A symplectic mapping for the synchronous spin-orbit problem

Abstract

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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Notes

  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 \(A\simeq B\simeq C\).

References

  1. Abdullaev, S.S.: A new integration method of Hamiltonian systems by symplectic maps. J. Phys. A 32, 2745–2766 (1999)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  2. Baland, R.M., Van Hoolst, T., Yseboodth, M., Karatekin, O.: Titan’s obliquity as evidence for a subsurface ocean? A &A 530, A141 (2011)

    ADS  Google Scholar 

  3. Beletskii, V.V.: Resonance rotation of celestial bodies and Cassini’s laws. Celest. Mech. dyn. Astron. 6, 356–378 (1972)

    MathSciNet  MATH  Google Scholar 

  4. Bills, B.G., Nimmo, F.: Rotational dynamics and internal structure of Titan. Icarus 214, 351–355 (2011)

    ADS  Article  Google Scholar 

  5. Borderies, N., Yoder, C.F.: Phobos’ gravity field and its influence on its orbit and physical librations. A & A 233, 235–251 (1990)

    ADS  Google Scholar 

  6. Bouquillon, S., Kinoshita, H., Souchay, J.: Extension of Cassini’s laws. Celest. Mech. Dyn. Astron. 86, 29–57 (2003)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  7. Breiter, S., Buciora, M.: Explicit symplectic integrator for rotating satellites. Celest. Mech. Dyn. Astron. 77, 127–137 (2000)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  8. Cachucho, F., Cincotta, P.M., Ferraz-Mello, S.: Chirikov diffusion in the asteroidal three-body resonance (5–2-2). Celest. Mech. Dyn. Astron. 108, 35–58 (2010)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  9. Celletti, A., Chierchia, L.: Measures of basin of attraction in spin-orbit dynamics. Celest. Mech. Dyn. Astron. 101, 159–170 (2008)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  10. Chirikov, B.V.: A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263 (1979)

    MathSciNet  ADS  Article  Google Scholar 

  11. Colombo, G.: Cassini’s second and third laws. Astron. J. 71, 861–891 (1966)

    Google Scholar 

  12. Deprit, A.: Free rotation of a rigid body studied in the phase plane. Am. J. Phys. 35, 424–428 (1967)

    ADS  Article  Google Scholar 

  13. D’Hoedt, S., Lemaitre, A.: The spin-orbit resonant rotation of Mercury: a two degree of freedom Hamiltonian model. Celest. Mech. Dyn. Astron. 89, 267–283 (2004)

    Google Scholar 

  14. Ferraz-Mello, S.: A symplectic mapping approach to the study of the stochasticity in asteroidal resonances. Celest. Mech. Dyn. Astron. 65, 421–437 (1996)

    MathSciNet  ADS  Article  Google Scholar 

  15. Froeschlé, C.: Mappings in astrodynamics. In: Ferraz-Mello, S. (ed.) Chaos, Resonance and Collective Dynamical Phenomena in the Solar System, pp. 375–390 (1992)

  16. Goldreich, P., Peale, S.: Spin-orbit coupling in the solar system. Astron. J. 71, 425–437 (1966)

    ADS  Article  Google Scholar 

  17. Hadjidemetriou, J.: Mapping models for Hamiltonian systems with application to resonant asteroid motion. In: Roy, A. (ed) Predictability Stability and Chaos in N-Body, Dynamical Systems, pp. 157–175 (1991)

  18. Hadjidemetriou, J.: A hyperbolic twist mapping model for the study of asteroid orbits near the 3:1 resonance. ZAMP 37, 776–792 (1992)

    MathSciNet  ADS  Article  Google Scholar 

  19. Hadjidemetriou, J.: Symplectic mappings. In: Ferraz-Mello, S. et al. (eds.) Dynamics, Ephemerides and Astrometry of the Solar System, IAU Symp 172, 255–266 (1996)

  20. Hadjidemetriou, J.: A symplectic mapping model as a tool to understand the dynamics of 2/1 resonant asteroid motion. Celest. Mech. Dyn. Astron. 73, 65–76 (1999)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  21. Henrard, J.: The rotation of Io. Icarus 178, 144–153 (2005)

    ADS  Article  Google Scholar 

  22. Henrard, J., Schwanen, G.: Rotation of synchronous satellites. Celest. Mech. Dyn. Astron. 89, 181–200 (2004)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  23. Iess, L., Rappaport, J., Jacobson, A., Racioppa, P., Stevenson, D.J., Tortora, P., Armstrong, J.W., Asmar, S.W.: Gravity field shape and moment of inertia of Titan. Science 327, 1367–1369 (2010)

    ADS  Article  Google Scholar 

  24. Kinoshita, H.: Theory of the rotation of the rigid earth. Celest. Mech. Dyn. Astron. 15, 277–326 (1977)

    MathSciNet  Google Scholar 

  25. Laskar, J., Robutel, P.: High order symplectic integrators for perturbed Hamiltonian systems. Celest. Mech. Dyn. Astron. 80, 39–62 (2001)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  26. Lemaitre, A., D’Hoedt, S., Rambaux, N.: The 3:2 spin-orbit resonant motion of mercury. Celest. Mech. Dyn. Astron. 95, 213–224 (2006)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  27. Lhotka, C.: Dynamics expansion points: an extension to Hadjidemetriou’s mapping method. Celest. Mech. Dyn. Astron. 104, 175–189 (2009)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  28. Noyelles, B., Lemaitre, A., Vienne, A.: Titan’s rotation—a 3-dimensional theory. A &A 478, 959–970 (2008)

    ADS  Google Scholar 

  29. Peale, S.J.: Generalized Cassini’s laws. Astron. J. 74, 483–489 (1969)

    ADS  Article  Google Scholar 

  30. Stumpff, K.: Himmelsmechanik Band I. Deutscher Verlag der Wissenschaften, Berlin (1959)

  31. Touma, J., Wisdom, J.: Lie-Poisson integrators for rigid body dynamics in the solar system. Astron. J. 107, 1189–1202 (1994)

    ADS  Article  Google Scholar 

  32. Vienne, A., Duriez, L.: TASS1.6: ephemerides of the major Saturnian satellites. A &A 297, 588–605 (1995)

    ADS  Google Scholar 

  33. Ward, W.R.: Tidal friction and generalized Cassini’s laws in the solar system. Astron. J. 80, 64–70 (1975)

    ADS  Article  Google Scholar 

  34. Wisdom, J.: The origin of the kirkwood gaps: a mapping for asteroidal motion near the 3/1 commensurability. Astron. J. 87, 577–593 (1982)

    MathSciNet  ADS  Article  Google Scholar 

  35. Wisdom, J.: Chaotic behaviour and the origin of the 3/1 Kirkwood gap. Icarus 56, 51–74 (1983)

    ADS  Article  Google Scholar 

  36. Wisdom, J., Holman, M.: Symplectic maps for the n-body problem. Astron. J. 102, 1528–1538 (1991)

    ADS  Article  Google Scholar 

  37. Wolfram, S.: Mathematica, V8. Wolfram Research, Champaign, IL USA (2012)

  38. Yoshida, H.: Recent progress in the theory and application of symplectic integrators. Celest. Mech. Dyn. Astron. 56, 27–43 (1993)

    ADS  MATH  Article  Google Scholar 

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Acknowledgments

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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Keywords

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