig. 5.. | \citet{zeh2006} and \citet{wats2006} have studied the distribution of opening angles, which is reproduced in Fig. \ref{fig5}. |
If the LGRD event rate was estimated considere this distribution of beam aueles. the 2p(0) probability factor in relation (9)) had to be replaced by the integrated probability P eiveu by where pt)=1cos? is given by (8)). and Q(0) is the distribution shown in Fie. 5.. | If the LGRB event rate was estimated considering this distribution of beam angles, the $2p(\theta)$ probability factor in relation \ref{brpf}) ) had to be replaced by the integrated probability $P$ given by where $p(\theta)\!=\!1-\cos\theta$ is given by \ref{prob}) ), and $Q(\theta)$ is the distribution shown in Fig. \ref{fig5}. |
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