## 2. DefinitionsThe purpose of this section is to rigorously define the
relativistic time-dilation integral that is used to transform between
terrestrial time, , or geocentric
coordinate time, , and the
(solar-system) barycentric coordinate times,
and .
, ,
and are defined by
Recommendations III and IV of IAU Resolution A4 (1992; see also
Paper I for further discussion).
is a linear transformation (Standish 1998c) of
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 "" or " ,
, and
are usually expressed in Julian day
numbers with a common epoch (if we ignore the effects of observer
location) of . However, for the
present paper The linear transformation between and is defined by is a rate coefficient defined (Eq. [12]) so that the mean rate (in
a particular sense, see discussion after Eq. [15]) of
and
are the same. (We discuss the recommended values for Eq. (1) implies the ephemeris unit of a second is equivalent to
Kto convert to the SI units appropriate
for the coordinate systems associated with
, ,
or . Ephemeris and SI units are
the same for velocity and .We define the relativistic time-dilation integral, Here, The current definition of has the linear trend conveniently removed by subtracting from the integrand. This is an important difference from the previous definition in Paper I which includes the linear trend. 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 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 formally excludes post-Newtonian and asteroid effects. However, we define With this definition our calculations implicitly include the mean rate effect of the post-Newtonian and asteroid corrections, and (Sect. 5). Our definition of is equivalent to one where these corrections are added to the integrand and (rather than ) subtracted from the integrand. The relationship between and is expanded as where is the barycentric position
vector of the observer in the ephemeris coordinate system, the
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 frame when observed from the 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 (Sect. 3). The relationship between and is defined by or where is the dimensionless constant, and is the gravitational plus spin potential of the Earth at the geoid. (We discuss the recommended values of and in Sect. 5.) If we combine Eqs. (1), (7), and (8), then after some manipulation we obtain To force the mean rate of and to be equal in a particular sense (see below) we define In Eqs. (11) and (13) we have ignored small correction terms (see the discussion after Eq. [7]). We have kept the factor, , 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: 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 (see discussion of
Eq. [3]). Thus, in this particularly defined sense 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
and
) includes a term
(where the primes distinguish his
With this identification, the two equations give identical results (once the difference in offset formulation is taken into account) except that the factor appearing in our Eq. (13) is replaced by the factor in the Standish equation. This difference is negligible. We have formulated our equations in such a way that any arbitrary
value of the offset, , 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
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, , 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 yields
(For offsets with this small a magnitude, dropping the constant from Eq. [13] would cause a negligible error.) The right-hand-side of Eq. (13) is a function of so, strictly speaking, iterations should be performed to calculate from using this equation. However, because is less than 2 ms, evaluating the right-hand-side of Eq. (13) using the approximation introduces a negligible error. © European Southern Observatory (ESO) 1999 Online publication: July 26, 1999 |