Hmm - doesn't show the image for some reason. (I'm going to do a feature request, I guess ^_^)
Anywhoo... as you can see in my ahm.. masterpiece ^^
The "effective potential Energy" is a sum of the (green) potential gravitational energy plus the centrifugal one. Now there are various cases - depending on the initial conditions.
If the resulting energy is negative but there is radial-kinetic-energy, then that's an ellipse. The blue line I shows that case. r1 and r2 are the minimal and the maximal distance of the planet to the sun.
If you do not have radial-kinetic-energy, you are going to have a circle. This is the minimal effetive energy you can possibly have - labeled II. The distance is a constant r0.
If your energy is positive, then you are going to have a hyperbola. Meaning your celestial body will escape the gravitational field of the sun. (labelled III) Note here, that the point C is the minimal distance to the sun, the body is going to have. Also note that the transistion from elliptical to hyperbola (E = 0) is called a parabola.
Please ask, what you don't understand yet, I'll elaborate.
\[E_{kin}^{\tan} = \frac{1}{2}\, m\, r^2\, \dot \phi ^2 = \frac{L^2}{2\, m\, r^2}\]\[E_{pot}^{eff} = E_{pot}(r) + \frac{L^2}{2\, m\, r^2} = -G \frac{m\, M}{r} + \frac{L^2}{2\, m\, r^2}\]\[E_{kin}^{rad} = \frac{1}{2}\, m \, \dot r ^2 = E - E_{pot}^{eff}\]\[\frac{d\, E_{pot}^{eff}}{dr}= 0 \rightarrow r_0 =\frac{L^2}{G \, m^2 \, M}\]