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5 Comparison


5.1 General remarks

 This section is dedicated to the global comparison of H96NUT with existing nutation series. First of all one has to choose the most suitable model for a comparison. Obviously, the older nutation models (like K77, ZG89 or KS90) have - although they are well established - a threshold which is not comparable to that of H96NUT. The recent theories of the nutation for the rigid Earth model SK96.2, RDAN97, SMART97 as already mentioned in Sect. 1 have a threshold smaller than our model and therefore we have only to decide which of them is the most convenient one for comparison. Unfortunately, there is no suitable benchmark nutation series, that is the total nutation angles computed over some time interval at discrete times, in contrast to the situation with tidal models where at least two reference series exist. The benchmark RDNN97 mentioned in Roosbeek & Dehant (1997) is still under testing and not yet prepared for public use. What makes the comparison more complicated is that in the particular nutation models different variables in the arguments are used. We use in H96NUT the mean longitude of planets referred to the mean dynamical ecliptic and equinox of date. Roosbeek & Dehant (1997) use the mean longitude referred to the mean dynamical ecliptic and equinox J2000. In both theories the values are taken from Simon et al. (1994) and differ by the value of the general precession in longitude $p_{\rm A}$. Williams (1995), Kinoshita & Souchay (1990), Souchay & Kinoshita (1996, 1997a,b), use the mean longitude referred to the mean dynamical ecliptic and equinox J2000 and moreover, the general precession $p_{\rm A}$ as an additional fundamental argument. In Bretagnon et al. (1998) the mean longitudes of planets are reckoned from the equinox of date. Their values differ from those of Simon et al. (1994) due to modifications of the tidal model in the lunar theory and the new inertial ecliptic and dynamical equinox defined by DE403/LE403 (Standish et al. 1995). It must be mentioned that also the precession theories used in the computations sometimes differ: Lieske et al. (1977), Williams (1994) or Simon et al. (1994) are commonly in use. Therefore, the scaling factor $H_{\rm dyn}$ also takes different values but in recent works the agreement is better than the relative uncertainty (about 10-5) due to that of the precession constant $p_{\rm A}$.It is also doubtful whether a simple rescaling of the nutation amplitudes makes much sense. To close these general remarks it should be noted that little is said about the consistency of the various nutation models, i.e. whether all theories and numerical constants (e.g. $H_{\rm dyn}$) used in the computations are compatible with each other or not. Except for the older expression of Aoki et al. (1982) used in the computation of the tidal potential HW95 and the relationship between $H_{\rm dyn}$ and $p_{\rm A}$ taken from KS90, everything else is compatible here. =.4
Table 11: Differences larger than 5 $\mu$as in longitude $\Delta\psi$ and obliquity $\Delta\varepsilon$ (in $\mu$as) between H96NUT and RDAN97 (t in J cy)

{rrrrrrrrr\vert r\vert rrrr\vert rrrr}
$l_{\rm V}$\space ...
 ... &
 2 & 0 & 2 & 
 9.133 & $-$9.05 & 0.34 &
 2.94 & 0.00 \\  

5.2 Comparison in detail

  For the term by term comparison we have chosen the model RDAN97 of Roosbeek & Dehant (1997) which appeared recently. This model employed a threshold of 0.1 $\mu$as and contains 1553 terms with similar arguments to that used in H96NUT. While the expressions of Delaunay's arguments D, F, l, $l_{\rm s}$ and $\Omega$are the same in both theories, different expressions for the mean longitudes of the planets were used. The mean longitude referred to the mean dynamical ecliptic and equinox of date is used in H96NUT in comparison with the mean longitude referred to the mean dynamical ecliptic and equinox J2000 used in RDAN97. Both systems were taken from Simon et al. (1994) and they differ by the value of the general precession in longitude $p_{\rm A}$. This difference appears in the direct and indirect planetary terms only and is vanishing for J2000 when the longitudes are the same in both systems. The compensation of the difference in the arguments should appear in the secular term of amplitude in the other phase. For example, for the largest of concerning terms in $\Delta\psi_{\sin}$ having in argument $8l_{\rm V}$ with an amplitude close to 300 $\mathrm \mu$as and period 91505.1 days the difference between the series should appear in $t\Delta\psi_{\cos}$ with amplitude about 50 $\mathrm \mu$as (for t not too large) as we can see in the first term of Table 11. In the comparison we have found 27 terms of H96NUT which are not involved in RDAN97. 14 of them are indirect planetary terms coming from the luni-solar potential, one is a direct planetary term of Venus and remaining 12 ones are luni-solar terms coming from the Moon.
\resizebox {\hsize}{!}
{\includegraphics{ds7434f2a}\includegraphics{ds7434f2b}}\end{figure*} Figure 2: Comparison in time domain H96NUT - SMART97 for short-periodic terms only $\Delta\psi$ a) $\Delta\varepsilon$ b): values in $\mu$as

The nutation model SMART97 (Bretagnon et al. 1998) is not convenient for the term by term comparison with H96NUT in spite of having the threshold more than one order smaller because of the different fundamental nutation arguments. To take advantage of the quality of this model we have utilized it for the comparison in the time domain.

5.3 Results of the comparison

Table 12: Overall and short period only comparison in time domain: H96NUT - SMART97, values in $\mathrm \mu$as

{l \vert r r \vert r r}
 & \multicolumn{2}{c\vert}{Overal...
 ...$1\,369 & $-$660 & $-$220\\  rms & 2\,815 & 624 & 186 & 74\\ \hline\end{tabular}

First, a term by term comparison for the nutation of the figure axis between H96NUT and RDAN97 has been carried out. The maximum difference, the sum of all absolute differences and the rms value are given in Table 9. The percentage distribution of the differences in the various intervals is shown in the Table 10. The list of the differences which are bigger than 5 $\mathrm \mu$as in longitude and obliquity is presented in Table 11. These three tables clearly demonstrate that more than 90% of the terms can be computed with comparable accuracy as in other nutation series. However significant differences occur for the largest nutation terms (with periods 18.6 y, 9.3 y, 365 d, 182 d, 13 d) and for the long-periodic nutation terms, say above 18.6 years. Both should be expected due to the reasons explained in Sect. 4.

Next, a comparison between H96NUT and SMART97 in time domain was performed. Thus the nutation angles $\Delta\psi$ and $\Delta\varepsilon$were evaluated numerically starting at JED = 2396931.666 with a time step of 20 hours and a total step number of 131072 (217) covering approximately 300 years. It follows from Table 9 that the largest difference of about 5500 $\mathrm \mu$as at 6786 days would dominate the comparison in time domain. Therefore, instead of showing the figures in that case another comparison in time domain was carried out where all nutation terms with periods larger than 6700 days were omitted in order to compare in a better manner the short-periodic terms. Indeed, the differences in time domain drop by one order of magnitude (see Table 12). They are shown for the time interval 1850-2150 in Fig. 2.

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