3 The spin-orbit coupling effect

In this chapter our aim is to calculate the coefficients of the nutation related to the spin-orbit effect with a better accuracy than previously (Kinoshita & Souchay 1990), and by picking up all the coefficients larger than $0.1 ~\rm\mu as$. Kubo (1982) showed that the Earth flattening is perturbing the orbital motion of the Moon, and this perturbation itself is modifying the motion of nutation of the Earth. The determination of the perturbation due to this reciprocical influence can be tackled when considering the global Earth-Moon system, not the system formed by the Earth itself, as it is the case in classical theories not involving the Hamiltonian (Woolard 1953). Kinoshita & Souchay (1990) included this effect in their second-order calculations involving the Delaunay canonical angular variables l', g' and h', and action variables L', G' and H'. l' is the mean anomaly of the Moon, g' is the argument of the perigee and h' is the longitude of the node, with respect to the ecliptic. The action variables have the following expressions:

$$L' = \Bigl[ {M_{\rm E} M_{\rm M} \over M_{\rm E} + M_{\rm M} } \Bigr] \times \sqrt{\mu a_{\rm M}}$$ (17.1)

$$G' = L' \times \sqrt{1 - e_{\rm M}^{2}}$$ (17.2)

$$H' = G' \cos I_{\rm M}.$$ (17.3)

For the calculations to be achieved properly, the spherical coordinates $r_{\rm M}$, $\lambda_{\rm M}$ and $\beta_{\rm M}$ must be replaced by their expressions in function of the canonical variables in the Eq. (3) giving the expression of the potential. $\beta_{\rm M}$ is related to the canonical variables H' and G' by the intermediary of the $I_{\rm M}$ variable which represents the inclination of the Moon'sorbit on the ecliptic. We have:

$$\sin \beta_{\rm M} = \sin I_{\rm M} \sin(f'+ g').$$ (18)

Moreover $\lambda_{\rm M}$ is the sum of the three angular variables:

$$\lambda_{\rm M} = f'+ g'+ h'$$ (19)

f' being the true anomaly of the Moon. For reasons of commodity, the indices M will be omitted, in the following, concerning the variables $a_{\rm M}$, $r_{\rm M}$ and $e_{\rm M}$.By substituing the values of $\sin \beta_{\rm M}$ and $\lambda_{\rm M}$ in equation (3) we find the following development for $U_{1,{\rm M}}$, which is the expression of the lunar potential at the first order:

$$\begin{eqnarray} &&U_{1,M} = k_{\rm M} G \Bigl( {a \over r} \Bigr)^{3} \times \... ...{\rm M})^{2} \over 4} \biggr) \cos(2f' + 2g' - 2h' + 2h) \Biggr .\end{eqnarray}$$ (20)

Where $\bigl( {a \over r} \bigr)^{3}$ and f' are themselves a function of the mean anomaly of the Moon l' and of the eccentricity $e_{\rm M}$:

$$\Bigl( {a \over r} \Bigr)^{3} = 1 + {3 e^{2} \over 2}+ 3e \cos l' + \Bigl({9 e^{2} \over 2}\Bigr) \cos 2l' + ...$$ (21)

$$f' = l' + 2e \sin l' + \bigl({5 e^{2} \over 4}\bigr) \sin 2l' + ...$$ (22)

The second-order potential $W_2^{\rm cp}$ characterizing the spin-orbit coupling effect has the same expression as in (1), but by substituting the Delaunay's variables h' and H' to the Andoyer's variables h and H:

$$\begin{eqnarray} &&W_2^{\rm cpl.} {=} \Biggl( {1 \over 2} \int\Bigl[ {\partial (... ...tial (W_{1}) \over \partial l'}\Bigr] {\rm d}t \Biggr)_{\rm per}.n\end{eqnarray}$$ (23)

Using the Eqs. (17.1-3) we obtain the derivative of a given function with respect to L',G' and H' starting from its derivatives with respect to a,e and $I_{\rm M}$:

$${\partial [...] \over \partial L'} = {2 L' \over \mu} \time... ... - e^{2}) \over e L'} \times{\partial [...] \over \partial e}$$ (24)

$${\partial [...] \over \partial G'} = {-\sqrt{1 - e^{2}} \ove... ...ot I_{\rm M} \over G'}{\partial [...] \over \partial I_{\rm M}}$$ (25)

$${\partial [...] \over \partial H'} = - {1 \over G' \sin I_{\rm M}} \times{\partial [...] \over \partial I_{\rm M}} \cdot$$ (26)

