Periodic Planar Collisionless Three-Body Orbits with Unequal Masses

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The three-body problem considers the orbits of three masses under their mutual gravitation interactions. Periodic solutions are very hard to find and only a small set of solutions is known. A set of 1349 families of periodic planar collisionless three-body orbits with unequal mass [1] is shown, labeled by . These orbits are divided into five classes according to their geometric and algebraic symmetries. The solutions are represented in two dimensions and on a "shape sphere" based on Jacobi three-body coordinates. 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 given by







and .

The labels give the initial conditions written as:

{"I.A1(0.5)", 0.2869236336, 0.0791847624, 4.176129219}.

The first entry is the class (the numerical quantity is the mass ); the second and the third quantities are and ; and the last is the period.


[1] X. Li, Y. Jing and S. Liao, "Over a Thousand New Periodic Orbits of a Planar Three-Body System with Unequal Masses," Publications of the Astronomical Society of Japan, 70(4), 2018 64. doi:10.1093/pasj/psy057.

[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, 2017 129511. doi:10.1007/s11433-017-9078-5.

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

[4] 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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