Comparison between first and second-order gauss’s variational equations under impulsive control

Due to the insufficient 20 page limit, this paper shows, just as an example, how to derive the 2nd-order Gauss’s variational equation of the inclination under impulsive control. The complete derivation of the 2nd-order Gauss’s variational equations for all classical and nonsingular orbital elements under impulsive control is done in Ref. [1]. In addition, this paper provides, for an assigned single impulsive velocity variation, the comparison between the orbital elements variations predicted by the 1st-order and 2nd-order with respect to the true Keplerian variations. The gain in accuracy using the 2nd-order over the 1st-order is quantified by numerical Monte Carlo tests. Least-squares estimate of the impulse vector is obtained by inverting the 1st-order and 2nd-order GVEs. The 2nd-order estimation is here proposed for the single-impulse orbital maintenance problem.