I am trying to create a differential equation with which I am can numerically solve to plot the orbit of an asteroid around Jupiter so far I have assumed the mass of jupiter is 0.001 of the mass of the sun. I am also assuming that the asteroids are massless so that they possess no gravitational force on Jupiter and Jupiter's orbits are circular.
Then, $GMm/r^2=mv_y^2/r$, thus $v_y=\sqrt{GM/r}$, where $v_y$ is the y component of the asteroids velocity, $r$ is the semi-axis diameter of the orbit. We can also write the total energy of the asteroid to be $E=1/2m(GM/r)-GmM/r=-GMm/2r$, which after some rearranging yields $r=-GMm/2E.$
These are all the equations I have thus far, basically I want to be about to plot the semi-axis diameter changing with respect to time. I am struggling with coming up with a differential equation with respect to time that I can integrate numerically to model this scenario. Any help would be much appreciated.
I have a running program using Grapher on the Macintosh. I'll put up a screenshot if you like. Basically I am assuming the two primary bodies, Jupiter and Sun are in circular orbit around each other and that the problem is two dimensional. The coordinate system is chosen so that the primary bodies are fixed on the X axis and the Asteroid's path is shown . The equations are in dimensionless units, the mass ratio is known mu1 , 1-mu1 (mu2) this is the basic setup of the Circular Restricted Three Body Problem .