# How come planets with different masses can orbit at the same velocity at the same altitudes?

Angular momentum is equal to r × p and angular momentum is also what gives planets with lower orbits a higher speed (because angular momentum is conserved). So as r decreases either m or v (p=mv) has to increase and as the mass can’t change the velocity has to increase. This I can understand, but the velocity of an orbit is the same for all planets at the same altitude. This doesn’t make sense to me as I would think that the velocity wouldn’t be as high if the mass is bigger, but the planets would really orbit at the same velocity, right? What am I missing here? How come the velocity can be the same even though the masses are different?

## 2 Answers

That the velocity of an orbiting body is independent of the orbiting body's mass is independent of mass is a consequence of Kepler's laws. Kepler's laws however are only approximately correct. Newtonian mechanics says otherwise: It says that ignoring the influences of other planets, the angular velocity of a planet orbiting the Sun is $$\sqrt{G(M_{\text{sun}} + M_{\text{planet}})/R^3}$$. For our solar system, even the largest planet is less than 1/1000th of the mass of the Sun. This means Kepler's laws are good to two or three decimal places. Beyond the third decimal place, Kepler's laws are not so good.

A specific example: The Moon's mass is about 0.0123 times that of the Earth. This means that a tiny object at the Moon's position would orbit the Earth a bit slower than does our Moon.

• One way to look at it is that in a binary system (one object orbiting another), the two objects orbit around their common center of mass. The centripetal acceleration depends on the square of the orbital velocity divided by the distance to the center of the orbit, whereas the gravitational acceleration depends inversely on the squared distance between the two objects. – S. McGrew Feb 14 at 16:29

There is no law that states that the angular momentum of two different planets has to be the same. The velocity of a planet that is on a certain orbit is independent of it's mass, as you can see in Kepler's Laws that don't feature the mass of the planets anywhere. That's because the gravitational force acting on the planet scales linearly with the mass in $$\vec{F} = G\cdot \frac{m_{Planet}M_{Star}}{r^2}$$which then dictates the acceleration of the planet through Newtons $$\vec{F}=m_{Planet}\cdot\vec{a}$$. The mass appears on both sides and can be cancelled out.