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I'm a beginner in astronomy and I'm trying to figure out how to model the positions of stars in the night sky mathematically (ie a function which spits out azimuth and altitude as a function of time). I was wondering if anyone knew of any such function, I understand it would vary with your position on earth.

Furthermore, does anyone know anything about the mathematics of star movements, I'm trying to learn all I can.

Thanks!

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    $\begingroup$ I think you could write a more detailed problem, the correct answer to your question would be probably too long. Typically, a good question has an answer at most a page length. But I am not sure, maybe also yours belongs to them. Good luck! $\endgroup$ – peterh - Reinstate Monica May 26 '18 at 1:27
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    $\begingroup$ Are you asking about the apparent movement of the stars in the sky (mostly due to the Earth/'s rotation) or are you asking about the much, much smaller proper motions of the stars with respect to one another? Or both? $\endgroup$ – Mark Olson May 26 '18 at 11:23
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    $\begingroup$ astronomy.stackexchange.com/questions/14492/… is my answer to this, but I'm actually not that happy with it, and hope to rewrite it. Also astronomy.stackexchange.com/questions/8390/… with same caveats. Basically, coordinate transforms in astronomy are rigid rotations and the matrix form is usually the best way to deal with those. $\endgroup$ – user21 May 26 '18 at 17:32
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    $\begingroup$ Also see Wikipedia: Celestial coordinates for conversion between equatorial and horizontal systems. $\endgroup$ – Mike G May 26 '18 at 21:57
  • $\begingroup$ Smarte's Spherical Astronomy is archived here, it's in there somewhere, but I can't point to one single page. $\endgroup$ – uhoh May 27 '18 at 0:53
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It's difficult to answer this question with any confidence, since it is so wide ranging, and the original poster has not (yet) provided any clarification.

However, it seems likely that the book "Astronomical Algorithms" by Jean Meeus, published by Willmann-Bell, will provide the answer(s). Failing that, the publisher's website contains several other reference works on this topic. I have no affiliation with the publisher or author.

Finally, as the author of the question is a beginner, I wonder whether in fact a simple introduction to astronomy, of which there are hundreds, might be a better place to start. Jumping in by trying to model the positions of stars looks hard, particularly as there are plenty of good applications already available, some of which are free.

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You can easily use something like Astropy. It already has the capacity to accept time and location as inputs for an Alt/Az calculation. You can just output a stream of data in a table with a varying variable: time or location. It's been well-tested and is the recommended solution for this kind of thing.

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The planet beneath us is providing the majority of movement. The simplest possible model to figure out where a star should be is to just rotate the celestial sphere by 360 degrees every 24 hours. This is what a clockdrive on an equatorial mount does, to track a particular star's position as the planet rotates. If you know a star's coordinates in right ascension and declination, you can convert this to altitude and azimuth by rotating the spherical coordinate system. Over longer periods of time though, there are many other motions that have to be included for better accuracy.

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    $\begingroup$ 360$^\circ$/24h is the solar rate; the sidereal rate is 1/365 faster. $\endgroup$ – Mike G May 31 '18 at 18:16

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