Changeset 40135

Show
Ignore:
Timestamp:
09/13/17 14:11:35 (3 months ago)
Author:
eugene
Message:

minor text edits

Files:
1 modified

Legend:

Unmodified
Added
Removed
  • trunk/doc/release.2015/systematics.20140411/diffusion.tex

    r40134 r40135  
    9696$3\pi$ survey to characterize the behavior of the deep-depletion 
    9797devices used in the Pan-STARRS\,1 Gigapixel Camera.  We have 
    98 identified systematic spatial variations in the photometric behavior and 
    99 stellar profiles which are similar to the so-called ``tree rings'' 
    100 identified in devices used by other wide-field cameras (DECam and 
    101 Hypersuprime Camera).  The tree-ring features identified in these 
    102 other cameras result from lateral electric fields which displace the 
    103 electrons as they are transported in the silicon to the pixel 
    104 location.  In contrast, we show that the photometric and morphological 
    105 modifications observed in the GPC1 detectors are caused by variations 
    106 in the vertical charge transportation rate and resulting charge 
    107 diffusion variations. 
     98identified systematic spatial variations in the photometric 
     99measurements and stellar profiles which are similar in pattern to the 
     100so-called ``tree rings'' identified in devices used by other 
     101wide-field cameras (e.g., DECam and Hypersuprime Camera).  The 
     102tree-ring features identified in these other cameras result from 
     103lateral electric fields which displace the electrons as they are 
     104transported in the silicon to the pixel location.  In contrast, we 
     105show that the photometric and morphological modifications observed in 
     106the GPC1 detectors are caused by variations in the vertical charge 
     107transportation rate and resulting charge diffusion variations. 
    108108\end{abstract} 
    109109 
     
    125125trap electrons, specifically those generated by absorbed photons.  The 
    126126thick silicon substrate required illumination from the ``front'' side 
    127 with the thin gate structures to allow the photons to reach the 
     127containing the thin gate structures to allow the photons to reach the 
    128128depletion region and be detected.  These early CCDs had modest quantum 
    129 efficiency as photons were easily absorbed by the several micron thick 
     129efficiency as photons were easily absorbed by the several-micron-thick 
    130130gate structures.  For an excellent review of the history of CCD 
    131131development, see \cite{1992ASPC...23....1J}. 
     
    137137delicate device only \approx 10 - 20\micron\ thick, exposing the 
    138138depletion region on the backside.  Photons entering the backside of 
    139 the device are not blocked by the gate structures and thus more easily 
    140 absorbed and detected.  Thinned backside-illuminated CCDs have high 
    141 quantum efficiency to blue photons.  However, as the wavelength 
     139the device are not blocked by the gate structures and are thus more 
     140easily absorbed and detected.  Thinned backside-illuminated CCDs have 
     141high quantum efficiency to blue photons.  However, as the wavelength 
    142142increases beyond \approx 800 nm, the silicon becomes more transparent 
    143 to the photons, with a corresponding drop in quantum efficiency for 
    144 red photons.  In addition, thin film interference between the entering 
     143to the photons with a corresponding drop in quantum efficiency for 
     144red photons.  In addition, thin-film interference between the entering 
    145145photons and those reflecting off the front side of the CCD result in 
    146146``fringe'' patterns for redder photons. 
     
    167167 
    168168While these deep-depletion CCDs seem to be ideal, they do have 
    169 features which can cause challenges for precise measurements.  As a 
    170 result of the ``Brighter-Fatter Effect'' 
     169features which can cause challenges for precise measurements.  For 
     170example, as a result of the ``Brighter-Fatter Effect'' 
    171171\citep{2014JInst...9C3048A,2015JInst..10C5032G}, the profile of bright 
    172172stars are measured to be wider than the profiles of faint stars.  The 
     
    177177The effects of lateral electric fields are likewise identified as the 
    178178cause of the so-called ``tree rings'' observed in the flat-field, 
    179 astrometry, and photometry response of thick deep depletion detectors 
     179astrometry, and photometry response of thick deep-depletion detectors 
    180180\citep{2014PASP..126..750P}.  These tree-ring patterns have been noted 
    181181in the flat-field response of deep depletion devices since their early 
     
