Constant tangential low-thrust trajectories near an oblate planet

The problem of changing spacecraft energy in minimum time near an oblate planet by means of tangential low thrust is considered. The mass flow rate and the exhaust velocity are assumed constant. Simple approximate formulas allowing calculation of the spacecraft three-dimensional spiral trajectory are obtained. Simple first-approximation formulas and more accurate second-approximation formulas are given. All of the formulas are valid both for ascent and descent of the spacecraft orbit. The accuracy of the formulas is estimated; it is shown that they provide high accuracy when the thrust acceleration is much smaller than the gravitational acceleration.