Collisionless Periodic Orbits in the Free-Fall Three-Body Problem

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The three-body problem considers the orbits of three masses under their mutual gravitational interactions. Periodic solutions are very hard to find and only a small set of solutions are known. A set of 316 families of collisionless periodic orbits in the free-fall three-body problem [1] are shown, labeled by . The orbits are shown in two-dimensional space and on a "shape sphere," based on Jacobi three-body coordinates. These can be divided into seven classes according to their geometric and algebraic symmetries. The red dots on the shape sphere are related to the three two-body collision points.

Contributed by: Enrique Zeleny (November 2019)
Open content licensed under CC BY-NC-SA


Initial conditions are





with .

The labels give the initial conditions, for example:

{F1(1, 1, 1), 0.0207067154, 0.3133550361, 2.1740969264}.

The first entry is the class (the numbers are the masses), the second and the third are the values of and , and the last is the period.


[1] X. Li and S. Liao, "Collisionless Periodic Orbits in the Free-Fall Three-Body Problem."

[2] X. Li and S. Liao, "More than Six Hundred New Families of Newtonian Periodic Planar Collisionless Three-Body Orbits," Science China Physics, Mechanics & Astronomy, 60(12), 2017 129511. doi:10.1007/s11433-017-9078-5.

[3] M. Šuvakov and V. Dmitrašinović, "Three Classes of Newtonian Three-Body Planar Periodic Orbits," Physical Review Letters, 110(11), 2013 114301. doi:10.1103/PhysRevLett.110.114301.


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