WFPC2 Instrument Science Report 95-02 The Geometric Distortion Correction for the WFPC Cameras 27 February 1995 Roberto Gilmozzi, Shawn Ewald, and Ellyne Kinney Introduction The WF/PC-1 and the WFPC2 are both subject to a geometric distortion eect which stems from optical and mechanical design constraints (wide eld of view and small optical bench volume) and o-axis optics. In both cameras, the distortion is exaggerated by the telescope's spherical aberration. The distortion ranges from a few tenths of a pixel in the center of each chip, up to 2-3 pixels at the edges of each chip. The plate scales for the WFCs are approximately 0.1 arcseconds/pixel, so the distortion eect can cause positional errors anywhere from 20 mas to 300 mas. For the PCs, which have plate scales of 0.043 arcseconds/pixel, these errors range from 8 mas to over 100 mas. For both scientic and operational purposes it is desirable to have more accurate measurements. An example of a scientic use of the distortion correction would be to derive relative positions of stars in a star cluster. On the operational side, a major use of the distortion correction is for WFPC{ assisted Target Acquisition for one of the small aperture SIs such as FOS or HRS. This stategy is especially useful for acquiring faint objects in crowded or high background elds. A WFPC image is taken beforehand, and is used to select a bright and/or isolated object near the fainter one. An accurate oset between the two objects is derived. The small aperture SI then peaks up on the brighter target, and then the slew to the intended target is performed. The WFC is usually used since the PC, while having a higher pixel resolution, has a FOV that is insucient for most oset pointing operations. It should be stressed that the astrometry derived from WF/PC-1 and WFPC2 images have a very high relative accuracy (4-10 mas). In an absolute sense, however, the coordinates derived from it are only as accurate as the coordinates of the guide stars used, and the knowledge of the location of the camera in the focal plane. For WFPC-assisted Target Acquisition, this means it is necessary to peak up on a nearby object, which in essence zeros out these uncertainties, rather than trying to acquire the object directly from the derived coordinates. If one wanted to derive accurate absolute astrometry, it would be necessary to do one of two things. Either use guide stars with astrometric coordinates, or include one or more stars with well known coordinates within the FOV. The Data The distortion correction was derived from a series of overlapping images of the globular cluster NGC1850, taken in the V bandpass, as well as a handful of images in the B , I 1, and UV 2 bandpasses. (See Table 1) Each set of V images consisted of one central pointing and four outer pointings. The four outer pointings were oset from the from the central one by one quarter the FOV in both x and y (20 arcseconds for the WFC chips and 8.5 arcseconds for the PC). (See Figure 1) The standard pipeline calibration system at STScI was used to remove instrumental eects from the images. The IRAF package DAOphot was used for the analysis of the data. The daond utility was used to nd all of the stars in the central pointing. The initial coordinate list was then edited, by examining the image at each of the coordinate values. Spurious entries such as diraction spikes, cosmic rays, or noise in the background were eliminated. This edited coordinate list was then used to create the coordinate lists for each of the overlapping oset pointings, by applying the appropriate oset values to the coordinates in the initial list. These coordinate lists were run through the