6 Analysis and discussion

The residuals shown in Table 4 include systematic orbital and observational errors (systematic and accidental). The systematic residuals probably arise from two causes. First, we used the two different ephemerides, which were produced by GUST86 and numerical integration respectively, for determining the calibration parameters. Second, the mean residuals mainly reflect the imperfection of the analytical theory. This implies that the use of the satellites themselves to calibrate the scale and orientation of the CCD frame, unavoidably introduces the systematic errors in the satellite positions. In order to illustrate this better, in Table 5 we deduced the mean residuals ( $\mu{\hbox{$^{\prime\prime}$ }}$) and the standard deviations ( $\sigma{\hbox{$^{\prime\prime}$ }}$) of numerical integration - GUST86 about the mean, for Ariel, Umbriel, Titania and Miranda. The bias in fact demonstrates the imperfection of the analytical theory. However the small residuals about the mean (generally, less than $0\hbox{$.\!\!^{\prime\prime}$ }01$) in Table 5, show that GUST86 has still very good internal consistency.

 

 

Table 5: The mean residuals $\mu$ and standard deviations $\sigma$ of the residuals (from numerical integration - GUST86) about the mean. Nu is the number of observations used.

		Position angle		Separation
Satellites	Nu	$\mu(\hbox{$^{\prime\prime}$ })$	$\sigma(\hbox{$^{\prime\prime}$ })$	$\mu(\hbox{$^{\prime\prime}$ })$	$\sigma(\hbox{$^{\prime\prime}$ })$
Ariel	114	-0.0002$~\pm~$0.0007	0.0075	-0.0089$~\pm~$0.0010	0.0060
Umbriel	113	-0.0001$~\pm~$0.0007	0.0073	0.0107$~\pm~$0.0013	0.0080
Titania	122	-0.0043$~\pm~$0.0010	0.0102	0.0147$~\pm~$0.0019	0.0118
Miranda	83	-0.0003$~\pm~$0.0009	0.0080	-0.0088$~\pm~$0.0012	0.0072

In Table 4 the mean residuals and standard deviations of Miranda are about twice as large as those of other satellites (they reach the value of $0\hbox{$.\!\!^{\prime\prime}$ }08$). Miranda is a faint satellite close to a bright primary. Because of the proximity of Uranus and the unavoidable effect from halo light, the image of the faint satellite Miranda is generally difficult to measure. This effect has also been noted by Harper et al. (1997) for Mimas, the innermost of the major satellites of Saturn.

In addition, Table 5 shows that the poorest residuals are those of Titania, which really corresponds to systematic errors in the ephemeris. The theory of this satellite seems to need further improvement. In order to find out whether Titania as a calibration satellite introduced a systematic error, we rerun the calibration without Titania. Table 6 gives the O-C residuals after the re-reduction only for GUST86. The results seem to indicate that the use of Titania for calibration might introduce a systematic error of about $0\hbox{$.\!\!^{\prime\prime}$ }01$, which is consistent with the results given in Table 5. In connection with that we had to run a separate calibration determination for each night, as pointed out in Sect. 3. However, after this investigation, we still prefer to include Titania in our calibration. Otherwise, for some observing nights there would be very few calibration satellites (as shown in Table 2), and this might result in unreliability in determining the calibration parameters.

 

 

Table 6: Statistics of O-C residuals from GUST86 in arcseconds calculated when using only Ariel, Umbriel and Oberon as calibration satellites.

		Position angle		Separation
Satellites	Nu	$\mu(\hbox{$^{\prime\prime}$ })$	$\sigma(\hbox{$^{\prime\prime}$ })$	$\mu(\hbox{$^{\prime\prime}$ })$	$\sigma(\hbox{$^{\prime\prime}$ })$
Ariel	114	-0.0019	0.0349	0.0042	0.0444
Umbriel	113	-0.0005	0.0414	0.0038	0.0451
Titania	122	-0.0052	0.0522	0.0029	0.0492
Miranda	83	-0.0034	0.0802	0.0028	0.0831