Zero-Velocity Surface in the General Three-Body-Problem

Abstract

The zero-velocity surfaces of the general planar three-body problem are constructed in form space, i.e., the factor space of the configuration space by transfer and rotation. Such a space is the space of congruent triangles, and the sphere in this space is similar triangles. The integral of energy in form space gives the equation of a zero-velocity surface. These surfaces can also be obtained based on the Sundman inequality. Such surfaces separate areas of possible motion from areas where motion is impossible. Without loss of generality, we can assume that the constant energy is −1/2 and the sought surfaces depend only on the magnitude of the angular momentum of the problem, J. Depending on this value, five topologically different types of surfaces can be distinguished. For small J, the surface consists of two separate surfaces, internal and external ones, motion is possible only between them. With J increasing the inner surface increases, the outer surface decreases, the surfaces first have a common point at some value of J, with a further increase in J, their topological type changes and finally the zero-velocity surface splits into three nonintersecting surfaces, and motion is possible only inside them. Examples of the corresponding surfaces are given for each of these types, their cross sections in the plane xy and in the plane xz and the surfaces themselves are constructed, and their properties are studied.