Periodic solutions in the commensurable three-body problem

Abstract

Various families of periodic solutions are shown to exist in the three body problem, in which two of the bodies are close to a commensurability in mean motions about the third body, the primary, which is considerably more massive than the other two. The cases considered are

• (a) The non-planar circular restricted problem (in which one of the secondary bodies has zero mass, and the other moves in a fixed circular orbit about the primary).

• (b) The planar non-restricted problem (in which the three bodies move in a plane, and both secondaries have finite mass).

• (c) The planar elliptical restricted problem (in which the three bodies move in a plane, one of the secondary bodies has zero mass, and the other moves in a fixed elliptical orbit about the primary).

The method used is to eliminate all short period terms from the Hamiltonian of the motion by means of a von Zeipel transformation, leaving only the long period terms which are due to the commensurability. Hence only the long period part of the motion is considered, and the variables used differ from the variables describing the full motion by a series of short-period trigonometric terms of the order of the ratio of the mass of the secondaries to that of the primary body. It is shown that solutions of the long-period problem in which the variables remain constant are equivalent to solutions in the full motion in which the bodies periodically return to the same configuration, and these are the types of periodic solution that are shown to exist.

The form of the disturbing function, and hence of the equations of motion, is found up to the fourth powers of the eccentricities and inclination by considering the d'Alembert property. The coefficients of the terms appearing in this expansion are functions of the semi-major axes of the orbits of the secondary bodies. Expressions for these coefficients are not worked out as they are not required.

Lete, n, m be the orbital eccentricity, mean motion and mass of one of the secondary bodies, and lete′, n′, m′ be the corresponding quantities for the other. (The mass of the primary is taken as unity). In cases (a) and (c) we will havem=0. In case (a)e′ will be zero, and in case (c) it will be a constant. Leti be the mutual inclination of the orbits of the secondary bodies. Suppose the commensurability is of the form(p+q) n =pn′, wherep andq are relatively prime integers, and put γ=(p+q) n/n′−p. The families of periodic solutions shown to exist are as follows.

For q=1

No periodic solutions are found withi≠0 in case (a), and none withe′≠0, in case (c). In case (b) periodic solutions are found in whiche=0 (m′/γ),e′=0 (m/γ) for values of γ away from the exact commensurability. As γ approaches zero thene ande′ become 0 (1).

For q≠1

Case (a). Families of periodic solutions bifurcating from the family withe=0, i=0 are shown to exist. Families in whichi=0 ande becomes non-zero exist for all values ofq. Families in whiche=0 andi becomes non-zero exist for even values ofq. Families in whiche andi become non-zero simultaneously exist for odd values ofq.

Case (b). No families are found other than those withe=e′=0.

Case (c). Families are found bifurcating from the familye=e′=0 in whiche ande′ become non-zero simultaneously.

For all these solutions existence is only demonstrated close to the point of bifurcation, where all the variables are small, as the method uses series expansions ine, e′ andi. From the form of the solutions it is clear that the non-zero variables will become large for values of γ away from the bifurcation point.