On the stability of hierarchical four-body systems

Details On the stability of hierarchical four-body systems On the stability of hierarchical four-body systems

Nov 1983

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1983cemec..31..241m&link_type=abstract

Celestial Mechanics (ISSN 0008-8714), vol. 31, Nov. 1983, p. 241-291. Sponsorship: Consiglio Nazionale delle Ricerche.

Statistics

Computation

23

Celestial Mechanics, Dynamic Stability, Four Body Problem, Coordinates, Hierarchies, Perturbation

Scientific paper

A four-body system is decomposed into three-body subsystems whose integral z (angular momentum squared times energy) at t = 0 are assumed to be smaller than the critical values corresponding to L2, so that both the subsystems are initially hierarchically stable. The duration of the stability is then estimated in two steps. The perturbing potentials are developed in series using the theory of Roy (1979) and Walker (1980) which is based on the relevant combinations of mass and length ratios, and the time derivatives of z are computed for the planar case. Only the long-periodic and secular perturbations are discussed in assessing the long-term behavior of the system. A Poisson bracket formalism, a generalization of the Lagrange theorem for semimajor axes, and a generalization of the classical first order theories for eccentricities and pericenters are used to prove that the z integrals do not undergo any secular perturbation. After the long-periodic perturbations have been accounted for, only the second order terms have to be considered in the computation of the timescales for the breakup of the three-body hierarchies. An investigation of the sun, Mercury, Venus, and Jupiter system determined a duration of stability of at least 110 million years.

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