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The epic observation data files CODES) were processed. using the XALAL-Scieuce Analysis Svsteii (SAS version 7.0.0). | The epic observation data files (ODFs) were processed using the -Science Analysis System (SAS version $7.0.0$ ). |
Using he latest calibration coustitueut files curreutly available. we processed the raw data with the aud ools to generate files with adequate eveut ists. | Using the latest calibration constituent files currently available, we processed the raw data with the and tools to generate files with adequate event lists. |
After screening with the standi criteria. as reconunended hy he Science Operation Ceutre technical note NADLPS-TN-13 v3.0. we rejected any time period. aft"med by soft xoton flares. | After screening with the standard criteria, as recommended by the Science Operation Centre technical note XMM-PS-TN-43 v3.0, we rejected any time period affected by soft proton flares. |
The remaining time intervals resulted iu effective exposures of &96 ks. &98 ks. aud &91 ks for MOS 1. MOS 2. and PN. respectively. | The remaining time intervals resulted in effective exposures of $\simeq 96$ ks, $\simeq 98$ ks, and $\simeq 94$ ks for MOS 1, MOS 2, and PN, respectively. |
The coordinates of the μ.ο... IRNS J150131.1- as determined by tool. are aqoyyy=[95302ts aud Agoqgg—21739/32776.. | The coordinates of the source 1RXS J180431.1-273932, as determined by tool, are $\alpha_{\rm J2000}=18^{\rm h}04^{\rm m} 30\fs{}48$ and $\delta_\textrm{J2000}=-27^\circ 39' 32\farcs 76$. |
A systeniatic shift (N-ravoptical) of 92.52"— (le) aud —3.09" (19) exists in right ascension for MOS 1 and MOS 2 aud for 1.19" aud 0.117 in declination. respectively. | A systematic shift (X-ray–optical) of $-2.52\arcsec$ $1\sigma$ ) and $-3.09\arcsec$ $1\sigma$ ) exists in right ascension for MOS 1 and MOS 2 and for $-1.19\arcsec$ and $0.41\arcsec$ in declination, respectively. |
We ας in the following that the error associated to the source coordinates. as determined by the X-rav observation. is at least ~2" on both celestial coordinates. | We assume in the following that the error associated to the source coordinates, as determined by the $X$ -ray observation, is at least $\sim 2\arcsec$ on both celestial coordinates. |
The source spectra were extracted in a circular region centered on the nominal position of the tareet im the three EPIC cameras. while the backgrouud spectra were accunnulated in auuuli ou the same coordinates. | The source spectra were extracted in a circular region centered on the nominal position of the target in the three EPIC cameras, while the background spectra were accumulated in annuli on the same coordinates. |
The resulting spectra were rebinued to have at least 25 as the nini uuuiberofcouuts per energy bin. | The resulting spectra were rebinned to have at least 25 as the minimum number of counts per energy bin. |
The EPIC PN iuaee has to be taken with caution since the source is on a chip eap. hereby reducing the uct uuuber of collected X-ray photons. | The EPIC PN image has to be taken with caution since the source is on a chip gap, hereby reducing the net number of collected $X$ -ray photons. |
The spectra were simultancously fitted with NSPEC (version 12.0.0). | The spectra were simultaneously fitted with XSPEC (version 12.0.0). |
In Fig. 1.. | In Fig. \ref{fig2}, |
we show the MOS 1. MOS 2, and PN spectra for IRNS J150131.1-273932 and. the respective fits. | we show the MOS 1, MOS 2, and PN spectra for 1RXS J180431.1-273932 and the respective fits. |
Note the clear evidence of au iron emission line in the spectra. | Note the clear evidence of an iron emission line in the spectra. |
The best-Stting model was an absorbed | The best-fitting model was an absorbed |
Alunooz. Trac. and Next. we consider the one-dimensional (1D). galaxy power spectrum measured along the lDinc-ofsight of a "pencil-beam" survey at high redshift with the same 3.43.4. field-ofview considered. in the previous section. | oz, Trac, and Next, we consider the one-dimensional (1D) galaxy power spectrum measured along the line-of-sight of a “pencil-beam" survey at high redshift with the same $3.4'\times3.4'$ field-of-view considered in the previous section. |
We explore what the information along the radial direction can or cannot tell us about the nature of high-redshift. &alaxies eiven the narrow field-of-view. | We explore what the information along the radial direction can or cannot tell us about the nature of high-redshift galaxies given the narrow field-of-view. |
