Stability of hierarchical triples - I. Dependence on inner eccentricity and inclination

Abstract

In simulations it is often important to decide if a given hierarchical triple star system is stable over an extended period of time. We introduce a stability criterion, modified from earlier work, where we use the closest approach ratio Q of the third star to the inner binary centre of mass in their initial osculating orbits. We study by numerical integration the orbits of over 1000 000 triple systems of the fixed masses and outer eccentricities eout, but varying inner eccentricities ein and inclinations i. 12 primary combinations of masses have been tried, representing the range encountered in stellar systems. The definition of the instability is either the escape of one of the bodies, or the exchange of the members between the inner and outer systems. An analytical approximation is derived using the energy change in a single close encounter between the inner and outer systems, assuming that the orbital phases in subsequent encounters occur randomly. The theory provides a fairly good description of the typical Q(st), the smallest Q value that allows the system to be stable over N = 10 000 revolutions of the initial outer orbit. The final stability limit formula is Q(st) = 10(1/3)A[(f g)(2)/(1 - e(out))](1/6), where the coefficient A similar to 1 should be used in N-body experiments, and A = 2.4 when the absolute long-term stability is required. The functions f (e(in), cos i) and g(m(1), m(2), m(3)) are derived in the paper. At the limit of e(in) = i = m(3) = 0, f g = 1.