## 4. Corrections to the FB time ephemeris seriesAnalytical 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 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
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 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).
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 |