4 Error estimation

  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  

$$A_{\rm nut} = \vert\sin \varepsilon_0 \, \Delta\psi\vert = ... ...\Delta\tilde\omega_i}} \frac{H_{\rm dyn}}{{\omega_i}} + \ldots$$ (55)

Due to the resonance denominator ${\Delta\tilde\omega_i} = {\omega_i} - \tilde{w}_{30}$ nutation terms might arise even when the tidal amplitude S²¹_i (see Eq. (8)) is small (or even below the chosen threshold for the tidal potential) provided the tidal frequency $\omega_i$ is close enough to the K₁-tide for which ${\Delta\tilde\omega_i} \equiv 0$.Using the numbers for the smallest tidal amplitudes from the HW95 model one finds that nutation terms of the order 0.45 $\mathrm \mu$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 recent tidal model RATGP95 of Roosbeek (1996). Since the factor $\tilde{w}_{30} / {\Delta\tilde\omega_i}$ 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 $\omega_i$ in (55) by $\delta\omega$this yields an error for the nutation amplitude of about  

$$\delta A_{\rm nut} \approx A_{\rm nut} \frac{-\delta \omega}{\Delta\omega_i} = A_{\rm nut} \, \frac{-\delta\omega}{2\pi}P_i,$$ (56)

where P_i 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 Souchay (1993) that 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 $2 p_{\rm s}$where $p_{\rm s}$ 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 $2 p_{\rm s}$,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 $15 \hbox{$.\!\!^\circ$}043\,278\,97$ in HW95 produces a nutation term in H96NUT with a period of 6786 days and an amplitude of about 5200 $\mathrm \mu$as in longitude 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 $H_{\rm dyn}$, is currently deduced from the precession constant $p_{\rm A}$.Based on the theory of KS90, chapter eight, and the precession constant of Simon et al. (1994), $p_{\rm A}= 5\,028\hbox{$.\!\!^{\prime\prime}$}82$, the value used for H96NUT is $H_{\rm dyn} = 3.273\,792\,489~10^{-3}$. Comparing this with the value found by Williams (1994) $p_{\rm A}= 5\,028 \hbox{$.\!\!^{\prime\prime}$}77$ 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 $\mathrm \mu$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 $\mathrm \mu$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 can be computed precisely from the tidal potential HW95 while the long-periodic ones have larger uncertainties.