# 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? Commented Nov 12, 2017 at 9:49

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
Commented May 2, 2014 at 7:37

The quantity $$b$$ in these equations is the impact parameter and is defined as $$L/E$$ (in units where $$c=1$$). It is called this because $$L/E$$ is the perpendicular distance between the trajectory of a photon at $$r \gg r_s$$ and a radial line through the origin, where the photon has linear momentum $$E$$ and angular momentum $$bE$$.

To see how this relates to the clostest approach it is better to write out the equation for the coordinate speed of light as $$\frac{dr}{dt} = \pm \frac{b}{r_s} \left(1- \frac{r_s}{r}\right)\left[ \left(\frac{r_s}{b}\right)^2 - \left(1 - \frac{r_s}{r}\right)\frac{r_s^2}{r^2}\, \right]^{1/2}\ .$$

The "turning points" are when the right hand side equals zero. One of these is when $$r=r_s$$, because light cannot cross he event horizon in Schwarzschild coordinates. Equating the contents of the square bracket to zero then gives a cubic equation in $$r$$ that determine the location of any other possible turning points $$r_{0}^3 - b^2 r_{0} + b^2 r_s = 0\ .$$

If you are interested in light approaching from $$r \gg r_s$$ with $$dr/dt<0$$ then it is the largest of the three possible roots, with $$r_0 \geq 3r_s/2$$, that is of interest (the middle one corresponds to light travelling outwards from $$r < 3r_s/2$$, initially with $$dr/dt>0$$, and then falling back; the smallest root is unphysical). Real roots with $$r_0 > r_s$$ only exist for $$b \geq 3\sqrt{3}r_s/2$$. Smaller $$b$$ values mean that $$dr/dt$$ is always negative until $$r \rightarrow r_s$$.

The cubic equation above can also be rearranged to give $$b$$ as a function of $$r_{0}$$: $$b = \pm \frac{r_{0}}{\sqrt{1 - r_s/r_{0}}}\ .$$ The plus and minus signs here can be interpreted as clockwise or anticlockwise orbits.

When $$r_{0} \gg r_s$$ then you can see that $$b \simeq \pm r_{0}$$ and it would be ok to use $$b$$ and $$r_0$$ interchangeably in these circumstances (e.g. the bending of light from stars around the limb of the Sun, where $$R_\odot \gg r_s$$).