There are two ways to analyse what happens to the orbit of a satellite when its orbital velocity is suddenly reduced (halved).
By considering the energy, when the velocity is halved, kinetic energy is quartered, so we have
\begin{equation}
KE_{new}=\dfrac{1}{4}KE_{old}=\dfrac{1}{4}\cdot\dfrac{GMm}{2r}=\dfrac{GMm}{2(4r)}\implies r_{new}=4r_{old}
\end{equation}
So the radius doubles. However this doesn't fit with intuition, as decreases in velocity results in orbital decay, which decreases the radius of orbit. Obviously from the calculation we have increase in radius. What mistakes did I make two produce these two faulty results?
P.S. After some research, it seems like I have to consider the total energy instead. So the question now becomes, why is it wrong to consider the kinetic energy alone?
Best Answer
The missing piece is in the "suddenly reduced" bit. How it gets reduced is very important. If you say it remains in a circular orbit, at half of the velocity, the only valid solution is for it to "suddenly" be at a higher orbital radius.
In reality there are more orbits than just circular ones. The actual collection of available orbits are the Keplerian orbits, which are conic sections. If you "suddenly" reduce the velocity but keep the same position, that will typically put you on an elliptical orbit, not a circular one. These orbits should have the natural intuitive feel of decreasing orbital radius. In effect, you did a substantial retrograde burn, and that takes you down towards Earth.
(Of course, if you don't hit the atmosphere, you'll keep following the elliptical orbit and return to the point where you did your retrograde impulse burn)