Because of the expected relative smallness of the nutation coefficients coming from the spin-orbit coupling effect, we can initially restrict ourselves to the leading terms of the potential as given by (20). Practically we can keep the components which remain large enough after integration, that is to say those whose the product of the amplitude and of the inverse of the frequency are the largest ones. As a result of the procedure, we retain in fact 6 terms with the argument $l_{\rm M}$, $\Omega$, $2 \Omega$, $2F+\Omega$, $2F+2\Omega$ and $l_{\rm M}+2F+2\Omega$ ($l_{\rm M}$ is the mean anomaly of the Moon, $\Omega$ the mean longitude of the node, and F is given by: $F=L_{\rm M}-\Omega$, where $L_{\rm m}$ is the mean longitude of the Moon).

To have an idea of their respective values, we can refer to the tables of the potential listed in Kinoshita (1977). Each of these components can be represented as a product $H_{i}(I,I_{\rm M},e_{\rm M}) \times U_{i}(l',g',h',h)$. This makes the calculations easier for H_i (i=1,6) depends only on the canonical action variables, whereas U_i(l',g',h',h) (i=1,6) only depends on the angle canonical variables. We can thus adopt for the potential the following development:

$$U = k_{\rm M} G \sum_{i=1}^{6} H_{i}(I,I_{\rm M},e) \times u_{i}(l',g',h',h).$$ (27)

With:

$$\begin{eqnarray} H_{1} &=& \Bigl( {3 \cos^{2}I -1 \over 16} \Bigr) (1 + 3 \cos 2... ...& - \Bigl({ 7 \sin^{2}I \over 32}\Bigr) (1+ \cos I_{\rm M})^{2} e \end{eqnarray}$$ (28)

$$\begin{eqnarray} u_{1} &=& \cos l' \nonumber\\ u_{2} &=& \cos(h'-h) \nonumber\\... ...\cos(2l'+2g'+2h'-2h) \nonumber\\ u_{6} &=& \cos(3l'+2g'+2h'-2h). \end{eqnarray}$$ (29)

By combining the Eqs. (23), (24), (25) and (26) with the help of the form given by (27), (28) and (29), then we can get a rather straightforward final expression for the second-order determing function $W_2^{\rm cp}$ related to the coupling effect that we are dealing with here, that is to say:

$$\begin{eqnarray} && W_{2}^{\rm cp} = - \Bigl({k_{\rm M}^{2} G^{2} \over 2 G' \si... ...({3 k_{\rm M}^{2}G^{2} \over \sqrt{\mu a}} \Bigr) \times A_{5}. \end{eqnarray}$$ (30)

With the following developments for A_i:

$$\begin{eqnarray} A_{1} &=& \sum_{i=1}^{6} \sum_{j=1}^{6} H_{i} {\partial H_{j} \... ... - w_{j} \Bigl( {\partial u_{i} \over \partial l'} \Bigr) \Bigr] \end{eqnarray}$$ (31)

where w_i (i=1,6) is obtained with a simple integration of u_i:

$$\begin{eqnarray} w_{1} &=& {\sin l' \over \dot{l'}} \nonumber\\ w_{2} &=& {\sin... ...2g'+2h'-2h) \over (3\dot{l'}+2\dot{g}'+2\dot{h}'-2\dot{h}) }\cdot \end{eqnarray}$$ (32)

Then the nutations in longitude $\Delta \psi_{\rm cp}^{W_{2}}$ and $\Delta \varepsilon_{\rm cp}^{W_{2}}$ coming from $W_2^{\rm cp}$ are given by:

$$\Delta \psi_{\rm cp}^{W_{2}} = -\Delta h = \Bigl( {1 \over G \sin I} \Bigr) {\partial W_{2}^{\rm cp} \over \partial I}$$ (33)

and:

$$\Delta \varepsilon_{\rm cp}^{W_{2}} = -\Delta I = - \Bigl[ {... ...\sin I} \Bigr] {\partial W_{2}^{\rm cp} \over \partial h}\cdot$$ (34)