We also examine whether or not it can be used to distinguish between luminous ealaxies hosted by dillerent. mass halos. and ultimately to probe reionization. | We also examine whether or not it can be used to distinguish between luminous galaxies hosted by different mass halos, and ultimately to probe reionization. |
We are considering the "galaxy power spectrum in the sense that we apply a duty evele for LAs and translate halo masses into hosted Lyman-a buminosities. but our results do not include the effects of reionization on the power spectrum that we expect to observe in real surveys (c.g.Babich&Loeb2006:MeQuinnctal.2007:WyitheLoeb2007 ). | We are considering the “galaxy" power spectrum in the sense that we apply a duty cycle for LAEs and translate halo masses into hosted $\alpha$ luminosities, but our results do not include the effects of reionization on the power spectrum that we expect to observe in real surveys \citep[e.g.][]{dBL06, McQuinn07, WL07}. |
. When analyzing the 1D power spectrum. we do not attempt to produce mock surveys viewed. along the since we wish to study the bias of the halos hosting hieh-redshift galaxies ancl compare our results to analytic. linear. theory. | When analyzing the 1D power spectrum, we do not attempt to produce mock surveys viewed along the since we wish to study the bias of the halos hosting high-redshift galaxies and compare our results to analytic, linear theory. |
Rather. we consider snapshots at 2=6 and z=10. | Rather, we consider snapshots at $z=6$ and $z=10$. |
For this purpose. it is preferrable that the evolution of the halo mass and correlation functions will not introduce features in the power spectrum (Munoz&Loeb 2008).. | For this purpose, it is preferrable that the evolution of the halo mass and correlation functions will not introduce features in the power spectrum \citep{ML08b}. |
Xdeditionallv. stitching together simulation slices to xocduce a view along the light-cone would unnecessarily add subtract (depending on the prescription used) power from 10 spectrum on scales larger than the size of the simulation σον and on the scale of the separation between simulation slices. | Additionally, stitching together simulation slices to produce a view along the light-cone would unnecessarily add or subtract (depending on the prescription used) power from the spectrum on scales larger than the size of the simulation box and on the scale of the separation between simulation slices. |
2 shown in Figures 5. and 6.. the 143Alpe comoving size of the simulation box is sullicicntly large to reproduce 1e linear results on the largest scales. | As shown in Figures \ref{fig:bias1D_z6} and \ref{fig:bias1D_z10}, the $143\,\rm{Mpc}$ comoving size of the simulation box is sufficiently large to reproduce the linear results on the largest scales. |
We plot our results only for⋅ values of & larger than 0.1Alpe1 to avoid. confusion⋅⋠ rom power loss due to the finite box size. | We plot our results only for values of $k$ larger than $0.1\,\mpc^{-1}$ to avoid confusion from power loss due to the finite box size. |
We calculate the spectrum in. terms. of. real-space wave-numboers. as opposed. to ones in redshift-space. and ignore contamination of distance measurements by peculiar velocities (Monacoetal.2005). since they are expected to be small in the early. universe. | We calculate the spectrum in terms of real-space wave-numbers, as opposed to ones in redshift-space, and ignore contamination of distance measurements by peculiar velocities \citep{Monaco05} since they are expected to be small in the early universe. |
The redshifts of LAEs can be constrained to within a few hundredkm/s.. corresponding to a wave-number of &—3Alpe|. while the distances to high-redshift Ls can be uncertain to hundreds of comoving (Bouwens&IHllingworth2006). | The redshifts of LAEs can be constrained to within a few hundred, corresponding to a wave-number of $k\sim3\,\rm{Mpc^{-1}}$, while the distances to high-redshift LBGs can be uncertain to hundreds of comoving \citep{BI06}. |
. We compare the power spectrum caleulated. from. the simulation using a 1D Fast Fourier. ‘Transform CEET) to that determined. analytically from. linear perturbation theory. | We compare the power spectrum calculated from the simulation using a 1D Fast Fourier Transform (FFT) to that determined analytically from linear perturbation theory. |
