Eccentricity estimates in hierarchical triple systems

Perturbation theory in the three body problem has greatly advanced our ability to understand and model a variety of systems ranging from artificial satellites to stars and from extrasolar planets to asteroid-Jupiter interactions. In a series of papers, we developed an analytical technique for estimating the orbital eccentricity of the inner binary in hierarchical triple systems. The method combined the secular theory with calculations of short period terms. The derivation of the short term component was based on an expansion of the rate of change of the Runge-Lenz vector by using first order perturbation theory, while canonical perturbation theory was used to investigate the secular evolution of the system. In the present work we extend the calculation to the orbit of the outer binary. At the same time, we provide an improved version for some previous results. A post-Newtonian correction is included in our model. Our analytical estimates are compared with numerical and analytical results on the subject and applications to stellar triples and extrasolar planets are discussed.