2 The crossed-nutation coupling effect

The second-order potential characterizing this effect is involving the Andoyer variables h and H of the rotation of the Earth (Kinoshita 1977), and we can use the same formulation as in Kinoshita & Souchay (1990) for the expression of the second-order determining function involved, that is to say:

$$W_2^{\rm cr.}= {1 \over 2} \int \bigl[ {\partial (U_{1}^{\rm... ...H} \times {\partial (W_{1}) \over \partial h}\bigr] {\rm d}t$$ (1)

where $U_{1}^{\rm per}$ is the periodic part of the first-order potential U₁ due to the combined action of the Moon and of the Sun, and W₁ is the first-order determining function which is determined from a simple integration:

$$W_1 = \int U_{1}^{\rm per} {\rm d}t$$ (2)

U₁ can be easily expressed as in the following way:

$$\begin{eqnarray} &&\hspace*{-1em}U_{1}= {\kappa^{2} M_{\rm M} \over a_{\rm M}^{3... ...P_{2}^{2}(\sin \beta_{\rm S}) \cos 2(\lambda_{\rm S} - h) \Bigr] \end{eqnarray}$$ (3)

h is the canonical variable representing the general precession in longitude ($h = -p_{\rm A}$).

$M_{\rm M}$ and $M_{\rm S}$ are respectively the masses of the Moon and the Sun,($\lambda_{\rm M}$, $\beta_{\rm M}$, $r_{\rm M}$) and ($\lambda_{\rm S}$,$\beta_{\rm S}$, $r_{\rm S}$) are their respective set of spherical coordinates with respect to the mean ecliptic and the mean equinox of the date. Notice that in a first approximation $\beta_{\rm S}$ which represents the latitude of the Sun with respect to the mean ecliptic of the date, can be set to $\beta_{\rm S}=0$, and that the terms of nutation due to the small displacements of the Sun with respect to the mean ecliptic have been calculated by Souchay & Kinoshita (1996). $a_{\rm M}$ and $a_{\rm S}$ are the basic constant values for the semi-major axes of the Moon and of the Earth, considering the keplerian motion. I is the obliquity angle ($I = - \varepsilon$) associating the two canonical variables H and G, by the trivial equation (Kinoshita 1977):

$$H = G \cos I$$ (4)

where G is the amplitude of the angular momentum of the Earth. G being constant, any partial derivative with respect to H is such as (Kinoshita 1977):

$${\partial [...] \over \partial H} = - {1 \over G \sin I} {\partial [...] \over \partial I}\cdot$$ (5)

The combination of Eqs. (1) to (5) leads to the following expression for W₂:

$$W_{2}^{\rm cr.} = {1 \over 2} \int [{\bf A}] {\rm d}t$$ (6)

with:

$$[{\bf A}] = (\sin I \cos^{2}I) A_{1} + \left( {\sin^{2}I \co... ...\right) A_{5}+ \left({\sin^{2}I \cos I \over 4}\right) A_{6}$$ (7)

and:

$$A_{1} = B_{1}^{\rm M} \times (C_{0}^{\rm M} + C_{0}^{\rm S})- C_{1}^{\rm M} \times (B_{0}^{\rm M} + B_{0}^{\rm S})$$ (8.1)

$$A_{2} = (C_{2}^{\rm M} + C_{2}^{\rm S}) \times (B_{0}^{\rm ... ...} + B_{2}^{\rm S}) \\ \times (C_{0}^{\rm M} + C_{0}^{\rm S})$$ (8.2)

$$A_{3} = B_{1}^{\rm M} \times C_{3}^{\rm M}- B_{3}^{\rm M} \times C_{1}^{\rm M}$$ (8.3)

$$A_{4} = B_{3}^{\rm M} \times (C_{2}^{\rm M} + C_{2}^{\rm S})- C_{3}^{\rm M} \times (B_{2}^{\rm m} + B_{2}^{\rm S})$$ (8.4)

$$A_{5} = (B_{4}^{\rm M} + B_{4}^{\rm S}) \times (C_{2}^{\rm ... ...rm M} + C_{4}^{\rm S}) \times (B_{2}^{\rm M} + B_{2}^{\rm S})$$ (8.5)

$$A_{6} = (C_{4}^{\rm M} + C_{4}^{\rm S}) \times B_{1}^{\rm M}- (B_{4}^{\rm M} + B_{4}^{\rm S}) \times C_{1}^{\rm M} .$$ (8.6)

