A geometric view of orbital element perturbations
Many systems commonly encountered in celestial mechanics and astrodynamics can be effectively modeled as a two-body problem subject to relatively small perturbations. Under these circumstances, the deviations from the two-body orbit due to the perturbative acceleration can be described in terms of time-variable orbital elements . In this post I will follow the elegant geometric treatment of the perturbation of the orbital elements given in Chapter VIII of Moulton (1902), from which I borrowed the key figures. This approach is complementary to the analytical derivation, usually leading to Lagrange's planetary equations. Having its roots in the first-ever treatment of orbital perturbations, which appeared in Newton's Principia , the geometric picture emphasizes intuitive, qualitative understanding over quantitative results. We shall focus on elliptic (i.e., bound) orbits, and restrict the discussion to the five classical orbital elements that describe the shape and o...