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ISSN 0038-0946, Solar System Research, 2007, Vol. 41, No. 1, pp. 4245. © Pleiades Publishing, Inc., 2007. Original Russian Text © I.S. Izmailov, S.A. Korotkii, M.V. Ereshko, A.V. Stepura, 2007, published in Astronomicheskii Vestnik, 2007, Vol. 41, No. 1, pp. 4750.

Astrometric Observations of the Second, Third, and Fourth Satellites of Uranus
I. S. Izmailova, S. A. Korotkiib, M. V. Ereshkob, and A. V. Stepurab
a

Pulkovo Observatory of the Russian Academy of Sciences, Pulkovskoe shosse 651, St. Petersburg, 196140 Russia b Ka-Dar Observatory, 1-i Neopalimovskii per., 16/13, Moscow, 119121 Russia
Received April 28, 2006

Abstract--The results of astrometric observations of three Uranian satellites, performed at the Ka-Dar observatory from August to October 2005, are presented. In total, 20 satellite positions in the equatorial frame and 14 "satellite minus satellite" relative positions were obtained. PACS numbers: 96.30.Qk DOI: 10.1134/S0038094607010042

INTRODUCTION It is known that regular observations of planetary satellites are necessary to maintain and refine the theories of their motion. The theories of motion are used to study the physical parameters of satellites, to calculate the circumstances of occultations, and to maintain space research. In this paper, we present the results of the observations of Uranus's satellites. OBSERVATIONS Uranian satellites were observed at the Ka-Dar observatory, located in the Moscow region at 37°52'59'' E, 55°14'52'' N. A 356-mm SchmidtCassegrain telescope with a 3325-mm focal length was used, equipped with a SBIG STL 6303E CCD camera. The main CCD specifications are as follows (see www.sbig.com): Total pixels: 3072 в 2048 Pixel size: 0.009 в 0.009 mm Field of view (when installed in the reflector focus): 28.6 в 19.1 arcmin Dark current: 0.3 e/pixel/s Read noise: 13.5 e Peak sensitivity wavelength: 5800 е Peak quantum efficiency: 68% During nine nights in AugustOctober 2005, we obtained 146 processable CCD images of the Uranian system. The exposure times varied between 10 and 60 s, depending on the observing conditions. The images were taken using a series of 510 exposures. They harbor the second (Umbriel), the third (Titania), and the fourth (Oberon) Uranian satellites; they also usu42

ally harbor several dozen field stars of up to stellar magnitude 17, making it possible to use reference stars for reduction. This is an advantage of our study over a number of works performed in the previous decade (Veiga and Vieira, 1995; Harper et al., 1997, 1999), in which this technique could not be applied because of the small fields of view. Our images of Uranus are heavily saturated and cannot be measured. DATA REDUCTION In our opinion, one of the main problems in the reduction of CCD observations of planetary satellites is the presence of a halo of scattered light from the planet. To account for this effect, we approximated the halo in rings around the images of satellites with second-order polynomials (Izmailov et al., 1998) and then subtracted the polynomial value for each element relevant to a satellite image. Images of both the satellites and reference stars were processed assuming the Lorentz profiles (Franz, 1973): C I ( x, y ) = ---------------------- + D ; ( 1 + Ar ) r = ( x j x0 ) + ( 1 + B ) ( y j y0 )
2 2 2

(1)

+ E ( x j x0 ) ( y j y0 ) ;

(2)

where I(x, y) is the brightness count for the element with coordinates x and y; x0 and y0 are the coordinates of the image center; and , A, B, C, D, and E are the model parameters.

