Quadrupling radius of satellite in orbit

[widderjoos]

Homework Statement

If a satellite in orbit changes it's orbiting radius to 4 times its initial one, how does it's velocity change?

I get different answers by using Newton's Law of gravitation and conservation of angular momentum.

Homework Equations

$F = \frac{G M m}{R^2}$

$a_c = \frac{v^2}{R}$

$L=\mathbf{r} \times \mathbf{p}$ is conserved.

The Attempt at a Solution

one way uses Newton's gravitation equation to get v=$\sqrt{\frac{GM}{R}}$ so that we see that quadrupling R halves the speed. However, using conservation of angular momentum $m v_i R_i = m v_f R_f$ we see that by setting the final radius to 4 times it's initial, the final speed decreases by a factor of 4. Why don't these results agree?

 

[widderjoos]

It was a problem on a practice MCAT thing and the answer used Newton's gravity equation. (I guess they were assuming uniform circular motion as well)

 

[Pengwuino]

There is a bit of a subtly going on here. Conservation of angular momentum tells you what happens when an orbiting body goes through its motions. You start your system up, the mass has a given angular momentum, and the conservation law tells you how it's conserved. However, this is clearly only useful when you're dealing with elliptical orbits - that is, when the radius is changing dynamically.

The Newton's laws and the form of centripetal acceleration you used in the first part were specifically for circular orbits. You must talk about 2 complete separate situations since something cannot go from 1 circular orbit to another naturally and still conserve momentum and only be acted upon by the Earth's gravity. This is the correct one you want to use (so the velocity halves) because you're talking about 2 distinct situations whereas the angular momentum argument would be talking about 1 situation where you're dealing with a highly highly elliptical orbit.

 

[Pengwuino]

And I just realized my explanation was probably pretty bad.

Use v^2/r = GMm/r^2. Compare 2 different circular orbits.

 

[widderjoos]

Thanks for the helpful reply.

There's one thing still bothering me though. Suppose we had circular motion of some satellite and for some reason, it wanted to adjust it's radius using it's with 2 well placed thrusts: one radially outward to get it to a larger orbit, and a second to keep it there. Since there's no external torque, shouldn't angular momentum conservation still work before, during, and after the process?

I'm not sure if I'm correct in saying this, but the Newton method doesn't care about the dynamics during the radius changing process right? So it seems that this should still work.

 

[Pengwuino]

Yes it would be conserved but you'd never be able to get it into a circular orbit. Given a certain angular momentum, an objects stable circular orbit is uniquely determined.

And yes in a sense it doesn't care about the dynamics, but you have to compare apples to apples. With the Newton method, you're comparing stable circular orbit to stable circular orbit. With angular momentum, you can't compare two stable circular orbits with the same angular momenta since conservation of angular momentum tells you how a system evolves and a system can't evolve from stable circular orbit to a different stable circular orbit.

What conservation of angular momentum COULD tell you is what the change in velocity would be when you change the radius of an object in an elliptical orbit.

 

[widderjoos]

Yes it would be conserved but you'd never be able to get it into a circular orbit. Given a certain angular momentum, an objects stable circular orbit is uniquely determined.

Yay, I understand it, thanks!