Three Classes of Newtonian Three-Body Planar Periodic Orbits

We present the results of a numerical search for periodic orbits of three equal masses moving in a plane under the influence of Newtonian gravity, with zero angular momentum. A topological method is used to classify periodic three-body orbits into families, which fall into four classes, with all three previously known families belonging to one class. The classes are defined by the orbits’ geometric and algebraic symmetries. In each class we present a few orbits’ initial conditions, 15 in all; 13 of these correspond to distinct orbits.

Figure 1

The (translucent) shape-space sphere, with its back side also visible here. Three two-body collision points (bold red circles)—punctures in the sphere—lie on the equator. (a) The solid black line encircling the shape sphere twice is the figure-8 orbit. (b) Class I.A butterfly I orbit (I.A.1). Note the two reflection symmetry axes. (c) Class I.B moth I orbit (I.B.1) on the shape-space sphere. Note the two reflection symmetry axes. (d) Class II.B yarn orbit (II.B.1) on the shape-space sphere. Note the single-point reflection symmetry. (e) Class II.C yin-yang I orbit (II.C.2) on the shape-space sphere. Note the single-point reflection symmetry. (f) An illustration of a real space orbit, the yin-yang II orbit (II.C.3a).