two_body_ode, an Octave code 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.
The information on this web page is distributed under the MIT license.
two_body_ode is available in a MATLAB version and an Octave version and a Python version.
octave_ode, an Octave code which sets up various ordinary differential equations (ODE).