Explicit evolution relations with orbital elements for eccentric, inclined, elliptic and hyperbolic restricted fewbody problems
Abstract
Planetary, stellar and galactic physics often rely on the general restricted gravitational body problem to model the motion of a smallmass object under the influence of much more massive objects. Here, I formulate the general restricted problem entirely and specifically in terms of the commonly used orbital elements of semimajor axis, eccentricity, inclination, longitude of ascending node, argument of pericentre, and true anomaly, without any assumptions about their magnitudes. I derive the equations of motion in the general, unaveraged case, as well as specific cases, with respect to both a bodycentric and barycentric origin. I then reduce the equations to threebody systems, and present compact singly and doublyaveraged expressions which can be readily applied to systems of interest. This method recovers classic LidovKozai and LaplaceLagrange theory in the test particle limit to any order, but with fewer assumptions, and reveals a complete analytic solution for the averaged planetary pericentre precession in coplanar circular circumbinary systems to at least the first three nonzero orders in semimajor axis ratio. Finally, I show how the unaveraged equations may be used to express resonant angle evolution in an explicit manner that is not subject to expansions of eccentricity and inclination about small nor any other values.
 Publication:

Celestial Mechanics and Dynamical Astronomy
 Pub Date:
 April 2014
 DOI:
 10.1007/s1056901495378
 arXiv:
 arXiv:1401.4167
 Bibcode:
 2014CeMDA.118..315V
 Keywords:

 Perturbation equations;
 Restricted problems;
 Planetary systems;
 Averaged equations;
 Gravitational Nbody problem;
 Circumbinary case;
 Astrophysics  Earth and Planetary Astrophysics;
 Mathematical Physics
 EPrint:
 Accepted for publication in Celestial Mechanics and Dynamical Astronomy