Abstract

Spacecraft starting from circular orbits and using a low thrust, tangentially directed, will follow a spiraled flight path as illustrated in Fig. 1. Whether or not the thrust is constant, if it remains less than one per cent of local vehicle weight the mean path of these trajectories will maintain the familiar circular velocityaltitude relationship for some time. Complete trajectory equations for this mean path and approximate equations for the oscillations about it are derived and presented here. For spacecraft using constant thrust acceleration, parameters are presented for generalizing all differential equations of motion to make them independent of the thrust acceleration and the gravity constant (choice of planet). In the case where this thrust is directed tangentially throughout flight, generalized plots are presented relating velocity, radius distance, and time. When the constant tangential thrust acceleration is less than 10 ~ initial g, single curves independent of the value of the thrust acceleration result, relating the parametric velocity, radius distance, and time. Plots of mean path velocity, distance from the center of the gravity source, and time may be constructed for any space craft of this type regardless of what planet, what initial altitude, and what thrust acceleration is employed, by simply multiplying the scales of the parametric plots by suitable constants. Simple algebraic expressions for speed, altitude, and time to achieve any specific point on the trajectory are available from the generalized plots. Examples of these are presented for the parabolic escape velocity point.