Ascent or Descent from Satellite Orbit by Low Thrust

Abstract

The problem of ascent or descent from an initially Keplerian orbit by a constant low thrust is investigated by the two-variable expansion procedure. The slowly varying elements of the Keplerian orbit are explicitly computed, and a simple integral relating the eccentricity and semimajor axis is derived and used to justify the omission of higher order terms in the eccentricity for ascent from almost circular initial orbits. The corresponding assumption for descent from an almost circular orbit is shown to be inconsistent. The present higher order theory that is valid for arbitrary initial eccentricities is shown to contain earlier results for circular starting orbits and gives simple explicit formulas for the solution at small eccentricities. The present results do, however, suffer from the deficiency of earlier solutions in that the validity is restricted to periods before escape velocity is reached for the case of ascending orbits. HE problem of ascent from circular orbit by a small thrust in either the radial or circumferenti al directions was first investigated by Tsien. 1 Tsien gave an exact solution for the case of radial thrust and developed an approximate first-order solution for the circumferential thrust case. Recently, Ting and Brofman2 extended Tsien's solution to one higher order and allowed the thrust to be arbitrarily oriented while remaining constant. Their approach consists of splitting the dependence of the radius into oscillatory and non-oscillatory parts and using an expansion procedure combining the features of singular perturbations and the method of averaging. More recently, Nayfeh3 also developed a solution for the case treated by Ting and Brofman, using a more systematic asymptotic method. The quoted results were all concerned with the case of circular initial orbits and do not shed any light on the more interesting case of an arbitrary elliptic starting orbit. In the closely related problem of ascent by tangential thrust, Zee4 and Cohen5 attempted to solve for the oscillatory terms in the radius contributed by the ellipticity of the initial orbit. Their solutions suffer from the presence of unnecessary assumptions that restrict the validity of the results to small eccentricity in spite of the complicated approaches proposed. In another recent paper, King6 considered the question of descent from a parking orbit by low thrust, but did not discuss the analytical implications of this reversal of the usual role of low thrust. In this paper, the following generalizations are made. The initial elliptic orbit is not restricted in any way, and the implications of negative thrust (i.e., thrust opposing the motion) are fully investigated. The solution is developed to second order by the two-variable expansion procedure developed by Cole and Kevorkian7 and Kevorkian.8 This method, which has been applied to two other problems in celestial mechanics by Eckstein, Shi and Kevorkian,9'10 is a generalized asymptotic expansion procedure that provides uniformly valid developments for the coordinates appropriate to satellite problems. At this point, a brief review of the diverse methods applied to the solution of this problem is appropriate. In the dis