Note on principles of the three-body problem in gravitation and Lagrangian points

Abstract

The three-body problem in gravitation is one of the oldest problems in celestial mechanics, having beenstudied since the end of the 18th century. Although there is no general closed-form solution to this problem,its analysis based on simplifications– as done by Lagrange and Euler– raises results of great importance formodern-day space exploration. This relevance is corroborated with the elucidation of stable points, known asLagrangian points, in systems comprised by an object with a considerably smaller mass (a satellite, for example)and two bodies with much greater mass (such as the Sun and the Earth). This report presents a review of thetheory of the three-body problem, a topic often covered without due depth in books on orbital mechanics. Theequations used to determine the location of Lagrange points are found. Computational methods, MATLABand POV-Ray codes, are also used to draw gravitational equipotential curves and forbidden regions definedby the Jacobi constant, both in the context of the restricted three-body problem. This provides a visualizationnot generally available in the existing bibliography on the subject, so important for the space sector and thedevelopment of new projects.