SPACE TELESCOPE ASTROMETRY FROM CCD IMAGES* LIANG-TAI GEORGE CHIU Yale University Observatory, New Haven, Connecticut 06520 Abstract. The astrometric application of the Wide Field Camera and the Planetary Camera is reviewed. It is shown that the digital image centering algorithms can yield a positional accuracyof 0.1 milli-arcsecond. Deconvolution of CCD's sensitivity,non-flatness of the filters, and crinkling of the CCD surface may limit the positional accuracy to 1 milli-arcsecond. 1. Introduction The series of articles on the Space Telescope A s t r o m e t r y in this issue of the Celestial M e c h a n i c s and a review p a p e r by van Altena (1978) are r e c o m m e n d e d for an overall picture of the subject. The Fine Guidance System (FGS), responsible for guiding the Space Telescope (ST), is also the prime astrometric instrument. The expected positional accuracy of the F G S is 0'.'002 (van Altena 1978). However, the F G S requires no digital image centering techniques and therefore it is outside the scope of this review. The Faint Object C a m e r a (FOC), supplied by ESA, uses very high resolution (0':022 pixel) B o k s e n b e r g device as the detector. The F O C data can be reduced with digital image centering algorithms, provided the electron-imaged field can be maintained geometrically stable. In this review, we will concentrate on the Wide Field C a m e r a (WFC) and the Planetary C a m e r a (PC) which use solid state C C D chips as detectors. In contrast to the F G S which uses the star-to-star measurement, the C C D chips record the two-dimensional picture simultaneously. Also the geometric stability of the C C D detectors is unsurpassed a m o n g electronic detectors. 2. WFC/PC The W F C or PC consists of four 800 • 800 C C D chips which form a contiguous field of 1600 • 1600 pixels in object space. The incoming b e a m is split into four symmetrical quadrants by a field mosaic pyramid. An axial rotation of the pyramid by 45 ~ switches the telescope light path from the W F C to the PC or vice versa. Although the relative positions of the four C C D ' s may be held to a few milli-arcseconds, it may be necessary to treat the four chips separately for astrometric applications. Table I illustrates the characteristics of the W F C / P C . For comparison, we have also included the ground-based astrometry work using the Hale 5-m and the K P N O 4-m reflectors. The first three rows of Table I list the f ratios, scales, and the sizes of the field respectively. The remaining rows are explained in the rest of the article. * Presented at the Symposium 'Star Catalogues, Positional Astronomy and Celestial Mechanics', held in honor of Paul Herget at the U.S. Naval Observatory, Washington, November 30, 1978. CelestialMechanics 22 (1980) 191-196. 0008-8714/80/0222-0191 $00.90. Copyright 9 1980 by D. ReideI Publishing Co., Dordrecht, Holland and Boston U.S.A. LIANG-TA1 GEORGE CHIU 192 TABLE I Ground-based astrometry versus space telescope astrometry from CCD images Space telescope f ratio Scale Field of view Pixel size S / N per 225 wm 2 Astrometric accuracy Hale 5-m Ross corrector KPNO 4-m with UBK-7 3.67 11'.'1/mm 18' in diam. 0':11 -5 (103a) o'l = 0'.'01 (PDS) 2.7 18~'6/mm 50' in diam. 0'.'19 -22 (IIIa-J) o-1 = 0"007 (PDS) WFC PC 12.9 6':66/mm 2'. 67 x 2'. 67 0'.'1 300 30 2('86/mm 68'57 • 68"7 0'.'043 300 ~x = 0':001 ~rx = 0"001 3. Digital Image Centering A. G R O U N D - B A S E D W O R K Since the astrometric data from the W F C / P C will be in the form of digitized pictures, it is appropriate here to review ground-based photographic astrometry using the PDS microdensitometer as the measuring device. This section serves two purposes. First, it gives us a chance to review the most accurate ground-based astrometry achieved so far which in turn can be compared with the expected accuracy of the ST Astrometry. (For an estimate of the accuracy achievable from the ground, see King (1978).) Second, it forms the basis for extrapolating our ground-based experience to the W F C / P C case. The Berkeley PDS was used to scan a set of prime-focus plates taken with the Hale 5-m and the K N P O 4-m reflectors. Locating the image center of a star from the two-dimensional digitized array is a crucial step in this research. Algorithms for digital image centering have been developed by van Altena and A u e r (1975), Auer and van Altena (1978), and Chiu (1977). Figure 1 of van Altena and Auer (1975) shows a typical point spread function (PSF) of a seeing-limited density profile. The pixel size of our ground-based PDS data is 10 ixm, not too different from the pixel size of the CCD's (15 ~xm). For convenience, we have listed in Table I the peak signal to noise ratios for the CCD cameras and for the IIIa-J and 103a-O emulsions which were computed from the data of Furenlid et al. (1977) and