Improved initialization conditions and single impulsive maneuvers for $$\hbox {J}_{2}$$ J 2 -invariant relative orbits

Abstract

The determination of the initial conditions for long-term bounded relative motion under natural perturbations is an important theme in satellite cluster flight. Considering the most significant perturbation of the geopotential, namely, the \(\hbox {J}_{2}\) term, many researchers have proposed \(\hbox {J}_{2}\)-mitigating initial conditions for satellite-bounded relative motion. To improve the existing \(\hbox {J}_{2}\)-invariant conditions, two new methods for finding the correction factor are presented in this paper. In these two methods, Method 1 is obtained by minimizing the possible maximum drift in the along-track relative motion. However, Method 2 is designed by nullifying the rates of change of the bounds of the relative motion. Then the geometric character, such as the self-intersection of the \(\hbox {J}_{2}\)-invariant relative orbits, is discussed. The conditions of 0, 1 and 2 (the possible maximum number) self-intersection points are also derived. Then, using Gauss’s equations of planetary motion, an analytical optimal single-impulsive maneuver is deduced to guarantee the secular bounded relative motion under \(\hbox {J}_{2}\), too. Some numerical simulations are performed to validate the corresponding theoretical predictions. The results demonstrate that the proposed methods enhance performance for achieving the bounded relative motion under \(\hbox {J}_{2}\) effects and can be used for future satellite cluster flight missions.