The expressions $\Delta \psi_{\rm cp}$ and $\Delta \varepsilon_{\rm cp}$ which characterize the total spin-orbit coupling effect are then given by (Kinoshita & Souchay 1990):

$$\Delta \psi_{\rm cp} = \Delta \psi_{\rm cp}^{W_{2}} - {1 \ov... ...{\partial W_{1} \over \partial H},W_{1} \right\rbrace_{\rm cp}$$ (35.1)

$$\Delta \varepsilon_{\rm cp} = \Delta \varepsilon_{\rm cp}^{W... ...W_{1},{\partial W_{1} \over \partial h} \right\rbrace_{\rm cp}$$ (35.2)

we insist on the fact that as long as we dealt with crossed-nutation, for instance in (14.1) and (14.2) the Poisson brackets $\lbrace ... \rbrace_{\rm cr}$ were calculated with respect to the Andoyer canonical variables l, g, and h. In this section which concerns the coupling effect, the Poisson brackets $\lbrace...\rbrace_{\rm cp}$ are calculated with respect to the Delaunay canonical variables l', g' and h'. It is also important to keep in mind that the derivatives with respect to a in the u_i's and the w_i's (where a is the semi-major axis for the keplerian motion) is not 0, for the coefficient $k_{\rm M}$in the expression of the potential $U_{\rm 1,M}$ in Eq. (20) contains (a³)^-1 at the denominator. Then these derivatives have to be taken into account when calculating the derivatives with respect to L', according to (24). This explains the presence of the coefficient A₅ in (30) and (31).

Let us now introduce the following quantities:

$$K_{i} = - \Bigl({1\over \sin I}\Bigr) {\partial H_{i} \over \partial I} ,$$

and

$$z_{i} = \Bigl( {\partial w_{i} \over \partial h} \Bigr)$$

that is to say:

$$K_{1} = \Bigl({3 e \cos I \over 8} \Bigr) \times (1 + 3 \cos 2 I_{\rm M})$$ (36.1)

$$K_{2} = - \Bigl({ \cos 2I \over 4 \sin I} \Bigr) \times \sin 2 I_{\rm M}$$ (36.2)

$$K_{3} = \Bigl({ \cos 2I \over 8 \sin I} \Bigr) \times (\sin 2 I_{\rm M} + 2 \sin I_{\rm M})$$ (36.3)

$$K_{4} = \Bigl({\cos I \over 4} \Bigr) \times \sin^{2} I_{\rm M}$$ (36.4)

$$K_{5} = \Bigl({\cos I \over 8} \Bigr) \times (1 + \cos I_{\rm M})^{2}$$ (36.5)

$$K_{6} = \Bigl({7 e \cos I \over 16} \Bigr) \times (1 + \cos I_{\rm M}) ^{2}$$ (36.6)

and:

z₁=0

$$z_{2} = -{\cos(h'-h) \over (h' - h)}$$ (37.2)

$$z_{3} = -{\cos(2l'+2g'+h'-h) \over (2l'+2g'+h'-h)}$$ (37.3)

$$z_{4} = -{\cos(2h'-2h) \over (h'-h)}$$ (37.4)

$$z_{5} = -{\cos(2l'+2g'+2h'-2h) \over (2l'+2g'+2h'-2h)}$$ (37.5)

$$z_{6} = -2{\cos(3l'+2g'+2h'-2h) \over (3l'+2g'+2h'-2h)}\cdot$$ (37.6)

Then the complementary term of the nutation in longitude, which corresponds to the part inside the Poisson brackets in (35.1), is given by:

$$\begin{eqnarray} &&\Delta \psi_{\rm cp}^{\rm comp} = - { 1 \over 2} \left\lbrac... ...r] \times \left({\partial w_{i} \over \partial l'}\right) w_{j} .\end{eqnarray}$$ (38)

 

Table 2: List of the coefficients of rigid Earth nutation coming from the spin-orbit coupling effects

And the complementary term of the nutation in obliquity in (28.2) can be expressed in a similar way by:

$$\begin{eqnarray} &&\Delta \varepsilon_{\rm cp}^{\rm comp} = \left[{1 \over 2G\s... ...\partial l'} - z_{j}{ \partial w_{i} \over \partial l'} \right].\end{eqnarray}$$ (39)

For our present computations, the parameter ${G \over G'}$ is necessary. It represents the ratio of the spin angular-momentum of the Earth to the orbital angular momentum of the Moon, and can be expressed as follows:

$${G \over G'} = {J_{2} \over H_{\rm d}} \times \left({M_{\rm... ...^{2}}\right) {\times }{1 \over \sqrt{1 - e_{\rm M}^{2}}}\cdot$$ (40)

Its value is: ${G \over G'} = 0.206971306$.