Latham (1978). In order to understand the meaning of the astrometric accuracy given in the last row of Table I, we must distinguish two types of rms positional errors. Following the notation used by Auer and van Altena (1978), we define o'x: This is the local positional rms error caused by the noise in the image profile. If, for example, the PSF is Gaussian, then o'x can be derived from the residuals of the Gaussian fit and the covariance matrix of the fitting parameters (see Equation (6) of Auer and van ARena (1978)). o-1: This is the global rms positional error of the least-squares plate solutions. Notice that o-x does not include that part of the global error due to, e.g., way corrections, emulsion shifts, SPACE TELESCOPE ASTROMETRY FROM CCD IMAGES 193 etc., whereas o-a does. For photographic astrometry, o-1 is used as a measure of astrometric accuracy. For ST astrometry, emulsions shifts and way corrections are eliminated by not using photographic emulsions; therefore we will rely on o-x as an indicator of the attainable astrometric accuracy. We may face some unexpected problems which make o-1 >>o'x, although the discussions below assume that o-1 - o-x. Incidentally, o-1 for the ground-based work should be free of the systematic effects of quadratic terms, magnitude terms, color terms, color-magnification terms, coma, and distortion, since we have used many reference stars ( - 5 0 ) to look for these systematic effects and remove them with plate solutions using a dozen terms in each coordinate if necessary (Chiu 1977, van Altena and Vilkki 1975). From this kind of investigation, we found that three factors govern the accuracy of the photographic image locations: (1) The larger the scale of the telescope, the better the accuracy; (2) The imhge should be well exposed so that a well developed image core exists (in practice this means that the image should be about 2.5 magnitudes above the plate limit); (3) The fine-grained emulsion yields higher astrometric accuracy (i.e., o-1 is smaller). From Table I we see that the IIIa-J has a better accuracy than the 103a-O even though the former is taken with a shorter focal length telescope! This is due to the fact that the signal to noise ratio of the IIIa-J is better than that of 103a-O. All three factors are of course relevant to the ST applications. B. W F C / P C IMAGES Turning now to the ST ease, we see from Table I that the scales of the W F C / P C are larger than those of the ground-based reflectors. For m v <~23, the stellar images are well exposed within reasonable interval of integration time. The signal to noise ratio is an order of magnitude better than with the photographic emulsions. However, we have far fewer pixels per image coverage than the ground-based work. Does this cause any problem? I will try to argue that it does not. We can estimate the local positional rms error o-x if we know the PSF of the ST and can calibrate the interference effects and the pixel-to-pixel variation in sensitivity of the CCD's as a function of wavelength. The deconvolution of the CCD's sensitivity will be performed by the ST Science Institute a n d / o r the 'observer'. The deconvolution uncertainty depends on how the sensitivity of the CCD varies with the time. Together with the photon noise, it may limit the positional accuracy to 0'.'001 Westphal 1977). The expected PSF's at A = 6328 A, 3250 A, and 1215 A are given in Figures 3-12 of the Announcement of Opportunity for Space Telescope. For A >I 6328 ~ , the telescope is diffraction-limited. The blur circle or the encircled energy function is approximately such that 70% of the energy is contained within 0"1 radius, 95% of the energy is contained within 0'.'2 radius. The PSF longward of A - 6328 A departs from the Airy pattern of Fraunhofer :liffraction for several reasons: (1) obscuration of the secondary mirror; 194 LIANG-TAI GEORGE CHIU (2) wavefront error (expected to be 0.05 at A = 6328 ~); (3) jittering of the ST (rms amplitude of the pointing error is expected to be 0"007). Nevertheless, the Airy pattern is a good approximation of the PSF for A/> 6328/~ with no pointing error and no wavefront error. Figure 1 illustrates that a Gaussian profile (denoted with circles) V 1.0 w 0.8" 0.6. 0.4- 0.2. X 0 Fig. 1. 