Long-term passive distance-bounded relative motion in the presence of $$J_2$$ perturbations.

This paper presents closed-form solutions for the problem of long-term satellite relative motion in the presence of $$J_2$$ perturbations, and introduces a design methodology for long-term passive distance-bounded relative motion. There are two key ingredients of closed-form solutions.One is the model of relative motion; the other is the Hamiltonian model and its canonical solution of the $$J_2$$ -perturbed absolute motion. The model of relative motion makes no assumptions on the eccentricity of the reference orbit or on the magnitude of the relative distances. Besides, the relative motion model is concise with straightforward physical insight, and consistent with the Hamiltonian model. The Hamiltonian model takes into account the secular, long-periodic and short-periodic effects of the $$J_2$$ perturbation. It also remains separable in terms of spherical coordinates to ensure the application of the Hamilton-Jacobi theory to derive the canonical solution. When deriving the canonical solution, pseudo-circular and pseudo-elliptical orbits are treated separately and Carlson's method is employed to calculate elliptic integrals, which takes advantage of the symmetry of the integrand. These symmetry properties hold physical insights of the $$J_2$$ -perturbed absolute motion. To design the long-term distance-bounded relative motion, the nodal period and the drift of right ascension of the ascending node (RAAN) per nodal period are, respectively, matched non-instantaneously. Even though the nodal period and the drift of RAAN per nodal period can be obtained via the canonical solution, action-angle variables are used to obtain the frequency of the system without finding the complete solution to the perturbed orbital motion. [ABSTRACT FROM AUTHOR]