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To determine the center coordinates and other parameters of a satellite or a star image, we solved a redundant system with the nonlinear least-squares method. In this way, we determined the CCD image center coordinates of both the Uranian satellites and the field stars. Note that the standard reduction procedure suggests a subsequent application of the method of six constants. However, analysis of reference star residuals revealed radial shifts of the measured coordinates towards the centers of the field of view, a socalled generalized distortion (Kiselev, 1989), which may be as large as several tenths of an arcsecond depending on the distance from the center of the field of view. The figure demonstrates the dependence of this radial shift (r) on the distance from the distortion center (r). The distance was calculated by applying a formula analogous to Eq. (2). In this case, the lines with an equal distortion may be ellipses with arbitrary orientations. Therefore, before applying the method of six constants for the calculation of equatorial coordinates, allowance for the generalized distortion was made, resulting in a significant decrease in the reference star residuals. We chose UCAC2 (Zacharias et al., 2004) as a reference catalog, because it is the most accurate catalog today and has a sufficient star density. The processing was performed with the Izmccd package (Izmailov, 2005). The typical error of unit weight, which is one of the results of the six-constant method and characterizes the quality of measurements and that of the catalog star coordinates, is approximately equal to 0.05, which is a satisfactory value. If more than one satellite fell into the field of view, then at the last stage of data reduction we calculated the differences in right ascension and declination for pairs of satellites. Finally, the observations were compared with theory by means of (O-C) calculations. For this purpose, we used the theory of motion by Laskar et al., 1987), coded by N. V. Emel'yanov (http://lnfm.sai. msu.ru/neb/nss/nsso-c0hr.htm). OBSERVATIONAL RESULTS The results of the observations are presented in Tables 1 and 2. Table 1 presents the J2000 topocentric equatorial coordinates of the satellites, and Table 2 demonstrates the "satellite minus satellite" coordinate differences. The tables provide the mean moments of the observations (month and day with a fractional part) on the UTC scale; the satellite number (in Table 2, the numbers of the satellites for which the coordinate differences are determined); , the right ascension (h,
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r, arcsec 0.2 0.1 0 0.1 0.2 0.3 0 200 400 600 800 1000 1200 1400 1600 1800 r, [px]

Distortion: the dependence of the radial shift of a star (r) on the distance from the distortion center (r).

min, s) in Table 1 or X = cos, arcsec, in Table 2; , the declination (degrees, arcmin, arcsec) in Table 1 or Y = , arcsec, in Table 2; K, the number of used CCD images; (OC)x and (OC)y, arcsec, the differences between the observed and ephemeris positions, averaged over a series of CCD images; and x and y, the errors calculated from the convergence of (O-C) in the series (arcsec; internal errors). The tables show that the mean intrinsic accuracy of our observations is about 0.03''. The mean (O-C) of equatorial coordinates are equal to (O-C)x = +0".042 + 0.017, (O-C)y = 0".027 + 0.034. The error of a single observation, calculated from the (O-C) convergence in the series of observations, i.e., the external error averaged over О and Y, equals 0".12. Its value is determined by the accuracy of measurements of star and satellite positions, by the accuracy of the reference star coordinates, by the accuracy of the reduction, and by the accuracy of the theory of motion for satellites. The facts that the intrinsic error is mostly determined by the accuracy of measurements, the accuracy of star coordinates is significantly better than the external error, and the external error is larger than the intrinsic error by a factor of nearly four indicate that the reduction technique and the theory of satellite motion should be refined. The authors will continue observing the Uranian satellites and developing the data reduction technique.

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Table 1. Topocentric equatorial coordinates of the satellites of Uranus, derived from the observations Date (UTC), 2005 No. 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 , h, min, s 44 42 41 41 41 41 41 41 41 41 41 38 38 38 37 37 37 37 36 36 53.0484 49.6434 46.8433 29.4075 28.5246 28.2176 3.0907 2.4856 2.4366 2.9988 2.3868 17.6730 18.0732 18.1253 35.0488 9.5102 8.6522 9.9480 56.5544 57.6000 , degree, arcmin, arcsec 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 49 0 7 9 9 8 12 11 12 12 11 27 28 28 31 34 34 35 35 36 26.103 58.294 28.032 50.638 2.956 57.450 26.957 45.280 0.508 27.712 45.902 55.492 34.012 42.353 40.221 59.747 39.865 13.803 24.260 34.836 K 10 5 6 10 10 10 9 9 5 5 5 8 8 8 10 6 6 6 5 5 (O-C)x, arcsec 0.095 0.117 0.049 0.158 0.100 0.033 0.014 0.081 0.016 0.004 0.065 0.184 0.045 0.007 0.044 0.015 0.065 0.003 0.040 0.208 (OC)y, x, arcsec y, arcsec arcsec 0.143 0.106 0.206 0.223 0.099 0.075 0.122 0.056 0.196 0.105 0.022 0.426 0.014 0.004 0.109 0.251 0.031 0.020 0.062 0.100 0.005 0.049 0.027 0.040 0.013 0.014 0.023 0.014 0.075 0.055 0.022 0.019 0.028 0.017 0.067 0.021 0.027 0.018 0.036 0.029 0.018 0.018 0.043 0.042 0.018 0.021 0.022 0.015 0.094 0.054 0.046 0.032 0.019 0.028 0.046 0.028 0.021 0.019 0.026 0.020

