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

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5.1. [FORMULA] and [FORMULA]

The determination of [FORMULA] from numerical quadrature results alone is not recommended. One could adjust [FORMULA] so the mean of the integrand of Eq. (3) was zero, but then the definition of [FORMULA] would be ambiguous because it would depend strongly on the epoch range chosen for the integration. A much better approach is to define [FORMULA] as the constant term (at J2000) in a series approximation for [FORMULA] or the corresponding linear term in a series approximation to the integral of this same quantity. (This integral is the same as the time ephemeris defined by Eq. [3] except that the [FORMULA] term is not subtracted from the integrand.) A series approach inevitably has some degree of ambiguity about how the periodic and secular terms are separated since a compromise must be made between the accuracy of the result and the longest period allowed in the analysis (FB). Beyond this period limit the periodic terms are approximated as secular perturbations which in turn affect the derived [FORMULA] value. However, the degree of ambiguity is much smaller than in the strictly numerical approach because the period limit is usually quite long. For example, FB treat a sinusoidal term with a [FORMULA]-yr period (and by implication all terms of longer period) as a secular perturbation in their work.

A drawback of the analytical approach for determining [FORMULA] is the current series are not as accurate as the numerical time ephemeris results (see previous section). Thus, the best current method of determining [FORMULA] is with a hybrid approach (Paper I) that combines results from the numerical and series methods. The hybrid method uses a linear least-squares fit of the residuals between series results (with the [FORMULA] term removed) and a numerical time ephemeris that is calculated with a preliminary value of [FORMULA]. The fit gives a correction to the numerical [FORMULA] value that is completely independent of the the series [FORMULA] value. The hybrid method removes the slope in the residuals of the fit to a high degree of precision (see next subsection) by averaging out the errors in the series which have time scales less than the epoch range of the numerical ephemeris.

The [FORMULA] results calculated with the hybrid technique for the TE200 and TE405 versions of the time ephemeris and for the corrected and supplemented series results of the last section are




We have presented rounded values here that are consistent (aside from two guard digits) with the accuracy discussed in the next subsection. For highest numerical precision of the time ephemeris we use machine-precision values of [FORMULA] to remove the slope. These precise [FORMULA] values are also stored in the ephemeris header to facilitate future rate adjustments to the time ephemeris when more accurate values of [FORMULA] become available.

The present results supersede the [FORMULA] and [FORMULA] results presented in Table 7 of Paper I. (We have changed our notation from Paper I; currently we prepend a "[FORMULA]" to the [FORMULA] symbol to indicate whenever post-Newtonian and asteroid effects are not corrected.) Our present [FORMULA] value is [FORMULA] larger than the previous hybrid value derived from the FB2 series. We ascribe this difference to the previously discussed errors in the FB2 series. Our present [FORMULA] value is [FORMULA] smaller than the previous hybrid value of [FORMULA] derived from the FB2 series. We ascribe most of this difference to differences between the DE245 and DE405 ephemerides. For comparison, note that our present [FORMULA] value is [FORMULA] smaller than the present value of [FORMULA].

5.2. Errors in [FORMULA] and [FORMULA]

The [FORMULA] values derived in the last subsection are used for both adjusting the rate of time-ephemeris results and calculating the ratio of ephemeris units to SI units. These two different uses have different error definitions which require separate error discussions.

5.2.1. Error in [FORMULA] associated with the rate adjustment of the time ephemeris

We have discussed in Sect. 3 the time-ephemeris errors caused by the JPL ephemeris errors (assuming a perfect rate adjustment), and here we discuss errors in the rate adjustment (assuming a perfect JPL ephemeris) and the associated time-ephemeris errors. Assuming there are no important missing terms in the analytical integration of the VSOP82/ELP200 ephemeris to form the series for the time ephemeris, the errors in the rate adjustment must be ultimately caused by inconsistencies between the VSOP82/ELP200 ephemeris and the particular JPL ephemeris that forms the basis for a numerical time ephemeris. These inconsistencies propagate to differences (e.g., Fig. 3) between the mass-corrected series results for the time ephemeris and the numerical time ephemeris. Furthermore, even when the present analytical time ephemeris is supplemented by additional terms, there are remaining differences (e.g., Fig. 4) which are propagated via the hybrid technique to errors in the derived [FORMULA] value and rate adjustment for the numerical time ephemeris.

One measure of the rate-adjustment errors is the formal least-squares errors of the [FORMULA] values determined by the hybrid technique. These errors are respectively [FORMULA] and [FORMULA] for [FORMULA] and [FORMULA]. These fitting errors greatly underestimate the actual errors in the rate adjustment because they only account for residuals with time scales shorter than the epoch range of the time ephemeris.