    189189silicon.  The changes in the effective area result in changes to the 
    190190apparent flat-field response as well as the astrometric response of 
    191 the detector.  More subtly, the flat-field response changes, since 
    192 they do not reflect actual variations in sensitivity, can lead to 
    193 systematic photometry errors for astronomical sources if the 
    194 flat-field images are used in the standard fashion. 
     191the detector.  More subtly, the changes in the flat-field response, 
     192since they do not reflect actual variations in sensitivity, can lead 
     193to systematic photometry errors for astronomical sources if flat-field 
     194images are used in the standard fashion. 
    195195 
    196196In this paper, we examine the behavior of an apparently-similar kind 
    197 of tree ring observed in the Pan-STARRS GPC1 CCDs.  Although we also 
    198 observe the pixel effective area changes caused by lateral electric 
    199 fields as described by \cite{2014PASP..126..750P}, we show below a 
    200 second effect which is more important in driving systematic photometry 
     197of tree-ring pattern observed in the Pan-STARRS\,1 Gigapixel Camera 1 
     198CCDs.  Although we also observe the changes in effective pixel area 
     199caused by lateral electric fields as described by 
     200\cite{2014PASP..126..750P}, we show below a second effect which is 
     201more important in these devices in driving systematic photometry 
    201202errors.  We find that variations in charge diffusion, also resulting 
    202203from changes in the silicon doping structures, affect both the 
    203204observed stellar profiles as well as the photometry measured with 
    204205profile fitting techniques.  In Section~\ref{sec:PS1}, we discuss the 
    205 Pan-STARRS telescope, camera, and survey data used in this analysis. 
    206 In Section~\ref{sec:tree.rings}, we present the tree-ring 
    207 patterns as observed in several different types of measurements: 
    208 flat-field response, systematic photometry residuals, systematic 
    209 astrometric residuals, and stellar profile shape variations.  In 
     206Pan-STARRS\,1 telescope, camera, and survey data used in this analysis. 
     207In Section~\ref{sec:tree.rings}, we present the tree-ring patterns as 
     208observed in several different types of measurements: flat-field 
     209response, systematic photometric residuals, systematic astrometric 
     210residuals, and stellar profile shape variations.  In 
    210211Section~\ref{sec:discussion}, we discuss the interpretation of 
    211212patterns we observe and present a simple model to explain the observed 
     
    219220Haleakala on the Hawaiian island of Maui, has been surveying the sky 
    220221regularly since May 2010 \citep{chambers2017}.  From May 2010 through 
    221 March 2014, PS1 was run under the aegis of the Pan-STARRS Science 
    222 Consortium to perform a set of wide-field science surveys; since March 
    223 2014, operations have been supported primarily by NASA's Near Earth 
    224 Object Observation program, see \cite{2015IAUGA..2251124W}.  Under the 
    225 PS1SC, the largest survey, both in terms of area of the sky covered 
    226 ($3\pi$ steradians) and fraction of observing time (56\%), was the 
    227 \TPS\ in which the entire sky north of Declination $-30$\degrees\ was 
    228 imaged up \approx 80 times over 4 years.  These observations were 
    229 distributed over five filters, \grizy, and have been astrometrically 
    230 and photometrically calibrated to good precision 
    231 \citep{magnier2017.calibration}. 
     222March 2014, PS1 was run under the aegis of the Pan-STARRS\,1 Science 
     223Consortium (PS1SC) to perform a set of wide-field science surveys; 
     224since March 2014, operations have been supported primarily by NASA's 
     225Near Earth Object Observation program 
     226\citep[see][]{2015IAUGA..2251124W}.  Under the PS1SC, the largest 
     227survey, both in terms of area of the sky covered ($3\pi$ steradians) 
     228and fraction of observing time (56\%), was the \TPS\ in which the 
     229entire sky north of Declination $-30$\degrees\ was imaged \approx 80 
     230times over 4 years.  These observations were distributed over five 
     231filters, \grizy, and have been astrometrically and photometrically 
     232calibrated to good precision \citep{magnier2017.calibration}. 
    232233 
    233234% 2004SPIE.5489..667H == PS1.optics 
     
