Orbital velocity of a circular planet is a R , where a is the centripetal acceleration, and R is radius of the planet. With v_1 as the tangential velocity of the rotating planet at the equator. On the non rotating body, suppose that the orbital velocity is v_0 , and, for an object launched on the rotating body's "equator", that the orbital velocity will be in the form of v_1 + v_2 (the body and the object both going counterclockwise). Now, I half-hypothesized v_0=v_1+v_2=sqrt(aR) and v_2 =sqrt(a′R) where a ′ is the "true" rotating body's acceleration, able to be calculated from the rotating frame of reference as a′=a−(v_1^2)/R

Brenton Dixon 2022-07-22 Answered
Orbital velocity of a circular planet is a R , where a is the centripetal acceleration, and R is radius of the planet. With v 1 as the tangential velocity of the rotating planet at the equator.
On the non rotating body, suppose that the orbital velocity is v 0 , and, for an object launched on the rotating body's "equator", that the orbital velocity will be in the form of v 1 + v 2 (the body and the object both going counterclockwise). Now, I half-hypothesized v 0 = v 1 + v 2 = a R and v 2 = a R where a is the "true" rotating body's acceleration, able to be calculated from the rotating frame of reference as
a = a v 1 2 R
The logic was that from rotating body's reference frame, the object would be traveling at v 2 , less than v 0 because of the centrifugal force, so v 2 has to be the orbital velocity if the gravity was "weakened" by centrifugal force.
Tried to solve for a and comparing it to the value, got from rotating frame of reference, ending up with
a = a ( v 1 ( v 0 + v 2 ) R )
Something's not right, and if I had to choose, I would guess the v 0 = v 1 + v 2 , that acceleration is not the same on those two planets, but I don't know how it would change, or why.
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Answers (1)

edgarovhg
Answered 2022-07-23 Author has 12 answers
You had some problems of frames of reference when making your hypothesis. Say that the non rotating planet is A and the other is B. The orbit velocity is v. So an object orbiting A have a velocity with respect to the center of A and to the surface is the same, and is v 0 = v. For B, say the object's velocity with respect to B's surface is v 1 , and B's surface with respect to B's center(the velocity of B's rotation at its equator), v 2 . Then the object's velocity with respect to B's center is v 1 + v 2 .
And since A and B are otherwise identical, v 0 = v 1 + v 2 = v.
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