The hierarchical stability of quadruple stellar and planetary systems using the Caledonian Symmetric Four-Body Model

¹ School of Computing and Mathematical Sciences, Glasgow Caledonian University, Cowcaddens Road, G4 0BA Glasgow, UK e-mail: a.szell@gcal.ac.uk
² Department of Astronomy, Eötvös University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

Received: 19 December 2003
Accepted: 27 July 2004

Abstract

The hierarchical stability and evolution of a symmetrically restricted four-body model called the Caledonian Symmetric Four-Body Problem (CSFBP) is studied. This problem has two dynamically symmetric pairs of masses m and M with mass ratio . The analytical stability criterion derived for the CSFBP by Steves & Roy (2001) is verified numerically for the coplanar case. It is shown numerically that there exists a direct relationship between the hierarchical stability of the system and the Szebehely constant C₀, a combination of the total energy and angular momentum of the system. For , where C_crit2 is a critical value dependent only on μ, the system undergoes no change in its hierarchical arrangement, and is therefore considered to be hierarchically stable. It is also shown that for large mass ratios μ the double binary configurations are the dominant hierarchical configuration, while for smaller mass ratios μ it is the configuration containing a single binary with two outer bodies that is the dominant configuration.