A symplectic mapping for the synchronous spin-orbit problem


We derive a symplectic mapping model based on Hadjidemetriou’s method for the synchronous spin-orbit problem with and without the additional precession of the nodes. The mapping is derived from the averaged potential of the spin-orbit dynamical model and includes the main spin-orbit interactions, i.e. the non-zero obliquity and wobble motion of the rotating body. In addition the orbit of the perturbing body allows non-zero inclination and eccentricity. To obtain the equilibrium configuration we calculate the position and stability of the fixed points in the 1:1 spin-orbit resonance and relate them to the equilibria of the continuous system. We use the mapping equations to investigate the long-term stability close to the fixed point solutions of the mapping. We also apply the mapping method to the case of the moon Titan and validate the mapping approach by means of numerical integrations. The mapping model reproduces all the characteristics of Deprit’s model of free rotation as well as the dynamical features of Henrard’s averaged model of spin-orbit interaction with great precision.

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  1. 1.

    We adapt the notation introduced in D’Hoedt and Lemaitre (2004) and also used in Lemaitre et al. (2006)

  2. 2.

    The superscripts \((o,s)\) which we are going to use separate the (o)rbital and (s)pin variables, respectively.

  3. 3.

    In contrast to D’Hoedt and Lemaitre (2004) we use the modified set of action-angle variables for both basic dynamical models since the modified actions are small quantities for small values of \(e,i\) and \(J,K\), thus suitable for perturbation theory.

  4. 4.

    The signs appearing in front of the angles in (14) are opposite to the signs given e.g. in Lemaitre et al. (2006). This is due to the different definition of the rotation matrices \(R_i\). The present definition of \(R_i\) is the one used in Noyelles et al. (2008) also given in the appendix.

  5. 5.

    A low order expansion of \(\bar{V}_G^{(2)}(P,p;P_4)\) can be found in the appendix B (in Supplementary Material appended to the online version of this article).

  6. 6.

    To be precise the average should be replaced by a canonical transformation from original to new mean variables. The transformation will be such to remove the dependency of the potential on the angle \(p_4\) to higher orders in a suitable small parameter.

  7. 7.

    In the Supplementary Material appended to the online version of this article.

  8. 8.

    Note, that the formulae are only correct up to order \(O(\gamma _1,\gamma _2)^2\), i.e. their product equals to one only up to \(O(\gamma _1,\gamma _2)^2\). We also checked the agreement of the second order formulae with the eigenvalues obtained with a numerical method in the range of values of the parameters \(A\simeq B\simeq C\).


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The author thanks A. Lemaitre and B. Noyelles for fruitful discussions and for reading the manuscript carefully. B. Noyelles provided an independent software to integrate the un-averaged problem to check the numerical integrations. C. Lhotka was financially supported by the contract Prodex C90253 ROMEO from BELSPO.

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Correspondence to Christoph Lhotka.

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Appendix A

Appendix A

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

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

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

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Lhotka, C. A symplectic mapping for the synchronous spin-orbit problem. Celest Mech Dyn Astr 115, 405–426 (2013). https://doi.org/10.1007/s10569-012-9464-5

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  • Synchronous spin-orbit coupling
  • Hadjidemetriou mapping
  • Cassini state problem
  • Titan
  • Regular satellites