2 4 6 8 Fraunhofer diffraction at a circular aperture. The continuous curve is the function y = (2Jl(x)/x) 2. The Gaussian approximation, denoted with circles, is exp (-(x/1.92)2). approximates closely the Airy pattern within its first minimum. The Gaussian approximation is even better with the jittering of the telescope included since the troughs of the PSF are smoother in that case (compare Figure 3 and Figure 4 in the Announcement of Opportunity for Space Telescope). In view of these simplifications, we have carried out the following numerical experiments. We have artificially created Gaussian profiles with known image centers and with a peak signal to noise ratio of 300. The half width or of the Gaussian profiles is chosen so that either or/l = 1, or o'/l = 2, where l = 15 ixm is the pixel size. o-/l = 1 and or/l = 2 simulate the resolutions of the WFC and the PC respectively. (For ground-based work the parameter o-/l is usually larger than 3.) We then use the least-squares Gaussian'~fit (see van Altena and Auer 1975, Auer and van Altena 1978, Chiu 1977) to find the image centers and compare them with the input (known) image centers. The experiment is repeated with the center of the theoretical SPACE TELESCOPE ASTROMETRY FROM CCD IMAGES 195 Gaussian profile displaced with respect to the pixel boundary in a 0.1 x l step, up to half of the pixel size. The result of such an exercise shows that with the high signal to noise ratio (=300) of the CCD, the centers of such profiles can be estimated to 0.1 milli-arcsecond. This numerical experiment is not meant to be the optimum way to locate the image centers for the W F C / P C . For A < 6328 ~ , the PSF could be very non-Gaussian. In that case, as long as the PSF is calibrated accurately, we can cross-correlate the known PSF with a stellar image to locate the image center. Lindegren (1978) has reviewed the relative merits of several digital image centering algorithms recently. We will have a much better idea of which is the best numerical centering algorithm to use when the actual PSF of the ST is available. Incidentally, because of the Poisson photon noise, the weight of each pixel is simply 1 1 WOC 2 Noise -N+eZ where N, the readout value of the pixel (which is proportional to the number of photons registered in that pixel), is the square of the photon noise and e is due to the dark current of the CCD which is usually less than 15 electrons. In practice, the accuracy of image locations is limited by three factors listed below rather than the digital image centering algorithms. (1) deconvolution uncertainty of the CCD pictures and photon noise as explained before. (2) non-flatness of the filters. The positional error, caused by a non-flat filter, can be calculated with the formula given by van Altena and Monnier (1968). At f/30 (PC), the flatness specification (1/10 to 1/15 wave/era) introduces a positional error of about 0"002. This type of systematic error is unimportant if one sticks to the same filter for differential astrometry. It does, however, limit the accuracy of positional work for combining locations from different passbands. Tightening the flatness specification by a factor of 2 is under consideration. (3) non-repeatable crinkling of the CCD surface under thermal stress. We refer to Dr. Douglas Currie's article in this volume for further details about this problem. We conclude that each of three problems above is expected to introduce a positional uncertainty of about 0'.'001. The astrometric accuracy given in Table I includes these uncertainties. 4. Summary The W F C / P C is well suited for astrometric projects in view of its potential accuracy ( - 1 milli-second). In fact, its positional accuracy is almost an order of magnitude better than that of ground-based work so far. Their astrometric usage will undoubtedly give us very valuable information about relative positions, proper motions, and parallaxes in the future. 196 LIANG-TAI GEORGE CHIU Acknowledgments I w o u l d l i k e to t h a n k D r . W i l l i a m F. v a n A l t e n a f o r m a n y u s e f u l c o n v e r s a t i o n s . T h a n k s a r e d u e to D r . I v a n R . K i n g , D r . J o h n W a r n e r a n d M r . P e t e r S t e t s o n f o r h e l p f u l c o m m e n t s . T h i s w o r k has b e e n s u p p o r t e d b y Y a l e U n i v e r s i t y , N A S A and NSF. References Auer, L. H. and van Altena, W. F.: 1978, Astron. Z 83, 531. Born, M. and Wolf, E.: 1975, Principlies of Optics, 5th edn., Pergamon Press, Oxford, pp. 395. Chiu, L.-T. G.: 1977, Astron. Y. 82, 842. Furenlid, I., Schoening, W. E., and Carder, B. E., Jr.: 1977, Amer. Astron. Soc. Photo-Bull. 16, 14. King, I. R.: 1978, 'Colloquium on European Satellite Astrometry', Padova, Italy, pp. 151. Latham, D. W.: 1978, Amer. Astron. Soc. Photo-Bull. 18, 3. Lindegren, L.: 1978, in F. V. Prochazka and R. H. Tucker (eds.), 'Modern Astrometry', IAUColloq. 48, 197. van Altena, W. F.: 1978, in F. V. Prochazka and R. H. Tucker (eds.), 'Modern Astrometry', IAUColloq. 48, 561. van Altena, W. F. and Auer, L. H.: 1975, Image Processing Techniques in Astrometry, ed. by C. de Jager and H. Nieuwenhuijzen, Reidel, Dordrecht, pp. 411. van Altena, W. F. and Monnier, R. C.: 1968, Astron. Y. 73, 649. van Altena, W. F. and Vilkki, E. U.: 1975, Astron. J., 80, 647. Westphal, J. 1977, Technical Proposal Investigation Definition Team Wide Field/Planetary Camera for Space Telescope.