Planetary motion around blackhole 

I have plotted the graph as shown.But I don't quite see the connection between potential energy plot and orbit. For example, to get a circular orbit, the first term $\dot{r}$ will be zero. Then $V_{eff}$ = E. But how do I connect it with the graphs? 
Also, doesn't positive potential means they repel each other? How can it happen between a black hole and a planet?
 A: In angular momentum terms, $V_{eff} = -\frac {m}{r}+ \frac{l^2}{2r^2}-\frac{l^2m}{r^3}$
Please look up the difference between potential, and effective potential.
The first two terms are what you would find in Newtonion physics, but the last term $ -\frac{l^2m}{r^3}$ arises from general relativity and is related to the speed  of light constraint on any particle.
The graphs and the above equation are telling you in effect, the difference between the angular velocity of a particle in Newtonion mechanics, where there is no limit on the speed of a particle and the real situation described by GR, and shown in the graphs above,  which says there is a top speed of a particle, that is the speed of light.
If you differentiate $V_{eff}$ with respect to r, and then you set $dVeff/dr =0$, you can obtain the radii of the orbits using the quadratic formula and binomial expansion. 
You will end up with a stable and an unstable orbit.

Doesn't positive potential means they repel each other? How can it happen between a black hole and a planet?

Ok, you mean particle not planet. Notice  the  minus sign in front of $V_{eff}$, physically this term means there is a limit to how near a particle can get near a black hole and stay in a stable orbit. That's what the graphs are telling you as well. 
