Homework Help: Satellite moving around the Earth

The way I would look at this question is that a change in G is no different from any other change in the central force. A larger force would pull the satellite further in, possibly causing a collision. And a smaller force would allow the satellite to move away, possibly out of orbit.

 

If it starts in a circular orbit it must move out into an elliptical orbit. I don't see how it could move out into another circular orbit.

Trickier than at first sight, perhaps.

 

can you quote/display the approprate mathematicas...equations etc

 

I nearly added something to that effect, as I've been imagining a more significant change in G.

But, I wonder whether the new orbit will always intersect the original? And whether, even with a slow change, the orbit will eventually become more eccentric?

 

Velocity decreases as radius increases since angular momentum remains constant, but as long as G decreases, there will be an imbalance between gravitational force and the fictitious centrifugal force.

As the object spirals outwards, a component of the objects path is radially outwards, which is opposed by gravitational force, so the object is slowed down as it moves outwards, but angular momentum remains conserved, since the only real force is radial.

An alternative example would be an object initially sliding in a circular path on a frictionless surface, attached to a massless string that goes through a hole in the surface at the center of the initial orbit. Then the tension in the string is reduced at a constant rate.

I'm wondering if there's a reasonably simple equation (polar coordinates?) to describe the path of the object with G decreasing at a constant rate. Velocity would decrease at a constant rate ## v = G \ M \ / \ (r_0 \ v_0) ## and radius would increase at constant rate ## r = (v0 \ r0)^2 / (G \ M)## .

 

Really? When G reaches zero, no matter how slowly, the satellite will move in a straight line, no? And it must still have angular momentum about the Earth.