The Role of Chaos in the Circularization of Tidal Capture Binaries. I. the Chaos Boundary

Abstract

A self-consistent model for binary evolution devised by Gingold \& Monaghan (1980) is used to show that two distinctly different types of behaviour are possible for close eccentric binaries. The model is based on a linear adiabatic normal mode analysis of the problem, which allows detailed examination of the transfer of energy from the orbit to the tides. We show that for most binaries, energy is exchanged quasi-periodically, with the system regulating itself so that the maximum tidal energy always remains small, and no circularization takes place. In contrast, for a range of eccentricities and periastron separations, {\it chaotic} behaviour prevails, with the eccentricity following a random walk and with the energy transferred to the tides during a single periastron passage being up to an order of magnitude larger than that transferred during the initial periastron passage. These results have important consequences for the study of tidal capture binaries. The standard model (see, for example, McMillan, McDermott \& Taam 1987) assumes that the energy transferred to the tides during a periastron encounter is independent of the oscillatory state of the stars, and that the amount of tidal energy present at any time can be calculated using a formula which (accurately) gives the amount deposited after the first encounter (Press \& Teukolsky 1977). The calculations presented here show that a self-consistent treatment is necessary in order to study the dynamical evolution of tidal capture binaries. We conclude that the tidal capture process would not be possible were it not for the existence of chaotic behaviour.