Fuel-optimal eccentricities for precise formation keeping of spacecraft on Keplerian orbits

In this paper, the problem of precise relative position keeping associated with the formation flying of two spacecraft is discussed. Taking into account of the applications to astronomic observation, the relative position of spacecraft is controlled to be precisely retained on a part of an orbit rather than on the entire orbit; moreover, the relative position is retained in the inertial coordinates rather than in the local vertical local horizontal coordinates. The fuel-optimal eccentricities are computed for each length of control time. Then, it is shown that the fuel-optimal eccentricity is close to 1 when the control time is shorter than approximately 90% of the orbit period; on the contrary, the fuel-optimal eccentricity is in the neighborhood of 0 when the control time is close to the orbit period. Moreover, several analytic expressions including the propellant consumptions at the eccentricities of 0 and 1 are obtained.