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This section tries to give some error estimates
for the individual nutation terms
while the next section is devoted to
the error on the nutation given by the whole nutation series.
Clearly, one has to consider the largest terms only.
The first point to be mentioned is that the tidal
potential HW95 has been computed
with a fixed amplitude threshold.
The approximate amplitude of a
(first order) nutation term is given by
Due to the resonance denominator nutation
terms might arise even when the tidal amplitude S21i (see
Eq. (8)) is small
(or even below the chosen threshold for the tidal potential)
provided the tidal frequency is close enough
to the K1-tide for which .Using the numbers for the smallest tidal amplitudes
from the HW95 model one finds that nutation terms
of the order 0.45 as might be missed
for periods longer than 350 days for the Moon
and 3.6 years for the Sun.
Obviously, that problem could be avoided
if the threshold for the tidal potential
would have been chosen to be frequency dependent
as was suggested by
Dehant et al. (1997).
Moreover, the transfer function between the rigid Earth nutation
and a more realistic Earth nutation has some other resonances
(like the free core nutation (FCN) at about 430 days)
in the diurnal tidal band which might also amplify small
tidal rigid Earth nutation amplitudes to give
a real nutation term above the threshold.
Originally we hoped to achieve such a small threshold for the tidal
amplitudes that any frequency dependence is unnecessary.
Unfortunately, due to the numerical computation method
for the tidal potential HW95 that could not be achieved.
The frequency resolution was - in spite of all efforts -
not high enough to resolve tides which are very close
to the K1-tide or to other large tidal terms.
It should be noted that the threshold for the tidal amplitudes
is smaller by a factor of about 3-5 in comparison with accurate
tidal model RATGP95 of
Since the factor can reach values
up to 106 this is - besides the HW95 tidal model -
the only one which is precise enough for the computation
of nutation series to highest accuracy
using the method outlined in this paper.
Next, the influence of possible errors in the HW95 model
on the corresponding nutation series H96NUT is investigated.
Assuming the numerical analysis procedure missed the correct
tidal frequency in (55) by this yields an error for the nutation amplitude of about
where Pi is the period of the i-th term in Julian centuries.
Therefore, this error grows with the period of the nutation term thus
confirming the statement of ZG89 and
long-periodic nutation terms have larger errors than the
short-periodic ones. However, this affects only the indirect
planetary tides, and corresponding nutation terms. As for the other
tides, the correct frequency could be found. Estimating the probable
frequency uncertainty not to be larger than 1.35 rad cy-1, it follows that an
indirect planetary nutation term with a period of 18.6 and 250 years has a relative
uncertainty of 4% and 50%, respectively.
In addition, there are some tides with very small tidal
amplitudes which lead to nutation terms only because
the frequency is close enough to the K1-tide.
Since those tidal amplitudes are of the order of the
truncation threshold for the tidal potential,
it is doubtful whether these nutation terms
are real or appear only because of numerical reason.
For the tidal potential HW95 and the nutation series H96NUT
this applies again mainly to the long-periodic nutation terms.
There is also another point to be mentioned.
In all tidal and nutation models there are terms
which differ in argument only by where is the perigee of the Sun
with a period of about 20000 years.
By comparison with other tidal models it turned out
that HW95 shows some differences concerning
the tidal amplitudes of the terms whose argument differs by ,especially when one of the tidal amplitudes is rather large.
It seems that the numerical procedure used during the computations,
and the final least squares fit of the tidal amplitudes
is somewhat critical to these terms.
In particular, the rather small tide at
in HW95 produces a nutation term in H96NUT with a period
of 6786 days and an amplitude of about 5200 as in
which is too large in comparison with other nutation models
by nearly 5 mas.
Finally, let us consider the accuracy of the constants
involved in the computation of the nutation.
Some discussion about that topic can be found in
Souchay & Kinoshita (1996, 1997a).
The scaling factor for the nutation, namely the
dynamical ellipticity , is currently deduced
from the precession constant .Based on the theory of KS90, chapter eight,
and the precession constant of
Simon et al. (1994),
, the value used for H96NUT is .
Comparing this with the value found
and used also in
RDAN97 and SMART97, one finds that
the relative accuracy is not better than 10-5 which therefore
produces an error in the largest nutation term of about 173 as.
The next largest nutation term is smaller
by almost one order of magnitude.
Therefore, one can doubt whether it presently makes sense to compute
nutation amplitudes much smaller than 1 as.
According to the previous discussions, the conclusion is
that short-periodic nutation terms with periods shorter than say 18.6
years except for the largest ones
from the tidal potential HW95 while the long-periodic ones
have larger uncertainties.
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