Restricted Problem of Three Bodies.

Abstract

The equations of motion of a sputnik in the gravitational field of two other bodies of finite mass have been integrated numerically, using a modified Runge-Kutta-Gill procedure. If the masses are m1 and 1n2, and ~= (n~1-n~2)/ (m1+rn2), K is the Jacobi constant and T=K-72, and if F and F are the Thiele regularizing variables, then a periodic solution may be represented by a plot of F vs F for given values of T and 7. A continuous family of such solutions forms a surface in (T, F, F, 7) space, which may be termed an eigensurface and which is characteristic of a class in the terminology of Stro~mgren. More than 9000 periodic solutions have been determined, and the evolution of the simple symmetric classes (a), (f), and (n) has been traced over the whole mass-ratio range -1 ~ 7 ~ 1. The class (g-f) and the new symmetric classes (p), (~), and (CL-~) have been studied over the range - 1 ~ 7 ~0.93. Asymptotic periodic orl~its have been calculated as a function of mass ratio, and found to appear and disappear in pairs. Further, some new asymmetric classes have been discovered. For a given mass ratio 7, and E=0, the section of the eigensurface is a plot of T vs F and may consist of more than one curve. As 7 varies, these curves may move toward each other, touch, and split into two other curves which then move apart. When the eigensurface touches the zero-velocity surface, a change of symmetry (reversal of velocity) occurs, as from g to f and from CL to ~. The g class at ~= - 1 consists of at least two branches which cross, one with eccentricity not zero and the other with eccentricity zero. As 7 increases, these two brand~es split apart sideways. Since the curves at ~= -1 do not appear to possess any single quantity which is invariant, it is unlikely that the curves for 7#- 1 have an invariant, either.