Analytical stability in the Caledonian Symmetric Five-Body Problem
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Analytical stability in the Caledonian Symmetric Five-Body Problem B. A. Steves1
· M. Shoaib2 · Winston L. Sweatman3
Received: 27 August 2019 / Revised: 22 September 2020 / Accepted: 8 November 2020 / Published online: 25 November 2020 © The Author(s) 2020
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 C0 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 C0 > 0.065946. This criterion can be applied in the investigation of the stability of quintuple hierarchical stellar systems and symmetrical planetary systems. Keywords Few-body problem · Five-body problem · Hierarchical stability · Celestial mechanics · Stellar dynamics
1 Introduction The five-body system considered in this paper is frequently hierarchical in structure. In hierarchical N -body systems, the masses involved can be divided into subgroups. The relative
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B. A. Steves [email protected] M. Shoaib [email protected] Winston L. Sweatman [email protected]
1
The Graduate School, Glasgow Caledonian University, Glasgow G4 0BA, Scotland, UK
2
Higher Colleges of Technology, P.O. Box 25035, Abu Dhabi, UAE
3
School of Natural and Computational Sciences, Massey University, Private Bag 102-904, Auckland 0745, New Zealand
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motion of the subgroups is dominated by their gravitational interaction with one another. Within each subgroup, the relative motion of the masses is dominated by their gravitational interaction with other masses in that subgroup. The subgroups themselves may be likewise divisible into further subgroups in a recursive fashion. Analytical studies of the stability of hierarchical systems, of three or more bodies, are challenging because of the greater number of variables involved with increasing numbers of bodies and the limitation of just 10 integrals that exist in the gravitational N -body problem. 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. These symmetric and restricted few-body systems with their analytical stability criteria can then provide useful information on the stab
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