Periodic Planar Three-Body Orbits

Read this notebook ➡️ on my personal website via Observable Notebook Kit.

This notebook collects 2,703 periodic solutions of the planar three-body problem in which three bodies experience mutual gravitational attraction according to Newton's law of universal gravitation, F=Gm1m2r2.F=G{\frac {m_{1}m_{2}}{r^{2}}}.F=Gr2m1​m2​​.

Three-body orbits don't have a closed-form analytical solution and aren't in general periodic, but by reducing the dimensionality of the problem through symmetries and performing a lot of numerical searching, a number of authors have found and published large sets of initial conditions for periodic solutions.

In this notebook we collect and display the orbits, then discuss methods for classification. It's largely a rehashing of an old project, Periodic three-body orbits, though the format of Observable makes it easy and fun to include discussion instead of limiting ourselves to pretty pictures!

Governing equations

The vector form of Newton's law of gravitation for the force on body 2 exerted by body 1 is F2=m2x¨2=−Gm1m2(x2−x1)∣∣x2−x1∣∣3,\mathbf{F}2 = m_2\mathbf{\ddot{x}}{2} = -\frac{Gm_1m_2(\mathbf{x}_2 - \mathbf{x}_1)}{|\