Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury

Abstract

We investigate the stable Cassini state 1 in the p : q spin–orbit resonant problem. Our study includes the effect of the gravitational potential up to degree and order 4 and p : q spin–orbit resonances with $p,q\le 8$ and $p\ge q$. We derive new formulae that link the gravitational field coefficients with its secular orbital elements and its rotational parameters. The formulae can be used to predict the orientation of the spin axis and necessary angular momentum at exact resonance. We also develop a simple pendulum model to approximate the dynamics close to resonance and make use of it to predict the libration periods and widths of the oscillatory regime of motions in phase space. Our analytical results are based on averaging theory that we also confirm by means of numerical simulations of the exact dynamical equations. Our results are applied to a possible rotational history of Mercury.

Appendices

A Expressions

The expressions for A[k], $s_{k,j}$ defined in (17), (18) are summarized in Tables 3 and 4, respectively. Only terms that have been used in this study are shown. Expressions not appearing in the respective tables are assumed to be zero. The complete list of terms can be found in the electronic supplement material to be accessed on https://l-sgn.org/cmda-2017/.

Table 3 Expressions in (17), (18) for the 3 : 2 and 1 : 1 spin–orbit resonances

Table 4 Expressions in (17), (18) for the 3 : 1, 5 : 2, 2 : 1 spin–orbit resonances

B Rotation matrices

The rotation matrices, used in the present paper, are defined following the convention:

$$\begin{aligned} R_1=\left( \begin{array}{lll} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad c &{}\quad -s \\ 0 &{}\quad s &{}\quad c \\ \end{array} \right) \ , \ R_2=\left( \begin{array}{lll} c &{}\quad 0 &{}\quad s \\ 0 &{}\quad 1 &{}\quad 0 \\ -s &{}\quad 0 &{}\quad c \\ \end{array} \right) \ , \ R_3=\left( \begin{array}{lll} c &{}\quad -s &{}\quad 0 \\ s &{}\quad c &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \\ \end{array} \right) \ , \end{aligned}$$

where $R_j=R_j(\psi )$ with $j=1,2,3$ and where we used the abbreviations $c=\cos (\psi )$ and $s=\sin (\psi )$.

C Table of notations

See Table 5.

Table 5 Table of notations