homework and exercises - How to calculate photons/light trajectory under gravity

I'm aware many questions are out there asking similar questions about photons and gravity and I got the basic concept by searching through them. I will just call it light although it may not be the most accurate term for it, but I'm sure what I'm asking is apparent. (feel free to answer with photons, lightlike curves, light, radiation or anything you think is accurate)

My question is as follows:

Consider a light beam, initially having a straight line trajectory. If I put a mass $M$ at distance $r$, trajectory will bend slightly towards the objects side. If I put the mass even closer, light might even hit the mass with following a 'death spiral' path. At one specific mass and distance, one can even make light orbit the mass forever if no other interaction exists (or am I mistaken about that?). How do we anticipate which happens?

To discuss on more specific terms: Say there is a wall at a distance $2d$ from the source, at the middle point $d$ there is an object of mass $M$. We send light with frequency $f$, and the straight line photons would follow without gravity has closest distance of $r$ to the object. How much would light would deviate from its final destination, due to gravity?

Answer

Yes, you can calculate light trajectories in gravitational fields. The equation is $$ \left( \frac{1}{r^2} \frac{\mathrm{d}r}{\mathrm{d}\theta} \right)^2 = \frac{1}{b^2} - \left(1- \frac{2M}{r}\right)\frac{1}{r^2}$$

Here, $r$ is the distance between the light beam and the massive object, $\theta$ is the angle of the light beam, $M$ is the mass of the object, and $b$ is the impact parameter (the minimum distance between the light beam and mass if there was no bending). See the image below.

Integrating over this equation gives the total deflection of the light during its passage near the massive object. As given in a comment, it is:

$$\Delta \theta = \frac{4MG}{Rc^2}.$$

Deriving these equations requires some very basic General Relativity. You can find a derivation on this lovely website by Christian Magnan, who cites the book "Exploring Black Holes, Introduction to General Relativity" by Edwin F. Taylor and John Archibald Wheeler.