Can a satellite be raised to a new orbit with two impulses in this scenario?

[fayled]

Homework Statement

A satellite of mass m is in circular orbit about a planet of mass M, with orbital radius r. Show that with two appropriately located impulses, each directed along the satellite's instantaneous velocity at the moment of the impulse, it can be raised to a new circular orbit of radius 2r. Specify the locations and magnitudes of the two impulses and the shape of the orbit after the rst impulse.

Homework Equations

E=0.5m(dr/dt)^2+L^2/2mr^2-GMm/r
Conservation of E
Conservation of L

The Attempt at a Solution

I have a complete answer but I am very unsure about it so would appreciate a quick check :)

Step one:
Convert into an elliptical orbit with apogee 2r.
Let the velocity after the first impulse be v. Then the initial energy is 0.5mv^2-GMm/r. At apogee, the velocity is halved by conservation of angular momentum. The energy is then 0.125mv^2-GMm/2r. Conserving energy and solving for v gives v=2/√3(√(GM/r)). As the initial velocity was root(GM/r), we need an impulse of [(2√3-3)/3]√(GM/r)m.

Step two:
Convert to a circular orbit of radius 2r.
From conservation of angular momentum, at apogee the velocity is 0.5v=1/√3(√(GM/r)). We need a velocity of √(GM/2r) in the tangential direction (impulse is in this direction so that's fine). So we need an impulse √(GM/r)[(3√2-2√3)/6]m.

Thanks in advance.

 

[dauto]

Your logic is correct. I didn't double check the calculation very carefully but it seems correct as well. This is called a "Hohmann transfer orbit".

 

[D H]

Your logic and your values are correct.