IPP Software Navigation Tools IPP Links Communication Pan-STARRS Links

Changeset 40135


Ignore:
Timestamp:
Sep 13, 2017, 2:11:35 PM (9 years ago)
Author:
eugene
Message:

minor text edits

File:
1 edited

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}
Note: See TracChangeset for help on using the changeset viewer.