Instability of the 2 + 2 Body Problem
Abstract
The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m_{1} and m_{2} 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 SunJupitertwo asteroids system, timescales longer than the age of the Solar System.
 Publication:

Celestial Mechanics
 Pub Date:
 1988
 DOI:
 10.1007/BF01238759
 Bibcode:
 1988CeMec..41..153M
 Keywords:

 Celestial Mechanics;
 Jacobi Integral;
 Many Body Problem;
 Angular Momentum;
 Hill Method;
 Kepler Laws;
 Solar System;
 Astrophysics