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Astron. Astrophys. 348, 642-652 (1999)

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3. Numerical calculation of the time ephemeris

3.1. Procedure

Following 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 [FORMULA] value that was subsequently corrected for each time ephemeris (Sect. 5). The granule interpolation boundaries (Standish et al. 1992) that occur every 4th midnight for the JPL ephemeris necessarily have second and higher-order derivative discontinuities which would reduce the effective order and efficiency of our numerical quadrature method if it spanned the interpolation boundaries. To avoid this potential problem the results were calculated on half-day integration intervals with end points at noon and midnight (or nearest midnight to the common epoch of [FORMULA] for the special integration interval that is required to help establish the zero point).

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 + [FORMULA]) - [FORMULA], where the sinusoid is the derivative of the principal term of the series for the time ephemeris. These tests indicated the maximum numerical errors from the quadrature and significance loss were negligible compared to the maximum numerical errors of 0.3 ps in the time ephemeris and [FORMULA] in its derivative caused by interpolating the results with 7 polynomial coefficients per 4-day granule. This high degree of numerical precision is straightforward to achieve, requires modest disk space, interferes negligibly with the estimated level of accuracy (see below), and provides a numerically clean benchmark to compare with other work. These results supersede the time-ephemeris results described in Paper I which were based on DE200 and DE245 and which were presented with an interpolation precision of order 1 ns. (Detailed results from the Paper I had been lost in a disk crash, but we checked that our new TE200 does reproduce residual plots made with the old TE200 consistently with the 1 ns interpolation errors of the older calculation.)

3.2. Accuracy

Comparisons 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.

[FIGURE] Fig. 1. Differences of numerical time ephemerides (with units of ns) and their derivatives (with units of [FORMULA]) over their common epoch range and over a modern epoch range. TE200 and TE405 correspond to the DE200 and DE405 versions of the JPL ephemeris. The TE200X ephemeris has been mass-corrected to the DE405 mass system using the procedure given in Sect. 4.

The maximum mass-dependent errors of TE200 are 6 ns for the time ephemeris and [FORMULA] for its derivative. Of all the mass parameters of the JPL ephemerides, the Uranus and Neptune masses have the largest relative change from DE200 to DE405. Consequently, the largest terms in the mass-dependent errors of TE200 are 4 sinusoids with the Uranus and Neptune synodic periods (relative to Earth) and sidereal periods. Differentiation amplifies short-term changes so the mass-dependent errors of the TE200 derivative are dominated by the Uranus and Neptune synodic periods near a year. These short periods beat together with the synodic period (170 yr) of Uranus relative to Neptune.

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 [FORMULA] for its derivative.

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 [FORMULA] for its derivative.


Table 1. Propagation of mass-parameter errors to the TE405 time ephemeris and its derivative.
a) Mass parameter. [FORMULA], [FORMULA],[FORMULA], and [FORMULA] are the respective ratios of the Mars, Jupiter, Saturn, and Moon masses to the solar mass. We have not included in this table the calculated error budget associated with other mass parameters because their contributions are insignificant.
b) Relative error in the given mass parameter derived from Standish (1995).
c) Sum over amplitudes that are affected by the given mass parameter (see text).
d) Estimated maximum error in [FORMULA] (rounded to 10 ps) from the product of the relative error in the given mass parameter and [FORMULA].
e) Sum over amplitudes [FORMULA] angular frequencies that are affected by the given mass parameter (see text). f) Estimated maximum error in [FORMULA] (rounded to [FORMULA]) from the product of the relative error in the given mass parameter and [FORMULA].

We estimate the maximum mass-independent errors are 0.1 ns for TE405 and [FORMULA] for its derivative, i.e., an order of magnitude less than the corresponding errors for TE200. These order-of-magnitude estimates of the mass-independent TE405 errors are based on the observation that consistency between modern JPL ephemerides is roughly an order of magnitude better than the consistency between DE200 and the modern ephemerides (see Standish et al. 1995; Standish 1998b).

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 [FORMULA] for its derivative. This level of accuracy is directly attributable to the great care that has gone into the preparation of the JPL ephemerides and is an order of magnitude better than the accuracy currently required for reducing pulse-arrival times of pulsars and spacecraft ranging. (Sect. 1). This level of accuracy is also at least an order of magnitude better on all time scales than the accuracy of the best frequency standards (linear ion traps, see Tjoelker et al. 1996) now being installed (see http://horology.jpl.nasa.gov/research.html ) at the ground stations of the Deep Space Network.

Our estimate of the accuracy of the time ephemeris ignores the effect of the error in the [FORMULA] rate adjustment that is subtracted from the time ephemeris. We estimate the [FORMULA] value and error in Sect. 5 for each version of our time ephemeris.

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© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999