The results related to the spin-orbit effect as studied here are listed in Table 2. We can remark that the number of coefficients down to $0.1 ~\rm\mu as$ is much smaller than what was found in the previous section for the crossed-nutation contribution. Also we can remark that the leading coefficient is by far the 18.6y $\Omega$ component, both in longitude and in obliquity, with respective in-phase values of $-463.8 ~\rm\mu as$ and $133.9 ~\rm\mu as$.The analytical expressions for these two leading terms are given by the rather cumbersome following formulas:

$$\begin{eqnarray} &&(\Delta \psi)_{\Omega}^{\rm cp} = \left( {k_{\rm M}^{2} G \s... ...I} \left( {\partial W_{\Omega}^{\rm cp} \over \partial I} \right) \end{eqnarray}$$ (41)

and:

$$\begin{eqnarray} &&(\Delta \varepsilon)_{\Omega}^{\rm cp} = \left( {k_{\rm M}... ... G \sin I} \left( {\partial W_{\Omega}^{\rm cp} \over h} \right) \end{eqnarray}$$ (42)

where $W_{\Omega}^{\rm cp}$ itself is expressed as a function of the H_i:

$$\begin{eqnarray} &&W_{\Omega}^{\rm cp} = \left( \frac{k_{\rm M}^{2} G^{2} \sin ... ... 2\dot g' + \dot h' - \dot h)(\dot h' - \dot h)} \right] \right) ,\end{eqnarray}$$ (43)

with: $\Omega = h' -h$. In Eqs. (40), (41), and (42), we use the following substitutions, with the help of (27.1-6) and (36.1-6):

$$\left (K_{2} {\partial H_{4} \over \partial I_{\rm M}} - H_... ... { (1 + 3 \cos 2 I_{\rm M}) \sin^{2}I_{\rm M} \sin I \over 16}$$ (44.1)

$$\left ( K_{3} {\partial H_{5} \over \partial I_{\rm M}} - H_... ... 2}\right)^{6} \times {(2 - 3 \cos I_{\rm M}) \sin I \over 4}$$ (44.2)

$$H_{3} K_{5} - H_{5} K_{3} = {1 \over 4} \left(\cos {I_{\rm M... ...ver 2}\right)^{7} \sin \left({I_{\rm M} \over 2}\right) \sin I$$ (44.3)

$$H_{2} {\partial H_{4} \over \partial I_{\rm M}} + 2 H_{4} {... ... (1 + 3 \cos 2 I_{\rm M}) \sin^{2}I_{\rm M} \sin^{3} I \cos I$$ (44.4)

$$H_{3} {\partial H_{5} \over \partial I_{\rm M}} + 2 H_{5} {... ...M} -8 \cos 2 I_{\rm M} -3 \cos 3 I_{\rm M} ) \sin^{3} I \cos I$$ (44.5)

$$H_{3} {\partial H_{5} \over \partial I_{\rm M}} + H_{5} {\p... ... M} -2 \cos 3 I_{\rm M} - 5 \cos 2 I_{\rm M}) \sin^{3} I \cos I$$ (44.6)

$$H_{3} H_{5} = - {1 \over 8} (\cos {I_{\rm M} \over 2})^{6} \sin I_{\rm M} \sin^{3} I \cos I.$$ (44.7)

Remark that the second largest component in Table 2 with argument $2F+2\Omega$ and with period 13.66d (fortnightly), was not found in previous calculations (Kinoshita & Souchay 1990), because of their incompleteness. On the opposite the two terms of argument $2F+\Omega$ and $2F + 3\Omega$ were already present in these calculations. An important part of the present ones were carried out by the help of Mathematica, a precious tool to compute formal analytical expressions.

The numerical values of the constant terms used in the present section are taken from ELP2000 (Chapront-Touzé & Chapront 1988) except for $k_{\rm M}$ (Souchay & Kinoshita 1996) and the ratio ${G \over G'}$ calculated above. They are given as follows: