How to establish and keep orthogonal constellation by two orbiters under perturbation

This paper describes how to establish and maintain an orthogonal constellation by two orbiters. We assume the constellation where one orbiter looks down upon the orbital plane of another from the apoapsis, or the normal vector of one orbiter and the eccentricity vector of another are parallel. There have been demands and requests for such orthogonal constellation composed by an in-situ observation orbiter and a remote-sensing orbiter in studies of planetary atmosphere. Since their orbits can be perturbed by several external forces, they must be designed carefully so as to keep orthogonality. In this paper, we introduced the "Orthogonality Index" by using orbital elements of two orbiters, which provides us a quantitative index for orthogonality of the two orbits. By differentiating the index with respect to time, we derived several conditions to keep orthogonality via a bottom-up approach. They were well coincident with the top-down results derived in our former approach. Four solutions for the inclination of the remote-sensing orbiter were derived, which is independent from orbit shapes or central celestial bodies. If we assume planets with a relatively large J2 term such as the earth or the mars, the orbital elements for the in-situ orbiter can also be calculated easily.