Restricted Three-Body Problem
J. C. Sprott
Department of Physics, University of Wisconsin,
Madison,
WI 53706, USA
September 13, 2007
The three-body problem (a planet orbiting a pair of stars, for example)
is one of the oldest problems in all of mathematical physics. It has
never been solved in the usual sense of solving a mathematical problem,
namely finding an equation to predict where the planet will be at some
future time for a given set of initial conditions. The reason is that
the motion is chaotic and exhibits sensitive dependence on initial
conditions. The problem can, of course, be solved numerically, and the
equations are not difficult. Shown here is the solution for a highly
restricted version of the 3-body problem in which the two point-like
stars are fixed and equally massive relative to the planet, which
orbits the stars in a plane. Since the orbit typically approaches
arbitrarily close to the stars, to make the problem numerically
tractable, the stars are given a small diameters (5% of their
separation),
inside of which the gravitational force varies linearly with the
distance from their centers rather than as the inverse square. The
calculations were done in PowerBASIC
whose source and object
code are available for inspection. A high resolution plot of the
resulting orbit is shown below:
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