Some special restricted four-body problems—I. Modelling the Caledonian problem

Abstract

A four-body system analogous to the Copenhagen problem (the restricted circular coplanar three-body problem with two finite equal masses) is modelled. The coplanar, restricted, circular equal mass four-body problem, or the Caledonian problem as it is named, involves four point-like bodies of equal mass moving in coplanar orbits about their centre of mass. Initially, the bodies are collinear and move co-rotationally with their velocity vectors perpendicular to the line they form. Models are developed for both a linear hierarchical dynamical system (the smallest binary's centre of mass forms a binary with the third body, while their centre of mass forms a binary with the fourth body) and a double binary hierarchical dynamical system (two binaries whose centres of mass form an additional binary). This simplified four-body model reduces the four-body problem from 24th-order with 28 initial parameters to 16th-order with only four independent initial parameter values, viz the mass m and three distance parameters a, b, c marking the initial separations between the four bodies. The advantages in minimisation of the numbers of variables and boundary condition parameters are obvious, yet, despite its simplifications, the Caledonian problem is shown to be suitable for application to the study of real four-body stellar systems. An exhaustive study of the model utilising ranges of the four parameters should provide: insight into why real four-body stellar systems are always found in either a linear or double binary hierarchy; an understanding of the part played by the parameters in the stability lifetimes of this model; a measure of the reliability of the c 2 H criteria applied to the triple subsets and the Roy-Walker empirical stability criteria in predicting the stability of the four body systems.