The functions $B_{i}^{\rm M}$ and $C_{i}^{\rm M}$ are parts of the potential due to the Moon. They are expressed as follows:

$$B_{0}^{\rm M} = \left({k_{\rm M} \over 2}\right) \left({a_{\rm M} \over r_{\rm M}}\right)^{3}(-1 + 3 \sin^{2} \beta_{\rm M})$$ (9.1)

$$B_{1}^{\rm M} = k_{\rm M} \left({a_{\rm M} \over r_{\rm M}}\... ...} \sin \beta_{\rm M} \cos \beta_{\rm M} \cos(\lambda_{\rm M}-h)$$ (9.2)

$$B_{2}^{\rm M} = k_{\rm M} \left({a_{\rm M} \over r_{\rm M}}\right)^{3} \cos^{2} \beta_{\rm M} \sin 2(\lambda_{\rm M}-h)$$ (9.3)

$$B_{3}^{\rm M} = k_{\rm M}\left({a_{\rm M} \over r_{\rm M}}\r... ...3} \sin \beta_{\rm M}\cos\beta_{\rm M} \sin(\lambda_{\rm M}-h)$$ (9.4)

$$B_{4}^{\rm M} = k_{\rm M} \left({a_{\rm M} \over r_{\rm M}}\right)^{3} \cos^{2} \beta_{\rm M} \cos 2(\lambda_{\rm M}-h)$$ (9.5)

$$C_{0}^{\rm M} = \int B_{0}^{\rm M} {\rm d}t$$ (9.6)

$$C_{1}^{\rm M} = \int B_{1}^{\rm M} {\rm d}t$$ (9.7)

$$C_{2}^{\rm M} = \int B_{2}^{\rm M} {\rm d}t$$ (9.8)

$$C_{3}^{\rm M} = \int B_{3}^{\rm M} {\rm d}t$$ (9.9)

$$C_{4}^{\rm M} = \int B_{4}^{\rm M} {\rm d}t .$$ (9.10)

The functions $B_{i}^{\rm S}$ and $B_{i}^{\rm S}$ are the corresponding parts of the potential due to the Sun, by taking into account: $\sin \beta_{\rm S} = 0$

$$B_{0}^{\rm S} =- \left({k_{\rm S} \over 2}\right) \left({a_{\rm S} \over r_{\rm S}}\right) ^{3}$$ (10.1)

$$B_{2}^{\rm S} = k_{\rm S} \left({a_{\rm S} \over r_{\rm S}}\right)^{3}\sin 2(\lambda_{\rm S}-h)$$ (10.2)

$$B_{4}^{\rm S} = k_{\rm S} \left({a_{\rm S} \over r_{\rm S}}\right) ^{3}\cos 2(\lambda_{\rm S}-h)$$ (10.3)

$$C_{0}^{\rm S} = \int B_{0}^{\rm S} {\rm d}t$$ (10.4)

$$C_{2}^{\rm S} = \int B_{2}^{\rm S} {\rm d}t$$ (10.5)

$$C_{4}^{\rm S} = \int B_{4}^{\rm S} {\rm d}t$$ (10.6)

$k_{\rm M}$ and $k_{\rm S}$ are the scaling factors used to compute the nutation, their expression is (Kinoshita 1977):

$$k_{\rm M} = 3 \times H_{\rm d}\times \Bigl({M_{\rm M} \over ... ...Bigr) \times \Bigl({n_{\rm M}^{2} \over \omega_{\rm E}} \Bigr)$$ (11.1)

$$k_{\rm S} = 3 \times H_{\rm d}\times \Bigl({M_{\rm S} \over ... ...Bigr) \times \Bigl({n_{\rm S}^{2} \over \omega_{\rm E}} \Bigr)$$ (11.2)

where $M_{\rm S}$, $M_{\rm E}$ and $M_{\rm M}$ are respectively the mass of the Sun, the Earth and the Moon and $\omega_{\rm E}$ the angular speed of rotation of the Earth. $n_{\rm M}$ and $n_{\rm S}$ are respectively the relative mean motions of the Moon and of the Sun. Souchay & Kinoshita (1996) calculated the values of $k_{\rm M}$ and $k_{\rm S}$ by choosing an up-to-date value of the general precession in longitude from which they depend directly, by a relationship explained in detail by Kinoshita & Souchay (1990). We keep these values, that is to say: $k_{\rm M} = 7546.71733''$/ J cy and: $k_{\rm S} = 3475.41352''$/J cy. The coefficients of nutation coming from W₂ as given by Eqs. (6) to (10) are determined by the following formula:

$$\begin{eqnarray} &&\Delta \psi^{W2}_{\rm cr} = - \Delta h = - {\partial W_{2}^{\... ...\cos I \sin^{2}I \over 2 \sin I}\Bigr] \times \int A_{6} {\rm d}t \end{eqnarray}$$ (12)

$$\begin{eqnarray} &&\Delta \varepsilon^{W2}_{\rm cr} = - \Delta I =-\Bigl[ {1 \ov... ...- {\sin 2I \over 16}\Bigl[{\partial A_{6} \over \partial h}\Bigr].\end{eqnarray}$$ (13)

