U 1 v2x2 y2

2 r13 r23

then Eqn (11.7) reduces to r3 - 2vy i + 2mx j = WU. (11.8)

This is a system of order 6, but Jacobi showed that the assumption of circular orbits for the primaries (in which case their coordinates do not change with time) allows us to reduce the order by one.

We choose the origin at the centre of gravity of primaries, which are a distance a apart, with the x axis passing through the primaries (and rotating with them) so that m2 m1 r1 =--ai, r2 =-ai, m1 + m2 m1 + m2

and a is the angular rotation rate of the primaries determined from the two-body problem. If we dot Eqn (11.8) with r3, we get r3-r3 = r3- V U which can be integrated since U is a function of x, y, z only to give x2 + y2 + z2 = 2U - C, (11.9)

where C is a constant of integration. This is Jacobi's integral, sometimes referred to as the 'integral of relative energy'. In terms of non-rotating coordinates (%, n, Z) defined by x = % cos at + n sin at, y = -% sin at + n cos at, z = Z,

0 0

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