three_body_ode, a Python code which defines a set of ordinary differential equations (ODE) which simulates the behavior of three planets, constrained to lie in a plane, and moving under the influence of gravity, by Walter Gander and Jiri Hrebicek.

Three bodies, regarded as point masses, are constrained to lie in a plane. The masses of each body are given, as are the positions and velocities at a starting time T = 0. The bodies move in accordance with the gravitational force between them.

The force exerted on the 0-th body by the 1st body can be written:

        F = - m0 m1 ( p0 - p1 ) / |p0 - p1|^3
assuming that units have been normalized to that the gravitational coefficient is 1. Newton's laws of motion can be written:
        m0 p0'' = - m0 m1 ( p0 - p1 ) / |p0 - p1|^3 
                  - m0 m2 ( p0 - p2 ) / |p0 - p2|^3
        m1 p1'' = - m1 m0 ( p1 - p0 ) / |p1 - p0|^3 
                  - m1 m2 ( p1 - p2 ) / |p1 - p2|^3
        m2 p2'' = - m2 m0 ( p2 - p0 ) / |p2 - p0|^3 
                  - m2 m1 ( p2 - p1 ) / |p2 - p1|^3


        y1 = p0(x)
        y2 = p0(y)
        y3 = p0'(x)
        y4 = p0'(y)
and using similar definitions for p1 and p2, the 3 second order vector equations can be rewritten as 12 first order equations. In particular, the first four are:
        y1' = y3
        y2' = y4
        y3' = - m1 ( y1 - y5  ) / |(y1,y2) - (y5,y6) |^3 
              - m2 ( y1 - y9  ) / |(y1,y2) - (y9,y10)|^3
        y4' = - m1 ( y2 - y6  ) / |(y1,y2) - (y5,y6) |^3 
              - m2 ( y2 - y10 ) / |(y1,y2) - (y9,y10)|^3
and so on. This first order system can be integrated by a standard ODE solver.

Note that when any two bodies come close together, the solution changes very rapidly, and very small steps must be taken by the ODE solver. For this system, the first near collision occurs around T=15.8299, and the results will not be very accurate after that point.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


three_body_ode is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and an Octave version and a Python version.

Related Data and codes:

python_ode, Python codes which sets up various systems of ordinary differential equations (ODE).


Original MATLAB version by Dominik Gruntz, Joerg Waldvogel.


  1. Dominik Gruntz, Joerg Waldvogel,
    Orbits in the Planar Three-Body Problem,
    Walter Gander, Jiri Hrebicek, editors,
    Solving Problems in Scientific Computing using Maple and Matlab,
    Springer, 1997,
    ISBN: 3-540-61793-0,
    LC: Q183.9.G36.

Source Code:

Last modified on 11 November 2020.