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I'm trying to understand how the parallax angle is calculated. I alredy read this explanation.

enter image description here

So we got that distance between the sun and the star is d = tan(α) * 1 AU.

That said my doubt is about the angle α. We can calculate this based on the distance of the star that we got measuring it at the distance of 2 AU (based on the length of ω in the image) ? The only thing I can think about is the arc of the circle but I don't think we can draw a circle using that points. enter image description here

Also why we need two measurements? Why can't we get the value of α from the triangle created by the parallel as shown in the next figure?enter image description here

We just need to know where we are compared to the sun. And since (as far as I understand from here) this is purely a measurement of angle I don't understand why we need parallax.

I'm a newbie so I'm sorry in advance if I made some mistakes, just try to learn more. Also sorry about the quality of the images, hope that at least they are clear.

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After few days of studying and asking I find my answer in this reddit thread

The calculation of the angle, as said here is a pure angle measure. The reason we need two measurement is because you can't get a point from a single line. So with a single measure we'll end up with something like this:

enter image description here

So we don't know where the star could be. My wrong assumption was that we know if the star is right in front of the sun, but we can't know that.

So we need the next measure 6 month apart to identify a single point where the star is. Now we end up with two angle: the first measure α and the second measure β. Our parallax angle will be (α+β)/2

Also there is always one time during the orbit where the star is right in front of our star, so we will use that measure (that we verify with the second 6 months apart) to create the right angle triangle.

enter image description here

We will not use this measure (image on the top) but instead we will wait the right time when the star is right in front of the sun so that we can build up a right angle triangle and calculate the distance using tri.

enter image description here

Hope that this could help someone else to understand a little bit better this argument.

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