    237238The wide-field PS1 telescope optics \citep{2004SPIE.5489..667H} image 
    238239a 3.3 degree field of view on a 1.4 gigapixel camera 
    239 \citep[GPC1][]{2009amos.confE..40T}, with low distortion and generally 
     240\citep[GPC1;][]{2009amos.confE..40T}, with low distortion and generally 
    240241good image quality.  The median seeing for the \TPS\ data vary 
    241242somewhat by filter: (\grizy) = (1.31, 1.19, 1.11, 1.07, 1.02) 
     
    244245University of Hawaii's Institute for Astronomy operations on Maui. 
    245246 
    246 GPC1 \citep{2009amos.confE..40T}, currently the largest astronomical 
    247 camera in terms of number of pixels, consists of a mosaic of 60 
    248 edge-abutted $4800\times4800$ pixel detectors, with 10~$\mu$m pixels 
    249 subtending 0.258~arcsec. These CCID58 detectors, manufactured by 
    250 Lincoln Laboratory, are 75\micron-thick back-illuminated CCDs 
    251 \citep{2006amos.confE..47T,2008SPIE.7021E..05T}.  Initial performance 
    252 assessments are presented in \cite{2008SPIE.7014E..0DO}. The active, 
    253 usable pixels cover \approx 80\% of the FOV. 
     247GPC1, currently the largest astronomical camera in terms of number of 
     248pixels, consists of a mosaic of 60 edge-abutted $4800\times4800$ pixel 
     249detectors, with 10~$\mu$m pixels subtending 0.258~arcsec. These CCID58 
     250detectors, manufactured by Lincoln Laboratory, are 75\micron-thick 
     251back-illuminated CCDs \citep{2006amos.confE..47T,2008SPIE.7021E..05T}. 
     252Initial performance assessments are presented in 
     253\cite{2008SPIE.7014E..0DO}. The active, usable pixels cover \approx 
     25480\% of the FOV. 
    254255 
    255256\subsection{Data Processing and Calibration} 
     
    268269objects).  In addition, the \TPS\ dataset has been re-processed 
    269270several times with improved calibration and analysis techniques.  To 
    270 date (2017 July), 3 re-processings starting from raw pixel data have 
    271 been performed.  The labels PV0, PV1, PV2, PV3 are used identify the 
    272 nightly processing and successive re-processing versions.  PV3 has 
     271date (2017 September), 3 re-processings starting from raw pixel data 
     272have been performed.  The labels PV0, PV1, PV2, PV3 are used identify 
     273the nightly processing and successive re-processing versions.  PV3 has 
    273274been used for the public release of the Pan-STARRS \TPS\ data via the 
    274275{\it Barbara A. Mikulski Archive for Space Telescopes} (MAST) at the 
    275 Space Telescope Science Institute.\footnote{http//panstarrs.stci.edu} 
     276Space Telescope Science Institute.\footnote{http//panstarrs.stsci.edu} 
     277The process of the construction of this database and the schema 
     278details are discussed in detail by \cite{flewelling2017}. 
    276279 
    277280The data processing and calibration operations are discussed in detail 
     
    328331photometry is re-calibrated within the databasing system based on the 
    329332properties of the measured photometry.  The calibration process is 
    330 discussed by 
    331 \cite{2012ApJ...756..158S,2013ApJS..205...20M,magnier2017.calibration}. 
    332 As part of this process, several flat-field corrections have been 
    333 determined.  For the PV2 analysis discussed here, a flat-field 
    334 correction determined during the ubercal analysis 
    335 \citep[see][]{2012ApJ...756..158S} consisted of an $8\times 8$ grid of 
    336 corrections for each GPC1 chip, corresponding to a correction for each 
    337 OTA ``cell'' and filter for each of 4 seasons.  The boundaries of 
    338 those seasons are tentatively identified with modifications to the 
    339 baffle structures or the system optics.  The critical point here is 
    340 that the final effective flat-field image for the PV2 dataset is based 
    341 on a dome-flat at the highest resolution, with very low resolution 
    342 corrections based on photometry, resulting in photometric systematic 
    343 uncertainties in the range 7 - 12 millimagnitudes, depending on the 
    344 filter \citep{2013ApJS..205...20M}. 
     333discussed by \cite{2012ApJ...756..158S} and 
     334\cite{2013ApJS..205...20M,magnier2017.calibration}.  As part of this 
     335process, several flat-field corrections have been determined.  For the 
     336PV2 analysis discussed here, a flat-field correction determined during 
     337the ubercal analysis \citep[see][]{2012ApJ...756..158S} consisted of 
     338an $8\times 8$ grid of corrections for each GPC1 chip, corresponding 
     339to a correction for each OTA ``cell'' and filter for each of 4 
     340seasons.  The boundaries of those seasons are tentatively identified 
     341with modifications to the baffle structures or the system optics.  The 
     342critical point here is that the final effective flat-field image for 
     343the PV2 dataset is based on a dome-flat at the highest resolution, 
     344with very low resolution (hundreds of pixels) corrections based on 
     345photometry, resulting in photometric systematic uncertainties in the 
     346range 7 - 12 millimagnitudes, depending on the filter 
     347\citep{2013ApJS..205...20M}. 
    345348 
    346349For all objects, positions are measured from the PSF model for the 
     
