Time-optimal formation establishment around a slowly rotating asteroid

A study is conducted to find a solution to the problem of time-optimal formation establishment around a slowly rotating asteroid, whose gravity is approximated as a second-degree and second-order gravitational field (SDSOGF). Similar to the methodology used for identifying the classical J2-invariant relative orbits, two necessary conditions are analytically derived to guarantee bounded relative motion in an SDSOGF. In particular, it is shown that when the nonspherical harmonic coefficients of the asteroid gravity are first-order small, the resulting necessary conditions are consistent with the recent literature results. Based on the analytically obtained constraints, the problem of time-optimal formation establishment is then emphasized via an indirect approach, in which the initial unknown costate vector is calculated with a scaling technique to alleviate its sensitivity to the initial guess problem.