R 13q t r23q t

Note that H depends on t explicitly since r13 and r23, when written in terms of %, n, Z, do. Hence, H is not the total energy of the system and is not a constant of the motion. However, if we apply the canonical transformation generated (see p. 403) by

W3 = -p1(Q1 cos at — Q2sin at) — p2(Q1 sin at + Q2 cos at) - P3 Q3, where P = (x - ay, y + ax, z), Q = (x, y, z), which is equivalent to the change of variables given in Eqn (11.10), we obtain a new Hamiltonian H* = H + d W3/dt that does not depend explicitly on t and is therefore constant. In fact,

H* = 2(x2 + y2 + z2) - U, and so we recover Jacobi's integral.

26 This name is somewhat misleading as the total energy of the two primaries is constant, but the energy associated with the small mass is not.

First given by Jacobi in Sur le mouvement d'un point et sur un cas particulier du problème des trois corps (1836).

Tisserand's criterion, and Hill curves

An example of the application of Jacobi's integral is in Tisserand's criterion for the identification of comets. In their passage round the Sun, comets are often perturbed significantly by the actions of the planets, particularly Jupiter. If we make the approximation that the orbit of Jupiter is a circle, then C must remain constant for the comet during its encounter with the planet. With certain other simplifying assumptions, none of which introduce any significant numerical errors provided data for the comet are obtained when it is far from Jupiter, it can be shown that the instantaneous elements of the comet's orbit must satisfy

—+ 2^a(1 - e2) cos i = constant, a in which the unit of length is the Sun-Jupiter distance. The quantity on the left-hand side thus represents a defining characteristic of a cometary orbit and can be used to check whether two comets observed at different times are one and the same object.

Jacobi's integral also can be used to restrict the region of space in which the small mass can move. It is clear from Eqn (11.9) that if 2U = C, the speed of the small mass (with respect to the two primaries) will be zero and thus, for a given C, the surface 2U = C forms a boundary to the motion. This idea was used by Hill in his ground-breaking work on the lunar theory to show that (neglecting the eccentricity of the Earth) the Moon could never escape from its orbit around the Earth. The nature of some of the surfaces that Hill described is illustrated in Figure 11.1, where we have further simplified the problem by assuming that all three masses move in the same plane. In this case, the Hill surfaces reduce to curves.

If we fix units so that mi + m2 = a = a>2 = 1, then the Hill curves are given by

0 0

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