phot program, with the centroid centering algorithm turned on, in order to get accurately centered coordinates and an estimate of the magnitudes. The photometry les were further processed to eliminate stars with poor/no magnitude estimate, which indicated that the star was too faint to get an accurate magnitude, and therefore was not accurately centered, or that the coordinates were outside the region that overlapped with the central pointing. The observed positions of the each star which appeared in overlapping central and oset exposures were tabulated. The dierence between the expected and observed positions were computed and used as the basis of the distortion mapping. Distortion Fitting We assume throughout that the positions of the stars are invariant across all exposures and that any changes from one to another are attributable to: a) the distortion being mapped, b) the oset and possible rotation of the detector from one exposure to another due to the telescope manoeuvre, c) dierences in the position determination of the stars in the images due to digitization of the image in space (pixels) and intensity (counts). The rst two changes are determined by this procedure, while the third is one the limits to the precision which can be obtained. This limitation is greater in the WFPC2 than in WF/PC-1 due to the undersampled nature of the unaberrated PSF. The nal product of the tting procedure is a pair of functions dx(x; y) and dy(x; y), which return the geometric distortion at any given pixel position (x; y). The corrected star position, (xc; yc), would be derived from its measured position (xm; ym) by: xc = xm dx(xm; ym) (1) yc = ym dy(xm; ym) WF/PC-1 I images courtesy of Jim Westphal and the WF/PC-1 IDT. (Prop. 3008) images were available for WFPC2 only. 1 2U V 2 If j represents the pointing (0 for the central pointing and 1-4 for the outer pointings) and i represents the index of the set of stars in an image, then the stars in two overlapping exposures can be related by the expression: xc 0 = xc + xj j > 0 (2) where xj is the x component of the telescope pointing dierence between the central pointing and the j th oset image. Substituting equation (1) into (2), and showing only the equations for the x terms, since y are identical, xm 0 dx(xm 0 ; ym 0 ) = xm dx(xm ; ym ) + xj j > 0 (3) xm 0 xm = dx(xm 0 ; ym 0 ) dx(xm ; ym ) + xj j > 0 (4) i; i; i; i; i:j i; i;j i;j i; i; i;j i;j i;j i;j The distortion, dx, is ' 1 pixel and the oset size is 200 pixels. Since dx xm and dx xj we may omit the dx terms in (4) above and average xj over all the measured stars in the image. X xm N xj = N 1 stars stars i=0 xm 0 xm i; i;j 0 i; xm (5) i:j xj = dx(xm 0 ; ym 0 ) dx(xm ; ym ) i; i: i;j i;j j >0 (6) Which gives the dierence between the distortions in both pointings. A least-squares best t is obtained for the measured distortions, using Legendre Polynomials. For PC-1 and WFPC2 a third order t was obtained, and a sixth order t was used for all the detectors of WFC-1. 3 3 anPn (x)bmPm(y) (7) dx(x; y) = XX n=0 m=0 The residuals between computed and measured distortion is added to the measured x position (central pointing only) and the manoeuvre and distortion is recomputed. This iterative process continues until the convergence criterion is met. The convergence criterion is expressed as xi: xi = X (xi; + dx(xi; ; yi; )) N stars i=0 0 0 0 (xi;j + dx(xi;j ; yi;j )) ! 0 (8) The scatter of xi about zero is an estimate of the errors of the t. (See Figure 2) The solutions derived from each iteration were applied to a set of independent control elds comprised of stars from the overlapping regions of adjacent oset pointings. (See Figure 3) The iterative process was terminated when both the r.m.s. of the t and of the control elds reached a similar value, typically 0.07-0.10 pixels. This was necessary to check that systematisms introduced in the data by the manouver calculation in each iteration were properly removed. 