    400403For all of these examples, we use a single GPC1 CCD (XY40) to 
    401404illustrate the effects in detail, but a similar set of effects are 
    402 seen in many of the GPC1 detectors.  First, we show the residual PSF 
    403 photometry.  Second, we show the residual aperture photometry.  Third, 
    404 we show the astrometric residual patterns.  Fourth, we show the 
    405 patterns observed in the flat-field images.  Finally, we show 
    406 measurements derived from the second-moments of the stars. 
     405seen in many, if not all, of the GPC1 detectors with varying 
     406strengths.  First, we show the residual PSF photometry.  Second, we 
     407show the residual aperture photometry.  Third, we show the astrometric 
     408residual patterns.  Fourth, we show the patterns observed in the 
     409flat-field images.  Finally, we show measurements derived from the 
     410second-moments of the stars. 
    407411 
    408412For all effects discussed below, we are measuring the mean value of 
     
    486490aperture photometry instead of PSF photometry.  The finging 
    487491pattern again dominates the plot for \yps, but the tree rings are not 
    488 seen in any of the filters.  A diagonal pattern is visible in \gps 
     492seen in any of the filters.  A diagonal pattern is visible in \gps\ 
    489493which is not observed in the PSF magnitudes.  While the per-pixel 
    490494scatter is somewhat (10\% to 20\%) higher for these aperture 
     
    523527superpixel.  We have determined the approximate center of the circular 
    524528tree-ring pattern as (-5,4960) for this particular chip based on the 
    525 pattern of the X astrometry displacements.  Using this coordinate as the center 
    526 of the pattern, we have converted the $\delta X,\delta Y$ offsets into 
    527 $\delta R,\delta \theta$ measurements ($\delta R$ : radial component 
    528 away from the center, $\delta \theta$ : tangential component). 
     529pattern of the X astrometry displacements.  Using this coordinate as 
     530the center of the pattern, we have converted the $\delta X,\delta Y$ 
     531offsets into $\delta R,\delta \theta$ measurements ($\delta R$ : 
     532radial component away from the center of the pattern, $\delta \theta$ 
     533: tangential component). 
    529534 
    530535Figure~\ref{fig:astrom.by.filter} shows the 2D patterns of $\delta R$ 
     
    534539following a circular pattern centered on the chip corner; the finging 
    535540pattern is not apparent in the \yps\ astrometry.  The per-pixel 
    536 standard deviations of these plots area listed in 
     541standard deviations of these plots are listed in 
    537542Table~\ref{table:sigmas.by.filter}.  The signal-to-noise of these 
    538543structures is again somewhat weak, but the pattern is clearly visible 
     
