# Overlap of Planets in Elliptical Orbit

I'm investigating further into my orbital overlap problem. I've already looked into the overlap ($0°$ angle between the two orbits) of two planets in a circular orbit around the sun. I'm now trying to find a method of doing the same for an elliptical orbit. For the circular orbit I used angular velocity as a function of time to calculate the time at which one orbit is exactly $2\pi$ ahead of the other orbit. How can I do this in an ellipse?

The data I have available:

Orbital Periods, Distance at perihelion and aphelion, Velocity at perihelion and aphelion

• You are in 2d, not 3d, right? How do you describe your orbits? Or in other words, what data do you have on your orbiting bodies to compute the orbit (as a function of time)? You are then looking for all times $t$ at which the angular component of the polar coordinates for two bodies orbiting the same sun is equal, modulo $2\pi$, right? Both of them are to be ellipses in your new statement? – MvG Dec 7 '14 at 11:23
• Yes i'm in 2D. 3D would be way beyond my ability. I have their orbital periods, orbital distance, apogee distance, perigee distance, apogee and perigee speed. Yes, that's what I'm looking for. In circular motion it was w1t=w2t+2π. Yes, they are both ellipses. – eagerlearner96 Dec 7 '14 at 11:30
• Do you have their position at $t=0$? I don't know what exactly “orbital distance” is, but I guess I could compute anything that might be needed from the rest. However, I remember from this question that expressing angle as a function of time will not be possible using a closed form due to the transcendental nature of Kepler's equation. So I'd not be surprised if numerical approximations are the best you can expect for this kind of question. – MvG Dec 7 '14 at 11:52
• You might want to edit your question to include all the details about what data you have. People shouldn't have to sift through the comments to find out what you're asking. – MvG Dec 7 '14 at 12:02
• Okay i'll edit the question with your suggestion. Will it be easier if we assume that at t=0, the planets are overlapping. Orbital distance is basically the distance covered by one orbit. I guess that's not important because in angular terms 2π can be used as the distance covered. Numerical approximation would be great as well. Any help would be very appreciated! – eagerlearner96 Dec 7 '14 at 12:21