3 Camera Head Osets and Rotations Once the Distortion Correction was determined it was possible to derive the relative camera head osets and rotations for each of the chips. This was done by matching stars in chip 2 of the central pointing with stars in the four chips of one of the outer pointings. (See Figure 4) The coordinates of the stars in these data sets were corrected for geometric distortions and the outer pointings were adjusted for the telescope oset from the central pointing. The resulting residuals were t to yield Camera Head Osets and Rotations, and plate scale changes across chips. Once these were determined it was possible to transform the coordinate systems of the four chip in each camera to one metachip coordinate system with chip 2 (or chip 6 in WF/PC-1 in its PC mode) as the reference. The following transformation equations were used: x0 = xi cos i yi sin i + xoff y0 = xi sin i + yi cos i + yoff x = x0 scalei y = y0 scalei Where i is the number of the chip being transformed. Tables 2-4 give the values of xoff , yoff , i, and scalei for each of the three cameras. (In the WF1 detector of WF/PC-1, the camera head had a slight tilt out of the focal plane and thus needed dierent scale factors for x and y.) i i i i Wavelength Dependance of the Distortion In addition to the images taken in V three other images were taken with the UV, B, and I lters at the same pointing as the central pointing. These images were used to check if there was any wavelength dependance of the geometric distortion. The only eect that was seen was the expected change in plate scale and a slight oset between the dierent lters for WFPC2. In WF/PC-1 some lters had shifts of their metachip coordinate systems with respect to the F555W metcachip coordinates. This was probably due to small, xed, tilts of the lters in their lter wheel mounting. Tables 5 and 6 show the wavelength dependance. We stress again that the solutions presented here were derived from observations at 5500 A, which is the one we recommend be used for astrometric purposes. Images taken with UV lters have PSF that are highly aected by the position of it's position in the FOV. The Wood's lter (f 160bw) shows a strong variation of the PSF across the eld of view, while in the distortion is not as sever f 170w lter. This eect is not addressed in this distortion correction, but is very prominent, in particular near the apex of the pyramid. This distortion may be removed through the use of a variable PSF in one's astrometric centrioding program. 4 References Ewald, S.P., Kinney, E.K., and Gilmozzi, R., 1994, BAAS, in press. Fosbury, R., in "The Future of Space Imaging", Brown, R., ed., 1993. Gilmozzi, R., Ewald, S.P., 1995, in press. Holtzman, J.A., et. al., The Performance and Calibration of the WFPC2, PASP, 1995, in press. Wide Field and Planetary Camera Final Orbital/Science Verifcation Report, WF/PC-1 IDT, 1991. Wide Field and Planetary Camera Instrument Handbook, J.W. MacKenty (Ed.), STScI publication (April 1992). Wide Field and Planetary Camera 2 Instrument Handbook, C.J. Burrows (Ed.), STScI publication (May 1994). 