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IssuesArchive of Issues2014-2pp.127-135

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T.E. Churkina, "Satellite Rotation Stability at a Mercurian Type Resonance," Mech. Solids. 49 (2), 127-135 (2014)
Year 2014 Volume 49 Number 2 Pages 127-135
DOI 10.3103/S0025654414020022
Title Satellite Rotation Stability at a Mercurian Type Resonance
Author(s) T.E. Churkina (Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe†sh.†4, GSP-3, A-80, Moscow, 125993 Russia, taniana802@mail.ru)
Abstract We consider the satellite plane motion about the center of mass in a central Newtonian gravitational field in an elliptic orbit. This motion is described by a second-order differential equation known as the Beletskii equation. In the framework of the plane problem (under the assumption that the body vibrates in the unperturbed orbit plane), there exists a family of periodic solutions of the Beletskii equation near the $3:2$ resonance between the orbital revolution and axial rotation periods. A nonlinear stability analysis of these periodic solutions is carried out both in the presence of third- and fourth-order resonances and in their absence as well as on the boundaries of the stability regions in the first approximation. The problem is solved numerically. For fixed parameter values (the eccentricity of the center-of-mass orbit and the inertial parameter), the construction of a symplectic mapping of the equilibrium into itself is used to calculate the coefficients of the mapping generating function, which are further used to conclude whether the equilibrium is stable or not.
Keywords satellite, Beletskii equation, stability, Hamiltonian, generating function, resonance
References
1.  V. V. Beletskii, Artificial Satellite Motion about Center of Mass (Nauka, Moscow, 1965) [in Russian].
2.  V. V. Beletskii, Artificial Satellites of the Earth (Izd-vo AN SSSR, Moscow, 1959) [in Russian].
3.  V. V. Beletskii and E. K. Lavrovskii, "On the Theory of the Resonance Rotation of Mercury," Astron. Zh. 52 (6), 1299-1308 (1975) [Sov. Astron. (Engl. Transl.) 19 (6), 776-781 (1976)].
4.  A. P. Markeev, Linear Hamiltonian Systems and Several Problems of Stability of Satellite Motion about the Center of Mass (NITs "Regular and Chaotic Dynamics," Institute for Computer Studies, Moscow-Izhevsk, 2009) [in Russian].
5.  A. P. Markeev, "Stability of Equilibrium States of Hamiltonian Systems: a Method of Investigation," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 6, 3-12 (2004) [Mech. Solids (Engl. Transl.) 39 (6), 1-8 (2004)].
6.  I. G. Malkin, Theory of Stability of Motion (Nauka, Moscow, 1966) [in Russian].
7.  V. A. Sarychev, V. V. Sazonov, and V. A. Zlatoustov, "Periodic Rotations of a Satellite in the Plane of an Elliptic Orbit," Kosmich. Issled. 17 (2), 190-207 (1979) [Cosmic. Res. (Engl. Transl.) 17 (2), 159-173 (1979)].
8.  V. P. Varin, "Degeneration of Periodic Solution of the Beletskii Equation," Regular and Chaotic Dynamics 5 (3), 313-328 (2000).
Received 12 May 2011
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