Abstract
In this paper, we develop an analytical stability criterion for a five-body symmetrical system, called the Caledonian Symmetric Five-Body Problem (CS5BP), which has two pairs of equal masses and a fifth mass located at the centre of mass. The CS5BP is a planar problem that is configured to utilise past–future symmetry and dynamical symmetry. The introduction of symmetries greatly reduces the dimensions of the five-body problem. Sundman’s inequality is applied to derive boundary surfaces to the allowed real motion of the system. This enables the derivation of a stability criterion valid for all time for the hierarchical stability of the CS5BP. We show that the hierarchical stability depends solely on the Szebehely constant C_0 which is a dimensionless function involving the total energy and angular momentum. We then explore the effect on the stability of the whole system of varying the relative sizes of the masses. The CS5BP is hierarchically stable for C_0 > 0.065946. This criterion can be applied in the investigation of the stability of quintuple hierarchical stellar systems and symmetrical planetary systems.
Highlights
The five-body system considered in this paper is frequently hierarchical in structure
Through the projections in the ρ1ρ2 plane given by the maximum extensions and the minima of the boundaries of real motion in ρ1ρ2ρ12 space, we can study the topology of the boundary surfaces and gain knowledge on the hierarchical stability of the system
We have investigated the hierarchical stability of the Caledonian Symmetric Five-Body Problem (CS5BP)
Summary
The five-body system considered in this paper is frequently hierarchical in structure. The utilisation of symmetries and/or neglecting the masses of some of the bodies compared to others can simplify the dynamical problem and enable global analytical stability conditions to be derived. The CSFBP is a planar problem with time symmetry and rotational symmetry about the centre of mass These authors derived an analytical stability criterion valid for all time, showing that the hierarchical stability of the CSFBP depends solely on a parameter which is a dimensionless function of energy, angular momentum, total mass of the system and the gravitational constant. They called this parameter the Szebehely Constant, C0.
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