Extremal Analytical Solutions for Intermediate-Thrust Arcs in a Newtonian Field

The variational problem of determining optimal trajectories of motion with constant exhaust velocity and limited mass-flow rate in a central Newtonian field is considered. The first-order necessary conditions of optimality reduce the problem to a Hamiltonian canonical system of equations for intermediate- and maximum-thrust arcs, both of which have no complete analytical solutions to date. The approach used in this work is based on the analytical integration of the canonical system by employing its first integrals and invariant expressions. Several new classes of extremal analytical solutions for planar intermediate-thrust arcs with free and fixed flight times are presented. The solutions describe families of spiral trajectories around the center of attraction. The main result of the paper is that, in their current form with known integrals, the differential equations of the variational problem for intermediate-thrust arcs are integrable in elementary functions and quadratures, and the solution of this problem with such arcs can be reduced to a system of algebraic continuity equations formed for each junction point. These solutions can be used as representative reference trajectories for guidance algorithms and to compute initial values of Lagrange multipliers for high-fidelity trajectory optimization software. As an illustrative example, the transfer maneuver to a given elliptical parking orbit using an intermediate-thrust arc is discussed. Results of simulations for three study cases containing the change of eccentricity and semiparameter of the parking orbit and specific impulses are presented.