# What is the difference between $\theta^{0}_{n}$ and $\theta_n$ in Strong gravitational lensing?

I am currently studying the strong gravitational lensing effects by general relativity.

However, when studying these effects I came across several paper (especially these two papers, Paper One and Paper Two, in particular). Where the authers use the variables $\theta^{0}_{n}$ and $\theta_n$ to indicate the position of the $n^{th}$ image however they never cleary explain the difference between the two.

Could someone please explain to me what is the difference between $\theta^{0}_{n}$ and $\theta_n$?

It's simple: $\theta^0_n$ are the image positions for which the deflection angle $\alpha$ is a multiple of $2\pi$. In other words, it corresponds to a light ray that winds around the black hole an integer number of times, and comes out in the same direction: as far as angles are concerned, this ray just kept on going straight. Bozza et al say as much:
We see that, when $\beta$ equals $\theta^0_n$, there is no correction to the position of the $n^\text{th}$ image, that remains in $\theta^0_n$ simply. In this particular case, the image position coincides with the source position.
In general, this won't be the case: if you're looking at a lensed star, you'd have to be pretty lucky to be standing precisely at one of the rays that winds around a multiple of $2\pi$; assuming high alignment, $\alpha$ will be close but not equal to $2\pi n$, so the actual images $\theta$ will be close to $\theta^0_n$. Indeed, I can cite Bozza again:
For practical purposes, $\theta^0_n$ are already a good approximation for the position of relativistic images.