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<html>
<head>
<title>
THREE_BODY_SIMULATION - Planar Three Body Problem Simulation
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
THREE_BODY_SIMULATION <br> Planar Three Body Problem Simulation
</h1>
<hr>
<p>
<b>THREE_BODY_SIMULATION</b>
is a FORTRAN90 program which
simulates the solution of the planar three body problem.
</p>
<p>
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.
</p>
<p>
The force exerted on the 0-th body by the 1st body can be written:
<pre>
F = - m0 m1 ( p0 - p1 ) / |p0 - p1|^3
</pre>
assuming that units have been normalized to that the gravitational
coefficient is 1. Newton's laws of motion can be written:
<pre>
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
</pre>
</p>
<p>
Letting
<pre>
y1 = p0(x)
y2 = p0(y)
y3 = p0'(x)
y4 = p0'(y)
</pre>
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:
<pre>
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
</pre>
and so on.
This first order system can be integrated by a standard ODE solver.
</p>
<p>
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 the ODE solver will not be very accurate after
that point.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>THREE_BODY_SIMULATION</b> is available in
<a href = "../../c_src/three_body_simulation/three_body_simulation.html">a C version</a> and
<a href = "../../cpp_src/three_body_simulation/three_body_simulation.html">a C++ version</a> and
<a href = "../../f77_src/three_body_simulation/three_body_simulation.html">a FORTRAN77 version</a> and
<a href = "../../f_src/three_body_simulation/three_body_simulation.html">a FORTRAN90 version</a> and
<a href = "../../m_src/three_body_simulation/three_body_simulation.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/brownian_motion_simulation/brownian_motion_simulation.html">
BROWNIAN_MOTION_SIMULATION</a>,
a FORTRAN90 program which
simulates Brownian motion in an M-dimensional region.
</p>
<p>
<a href = "../../f_src/duel_simulation/duel_simulation.html">
DUEL_SIMULATION</a>,
a FORTRAN90 program which
simulates N repetitions of a duel between two players, each of
whom has a known firing accuracy.
</p>
<p>
<a href = "../../f_src/fair_dice_simulation/fair_dice_simulation.html">
FAIR_DICE_SIMULATION</a>,
a FORTRAN90 program which
simulates N tosses of 2 dice, making a histogram of the results.
</p>
<p>
<a href = "../../f_src/ising_2d_simulation/ising_2d_simulation.html">
ISING_2D_SIMULATION</a>,
a FORTRAN90 program which
carries out a Monte Carlo simulation of an Ising model.
a 2D array of positive and negative charges,
each of which is likely to flip to be in agreement with neighbors.
</p>
<p>
<a href = "../../f_src/ising_3d_simulation/ising_3d_simulation.html">
ISING_3D_SIMULATION</a>,
a FORTRAN90 library which
carries out a Monte Carlo simulation of a 3D Ising model.
</p>
<p>
<a href = "../../f_src/poisson_simulation/poisson_simulation.html">
POISSON_SIMULATION</a>,
a FORTRAN90 library which
simulates a Poisson process in which events randomly occur with an
average waiting time of Lambda.
</p>
<p>
<a href = "../../f_src/reactor_simulation/reactor_simulation.html">
REACTOR_SIMULATION</a>,
a FORTRAN90 program which
is a simple Monte Carlo simulation of the shielding effect of a slab
of a certain thickness in front of a neutron source. This program was
provided as an example with the book "Numerical Methods and Software."
</p>
<p>
<a href = "../../f_src/rkf45/rkf45.html">
RKF45</a>,
a FORTRAN90 library which
implements the Runge-Kutta-Fehlberg ODE solver.
</p>
<h3 align = "center">
Author:
</h3>
<p>
Original MATLAB version by Dominik Gruntz, Joerg Waldvogel;
FORTRAN90 version by John Burkardt.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Dominik Gruntz, Joerg Waldvogel,<br>
"Orbits in the Planar Three-Body Problem",<br>
Walter Gander, Jiri Hrebicek,<br>
Solving Problems in Scientific Computing using Maple and Matlab,<br>
Springer, 1997,<br>
ISBN: 3-540-61793-0,<br>
LC: Q183.9.G36.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "simple_rkf45.f90">simple_rkf45.f90</a>, the source code.
</li>
<li>
<a href = "simple_rkf45.sh">simple_rkf45.sh</a>,
BASH commands to compile the source code.
</li>
<li>
<a href = "simple_rkf45_output.txt">simple_rkf45_output.txt</a>,
printed output from the run.
</li>
<li>
<a href = "simple_rkf45_t.txt">simple_rkf45_t.txt</a>,
the values of time T at every 0.1 seconds.
</li>
<li>
<a href = "simple_rkf45_y.txt">simple_rkf45_y.txt</a>,
the values of the 12 state variables at every time.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<b>SIMPLE_RKF45</b> simulates the problem by calling the ODE integrator RKF45.
This approach loses accuracy when the bodies come close to colliding, which is
likely to happen often.
<ul>
<li>
<b>MAIN</b> is the main program for SIMPLE_RKF45.
</li>
<li>
<b>SIMPLE_RUN</b> runs the simple three body ODE system.
</li>
<li>
<b>SIMPLE_F</b> returns the right hand side of the three body ode system.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last modified on 04 April 2011.
</i>
<!-- John Burkardt -->
</body>
</html>