THREE_BODY_SIMULATION
Planar Three Body Problem Simulation

THREE_BODY_SIMULATION is a MATLAB library which simulates the solution of the planar three body problem.

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
      

  
        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
      

Letting

        y1 = p0(x)
        y2 = p0(y)
        y3 = p0'(x)
        y4 = p0'(y)
      

        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
      

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 produced by MATLAB's ode113 will not be very accurate after that point.

Licensing:

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

Languages:

THREE_BODY_SIMULATION is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

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Author:

Dominik Gruntz, Joerg Waldvogel

Reference:

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

Source Code:

SIMPLE simulates the problem by immediately calling an ODE integrator. This approach loses accuracy when the bodies come close to colliding, which is likely to happen often.

• simple_ode113.m, is a MATLAB script which simulates the problem by calling ODE113.
• trajectory1.png, an image of the first third of the trajectory.
• trajectory2.png, an image of the second third of the trajectory.
• trajectory3.png, an image of the last third of the trajectory.
• trajectory.png, an image of the entire trajectory.
• simple_rkf45.m, is a MATLAB script which simulates the problem by calling RKF45.
• simple_f.m, evaluates the right hand side of the ODE system.
• timestamp.m, prints the current YMDHMS date as a time stamp.

THREEBP eliminates some variables by working in the center of mass coordinate system, and uses an approach suggested by Levi-Civita in which a new time scale is devised. The transformed system of ODE's can be solved with greater accuracy.

• data3.mat, a MAT data file containing the computational results T1 and Y1 for the simple approach, and S and Z for the transformed approach.
• f.m, evaluates the right hand side of the simple ODE system.
• fig_data.m, creates plots from the book, using the data file.
• gen_data.m, creates the data file, by solving the simple and transformed systems.
• proc.m, computes the regularized Hamiltonian function K.
• threebp.m, evaluates the right hand side of the transformed ODE system.

You can go up one level to the MATLAB source codes.