4. Corrections to the FB time ephemeris series
Analytical time ephemerides are the result of integration of Eq. (3) using an integrand determined from analytical planetary and lunar ephemerides such as VSOP82/ELP2000 (Bretagnon 1982; Chapront-Touzé & Chapront 1983). Analytical and numerical time ephemerides complement each other. The analytical approach is essential for identifying the important terms of the numerical results, for error analysis and mass correction of the numerical results (Sect. 3), and for calculating the and K values (Sect. 5). On the other hand, the current numerical results are almost 3 orders of magnitude more accurate than the published series results. In this section we present corrections to the analytic results that reduce their errors by an order of magnitude. In addition we suggest a plausible explanation for the remaining errors of the analytical results.
We believe a 1705-term version (Bretagnon 1995) of the FB series has the best potential for correction so we have concentrated exclusively on it in the present investigation. We designate this series "FB3" to distinguish it from the (shorter) series that were investigated in Paper I.
The FB3 series includes all known terms whose amplitudes are 10 ps or more. Fig. 2 shows the maximum truncation errors over the epoch range from 1600 to 2200 as a function of truncation limit. From this figure, the truncation limit of 10 ps for the FB3 series should correspond approximately to 0.5 ns error which is an acceptable result. However, truncating the series at larger limits than the FB3 series cutoff (as occurs, for example, for many of the series which were investigated in Paper I) would produce substantial truncation errors.
Fig. 3 illustrates how removing the angular-frequency transformation that was incorrectly applied to the published results can improve the FB series. The original FB3 series as received from Bretagnon is expressed in terms of the independent time variable of the VSOP82/ELP2000 ephemeris. The most-important angular frequencies (, , and ) published in FB are actually the original FB3 series angular frequency values multiplied by the factor . Apparently this frequency transformation was done to compensate for a mean rate difference that was thought to occur between the independent variable of the VSOP82/ELP2000 ephemeris and . However, the independent variable of the VSOP82/ELP2000 ephemeris (like the independent variable of numerical ephemerides, see Sect. 2) has no mean rate difference with . The results in Fig. 3 serve as numerical proof of this because we obtain much better agreement with TE405 if we do not apply the angular frequency transformation to the FB3 series. The invalid angular frequency transform was also present for the FB (127 terms truncated at 10 ns) and FB2 (791 terms truncated at 0.1 ns) forms of the series investigated in Paper I. The invalid angular frequency transformation contributes to the errors found for the FB series (although the FB series truncation errors should be larger, see Fig. 2) and should provide the principal source of the errors found for the FB2 series (see Table 7 of Paper I).
Fig. 3 also illustrates the modest but important improvements possible with mass transformation of series. For this transformation (and also for the mass transformations and error analyses in Sects. 3 and 5) we use parameters defined as the ratio of various planetary masses or the lunar mass to the solar mass. We also use the mass parameter, , the square of the number of meters per AU. We use rather than (which is proportional to ) as a mass parameter because with the planetary and lunar mass ratios fixed, the other quantities that appear in Eqs. (3) and (4) are and which are both proportional to . The VSOP82/ELP2000 ephemeris and the resulting time-ephemeris series are mostly based on the 1976 IAU system of masses. (Compare Bretagnon 1982, Table 5 with Seidelmann 1992. See also FB.) Through the fit to DE200, the DE200 masses also indirectly affect the VSOP82/ELP2000 ephemeris and the resulting time-ephemeris series, but this inconsistency does not seem to introduce appreciable errors to the mass-correction procedure (see Fig. 1).
To transform from the IAU 1976 system to the DE405 system, we multiply all important amplitudes that are affected by a particular planet or the Moon by the ratio of the associated mass-ratio parameters. Similarly, all important amplitudes are multiplied by the ratio of the parameters. We transform TE200 (Sect. 3) to the DE405 mass system using similar procedures. The various forms of the Fairhead & Bretagnon series have convenient identifications of the solar, planetary (excluding the Earth) or lunar source of each term. The mass-transformation procedure necessarily excludes Pluto because no analytical theory of Pluto is yet available. However, this exclusion should not make much difference since from Eq. (13) of Paper I the time-ephemeris amplitude of the principal term of the Pluto effect should be only 0.6 ns.
Work on an improved analytical theory is in progress (Bretagnon & Moisson 1998), but until that is completed it is difficult to identify the exact source of the remaining discrepancies between the mass-corrected FB3 series and TE405. However, since the 1-ns discrepancies (Fig. 1) between the mass-corrected TE200 and TE405 results are substantially smaller than the 15-ns discrepancies between the mass-corrected analytical results and TE405 (Fig. 3), we suggest that a possible cause of these latter discrepancies could be the deviations between VSOP82/ELP2000 results and DE200 (and DE405) results. The analytical ephemerides were only fit to the DE200 positions over a limited time interval (1890 to 2000) with masses that are inconsistent with the DE200 masses. The maximum residuals in the fitting interval of the Earth's latitude, longitude, and heliocentric distance (Bretagnon 1982, Table 6) correspond respectively to 114, 240, and AU. If we propagate these maximum position errors to the term proportional to the square of the earth speed in Eq. (3) and corresponding time-ephemeris error ignoring phase effects we derive an upper limit of 2.6 ns independent of assumed period for the error perturbation. If we propagate the maximum error of the heliocentric distance of the Earth to the gravitational potential term in Eq. (3) and corresponding time-ephemeris error ignoring phase effects we derive an upper limit of 0.79 ns where P is the assumed period (in years) of the perturbation. Inside the fitting interval the VSOP82/ELP2000 model and mass inconsistencies with DE200 are partially compensated by the fit. Outside the fitting interval the position discrepancies and associated mass-corrected time-ephemeris errors should grow substantially. When this additional factor is combined with the calculated upper limits for the time-ephemeris error due to the speed and gravitational potential terms it seems plausible that VSOP82/ELP200 fitting errors are responsible for the errors of the mass-corrected series illustrated in Fig. 3.
The final correction we add to the mass-corrected FB3 series is two long-period sinusoids with amplitudes significantly above the FB3 truncation limit. Table 2 presents sinusoidal coefficients which were determined by the method of non-linear least squares using a fit to the residuals between the mass-corrected series and TE405 results. Fig. 4 presents the residuals which are smaller (when compared over the same limited epoch ranges) than the best residuals between numerical and analytical work that have been previously published (see Fig. 3 of FB and Fig. 6 of Paper I).
Table 2. Coefficients of the correction to the FB3 series.
Comparison of Figs. 3 and 4 shows how the long-term residuals are improved by adding the sinusoidal corrections to the series. (Similar sinusoidal corrections and residuals were obtained from a fit to the residuals between mass-corrected series and TE200 results.) From these results it should be worthwhile to look for additional terms in the analytical results with periods near 200 and 600 years. However, we caution that although these sinusoids give a good representation of the long-term residuals in Fig. 3 the actual deviations may be caused by a correction of different functional form (e.g., a sum of mixed secular and sinusoidal terms). Because of this uncertainty and because of general extrapolation uncertainties, the correction sinusoids should not be used outside the epoch range of TE405. Even within the epoch range the correction sinusoids only reduce the maximum residuals to 5 ns. Thus, for the most accurate results we recommend using the numerical time-ephemeris rather than the corrected series results. Nevertheless, the corrected series results are useful for determining with the hybrid technique (see next section).
© European Southern Observatory (ESO) 1999
Online publication: July 26, 1999