Rj ri i 1 2 392

j=1 rij where mi and ri are, respectively, the mass and position vector of body i, rij is the distance between bodies i and ■ , and the prime on the summation sign indicates that the term j = i should be omitted. One approach, which might sensibly be pursued for some problems, is to consider the three-body problem as a perturbation of the two-body problem. Thus, when considering the motion of the Moon due to the interactions of the Sun and Earth, we might consider the Sun as a perturbing influence on the solution to the Earth-Moon two-body problem. Similarly, the effect of Saturn on the orbit of Jupiter could be considered as a perturbation of a known solution. It is appropriate for such problems to generate the equations of motion relative to one of the three bodies, which we will label as body 1. The equation corresponding to i = 1 in Eqn (9.2) can be subtracted from that corresponding to i = 2 to give

r12 r23 r13

where /xij = G (mi + mj) and rij = ri - rj. An equation for r3i can be obtained similarly. If the right-hand side were zero, then the above equation is identical to Eqn (8.8) (p. 274) and we would simply have the two-body problem with its resultant elliptic orbits. The two terms on the right-hand side thus represent the perturbing influence of body 3 on this Keplerian motion; the first represents the effects of body 3 on body 2, while the second represents the effects of body 3 on body 1.

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