ENERGY ANALYSIS OF HYPERBOLIC MOTION IN THE RESTRICTED THREE-BODY PROBLEM: APPLICATIONS TO PLANETARY ENCOUNTERS AND JUPITER

Abstract

Abstract: The Restricted Three-Body Problem (RTBP) provides an important theoretical framework for studying the motion of a comparatively small body under the gravitational influence of two massive primary bodies. When the small body approaches a planetary system on an unbound or hyperbolic trajectory, its energy characteristics become particularly important for understanding planetary encounters, escape processes, gravitational scattering, and spacecraft trajectories. This paper investigates the energy behavior of hyperbolic motion within the RTBP, with particular emphasis on encounters involving Jupiter. The analysis considers the relationship between incoming and outgoing trajectories, variations in kinetic and gravitational potential energy, the role of the rotating reference frame, and the influence of the two primary masses. Jupiter is especially suitable for such an investigation because of its large mass, strong gravitational field, and important role in modifying the trajectories of comets, asteroids, and spacecraft. A summary table is included to compare major energy characteristics associated with different stages of a planetary encounter. The study shows that although the restricted model assumes a negligible mass for the third body, the energy exchange associated with the gravitational configuration of the two primaries can produce significant changes in the heliocentric trajectory of the small body. The results have applications in planetary science, celestial mechanics, asteroid dynamics, and gravity-assist mission design. Keywords: Restricted Three-Body Problem, hyperbolic motion, energy analysis, Jupiter, planetary encounters, gravitational scattering, celestial mechanics, gravity assist.