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Astron. Astrophys. 348, 642-652 (1999) 3. Numerical calculation of the time ephemeris3.1. ProcedureFollowing Paper I we have used a 10th-order numerical
quadrature method (a double-precision [8 bytes total floating-point
word length] version of subroutine QROMB from Press et al. 1992) to
evaluate Eq. (3). The integrand was evaluated using the JPL
ephemerides for the Sun, Moon, and all planets in the solar system. We
ignored the 15-ps periodic effect of the asteroids (Paper I)
because it is negligible compared to the 0.1-ns accuracy of our final
result (see later discussion). We performed the time-ephemeris
calculation for both the DE200 and DE405 versions of the JPL ephemeris
and we designate the corresponding time-ephemeris results as TE200 and
TE405. The integrations were performed with a preliminary
We have interpolated the time-ephemeris results using Chebyshev polynomials on 4-day granules. These polynomials fit the time ephemeris and its derivative exactly at the end points of the granule (thus providing overall continuity in these quantities) and fit these quantities by least squares at the 7 interior points. The Chebyshev coefficients consistent with these requirement have been calculated with the CHEBFIT routine (described by Newhall 1989and communicated by Standish 1998a) that is used to calculate the Chebyshev coefficients of the JPL ephemeris. We have used 7 polynomial coefficients per 4-day granule to represent the TE200 and TE405 versions of the time ephemeris. This choice results in a gzip-compressed ASCII file size of 5 Mbytes and binary file size of 3 Mbytes for the full 600-yr epoch range of either version of the time ephemeris. These disk-space requirements are modest. Tests of the quadrature error and significance-loss error were
obtained by comparing complete time ephemerides from 1600 to 2200
based on half-day and one-day integrations and based on a numerical
quadrature of an integrand of the form, (sinusoid +
3.2. AccuracyComparisons between TE200 and TE405 (Fig. 1) help us estimate the accuracy (maximum value of the actual errors rather than the numerical precision) of our results. Because of the good numerical precision of our calculations, the accuracy of the time ephemerides is essentially determined only by the accuracy of the JPL mass parameters and ephemerides. The comparisons between TE200 and TE405 show there are both mass-dependent and mass-independent components to the time-ephemeris errors. The mass-independent errors still remain after mass correction according to the method of Sect. 4 and are presumably the result of different models and starting conditions for the JPL ephemerides extrapolating in different ways to epochs far removed from the modern era where astrometry constrains the solutions. DE405 is a recent JPL ephemeris (Standish 1998b; see also Standish et al. 1995) that fits a substantial amount of additional precise astrometry compared to DE200 which was prepared in 1981. Therefore, the errors of TE200 should be substantially larger than the errors of TE405 so that Fig. 1 essentially illustrates both the mass-dependent and mass-independent errors of TE200 and its derivative.
The maximum mass-dependent errors of TE200 are 6 ns for the
time ephemeris and As expected, the envelope of the mass-independent errors of TE200
grows substantially larger as we extrapolate from the modern era where
DE200 was calibrated by the astrometry that was available when it was
created. The maximum values of these errors within the epoch range are
1 ns for the time ephemeris and For TE405, the mass-dependent and mass-independent error components are more difficult to estimate since we have no time ephemeris with higher accuracy for comparison. We estimate the maximum mass-dependent errors (see Table 1)
for TE405 by propagating the relative mass-parameter errors taken from
Standish (1995) to the relevant amplitudes of the series results using
the same model that is used in Sect. 4 to correct for changes in mass.
For each mass parameter we have summed amplitudes that are affected by
that mass parameter using the extended form of the FB series. Summing
amplitudes ignores phase information, but this approximation should be
excellent because usually the sums are dominated by just one
component, and even in the case of two dominant components the two
error sinusoids should be in phase for a number of epochs within the
large range we are considering here. We evaluate the mixed
secular-periodic amplitudes at 1600 to maximize their contribution to
the sums and error estimates. The total mass-dependent error estimates
from Table 1 are obtained by simply adding the individual results
(again ignoring phase information). The maximum mass-dependent error
estimates are 80 ps for TE405 and
Table 1. Propagation of mass-parameter errors to the TE405 time ephemeris and its derivative. We estimate the maximum mass-independent errors are 0.1 ns for
TE405 and Combining the mass-independent and mass-dependent errors estimates
together we conclude that the accuracy is the order of 0.1 ns for
TE405 and Our estimate of the accuracy of the time ephemeris ignores the
effect of the error in the ![]() ![]() ![]() ![]() © European Southern Observatory (ESO) 1999 Online publication: July 26, 1999 ![]() |