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

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2. Definitions

The purpose of this section is to rigorously define the relativistic time-dilation integral that is used to transform between terrestrial time, [FORMULA], or geocentric coordinate time, [FORMULA], and the (solar-system) barycentric coordinate times, [FORMULA] and [FORMULA]. [FORMULA], [FORMULA], and [FORMULA] are defined by Recommendations III and IV of IAU Resolution A4 (1992; see also Paper I for further discussion). [FORMULA] is a linear transformation (Standish 1998c) of [FORMULA] that represents the independent variable of a modern ephemeris such as the JPL ephemeris. Our development follows Seidelmann & Fukushima (1992), Seidelmann et al. (1992), and Paper I, but we have replaced the ambiguous time designations "[FORMULA]" or "t" that occurred in the previous work with explicit time scales. We prefer the current approach because of its definiteness although the removed ambiguities are only the order of the post-Newtonian corrections.

[FORMULA], [FORMULA], and [FORMULA] are usually expressed in Julian day numbers with a common epoch (if we ignore the effects of observer location) of [FORMULA]. However, for the present paper we subtract the common epoch from these time scales and also [FORMULA]. This change in zero point simplifies the exposition so that we have been able to use units of seconds (s) for all time scales in our equations.

The linear transformation between [FORMULA] and [FORMULA] is defined by

[EQUATION]

where

[EQUATION]

is a rate coefficient defined (Eq. [12]) so that the mean rate (in a particular sense, see discussion after Eq. [15]) of [FORMULA] and [FORMULA] are the same. (We discuss the recommended values for K and the equivalent [FORMULA] in Sect. 5.) [FORMULA] is the value of [FORMULA] corresponding to the instant when [FORMULA] (and [FORMULA] and [FORMULA] if we ignore the effects of observer location) is zero. [FORMULA] may vary from ephemeris to ephemeris depending on the treatment of the integration constant for the relativistic time-dilation integral (see discussion prior to Eq. [17]).

Eq. (1) implies the ephemeris unit of a second is equivalent to K SI seconds and similarly for meters because the speed of light in vacuum is identical in all coordinate systems. Thus, all times (s), positions (m), and values ofGM(gravitational constant times mass in units of m3 s-2) associated with, for example, the JPL ephemeris and header values must be multiplied byKto convert to the SI units appropriate for the coordinate systems associated with [FORMULA], [FORMULA], or [FORMULA]. Ephemeris and SI units are the same for velocity and [FORMULA].

We define the relativistic time-dilation integral,

[EQUATION]

where

[EQUATION]

and

[EQUATION]

Here, c is the speed of light in vacuum, [FORMULA] and [FORMULA] are the position and velocity of the geocenter relative to the solar-system barycenter (SSB), the sum in Eq. (4) is over all solar-system objects excluding the Earth that are more massive than the asteroids, and [FORMULA] and [FORMULA] are the mass and SSB position vector of the ith object in the solar system. Note that all quantities in Eqs. (3) through (5) are expressed in terms of ephemeris meters and seconds rather than SI meters and seconds.

The current definition of [FORMULA] has the linear trend conveniently removed by subtracting [FORMULA] from the integrand. This is an important difference from the previous definition in Paper I which includes the linear trend. [FORMULA] is determined (Sect. 5) by finding the zero of the slope of the residuals between the numerical time ephemerides which approximate the integral and a corrected secular + sinusoidal series (with origin at J2000 but no linear term) for the time ephemeris. In this way the separation of the linear trend from the time ephemeris is done similarly to the series results, but the exact value of [FORMULA] depends on the underlying planetary and lunar ephemeris used to create a time ephemeris and is partially insulated from the remaining errors in the corrected series results.

Our definition of [FORMULA] formally excludes post-Newtonian and asteroid effects. However, we define

[EQUATION]

With this definition our calculations implicitly include the mean rate effect of the post-Newtonian and asteroid corrections, [FORMULA] and [FORMULA] (Sect. 5). Our definition of [FORMULA] is equivalent to one where these corrections are added to the integrand and [FORMULA] (rather than [FORMULA]) subtracted from the integrand.

The relationship between [FORMULA] and [FORMULA] is expanded as

[EQUATION]

where [FORMULA] is the barycentric position vector of the observer in the ephemeris coordinate system, the K factor converts from ephemeris units to SI units (see previous discussion), and the recommended value of [FORMULA] is discussed in Sect. 5. (For consistency in notation we have expressed the position vector of the observer relative to the geocenter, [FORMULA], in ephemeris units, but if this vector is expressed in SI units, as is normally the case, then the K factor should not multiply the first term.)

