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.

 

[HalfLostAndSearching]

Your function looks correct, but there are a few things to consider when determining the amount of force needed to keep an object in a stable circular orbit.

Firstly, it is important to note that an object in a stable circular orbit is constantly accelerating towards the center of its orbit due to the force of gravity. This acceleration is known as the centripetal acceleration and is given by the equation a = v^2/r, where v is the orbital velocity and r is the orbital radius.

When an object loses altitude due to a collision, it will experience a decrease in orbital radius. This means that the required force to keep the object in a stable orbit will increase, as shown in your function.

However, it is also important to consider the force of gravity from the object it collided with. This force will also affect the object's orbital path and may require additional force to maintain a stable orbit.

Furthermore, the type of collision and the resulting change in velocity will also affect the required force. If the collision results in a decrease in orbital velocity, the required force will increase. On the other hand, if the collision results in an increase in orbital velocity, the required force will decrease.

In summary, your function provides a good starting point for determining the required force to maintain a stable circular orbit after a collision. However, it is important to consider other factors such as the force of gravity from the object it collided with and the change in velocity resulting from the collision.