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Suppose we have our approximate position only but it is a clear night
at sea with lots of stars visible. We have a camera with good low
light performance but the ship is of course pitching and rolling a
bit, We take a photo of the horizon and this records stars up to a
certain magnitude sensor pixel accuracy, we run astronomy.net's
solve-field or similar on the photo so we now know the Ra/DEC of the
camera  pixel coordinates. We know the stars we see are above the
horizon (assume no reflection), we also know there are stars that we
cannot see below the horizon, as the star catalogue tells us where we
are looking. So we can estimate the sea horizon and we can do it
better the more stars we can see. Theoretically  what is the
relationship between the apparent magnitude of the stars we can see to
the accuracy (say standard deviation) of our fit to the horizon?  We
could use multiple cameras on the same mounting to see up to the whole
horizon.

Roughly I am asking something like the number of stars within a given
number of minutes of a typical great circle in the celestial sphere
(below a certain apparent magnitude). Presumably such statistical
properties of the distribution of stars has been studied to death?

From an image processing point of view a "great circle Hough
transform" would be the typical way to estimate the horizon, but we
can also frame it probabilistically as a kind of regression problem.

I suppose this will work better when you are closer to the galactic
poles and there are more stars near the horizon (I am no great expert
on astronomy as you can tell)

   

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