The Caledonian Symmetrical Double Binary Four-Body Problem I: Surfaces of Zero-Velocity Using the Energy Integral

Abstract

The Caledonian four-body problem introduced in a recent paper by the authors is reduced to its simplest form, namely the symmetrical, four body double binary problem, by employing all possible symmetries. The problem is three-dimensional and involves initially two binaries, each binary having unequal masses but the same two masses as the other binary. It is shown that the simplicity of the model enables zero-velocity surfaces to be found from the energy integral and expressed in a three dimensional space in terms of three distances r ₁, r ₂, and r ₁₂, where r ₁ and r ₂ are the distances of two bodies which form an initial binary from the four body system’s centre of mass and r ₁₂ is the separation between the two bodies.