    588593strong in the (\gps,\rps,\ips) images, but nearly swamped by fringing 
    589594in \zps, and completely lost to finging in \yps.  A diagonal banding 
    590 pattern is seen in \gps: this features is thought to be due to the 
    591 lithography process used to generate the CCD.  A blob can also been 
    592 seen covering 4 cells near the center of this chip; this is apparently 
    593 a deposit of some kind on the detector.  Both of the latter two 
    594 effects behave like quantum efficiency variations and are removed well 
    595 by standard flat-field techniques.  Note that a small amount of the 
    596 diagonal banding pattern remains in the aperture magnitude residuals 
    597 for \gps.  For the rest of this article, we ignore these features and 
    598 concentrate on the tree ring features. 
     595pattern is also seen in \gps: this feature is thought to be due to 
     596the lithography process used to generate the CCD.  A blob can also 
     597been seen covering 4 cells near the center of this chip; this is 
     598apparently a deposit of some kind on the detector.  Both of the latter 
     599two effects behave like quantum efficiency variations and are removed 
     600well by standard flat-field techniques.  Note that a small amount of 
     601the diagonal banding pattern remains in the aperture magnitude 
     602residuals for \gps.  For the rest of this article, we ignore these 
     603features and concentrate on the tree-ring features. 
    599604 
    600605In order to suppress the large-scale structures for a quantitative 
     
    645650$\sigma_{w}$.  (Note that, since the measured $\sigma$ of stellar 
    646651objects is biased down by the weighting function, this is not quite 
    647 the same as having $\sigma_{w} = 1.6$ times the true PSF $\sigma$, see 
     652the same as having $\sigma_{w} = 1.6$ times the true PSF $\sigma$; see 
    648653discussion in \citealt{magnier2017.analysis}).  For each stellar 
    649654detection, we extract the values $M_{xx,xy,yy} = \sum F_i w_i (x^2, x 
     
    677682PSF ellipticity from the $e_1$ term. 
    678683 
    679 Figure~\ref{fig:smear.by.filter} shows the spatial trend of $e_0$, the {\em 
    680   smear}.  This value corresponds to the increase or decrease in 
    681 the circularly-symmetric component of the image size.  The dynamic 
    682 range of these images is -0.3 to +0.3 pixel$^2$. A tree-ring 
    683 pattern is visible for all 5 filters, though \yps is dominated by the 
    684 fringing pattern.  Structures with relatively low spatial frequencies 
    685 can also be seen. 
    686  
    687 Figure~\ref{fig:shear.by.filter} shows the spatial trend of $e_2$, the 
    688 {\em shear}.  This value is positive definite and is plotted with a 
    689 color scale ranging from -0.02 to 0.22 pixel$^2$.  We can also 
    690 determine the orientation of the corresponding ellipse.  Overlayed on 
     684Figure~\ref{fig:smear.by.filter} shows the spatial trend of the smear, 
     685$e_0$.  The dynamic range of these images is -0.3 to +0.3 pixel$^2$. A 
     686tree-ring pattern is visible for all 5 filters, though \yps\ is 
     687dominated by the fringing pattern.  Structures with relatively low 
     688spatial frequencies can also be seen. 
     689 
     690Figure~\ref{fig:shear.by.filter} shows the spatial trend of the shear, 
     691$e_2$.  This value is positive definite and is plotted with a color 
     692scale ranging from -0.02 to 0.22 pixel$^2$.  Overlayed on 
    691693Figure~\ref{fig:shear.by.filter} is a set of vectors representing the 
    692694ellipse orientation as a function of postion.  The length of the 
    693 vectors corresponds to the value of $\sigma^2_{major} - 
    694 \sigma^2_{minor}$.  The tree-ring structure is {\em not} apparent 
    695 in this figure for any filter.  The spatial variations are 
    696 low-frequency and unrelated to the radial trend from the upper-left 
    697 corner. 
     695vectors corresponds to the value of $e_2$.  The tree-ring structure is 
     696{\em not} apparent in this figure for any filter.  The spatial 
     697variations are low-frequency and unrelated to the radial trend from 
     698the upper-left corner. 
    698699 
    699700\subsection{Correlations Between Tree-Ring Patterns} 
     
