# Analytical, circle-to-circle low-thrust transfer trajectories with plane change

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## Abstract

Orbit averaging techniques are used to develop analytical approximations of circle-to-circle low-thrust trajectory transfers with plane-change about the Sun. Separate expressions are developed for constant acceleration, or thrust, electric propulsion, solar sail propulsion and combined, or hybrid electric (constant acceleration or thrust) / solar sail propulsion. The analytical expressions uniquely allow the structure of circle-to-circle low-thrust trajectory transfers with plane-change about the Sun to be understood, and the optimal trajectory structure is analytically derived for each propulsion system considered. It is found that the optimal fixed thrust electric propulsion transfer reduces the orbit radius with no plane change and then performs the plane-change, while the optimal solar sail and hybrid transfers combine the reduction of orbit radius with some plane change, before then completing the plane change. The optimal level of plane change during the reduction of orbit radius is derived and it is found the analytically-derived minimum time solar sail transfer is within 1% of the numerically-derived optimal transfer. It is also found that, under the conditions considered, a sail characteristic acceleration of less than 0.5 mm/s2 can, in 5-years, attain a solar orbit that maintains the observer-to-solar pole zenith angle below 40 degrees for 25 days; the approximate sidereal rotation period of the Sun. However, a sail characteristic acceleration of more than 0.5 mm/s2 is required to attain an observer-to-solar pole zenith angle below 30 degrees for 25 days within 5-years of launch. Finally, it was found that the hybridization of electric propulsion and solar sail propulsion was, typically, of more benefit when the system was thrust constrained than when it was mass constrained.

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