The first term of Eq. (7) depends on observer location and can be derived from the Principle of Equivalence; in the limit of small fields and accelerations, this term is required by the theory of special relativity to correct for the lack of synchronization of clocks (see discussion on p. 25 of Eisberg 1961) in the moving [FORMULA] frame when observed from the [FORMULA] frame. More generally, this term is the first term in a Taylor series with both Newtonian and post-Newtonian higher order terms (Thomas 1975; Moyer 1981a). Near the geoid, these higher order terms are smaller than the periodic post-Newtonian and asteroid corrections to the second term of Eq. (7). These latter corrections have a maximum magnitude of 33 ps for the post-Newtonian correction and 15-ps for the asteroid correction. In practice we ignore all these corrections (indicated as "small correction terms" in Eq. [7]) because they are smaller than the maximum errors, of order 0.1 ns, in TE405, our best approximation to [FORMULA] (Sect. 3).

The relationship between [FORMULA] and [FORMULA] is defined by

[EQUATION]

or

[EQUATION]

where [FORMULA] is the dimensionless constant,

[EQUATION]

and [FORMULA] is the gravitational plus spin potential of the Earth at the geoid. (We discuss the recommended values of [FORMULA] and [FORMULA] in Sect. 5.)

If we combine Eqs. (1), (7), and (8), then after some manipulation we obtain

[EQUATION]

To force the mean rate of [FORMULA] and [FORMULA] to be equal in a particular sense (see below) we define

[EQUATION]

and Eq. (11) reduces to

[EQUATION]

In Eqs. (11) and (13) we have ignored small correction terms (see the discussion after Eq. [7]). We have kept the factor, [FORMULA], in Eq. (13) to be exactly consistent with our previous equations, but replacing this factor by unity would cause a maximum error of only 30 ps.

From Eqs. (7) and (13) we derive the following mean rates:

[EQUATION]

and

[EQUATION]

where the angle brackets correspond to taking a specially defined mean. This mean is formally defined by determining a secular plus periodic series with origin at J2000 and ignoring all but the linear term before taking the derivative. By definition no such mean rate exists for [FORMULA] (see discussion of Eq. [3]). Thus, in this particularly defined sense the mean rate of the [FORMULA] and [FORMULA] are equal. We emphasize that definition (12) is necessary to insure this condition.

Aside from a difference in the way the offset is formulated (to be discussed below) our Eq. (13) also differs slightly from Eq. (3) of Standish (1998c) in one other particular. His equation (with [FORMULA] and [FORMULA]) includes a term [FORMULA] (where the primes distinguish his K and [FORMULA] constants from ours). Our formulation replaces this term by [FORMULA] because we prefer the relationship between [FORMULA] and [FORMULA] to be defined independently of [FORMULA] (see Eq. [8]). To keep [FORMULA] (and therefore the mean rate of [FORMULA] exactly the same in both formulations) we must have

[EQUATION]

With this identification, the two equations give identical results (once the difference in offset formulation is taken into account) except that the factor [FORMULA] appearing in our Eq. (13) is replaced by the factor [FORMULA] in the Standish equation. This difference is negligible.

We have formulated our equations in such a way that any arbitrary value of the offset, [FORMULA], may be used for the construction of future planetary and lunar ephemerides. Nevertheless, when interpreting observations with a presently existing ephemeris, it is necessary to employ the same effective [FORMULA] value that was used for the construction of that ephemeris. In the case of the recent JPL ephemerides (DE143, DE145, and DE403-DE406), for example, the quantity, [FORMULA], in our present notation (see also Eq. [3] from Standish 1998c) was approximated by a series (from Hirayama et al. (1987) as corrected by Paper I). Evaluation of this series at [FORMULA] yields [FORMULA] µs (Standish 1999). Therefore, to interpret observations correctly with the indicated recent JPL ephemerides and our formulation it is necessary to adopt

[EQUATION]

(For offsets with this small a magnitude, dropping the constant [FORMULA] from Eq. [13] would cause a negligible error.)

The right-hand-side of Eq. (13) is a function of [FORMULA] so, strictly speaking, iterations should be performed to calculate [FORMULA] from [FORMULA] using this equation. However, because [FORMULA] is less than 2 ms, evaluating the right-hand-side of Eq. (13) using the approximation [FORMULA] introduces a negligible error.

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© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999
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