4 Final values for the 18.6 Y. Leading nutation term

The computation of the 18.6 y leading coefficients of nutation, both in longitude and in obliquity, is rather delicate, so that this chapter is devoted to it. As explained in detail by Williams (1994) and Souchay & Kinoshita (1996), these coefficients are the result of the combination of various contributions, which are, for the in-phase coefficient: the first-order component related to the main problem of the Moon, the crossed-nutation effect, the spin-orbit coupling effect, the planetary- tilt effect.

For the out-of-phase components, contributions come from the planetary-tilt effect (Williams 1994) and from the secular variation of the mean obliquity with respect to the moving ecliptic of the date. This last contribution, clearly pointed out and accurately calulated recently by Bretagnon et al. (1997), was not considered in previous works (Kinoshita & Souchay 1990) although already Kinoshita (1977) mentioned that strictly speaking, in the integration of the potential which serves to give the expression of the determining function in Hamiltonian theory, we shoud have to take into account the secular change of the obliquity $I^{\star}$ (which is the value of the obliquity when not considering its periodical variations, that is to say the mean obliquity itself including its secular variation). By taking into account this effect, we find a correction for the out-of-phase coefficient in longitude of which matches very well the difference found by Bretagnon et al. (1997) when comparing his value with Souchay & Kinoshita (1997), that is to say 0.250 $\mu$as.

The calculation related to the correction above consists in replacing the constant value of $I_{0}= - \varepsilon_{0}$ at J2000.0 in the expression of the potential by $I_{0} + \dot I$ where $\dot I$ is the secular variation of the mean obliquity of the date ($I = - \varepsilon$) with respect to the mean ecliptic of the date. The conventional value of $\dot I$ can be found in Lieske et al. (1977), that is to say: $\dot I = 468.150''/$1000 yrs. = 0.0022696 rd /1000 yrs. Then after integrating the potential and applying the canonical equations which serve to the determination of the coefficients of nutation at the first order (Kinoshita 1977), we find that the ratio of the out-of-phase $\Omega$ component with respect to the in-phase one is $\rho_{\psi}^{\Omega} = - \biggl({ \dot I \over \dot \Omega } \biggr)(2 \tan 2 I_{0} + \cot I_{0})$ in longitude, and: $\rho_{\varepsilon}^{\Omega} = \biggl( {\dot I \over \dot \Omega } \biggr) \tan I_{0}$ in obliquity.

Thus the values in milliarcsecond obtained for the out-of-phase component are $0.5161 \cos \Omega$ for $\Delta \psi$ and $0.0267 \sin \Omega$ for $\Delta \varepsilon$. Notice that these values are the same as those found Roosbeek & Dehant (1997).

Moreover, in longitude only, the secular variation of the obliquity produces an additional out-of-phase component coming from the expression of the complementary part of the Hamiltonian E (Kinoshita 1977) which is related to the change of canonical variables from the fixed ecliptic of the epoch to the moving ecliptic of the date. E can be written, by using the same notations as in Kinoshita (1977):

$$\begin{eqnarray} &&E = 2 \sin^{2}{\pi_{\rm A} \over 2} \left[ H \times {{\rm d} ... ...}^{2}) \nonumber\\ &&= G \sin I' (-q \sin h' + p \cos h') + o(t) .\end{eqnarray}$$ (45)

Where h' is the combination of the general precession in longitude $(-p_{\rm A}$) and of the nutation in longitude: $h' = -p_{\rm A} - \Delta \psi$.By applying the canonical equations:

$${{\rm d} \Delta h' \over {\rm d}t} = - {1 \over G \sin I'} {\partial E \over \partial I'}$$ (46.1)

$${{\rm d} \Delta I' \over {\rm d}t} = {1 \over G \sin I'} {\partial E \over \partial h'} ,$$ (46.2)

we find the following expressions for the derivatives:

$${{\rm d }\Delta h' \over {\rm d}t} = q \cot I_{0} \Delta h'$$ (47.1)

$${{\rm d} \Delta I' \over {\rm d}t} = -p \Delta h'$$ (47.2)

where p and q are respectively the linear trends of the functions $\sin \pi_{\rm A} \sin \Pi_{\rm A}$ and $\sin \pi_{\rm A} \cos \Pi_{\rm A}$, and I₀ is the value of the obliquity ($I_{0}= - \varepsilon_{0}$) at the epoch (J2000.0).

By taking: $q = -46\hbox{$.\!\!^{\prime\prime}$}82/$J cy. and p = 5.341''/J cy. integration of equations (47.1) and (47.2) leads to the following values:

$$\Delta \psi = -268.1 ~\mu {\rm as} \cos \Omega ~~~ \Delta \varepsilon = 13.3 \cos \Omega.$$

Notice that the results are an out-of-phase component in longitude and an in-phase component in obliquity. The ratio of the out-of-phase term to the in-phase leading term in longitude can be approximated as follows:

$$\rho_{\rm \psi ,b}^{\Omega} \approx \biggl({ \dot I \over \dot \Omega } \biggr) \cot I_{0}.$$ (48)

Notice that the ratio $\rho_{\psi}^{\Omega}$ found previously can be split into two parts and one of them is exactly the opposite of the ratio $\rho_{\rm \psi ,b}^{\Omega}$, so that the two terms nearly annulate when mixed together. Then the final ratio in longitude of the out-of-phase component with respect to the in-phase one, due to the secular variation of the obliquity, can be written:

$$\rho_{\psi}^{\Omega} + \rho_{\rm \psi ,b}^{\Omega} \approx -... ...iggl({ \dot I \over \dot \Omega } \biggr) \times \tan 2 I_{0}.$$ (49)

And its value is: $\Delta \psi = 0.5161 - 0.2681 = 0.2480~\mu$as. This value matches quite well the difference of 0.250 $\mu$as already noticed by Bretagnon et al. (1997) when comparing the value of the out-of-phase component with Souchay & Kinoshita (1996), for the contribution above was not included.

Notice that the second-order tilt-effect (Souchay & Kinoshita 1996) gives 0.1351 $\mu$as and $-0.0298~\mu$as for the out-of-phase part respectively for $\Delta \psi$ and $\Delta \varepsilon$, so that the final values are respectively 0.3831 $\mu$as and $-0.0031~\mu$as.

The same remark which leads to the calculation of out-of-phase terms depending on $\dot I$ can be done for the other coefficients, but these out-of-phase terms are respectively much smaller, so that even the semi-annual term gives a component smaller than 1$~\rm\mu as$, that is to say $-0.4 ~{\rm \mu as} \cos 2\ L_{\rm S}$ for $\Delta \psi$and $-0.5 ~\mu {\rm as} \sin 2\ L_{\rm S}$ for $\Delta \varepsilon$, where $L_{\rm S}$ is the mean longitude of the Sun.

 

Table 3: 1.  Summary of the various contributions to the 18.6 y leading coefficient of the rigid Earth nutation in longitude, figure axis

 

Table 3: 2.  Summary of the various contributions to the 18.6 y leading coefficient of the rigid Earth nutation in obliquity, figure axis

In Tables 3.1 and 3.2 we present our final values for the rigid Earth nutation leading 18.6y component, respectively in longitude and in obliquity, with the detailed account of all the effects. The difference with Souchay & Kinoshita (1997) is, in milliarcsecond: $\delta(\Delta \psi) = -0.018 \sin \Omega + 0.248 \cos \Omega$ and: $\delta(\Delta \varepsilon) = 0.015 \cos \Omega + 0.027 \sin \Omega$.The big differences for the out-of-phase components come from the new contribution related to the secular variation of the obliquity, as detailed above.