The theoretical LD power spectrum is obtained by integrating the 3-dimensional power spectrum over a window function corresponding to the field-of-view | The theoretical 1D power spectrum is obtained by integrating the 3-dimensional power spectrum over a window function corresponding to the field-of-view |
The theoretical LD power spectrum is obtained by integrating the 3-dimensional power spectrum over a window function corresponding to the field-of-view. | The theoretical 1D power spectrum is obtained by integrating the 3-dimensional power spectrum over a window function corresponding to the field-of-view |
first. inhomogeneity can changee the viewing-anglee> averagedὃν light curve comparedὃν to that of a spherically svnunictric model. | first, inhomogeneity can change the viewing-angle averaged light curve compared to that of a spherically symmetric model. |
Secondly. inhomogeneity could. also [ead to angular dependence of the light. curve the deeree of angular dependence may be reasonably expected. to depend on a combination of the dynamic range of the inhomogencity in clensity ancl the physical length scale of the variations. | Secondly, inhomogeneity could also lead to angular dependence of the light curve – the degree of angular dependence may be reasonably expected to depend on a combination of the dynamic range of the inhomogeneity in density and the physical length scale of the variations. |
The length scale is relevant since if only small scale inhomosgeneity is present it will tend to be averaged: out when integrating over solid angle and so not introduce significant angular variation of the lisht curve. | The length scale is relevant since if only small scale inhomogeneity is present it will tend to be averaged out when integrating over solid angle and so not introduce significant angular variation of the light curve. |
Given that the eode used here adopts several simplifving assumption — most importantly. perhaps. that of a grey absorption coellicient. for radiation — the emphasis here is on understanding ancl assessing the dillercntial ellect of introducing 3D. structure by comparing with an equivalent. 1D. model using a fixed. set. of well-understood approximations. | Given that the code used here adopts several simplifying assumption – most importantly, perhaps, that of a grey absorption coefficient for radiation – the emphasis here is on understanding and assessing the differential effect of introducing 3D structure by comparing with an equivalent 1D model using a fixed set of well-understood approximations. |
Such a calculation is an important first step in understanding the role of 3D structure and is a useful starting point for further work where the micro-physics is improved (sec Section 5 for further discussion) | Such a calculation is an important first step in understanding the role of 3D structure and is a useful starting point for further work where the micro-physics is improved (see Section 5 for further discussion). |
The model used. for this test calculation is based. on the 3D explosion model computed by BRóppke (2005). | The model used for this test calculation is based on the 3D explosion model computed by Röppke (2005). |
For this model. Itóppke (2005) followed. the hvdrodynamics of an exploding white dwarf star. (with total mass 1.4. Al.) for 10 seconds on a schematic grid of 512% Cartesian cells. | For this model, Röppke (2005) followed the hydrodynamics of an exploding white dwarf star (with total mass 1.4 $M_{\odot}$ ) for 10 seconds on a schematic grid of $^{3}$ Cartesian cells. |
Only one spatial octant was simulated and symmetry under reflection was assumed. to describe the remaining octants (thus numerically only 256° erick cells. were used) | Only one spatial octant was simulated and symmetry under reflection was assumed to describe the remaining octants (thus numerically only $^{3}$ grid cells were used). |
The distribution of mass-censity ancl mass-fraction of iron-group elements in cach erid cell at the end of their simulation was mace available for this work. | The distribution of mass-density and mass-fraction of iron-group elements in each grid cell at the end of their simulation was made available for this work. |
Light curves have previously been simulated. from 1D representations of this model by Dlinnikov et al. ( | Light curves have previously been simulated from 1D representations of this model by Blinnikov et al. ( |
2006). | 2006). |
In order to make the calculations here tractable in terms of both computer memory. required. aid. photon statistics in each grid cell the model adopted. here uses only 170 grid cells. | In order to make the calculations here tractable -- in terms of both computer memory required and photon statistics in each grid cell – the model adopted here uses only $^3$ grid cells. |
