For a satellite given an initial speed and position above a planet, I'd like to know whether it will experience an elliptical, circular, parabolic or hyperbolic trajectory, or crash into the planet. The intitial position $(x,y)$ relative to the centre of Mars, and the velocity $(v_x,v_y)$ are known.
I feel as though the orbit will be circular if effective potential $U^*(r) = L^2/{2mR^2} - GMm/R $ is at a minimum (with respect to r), where L=angular momentum. Differentiating, $\frac{dU^*}{dr} = \frac{GMm} {r^2} - \frac{L^2}{mr^3} = 0$. And hence if it undergoes a circular orbit, radius is $\frac{L^2}{GMm}%$, is this correct?
If air resistance is ignored, how much information is necessary to determine whether a satellite will crash into a planet or not, is there a minimum energy below which it will always spiral in and collide?