Constrained Circularization in Elliptic Orbit Using Low Thrust with Shadowing Effect

The optimal thrust pitch angle variation that results in the maximum change in the eccentricity in general elliptic orbit using continuous, constant, low-thrust acceleration while keeping the orbit energy unchanged after a full thrust cycle is determined by direct use of the theory of maxima and through numerical quadrature and search techniques. The analysis takes into account the presence of a shadow arc arbitrarily positioned along the elliptic orbit, where thrust is cut off. Unlike the well-known nonoptimal scheme that uses a thrust orientation perpendicular to the line of apsides at all times, the present optimal scheme allows for the maximum change in eccentricity for a more efe cient orbit circularization. Approximate but highly accurate analytic expressions for the changes in the eccentricity and semimajor axis of a general elliptic orbit, perturbed by a constant low-thrust acceleration applied along the e xed inertial direction normal to the orbit major axis, are also derived for general use and rapid calculations. HE problem of the maximization of the change in the eccen- tricityofa generalelliptic orbitusingcontinuousconstantlow- thrust acceleration while constraining the semimajor axis a to stay constant after one full cycle of thrust is analyzed, by also taking into account the presence of the Earth shadow arc where the thrust is turned off. Elliptic orbit circularization with electric thrusters is presently performed by geostationary communications spacecraft that are initially released into highly elliptic supersynchronous or- bits, subsequently circularized by also maintaining orbital energy constant. The optimization method is similar to the one used in Refs. 1- 3, which was also applied in Ref. 4 under the same assumption of the initially circular orbit model. In particular, in Refs. 2 and 3, the problem of transferring a spacecraft between two inclined circular orbits, of different size and inclination, in minimum time, using discontinuous low-thrust acceleration is considered. The two-body thrust-perturbed orbit is, thus, constrained to remain circular dur- ing the transfer, and the optimal control law for the thrust direction derived for the fast timescale problem of maximizing the inclina- tion change for a given change in the semimajor axis is used in an averaging procedure to solve the overall slow timescale trans- fer problem. The present paper tackles the fast timescale planar problem in the more general elliptic case, by deriving the optimal thrust pitch proe le that maximizes the change in eccentricity with- out changing semimajor axis and by also extending the analysis to the dual problem of maximizing the orbit semimajor axis while keeping the eccentricity e unchanged after one cycle of intermittent thrust. The change in eccentricity over a single orbit can then be used in conjunction with an inclination change to produce averaged rates of change in the eccentricity and inclination to solve the over- all slow timescale minimum-time transfer problem of circularizing and rotating an initial elliptic orbit without changing its orbital en- ergy. The theory of maxima is, thus, employed in the present planar problem, and the value of a certain constant Lagrange multiplier is