Three-body problem, planetary

The contents of this paper include, with variable emphasis, the topics of a series of lectures whose main title was Routes to Order Capture into resonance . This was indeed the subject of the last section above. The study of this subject has, however, shown that - unlike the restricted three-body problem - capture into resonance drives the system immediately to stationary solutions known as Apsidal corotations . The whole theory of these solutions was also included in the paper from the beginning - that is, from the formulation of the Hamiltonian equations of the planetary motions and the expansion of the disturbing function in the high-eccentricity planetary three-body problem. The secular theory of non-resonant systems was also given. Motions with aligned or anti-aligned periapses, resonant or not, resulting from non-conservative processes (tidal interactions with the disc) in the early phases of the life of the system, seem to be frequent in extra-solar planetary systems. 

Beauge, C. and Michtchenko, T.A. (2003), Modelling the high-eccentricity planetary three-body problem. Application to the GJ876 planetary system. Month. Not. Roy. Astron. Soc. 341, 760-770. 

The restricted three-body problem Two bodies of finite masses, called primaries, revolve around their common center of mass in circular orbits and a third body with negligible mass moves under their gravitational attraction, but does not affect the orbits of the two primaries. In most astronomical applications the second primary has a small mass compared to the first primary, and consequently the motion of the third, massless, body is a perturbed Keplerian orbit. This is a model for the study of an asteroid (Jupiter being the second primary), a trans-Neptunian object (Neptune being the second primary) or an Earth-like planet in an extrasolar planetary system. 

The general three-body problem Three bodies with finite masses moving under their gravitational attraction. This is a model for a triple stellar system. In many astronomical applications one of the three bodies has a large mass and the other two bodies have small, but not negligible masses. This is a model for an extrasolar planetary system, or a system of two satellites moving around a major planet. In the latter two cases the two small bodies move in perturbed Keplerian orbits. 

Michtchenko, T. and Malhotra, R. (2004), secular dynamics of the three-Body problem Application to the v Andromedae planetary system. Icarus (in press). 

---