Gravitational Force: Determining Relative Accelerations

[sheriefer]

(This isn't a homework problem, just something I've been thinking about.) Suppose we have two planets isolated in space separated by a distance "D." They exert a gravitational force on each other which changes as they get closer. If we were in the accelerating reference frame of one of the planets, how would one determine the acceleration of the other planet? Of course the acceleration is not constant and would be dependent on their distance. Relative accelerations don't add like relative velocities in the Galilean transformations, correct? Can you satisfy my curiosity?

 

[Sciurus]

Hi, sheriefer.

In an accelerating reference frame, you can add a pseudo-force to everything to compensate for your acceleration -- otherwise Newton's equations are not valid. Essentially, if you wanted to work in the planet's reference frame, you would have to add a pseudo accelerating force to the other planet to accelerate it toward you at the rate you were accelerating toward it, then also add the gravitational acceleration.

If you have an accelerating object from the point of view of an inertial reference frame, then look at it from an accelerating reference frame, let's say accelerating in the opposite direction, the acceleration of the frame and object would add for the apparent acceleration in the accelerating reference frame, if that's what you mean.