On the Milankovitch orbital elements for perturbed Keplerian motion

• Published in: Celestial mechanics and dynamical astronomy Vol. 118; no. 3; pp. 197 - 220
• Main Authors: Rosengren, Aaron J., Scheeres, Daniel J.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2014; Springer Nature B.V
• Subjects: Aerospace Technology and Astronautics; Astrophysics; Astrophysics and Astroparticles; Classical Mechanics; Dynamical Systems and Ergodic Theory; Geophysics/Geodesy; Orbits; Original Article; Physics; Physics and Astronomy; Solar physics; Solar radiation; Nonsingular elements; Variation of parameters; Averaging; Solar radiation pressure; Orbit perturbation; Kepler problem
• ISSN: 0923-2958; 1572-9478
• DOI: 10.1007/s10569-013-9530-7

We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace–Runge–Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution.