Wide Field and Planetary Camera 2 Instrument Handbook for Cycle 14
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5.1 Effects of OTA Spherical AberrationChapter 5:
Point Spread
Function
5.2 Aberration Correction
5.3 Wavefront Quality
5.4 CCD Pixel Response Function
5.5 Model PSFs
5.6 PSF Variations with Field Position
5.6.1 Aperture Corrections vs. Field Position
5.7 PSF Variations with Time / OTA Focus
5.8 PSF Anomaly in F1042M Filter
5.9 Large Angle Scattering
5.10 Ghost Images
5.11 Optical Distortion
5.1 Effects of OTA Spherical Aberration
The OTA spherical aberration produces a Point Spread Function (PSF-the apparent surface brightness profile of a point source), as presented to the instruments, with broad wings. Briefly, the fraction of the light within the central 0.1" was reduced by a factor of about 5. The resulting PSF had "wings" which extended to large radii (several arcseconds), greatly reducing the contrast of the images and degrading the measurements of sources near bright objects or in crowded fields. Burrows, et al. (1991, Ap. J. Lett. 369, L21) provide a more complete description of the aberrated HST PSF. Figure 5.1 shows the PSF in three cases.
Figure 5.1: PSF Surface Brightness. The percentage of the total flux at 4000Å falling on a PC pixel as a function of the distance from the peak of a star image.
It shows the aberrated HST PSF, the WFPC2 PSF, and for comparison the PSF that would be obtained from a long integration if HST were installed at a ground based observatory with one arcsecond seeing. All of the PSFs were computed at 4000Å. The FWHM of the image both before and after the installation of WFPC2 is approximately proportional to wavelength, at least before detector resolution and MTF effects are considered. (The WF/PC-1 core was approximately 50% broader than the core that is obtained with WFPC2). Figure 5.2 shows the encircled energy (EE), the proportion of the total energy from a point source within a given radius of the image center, for the same three cases.
Figure 5.2: Encircled Energy. The percentage of the total flux at 4000Å within a given radius of the image peak.
The WFPC2 curve shown is the average of measurements taken with F336W and F439W. It can be seen that the core of the image in WFPC2 contains most of the light. At this wavelength, 65% of the light is contained within a circle of radius 0.1". However, this proportion is considerably less than the optics deliver. The reason for this is discussed in CCD Pixel Response Function. Encircled energy curves for other filters are shown in Figure 5.3 and Figure 5.4; note that these curves are normalized to unity at 1.0" radius.
Figure 5.3: Encircled Energy for CCD PC1. The fraction of energy encircled is plotted vs. aperture radius for several filters. Curves are normalized to unity at a radius of 1.0". From Holtzman, et al. 1995a.
Figure 5.4: Encircled Energy for CCD WF3. The fraction of energy encircled is plotted vs. aperture radius for several filters. Curves are normalized to unity at a radius of 1.0". From Holtzman, et al. 1995a.
5.2 Aberration Correction
WFPC2 has corrective figures on the relay secondary mirrors where the primary mirror is imaged; this optical correction recovers near-diffraction limited images over the entire CCD fields-of-view. Proper correction requires tight optical alignment tolerances, which are facilitated on-orbit by actuated optics. The corrective optics enable essentially all of the scientific objectives of the original WF/PC-1 to be met.
Table 5.1: Wavefront Error Budget.Camera WFC (F/12.9) PC(F/28.3) Design error /143
Fabrication and alignment error Alignment stability error Total wavefront error
Through a number of independent analyses, based on investigations of star images obtained on-orbit, and the examination of fixtures used during the figuring of the primary mirror, the aberrations of the HST optics were accurately characterized. The primary mirror was figured to an incorrect conic constant: -1.0139±0.005 rather than the -1.0023 design requirement, resulting in a large amount of spherical aberration. The optical design of WFPC2 creates an image of the OTA primary mirror near the surface of the relay Cassegrain secondary mirror in each of its channels. This design minimizes vignetting in the relay optics, but more importantly, facilitates correction of spherical aberration in the OTA primary by application of the same error (but with opposite sign) to the relay secondary. The optical figure of the WFPC2 secondary mirrors contains a compensating "error" in the conic constant. By adopting a prescription within the error bars for the HST primary mirror, corrective secondary mirrors were made with sufficient accuracy that the residual spherical aberration in the WFPC2 wavefront is small compared to other terms in the WFPC2 optical wavefront budget.