    741742signal further. 
    742743 
    743 To quantatatively compare the tree-ring trends between 
    744 filters and between the types of measurements, we need to measure the 
    745 tree-ring structure explicitly and filter out the other effects if 
    746 possible.  To do this, we have applied a high-pass filter to all of 
    747 the relevant images (PSF photometry residuals, astrometric residuals 
    748 in the radial direction, flat-field residuals, and second moment smear 
    749 terms) to remove unrelated spatial structures.  We have then measured 
    750 the median of the signal in radial bins centered on (-5,4960) across 
    751 an arc from $\phi$ = -20\degrees\ to -50\degrees (as measured relative 
    752 to the top row of the images.  We have selected a small fraction of 
    753 the arc to minimize the error associated with the choice of the 
    754 pattern center and to avoid several bad cells near the bottom of the 
    755 chip. 
     744To quantitatively compare the tree-ring trends between filters and 
     745between the types of measurements, we need to measure the tree-ring 
     746structure explicitly and filter out the other effects if possible.  To 
     747do this, we have applied a high-pass filter to all of the relevant 
     748images (PSF photometry residuals, astrometric residuals in the radial 
     749direction, flat-field residuals, and second moment smear terms) to 
     750remove unrelated spatial structures.  We have then measured the median 
     751of the signal in radial bins centered on (-5,4960) across an arc from 
     752$\phi$ = -20\degrees\ to -50\degrees (as measured relative to the top 
     753row of the images).  We have selected a small fraction of the arc to 
     754minimize the error associated with the choice of the pattern center 
     755and to avoid several bad cells near the bottom of the chip. 
    756756 
    757757% \note{include the arc on one of the figures?} 
     
    852852astrometric residual is anti-correlated with the flat-field residual 
    853853errors: $\frac{\partial \delta R}{\partial radius} \sim \delta flat$ 
    854 (see Figure~\ref{fig:dastrom.vs.flat}.  This last relationship is 
    855 somewhat weakly measured.  Because of the periodic nature of the Tree 
    856 Rings, it is also difficult to be completely certain that the 
     854(see Figure~\ref{fig:dastrom.vs.flat}).  This last relationship is 
     855somewhat weakly measured.  Because of the periodic nature of the tree 
     856rings, it is also difficult to be completely certain that the 
    857857flat-field is proportional to the derivative of the astrometry 
    858858residual, rather than the astrometry residual being proportional to 
     
    862862residual values without a derivative.  We are convinced that we have 
    863863the sense of the derivative correct by examination of specific 
    864 features in each imaage. 
     864features in each image. 
    865865 
    866866\begin{table} 
     
    988988below the pixel-to-pixel noise in the aperture magnitude residuals. 
    989989It is likely in our opinion that the plate-scale changes causing the 
    990 flat-field and astrometry effects is affecting both the ellipticity 
     990flat-field and astrometry effects are affecting both the ellipticity 
    991991and the aperture magnitudes, but the level of the effect is too small 
    992992to see given the other systematic structures (in the shear plot) and 
     
    996996astrometry residuals shows that these two effects are connected. 
    997997Although the correlation is weak in Figure~\ref{fig:dsmear.vs.astrom}, 
    998 careful inspection of the location of the these two tree ring patterns 
     998careful inspection of the location of these two tree ring patterns 
    999999shows that the locations of the rings in the radial astrometric 
    10001000residual images occurs at the boundaries between regions with 
     
