Phase-Space Analysis of a Test Particle Dynamics in TRAPPIST-1 Using a Gravitational Model

Abstract: Using the Circular Restricted Three-Body Problem (CRTBP) framework, this study examines the dynamics of a small test particle in the TRAPPIST-1 system, concentrating on the gravitational effect between the star and the planet TRAPPIST-1e. The equations of motion are represented using dimensionless coordinates in a rotating reference frame in both Lagrangian and Hamiltonian forms, beginning with the general Newtonian N-body equations. The Runge–Kutta–Fehlberg (RKF45) method and root-finding techniques are used to solve the nonlinear equations in extensive numerical simulations carried out in MATLAB in order to determine equilibrium points. The intricate nonlinear nature of the system and the sensitivity of motion to initial circumstances are further highlighted by phase-plane analysis. The findings show how resonance structures, stability regions, and orbital confinement in compact exoplanetary systems can be better understood by combining classical CRTBP theory with contemporary numerical integration methods. As a result, the TRAPPIST-1 system functions as a representative dynamical laboratory for researching resonant motion and nonlinear gravitational interactions in densely populated planetary structures.