On the other hand, new and stringent alignment requirements were created by the steep optical figure on the corrective relay secondary mirrors. The primary mirror image must be accurately centered on the corrective mirror, and must have the correct magnification. Centering is the most demanding requirement. A failure to center accurately would create a new aberration in the form of coma. A misalignment of 7% of the pupil diameter introduces as much RMS wavefront error as was present in the form of spherical aberration prior to the introduction of corrective optics. The new requirements for alignment accuracy and stability led to the introduction of a tip-tilt mechanism on the pick-off mirror, to compensate for camera alignment uncertainties with respect to the OTA, and actuated fold mirrors which can compensate for internal misalignments. There was an additional term in the CEIS specification of the overall instrument wavefront error budget for alignment stability. It is
"Design error" refers to the aberrations inherent in the design itself, which would be seen if the optics conformed perfectly to their specifications. All of the optics were fabricated and integrated into the WFPC2 optical bench. It was established on the basis of component tests, end-to-end optical interferometry, and through focus phase retrieval, that the WFPC2 optical system performed within the stated tolerances for "fabrication and alignment" in the laboratory environment. What remained was to demonstrate the stability of the optical alignment after launch vibration and in response to the thermal environment on-orbit. The "stability" line anticipated these uncertainties, and was verified during early science operations.
5.3 Wavefront Quality
The conclusion of the extensive optical testing in Thermal Vacuum was that the cameras are well corrected to within the specifications. The dominant problem was a small difference in focus between the four cameras (Krist and Burrows 1995). The actuated fold mirrors and pick-off mirror mechanism performed flawlessly in correcting residual coma aberrations in the image, and enabled the on-orbit alignment procedures. Using out-of-focus images, a very accurate alignment of the cameras was accomplished. A side product was that the aberrations in each camera were measured (Krist and Burrows, Applied Optics, 1995). The results are given in Table 5.2. These values were used in generating the simulated PSFs given in Model PSFs. The WF3 wavefront error is higher than that of the other chips because it is the most out-of-focus relative to the PC (which is assumed to be in focus). It is the equivalent of about 10 microns of breathing out-of-focus.
Table 5.2: Aberrations in Each Camera. The numbers quoted are RMS wavefront errors in microns over the HST aperture (Zernike coefficients).
5.4 CCD Pixel Response Function
From Thermal Vacuum testing, there was evidence that the images are not as sharp as expected, despite the good wavefront quality. The decrease in sharpness corresponds to a loss in limiting magnitude of about 0.5 magnitudes in the WF cameras, and less in the PC.
Further testing, by covering a flight spare CCD with a 2µm pinhole grid in an opaque metallic mask and illuminating it with a flat field source, showed that even when a pinhole was centered over a pixel only about 70% of the light was detected in that pixel.
For practical purposes, the effect can be modeled as equivalent to about 40 mas RMS gaussian jitter in the WFC, and 18 mas in the PC (as compared with the typical real pointing jitter of ~3 mas delivered by the excellent HST pointing control system). Alternatively, at least in the V band, it can be modeled by convolving a simulated image by the following kernel, which gives the pixel response function averaged within pixels:
One clue is the wavelength dependence of the observed sharpness: the results from the 2µm pinhole grid test get worse at longer wavelengths. This may reflect the greater penetration into the silicon of low energy photons, which facilitates the diffusion of photoelectrons across the pixel boundaries defined by the frontside gate structure.
There is also evidence for sub-pixel QE variations at the 10% level. There is an implied dependence on pixel phase for stellar photometry. This has been seen at about the 1-3% level in on-orbit data. The work of Jorden, Deltorn, and Oates (Greenwich Observatory Newsletter 9/93) has yielded quite similar results, and suggests that sub-pixel response must be taken into account when seeking to understand the behavior of all CCD detectors forming undersampled images.
