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Imagine you have 3 star points in 3D space A,B and C. From your unknown position P in space you can measure the angles APC, APB, BPC and you know coordinates of A,B and C and therefore the distances between AB,BC,CA. Is there a way to calculate your own position P? Imagine you are floating in space and you're not on a sphere like earth.

Perhaps this is a frequent problem in astronomy and am wondering if there are any techniques to solve it. If you think there is, please point me in the right direction!

Greetings, Matlab M.

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  • $\begingroup$ This is straightforward geometry (although it is a little tedious). But in astronomy, you're unlikely to have good 3D coordinates for A, B, & C. Plus, everything's moving. And if the distance to your reference points is large, you're going to need very good angle measurements to narrow down the region of your location. $\endgroup$
    – PM 2Ring
    Commented Apr 22, 2019 at 11:30
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    $\begingroup$ Okay, in the situation I'm curious about, the distances to the stars is about 1000 times the distance between stars which are well known, the angles are about ~50 millirad with a accuracy of 1 microrad, I think it might be possible. I'm looking for the straightforward geometry, but am unsuccessful in finding it. $\endgroup$
    – Matlab M.
    Commented Apr 22, 2019 at 11:43
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    $\begingroup$ Hi @MatlabM. go ahead an edit your question and include all of the information that you put in the comment. Stack Exchange runs on question posts and answer posts, not "threads". Many people will answer based only on the question, and not bother to read comments, and comments should be considered temporary and can be unpredictably/unexpectedly cleaned up. $\endgroup$
    – uhoh
    Commented Apr 23, 2019 at 22:30

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The related observation in astronomy is stellar parallax: the observer knows the Earth's position relative to the Sun, measures tiny annual oscillations in the direction to another star, and estimates the distance to the star. A star at a distance of 1 parsec (pc) appears to shift by 1 arcsecond (4.85 μrad) per astronomical unit (au) of lateral displacement. Since parallax is inversely related to distance, the relative error of a parallax-based distance estimate increases with distance. For any given star, the position uncertainty ellipsoid is shaped like a needle pointing toward the Sun.

Aboard an interstellar spacecraft with limited observational facilities, it may not be practical to measure angles between nearby stars directly. Instead you might individually measure their positions relative to several distant reference stars near them in the sky. For best results, you would need to account for the stars' proper motion over the duration of your journey.

After doing that data reduction, you can compute angles APB, BPC, APC fairly precisely. However, the Sun distances SA, SB, SC have significant uncertainties which propagate to other quantities computed from them, e.g. distances AB, BC, AC. You can estimate distances PA, PB, PC by the law of sines, but these are uncertain by similar amounts. Alternatively you can estimate PA, PB, PC by comparing apparent magnitudes to absolute magnitudes, which are subject to the same uncertainties. Then you can find the intersection of spheres around A, B, C to estimate position P, but still no more precisely than distances SA, SB, SC. P is not so much a point as a probability distribution cloud which expands as you move away from the Sun.

In the real world, our most distant spacecraft Voyager 1 is only 145 au (0.0007 pc) away. Within about 400 times that distance, or 1/5 the distance to α Cen, the Sun is the brightest star in the sky by far and would remain your primary point of reference.

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