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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:

Parallax illustration

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:

Actual angle measured

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?

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  • $\begingroup$ The measurement of parallax is achieved by modelling the apparent position of the nearby star over the course of at least 18 months. $\endgroup$
    – ProfRob
    Commented Jun 30, 2022 at 4:10

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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...

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  • $\begingroup$ My question is not so much how you calculate the angle, it's about which angle you measure. Is it correct that you measure the angle $r$ and that this angle is the same as the angle $q$? $\endgroup$
    – bgst
    Commented Jun 30, 2022 at 9:38
  • $\begingroup$ @bgst I actually have problems understanding your second plot. In this case you would see the star only at B, not at A. You need to wait half a year and take an image of the exactly same part of the sky again. If you combine both images, the stars far away are at the same position, but your star as being closer to you appears at two positions, A and B. $\endgroup$
    – Charles Tucker 3
    Commented Jun 30, 2022 at 10:58
  • $\begingroup$ The second plot shows the angle between the positions A and B from the perspective of the earth at some time of year. The positions are measured 6 months apart of course. $\endgroup$
    – bgst
    Commented Jun 30, 2022 at 12:26
  • $\begingroup$ Ok, but what is the advantage? You can lift up your photo and say "Ah, the star is now at B, but if I wait six months it will be at A"... The important point is to get q on your first image, and also not being an astronomer I tried to explain how I would do that. If you have q, and you know the distance sun-earth, you can calculate the distance sun-star (approximately this is also earth-star). $\endgroup$
    – Charles Tucker 3
    Commented Jun 30, 2022 at 14:21
  • $\begingroup$ But when I read your first explanation, it seems to me that your telescope has to be placed on the near star. The vertex of $q$ is on the star, not on the earth. That's what's bothering me. $\endgroup$
    – bgst
    Commented Jun 30, 2022 at 14:38

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