5.5 Model PSFs
Considerable effort has gone into the modeling of the HST point spread function (PSF), both in order to measure the optical aberrations in support of the WFPC2, COSTAR, and advanced scientific instruments, and to provide PSFs for image deconvolution in the aberrated telescope. Such PSFs are noise free and do not require valuable HST observing time. Software to generate model PSFs for any filter and at any location within the field-of-view is available from the STScI (TIM package, Hasan and Burrows 1993; TinyTIM package, Krist 1995). The results are illustrated in Table 5.3 and Table 5.4 for the PC1 and WF2 cameras, respectively. A representative PSF is on the left in each panel. It meets the wavefront error budget, with the measured mix of focus, coma, astigmatism, and spherical aberration. It has been degraded by the pixel response function as discussed in Section CCD Pixel Response Function. On the right is the diffraction limited case for comparison. In each case the percentage of the total flux in a central 5x5 pixel region of a point source is displayed. The peak of the star image can be at an arbitrary point relative to the boundaries of the CCD pixels. Two cases are shown: one where the star is approximately centered on a pixel, and one where it is approximately centered at a pixel corner. As a consequence of the under-sampling in the WFPC2, the limiting magnitude attainable in the background limit varies by about 0.5 magnitude, depending on the position of the source within the CCD pixel. This point is discussed in more detail in Chapter 6.
Neither observed nor modeled PSFs will provide a perfect match to the PSF in actual science observations, due to modeling uncertainties, the "jitter" in the HST pointing, and orbit to orbit variations in telescope focus ("breathing"-which seems to be generally limited to about 1/20 wave peak-to-peak). Jitter is not predictable but can be recovered to a reasonable extent for observations obtained in Fine Lock. In long exposures, up to about 10 mas of apparent pointing drift may occur as a result of the breathing effects in the FGS, although smaller variations of ~3 mas are typical.
5.6 PSF Variations with Field Position
The WFPC2 PSFs vary with field position due to field-dependent aberrations, obscuration shifting, and scattering. This complicates photometry, PSF subtraction, and deconvolution (Krist, 1995).
The coma and astigmatism aberrations vary significantly within a camera across the field-of-view. These variations are simply part of the optical design. At the extreme corners of the WFC CCDs, away from the OTA axis, there is about 1/5 wave of astigmatism (referenced at 633 nm), which decreases to nearly zero at the CCD centers. Astigmatism at this level causes the PSF core to become elliptical and slightly less sharp; note the flattening of the PSF at pixel positions (54,777) and (605,148) in Figure 5.5. Coma also varies, but to a much lesser extent. Coma and astigmatism variations are considerably smaller in PC1 (though we note the astigmatism at the center of PC1 is fairly significant - see Table 5.2).
Figure 5.5: PSF Variations with Field Position - Aberrations. Nine observed PSFs (filter F814W) are shown from a widely spaced grid on WF3. CCD pixel positions are labeled. Note the flattening of the PSF in the (54,777) and (605,148) positions.
The obscuration patterns due to the camera optics (relay secondary mirror and spiders) appear to shift with respect to the OTA obscurations, depending on field position. The interacting diffraction patterns of the WFPC2 and OTA spiders cause ripples in the spider diffraction spikes, which vary with field position as the two spiders shift relative to each other. In Figure 5.6 the OTA spider is hidden behind the WFPC2 spider at the field center and hence the diffraction spikes there have a simple, smooth appearance (c.f. position 446,425). At the CCD corners, however, one or more vanes of the OTA spider move out from behind the WFPC2 spider, and the double set of obscurations causes a "beating" pattern in the diffraction spikes.
The spiders also interact with light diffracted from zonal errors in the OTA mirrors, causing streaks in the scattering halo which vary in position and intensity.
Figure 5.6: PSF Variations with Field Position - Obscuration Shifts. Five saturated PSFs observed in F814W are shown from a widely spaced grid on WF4. Note the changes in the spider diffraction spikes. CCD pixel positions are labeled. The vertical feature is caused by saturation and blooming (see Blooming).
5.6.1 Aperture Corrections vs. Field Position
The amount of energy encircled by an aperture used for stellar photometry will depend on the aperture size, and on any variations in the PSF with field position, time, etc. In general, larger apertures will provide more stable results in the presence of PSF variations. However, large apertures will also exacerbate many problems: contamination from residual cosmic rays, scattered light from nearby stars, and the lower signal-to-noise (S/N) that typically results.
Gonzaga et al. (1999) have measured aperture corrections and characterized their change as a function of field position and filter. The differences in photometric magnitude between apertures with various radii (i.e. aperture corrections), and their mean and standard deviations for the F555W filter, are presented in Table 5.5. For example, the first row of the table indicates that stars measured with a 1 pixel radius aperture will be about 0.887 magnitude fainter than if a 5 pixel radius aperture were used (averaged over entire PC CCD), and this difference will vary by about 0.054 magnitudes RMS across the CCD.
