*Astron. Astrophys. 348, 642-652 (1999)*
## 5. , , , , and *K*
### 5.1. and
The determination of from
numerical quadrature results *alone* is not recommended. One
could adjust so the mean of the
integrand of Eq. (3) was zero, but then the definition of
would be ambiguous because it would
depend strongly on the epoch range chosen for the integration. A much
better approach is to define as the
constant term (at J2000) in a series approximation for
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 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 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
-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
is the current series are not as
accurate as the numerical time ephemeris results (see previous
section). Thus, the best current method of determining
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
term removed) and a numerical time
ephemeris that is calculated with a preliminary value of
. The fit gives a correction to the
numerical value that is completely
independent of the the series 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 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
and
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 to
remove the slope. These precise
values are also stored in the ephemeris header to facilitate future
rate adjustments to the time ephemeris when more accurate values of
become available.
The present results supersede the
and
results presented in Table 7
of Paper I. (We have changed our notation from Paper I;
currently we prepend a "" to the
symbol to indicate whenever
post-Newtonian and asteroid effects are not corrected.) Our present
value is
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
value is
smaller than the previous hybrid
value of 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
value is
smaller than the present value of
.
### 5.2. Errors in and
The 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 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
value and rate adjustment for the
numerical time ephemeris.
One measure of the rate-adjustment errors is the formal
least-squares errors of the values
determined by the hybrid technique. These errors are respectively
and
for
and
. 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 , and we take this as
a speculative estimate of the rate error in the present series results
(without 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
term) and numerical time
ephemerides. The speculative estimate of the rate error corresponds to
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
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 associated with the conversion from ephemeris to SI units
The total errors of and
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
error should be close to
. The mass-dependent part of this
difference can be estimated using the components of
from Paper I and the known
differences in mass parameters between DE200 and DE405 (Table 3).
For this calculation we have assumed that all
components are proportional to
(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 . Thus, by subtraction the
mass-independent part of the ephemeris component of the
error should be close to
.
**Table 3.** Propagation of mass-parameter differences and errors to .
Notes:
a) Mass parameter. is the square of the number of meters per AU, and , , , and 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 differences and errors associated with other mass parameters because their contributions are insignificant.
b) Component of 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 from the product of the given component and associated relative mass difference.
e) Relative error in the given mass parameter derived from Standish (1995).
f) Estimated maximum error in from the product of the given component and the associated relative error in the mass parameter.
The ephemeris component of the
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 the
resulting mass-dependent error is
(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
is even more uncertain; following an argument in Sect. 3 we estimate
it as , i.e., an order of magnitude
less than the mass-independent error for
.
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 associated
with the conversion from ephemeris to SI units is
.
### 5.3. Recommended values and errors of , , , and *K*
Our recommended 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
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 and
, are probably only reliable in
order of magnitude and similarly for the resultant uncertainties given
for , *K*, and
which we have estimated by simply
adding the component uncertainties and rounding.
© European Southern Observatory (ESO) 1999
Online publication: July 26, 1999
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