Astron. Astrophys. 348, 642-652 (1999)
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.
![[FIGURE]](img97.gif) |
Fig. 2. Maximum difference over the epoch range from 1600 to 2200 of results from truncated FB3 series compared with results from the original FB3 series (which includes terms with amplitudes of 10 ps or more). The truncation limit is the minimum amplitude term included in the truncated series.
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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).
![[FIGURE]](img103.gif) |
Fig. 3. Differences of time ephemerides (with units of ns) over the complete TE405 epoch range and over a modern epoch range. Three versions of the FB3 series are compared with TE405 results. The FB3A series is the original FB3 series transformed to the published angular frequencies (see text). The FB3B series is the original FB3 series. The FB3C series is the original FB3 series transformed to the mass parameters of DE405 (see text).
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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]](img114.gif)
Table 2. Coefficients of the correction to the FB3 series.
Note:
These coefficients are defined in the same way as the coefficients of the original FB3 series (aside from the difference of the amplitude unit). The correction to be added to the FB3 series is .
![[FIGURE]](img117.gif) |
Fig. 4. Differences of a corrected and supplemented FB3 series and TE405 results (with units of ns) and their derivatives (with units of ) over the complete TE405 epoch range and over a modern epoch range. The original FB3 series has been corrected to the DE405 masses (see text) and also supplemented by two long-term sinusoids. The coefficients of these sinusoids are given in Table 2.
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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
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