Instability of the 2 + 2 Body Problem

Details Instability of the 2 + 2 Body Problem Instability of the 2 + 2 Body Problem

Mar 1987

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1987cemec..41..153m&link_type=abstract

Celestial Mechanics, Volume 41, Issue 1-4, pp. 153-160

Physics

Scientific paper

The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m1 and m2 in circular keplerian orbit) have an integral analogous to the Jacobi integral of the circular 2+1 body problem. We show here that with 2+2 bodies this integral does not give rise to Hill stability, i.e. to confinement for all time in a portion of the configuration space not allowing for some close approaches to occur. This is because all the level manifolds are connected and all exchanges of bodies between the regions surroundingm 1,m 2 and infinity do not contradict the conservation of the integral. However, it is worth stressing that some of these exchanges are physically meaningless, because they involve either unlimited extraction of potential energy from the binary formed by the small bodies (without taking into account their physical size) or significant mutual perturbations between the small masses without close approach, a process requiring, for the Sun-Jupiter-two asteroids system, timescales longer than the age of the Solar System.

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