On retrograde and prograde orbits

This page is served as a supplement of my joint work with Yu-Chu Lin [1] on retrograde and prograde orbits of the three-body problem. In [1] we extend major results in [2] and showed many examples of retrograde and prograde orbits.

Roughly speaking, a retrograde orbit of the planar three-body problem is a relative periodic solution with two adjacent masses revolving around each other in one direction while their mass center revolves around the third mass in the other direction. The orbit is said to be prograde or direct if both revolutions follow the same direction. Let T>0 and phi in [0,2pi) be fixed, and consider the rotating frame which rotates the inertia frame about the origin with angular velocity phi/T. In [2] we proved the existence of action-minimizing retrograde orbits which are T-periodic on this rotation frame for a large class of masses and for a continuum of phi. In [1] we refine the estimates in there so that it works for a much wider range of masses and angles. Apart from that, we also provide some quantitative estimates for admissible masses and mutual distances, and show miscellaneous examples of action-minimizing retrograde orbits as well as some prograde and retrograde solutions with additional symmetries.

There is a total of 138 examples in the paper, their initial data are given but some of them require initial data with higher precision in order to produce satisfactory numerical graphics. We put these initial data in the file

retr-prog-initial-data.txt. You may reproduce most of the figures in [1] by using these initial data. This file contains initial data and action values (from t=0 to t=1) of retrograde and symmetric prograde orbits. Unit of rotation angle is pi, and initial position (x,y,z), initial velocity (u,v,w) are
    m_1 x_1 y_1 z_1   u_1 v_1 w_1
    m_2 x_2 y_2 z_2   u_2 v_2 w_2
    m_3 x_3 y_3 z_3   u_3 v_3 w_3

The paper [1] includes very few examples with masses m_1, m_2, m_3 satisfying

m_1 < m_3 = 1 < m_2 < 2 or m_3 = 1 < m_1 < m_2 < 2 because the deformation of figures for action minimizers is usually not significant. Many examples of this type are included in retr-prog-initial-data.txt. The above file contains 127 examples. It includes a fairly comprehensive catalog of action minimizers with phi = 0.6pi, 1.1pi. Another 50 examples are in retr-prog-initial-data.txt. The precision of initial data in these two files is in general lower than retr-prog-initial-data.txt.

References

K.-C.Chen and Y.-C.Lin, On action-minimizing retrograde and prograde orbits of the three-nody problem. Preprint 2008. [PDF]
K.-C.Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses. Annals of Math. 167 (2008), 325-348.

Last modified: October 6, 2008