The General Three-Body Problem

It might be thought that, apart from the known integrals and the virial theorem, no general statements can be made on the three-body problem, especially since the totality of solutions in even the restricted problem is not yet explored. In fact, when the restriction that the two finited masses in the restricted problem move in circular orbits about their common centre of mass is relaxed to the extent that they may move in Keplerian ellipses, we also lose the Jacobi integral. Nonetheless, work in recent years—

mainly in extended numerical integrations of the general three-body problem utilizing wide spectra of starting conditions and masses—has enabled certain statements to be made about three-body systems in general. In a sence, we now have the actuarist’s ability to make precise statements about the popu-lation of human beings as time passes—what percentage will die within the next year, and so on. We

have his limitations too in his inability to single out the individual human beings who will make up that percentage. We will see also, that by a suitable combination of the angular momentum and energy in-tegrals, a time-invariant statement analogous to the Jacobi integral in the restricted three-body problem may in fact be made in the case of certain general three-body problems.

Szebehely (1967) introduced a useful system of clarification of the dynamic behaviour of the gen-eral three-body problem. Before using it, we set up the equations of motion and define certain quanti-ties.

Let i = 1, 2, 3 denote the three bodies. Let I be the moment of inertia of the system, T the total ki-netic energy, U the force function, C the total energy of the system. Take r_ias the position vector of the ith body, of mass m_i, and take r_ij= r_j− r_i, as the position vector of the jth body with respect to the ith.

The equations of motion are then

where the force function U is defined in the usual way by

G being the constant of gravitation and _i, the grad operator of the ith body.

From these equations we have the 10 integrals, including the energy relation The moment of inertia I is given by

Now we know by the virial theorem that for positive energy (C > 0) the system must split up, since in this case

or

Then either one mass recedes to an infinite distance (the other two forming a binary), or all three de-part on hyperbolic orbits. Szebehely terms the former occurrence escape (sometimes called hyperbolic-elliptic); the latter he calls explosion.

5.12.1 The case C < 0

The case of total negative energy (C < 0) is more complicated and is best split into a number of classes, though it may be remarked that in any system one class of dynamic behaviour does not necessarily

preclude another. In interplay the masses follow complicated trajectories, including close approaches to each other so that on many occasions |r_ij| < r, a small distance. This may be followed by ejection, when two bodies form a binary while the third departs with elliptic velocity relative to the centre of mass of the binary. If the third body achieves escape velocity it will recede indefinitely, so that this event may also be classed as escape (or hyperbolic-elliptic).

If the semimajor axis of the third body’s perturbed elliptic motion about the binary’s centre of mass is large compared with the binary component’s separation, the configuration is relatively stable; we may recall that such a configuration is common in triple stellar systes. Szebehely classifies this as rev-olution.

The Lagrange special solutions are of course equilibrium configurations but all are unstable (none of the masses is infinitesimal apart from the unlikelihood of the other two having aµ value below Routh’s value). Hence if a triple system was set up for any of the Lagrange solutions it would imme-diately pass into the interplay mode.

Periodic orbits are known in the general three-body problem for C < 0 but are unstable.

5.12.2 The case for C = 0

The case C = 0 is a special one. Separating the ranges of total positive from total negative energies, it is unlikely to occur in nature. It can give hyperbolic–parabolic (i.e. explosion), or hyperbolic–elliptic (i.e. escape) cases.

Summing up, we give a table modelled on one drawn up by Szebehely of the possible modes of be-haviour. If there is no escape or explosion the moment of inertia I remains bounded; otherwise I . What the table does not state is the established fact that the vast majority of initial triple configu-rations end up in escape (after a sufficiently long time) in the hyperbolic–elliptic class. This result is immediately relevant to the understanding of the ratios of the numbers of single, binary and triple stel-lar systems found in the Galaxy. It is also found that when a triple system breaks up it is the particle with the smallest mass that is usually ejected.

5.12.3 Jacobian coordinates

We introduce a form of the equations of motion of the general three-body problem that is found to be extremely useful in both a lunar problem (for example Earth–Moon–Sun) and the typical triple stellar system problem.

If we let C be the centre of mass of the particles P₁and P₂(figure 5.10), then the vector CP₃(ρρ) is taken with the vector P₁P₂(r) as the position vectors. This set of variables was first introduced by Ja-cobi and Lagrange.

Now the relative equations of motion of the three particles may be obtained from equations (5.94) by dividing each by m_i(i = 1, 2, 3) using the grad operator and using the fact that r_ij= r_j− r_i. We ob-tain

where and

Now r = r₁₂, and also ρρ = (m₁/µ)r + r₂₃= (−m₂/µ)r − r₃₁(where µ = m₁+ m₂), since the vector sum of the sides of a triangle is zero. Then from the first of equations (5.94), we have

Figure 5.10

or

Also

Hence, using the second of equations (5.94) and equation (5.96), we have after a little reduction

Following Szebehely we define the vector ƒ(x) by ƒ(x) = Gx|x|⁻³and write ν = m₁/µ, ν* = m₂/µ.

Then equations (5.96) and (5.97) may be written as

and

Equations (5.98) and (5.99) in the Jacobi coordinates form a 12th-order system, the reduction from 18th order to 12th having been essentially effected by the use of the six centre-of-mass integrals. There therefore remain the energy and angular momentum integrals. Their formulation using relations (5.98) and (5.99) is left as an exercise for the reader.

Equations (5.98) and (5.99) may be put in a neater form which will be of immediate use later when we consider the lunar problem (chapter 10) and the three-body stellar problem (chapter 15). Define

It is then readily seen that equations (5.98) and (5.99) take the form where