Invariant manifold and bounds of relative motion between heliocentric displaced orbits

In this paper, we establish a methodology for modeling relative motion between heliocentric displaced orbits by utilizing the Cartesian state variables in combination with a set of displaced orbital elements. Similar to classical Keplerian orbital elements, the newly defined set of displaced orbital elements has a clear physical meaning and provides an alternative approach to obtain a closed-form solution to the relative motion problem between displaced orbits, without linearizing or solving nonlinear equations. The invariant manifold of relative motion between two arbitrary displaced orbits is determined by coordinate transformations, obtaining a straightforward interpretation of the bounds, namely maximum and minimum relative distance of three directional components. The extreme values of these bounds are then calculated from an analytical viewpoint, both for quasi-periodic orbits in the incommensurable case and periodic orbits in the 1:1 commensurable case. Moreover, in some degenerate cases, the extreme values of relative distance bounds can also be solved analytically. For each case, simulation examples are discussed to validate the correctness of the proposed method.