# Analytical solutions of a class of optimum orbit modifications

- Paper ID
IAF-69-37

- author
- company
University of Michigan

- country
U.S.A.

- year
1969

- abstract
Consider a space vehicle initially in an orbit (E0) around a spherical planet with center of attraction at 0. The initial orbit is defined by its semi major-axis a0 and its eccentricity e0. It is proposed to bring the vehicle, by a series of orbital maneuvers, into a final orbit such that its elements, denoted by the subscript 1, satisfy a relation of the form /(«i > Ci) = 0- (1) We seek to minimize the total characteristic velocity for the maneuver. Since for a high-thrust propulsion system the characteristic velocity provides a direct measure of the fuel consumption, the optimal trajectory considered in this paper yields the minimum fuel expenditure. We assume the planet is surrounded by a spherical atmosphere with center at 0 and radius R (Fig. 1). In the search of the absolute minimum fuel consumption we further assume that the duration of the maneuver is unlimited, and the thrust provided by the rockets on board the space vehicle is not bounded, that is it can produce impulsive changes in the velocity. For the case where the thrust magnitude is limited, it can be made impulsive by the process of fractioning. Thus the problem is of the class of time-free, impulsive, orbital transfers.