Stability criteria in many-body systems

Abstract

The analytical stability criterion applicable to coplanar hierarchical three-body systems described in the first paper of this series, Walkeret al. (1980), is modified to give an exact representation ofHill-type stability in all such cases. The dependence of the stability on all orbital parameters (in the coplanar case) is taken into account. The criterion for stability is now dependant upon the participating masses, the elements of the initial osculating Keplerian orbits of the system (viz. the orbits ofm ₂ aboutm ₁ andm ₃ about the mass-centre of the (m ₁,m ₂) system) and the positions within these orbits.

The behaviour of the stability of such systems is demonstrated (both analytically and numerically) with respect to certain of the parameters involved to consider effects not dealt with in the above-mentioned paper. In particular two interesting real cases of triple systems in the Solar System are discussed, namely Sun-Jupiter-Saturn and Earth-Moon-Sun. The results of the present paper are compared with those of past authors who considered the same systems.

Finally some general features arising out of our analysis are discussed.