Families of Newtonian Periodic Planar Three-Body Orbits

The three-body problem is concerned with solutions for the orbits of three masses under their mutual gravitation interactions. Periodic solutions are very hard to find. A set of 695 families of Newtonian periodic planar three-body orbits with equal mass and zero angular momentum [1] are shown, labeled by . These are divided into seven classes according to their geometric and algebraic symmetries. The solutions are presented 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.

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DETAILS

Initial conditions are given by
,
and
,
,
with
and .
The labels on the graphic are initial conditions written as:
{"I.A1", 1.002427797, 0.0041695061, 0.3489048974, 0.53063051, 6.3490473929}.
The first entry is the class, the second and third are the values of and , the fourth and the fifth are and , and the last is the period.
References
[1] 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.
[2] 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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