Closed Orbit of Binary Planet System: Calculate Y-Velocity at (0,1)

[ehrenfest]

Homework Statement

I have two planets of equal mass at points (0,-1) and (0,1) on my axes and I want to find the y-velocity of the planet at (0,1) that will make this a closed orbit. We can uniquely determine the y-velocity of the with this because linear momentum must be zero. Any ideas? What formulas should I use? The center of mass is fixed at the origin.

Homework Equations

The Attempt at a Solution

 

[Dick]

In Newtonian gravity all two body orbits are 'closed' in the sense each body will return periodically to the same center of mass coordinates - if the system is bound. In general relativity no orbit is closed, because the system will radiate gravitational energy. Can you clarify this?

 

[ehrenfest]

This is a Newtonian problem, I think I just want the bodies to return to their initial position after one orbit if you define an orbit for the body as crossing the axis it started with the second time. This is a computational (computer program) problem so I need the parameter to input into the program. Let me play around with it a little. I think maybe the problem is that the orbits are not bounded.

 

[ehrenfest]

OK. I think I fixed it. I guess I was looking for a a unique solution which does not exist as long as the orbit bounded.

 

[Dick]

Good. They won't return unless the total momentum is zero so your computer coordinates are the same as center of mass coordinates.

 

[ehrenfest]

My program automatically calculates the initial velocity of the second mass to make the total momentum zero.

Something I do not understand now is that if I make the velocity too, small the planets diverge as well. It looks like they are crashing into each other first (which the program does not realize) and then fly away in opposite directions? Does that make sense? Maybe it is just a fluke in my program? Maybe this is just a parabolic orbit and they do not really crash into each other?

 

[Dick]

If you make the velocity too small, then they will almost crash into each other. At that point, if your time step size is too large, anything can happen. And it's not physical, it's just an integration error.

 

[ehrenfest]

That makes sense. But I don't know whether it is just a time step issue. Do you think this would not happen in the limit of small time steps? I think it might be that they really would crash into each other physically and as they get closer their velocities increase without bound since we divide by the distance squared. So the problem is that only the gravitational force is considered and not, for example, the electromagnetic force which would become considerable as the planets (and their atoms) collide.

 

[Dick]

Does your computer model include exotic electromagnetic forces? Have you considered what conservation of energy might have to say?