Invariant relative orbits for spacecraft formation flying in high-order gravitational field

This study addresses the boundary and fluctuation of invariant relative orbits for spacecraft formation flying in a high-order gravitational field. The invariant relative condition is analyzed to obtain the analytical boundary. To ensure the accuracy of invariant condition, zonal harmonic terms of the gravitational field, from J₂ to J₁₅, are considered. Short- and long-period terms of Delaunay orbital elements are individually eliminated via Hori–Lie transformations. The distribution and evolution of periodic orbits are revealed, along with their corresponding nodal period and drift of the right ascension of the ascending node (RAAN) via single-average Delaunay elements including long-period and secular terms. The corresponding double-average elements with only secular terms are used to match the invariant relative orbits. The boundary and fluctuation of invariant relative motion are derived analytically, which can be treated as new indices for the design of invariant relative orbits. Moreover, the accuracy of the analytical boundary and fluctuation is verified compared with the results of numerical integration.