    10191019between these regions. 
    10201020 
    1021 We interpret the changes in the {\em smear} term as changes in the 
    1022 amount of charge diffusion as the photoelectrons travel to the bottom 
    1023 of the pixel well.  The blue filters exhibit the strongest changes in 
    1024 the amount of smear.  These are also the filters for which the 
    1025 detected electrons have travelled the longest distance in the silicon, 
    1026 and are thus most affected by diffusion effects.  Charge diffusion (as 
    1027 opposed to the charge drift caused by the lateral electric fields) 
    1028 results in a Gaussian smearing of the stellar profile: as the 
    1029 photoelectrons migrate from the site where they were generated by the 
    1030 incoming photon to the bottom of the pixel well, they follow a random 
    1031 walk in the plane of the detector.  The longer the electrons take to 
    1032 make the journey down to the bottom of the pixel, the further they are 
    1033 able to wander from their creation coordinate in the detector. 
    1034 Following the discussion in \cite{Holland.2003}, the amount of charge 
    1035 diffusion is thus related to the velocity of the electrons in the 
    1036 direction of the optical axis: $\sigma \sim \sqrt{2Dt}$ where $\sigma$ 
    1037 is the size of the smearing kernel, $t$ is the time required for the 
    1038 electrons to traverse the thickness of the silicon wafer, and $D$ is 
    1039 the diffusion coefficient.  The velocity of the photoelectron, and 
    1040 thus the time to traverse the silicon, is related to the vertical 
    1041 electric fields in the silicon, which are caused by a combination of 
    1042 the applied voltages and the distribution of the space charges from 
    1043 the dopant.  As shown by \cite{Holland.2003}, the charge diffusion is 
    1044 related to the space charge density by $\sigma \sim 
    1045 \rho^{-\frac{1}{2}}$ (their equation 6).  Regions with high space 
    1046 charge densities increase the migration speed of the photoelectrons 
    1047 and reduce the amount of charge diffusion smearing; and vice versa for 
    1048 regions of low space-charge densities.  
     1021We interpret the changes in the smear term as changes in the amount of 
     1022charge diffusion as the photoelectrons travel to the bottom of the 
     1023pixel well.  The blue filters exhibit the strongest changes in the 
     1024amount of smear.  These are also the filters for which the detected 
     1025electrons have travelled the longest distance in the silicon, and are 
     1026thus most affected by diffusion effects.  Charge diffusion (as opposed 
     1027to the charge drift caused by the lateral electric fields) results in 
     1028a Gaussian smearing of the stellar profile: as the photoelectrons 
     1029migrate from the site where they were generated by the incoming photon 
     1030to the bottom of the pixel well, they follow a random walk in the 
     1031plane of the detector.  The longer the electrons take to make the 
     1032journey down to the bottom of the pixel, the further they are able to 
     1033wander from their creation coordinate in the detector.  Following the 
     1034discussion in \cite{Holland.2003}, the amount of charge diffusion is 
     1035thus related to the velocity of the electrons in the direction of the 
     1036optical axis: $\sigma \sim \sqrt{2Dt}$ where $\sigma$ is the size of 
     1037the smearing kernel, $t$ is the time required for the electrons to 
     1038traverse the thickness of the silicon wafer, and $D$ is the diffusion 
     1039coefficient.  The velocity of the photoelectron, and thus the time to 
     1040traverse the silicon, is related to the vertical electric fields in 
     1041the silicon, which are caused by a combination of the applied voltages 
     1042and the distribution of the space charges from the dopant.  As shown 
     1043by \cite{Holland.2003}, the charge diffusion is related to the space 
     1044charge density by $\sigma \sim \rho^{-\frac{1}{2}}$ (their equation 
     10456).  Regions with high space charge densities increase the migration 
     1046speed of the photoelectrons and reduce the amount of charge diffusion 
     1047smearing; and vice versa for regions of low space-charge densities. 
    10491048 
    10501049In summary, the variations in the space-charge density caused by 
     
    10751074\section{Conclusion} 
    10761075 
    1077 The tree rings observed in the Pan-STARRS GPC1 data show (at least) 
    1078 two effects, though they are related.  First, the images are 
    1079 experiencing circularly-symmetric changes in the PSF size correlated 
    1080 with the tree-ring pattern.  These PSF size changes drive errors in 
    1081 the PSF photometry on the scale of a few millimagnitudes, are also 
     1076The tree rings observed in the Pan-STARRS GPC1 data show two different 
     1077effects, though they are related.  First, the images are experiencing 
     1078circularly-symmetric changes in the PSF size correlated with the 
     1079tree-ring pattern.  These PSF size changes drive errors in the PSF 
     1080photometry on the scale of a few millimagnitudes, and are also 
    10821081correlated with the tree-ring pattern.  These PSF size changes are 
    10831082consistent with changes in the charge diffusion, which also introduces 
     
    10851084 
    10861085In addition, there are radial plate-scale changes correlated with the 
    1087 tree rings.  These plate-scale changes introduce a flat-field errors 
    1088 on the scale of \approx 1 millimagnitude and astrometric errors in the 
     1086tree rings.  These plate-scale changes introduce flat-field errors on 
     1087the scale of \approx 1 millimagnitude and astrometric errors on the 
    10891088scale of 2-3 milliarcseconds.  The observed relationship between the 
    10901089flat-field deviations and the radial derivative of the astrometric 
     
    11541153Lorand University (ELTE) and the Los Alamos National Laboratory. 
    11551154 
    1156 \note{Ken: please add NASA ops grants} 
     1155% \note{Ken: please add NASA ops grants} 
    11571156 
    11581157\bibliographystyle{apj}