Relation between impact parameter and distance of closest approach of a light ray in Schwarzschild Geodesics The following wikipedia articles are incompatible :


*

*Two body problem / bending of light by gravity

*Schwarzschild geodesics / bending of light by gravity
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 :


*

*Which one is correct? (What is $b$?)

*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 $r_0$)?
 A: 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.
A: 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 $R_\odot \gg r_s$).
