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

This paper considers the long-term behaviour of hierarchical triple stellar systems. To a zeroth-order approximation, the motion of such systems can b e regarded as consisting of two independent Keplerian binary orbits: one co mprising 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 ele ments. In the present paper we derive averaged equations to approximate the se 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 rati o. Two terms of the Hamiltonian of the interaction between the two orbital motions are averaged by the von Ziepel method, yielding second-order equati ons which describe the long-term behaviour of the orbital elements. We test the second-order theory by comparing its prediction for the inner e ccentricity modulation with that obtained from numerical integration of New ton's equations. The theory predicts very well the modulation amplitude and time-scale. This is demonstrated in three cases - low, intermediate and hi gh inclination. The series expansion of the interaction in Legendre polynomials, and the tw o terms that we keep in particular, enables us to distinguish between the c ontributions of the two different terms to the inner eccentricity modulatio n in the various configurations studied.