What is the Critical impact parameter for photons of a black hole?

Question

What is the "Critical impact parameter" for photons of a Black hole with a Radius $r$? Here I'm defing the Critical impact parameter $C$ as the value such that.

1. A photon with an impact parameter > C will be deflected by the black hole.

2. A photon with am impact parameter < C will be pulled into the black hole.

Answer 1

The impact parameter $b$ of a scattering orbit is given by (in units with $G=c=1$)

$$ b= \frac{L}{E}$$

A critical photon trajectory will have the same ratio $L/E$ as the photon orbit. This we can calculate by taking the expressions for $E$ and $L$ for circular orbits in Schwarschild spacetime:

$$ E= M\frac{r-2M}{\sqrt{r(r-3M)}}$$

and

$$ L= M^{3/2}\frac{r}{\sqrt{(r-3M)}}$$

Taking the ratio and the limit $r\to 3M$ (i.e. the photon radius) you find

$$ b= 3\sqrt{3} M$$

or (restoring $G$ and $c$),

$$ b= 3\sqrt{3} \frac{GM}{c^2} $$.

Answer 2

Using Schwarzschild metrics, the circular orbit implies $ \frac{dr}{d\varphi}=0 $ which means $ \frac{r^3}{b^2}-r+R_s=0 $, with $ R_s=\frac{2GM}{c^2} $ and $ b $ impact parameter.

The discriminant of this equation is $ \Delta=4b^2-27R_s^2 $, which is zero for $ b=\frac{3\sqrt{3}}{2}R_s=3\sqrt{3}\frac{GM}{c^2} $.

Hoping to have answered your question,

Best regards.