Mathematics
Scientific paper
Nov 1983
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1983cemec..31..213m&link_type=abstract
Celestial Mechanics (ISSN 0008-8714), vol. 31, Nov. 1983, p. 213-240. Research supported by the Consiglio Nazionale delle Ricerc
Mathematics
12
Celestial Mechanics, Dynamic Stability, Three Body Problem, Topology, Branching (Mathematics), Manifolds (Mathematics), Many Body Problem, Solar System
Scientific paper
A proof of the topological stability criterion for 3-D 3-body systems is given using only Lagrangian mechanics and elementary calculus. Then the formal relationship between the general and the restricted 3-body problem is explained. When (m3/m2)-squared is negligible the confinement curves for the general problem are the same as the zero velocity curves of the restricted circular one. But the offset between the integral c-squared times h (angular momentum squared times energy) and the Jacobi integral, which are the bifurcation parameters in the general and restricted problem, respectively, must be taken into account to determine the stability of the general problem. For small values of epsilon (mass ratio), the offset of the two bifurcation values is about one half the eccentricity squared of the orbits of the two larger bodies. This approximate stability criterion, quantifying the destabilizing effect of the eccentricity of the binary on the third body, is used to prove the instability of systems formed by the sun, Jupiter and Mercury, Mars, Pluto, or any one of the asteroids.
Milani Andrea
Nobili Anna M.
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