It was obtained from the 512 grid by first removing the outermost cell from both ends of the &erid in each of the three Cartesian directions resulting in a 510° erid. | It was obtained from the $^3$ grid by first removing the outermost cell from both ends of the grid in each of the three Cartesian directions resulting in a $^3$ grid. |
Phe mass in the cells removed in this process was negligible. | The mass in the cells removed in this process was negligible. |
Phe resolution was then reduced by a [actor of 3 by subdividing the 510 grid into 37 blocks and replacing each block with a single cell whose density was equal to the mean density of the original 27 cells. | The resolution was then reduced by a factor of 3 by subdividing the $^3$ grid into $^3$ blocks and replacing each block with a single cell whose density was equal to the mean density of the original 27 cells. |
Given the grey treatment of radiation currently adopted. in the code. it is not necessary to specify the detailed composition of the material in cach grid cell. | Given the grey treatment of radiation currently adopted in the code, it is not necessary to specify the detailed composition of the material in each grid cell. |
Llowever. it is necessary to specily the initial distribution of Ni whieh provides the source of radiative energy. | However, it is necessary to specify the initial distribution of $^{56}$ Ni which provides the source of radiative energy. |
The ivdrodynamies code used. by Hóppke (2005) provides as estimate of the fractional mass of ivon-group elements in each erid cell but. does not. give a reliable estimate of he breakdown of this material into specific isotopes and elements. | The hydrodynamics code used by Röppke (2005) provides as estimate of the fractional mass of iron-group elements in each grid cell but does not give a reliable estimate of the breakdown of this material into specific isotopes and elements. |
The 7 Ni mass-Lractions used here were obtained w adopting a constant ratio for the mass of NI to the otal mass of ivon-group elements in all eric cells. | The $^{56}$ Ni mass-fractions used here were obtained by adopting a constant ratio for the mass of $^{56}$ Ni to the total mass of iron-group elements in all grid cells. |
This ratio was fixed to vield a total ""Ni-mass of 0.28. Al... as derived. by Iwozma et al. ( | This ratio was fixed to yield a total $^{56}$ Ni-mass of 0.28 $M_{\odot}$, as derived by Kozma et al. ( |
2005) for this model: note hat the nucleosynthesis calculations described. by Kozma et al. ( | 2005) for this model; note that the nucleosynthesis calculations described by Kozma et al. ( |
2005) do not. produce compositional information in sullicient detail to reconstruct. the Lull 3D-clistribution of "Nianass owing to the modest (277) number of tracer particles they used in comparison to the number of erid cells in the 3D. moclel. | 2005) do not produce compositional information in sufficient detail to reconstruct the full 3D-distribution of $^{56}$ Ni-mass owing to the modest $^3$ ) number of tracer particles they used in comparison to the number of grid cells in the 3D model. |
ln order that the cillerential οσοι of the 3D structure could. be assessed. a LD comparison model was mace by averaging the 3D. model over spherical shells. taking care to conserve the radial distributions of total niass ancl 7Ni- mass. | In order that the differential effect of the 3D structure could be assessed, a 1D comparison model was made by averaging the 3D model over spherical shells, taking care to conserve the radial distributions of total mass and $^{56}$ Ni-mass. |
In order to illustrate the degree. of inhomogencity in the 3D model. Figure 7 shows the velocity. cistribution of density. in the spherically averaged model with points indicating the actual densities of individual grid. cells in the 3D model. | In order to illustrate the degree of inhomogeneity in the 3D model, Figure 7 shows the velocity distribution of density in the spherically averaged model with points indicating the actual densities of individual grid cells in the 3D model. |
This shows that across most of the velocity range of the model. there is a spread in density of at least a factor of three between dillerent cells with similar velocities. | This shows that across most of the velocity range of the model, there is a spread in density of at least a factor of three between different cells with similar velocities. |
