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

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1. Introduction

The relativistic time-dilation integral, [FORMULA] (Eq. [3]), is a quantity that is required to transform between terrestrial time, [FORMULA], and the (solar-system) barycentric time scales [FORMULA] or [FORMULA]. [FORMULA] is a linear transformation of [FORMULA] that represents (Standish 1998c) the independent variable of a modern ephemeris such as the JPL ephemeris. Thus, an accurate approximation of [FORMULA] (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 [FORMULA]. 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 104 such observations). Thus, the time-ephemeris derivative errors should be kept less than [FORMULA] for the interpretation of observed ranges of spacecraft. This limit requires some care to achieve because it is 5 orders of magnitude smaller than the maximum absolute value of the time-ephemeris derivative, [FORMULA].

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 µs and are beginning to approach 0.1 µs (Kaspi et al. 1994). We adopt this latter value as the nominal best error of a daily mean pulse-arrival epoch. Using the criterion that systematic errors should be 2 orders of magnitude less than the smallest random errors leads to a maximum acceptable error for a time ephemeris of 1 ns. This error limit requires some care to achieve because it is 6 orders of magnitude smaller than the maximum absolute value of the time-ephemeris, 2 ms.

Efforts to obtain analytical approximations for [FORMULA] (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 K value). We rigorously define in Sect. 2 the relativistic time-dilation integral that is used to transform between Earth-based and solar-system-based time scales. We present in Sect. 3 a numerical approximation of this integral, TE405, which has an unprecedented accuracy of 0.1 ns. We present in Sect. 4 angular-frequency and mass transformations of the FB time-ephemeris series which reduce its maximum errors to 15 ns from 1600 to 2200 and suggest the remaining long-term residuals (which can be fit by two sinusoids) and short-term residuals are due to errors in the fit of VSOP82/ELP2000 to DE200. We use in Sect. 5 the combination of TE405 and the corrected FB series to determine [FORMULA], the coefficient of the linear term that is subtracted from TE405. We also determine K which relates ephemeris units for time and distance and the corresponding SI units for the same quantities. We conclude the paper in Sect. 6.

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© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999