Planetary capture and escape in the planar four-body problem

The dynamics of a fourth body in the sun-planet-satellite system is used to study the planetary capture and escape. An expression to describe the energy exchange between the satellite and the fourth body is derived when the fourth body is in the vicinity of the planet. The analytic result can be obtained when the fourth body is within the sphere of influence of the satellite. The analytic expression gives an intuitive description of the energy exchange. It indicates that the energy change depends on the shape and direction of the fourth body's hyperbolic orbit with respect to the satellite. Based on the energy exchange expression, we can identify the bodies on the heliocentric orbits that can be captured by the planet after the satellite flyby. Similarly, the planet-centered bodies that will escape the planet after satellite flyby are identified. Several numerical examples are given to illustrate the capture and escape processes.