The Stability of the Caledonian Symmetric Four-Body Problem with Close Encounters

  • Anoop Sivasankaran

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


The central theme of the research presented in this thesis is an investigation of the stability of a symmetrically restricted four-body problem called the Caledonian Symmetric Four-Body Problem (CSFBP) (Steves and Roy, 2001) using a newly developed numerical integration scheme which enables the numerical exploration of the systems as they pass through two-body close encounters. A study of the hierarchical stability of the CSFBP system is made, followed by an empirical stability analysis of hierarchically stable regions in the phase space of the CSFBP. The study of the dynamics and stability of four-body systems like CSFBP is relevant in order to determine stable hierarchical arrangements which will be capable of hosting exoplanetary systems.

A comprehensive literature review of the key features of the CSFBP is presented. The collision manifold of the phase space of the CSFBP is explored for a whole range of CSFBP systems and the fundamental limitations of the existing numerical integration scheme (cf. Széll, Steves and Erdi (2004a); Széll, Erdi, Sândor and Steves (2004)) have been analysed. It was found that, neglecting the collision orbits in the phase space of the CSFBP is a major limitation in the numerical exploration of the global stability features of the CSFBP. A review of régularisation theory is given, highlighting the key stages needed to develop a régularisation method for a gravitational few-body problem. A global régularisation method (cf. Heggie (1974)) is then derived to handle various two-body close encounters.

An algebraic optimisation algorithm (Gruntz and Waldvogel, 1997) is adapted for numerically implementing the régularisation scheme. The numerical accuracy and the computational performance of the developed integration scheme were tested for a broad range of CSFBP orbits. Regardless of the nature of the orbits, it was found that the regularised integration scheme outperformed the standard non-regularised integration schemes in terms of computational performance and improved numerical accuracy characterized by stable energy profiles.

The hierarchical stability of the CSFBP is investigated using the developed integration schemes. Numerical simulations were conducted for a comprehensive set of CSFBP orbits. It was found that the analytical hierarchical stability criteria was satisfied even after the inclusion of orbits with two-body close encounters. An empirical stability investigation was also made and it identified regions of hierarchical stability in the phase space of the CSFBP for any value of Co < Ccrit
Date of Award2010
Original languageEnglish
Awarding Institution
  • Glasgow Caledonian University
SupervisorBonnie Steves (Supervisor)

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