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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?

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