3 Comparison with DE405

For Jupiter and Saturn, where two of their satellites were observed, the observations were transferred to the barycentres of the systems using their orbital theories. The ephemeris in the Connaissance des Temps (CdT), which is based on the G-5 theory (Arlot 1982), was used to reduce the observations of the satellites of Jupiter to the barycentre. For Titan and Iapetus, the theories of Taylor&Shen(1988) and Harper&Taylor(1993), respectively, were used. The tolerance limit of the ephemeris for Europa is 60mas, and for Titan it is 50mas. These should be adequate for the present investigation of possible systematic errors in DE405 because these orbital errors should enter randomly in forming opposition means in right ascension and declination, provided the orbit is well-sampled.

The theory for Iapetus is more complicated because of the importance of solar perturbations, and the accuracy may not be adequate. The asymmetry of the albedo of Iapetus also contributes to the uncertainty of the observed position. For these reasons we investigated the behaviour of the individual Carlsberg observations by plotting the residuals, observation minus theory, as a function of the phase of the orbital period of Iapetus. These are shown in Fig. 1. The amplitude of the fitted sinusoids are 0$.\!\!^{\prime\prime}$10 and 0$.\!\!^{\prime\prime}$08 in right ascension and declination, respectively. Whereas the phase of the sinusoids agree with the corrections expected from the asymmetry of the albedo (the dark hemisphere leads), the amplitudes are greater than expected ($\sim$0$.\!\!^{\prime\prime}$03). The largest contribution is likely to arise from deficiences in the Harper/Taylor theory, and a comparison with the theory of Vienne&Duriez (1995) could be instructive. From the standpoint of checking the DE405 ephemeris of Saturn, we have subtracted these sinusoids from the individual residuals before proceeding. The average offsets in Fig. 1 were retained since these may be interpreted as arising from DE405.

  

Figure 1: Comparison of observations of Iapetus with DE405 and the Harper&Taylor (1993) theory (upper, right ascension; lower declination). The differences, observation minus DE405+Harper/Taylor, are plotted against the phase of the 79.92-day synodic period of Iapetus, where zero phase is greatest eastern elongation and 0.25 phase is inferior conjunction

The differences, observation minus DE405, were formed for the four datasets: Carlsberg and Bordeaux opposition means, Hipparcos group solutions and Tycho individual observations. The differences in right ascension and declination and the estimates of the observational errors for Jupiter are plotted in Fig. 2, for Saturn in Fig. 3, for Uranus and Neptune in Fig. 4, and for Pluto in Fig. 5. The estimates of the errors of the Carlsberg opposition means were calculated from the scatter of the individual observations about the means. These do not include possible contributions from systematic errors.

  

Figure 2: Comparison of the observation positions of Jupiter with DE405 ephemerides (upper, right ascension; lower, declination). The Hipparcos normal points derived from observations of Europa are shown as filled squares. Individual Tycho positions derived from observations of Callisto are plotted as light error bars. The opposition means derived from Carlsberg observations of Callisto are plotted as heavy error bars

  

Figure 3: Comparison of the observation positions of Saturn with DE405 ephemerides (upper, right ascension; lower, declination). The Hipparcos normal points derived from observations of Titan are shown as filled squares with error bars. The opposition means derived from Carlsberg observations of Titan and Iapetus are plotted as heavy error bars

  

Figure 4: Comparison of the observation positions of Uranus and Neptune with DE405 ephemerides (upper, right ascension; lower, declination). The opposition means derived from Carlsberg observations are plotted as heavy error bars

  

Figure 5: Comparison of the observation positions of Pluto with DE405 ephemerides (upper, right ascension; lower, declination). The opposition means derived from Carlsberg observations are plotted as heavy error bars; those from Bordeaux as circles with error bars

3.1 Jupiter

The Hipparcos group solutions in right ascension and declination provide a completely independent check on DE405, since these observations were not used in the integration. The errors of the Hipparcos group solutions lie between 7 and 13mas, and are thus smaller than the size of the squares used to represent these points in Fig. 2. The individual Tycho observations in declination also provide a good constraint. The greater scatter of the Tycho observations in right ascension is a consequence of the scanning law of the Hipparcos mission.

The optical observations all show a positive bias in right ascension, regardless of the Galilean satellite that was observed. The Hipparcos points for Europa have an average value of +20mas and the Carlsberg for Callisto and Ganymede an average of +50mas. In declination the Hipparcos and Tycho observations have an average offset of -20mas, and the Carlsberg observations show a fluctuation of amplitude $\sim$100mas over the 12-year sidereal period of Jupiter's orbit. The offset in right ascension and fluctuation in declination could result from an offset of the optical reference frame from the radio frame. Only the radio frame was used in fixing the DE405 ephemeris of Jupiter. However, both the optical and radio observations are referred to the ICRF. So the disagreement implies that either the optical or radio observations, or possibly both (for different reasons), are not well linked to the ICRF.

