## 1. IntroductionThe relativistic time-dilation integral, (Eq. [3]), is a quantity that is required to transform between terrestrial time, , and the (solar-system) barycentric time scales or . is a linear transformation of that represents (Standish 1998c) the independent variable of a modern ephemeris such as the JPL ephemeris. Thus, an accurate approximation of (we denote such approximations as time ephemerides of the Earth or just as time ephemerides for short) is required to analyze all astronautical or astronomical measurements that are precisely reduced with the aid of a modern ephemeris. From the view point of large scale metrology, time-related measurements made on the Earth suffer the variation of topocentric time scales. Examples which depend on locally realized frequency standards are measurements of spacecraft ranges and pulse-arrival times from pulsars. The observational errors of spacecraft ranges pose a stringent
constraint on the maximum acceptable errors of a time-ephemeris
derivative. The typical error of a spacecraft range observed with the
Deep Space Network is 1 m (DSN 1999). At the distance of Pluto
this translates to a relative range error and corresponding
time-ephemeris derivative error of .
Systematic errors should ideally be at least two orders of magnitude
smaller than random errors in the best single observations (thus
making the systematic error equivalent to the random error of the mean
of 10 The observational errors of daily mean pulse-arrival epochs of
pulsars pose a stringent constraint on the maximum acceptable errors
of a time ephemeris. These epoch errors are less than
1 Efforts to obtain analytical approximations for (Eq. [3]) or its derivative go back to Aoki (1964) and Clemence & Szebehely (1967). This initial work has been followed by analytical approximations of increasing accuracy and complexity (e.g., Moyer 1981b; Hirayama et al. 1987) culminating in a 1705-term version (Bretagnon 1995) of a time-ephemeris series given by Fairhead & Bretagnon (1990, FB). This series is based on the VSOP82/ELP2000 analytical planetary and lunar ephemeris (Bretagnon 1982; Chapront-Touzé & Chapront 1983) that was fit to DE200 from 1890 to 2000. As an alternative to the series approach, one may directly calculate a time ephemeris using numerical quadrature of quantities supplied by a planetary and lunar ephemeris (see Backer & Hellings 1986). The mass-corrected series results and numerical results made privately available from the JPL group to Fairhead and Bretagnon agreed within 3 ns over the epoch range from 1900 to 2000 (see Fig. 3 of FB). The 3-ns level of agreement between the JPL and FB time ephemerides was not obtained by subsequent much more extensive comparisons of numerical and analytical time ephemerides (Fukushima 1995, Paper I). For example, the RMS deviation of TE200 (a numerical time ephemeris based on the JPL ephemeris, DE200) with the FB2 series (an extended form of the FB series containing 791 terms) is 26 ns (Table 7 of Paper I). Subsequently, we found the source of this large disagreement was an inappropriate angular-frequency transformation (see discussion in Sect. 4) that was used for the published FB coefficients. Presumably, this source of error was not present in the published comparison between the FB results and the numerical time ephemeris from the JPL group. The purpose of the current paper is to follow up Paper I by
presenting new results for a numerical time ephemeris, series
corrections, and the ratio of ephemeris units to SI units (the
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