5 Figure 1a. WFPC-2 takes four images each time the shutter is opened and the CCDs read out to the on-board tape recorder. The eld of view is divided into four sections, one per CCD. The sections are labelled here by the CCD names, PC1 (which has a higher magnication than the other CCDs), WF2, WF3 and WF4. The "L"-shaped area adjacent to PC1 is not observed by WFPC2. Figure 1b. For each CCD, the pattern of overlapping exposures taken in order to derive the geometric distortion of its optics is shown in this diagram. The dark outline indicates the position of the CCD's eld of view in the 'central pointing' exposure. The lighter outlines are the areas of the sky observed in the oset pointings. The cross-hatched area if the region where the same stars are seen in one part of the CCD in the central pointing and another part of the CCD in an oset pointing. 6 Figure 2. The two panels show the dierence between positions of the same 600+ stars observed in one of the control elds for chip 1, without the correction applied. The rst is a vector plot (magnied 25 times) and the second is a residuals plot (x vs y1). The bottom panel shows the residuals after correction.pThe r:m:s: is 0:11 pixels (indicating accuracies on the single measurements of 0:11= 2). 7 Figure 3. The full eld of view for two oset pointings is detailed in this diagram. The shaded region is the overlap of one CCD in one exposure with a dierent CCD in another exposure. After the geometric distortions associated with each CCD's optics are removed from the positions of the stars seen in the two images in the shaded region, the pairs of observed positions of each star matched. The dierence between these pairs of positions is the sum of the telescope manoeuvre between the two exposures and the relative displacement, plate scale and rotation between the two cameras. 8 Figure 4. The outline of the central pointing exposure's full eld of view has been added to the previous gure to show the relationship of the two oset pointings used to measure the camera head position dierences. If the shaded region contains WF2 in Oset2 and WF3 in Oset3, then portions of those CCDs' images overlap with dierent portions of the same CCDs in the central pointing. The upper-right quarter of WF2 in Oset2 overlaps the lower-left quarter of WF2 in the Central pointing, and the lower-right quarter of WF3 in Oset3 overlaps the upper-left quarter of WF3 in the Central pointing. The information in the overlaps between central and oset pointings contribute to the geometric distortion solution for each camera, but is independent of the information contained in the overlap between two oset pointings of two dierent CCDs. 9 Figures 5-8. Camera Head Osets and Rotations The camera head osets were calculated by comparing the position of a bright star that appeared in chip 2 of the central pointing and one of the chips in the outer pointing. Renements to theses values were made by looking at the plots of dx vs. x and dy vs. y, which helped rene the osets and relative scales. An oset of dx or dy from 0 indicated the amount of oset due to the camera heads. A slope in dx or dy indicated a scale change. A slope in plots of dx vs. y or dy vs. x indicated a rotation in the camera heads. 10 Figure 6. 11 Figure 7. 12 Figure 8. 13 Table 1: Camera Head Osets and Rotations for PC-1 Camera PC5 PC6 PC7 PC8 xoff yoff i scalei (pixels) (pixels) (degrees) i -0.15 0.0 63.75 51.61 i 71.36 0.0 3.70 64.58 269.815 1.00260 0.0 1.0 90.753 0.99935 180.407 1.00000 Table 2: Camera Head Osets and Rotations for WFC-1 Camera xoff yoff i (pixels) (pixels) (degrees) i i WF1 -14.054 55.884 WF2 WF3 WF4 0.0 45.262 36.585 0.0 4.933 61.145 scalei 269.449 x : 1.001671 y : 1.001276 0.0 1.0 90.187 1.000000 180.305 x : 1.000000 y : 1.000400 Table 3: Camera Head Osets and Rotations for WFPC2 Camera PC1 WF2 WF3 WF4 i yoff xoff (pixels) (pixels) (degrees) i i scalei 58.676 101.347 270.496 x : 0.45730450 y : 0.4571768 0.0 0.0 0.0 1.0 98.627 -6.509 90.327 