For future work it should be straightforward to calculate the time ephemeris associated with the long JPL ephemeris, DE406. This ephemeris has 10 times the epoch range of DE405 and the associated time ephemeris could be used to evaluate the long-term errors in the series results. For the present we have no such long-term comparison so a discussion of the long-term errors in the analytical time ephemeris must be speculative. However, it is not unreasonable to suppose that sinusoidal error terms with 10 times the amplitude and period of the 600-yr sinusoid from Table 2 exists in the present series results. The maximum rate adjustment corresponding to such an error term is [FORMULA], and we take this as a speculative estimate of the rate error in the present series results (without [FORMULA] term) for the time ephemeris caused by inconsistencies between the VSOP82/ELP200 and JPL ephemerides. The present numerical results for the time ephemeris also share this same rate error because the hybrid technique forces the average rates to be the same for the analytical (without [FORMULA] term) and numerical time ephemerides. The speculative estimate of the rate error corresponds to [FORMULA] ns over the 600-yr epoch range of the present time ephemeris.

This source of error is much larger than the 0.1 ns estimated accuracy (aside from rate-adjustment errors) of the TE405 time ephemeris (Sect. 3). However, a rate adjustment is a simple correction that can be made whenever improved [FORMULA] values become available without having to recalculate the numerical quadrature and Chebyshev interpolation coefficients of the present time ephemeris.

5.2.2. Total error in [FORMULA] associated with the conversion from ephemeris to SI units

The total errors of [FORMULA] and [FORMULA] can be split into the (just discussed) rate-adjustment component due to inconsistencies between VSOP82/ELP200 and JPL ephemerides and the component due to the errors in the JPL ephemerides themselves.

Because the DE405 ephemeris errors should be substantially smaller than the DE200 ephemeris errors, the ephemeris component of the [FORMULA] error should be close to [FORMULA]. The mass-dependent part of this difference can be estimated using the components of [FORMULA] from Paper I and the known differences in mass parameters between DE200 and DE405 (Table 3). For this calculation we have assumed that all [FORMULA] components are proportional to [FORMULA] (following an argument made in Sect. 4) while the individual components other than the solar component and the component proportional to the square of the Earth speed are proportional to the appropriate ratio of the lunar or planetary mass to the solar mass. The result is the mass-dependent part of [FORMULA]. Thus, by subtraction the mass-independent part of the ephemeris component of the [FORMULA] error should be close to [FORMULA].


Table 3. Propagation of mass-parameter differences and errors to [FORMULA].
a) Mass parameter. [FORMULA] is the square of the number of meters per AU, and [FORMULA], [FORMULA], [FORMULA], and [FORMULA] are the respective ratios of the Jupiter, Saturn, Uranus, and Neptune masses to the solar mass. We have not included in this table the calculated [FORMULA] differences and errors associated with other mass parameters because their contributions are insignificant.
b) Component of [FORMULA] from Table 8 of Paper I that corresponds to the given mass parameter (see text).
c) Relative difference (in the sense of DE200 - DE405) in the given mass parameter derived from data in the JPL ephemeris headers.
d) Calculated difference [FORMULA] from the product of the given [FORMULA] component and associated relative mass difference.
e) Relative error in the given mass parameter derived from Standish (1995).
f) Estimated maximum error in [FORMULA] from the product of the given [FORMULA] component and the associated relative error in the mass parameter.

The ephemeris component of the [FORMULA] error is difficult to estimate because we have no better value for comparison. If we propagate the estimated mass parameter errors for DE405 using the same mass model that was used to predict the mass-dependent part of [FORMULA] the resulting mass-dependent error is [FORMULA] (Table 3). This error estimate is quite uncertain so for simplicity we have simply added the various error components. The mass-independent error estimate for [FORMULA] is even more uncertain; following an argument in Sect. 3 we estimate it as [FORMULA], i.e., an order of magnitude less than the mass-independent error for [FORMULA].

If we add the speculative rate-adjustment error (previous sub-section) to the present results and round to one significant digit we find the total error in [FORMULA] associated with the conversion from ephemeris to SI units is [FORMULA].

5.3. Recommended values and errors of [FORMULA], [FORMULA], [FORMULA], and K

Our recommended [FORMULA] value is


where the post-Newtonian correction (Paper I) is


and the asteroid correction (Paper I) is


(For Eq. [21] we have corrected a sign error that occurred in front of the integral in Eq. [24] of Paper I that propagated to Eqs. [26], [30], [31], and [38] of that paper.) Even though the value of the asteroid correction is the same as its uncertainty we add this correction anyway because it is known to be positive.

A recent determination of the potential at the geoid yields (Bursa et al. 1997),


(Note there is a typographical error in the value of the [FORMULA] error stated in the abstract of the Bursa et al. paper. We have taken the error from Eq. [12] of that paper which is 10 times the abstract value but consistent with the other error values given in that paper.) We derive from this value and Eq. (10) our recommended value of


From Eqs. (12), (20), and (24) we derive our recommended value of


which from Eq. (2) is equivalent to


The uncertainties in [FORMULA] and [FORMULA], are probably only reliable in order of magnitude and similarly for the resultant uncertainties given for [FORMULA], K, and [FORMULA] which we have estimated by simply adding the component uncertainties and rounding.

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© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999