The expressions $\Delta \psi_{\rm cr.}$ and $\Delta \varepsilon_{\rm cr.}$ which characterize the crossed-nutation effect are given by (Kinoshita & Souchay 1990):

$$\Delta \psi_{\rm cr} = \Delta \psi^{W2}_{\rm cr} -{1 \over 2} \left\{{\partial W_{1} \over \partial H},W_{1}\right\}$$ (14.1)

$$\Delta \varepsilon_{\rm cr.} = \Delta \varepsilon^{W2}_{\rm ... ...G\sin I} \left\{W_{1},{\partial W_{1} \over \partial h}\right\}$$ (14.2)

which gives, after development:

$$\Delta \psi_{\rm cr} = \Delta \psi^{W2}_{\rm cr} -{1 \over 2... ...i \times {\partial (\Delta \psi) \over \partial h}\Bigr) \Bigr]$$ (15.1)

$$\Delta \varepsilon_{\rm cr} = \Delta \varepsilon^{W2}_{\rm c... ...\partial h} \Bigr)- \cot \varepsilon (\Delta \varepsilon)^{2}]$$ (15.2)

$\Delta \psi$ and $\Delta \varepsilon$ are the nutations at the first order given by the basic relationships:

$$\Delta \psi = - \Delta h = {1 \over G\sin I}{\partial W_{1} \over \partial I}$$ (16.1)

$$\Delta \varepsilon = - \Delta I = -{1 \over G\sin I}{\partial W_{1} \over \partial h}$$ (16.2)

where W₁ is calculated by the intermediary of Eqs. (2) and (3).

For the computations relative to the present effect, we catch all the coefficients up to a relative 10^-8 with respect to the largest term, in the expressions $B_{i}^{\rm M}$,$B_{i}^{\rm S}$, $C_{i}^{\rm M}$ and $C_{i}^{\rm S}$ which are used in Eqs. (8.1) to (8.6) and (9.1) to (9.10). In a similar way, we take all the coefficients of nutation in $\Delta \psi$ and $\Delta \varepsilon$ larger than 0.1 $\mu$as for the combinations inside (15.1) and (15.2), which is quite enough if we want to reach the $0.1 ~\rm\mu as$level for the resulting coefficients. These computations are made with the Broucke (1980) subroutines for manipulation of Fourier series, which are the same we used previously for the reconstruction of the theory of nutation for a rigid Earth model (Kinoshita & Souchay 1990).

 

Table 1: List of the coefficients of rigid Earth nutation coming from the crossed-nutation effects

The results are listed in Table 1. This table constitutes a big improvement with respect to previous computations carried out manually (Zhu & Groten 1989; Kinoshita & Souchay 1990) by picking up only the largest coefficients in the potential. It seems that it is much more difficult to select these terms with a theory of nutation based on the classical equation for the angular momentum (Bretagnon et al. 1997; Roosbeek & Dehant 1997), because there is no clear method to separate them from the spin-orbit effect described in the next chapter.

We can remark also that no less than 68 coefficients are present above the $0.1 ~\rm\mu as$ level, which demonstrates its rather big influence. The number of coefficients is still 24 for $\Delta \psi$, and 20 for $\Delta \varepsilon$ up to $1 ~\rm \mu as$.The by far largest component with argument $2 \Omega$ and amplitude 1.2206 $\mu$as results naturally from the interactions between the nutations of the leading component at the first order with argument $\Omega$. It was already calculated for the first time by Kinoshita & Souchay (1990). As it was not taken into account before, this explained the big 9.3 y signature when comparing the previous analytical nutation (Kinoshita 1977) with numerical integration (Schastok et al. 1989), which disappeared after the reconstruction of the theory, as was shown by Souchay & Kinoshita (1991). Notice also the 2 coefficients with amplitude $117.7 ~\rm \mu as$ and $-92.8 ~\rm\mu as$in longitude (respectively $-17.3 ~\rm\mu as$ and $73.1 ~\rm\mu as$ in obliquity) around the semi-annual period, and the 2 coefficients with amplitude $19.0 ~\rm\mu as$ and $-15.2 ~\rm\mu as$ in longitude (respectively $-2.8 ~\rm\mu as$ and $12.0 ~\rm\mu as$ in obliquity) around the fortnightly period. At last we can also remark the clustering of coefficients around these two fundamental periods.