Variations in the PSF with field position will, of course, cause a position dependence in the aperture corrections. Figure 5.7 illustrates how the aperture correction varies with distance from the CCD center, R, for different pairs of aperture sizes. The scatter in the plots is due to contamination from residual cosmic rays and nearby faint stars within the larger aperture. While the data are somewhat incomplete, a clear trend is present: the aperture correction generally increases linearly as a function of distance from the CCD center. For example, the aperture correction between 1 to 5 pixel radius is about 0.82 magnitudes at the PC center, and increases to about 0.94 magnitude at the far corners of the CCD. (The average correction is about 0.89 magnitude, as given in the first line of Table 5.5.) The other WFPC2 CCD chips show results similar to the PC chip.
Table 5.5: Magnitude differences produced by different aperture sizes. Results given for PC, WF2, WF3, and WF4 in F555W.PC F555W 1 vs. 5 116 0.887 0.054 PC F555W 2 vs. 5 115 0.275 0.028 PC F555W 2 vs. 10 115 0.401 0.075 PC F555W 5 vs. 10 115 0.106 0.055 WF2 F555W 1 vs. 5 558 0.608 0.130 WF2 F555W 2 vs. 5 558 0.160 0.085 WF2 F555W 2 vs. 10 544 0.310 0.257 WF2 F555W 5 vs. 10 548 0.133 0.204 WF3 F555W 1 vs. 5 660 0.680 0.133 WF3 F555W 2 vs. 5 656 0.188 0.076 WF3 F555W 2 vs. 10 649 0.376 0.308 WF3 F555W 5 vs. 10 647 0.154 0.233 WF4 F555W 1 vs. 5 828 0.672 0.129 WF4 F555W 2 vs. 5 831 0.198 0.115 WF4 F555W 2 vs. 10 815 0.386 0.350 WF4 F555W 5 vs. 10 814 0.160 0.252
1Magnitude difference averaged around CCD.
2RMS magnitude difference around CCD.
In practice, the aperture correction also depends on defocus. The interplay between aperture correction and defocus may be complex, since the optimal focus changes with field position. A full correction has not been established, but the TinyTIM PSF model (see Section 5.7) can be used to estimate the extent of the variation in the aperture correction. In general, we recommend that a minimum aperture radius of 2 pixels be used whenever possible, in order to reduce the impact of variations of the aperture correction with focus and field position. If the field is too crowded and a smaller aperture is needed, we recommend that users verify the validity of the corrections on a few well-exposed stars.
The following section includes a discussion of aperture corrections as a function of OTA focus.
Figure 5.7: Aperture correction (delta) between two given apertures within the PC chip versus radial distance of the target from the center of the chip. Open symbols indicate spurious data.
5.7 PSF Variations with Time / OTA Focus
The shape and width of observed PSFs varies slightly over time, due to the change in focus of the telescope. The focus variation consists of two terms: a secular change due to the ongoing shrinkage of the Metering Truss Assembly at an estimated rate of 0.25 µm per month in 1995 (Note: the shrinkage essentially stopped around 2000), and short-term variations, typically on an orbital time-scale (the so-called "breathing" of the telescope, see Figure 5.9). The breathing is probably due to changes in the thermal environment as the telescope moves through its orbit, and has a typical peak-to-peak amplitude of 4 µm; larger variations are occasionally seen.
These small focus shifts will impact photometry performed with small (few pixel radius) apertures. Typical ±2 µm focus shifts will result in photometric variations in the PC1 of 6.8%, 4.5%, 2.0%, and 0.2% for aperture radii of 1, 2, 3, and 5 pixels, respectively, in F555W. This is based on the focus monitoring data taken over the period from January 1994 to February 2003 (see Figure 5.9). Hence, "breathing" is often one of the major sources of errors for small-aperture photometry. However, relative photometry (i.e. the difference in magnitudes of stars in the same image) is less affected by this variation, since all the stars in an image tend to be impacted by the defocusing in a similar way.
Figure 5.8: HST Focus Trend over Mission Life.
