# Capture cross-section for massive particle in Schwarzschild solution

In the Schwarzschild solution, a massive particle starts with non-relativistic velocity $$v$$ relative to a stationary observer and at position $$r\to\infty$$. I need to show that the capture cross-section is approximately:

$$\sigma=16\pi M^2/v^2$$

where $$\sigma=\pi b_c^2$$

and $$b_c$$ is the critical impact parameter ($$b$$) value below which the particle will get captured by the black hole, with $$b$$ is defined by:

$$b=J/\sqrt{E^2-\kappa}$$

I've shown that:

$$b=J/v+O(v)$$

I also know that to reach the black hole we must have:

$$\frac{E^2-1}{2}>V(r_-)$$

with:

$$r_-=\frac{J^2}{2M}\left( 1-\sqrt{1-12\left(\frac{M}{J}\right)^2} \right)$$

But when I put it all together, the square roots in $$r_-$$ won't go away and I can't get the expected answer. How can I do that?