I wanted to look up the formula for orbital speed for a circular orbit on Wikipedia, and I found 2 formulas:
All bounded orbits where the gravity of a central body dominates are elliptical in nature. A special case of this is the circular orbit, which is an ellipse of zero eccentricity. The formula for the velocity of a body in a circular orbit (orbital speed) at distance r from the centre of gravity of mass M is $ v = \sqrt{\dfrac{GM}{r}}$.
I found this weird, because this leaves out the mass of the body orbiting $M$. I would think that this would have some effect on the orbital speed. I figured that perhaps the radius of the orbit indicates the mass of the orbiting body through some physical relation, but I wasn't sure so I continued looked up stuff on Wikipedia and I found:
The relative velocity is constant: $v = \sqrt{\dfrac{G(M+m)}{r}}$.
This was the equation I intially expected to see, but now I'm confused because these are 2 formulas for the same situation, right? Or does the word 'relative' indicate a difference?
Best Answer
I think these are two separate questions that should be approached separately.
1) "Why isn't $m$ in in the first equation?" The mass of a body does change the force acting on it. But the mass of a body also changes its acceleration. If you increase the mass of an object it feels a larger force, but it's also harder to move. The equation for gravitational force is $F = \displaystyle GmM/{r^2}$, while the equation for acceleration is $a = F/m$. Glue to two together and you get $a = GM/r^2$.
2) "Why is $m$ in the second equation?" Think about the moon and the earth. The earth is pulling on the moon, but the moon is also pulling on the earth! The two bodies actually orbit around their common center of mass. This is important for the relative velocity: we need to add how fast the earth is orbiting to how fast the moon is orbiting. That's why you have the $m$ term.