Up-to-date focus information is maintained on our web page at:
Figure 5.9: Measured OTA Focus Position (microns) as Function of Days since January 1, 1994. The focus position is defined as the difference between the optimal PC focus and the measured focus, in microns at the secondary mirror. Times and size of OTA focus adjustments are indicated along the bottom of the plot.http://www.stsci.edu/instruments/observatory/focus
Figure 5.10: Measured Aperture Correction, V(r) - V(r=10 pix), in Magnitudes as Function of Shift from Optimal Focus. Data are given for aperture radii r=1, 2, 3, and 5 pixels for F555W filter on CCD PC1.
Systematic errors due to the secular focus drift can be corrected using aperture corrections as a function of focus change (see Figure 5.10). The aperture correction adjusted for focus change is hence:
ap_corr = ap_corr_nominal + a(r) x dwhere ap_corr_nominal is the nominal aperture correction (mag) as derived from Table 2a in Holtzman et al. (1995a), a(r) is the flux variation per 1 µm of focus drift (mag per micron) using an aperture with radius r (pixels), and d (µm) is the focus shift from the nominal position. The monitoring data mentioned above yield for PC1 and F555W, the following values for a(r):
Suchkov and Casertano (1997) provide further information on aperture corrections. They find that the aperture correction varies with focus by up to 10% for a 1-pixel radius in the PC, and is generally well-fitted by a quadratic function of focus position (see Figure 5.11). A 10% change is measured only for 5 µm defocus, which is about the largest that is expected during normal telescope operations.
It is important to note that WF cameras can also have significant variations in their aperture corrections as the focus varies. While one would naively expect the larger pixels on the WFC to produce weaker variations in the aperture corrections, in practice, the focus offsets between cameras, and the fact that the overall OTA focus is usually optimized for PC1, can lead to significant corrections in the WFC.
Suchkov and Casertano provide formulae that estimate the change in the aperture correction due to defocus for a variety of circumstances.
Figure 5.11: Magnitude change for a 1 pixel radius aperture as function of focus position. Derived from quadratic fits to observed data. Note offset between optimal focus for PC1 (solid line) and WF3 (dashed lines). From Suchkov and Casertano (1997).
Large focus changes, with amplitudes up to 10µm, are seen occasionally (See Hasan and Bely 1993, Restoration of HST Images and Spectra II, p. 157). On May 1, 1994, and February 27, 1995, a short-lived defocusing of the telescope of up to 10µm was seen, probably due to extreme thermal conditions after the telescope was at an almost exact anti-sun pointing for an extended time. Such a defocusing causes an increase of the PSF width by about 5-10% and a significant change in its shape. This is especially evident in the PC both because of its higher resolution and its astigmatism, which makes the out-of-focus image appear elongated. The change in the PSF appears to be modeled adequately by the TinyTIM software. (See Hasan and Bely 1993, Restoration of HST Images and Spectra II, p. 157. Also see the sample PSF subtraction in Figure 7.2).
For more information, see the HST focus web site at
http://www.stsci.edu/instruments/observatory/focus
Two- Gyro Mode: At some future date HST may be operated with only two gyros, hence causing additional spacecraft jitter and degradation of the effective PSF. Please see the
Two-Gyro Mode Handbookfor discussion of these effects.5.8 PSF Anomaly in F1042M Filter
We note that the F1042M filter has an anomalous PSF containing additional light in a broad halo component. This is due to the CCD detector becoming transparent at these wavelengths, so that light is reflected and scattered by the back of the CCD producing a defocused halo. Figure 5.12 compares the F1042M PSF with the more normal PSF seen slightly blueward in F953N. This scattering will impact photometry in the F1042M filter relative to other filters, since a greater fraction of the counts will lie outside the 1 arcsecond diameter aperture used herein for photometry on standard stars.
Figure 5.12: Comparison of azimuthal averages for observed F1042M and F953N PSFs. Courtesy of John Krist.
5.9 Large Angle Scattering
Analysis of the WFPC2 saturated star images indicate that the large angle scattering (>3" from a star) is significantly higher than expected.
Three data sets were used to determine the WFPC2 scattering. The first set was from the SMOV Ghost Check Proposal 5615, in which 100-second images of
Cas (V=2.7) were obtained at the center of each chip in F502N. The second set was a series of 6-second exposures of Vega (V=0.0) centered on WF2 through F410M (WFPC2 GTO Proposal 5205). The third set was
Eridani (V=3.73) centered on the PC and taken through F631N (500s each) and F953N (2200s each). These were from GTO Proposal 5611.
