Energy-Conserving Planar Spacecraft Motion with Constant-Thrust Acceleration

Abstract

Orbital motion of spacecraft under the action of constant-thrust acceleration in the direction perpendicular to the velocity is studied. A spacecraft starting in an initial circular orbit obtains a minimum (perigee) radius along an inbound trajectory. If the thrust is maintained, the spacecraft returns to the initial circular orbit in an outbound trajectory. A maximum (apogee) radius is obtained when starting in a circular orbit along an outbound trajectory, after which the vehicle returns to its initial orbit. Only one integral of motion is known to exist in this system; therefore, a full analytical solution cannot be found. However, an analytical solution to the flight-direction angle (that is, the angle between the position and velocity vectors) is found in terms of the position magnitude. This solution leads to several insights, such as an analytical expression for the range of acceleration magnitudes that enables prograde motion. Periodic orbits are also found in this system for various values of acceleration, and their stability is studied. Even though a full analytical solution is not provided, a finite Taylor expansion about the acceleration magnitude yields an approximate analytical solution for the trajectory beginning on a circular orbit.