3 Observation reduction

In Paper I, we noted that the CCD device was remounted on the reflector at the Sheshan station each night, thus we had to run a separate calibration determination for each night. In another paper (Shen et al. 2001, hereafter called Paper II), a detailed description and analysis of the "brighter moon calibration'' method used by us was given. This method has been widely and efficiently used in calibration reduction of satellites (see, e.g. Harper et al. 1997), although at present it still is imperfect. We have successfully applied the method to the calibration of observations of Saturnian satellites over 1994-1996. The same procedure was used in the present calibration reduction, but here the satellite ephemerides were produced from GUST86 and a numerical integration. Only four better-known satellites Ariel, Umbriel, Titania and Oberon were used as calibration satellites. Because these satellites have better ephemerides and clearer images, the calibrations were affected by smaller systematic errors. The images of Miranda are poorer due to its proximity to Uranus, hence it was excluded from the calibration.

The calibration parameters reduced from the use of the two ephemerides are presented in Table 2. Different values were obtained from analytical theory and from numerical integration. The differences prove that the use of "brighter moon calibration'' method unavoidably introduces errors into the derived satellite positions. This is a difficult problem to overcome. In addition, it is also necessary to point out that such a reduction for calibration makes the determination of the mass of Uranus impossible because of the correlation between the scale factor and the mass. However, we can exploit the method while noting its weaknesses.

 

 

Table 2: Calibration parameters for each night of observations. Num gives the respective numbers of calibration satellites. Oberon was the reference satellite.

	Numerical integration		GUST86		Num
Dataset	$\delta P$(degree)	$\rho$(arcsec/pixel)	$\delta P$(degree)	$\rho$(arcsec/pixel)	Ariel	Umbriel	Titania
95.08.14	2.2935$~\pm~$0.0063	0.249815$~\pm~$0.000012	2.2945$~\pm~$0.0058	0.249749$~\pm~$0.000013	16	16	16
95.08.15	2.2133$~\pm~$0.0050	0.249802$~\pm~$0.000009	2.2149$~\pm~$0.0052	0.249771$~\pm~$0.000009	19	19	19
95.08.16	2.2340$~\pm~$0.0076	0.249861$~\pm~$0.000009	2.2334$~\pm~$0.0077	0.249824$~\pm~$0.000009	11	11	11
96.08.07	-3.3164$~\pm~$0.0142	0.251309$~\pm~$0.000016	-3.3447$~\pm~$0.0128	0.251188$~\pm~$0.000016	5	5	5
96.08.08	-3.1789$~\pm~$0.0081	0.251596$~\pm~$0.000008	-3.1897$~\pm~$0.0080	0.251301$~\pm~$0.000008	9	0	9
96.08.09	-3.2566$~\pm~$0.0075	0.251254$~\pm~$0.000014	-3.2581$~\pm~$0.0075	0.251161$~\pm~$0.000014	9	9	9
96.08.10	-3.3845$~\pm~$0.0062	0.251326$~\pm~$0.000008	-3.3690$~\pm~$0.0068	0.251231$~\pm~$0.000008	11	11	11
96.08.11	-3.1729$~\pm~$0.0048	0.251254$~\pm~$0.000009	-3.1595$~\pm~$0.0047	0.251173$~\pm~$0.000009	17	17	17
96.08.12	-3.3269$~\pm~$0.0108	0.251136$~\pm~$0.000018	-3.3071$~\pm~$0.0110	0.251070$~\pm~$0.000018	6	6	6
97.09.01	14.7824$~\pm~$0.0122	0.251343$~\pm~$0.000016	14.7934$~\pm~$0.0122	0.251181$~\pm~$0.000016	0	8	8
97.09.05	14.8314$~\pm~$0.0082	0.251103$~\pm~$0.000012	14.8248$~\pm~$0.0083	0.251026$~\pm~$0.000012	11	11	11

 

 

Table 3: The initial coordinates and velocities of the satellites in units of AU and AU/day respectively. The epoch is JED 2449948.5. The reference frame is the equator and ascending node of the equator of Uranus w.r.t. Earth's B1950.0.

	Miranda	Ariel	Umbriel	Titania	Oberon
X0	0.2361733945D-02	-0.8797002368D-03	0.2306591996D-02	0.1726616737D-02	0.6793133718D-03
Y0	-0.1708746181D-02	-0.9228496443D-03	-0.2201205161D-02	-0.4517485737D-03	0.2601083170D-02
Z0	-0.1259666501D-05	-0.1118937248D-05	0.4560861795D-06	-0.1410787960D-05	0.5653119845D-05
$\dot{X}_0$	0.1632335411D-02	0.1232334544D-02	0.3124808798D-02	0.1089842773D-02	0.7860181204D-03
$\dot{Y}_0$	0.3495616918D-02	0.1710611378D-02	-0.2338011911D-02	0.1458592286D-02	-0.3646703001D-03
$\dot{Z}_0$	0.2426606830D-03	-0.1317287544D-05	0.7809170238D-05	0.9112033377D-06	-0.3680678777D-04

Recently Vienne et al. (2001a, 2001b) and Peng et al. (2002) discussed astrometric reduction of CCD observations of planetary satellites, when no astrometric stars are present in the frame. They think that the values of the calibration parameters given by a reduction using the positions of the satellites are difficult to interpret. Their further study shows that these values are affected by some errors, which are often neglected.

In our reduction the effects of stellar aberration and topocentric parallax have been incorporated into the positions derived from the orbital models, as we did in Paper I. From the differential corrections listed in Table 2 of Vienne et al. (2001a), we can see that the effects of light-travel time between satellites (maximum = $0\hbox{$.\!\!^{\prime\prime}$ }001$ for Uranus), relativistic deflexion from the Sun and the planet are small enough to be negligible. We have also noted that computation made by Vienne et al. in an extreme case (a standard CCD frame $400\hbox{$^{\prime\prime}$ }$ large) for inter-satellites measurements shows that the refraction effect can reach up to $1\hbox{$^{\prime\prime}$ }$ for zenith distance less than $70\hbox{$^\circ$ }$. Therefore this correction has to be considered in the present work, unless the zenith distance is less than $30\hbox{$^\circ$ }$. As a result, we have used rigorous formulae given by Woolard & Clemence (1984) for the correction of separation and position angle. In practice, we have found that maximum correction for the refraction effect does not exceed $0\hbox{$.\!\!^{\prime\prime}$ }03$. Although this is a smaller quality, still astrometric consideration for it is obviously necessary.

The non-linearity of the projection of the celestial sphere on the tangential plane of the focal point requires a correction which can be as large as $0\hbox{$.\!\!^{\prime\prime}$ }03$ for fields of $240\hbox{$^{\prime\prime}$ }\times240\hbox{$^{\prime\prime}$ }$. Vienne et al. (2001b) have shown that this effect depends also strongly upon the declination of the observed bodies and can be approximated by $s^2\tan\delta_{\rm c}$. Our observations were made at a declination of $-20\hbox{$^\circ$ }$ and the maximum separation between the satellites is less than $55\hbox{$^{\prime\prime}$ }$, so the corresponding correction never exceeds  $0\hbox{$.\!\!^{\prime\prime}$ }005$. As a result, we may neglect this correction. However, we have published our observations as raw pixel coordinates so that the reader may perform a rigorous reduction if he wishes.