Stability criteria in many-body systems

The stability parameters developed and discussed in the first paper of this series (Walkeret al., 1980) are used to determine empirically, by means of numerical integration experiment, regions of stability for corotational, coplanar, hierarchical three-body systems. The initially circular case of these systems is studied: the components of the close binary are taken to move initially in circular orbits with respect to their common mass-centre, the third mass initially moving in a circular orbit with respect to the same mass-centre such that its orbit lies wholly outside those of the former two masses. The stability of these systems is then studied by reference to the empirical stability parameters and the initial ratio of the semi-major axes of the orbit of the close binary to that of the third mass about the binary's mass-centre, which is less than unity. For given values of the stability parameters it is determined how the stability of a system is affected by changes in the ratio of the semi-major axes. It is found that an upper limit to this ratio exists which determines the region of stability for such systems. It is also found possible, in the region of instability, to predict how unstable a system will be i.e. crudely speaking, the number of orbits it may be expected to execute before some gross instability sets in. The effect commensurabilities in mean motion have on the stability of these systems is also considered. It is generally found that these commensurabilities enhance the stability of these systems. The predictive powers of the method are then tested: using many test cases it is seen how accurately the stability or instability of a system may be predicted.