# 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. Jun 11, 2020 at 10:45
• I should of used Critical impact paramater. Jun 11, 2020 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. Jun 11, 2020 at 13:38
• – Void
Jun 11, 2020 at 13:47
• Does this answer your question? Black Hole Photon Sphere Jun 25, 2020 at 5:51

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. Jun 25, 2020 at 6:49

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