Attitude Dynamics of a Rigid Body around the Lagrangian Points

Abstract

This paper explores the attitude dynamics of a small rigid satellite acted upon by the gravitational field of two massive bodies. Attitude stability of the rigid satellite is studied when it is located near the collinear and triangular Lagrangian point. Lagrangian point satellites are typically placed in periodic orbits around the specified point. It is noted that resonant attitude motion can occur if the natural frequencies associated with the attitude motion and the frequency of the Lyapunov orbit satisfy certain relationship. Attitude motion of the satellite in a halo orbit is also studied. I. Introduction HE basic Circular Restricted Three-Body Problem (CRTBP) deals with the dynamics of a point mass under the gravitational influence of two massive primary bodies, M1 and M2, that are orbiting about each other in circular orbits. Five equilibrium solutions for the CRTBP exist if the problem is set in a rotating frame of reference. These five points are named the Lagrangian points and are labeled from L 1 to L 5. The L 1, L 2, and L 3 are called the collinear points as they lie on the line joining the primary masses. The other two points, L4 and L 5, are called the triangular points as they lie on the vertex of equilateral triangles with M1 and M2 at the other vertices. In this paper the L 1 point is between M1 and M2, and the L 2 point lies beyond M2. An extension of the CRTBP substitutes a rigid body for the point mass and considers its attitude dynamics. In our solar system many interesting objects, both natural and man-made, can be found near the different Lagnangian points. Several spacecraft have orbited around the Sun-Earth L 1 point and they include the ISEEE-3 and the Genesis solar sample collector. In the near future several new space telescopes are likely to orbit the Sun-Earth L 2 point. Further down the road, it is conceivable that a space station could be constructed near the Earth-Moon L 1 in order to facilitate future lunar explorations. Located near the L 4 and L 5 of the Sun-Jupiter system are Trojan asteroids, trapped there by the gravitational forces of the Sun and Jupiter. Some previous works have explored the attitude dynamics of Lagrangian point satellites. Kane and Marsh 1 considered the attitude dynamics of an axial symmetric satellite that is spinning about its axis of symmetry, with the symmetry axis normal to the orbital plane of the primary bodies. Robinson 2,3 first studied the attitude dynamics of a dumb-bell satellite located at a triangular Lagrangian point, and later investigated the attitude stability of a satellite of arbitrary shape located at either a collinear point or a triangular point and determined the regions of stability. Misra and Bellerose 4 studied the librational dynamics of a tethered satellite located at the Earth-Moon Lagrangian points and obtained the libration frequencies. In all of the previous studies the rigid body is assumed to be held at the