0.9994667 95.430 75.480 180.899 0.9999972 14 Table 4: Wavelength Dependance of the Distortion for WF/PC-1 Filter Chip f 439w 1 2 3 4 5 6 7 8 1 2 3 4 f 785lp scalei xoff 0.99987902 0.99990500 0.99999432 0.99996953 0.99985987 1.00019560 0.99985199 1.00016730 1.0009088 1.0008418 1.0009332 1.0009749 1.209 1.188 1.449 1.385 4.074 4.096 4.194 4.200 3.073 3.045 3.257 3.260 i yoff i 1.458 1.382 1.305 1.431 1.397 1.207 1.250 1.391 2.708 2.521 2.547 2.724 Table 5: Wavelength Dependance of the Distortion for WFPC2 Filter Chip f 791w f 439w f 170w 1 2 3 4 1 2 3 4 1 2 3 4 scalei xoff 1.00451 1.00209 1.00111 1.00259 1.00046 1.00019 1.00013 1.00018 0.999718 0.999734 0.999744 0.999848 0.2119613 0.248912 0.183698 0.193212 0.1506212 0.182048 0.153802 0.129359 0.5386210 0.646853 -0.227277 -0.170283 15 i yoff i 0.2405904 0.321801 0.303697 0.265783 0.0948799 0.124968 0.146408 0.110601 0.2327530 0.512240 0.793233 -0.200359 Table 6: Co-ecients for the PC-1 Geometric Distortion Correction Third Order Polynomial Fit Chip 5 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.276987E-01 1.80270 0.472875E-01 0.839371 -0.194826E-01 0.205235 -0.538380E-01 -0.585192E-01 0.102121 1.57496 0.996310E-03 -0.202905E-01 0.359101E-01 -0.263058E-01 0.515688E-01 0.556717E-01 0.887961E-01 -0.258017 0.187032E-01 -0.432581E-02 1.68511 0.169656 1.48879 0.166961 0.212737 -0.552163E-01 -0.114369 0.379805E-01 0.969871 0.426470E-02 -0.275033E-01 -0.228036E-01 Table 7: Co-ecients for the PC-1 Geometric Distortion Correction Third Order Polynomial Fit Chip 6 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.826583E-01 1.56328 0.112179 0.931775 0.149869 0.783824E-01 0.155961 -0.474787E-01 0.195649E-01 1.39522 -0.785402E-01 0.155761 0.736697E-01 -0.187339E-01 -0.149583E-01 -0.644191E-01 16 -0.192901E-01 0.954678E-01 0.260649E-01 0.713268E-01 1.64822 0.186032 1.29882 -0.104328 0.500916E-01 0.139403 0.640607E-01 -0.282517E-01 0.929756 -0.161168 -0.224210E-01 0.467203E-01 Table 8: Co-ecients for the PC-1 Geometric Distortion Correction Third Order Polynomial Fit Chip 7 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.123508 1.63335 0.262432 0.809902 -0.238150E-01 0.170262 -0.119149 -0.158564 -0.216911E-01 1.31328 -0.733799E-01 -0.725823E-01 -0.227318E-01 -0.993457E-01 -0.667001E-01 -0.523880E-01 0.716987E-01 -0.577290E-01 0.838524E-01 -0.418771E-01 1.69257 0.432265 1.60774 0.185271 0.942161E-01 -0.127476 -0.628792E-01 -0.131979 0.851954 0.350061E-01 0.468264E-01 0.184103 Table 9: Co-ecients for the PC-1 Geometric Distortion Correction Third Order Polynomial Fit Chip 8 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.826797E-01 0.963531 -0.985975E-01 0.963147 0.258219 0.246323 0.114268 0.890538E-03 -0.705783E-01 1.52589 0.160118 0.201436E-01 -0.168445 -0.141034 -0.370968 -0.339899 17 0.618644E-01 0.205632 0.181418E-01 -0.857678E-01 0.882170 -0.183242 1.54962 0.387620E-01 0.112948 0.233455 -0.186104E-01 -0.149138 1.02215 0.445508E-01 -0.387411E-01 -0.227995 Table 10: Co-ecients for the WFC-1 Geometric Distortion Correction Sixth Order Polynomial Fit Chip 1 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 -0.233644E-01 0.527532 0.160644 0.901084 -0.216011E-01 -0.140963E-01 -0.142361E-01 0.558357E-02 0.139696 -0.194561E-01 -0.111291 0.503659E-01 0.438467E-05 0.194187E-01 -0.263032E-01 1.48265 -0.760040E-01 0.325426E-01 -0.779825E-01 0.725623E-03 0.100961E-01 0.149592E-03 -0.170976 0.602980E-02 -0.123613 0.875220E-01 -0.115254 0.195115 -0.681991E-01 0.723664E-01 -0.414895E-01 -0.161967E-01 0.733362E-01 -0.982506E-01 -0.156723 -0.165164E-01 -0.220650 18 -0.229461E-02 -0.241854E-01 -0.392640E-01 -0.235227E-01 -0.642825E-02 -0.840310E-02 0.291674E-01 0.548335 0.387662 