3 Results

 

3.1 General remarks

 This section deals with the presentation of the results of our new rigid Earth nutation model called H96NUT computed from the tidal development HW95 using the formulas given in the previous section. The truncation threshold chosen for the nutation amplitudes is 0.45 $\mathrm \mu$as, which is smaller by one order of magnitude than that from the KS90 nutation model (Kinoshita & Souchay 1990). Thus the condition for keeping a term is:  

$$\vert\sin \varepsilon_0 \cdot \Delta\psi\vert \ge 0.45\;\hb... ... \vert\Delta\varepsilon\vert \ge 0.45\;\hbox{$\mathrm \mu$as} .$$ (52)

The factor $\sin \varepsilon_0$ has approximately the value $0.397\,777$and is introduced because VLBI measures the product $\sin \varepsilon_0 \cdot \Delta\psi$ (see also the discussion in Williams 1995). If only one of the two conditions in (52) is satisfied the other nutation component is also considered. This leads to a total number of 699 nutation terms, both in longitude and obliquity. According to Williams (1995) or any textbook on statistics the summation of the whole nutation series introduces some error due to the limited number of digits for the nutation amplitudes. The total error is at least  

$$\sigma_{\rm total} = \sqrt{\frac{N}{2}} \, \sigma_i .$$ (53)

where $\sigma_i$ is the individual error and N the number of terms. Since N=699 for H96NUT this error is larger by a factor of 19 than the individual error. To keep that error under the truncation threshold of 0.45 $\mathrm \mu$as all nutation amplitudes are given to a resolution of 0.01 $\mathrm \mu$as. The accuracy of the individual nutation terms is discussed in more detail below.

3.2 Planetary and luni-solar precession

 As indicated in the last section it is also possible to deduce values for the precession from the amplitudes of the K₁-tide. Using the tidal model HW95 (and the separately derived values for the K₁-tide due to Uranus and Neptune) results for the planetary effect on the general precession in longitude and on the obliquity rate are given in Table 1 showing excellent agreement with the values given by Williams (1995).

  

Table 1: Precession and obliquity rates at J2000 from direct planetary torques on the Earth's bulge. (in $\mathrm \mu$as yr^-1), $H_{\mathrm{dyn}} = 3.273\,792\,489~10^{-3}$. The subscript H denotes H96NUT, W Williams (1995), respectively

  

Table 2: Luni-solar precession in longitude and obliquity (in $\mu$as yr^-1, $H_{\rm dyn} = 3.273\,792\,489~10^{-3}$)

The results in longitude and obliquity for the luni-solar precession are shown in Table 2. The agreement in longitude is within the current accuracy of the observations (about five digits). However, for the precession in obliquity the numbers coming from the tidal potential are much less accurate. This is due to the fact that they arise due to a smaller tide very close to the K₁-tide and not to the K₁-tide itself. Note that the best value for the precession in obliquity is still under discussion.

3.3 The nutation terms at the first order

 Next, some of the results for the nutation are presented. Most of the numbers refer to the nutation of the figure axis. First there is an overview in Table 3 on the terms due to the different effects taken into account.

  

Table 3: Number of nutation (${\cal N}$) and precession (${\cal P}$) terms in H96NUT, truncation level at 0.45 $\mathrm \mu$as

3.3.1 Direct planetary nutation terms

 The direct planetary nutation terms have been first computed by Vondrák (1982) and then they have been included in most of the nutation theories computed in the nineties. Hartmann & Soffel (1994) computed the nutation due to the direct planetary effect using a simplified version of the theory outlined in the last section. The planetary tidal potential used at that time was based on the numerical ephemerides DE102 but differences to DE200 are very small. The threshold was 2.5 $\mathrm \mu$as and - as in Williams (1995) where it is 0.5 $\mathrm \mu$as - the same for $\Delta\psi$and $\Delta\varepsilon$ thus ignoring the factor $\sin \varepsilon_0$.For H96NUT the HW95 tidal potential based on DE200 and a threshold of 0.45 $\mathrm \mu$as was chosen. A term by term comparison gives for the maximum difference, the sum of the absolute differences and the root mean squares (rms) error the values indicated in Table 4. Now, computing a time series over 90 years starting from J2000 and evaluating the differences in time domain leads to the results in Table 5. It is important to note that the rms values are about one order of magnitude larger than those of the term by term comparison. Furthermore, the maximum and minimum differences (see Table 5) nearly reach the sum of the absolute differences of the individual terms (see Table 4). As already mentioned in Hartmann & Soffel (1994) the direct planetary nutation can be computed from a tidal potential with high accuracy (also due to the smallness of these terms).

  

Table 4: Term by term comparison: direct planetary nutation H96NUT - Williams (1995), values in $\mu$as

  

Table 5: Comparison H96NUT - Williams (1995) in time domain: direct planetary nutation, values in $\mu$as

3.3.2 Nutation term due to J₃ and J₄

 The nutation terms due to the higher order parts of the tidal potential have already been given in Hartmann et al. (1996). As an extension the largest solar term due to $J_3^{\oplus}$has the argument $F-D+\Omega$, a period of 365.229 days and amplitudes of -0.26 $\mu$as and -0.22 $\mu$as for $\Delta\psi$ and $\Delta\varepsilon$, respectively. Hence, this term is under our threshold of 0.45 $\mu$as.

