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To my knowledge, this question can be related to celestial navigation on one hand and to attitude (or orientation) determination from stars direction on the other hand. However the combination of this two methods does not solve the problem, so I am wondering if there is a different approach.

In celestial navigation, the altitude of the stars (elevation wrt to the horizon) at a given time can be used to determine the observer's position. The limit of this method is that the horizon must be visible, which is not necessarly the case in a photo of the night sky.

An image of the night sky can also provide information about the attitude (orientation) of the camera wrt to an inertial frame. The measured direction of a few stars (from their position on the image) and their known direction in an inertial frame give an estimate of the attitude. This method is used in devices called star trackers onboard satellites for instance.

The combination of this 2 approaches fails to solve the problem because the information about the horizon is still missing. Indeed, the orientation is found with respect to the inertial frame, not the local frame, and thus the horizon is still unknown.

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  • $\begingroup$ This is a really cool question! For clarification, I'm guessing by "time of the shot" you mean something like UTC, and a position accuracy of say a few kilometers would be okay, and a clear horizon such as an ocean or some plains known to be flat? $\endgroup$ – uhoh Aug 26 at 13:27
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    $\begingroup$ My guess: if you allow for rotation of the camera (which you do), the answer is no, except that you can check where in the world those stars were up at the time and perhaps do a little "cheating" by checking to see if some stars are faded enough that it's clear the image is taken close to the horizon. Reasoning: by rotating the camera, I believe you can obtain an image similar to any location on Earth, since you're limited by the relative position of the stars, which is fixed for everyone. $\endgroup$ – barrycarter Aug 26 at 14:51
  • $\begingroup$ Yep - you need some kind of reference in the "other axis," since a narrow FOV image could be pointed to zenith at 1AM , then rotated and tilted to get the same image at 4 AM. The "trick" here is that the planets & moon will have moved infinitesimally over those 3 hours, so with a high - enough angular accuracy and precision, you could retrieve the time of the image but not the location. $\endgroup$ – Carl Witthoft Aug 26 at 18:47
  • $\begingroup$ @CarlWitthoft If we are allowed to have the Moon in the image, won't it have a different apparent position relative to stars depending on our latitude? $\endgroup$ – Keith McClary Aug 27 at 2:45
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    $\begingroup$ @uhoh yes that is it :) However the horizon is not always visible on a picture, which is why it is necessary to somehow estimate the orientation of the camera with respect to the local horizontal plane. $\endgroup$ – LaTilz Oct 17 at 20:06
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What can you tell from a picture of some stars?

At the very least, you need a recognisable asterism

If the exposure is too low or the light pollution too high to identify unambiguously an asterism of three or more stars, you're simply not going to be able to tell much from the photo.

An asterism on its own can only tell you where the picture wasn't shot.

If a star in a picture has a declination of $60^o$, you can rule out all shooting locations south of $-30^o$ latitude. A star like Polaris can only really be seen in the northern hemisphere, while Sirius can be seen everywhere south of Svalbard.

An asterism plus the horizon is sufficient for determining latitude

If the edge of land in the picture is sufficiently close to being the actual horizon (and not a range of mountains) and it is present alongside a recognisable asterism in the photo, you can calculate the latitude. This is true for any magnification, as you can use the angular distance between stars (a known quantity) to measure the angle to the horizon. I can't think of any special locations where you wouldn't be able to determine the latitude from the stars + horizon.

I haven't really done the math for how accurately one would be able to determine the latitude, but here are some considerations: There are two sources of error, the position of the stars and the position of the horizon. The star image may spread out over several pixels, and if the picture was shot through a standard camera and not a telescope, there would be an error, at a wild guess, of around ten arcminutes. The horizon can appear lower than $0^o$ depending on your distance above ground, but the effect is small. Looking out from Mt Everest, the horizon will appear a twentieth of a degree lower down than it ought to. I believe the horizon's exact position can also be obscured through atmospheric refraction.

All in all, the error in the angle between an identified star and the horizon equals the error in latitude of your identified position. For a 10 arcminute error, that corresponds to ten arcminutes of latitude, which is about 18 kilometres.

For longitude, you need global, not local time

If you only have local time, the photo could have been taken anywhere along that latitude. If you have some form of global time (UTC, GMT, whatever, it doesn't matter which as long as you know what it is) you can figure out longitude from there.

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  • $\begingroup$ FWIW, there has been military R&D along these lines with a hope of creating partial backups for military systems if GPS fails. $\endgroup$ – ohwilleke Aug 27 at 20:17

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