# TWO_BODY_SIMULATION Planar Two Body Problem Simulation

TWO_BODY_SIMULATION is a MATLAB library which simulates the solution of the planar two body problem.

Two 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. One body is assume to be much more massive than the other. Therefore, the common motion of the two bodies about their center of mass can be approximated by assuming that the large body remains fixed.

Under these assumptions, Newton's equations for (x(t),y(t)), the positition of the lighter body with respect to the heavy body at (0,0), can be written as:

r(t) = sqrt ( x(t)^2 + y(t)^2 )
x''(t) = - x(t) / r^3
y''(t) = - y(t) / r^3

These two second order equations can be rewritten as a system of four first order equations using the variable u = [ x(t), x'(t), y(t), y'(t) ], resulting in the equations:

r = sqrt ( u(1)^2 + u(3)^2 )
u'(1) =   u(2)
u'(2) = - u(1) / r^3
u'(3) =   u(4)
u'(4) = - u(3) / r^3

By specifying some initial condition for u, the system can then be integrated in time using a standard ODE solver.

### Languages:

TWO_BODY_SIMULATION is available in a FORTRAN90 version and a MATLAB version.

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### Reference:

1. Charles VanLoan,
Introduction to Scientific Computing,
Prentice Hall, 1997,
ISBN: 0-13-125444-8,
LC: QA76.9.M35V375.

### Source Code:

• kepler.m, evaluates the right hand side of the ODE system.
• timestamp.m, prints the current YMDHMS date as a time stamp.

### Examples and Tests:

INITIAL_ORBIT simulates the problem over approximately two orbits.

ORBITAL_DECAY computes about twenty successive orbits, showing how the orbit gradually decays to a more elliptical form.

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