WFPC2 scattering was determined by computing the azimuthal average and azimuthal median profiles. The regions near the diffraction spikes and saturated columns were not used. The profiles were determined using images corrected for horizontal smearing.
The measurements indicate that the average scatter in WFPC2 is an order of magnitude greater than in WF/PC-1. The increase is due to scattering in WFPC2, not due to the OTA. In the WFPC2 images, the pyramid edge shadow is not visible in the scattered light; the light is spread out to the chip edges, indicating that most of the scattering occurs after the pyramid. However, the light level in adjacent channels is back down at the WF/PC-1 levels as shown in Figure 5.13.
The scattering does not show any strong dependence on wavelength between 410 nm and 953 nm, within the uncertainties of the measurements.
The scattered light is not uniform. There are high frequency spatial structures in the form of streaks radiating outwards from the star. These features are probably both wavelength and position dependent, and so cannot be readily subtracted.
The source of the WFPC2 scattering may be the CCDs. The WF/PC-l CCDs were back illuminated and had shiny surfaces. The electrode structure was not visible over most of the wavelength range. The WFPC2 CCDs, however, are front illuminated, so the electrode structure is visible and may be scattering the light. There was a large ghost in WF/PC-l due to a reflection between the CCD and filter, but no such feature has been seen in WFPC2. The flux from this missing ghost may instead constitute part of the scatter. (See also related material in Observing Faint Targets Near Bright Objects.)
Figure 5.13: Large Angle Scattering. The proportion of the total flux in F555W falling per square arcsecond as a function of the distance from the peak of a saturated stellar image. These curves are for a target in the PC. Note the large drop in the scattered light level when looking in an adjacent camera.
5.10 Ghost Images
Common ghost images result from internal reflections in the filters and in the field-flatteners. Two filter ghosts, caused by double (and quadruple) reflection inside the filter, are visible below and to the right of the star in Figure 5.14. The position and brightness of these ghosts varies from filter to filter, typically being most obvious in interference filters. The comatic shape of the ghost is caused by the camera optics being effectively misaligned for the light path followed by the ghost. The relative position of these ghosts does not vary much over the field.
An additional ghost is caused by an internal reflection inside the MgF2 field flattener lens immediately in front of each CCD (Figure 5.15). The field flattener ghost is doughnut shaped (image of OTA pupil) in the WFC, but is smaller and more disk-like on the PC. This ghost contains ~0.15% of the total energy of the star. It is positioned on a line through the CCD center and the bright star; the distance from the ghost to the CCD center is 1.25 to 1.4 times the distance from the bright star to the CCD center. This geometry results from curvature of the field flattener lens.
The large ghost image expected to be caused by reflection off the CCD back to the filter and then back to the CCD is not seen. It was deliberately eliminated in the PC by tilting the CCD slightly.
Figure 5.14: Saturated Stellar Image Showing Filter Ghosts. Intensity scale is logarithmic.
Figure 5.15: Saturated Stellar Image Showing Field Flattener Ghost on WF2.
5.11 Optical Distortion
The geometric distortion of WFPC2 is complex since each individual CCD chip is integrated with its own optical chain (including corrective optics), and therefore each chip will have its own different geometric distortion. Apart from this, there is also a global distortion arising from the HST Optical Telescope Assembly (Casertano and Wiggs 2001).
Early attempts to solve the WFPC2 geometric distortion were made by Gilmozzi et al. (1995), Holtzman et al. (1995), and Casertano et al. (2001), using third order polynomials for all chips in the PC system, i.e. the coordinates X,Y were transformed into one meta-chip coordinate system and fitted to find the offsets, rotation and scale for each of the four chips. These early meta-chip solutions failed to constrain the skew-related linear terms, which actually are responsible for ~0.25 pix residual distortion. These solutions did not have on-orbit data sets which were rotated with respect to each other.
In 2003, Anderson and King derived a substantially improved geometric distortion solution for WFPC2 in the F555W filter. First, the measured positions Xobs,Yobs were normalized over the range of (50:800) pixels excluding the pyramid edges (Baggett, S., et al. 2002) and adopting the center of the solution at (425,425) with a scale factor of 375, i.e:
The final solution was presented as a third-order polynomial:
where Xg and Yg are the corrected coordinates.
The coefficients of the polynomials for F555W filter are given in Table 5.6 (Anderson and King 2003).
Figure 5.16 shows the vector diagram of the geometric distortion in filter F555W.