The radio data are not all self-consistent to within their estimated errors. The Voyager1 data of 1979 and the VLA data of 1983 do not agree with the current ($\sim$1997) VLBI Galileo data (Standish 1998). However, the Galileo data, which are uni-directional, are internally consistent at the level of 10mas and this engenders a high degree of confidence, especially in the direction of right ascension, to which the data are most sensitive.

The Carlsberg optical data are subject to significant systematic errors, even in the the most recent period where the positions are measured directly with respect to the Hipparcos frame which is aligned to the ICRF to within 1mas. This arises because the nightly block adjustment of the instrumental frame is carried out using the observed positions of Hipparcos stars, but this adjustment does not allow for distortions which have a scale length less than about 30$^\circ$ on the sky. These characteristic distortions can persist for several nights, and thus influence significantly the mean opposition position of a planet which happens to be in that area of the sky. From the discussion of the observations of Uranus and Neptune below, it is estimated that the systematic errors of the opposition means are about $\pm$50mas, and that these systematic errors are not entirely independent from one opposition to the next. This produces spurious fluctuations with a characteristic period of several years which can be seen in all the figures. There is a strong correlation between the fluctuations in the Carlsberg residuals for Callisto and Ganymede, which demonstrates that the fluctuations are independent of any possible shortcomings of the theories of the satellites' orbits.

Notwithstanding, this still does not explain completely the fluctuation in the Carlsberg-DE405 residuals in declination which, suspiciously, has the 12-year sidereal period of Jupiter's orbit. The most likely explanation for this fluctuation is a misalignment of the radio and optical frames to which the observations are referred.

3.2 Saturn

The Hipparcos observations confirm that the accuracy of DE405 for the declination of Saturn is within 20mas around the epoch 1991. The constraint is not so tight in right ascension, where the residuals range from -40mas to +50mas. The apparent internal inconsistency of the Hipparcos points in right ascension could not be removed by changing the starting conditions of the DE405 integration, because any reasonable change would simply move the zero-point vertically in Fig. 3 and would not change the slope around 1991, as required by the observations. This suggests that the errors of the Hipparcos points in right ascension may have been underestimated. This could occur if corrections to the orbital elements of Titan (not considered here) were correlated with corrections to the planetary ephemerides. This may be the case because the Hipparcos observations tend to be clumped in time and are not well spread around the orbit.

The Carlsberg observations of Iapetus have particular problems due to the complexity of its theory and the asymmetry of its albedo. The predominant effect of these is to introduce $\sim$80-day periodic errors in the orbital longitude which affects the derived position of Saturn. We have corrected empirically for this, but it still may not be completely satisfactory. These observations may be of more use in the improvement in the theory of Iapetus than they are in checking the DE405 ephemeris of Saturn. This said, the Carlsberg observations do show a positive bias in right ascension, similar to that of Jupiter, and, also to those of Uranus and Neptune.

3.3 Uranus and Neptune

DE405 is dependent in recent years on the Carlsberg observations; so the overall agreement in Fig. 4 is to be expected. However, the right ascension still shows a positive bias. There is a high correlation between the residuals of Uranus and Neptune, particularly in right ascension. Uranus and Neptune were close together in the sky during the period of the Carlsberg observations. Therefore, the observations of the two planets are subject to the same systematic errors. A possible revision of DE405 could only lead to a smooth linear change in Fig. 4. So, the fluctuations are definitely due to the observations, and their magnitude provides an estimate of the effect of systematic errors on the opposition means which was considered above in the discussion of the Jupiter residuals.

Stone (1998) also reports a positive systematic bias in the right ascension of observations of Uranus and Neptune in 1996 and 1997 taken with the Flagstaff Astrometric Scanning Transit Telescope. The bias is not so pronounced for these years in Fig. 4, but the fluctuations due to systematic errors may have reduced the bias fortuitously.

3.4 Pluto

DE405 is in good agreement with the Carlsberg and Bordeaux observations, as shown in Fig. 5. Again, this is not surprising because these observations were important in generating DE405. However, they were not the only observations used for this purpose. Another important set was contributed by FASTT at Flagstaff (Stone 1996). The fluctuations in Fig. 5 are due to systematic errors in the Carlsberg observations, as in the case of the other planets.