1.41894 -0.105085 0.440937E-02 0.843286E-01 -0.497410E-02 0.700789E-01 -0.528973E-01 0.701003E-01 -0.154925 -0.264293E-01 -0.846450E-01 0.152589 0.877111 -0.132327 -0.106160 0.196699 -0.121649 -0.173855E-01 0.199421E-01 -0.122222E-02 -0.661174E-02 -0.325303E-02 -0.269248 0.614570E-01 -0.758538E-01 0.163976 0.282092E-01 -0.202283 Table 10: Co-ecients for the WFC-1 Geometric Distortion Correction Sixth Order Polynomial Fit Chip 1 a b 38 39 40 41 42 43 44 45 46 47 48 49 -0.267263E-01 -0.918068E-01 -0.647043E-02 0.218341 0.490153E-01 0.448921E-01 0.328306E-01 -0.735550E-01 0.674810E-01 -0.241444 0.688317E-02 -0.681448E-01 19 0.222040E-01 0.531590E-01 -0.127389 0.298571 0.379930E-01 -0.182698E-01 0.332307E-02 -0.123486 -0.427196E-01 0.277558 -0.104449 -0.152209 Table 11: Co-ecients for the WFC-1 Geometric Distortion Correction Sixth Order Polynomial Fit Chip 2 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 0.550319E-02 0.432135 0.105880 0.905788 0.949131E-02 0.274268E-01 0.265834E-01 -0.936640E-01 0.286223 -0.595855E-01 0.661712E-02 0.404042E-01 0.596573E-02 0.137155 -0.715509E-01 1.58508 0.483154E-02 -0.171728 0.548727E-01 0.129194 -0.119332 -0.311638E-01 -0.189211E-01 0.512064E-01 0.166421 0.146718 0.380757E-01 -0.349375E-02 0.226666E-01 -0.690692E-01 0.104447 0.128583 0.861257E-01 0.843517E-01 -0.143859 0.342341E-01 0.182830 20 0.176582E-01 -0.128684 -0.183566E-01 -0.233728E-01 0.174834E-01 0.102550 0.484545E-01 0.598599 0.277103 1.63174 0.148166E-01 -0.634429E-01 0.833309E-01 0.578246E-01 0.142916 0.189755E-01 0.332725E-01 0.883261E-01 0.421103E-01 0.181612 -0.106995E-01 0.937254 -0.183634E-01 -0.130179 0.809492E-01 0.879568E-01 -0.433506E-01 0.145550 0.665690E-01 0.897547E-01 0.208942E-01 0.245311 -0.754056E-01 -0.939822E-01 -0.124732 -0.508283E-01 0.454969E-01 Table 11: Co-ecients for the WFC-1 Geometric Distortion Correction Sixth Order Polynomial Fit Chip 2 a b 38 39 40 41 42 43 44 45 46 47 48 49 0.981716E-01 0.736910E-01 0.592615E-01 -0.351944 -0.141686 0.478025E-02 0.185782E-01 -0.363186E-01 0.122674E-02 -0.221890 0.344632 0.229562 21 0.145580E-01 0.217669E-02 -0.999051E-02 0.154702E-01 0.250810 0.102983E-01 0.778874E-01 0.285723E-01 0.516451E-01 -0.226409E-01 0.253033 -0.249701 Table 12: Co-ecients for the WFC-1 Geometric Distortion Correction Sixth Order Polynomial Fit Chip 3 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 -0.264266E-01 0.336880 0.104388 0.812891 0.599160E-01 0.688058E-01 -0.970845E-01 -0.261449E-01 0.292877 0.307018E-01 -0.353839E-01 0.833566E-02 -0.523576E-01 0.755655E-02 -0.423350E-01 1.43097 0.741312E-02 0.298880E-01 -0.129647 -0.159938E-01 0.143802E-01 -0.161040E-01 -0.350987E-01 -0.132448E-01 0.136141E-01 0.380449E-01 0.122190E-01 -0.198862E-01 -0.462074E-01 0.392152E-01 -0.794818E-01 -0.888443E-01 0.348593E-01 0.177125E-02 0.156821 0.402841E-02 -0.232923E-01 22 -0.295294E-03 -0.347180E-01 -0.369894E-01 -0.249304E-01 -0.593540E-02 0.993049E-02 0.281742E-01 0.345864 0.169266 1.44205 0.498738E-01 0.568238E-01 -0.180031 -0.246800E-01 0.203811 -0.133171E-01 -0.634889E-01 -0.593117E-02 0.420348E-02 -0.402736E-01 0.434928E-01 0.764973 0.460570E-01 0.586922E-01 -0.851932E-01 -0.396292E-01 0.164158 -0.569705E-01 0.502268E-01 0.588378E-01 -0.172513E-01 -0.336568E-02 0.102801 -0.814054E-01 0.192949E-01 0.664845E-02 -0.258540E-01 Table 12: Co-ecients for the WFC-1 Geometric Distortion Correction Sixth Order Polynomial Fit Chip 3 a b 38 39 40 41 42 43 44 45 46 47 48 49 -0.414563E-01 -0.777913E-01 -0.874515E-01 0.129185 0.157718 0.152659E-01 0.259774E-01 0.808500E-01 0.335988E-01 0.207165 -0.996049E-01 -0.264415 23 -0.323719E-01 0.348109E-02 -0.556088E-01 0.177595 -0.277053E-01 -0.637379E-01 0.496433E-01 -0.275070E-01 -0.904321E-02 0.288531E-01 -0.133526 -0.241295E-01 Table 13: Co-ecients for the WFC-1 Geometric Distortion Correction Sixth Order Polynomial Fit Chip 4 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 -0.201227E-01 0.422125 -0.787926E-02 0.712238 0.654487E-01 -0.105002 -0.136475E-01 0.523019E-01 0.459140 0.148413E-01 -0.725054E-01 -0.104992 0.140289 -0.191440 -0.927930E-01 1.38338 -0.921004E-02 0.669445E-01 0.128159E-01 0.530097E-01 0.869855E-01 -0.126787E-01 0.180397E-01 -0.166184 0.252978E-01 0.146418E-01 0.411792E-01 -0.760605E-01 -0.256522E-02 0.795980E-02 0.453194E-01 -0.742488E-01 0.609984E-01 -0.403499E-01 0.342597E-01 0.811365E-02 -0.175822E-01 24 0.691057E-01 0.156087 0.195515E-01 -0.526593E-02 0.839561E-02 -0.132709 0.101332 0.521700 0.617597E-02 1.44366 0.531254E-01 -0.155309 0.130997 0.507035E-02 0.309520 -0.106961 0.154650 -0.205095 0.226045 -0.280692 0.225150 0.811252 0.364762E-01 -0.814805E-01 0.135707 -0.729369E-01 0.138829 -0.222966 0.545485E-01 -0.174782 0.133649 -0.140565 0.381033 0.141552 -0.651064E-01 0.278417E-01 0.130193 Table 13: Co-ecients for the WFC-1 Geometric Distortion Correction Sixth Order Polynomial Fit Chip 4 a b 38 39 40 41 42 43 44 45 46 47 48 49 -0.659592E-01 -0.769669E-01 -0.295574 -0.238558E-01 0.498107 0.915286E-01 0.239522E-01 -0.984721E-02 0.560242E-01 -0.193011 -0.393026E-01 0.107539 25 0.547447E-01 0.255356 0.123642E-01 -0.556008 -0.399270 -0.164447E-01 -0.635381E-01 0.335267E-01 -0.471332E-01 -0.825712E-01 -0.834433E-01 0.284922 Table 14: Co-ecients for the WFPC2 Geometric Distortion Correction Third Order Polynomial Fit Chip 1 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.668149E-01 2.16326 0.380947E-01 0.940948 0.516275E-01 0.616919E-01 -0.468250E-02 -0.799010E-02 0.106746 1.48615 0.224224E-01 0.569497E-01 -0.871714E-02 0.343500E-01 0.151337E-01 -0.315461E-01 0.574160E-01 0.176289E-01 0.130385 0.169268E-01 2.36346 0.806029E-01 1.51212 0.224606E-01 -0.139148E-01 0.253915E-01 -0.169642E-01 0.759251E-01 0.965949 -0.443844E-01 0.131287 -0.532698E-01 Table 15: Co-ecients for the WFPC2 Geometric Distortion Correction Third Order Polynomial Fit Chip 2 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.834638E-01 1.93344 0.171628 0.846194 0.499745E-01 0.156076 0.267013E-02 -0.976506E-02 -0.122987E-01 1.46568 -0.151957E-01 0.423389E-01 -0.336837E-02 -0.113084E-01 0.325430E-01 -0.816739E-01 26 -0.961072E-02 -0.432376E-01 -0.572591E-01 0.234889E-02 1.91277 0.335549 1.45848 -0.272907E-01 0.414067E-01 0.239586E-01 0.673926E-02 0.207205E-01 0.834137 -0.613613E-01 0.326873E-01 -0.337791E-01 Table 16: Co-ecients for the WFPC2 Geometric Distortion Correction Third Order Polynomial Fit Chip 3 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -0.638637E-01 1.92847 -0.403292E-01 0.880849 -0.672139E-01 0.153023 0.588188E-02 -0.102037E-01 -0.861200E-01 1.46324 -0.330019E-02 0.756068E-01 0.863599E-03 -0.493526E-01 -0.144707E-02 -0.338435E-01 -0.376068E-01 -0.441374E-01 -0.768074E-01 0.420585E-02 1.95842 0.163623 1.48087 -0.272244E-01 0.197844E-01 0.251328E-01 0.241439E-02 0.335336E-01 0.868325 -0.235801E-01 0.584494E-01 -0.320786E-01 Table 17: Co-ecients for the WFPC2 Geometric Distortion Correction Third Order Polynomial Fit Chip 4 a b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.537544E-02 1.95195 0.754186E-01 0.877816 -0.533245E-01 0.171163 0.295295E-01 0.180753E-01 -0.578378E-01 1.46786 0.136570E-01 0.367823E-02 0.131553E-01 -0.137205E-01 0.340074E-01 -0.213247E-01 27 -0.209039E-01 0.458124E-01 -0.552349E-01 -0.150328E-01 2.01643 0.269249 1.46365 0.174260E-01 0.260297E-01 0.371601E-01 0.177419E-01 -0.374130E-01 0.886505 0.229406E-02 0.108477E-01 0.483584E-01