In my final year project I have been dealing a lot with gravitational lensing by a Schwarzschild black hole. The famous formula for the bending angle (in units of $c = G = 1$) is this:

This formula forms the basis for one of the key observational confirmations of Einstein’s theory of General Relativity (GR). The Newtonian prediction is only half of the GR result.

When I first started out on my project, I was a little confused about which distance $R$ was referring to.

There are 3 distances that are typically used as length measures:

1. $b$, the impact parameter, defined as the perpendicular distance between the light path and the black hole
2. $r_0$, the distance of closest approach between the light path and the black hole
3. $R$, the distance at turning point assuming the light ray was bent at a single point

Their differences between them can be seen in the diagram below. The pink path is the path the light ray actually took, where the distance of closest approach $r_0$ is defined, whereas the orange path is the light path assuming the bending occurs at a single point (which is a pretty good assumption, since the bending angle is small), from which $R$ is defined.

These distances are similar, but not the same. To be precise, they are all equal to one another to first order. This means that to first order in $M/R$, the bending angle can be written equivalently as

However, they can no longer be used interchangeably from second order onwards. It is easy to convert between them using the relationships between the three distance measures:

For example, here is a table of the second and third order coefficients for the different distance measures:

2nd order 3rd order
$M/R$ $15\pi/4$ $401/12$
$M/r_0$ $-4 + 15\pi/4$ $122/3 - 15\pi/2$
$M/b$ $15\pi/4$ $128/3$

This paper contains a detailed derivation of the lensing formula up to arbitrary order.