Numerical investigation of all possible equilibria of dual spin satellites

Abstract

This paper is the continuation of a previous work [6] in which we have obtained the set of all possible equilibria of a gyrostat satellite attracted by n points mass by solving two algebraical equations P₁=0 and P₂=0. It results that there is a maximum of 24 isolated equilibrium orientations for the satellite. Sufficient conditions of stability for these relative equilibria are given.

Here we consider only the elementary case n=1. We show that the coefficients of the two algebrical equations depend on four parameters j₁, j₃, K and v². The two first parameters depend only on the direction of the internal angular momentum$\overrightarrow k$ of the rotors, the third being only function of the principal moments of inertia of the satellite and the last parameter is a decreasing function of one of the components of$\overrightarrow k$. We show that the two polynomials P₁ and P₂ are unvariant within two transformations of the parameters j₁ and j₃. It is then possible to reduce the range of variation of these parameters.

For some particular values of the parameters, it is possible to give the minimum number of real roots of equations P₁=0 and P₂=0. In general cases, a computing program is written to obtain the number of real roots of these equations according to the values of the parameters. We show that among the roots found, few of them corresponds to stable equilibrium orientations.