Is this how we get the stellar parallax angle?

Question

I know this question has been asked before, but for me, something is missing in the answers. I think I might have it figured out though. So the parallax is usually explained with an illustration similar to this:

The parallax angle is the angle $p$. My question is, how do we get $p$? From the previous answers, I understand that the angle is calculated from the distance between the positions A and B. That distance corresponds to an angle, depending on the focal length of the telescope. But the telescope is placed on the earth, not on the near star, so what we get is the angle $r$ in this illustration:

But because the distance from the near star to the far stars is way bigger that the distance from the earth to the near star (and also way bigger than the distance from the sun to the earth of course), the angle $r$ is actually equal to $q$. So $$p=\frac{q}{2}=\frac{r}{2}$$ Is this how it works?

Answer 1

Please note, I'm not an astronomer, but I think you directly get the angle between A and B if you have an image with A and B, which is the first step if I get your question and the explanation in e.g. https://en.m.wikipedia.org/wiki/Stellar_parallax correct.

You need to know the focal length f of your system and use $\alpha = 2 \cdot arctan(d/2f)$, which is the equation for calculating the horizontal or vertical (or diagonal) angular field of view of your camera, if $d$ is the horizontal, vertical or diagonal of your image sensor (the equation comes from https://en.m.wikipedia.org/wiki/Angle_of_view, scroll down to the topic: Calculate a camera's angle of view). You could also use the pixel size as $d$, $\alpha$ is then the angle seen by one pixel. Then count the distance between A and B in your image in pixels and multiply...