Title:

The structure of symmetric periodic solutions of the restricted threebody problem

This thesis is concerned with the structure of symmetric periodic solutions of the restricted threebody problem, in the cases of planar and threedimensional motion of the massless third particle, and for circular and elliptic motion of the two massive primaries. Particular emphasis is placed on the relationships existing between families of periodic orbits in the different versions of the problem, and on numerical methods of continuing periodic orbits in the simplest version of the restricted problem, the planar circular case, into the more general versions of the problem. The restricted threebody problem is introduced in Chapter 1; applications to actual physical systems are discussed, and a derivation of the equations of motion in their usual form is offered. The Jacobi integral and the Lagrange equilibrium solutions are also obtained for later reference. Chapter 2 deals with periodic orbits and their significance from both the theoretical and practical points of view; symmetry properties and periodicity conditions are discussed in terms of the two possible types of "mirror configuration" in the restricted problem. The existence of monoparametric sets or families of periodic orbits in both the circular and elliptic versions of the restricted problem, for a fixed value of the mass parameter of the primaries, is discussed in this chapter, and Stromgren's and Henon's explorations of the planar circular problem are briefly reviewed, A firstorder treatment of variations in a periodic orbit resulting from small changes in the initial conditions is given in Chapter 3and this is used to establish the usual linear stability criterion. Variations resulting from small changes in the parameters mu (mass parameter) and e (orbital eccentricity) of the primaries are also introduced for use in subsequent chapters. The bifurcation, or branching, of families of periodic orbits (for fixed mu) is discussed in general terms, with a more detailed analysis in the particular case of "vertical" bifurcations, that is, bifurcations of planar with threedimensional periodic orbits, Numerical techniques for establishing families of threedimensional symmetric periodic orbits are described in Chapter 4, with particular reference to the numerical determination of "vertical branches", or families of threedimensional orbits generated from vertical bifurcations. The results of a numerical investigation of the vertical branches of Stromgren's family f, in the SunJupiter case (mu = 0.00095) of the circular problem, are given in Chapter 5 to illustrate the foregoing discussion. Examples corresponding to each kind of vertical bifurcation, and all possible symmetry classes, are given. The results confirm the prediction that families of threedimensional orbits bifurcating vertically from the plane occur in pairs, and the predicted relationship between symmetry properties and multiplicity is observed. With a single exception, the vertical branches investigated are found to connect the families of retrograde and of direct satellite orbits about the less massive primary (Jupiter). The continuation of periodic orbits of the circular restricted problem into the elliptic case is discussed in Chapter 6, Threedimensional as well as planar periodic orbits are considered, and it is shown that for any commensurability in the period with that of the primaries, two families of periodic orbits of the elliptic problem can always be obtained from a commensurable orbit of the circular problem. To illustrate the discussion, numerical examples are presented for each type of commensurability, including orbits of both simple and double symmetry. Chapter 7 deals with the numerical determination of series of vertical bifurcation orbits, for which the vertical stability index aV is kept constant and the mass parameter mu is allowed to vary. The importance of this type of series (from any orbit of which may be generated either one or two entire families of threedimensional periodic orbits), with regard to the structure of symmetric periodic solutions, is discussed, and numerical examples in both the circular and elliptic cases of the restricted problem are offered, together with an instance of the continuation of planar periodic orbits of the elliptic restricted problem into three dimensions. Chapter 8 presents the results of a preliminary investigation into the phenomenon of threedimensional bifurcation; that is, the intersection of two families of periodic orbits in three dimensions. Numerical examples given in this chapter include a family of threedimensional orbits which appears to terminate in a planar orbit with consecutive collisions, and a family originating from a quadruple bifurcation in three dimensions.
