3 Reduction of the observations

3.1 Centering of the images

The accurate measurement of astrometric position of the natural satellites on the CCD target is closely related to the center determination of the satellite images. The methods we used for center-finding was similar to that used by Beurle et al. (1993); it was accomplished by the aid of a software "centroid" algorithm with a Poisson noise model extracted from IRAF that isolates a square range (box) around the image for the purpose of suppressing or removing the effects of gradient in the background close to the heavily overexposed image of the planet and rings. In using the software the user is required to choose the parameters for himself, of which only three parameters will effect the position determination. We give 5 to FWHM, 8-12 to Cbox (centering box width) and 50-80 to Threshold.

3.2 Solution for calibration parameters

Our reduction shows the observations of 61 Cygni can provide calibration data with better internal consistency. The 1.56 m telescope we used at the Sheshan Station is an astrometric reflector with better stable focal length. The variations in scale for this telescope on different nights during the same observing campaign are quite small, see Table 2, consequently the scale can be much better determined. However, the orientation of the CCD device changes from one night to another because the device was remounted on different nights. Thus we had to have run calibration determination for each night. We made the similar reductions to those described by Colas & Arlot (1991) to obtain the orientation of the CCD device. In our implementation of this method only four well-known satellites Titan, Rhea, Tethys and Dione, which are expected to have the most accurate predictions with the lowest dispersions ($\sim$0$.\!\!^{\prime\prime}$10), were used to give the small corrections to the scale and orientation of the CCD target by an iterative least squares process for optimizing the (${\rm O}-{\rm C}$) residuals when compared to an orbital model. This technique was efficiently adopted in the work of Veillet & Ratier (1980), Veillet & Dourneau (1992) and Harper & Taylor (1997). However as Pascu (1987) and Colas (1991) point out, this technique presents a problem that the errors in the computed ephemerides of the brighter moons enter directly into the derived positions for the satellites. To minimize the systematic error induced by the use of one theory, we used the four orbital models, which are developed by Taylor & Shen (1988), Harper & Taylor (1993), Duriez & Vienne (1997) and Dourneau (1987, 1993) respectively. For each date of observation of the above-mentioned four satellites the differences between the observation data and computed values produced from their respective theories were incorporated into the sum of the squares of the (${\rm O}-{\rm C}$) residuals to form normal equations, then the improvement of the corrections were completed in an iterative solving process. Table 2 gives the calibration parameters for each night of observation.

3.3 Format of the data

The data in our measurements are adopted in the form of polar coordinates which are position angle P and separation S. Relevant calculated values are denoted as $P_{\rm c}$ and $S_{\rm c}$. Let ($\triangle x$, $\triangle y$) denote the measured relative coordinates of satellites B with respect to satellite A as a reference object. We use $\delta P$, $\rho$ to denote the corresponding calibration parameters. The relations between ($\triangle x$, $\triangle y$) and (P, S, $\delta P$, $\rho$) are given as follows:

$$\begin{eqnarray} P &= {\rm arctg} \, (\triangle x / \triangle y) + \delta P \\ S &= \rho (\triangle x + \triangle y)^{1/2}. \nonumber\end{eqnarray}$$

For separation the residual is $\triangle S$, but for position angle the residual is the product $S_{\rm c}\triangle P$ where $\triangle P$ is the (${\rm O}-{\rm C}$) in radians.

The use of such a format for the data has the advantage that the corrections to the scale and orientation may be solved individually.

3.4 Sources of systematic error

In addition to the systematic errors coming from the above-described theories used to define the reference system for calibration, such inter-satellite measurements are affected by differential parallax, aberration and refraction. The first two effects have been incorporated into the positions derived from the orbital models. For differential refraction a correction has been incorporated in our published data. For the effects of refraction our reduction indicates that the changes introduced by it can not exceed 0$.\!\!^{\prime\prime}$02 when satellites are observed at zenith distance less than $45^{\circ}$. Hence we can conclude that in such a small field atmosphere is not a major contributor to the systematic errors that affect the measurements. This is consistent with the conclusion given by Colas (1991) from his CCD observation study. He claimed that the correction of refraction can not be outdated a few hundredths of an arcsecond, so it should not be responsible for large residuals.

No position of catalogue star as the reference star is required in our method. The only errors come from the double star catalogue used for as calibration, which have been minimized in the iterative calibrating process. The errors induced by the flexure of the telescope are difficult to model. For this error we assume that it is sufficiently small to be considered as an accidental error.

3.5 Measurements of the faint satellites

Mimas and Enceladus are the two faint satellites close to the bright primary. In most case it was difficult to discern them from the bright background in the proximity of the planet and ring. Similarly, it was also not easy to obtain the discernible image of Hyperion due to its smallness and faintness. Fortunately, valuable observations of these faint objects have been reliably obtained in our observation, although their images do not seem to us good enough to be measured precisely. From Table 3 we can see that the precision attainable for the faint objects can be considered to be satisfactory. It seems to give an indication that the software "centroid" from IRAF is very effective in removing the effects of the steep background slope around the planet.

  

Table 3: Statistics of RMS of the (${\rm O}-{\rm C}$) residuals of the inter-satellite measurements including the Saturnian major satellites exclusive of Iapetus for each reduction made from the four contemporary orbital theories. Satellite numbers conform to the conventional IAU numbering system, so S1-S7 denote the satellites in increasing order of the distance from the primary. SA is referred to the reference satellite (Titan). T1-T4 denote the theory of Taylor & Shen, Harper & Taylor, TASS1.7 and Dourneau respectively