Introduction and Background
It is well known that spacecraft propelled by low-thrust electric propulsion (EP) can potentially deliver a greater payload fraction compared to vehicles propelled by conventional chemical propulsion. The increase in payload fraction for EP systems is due to its much higher specific impulse (Isp) or engine exhaust velocity, which is often an order of magnitude greater than the Isp for a chemical system. However, optimizing low-thrust orbit transfers is a challenging problem due to the low control authority of the EP system and the existence of long powered arcs and subsequent multiple orbital revolutions. Therefore, obtaining optimal transfers is sometimes tedious and time consuming. In his seminal paper, Edelbaum presented analytical solutions for optimizing continuous-thrust transfers between inclined circular Earth orbits. These results serve as an excellent preliminary design tool for estimating ΔV and transfer time for low-thrust missions with continuous thrust and quasi-circular transfers. Real solar electric propulsion (SEP) spacecraft, however, experience periods of zero thrust during passage through the Earth's shadow, and this major effect is not accommodated in Edelbaum's analysis. Colasurdo and Casalino have extended Edelbaum's analysis and developed an approximate analytic technique for computing optimal quasi-circular transfers with the inclusion of the Earth's shadow. Only coplanar transfers are considered, and the thrust-steering is constrained so that the orbit remains circular in the presence of the Earth's shadow.