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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, Volume 28, Issue 4, Pages 549–564 (Mi vuu656)

MECHANICS

On evolution of the planet's obliquity in a non-resonant planetary system

P. S. Krasil'nikova, O. M. Podviginab

a Moscow Aviation Institute, Volokolamskoe shosse, 4, Moscow, 125993, Russia
b Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, ul. Profsoyuznaya, 84/32, Moscow, 117997, Russia

Abstract: We investigate the evolution of the obliquity of a planet in the gravitational field of a star and other planets comprising a planetary system. The planet is assumed to be an axially symmetric rigid body ($A=B$). This planet and other planets move around the star along Keplerian ellipses with frequencies $\omega$ and $\omega_2,\ldots,\omega_N$, respectively, where $N$ is the number of celestial bodies (material points) affecting the planet.
We derive Hamiltonian for the problem in the Depri–Andoyer variables in the satellite approximation. The Hamiltonian is averaged over the fast variables of the rotational and orbital motions, assuming that the motions are not resonant. The averaged Hamiltonian involves, in addition to the classic parameters, parameters $D_i$, that can be considered as functionals on the family of orbits of celestial bodies comprising the planetary system. The averaged Hamiltonian admits separation of variables, which implies the existence of three first integrals in involution. Regarding the gravitational torques of the other planets as small perturbations, we obtain from the energy integral of the averaged equations explicit approximate expressions for obliquity of the planet and its perturbed period of precession.
We investigate numerically the amplitude of oscillations of the planet's obliquity and it's perturbed period of precession for a planetary system involving a star, the planet itself and another massive planet (similar to Jupiter), whose orbits satisfy certain symmetry conditions and orbital planes intersect at angle $\gamma$.

Keywords: planet's rotation, planetary system, averaged equations, planet's obliquity, precession period.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00820_à This work was supported by Russian Foundation for Basic Research under Grant 18-01-00820.

DOI: https://doi.org/10.20537/vm180408

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Bibliographic databases:

UDC: 521.92, 517.928.7
MSC: 70F15, 70K65

Citation: P. S. Krasil'nikov, O. M. Podvigina, “On evolution of the planet's obliquity in a non-resonant planetary system”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:4 (2018), 549–564

Citation in format AMSBIB
\Bibitem{KraPod18} \by P.~S.~Krasil'nikov, O.~M.~Podvigina \paper On evolution of the planet's obliquity in a non-resonant planetary system \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2018 \vol 28 \issue 4 \pages 549--564 \mathnet{http://mi.mathnet.ru/vuu656} \crossref{https://doi.org/10.20537/vm180408} \elib{https://elibrary.ru/item.asp?id=36873369}