If a satellite of mass $m$ is orbiting a planet of mass $M$ with radius $r_1$ and orbital speed $v_1$ and is brought to orbit at $r_2$ with speed $v_2$, its kinetic energy changes by a quantity
$$ \Delta E_k = \frac{1}{2} GMm \left(\frac{1}{r_2}-\frac{1}{r_1}\right)$$
Since the total energy in orbit 2 is equal to the total energy in orbit 1 plus the work done to change orbit:
$$E_{tot}^{(2)}=E_{tot}^{(1)}+W_{12}$$
$$\frac{1}{2}mv_1^2 – \frac{GMm}{r_1} = \frac{1}{2}mv_2^2 – \frac{GMm}{r_2} + W_{12}$$
I obtain that the work done to change orbit is
$$W_{12}= \Delta E_k – \Delta E_p = -\frac{1}{2} GMm \left(\frac{1}{r_2}-\frac{1}{r_1}\right)$$
I thought I could calculate this assuming that the minimum work required to change orbit is that done along a radius (either against or with gravity) by a force equal to gravity (in magnitude). Using the definition of work done:
$$W_{12} = \intop_{r_1}^{r_2} \frac{GMm}{r^2} dr$$
I don't see how to obtain the factor $1/2$. What am I doing wrong?
Best Answer
The framing of the problem makes it sound like a satellite is switching from one circular orbit to another.
You have the force due to gravity in the last integral, not the force due to whatever is changing the orbit. The central gravity alone will not change the object's orbit, so there must be another force that is doing it. When computing changes in total energy you must take in to account the external forces acting on the system.