Study of orbital stability for planar oscillations of a magnetized symmetric satellite in a circular orbit

Theoretical mechanics


Bardin B. S.*, Savin A. A.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



It is considered the motion of a satellite about its center of mass in a central Newtonian gravitational field. The satellite is a magnetized dynamically symmetrical rigid body, whose typical linear dimensions are small compared to the radius-vector of the mass center. The above assumption allows us to consider the problem in a restricted formulation, that is we assume that the motion of the satellite about the center of gravity does not affect the motion of the center of mass [1]. It is also assumed that the orbit of the center of mass is circular. The orientation of the satellite relative to the orbital coordinate system is described by Euler angles ψ, θ, φ.

In this paper it is assumed that magnetic moment arises from the magnetization of the material of satellite shell by earth’s magnetic field [1, 4, 5]. Earth’s geomagnetic field is described by dipole model [6]. It is assumed that the Earth’s magnetic dipole axis coincides with the axis of rotation (direct dipole), and the orbit of the satellite lies in the equator plane.

Under these assumptions, the equations of motion of the magnetized satellite about the center of mass can be written in Hamiltonian form. One of the generalized coordinates (proper rotation angle φ) is a cyclic coordinate, so the corresponding momentum pφ is the first integral of the motion. Planar periodic motions are possible only if pφ=0. It is assumed that the perturbations satisfy the above equality.

The problem has three parameters: the parameter β, which characterizes the geometry of the masses; the parameter ξ, characterizes the effect of the magnetic field of the Earth; doubled amplitude of planar oscillations Ψ.

Linear analysis of the orbital stability for periodic motions is performed. The above motions are planar pendulum-like oscillations such that the polar principle axis of inertia is located in the plane of the orbit of the mass center. To this end local coordinates are introduced in the neighborhood of the unperturbed periodic motion and linearized equations are written.

Separately, it was considered the special case when amplitudes of the oscillations are small. In this case small parameter (doubled oscillation amplitude) has been introduced. It allowed to study the orbital stability analytically. In particular, using the normal forms theory the explicit formulas describing the boundaries of parametric resonance domains were obtained.

For oscillations with arbitrary amplitudes linear stability analysis based on the numerical integration of the linearized system has been performed. In typical cross-sections of the three-dimensional parameter space the domains of orbital instability and stability have been obtained. The results of numerical analysis of orbital stability are in good agreement with the results of analytical study performed in the above mentioned special case.


Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, orbital stability


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