Acceleration of two bodies in circular orbit provided by own gravity

[Ry122]

If two bodies are moving about a common point, with one orbiting the other, in a completely circular orbit, from knowing just their periods (which are equal) and the graviational constants of each body times mass, how would you determine the radius of each of their circular orbits?

You can get the velocity as a function of the period and radius.
And for circular orbits vo=sqrt(GM^2/ro)

But since the inner body's position is constantly changing you can't straight forwardly apply the above equation to the outer body can you?

And for the inner body it would be in circular rotation about a point below its surface, so I'm not sure how you would apply the equation in that situation.

 

[Ry122]

I came up with this idea. Would it work?




centripetal force on outer body = m1v^2/r

Force due to gravity = m1*m2/dist^2


force due to gravity will be equal to the centripetal force since it's in circular motion so




m1*m2/dist^2 = m1v^2/r



model dist between object one and two as a function of their masses (with centre of mass equation) to reduce to a one variable function and solve for r

 

[Ken G]

Yes, that is just what you want to do. After you've solved it that way, look up how the concept of "reduced mass" can be used to replace a problem with two masses with an effective problem that is a lot simpler because it just has one effective mass going around in a circle with a radius of the distance between the real masses, rather than the actual outer mass going around the center-of-mass as you correctly describe the situation. Replacing the real problem with that effective problem does all the tricky arithmetic for you, but you should do it your way first to verify that it works.