This is comparable to the dynamic range of density for a eiven velocity in the original hvdrodynamical model: thus one may be confident that the model used here contains a fairly reliable representation of the inhomogeneity implied bv the hyvdrodvnanies. | This is comparable to the dynamic range of density for a given velocity in the original hydrodynamical model; thus one may be confident that the model used here contains a fairly reliable representation of the inhomogeneity implied by the hydrodynamics. |
We continue to. adopt a grey. UVOIR-absorption cross-section. | We continue to adopt a grey -absorption cross-section. |
However. to correctly evaluate the inlluence of 3D structure on the light curve. it is necessary {ο consider that compositional inhomogeneity would cause the cross-section. per gram to be a function. of position. | However, to correctly evaluate the influence of 3D structure on the light curve, it is necessary to consider that compositional inhomogeneity would cause the cross-section per gram to be a function of position. |
Lt goes bevond the scope of this paper to undertake full caleulations of the composition dependence of the opacity: instead. a simple one-parameter description of the opacity is adopted. | It goes beyond the scope of this paper to undertake full calculations of the composition dependence of the opacity; instead, a simple one-parameter description of the opacity is adopted, |
Figure 2 gives a sample of the results from this simulation. | Figure 2 gives a sample of the results from this simulation. |
In all of these images, the vertical (or y) coordinate is parallel to the disk's symmetry axis. | In all of these images, the vertical (or $y$ ) coordinate is parallel to the disk's symmetry axis. |
The top panel shows the intensity 7, of light polarized in the horizontal (or x) direction on a linear color The corresponding intensity I, of light polarized in the y direction appears in the middle panel. | The top panel shows the intensity $I_x$ of light polarized in the horizontal (or $x$ ) direction on a linear color The corresponding intensity $I_y$ of light polarized in the $y$ direction appears in the middle panel. |
The bottom panel is an image (using the same spatial scale as the first two) of the polarization fraction, defined as where F’, and Εν are the fluxes of polarized light calculated in the x and y directions, respectively, by integrating J, and J, over their respective solid angles, with all the usual gravitational area amplification—taken into account. | The bottom panel is an image (using the same spatial scale as the first two) of the polarization fraction, defined as where $F_x$ and $F_y$ are the fluxes of polarized light calculated in the $x$ and $y$ directions, respectively, by integrating $I_x$ and $I_y$ over their respective solid angles, with all the usual gravitational effects---e.g., area amplification—taken into account. |
The interpretation of these results is rather straightforward (see also Bromley et al. | The interpretation of these results is rather straightforward (see also Bromley et al. |
2001). | 2001). |
As we discuss in 3 above, the dominant magnetic field component in this region is expected to be azimuthal. | As we discuss in 3 above, the dominant magnetic field component in this region is expected to be azimuthal. |
The extraordinary emissivity dominates over the ordinary, so most of the y-polarized light comes from the front and rear of the disk, whereas most of the x-polarized light comes from the sides. | The extraordinary emissivity dominates over the ordinary, so most of the $y$ -polarized light comes from the front and rear of the disk, whereas most of the $x$ -polarized light comes from the sides. |
In addition, the disk is assumed to rotate counter-clockwise in this geometry, so the left is blue-shifted, whereas the right is red-shifted. | In addition, the disk is assumed to rotate counter-clockwise in this geometry, so the left is blue-shifted, whereas the right is red-shifted. |
All the general relativistic and emissivity effects together conspire to produce a shift to the left of J, relative to /,. | All the general relativistic and emissivity effects together conspire to produce a shift to the left of $I_x$ relative to $I_y$. |
However, the polarized fraction is not affected by redshift, so the bottom image is symmetric about the vertical axis, and more clearly demonstrates the spatial dependence of the extraordinary versus ordinary emissivities. | However, the polarized fraction is not affected by redshift, so the bottom image is symmetric about the vertical axis, and more clearly demonstrates the spatial dependence of the extraordinary versus ordinary emissivities. |
In principle, the polarization fraction varies over a broad range in values, from —0.7 (the negative sign meaning that J, is dominant over /,) to 0.7 (when J, dominates over {γ). | In principle, the polarization fraction varies over a broad range in values, from $-0.7$ (the negative sign meaning that $I_y$ is dominant over $I_x$ ) to $0.7$ (when $I_x$ dominates over $I_y$ ). |
What we see, however, is an integration of the intensity over the whole image, yielding a single polarized flux, so the net polarization fraction is never quite this high (see figure 4 below). | What we see, however, is an integration of the intensity over the whole image, yielding a single polarized flux, so the net polarization fraction is never quite this high (see figure 4 below). |
The polarimetric images at other energies, say 2—10 keV, are similar to these, and will be reported as part of a more complete survey of results in Falanga et al. ( | The polarimetric images at other energies, say $2-10$ keV, are similar to these, and will be reported as part of a more complete survey of results in Falanga et al. ( |
2011). | 2011). |
But we will continue our discussion of these characteristics in conjunction with figure 4 below. | But we will continue our discussion of these characteristics in conjunction with figure 4 below. |
Before doing so, however, let us examine the impact on our results of a scattering halo. | Before doing so, however, let us examine the impact on our results of a scattering halo. |
Let us assume now that, in addition to the synchrotron emitting ring described above, we also have a | Let us assume now that, in addition to the synchrotron emitting ring described above, we also have a |
The gravitational lens svstem D0218|357 (Patnaiketal.1993) has long been recognised as an excellent. svsteni for measuring the Llubble constant. (44) over cosmological distances using the method of Refsdal (1964). | The gravitational lens system B0218+357 \cite{patnaik93} has long been recognised as an excellent system for measuring the Hubble constant $H_0$ ) over cosmological distances using the method of Refsdal (1964). |
The system has a simple morphology comprising two compact images (X and 13) of a flat-spectrum radio core and a radio Einstein ring (Figs 1.. 2. and 4)). | The system has a simple morphology comprising two compact images (A and B) of a flat-spectrum radio core and a radio Einstein ring (Figs \ref{mer1}, \ref{mer3} and \ref{mervla}) ). |
There is also a kpe-seale radio jet that extends south [rom the bottom of the Einstein ring (seen most clearly in Fig 3)). | There is also a kpc-scale radio jet that extends south from the bottom of the Einstein ring (seen most clearly in Fig \ref{vla}) ). |
Both lens and source redshifts are well determined. at 0.6847. (Browneetal.1993). and. 0.96 (Lawrence1996) respectively. | Both lens and source redshifts are well determined, at 0.6847 \cite{browne93} and 0.96 \cite{lawrence96} respectively. |
In. addition. the time delay between the two compact components has been accurately measured (10.5+0.4 d) from VLA monitoring (Bigesetal. 1999). | In addition, the time delay between the two compact components has been accurately measured $10.5\pm0.4$ d) from VLA monitoring \cite{biggs99}. |
. Preliminary modelling of the mass distribution in the lensinggalaxy using a Singular Isothermal Ellipsoidal (SLE) parameterization (Ixormann.Schneider&Bartelmann19004) gave a value for {4ο of 69τὸ + +. | Preliminary modelling of the mass distribution in the lensinggalaxy using a Singular Isothermal Ellipsoidal (SIE) parameterization \cite{kormann94} gave a value for $H_0$ of $^{+13}_{-19}$ $^{-1}\,$ $^{-1}$. |
"The quoted. error is a 95 per cent confidence limit. (statistical). | The quoted error is a 95 per cent confidence limit (statistical). |
Constraints on this model come from the VLBI substructure of the compact images (Patnaik. Porcas Browne 1995) as well as the Fux densitv ratio measured. from the VLA monitoring. | Constraints on this model come from the VLBI substructure of the compact images (Patnaik, Porcas Browne 1995) as well as the flux density ratio measured from the VLA monitoring. |
A tree parameter in the model is the position of the lensing galaxy centre which is poork determined from(HST) optical and. infra-red observations. | A free parameter in the model is the position of the lensing galaxy centre which is poorly determined from optical and infra-red observations. |
Lehárr et al. ( | Lehárr et al. ( |
2000) have pointed out that the uncertainty in the position of the lensing galaxy implies an uncertainty in /fy that is much greater than that given in Diges ot al. ( | 2000) have pointed out that the uncertainty in the position of the lensing galaxy implies an uncertainty in $H_0$ that is much greater than that given in Biggs et al. ( |
1999). | 1999). |
A potential source of model constraints in DO21IS|357 is he radio brightness distribution in the Einstein ring which jw been neglected until now. | A potential source of model constraints in B0218+357 is the radio brightness distribution in the Einstein ring which has been neglected until now. |
Einstein rings have proved ο be a valuable source of constraints in other lens systems (c.g. MC. 1131|0456 and. PINS 1830-211) as they probe the mass clistribution in the lensing galaxy at many points. thus woviding many more constraints than are available [rom enses that do not contain images of large-scale. extended structure. (IXochanek.1990). | Einstein rings have proved to be a valuable source of constraints in other lens systems (e.g. MG 1131+0456 and PKS 1830-211) as they probe the mass distribution in the lensing galaxy at many points, thus providing many more constraints than are available from lenses that do not contain images of large-scale extended structure \cite{kochanek90}. |
.. Techniques that have. been developed. to. optimise lens models using radio maps of Einstein rings include the Ring €vele algorithm (Ixochaneketal.1989). and. LensClean (Ixochanek&Naravan 1992).. | Techniques that have been developed to optimise lens models using radio maps of Einstein rings include the Ring Cycle algorithm \cite{kochanek89} and LensClean \cite{kochanek92}. . |
The Einstein ring in DO218|357 is believed to be an image of part of the extended: emission. of the kpe-scale | The Einstein ring in B0218+357 is believed to be an image of part of the extended emission of the kpc-scale |
As a matter of fact, if we chose a regularly sampled and unweighted grid point distribution, the Voronoi diagram will describe an ordinary Cartesian mesh. | As a matter of fact, if we chose a regularly sampled and unweighted grid point distribution, the Voronoi diagram will describe an ordinary Cartesian mesh. |
One beneficial property of the Voronoi grid is that it automatically conserves the mass when mapping a density function. | One beneficial property of the Voronoi grid is that it automatically conserves the mass when mapping a density function. |
This is difficult to do on a Cartesian grid because the mass centroids of the cells are dependent on the cell orientation with respect to the underlying model. | This is difficult to do on a Cartesian grid because the mass centroids of the cells are dependent on the cell orientation with respect to the underlying model. |
Furthermore, because of the random orientation and shape of its cells, Voronoi grids do not suffer from the aliasing effects which are inherent to regular Cartesian grids. | Furthermore, because of the random orientation and shape of its cells, Voronoi grids do not suffer from the aliasing effects which are inherent to regular Cartesian grids. |
When the grid points have been distributed throughout the model domain, it is inevitable, due to the stochastic nature of the sampling method, that some points end up much closer than the local separation expectation value and some will be much further apart. | When the grid points have been distributed throughout the model domain, it is inevitable, due to the stochastic nature of the sampling method, that some points end up much closer than the local separation expectation value and some will be much further apart. |
This results in a Delaunay triangulation that is very irregular with some triangles being very long and narrow. | This results in a Delaunay triangulation that is very irregular with some triangles being very long and narrow. |
This irregularity can be remedied by applied what is known as Lloyds algorithm (??),, which iteratively moves a grid point slightly toward its Voronoi cells center of mass. | This irregularity can be remedied by applied what is known as Lloyds algorithm \citep{Lloyd1982,springel2010}, which iteratively moves a grid point slightly toward its Voronoi cells center of mass. |
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