Please check what I've derived. Involves force and orbital velocity.

[e^(i Pi)+1=0]

Basically, I'm trying to come up with a function that determines the amount of force need to keep an object in a stable circular orbit if it were to suddenly lose altitude due to a collision or what not. Basically required force as a function of orbital radius.

Here's what I've come up with:

http://imgur.com/N1FUmtN

 

[Simon Bridge]

What you are trying to achieve is under-specified.
I cannot tell from what you presented here what the situation is or what your assumptions are.
But it looks a bit odd to me even so - what acceleraton is a_r the acceleration of? due to what?

I'm reading it like this so far:
...an object starts out in a circular orbit with radius R1 in the central-force situation.
For some reason, at time t, it finds itself at radius r(t)<R1 ... so it is in a new orbit which need not be circular.
You want to do work to (a) turn the new orbit into a circular one at some new altitude R2<R1 or (b) restore the old orbit or (c) something else?

Generally, any old additional force will do the trick.
It's just a question of how long you are prepared to apply it for - and how cleverly you apply it.

 

[e^(i Pi)+1=0]

What you are trying to achieve is under-specified.
I cannot tell from what you presented here what the situation is or what your assumptions are.
But it looks a bit odd to me even so - what acceleraton is a_r the acceleration of? due to what?

I'm reading it like this so far:
...an object starts out in a circular orbit with radius R1 in the central-force situation.
For some reason, at time t, it finds itself at radius r(t)<R1 ... so it is in a new orbit which need not be circular.
You want to do work to (a) turn the new orbit into a circular one at some new altitude R2<R1 or (b) restore the old orbit or (c) something else?

Generally, any old additional force will do the trick.
It's just a question of how long you are prepared to apply it for - and how cleverly you apply it.

Yeah sorry. This is for someone else for who it seems English is not their first language. From what I can gather, the question is basically if an orbiting body loses energy due to a collision or some other event, how much force that is tangent to the new orbit is required to restore the original orbit? I tried considering dv/dr as the required acceleration, but I'm not sure if it's even valid to talk about a change in velocity with respect to something other than time as acceleration.

 

[Simon Bridge]

I don't think there is enough information.

I suppose it kinda looks like circular orbits are assumed.

Notice: the tangential force initially creates a torque ... thus angular acceleration.

The object goes faster then it's orbit radius increases.
But any amount of force may be used to do this - just depends on the time frame.

Note I said initially - does the force remain tangent to the trajectory (which is not circular when under acceleration) or is it to remain perpendicular to the radius vector?
I think a lot of the missing information is in the context.