Studies of multiple stellar systems - II. Second-order averaged Hamiltonian to follow long-term orbital modulations of hierarchical triple systems

Abstract

This paper considers the long-term behaviour of hierarchical triple stellar systems. To a zeroth-order approximation, the motion of such systems can be regarded as consisting of two independent Keplerian binary orbits: one comprising the two close stars, and another consisting of the centre of mass of the inner binary and the third distant body. The interaction between the two orbits results in slow variations of their instantaneous Keplerian elements. In the present paper we derive averaged equations to approximate these long-term variations. We use an expansion of the Hamiltonian of hierarchical triple systems with the small parameter characterizing these systems - the semi-major axis ratio. Two terms of the Hamiltonian of the interaction between the two orbital motions are averaged by the von Ziepel method, yielding second-order equations which describe the long-term behaviour of the orbital elements. We test the second-order theory by comparing its prediction for the inner eccentricity modulation with that obtained from numerical integration of Newton's equations. The theory predicts very well the modulation amplitude and time-scale. This is demonstrated in three cases - low, intermediate and high inclination. The series expansion of the interaction in Legendre polynomials, and the two terms that we keep in particular, enables us to distinguish between the contributions of the two different terms to the inner eccentricity modulation in the various configurations studied.