8 15.952333 3 8 29.847666 3 9 9 9 9 5.848575 2 7.877994 2 7.877994 3 7.877994 4

9 10.858932 3 9 10.858932 4 9 10.869320 2 9 10.869320 3 9 10.869320 4 9 30.797414 2 9 30.797414 3 9 30.797414 4 10 6.794935 4 10 10.847958 2 10 10.847958 3 10 10.847958 4 10 12.841250 3 10 12.841250 4

Table 2. Relative positions of the Uranian satellites derived from the observations Date (UTC), 2005 9 9 9 9 9 9 9 9 9 9 10 10 10 10 7.877994 7.877994 7.877994 10.858932 10.869320 10.869320 10.869320 30.797414 30.797414 30.797414 10.847958 10.847958 10.847958 12.841250 3 3 4 4 2 2 4 2 2 3 2 3 3 3 No. 2 4 2 3 3 4 3 3 4 4 4 2 4 4 X, arcsec 13.075 4.547 17.621 8.960 8.324 0.737 9.062 5.922 6.691 0.769 6.476 12.691 19.166 15.464 Y, arcsec 47.682 5.506 53.188 41.677 27.204 14.606 41.810 38.520 46.860 8.340 14.057 19.882 33.938 70.576 K 10 10 10 9 5 5 5 8 8 8 6 6 6 5 (O-C)x, arcsec 0.058 0.131 0.174 0.066 0.020 0.048 0.068 0.226 0.188 0.038 0.011 0.049 0.061 0.034 (O-C)y, arcsec 0.124 0.173 0.263 0.177 0.091 0.174 0.083 0.411 0.421 0.010 0.271 0.282 0.011 0.010 x, arcsec 0.034 0.012 0.040 0.016 0.044 0.072 0.045 0.030 0.021 0.032 0.015 0.025 0.019 0.009
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y, arcsec 0.043 0.017 0.062 0.024 0.089 0.053 0.063 0.042 0.047 0.017 0.020 0.025 0.020 0.009
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ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project no. 04-02-16157. REFERENCES
Franz, O.G., Observational Procedures for Visual DoubleStar Work, J. R. Astron. Soc. Can., 1973, vol. 67, pp. 8186. Harper, D., Beurle, K., Williams, I.P., et al., CCD Astrometry of Saturn's Satellites 19901994, Astron. Astrophys., Suppl. Ser., 1997, vol. 121, pp. 6569. Harper, D., Beurle, K., Williams, I.P., et al., CCD Astrometry of Saturn's Satellites in 1995 and 1997, Astron. Astrophys., Suppl. Ser., 1999, vol. 136, pp. 257259. Izmailov, I.S., Kiselev, A.A., Kiseleva, T.P., and Khrutskaya, E.V., Using a CCD camera in Pulkovo programs of observations of binary and multiple stars and satellites

of major planets with the 26-inch refractor, Pis'ma Astron. Zh., 1998, vol. 24, no. 10, pp. 772779 [Astron. Lett. (Engl. Transl.), no. 10, pp. 665672]. Izmailov, I.S., Izmccd - A Program Package for Astronomical Processing of Digital Images of Celestial Objects, http:izmccd.puldb.ru/izmccdrus, 2005. Kiselev, A.A., Teoreticheskie osnovaniya fotograficheskoi astrometrii (Theoretical Basis of Photographical Astrometry), Moscow: Nauka, 1989. Laskar, J. and Jacobson, R.A., GUST86 - an Analytical Ephemerides of the Uranian Satellites, Astron. Astrophys., 1987, vol. 188, pp. 212224. Veiga, C.H. and Vieira, M.R., CCD Astrometric Observations of Uranian Satellites, Astron. Astrophys., Suppl. Ser., 1995, vol. 113, pp. 557560. Zacharias, N., Urban, S.E., Zacharias, M.I., et al., The Second US Naval Observatory CCD Astrograph Catalog (UCAC2), Astron. J., 2004, vol. 127, pp. 30433059.

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