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

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.

• It does not, answer my question. I know that the "Effective Capture radius" is larger then the radius of teh photon sphere. – blademan9999 Jun 11 '20 at 10:45
• I should of used Critical impact paramater. – blademan9999 Jun 11 '20 at 10:58
• @PM2Ring The impact parameter is defined w.r.t. the trajectory at infinity. I.e. it is not the closest approach to the black hole of the actually trajectory, but of an imaginary straight line in flat space. – mmeent Jun 11 '20 at 13:38
• – Void Jun 11 '20 at 13:47
• Does this answer your question? Black Hole Photon Sphere – GiorgioP Jun 25 '20 at 5:51

## 1 Answer

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 $$c$$),

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

• This answers my question. – blademan9999 Jun 25 '20 at 6:49