Figure 5.16: The geometric distortion map for F555W filter using the Anderson and King solution (2003). The size of the longest arrows are 6.29 pixel for the PC (in PC pixels) and ~6 pixel for WF cameras (in WF pixels). The panels correspond to PC - upper right; WF2 - upper left; WF3 - lower left and WF4 - lower right. The size of the residuals are scaled by a factor of 10 relative to the pixel coordinates.
Table 5.6: Polynomial Coefficients of the Geometric Distortion for F555W.
Trauger et al. (1995) showed that the geometric distortion for WFPC2 also depends on wavelength. This is due to the refractive MgF2 field-flattener lens in front of each CCD. They computed the wavelength-dependent geometric distortion by analyzing the results of ray tracing, where the coefficients were represented as a quadratic interpolation function of the refractive index of the field-flattener lenses. Kozhurina-Platais et al. (2003), using the Anderson and King methodology (2003), derived the geometric distortion solutions for two other filters: F814W and F300W. Figure 5.17 presents the difference in distortion between F555W and F300W, which clearly indicates a large amount of distortion in F300W, especially at the corners of the chips. An average increase of distortion in the F300W filters is ~3%, or 0.18 pixels in PC and 0.25 pixels in WF cameras. In contrast, there is only a small ~1% difference in distortion between F555W and F814W. Figure 5.18 presents the difference between the filters F555W and F814W. The coefficients of the polynomials for filters F300W and F814W are given in Table 5.7 and Table 5.7, respectively.
Figure 5.17: Difference in the distortion correction between F555W and F300W (F555W-F300W). The size of the longest arrows are 0.18 pixel for PC (in PC pixel) and 0.25 pixel for WF cameras (in WF pixels). The panels are the same as in Figure 5.16, except the size of the residuals are scaled by a factor of 300.
Figure 5.18: Difference in the distortion correction between F555W and F814W. The small amount of these differences along with fairly random pattern changing from chip to chip indicate that the differences are very small, if present at all. The size of the longest arrow is 0.04 PC pixels for the PC, and ~0.05 pixels for the WF cameras. The panels are the same as in Figure 5.16, except the size of the residuals are scaled by a factor of 300.
Application of the distortion coefficients are straight forward. To correct for geometric distortion, the measured raw coordinates should be normalized as in the equation 5.1 above. Then equation 5.2 above should be used, employing the coefficients from Table 5.6, Table 5.7 or Table 5.8, depending on the filter used. Finally, the corrected coordinates Xg, Yg should then be shifted back to the natural system of the detector, with proper orientation and scale, specifically:
The constant terms a1 and b1 are offsets (or zero-points) between any two frames and can be ignored for most purposes. The linear coefficients a2 and b3 represent the plate scale and can be found in Anderson and King (2003) and Kozhurina-Platais et al. (2003). The FORTRAN code developed by Anderson which correct the measured coordinates X and Y can be down-loaded from
http://www.stsci.edu/instruments/wfpc2/Wfpc2_memos/anderson king_distortion_routine.txtThe same program could be used to correct for distortion in filters F300W and F814W, using the coefficients from Table 5.7 or Table 5.8, respectively.
Table 5.7: Polynomial Coefficients of the Geometric Distortion for F300W.
Table 5.8: Polynomial Coefficients of the Geometric Distortion for F814W.
Geometric distortion not only affects astrometry but photometry as well, since it induces an apparent variation in surface brightness across the field of view. The effective pixel area can be derived from the geometric distortion coefficients, and is presented in Figure 5.19. The pixel area map correction is necessary since the flat fields are uniformly illuminated, and do not explicitly conserve the total integrated counts for a discrete target, whereas the geometric distortion conserves the total counts and redistributes the counts on the CCD chip. Thus, for precise stellar photometry the raw flat fielded images require a correction for the pixel area -- raw flat-fielded images should be multiplied by the pixel area map so as to restore the proper total counts of the target. The pixel area map is available as a fits file in the HST archive. (Some additional discussion of the pixel area correction can be found in the ACS Instrument Handbook for Cycle 14.)
Figure 5.19: A map of the effective pixel areas of the WFPC2 chips. The areas are normalized to unity at the center of each chip. The contours are shown at half present level. The panel corresponds to PC -upper right; WF2 - upper left; WF3 - lower left and WF4 - lower right.
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