1
$\begingroup$

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.

$\endgroup$

1 Answer 1

1
$\begingroup$

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 .

$\endgroup$
4
  • $\begingroup$ Maybe I should to try to preface with my goal. I am trying to show the effect of mean motion resonance on the distribution of asteroids at various distances from the sun demonstrating the existence of Kirkwood Gaps. I have already calculated the distances r such that these gaps occur due to mean motion resonance using Kepler's third law. Can the setup you have effectively visually demonstrate these gaps at the various resonance ratios? $\endgroup$
    – user75514
    Commented Nov 24, 2013 at 22:54
  • $\begingroup$ No , no, it's a toy program. It should nonetheless be possible to write a program for the purpose you describe. I'm looking at " The origin of the Kirkwood gaps - A mapping for asteroidal motion near the 3/1 commensurability " by Jack Wisdom. $\endgroup$
    – Alan
    Commented Nov 24, 2013 at 23:59
  • $\begingroup$ I have seen that paper and some others that are similar. Most treatments of the problem use a form of the planar elliptic restricted three body problem, but a simplification can definitely be achieved if circular orbits are used. $\endgroup$
    – user75514
    Commented Nov 25, 2013 at 0:27
  • $\begingroup$ The equations for the elliptic restricted problem are given in Victor Szebehely's "Theory Of Orbits" Chapter 10, section 3. This isn't exactly a modern reference, but it might be worth a look. $\endgroup$
    – Alan
    Commented Nov 25, 2013 at 1:02

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .