Analytically exact non-Keplerian motion for orbital transfers

This paper presents a continuous thrust program for which the equations of two-dimensional motion of a space vehicle in orbit, viewed as a point mass, afford an exact analytic solution. The thrust program is proportional to an arbitrary throttling parameter, and arises from the optimization problem corresponding to minimum-fuel, power-limited, coplanar transfers between two arbitrary conics. It is shown that, for this problem, the assumption that the thrust is tangent to the flight path results in the complete analytic solution of the system of state-costate equations governing the optimal trajectories. This approximation of tangential thrust is made only in the costate equations, affecting only optimality, resulting in the elimination of the costates, and giving rise to the thrust program, for which the state equations can be solved analytically with no further approximations. The most striking aspect of the motion suggested by this solution is the fact that it is described by a trajectory equation identical in form to the one corresponding to Keplerian motion! The difference is that, in the latter the angular momentum is a constant, while in the former it is a linear function of time, with slope equal to the throttling parameter. This similarity allows one to speak of the two motions by using more or less the same mathematical vocabulary. The transverse (horizontal) component of the thrust program is inversely proportional to the radial distance from the inverse-square attractive center, while the radial (vertical) component is such that the thrust is in the flight-path direction. The constant of proportionality (throttling parameter) appearing in the thrust program can be appropriately selected to satisfy the boundary conditions corresponding to arbitrary coplanar transfer and escape problems. The time duration for such maneuvers however can be prespecified only for escape problems, and not for transfer problems. To document these facts, and demonstrate the existence of such particular solutions, several examples are given of such maneuvers. Optimality not being the main issue in this paper, relevant questions are dealt with in an elementary way, and hints are provided for both power-limited, and constant ejection velocity propulsion systems. The thrust program can be used not only with power-limited propulsion systems, but also with any propulsion system operating on continuous thrust, and, for *P~.D. Candidate, student member AIAA. Copyright @ 1994 by Nikos Markopoulos. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. propulsion types and classes of transfers for which it turns out to be sufficiently optimal, this paper suggests a means of maneuvering during planetocentric or heliocentric orbital operations, for which the exact analytic solution of the governing equations of motion is now known. Elementary Nomenclature t = time M = angular momentum per unit mass X = radial velocity component R = radial distance from attractive center 9 = argument of latitude ER = mdial thrust acceleration component E, = transverse thrust acceleration component z = nondimensional time h = nondimensional angular momentum p. u. m. x = nondimensional radial velocity component r = nondimensional radial distance V = nondimensional speed of the vehicle y = flightpath angle E, = nondimensional radial thrust acceleration ee = nondimensional transverse thrust acceleration a = semimajor axis of a conic e = eccentricity of a conic w = orientation of a conic (on a given plane) E = generalized eccentric anomaly H = generalized hyperbolic anomaly, Hamiltonian A = throttling parameter B = generalized eccentricity of transfer trajectory C = generalized orientation of transfer trajectory ( ), = value of a quantity at reference radius Rs ( ), = value of a quantity at the initial time z=O ( ), = value of a quantity at the final time T=T,