Effective stability around the Cassini state in the spin-orbit problem

Abstract

We investigate the long-time stability in the neighborhood of the Cassini state in the conservative spin-orbit problem. Starting with an expansion of the Hamiltonian in the canonical Andoyer-Delaunay variables, we construct a high-order Birkhoff normal form and give an estimate of the effective stability time in the Nekhoroshev sense. By extensively using algebraic manipulations on a computer, we explicitly apply our method to the rotation of Titan. We obtain physical bounds of Titan’s latitudinal and longitudinal librations, finding a stability time greatly exceeding the estimated age of the Universe. In addition, we study the dependence of the effective stability time on three relevant physical parameters: the orbital inclination, \(i\), the mean precession of the ascending node of Titan orbit, \(\dot{\varOmega }\), and the polar moment of inertia, \(C\).

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Fig. 1
Fig. 2

Notes

  1. 1.

    The analytical form of \(\alpha \) and \(\beta \) can be found in the Henrard and Schwanen (2004) [see Eqs. (16) and (17)].

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Acknowledgments

The work of C. L. was financially supported by the contract Prodex C90253 “ROMEO” from BELSPO, and partly by the Austrian FWF research grant P-J3206. The work of M. S. is supported by an FSR Incoming Post-doctoral Fellowship of the Académie universitaire Louvain, co-funded by the Marie Curie Actions of the European Commission.

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Correspondence to Marco Sansottera.

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Appendix A: Semi-analytical expansion of the Hamiltonian (Titan application)

Appendix A: Semi-analytical expansion of the Hamiltonian (Titan application)

We report here the explicit expansions of the relevant Hamiltonian functions related to the application to Titan. We recall that the Titan physical parameters adopted here are reported in Table 1.

The averaged potential in Eq. (5) takes the form

$$\begin{aligned} \langle V \rangle _{l_4}&=-1.01\times 10^{-2}\cos ^2 K-3.13\times 10^{-7} \sin ^2 K \\&\quad -1.12\times 10^{-4} \cos \left( \sigma _3\right) \cos K \sin K \\&\quad -\cos \left( 2 \sigma _1\right) \left( 3.14\times 10^{-3}+6.29\times 10^{-3}\cos K+3.14\times 10^{-3}\cos ^2 K\right) \\&\quad +\cos \left( 2 \sigma _3\right) \left( 1.56\times 10^{-7}\cos ^2 K-1.56\times 10^{-7}\right) \\&\quad -\cos \left( 2 \sigma _1+\sigma _3\right) \left( 3.51\times 10^{-5}\cos K+3.51\times 10^{-5}\right) \sin K \\&\quad +\cos \left( 2 \sigma _1+2 \sigma _3\right) \left( 4.90\times 10^{-8}\cos ^2 K-9.79\times 10^{-8}\sin ^2 K-4.90\times 10^{-8}\right) \\&\quad +\cos \left( 2 \sigma _1+3\sigma _3\right) \left( 2.73\times 10^{-10} \cos K-2.73\times 10^{-10}\right) \sin K \\&\quad -1.91\times 10^{-13} \cos \left( 2 \sigma _1+4 \sigma _3\right) \left( 1.00-1.00 \cos K\right) ^2\ . \end{aligned}$$

The quadratic part of the Hamiltonian (8), \(H_0\), reads

$$\begin{aligned} H_0&= 2.52\times 10^{-2}\sigma _1^2 + 1.08\times 10^{-6} \sigma _1\sigma _3 + 7.00\times 10^{-7} \sigma _3^2 \\&+ 7.20\times 10^1 \varSigma _1^2 - 2.08\times 10^{-2}\varSigma _1 \varSigma _3 + 2.01\times 10^2\varSigma _3^2\ , \end{aligned}$$

and the values of the parameter \(\alpha \) and \(\beta \) corresponding to the untangling transformation, see Eq. (9), are \(\alpha =-8.46\times 10^{-5}\) and \(\beta =2.14\times 10^{-5}\,\). Thus, the quadratic part of the Hamiltonian (10) in diagonal form reads

$$\begin{aligned} H_0=2.52\times 10^{-2} \sigma _1^2 +7.00\times 10^{-7}\sigma _3^2 +7.20\times 10^1 \varSigma _1^2 +2.01\times 10^2 \varSigma _3^2\ . \end{aligned}$$

The parameters \(U_{1}^*\) and \(U_{3}^*\) related to the rescaled polar coordinates, see Eq. (11), take the values \(U_1^*=5.348\times 10^{1}\) and \(U_{3*}=1.696\times 10^{4}\,\), and the quadratic part of the Hamiltonian in action-angle variables, see Eq. (12), reads

$$\begin{aligned} H_0=2.69\,U_1+2.37\times 10^{-2}U_3\ , \end{aligned}$$

while the term of order \(3\) in Eq. (12), \(H^{(0)}_1\,\), reads

$$\begin{aligned} H^{(0)}_1&= 2.86\times 10^{-7}\cos (u_1) \sqrt{U_1}^{3} -2.87\times 10^{-7}\cos (3 u_1) \sqrt{U_1}^{3} \\&+1.98\times 10^{-3}\cos (2u_1-u_3) U_1 \sqrt{U_3} -3.99\times 10^{-3}\cos (u_3) U_1 \sqrt{U_3} \\&+2.01\times 10^{-3}\cos (2u_1+u_3) U_1 \sqrt{U_3} -3.97\times 10^{-3}\cos (u_1) \sqrt{U_1} U_3 \\&+9.29\times 10^{-2}\cos (u_1-2u_3) \sqrt{U_1} U_3 -9.62\times 10^{-2}\cos (u_1+2u_3) \sqrt{U_1} U_3 \\&-2.19 \cos (u_3) \sqrt{U_3}^{3} -2.19 \cos (3u_3) \sqrt{U_3}^{3}\ . \end{aligned}$$

We provide the approximation of Eq. (12), up to order \(6\) in \(\left( \sqrt{U_1}\,,\ \sqrt{U_3}\,\right) \,\), in the electronic Supplemental Material (see Table 2) while we report below the number of coefficients in Eq. (12) at each order,

Order 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
#terms 2 10 19 28 44 54 70 84 93 105 112 125 130 143 145
Order 3 4 5 6 7 8 9 10 11 12 13 14 15 16
#terms 10 19 28 44 60 85 110 146 182 231 280 345 423 544

Finally, we report in the electronic Supplemental Material (see Table 3) the truncated normal form, up to order \(12\). In this case, the number of terms of order \(2r\) in \(\left( \sqrt{U_1}\,,\ \sqrt{U_3}\,\right) \) is just equal to \(r+1\,\), thus, at each order, we report in the table below the number of coefficients in the remainder term at different orders

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Sansottera, M., Lhotka, C. & Lemaître, A. Effective stability around the Cassini state in the spin-orbit problem. Celest Mech Dyn Astr 119, 75–89 (2014). https://doi.org/10.1007/s10569-014-9547-6

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Keywords

  • Spin-orbit resonance
  • Normal form methods
  • Cassini state
  • Titan
  • Long-time stability