Some integrals observed in spacecraft relative motion problem with circular reference orbit

In this paper, the original nonlinear relative motion problem for a pair of satellites in which the leader is in a circular orbit is studied with a new perspective. This problem is regarded as a degenerate form of the circular restricted three-body problem, where only one central body's mass is considered. By using the techniques that are often adopted in the three-body problem, four integrals are found. In these integrals, the first one is the degenerate form of the well-known Jacobi integral. The other ones correspond to the orbital integral constants of the classical two-body problem in a different way. A numerical simulation is conducted to validate these theoretical results. The simulation results show that the solutions of this paper are all correct. By using these integrals, one may understand the relative motion more thoughtful.