3.3.3 Nutation terms due to the triaxiality of the Earth

  The nutation terms due to the triaxiality of the Earth come from the difference between the equatorial moments of inertia A and B (or equivalently from the geopotential coefficients C₂₂ and S₂₂). Since that difference is small and the denominator in the nutation formulas (28-30) is not the small difference ${\Delta\tilde\omega_i}$but the large sum ${\Sigma\tilde\omega_i}$, only few terms play a role. Some authors do not consider them due to their short periods of about half a day. Nevertheless, they have been calculated by Kinoshita & Souchay (1990), with better accuracy by Souchay & Kinoshita (1997a,b) and by Roosbeek & Dehant (1997). The terms are given in Table 6, where $2 \tau$ defined in Eq. (9) has to be added to the argument, and a comparison with RDAN97 shows that most of the terms are in good agreement with respect to the thresholds.

  

Table 6: The triaxiality terms, values in $\mu$as, period in days. The cut-off level considered for the table corresponds to the condition (52). The subscripts H and RD stand for H96NUT and Roosbeek & Dehant (1997), respectively

3.3.4 Luni-solar nutation terms at first order due to J₂

 

The direct luni-solar nutation terms, i.e. those which can be represented solely by the Delaunay arguments $l_{\rm m}, l_{\rm s}, F, D$and $\Omega$, and the indirect planetary nutation terms which contain besides at least one mean longitude of a planet are calculated the same way from the tidal potential HW95. From Table 3 we can see that these terms make up more than 80% of all terms in H96NUT. The term by term comparison of H96NUT and RDAN97 is presented in Table 11.

For the transformation between the tidal argument and the nutational argument see ZG89, p. 1107. It must be mentioned that there is some ambiguity concerning the sign of nutation amplitudes and the nutation argument. Here, all signs were chosen in such a unique way that the resulting nutation period is positive. It is hoped that other authors will follow that convention to simplify comparisons between different nutation models.

3.4 The nutation terms at second order

 

The additional contributions not considered at first order are: the effect of the precession on the nutation as given in (37), the geodesic nutation from (48) and the second order terms according to (38) and (35). It must be stressed that these terms cannot be compared directly to second order terms of other nutation series since they differ by definition. To demonstrate this important fact let us consider the so-called J₂-tilt effect. The J₂ coefficient of the Earth slightly tilts the orbit of the Moon with respect to the ecliptic. This is implicitly included in the tidal potential HW95 and therefore also in the nutation model H96NUT (due to the inclusion of the corresponding force and the numerical integration of the equations of motion of all bodies in the solar system in the numerical ephemerides DE200). Other theories which use analytical ephemerides without this force have to include the J₂-tilt effect separately. However, the total nutation amplitudes should be directly comparable.

3.5 The influence of T² terms

 

Another point that deserves attention is the influence of T² terms. There are basically two points where they appear. First, there are the nutation amplitudes themselves. Since the tidal amplitudes are only linear with respect to time T the same applies to the Euler angles. However, the transformation from the fixed ecliptic to the ecliptic of date introduces higher powers of T which might not be neglected if high accuracy is required. For the model H96NUT T²-contributions to the nutation amplitudes are listed in Table 7. For small time intervals around J2000 they may be omitted.

  

Table 7: T²-contributions to the nutation amplitudes

  

Table 8: Corrections due to T²-terms in the nutation arguments

  

Table 9: Term by term comparison: H96NUT - RDAN97, values in $\mu$as. Max. are the absolute values

  

Table 10: Percentage distribution of residuals ($\Delta$) in intervals from the term by term comparison. The boundaries of intervals are in $\mathrm \mu$as. There is total number 580 of the compared terms

The second place where T²-contributions occur is in the nutation arguments. On the assumption that the arguments take the form $\arg(T) = a_0 + a_1 T + a_2 T^2$ in all nutation theories an integration with respect to time is required to solve the rotational equations of motion. Then, approximating the integral $\int \cos(\arg(T)) \, {\rm d}T\,$by $1/a_1 \sin(\arg(T))$results in an error

$$\begin{eqnarray} \int \cos( \arg(T) ) \, {\rm d}T - \frac{1}{a_1} \, \sin( \arg... ...t[ T \, \sin( \arg(T) ) + \frac{1}{a_1} \cos( \arg(T) ) \right] .\end{eqnarray}$$ (54)

For the model H96NUT only the 18.6 and 9.3 yr term (see Table 8) contribute significantly, (after a long time) thus proving the heuristic linear approximation is quite appropriate.

Using only the linearized nutation argument results in an error increasing quadratically with time. For the main nutation term in longitude (amplitude $17 \hbox{$^{\prime\prime}$}$)this can reach values up to $622\;\hbox{$\mu$as}$ after one century. Thus at least the T² coefficients - or as in H96NUT up to the T⁴ coefficient - for the astronomical arguments must be taken into account.