# Relation between impact parameter and distance of closest approach of a light ray in Schwarzschild Geodesics

The following wikipedia articles are incompatible :

According to both articles, the equation describing the path of a photon in a Schwarzschild metric is :

$$\frac{d\varphi}{dr} = \frac{1}{r^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{r}\right)\frac{1}{r^2}}}$$

with :

• $\varphi$ the deviation angle
• $r$ the distance between the photon and the mass
• $r_s$ the Schwarzschild radius

and $b$ which is the distance of closest approach in the first article and the impact parameter in the second one.

So I have two questions :

1. Which one is correct? (What is $b$?)

2. If $b$ is the impact parameter, what is the formula of the distance of closest approach $r_0$ as a function of $b$ (or inversely, the impact parameter $b$ as a function of $r_0$)?

• I have seen $b$ defined as $L/E$ where $L$ is the angular momentum and $E$ is the energy. e.g. p.188 in Introduction to Black Hole Physics by Frolov. How does this make sense for a photon? Why does the impact parameter, essentially a function of the trajectory, depend on the angular momentum or the energy when the equations of motion don't depend on these things? – supercoolphysicist Nov 12 '17 at 9:49

## 1 Answer

The relationship between $b$ and $r_0$ for the Schwarzschild metric is:

$$b = \frac{r_0}{\sqrt{1 - \tfrac{r_s}{r0}}}$$

where $r_s$ is the radius of the event horizon. See this paper for the gory details.

The equations given in the two articles are derived in the weak field limit i.e. $r \gg r_s$ so $b \approx r_0$ anyway and it doesn't make any real difference which you use. If I were writing the article I would describe $b$ as the impact parameter and not the distance of closest approach, but I don't feel strongly enough about it to want to edit the offending Wikipedia article.

• $b$ is certainly the impact parameter. As you mention, in the relevant limit $b\approx r_0$ and hence I would say it's probably OK for the wikipedia article that's wrong to state that "the length-scale b can be interpreted as the distance of closest approach", which is already worded somewhat carefully. – Danu May 2 '14 at 7:37