Restricted three body problems and lunar capture

An examination is made of the geometry of the two bottleneck openings formed by the zero velocity surfaces, at the L[lowered 1] and L[lowered 2] libration points on either side of the smaller primary mass, of the circular restricted three body problem. The size of an opening is interpreted in terms of the probability of the entrance of the massless particle into the vicinity of the smaller primary mass, and the energy dissipation necessary to then close the openings and reduce escape probability to zero. The derivations are applicable to any circular problem, both two and three dimensional, but the results are examined in detail for that problem where the primary masses model the earth and sun ([mu]=3 [multiplication dot] 10[raised -6]) . The bottleneck openings are found to be elliptical in shape and lie on a spherical surface passing through the L[lowered 1] and L[lowered 2] libration points. The eccentricity of the elliptical openings varies only from 0.5 (for [mu]=0) for to 0.632 (for [mu]=0.5). The area of an opening is a linear function of the Jacobian constant. For [mu]=3 [multiplication dot] 10[raised -6], the change in energy per unit mass is equal to the change in the Jacobian constant x 4.5[multiplication dot]10[raised 8] Joules/kg. A set of graphs relates change in Jacobian constant to escape probability. Equations of motion are derived in detail for the most general restricted problem (elliptical three-dimensional) in a dimensionless rotating coordinate system. This is done by a series of transformations from the familiar dimensional inertial system, to a rotating dimensional system, and then to the dimensionless rotating system. These equations are then simplified for the less general circular and two-dimensional problems. Fourth order trajectory propagation equations are derived for the elliptical three-dimensional restricted problem. Listings and output examples of three computer programs are given which (l) perform position and velocity transformations between coordinate systems, (2) propagate and automatically plot a trajectory in the elliptical three-dimensional problem, and (3) automatically plot a zero velocity surface of a circular two-dimensional problem.Physics, Department o