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

摘要

The variational problemof determining optimal trajectories of motion with constant exhaust velocity and limitednmass-flow rate in a central Newtonian field is considered. The first-order necessary conditions of optimality reducenthe problem to a Hamiltonian canonical system of equations for intermediate- and maximum-thrust arcs, both ofnwhich have no complete analytical solutions to date. The approach used in this work is based on the analyticalnintegration of the canonical system by employing its first integrals and invariant expressions. Several new classes ofnextremal analytical solutions for planar intermediate-thrust arcs with free and fixed flight times are presented. Thensolutions describe families of spiral trajectories around the center of attraction.Themain result of the paper is that, inntheir current formwith known integrals, the differential equations of the variational problemfor intermediate-thrustnarcs are integrable in elementary functions and quadratures, and the solution of this problem with such arcs can benreduced to a systemof algebraic continuity equations formed for each junction point. These solutions can be used asnrepresentative reference trajectories for guidance algorithms and to compute initial values of Lagrange multipliersnfor high-fidelity trajectory optimization software. As an illustrative example, the transfer maneuver to a givennelliptical parking orbit using an intermediate-thrust arc is discussed. Results of simulations for three study casesncontaining the change of eccentricity and semiparameter of the parking orbit and specific impulses are presented.