Calculation of Interplanetary Earth-Mars Low Thrust Transfer without Method of Gravispheres

Dynamics, ballistics, movement control of flying vehicles


Sinitsin A. A.

Keldysh Research Centre, 8, Onezhskaya str., Moscow, 125438, Russia



The article considers the spacecraft interplanetary transfer from circular low Earth orbit to circular low orbit near Mars by propulsion system with constant values of thrust and specific impulse. The main goal of the article consists in null length gravispheres method inaccuracy estimation.

Due to the nearness of planets’ phases, the simulation revealed substantial for heliocentric phase value of excessive hyperbolical velocity on the edge of gravispheres. while applying conventional technique, imposing the null value of excessive hyperbolical velocity for the trajectory conjugate phases and the planets’ heliocentric phase. These simplifications affect the transfer effectiveness figures, namely, transfer duration and final mass. Another factor, defining the value of gravispheres method inaccuracy consists in neglecting the long duration of gravitational attraction of the Sun on planet phases, as well as of the planets on heliocentric phase of spacecraft trajectory.

The article presents the developed technique for direct determination of spacecraft trajectory (without applying gravispheres method) including Earth, Sun and Mars gravitational attraction effect on a spacecraft (as Newton’s dynamic model). Thrust vector direction is defined from solving variation problem with transfer duration as performance index (minimum time problem) applying necessary optimality condition in the form of Pontryagin’s maximum principle. Maximum principle application allowed simplify variation problem to the three points boundary value problem, solved numerically.

Comparison of transfer performance indexes, i. e. duration and final spacecraft mass, obtained by the developed technique implementation, and application of gravispheres method is presented. In the last case on planet phases variants of parabolic velocity and hyperbolic excess velocity with value similar to the direct solution (without gravispheres method applying) is considered. Presented computation results revealed worsening of transfer duration and final mass in the case of applying gravispheres method vs. direct trajectory calculation (4-11% for duration and 2-6% for mass).⁠


interplanetary transfer trajectory, low thrust, method of gravispheres, optimization, maximum principle, response time problem


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