How would I measure that I'm at a pole?

How would a person measure that he is at a planetary pole? My first inclination is to use a sextant to ensure that the Sun remains at a constant inclination. However, due to the orbit around the Sun and a possible planetary axial tilt, the Sun's inclination may vary over the course of the year and that could affect measurements if the length of the day is a significant portion of the length of the year.

A second inclination is to try to identify a pole star or at least the point about which distant stars rotate during the course of a planetary rotation. The observer could then (at least in theory) measure the inclination of this point assuming that planetary precession takes a long enough time (as on Earth). However, in the most likely chance that there is no bright star near the pole, I don't see how a sextant could measure reliably the inclination of a blank point.

Perhaps the first method could be used, but with other objects in the sky (stars) in place of the Sun. Thus, if a given star's inclination does not change as it (apparently) revolves around the horizon, we are at the pole.

Are there any better methods, assuming a completely self-contained system (i.e. no external satellites)?

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You could confirm with taking an image of the star trails. They would form a circle with the apparent center at the zenith of the location. You do not need a pole star at all. Just a night of viewing.

You would also be able to tell based on the height above the horizon that the sun is. On the equinox, the sun would be on the horizon at noon (when it is at its maximum height). On the summer solstice, its angle will be above the horizon equal to the angle of precession. And below by the same amount on the winter solstice.

Measuring latitude using the sun

For any other day in the year, you will need to determine the max angle based where you are between the equinox and solstice.

Compensating for the Earth's tilt

Basically, you would determine if you were at a pole the same way you determine any other latitude. You are just looking for a result of 90 degrees.

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Of course, if you're at a Pole a "night of viewing" would last six months, right? –  User58220 Nov 20 '13 at 20:42
I meant just a long enough time so that an exposure will capture the arc in the star trails. –  Schleis Nov 20 '13 at 20:50
@User58220 A "night" of viewing will depend on the length of day of the planet you're on. On Earth, a 24-hour night at the North pole (only possible because our axis is tilted) would result in the star trails turning full circle. However you'd only really need several hours' worth of trails in order to make a good judgement as to the location of the pole. –  Moriarty Nov 20 '13 at 20:55
WRT the sun. From a pole, the sun doesn't rise or set, or have a "maximum height" daily. It just circles the horizon, moving slowly and steadily higher and higher, taking around $90$ days to reach a high point of around $23^\circ$, then spirals back down again... –  User58220 Nov 20 '13 at 21:21
@User58220 That is true, however measuring its angle above the horizon and accounting for date will provide the latitude which will tell you if you are at the pole or not. –  Schleis Nov 20 '13 at 22:22

The answers provided to this question so far seem to be good. The easiest way I'd say is to take a long exposure picture in one of two places:

1) At the zenith (directly overhead), if you see that your picture looks like this:

you are at either one of the poles. If the center of rotation of the stars is off by some angle (in degrees), you are that same amount of degrees south (or north) of the one of the poles.

2) The alternative is to take a long exposure picture due East or West. The angle between the direction of the star trails to that of your horizon gives you your latitude in degrees. The picture below is at some mid-latitude location on the Earth's surface. What you'd be looking for on the North or South pole are for the star trails to be parallel to your horizon.

To break the degeneracy about which pole you're at (if you're an alien and crash-landed on the planet Earth (or any planet for that matter)), ask a local if they're friendly. They may not define the North and South poles the same as we do. If you're a human being on planet Earth, you could use constellations to figure this out. Constellations like Draco, Ursa Major/Minor and Cassiopeia are constellations which should not be visible from the South Pole. One could also use GPS to figure this out.

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That is a beautiful last picture, especially how the trails and the roofline are parallel. –  dotancohen Nov 24 '13 at 15:57
Agreed. I can't take credit for it, though. –  astromax Nov 24 '13 at 16:12

To determine if you're at a pole:

1. Be there during that pole's half-year "night". Better around the solstice, when the sun will be well below the horizon
2. Pick some of the brighter stars visible, scattered at different azimuths.
3. Measure the altitude above the horizon of each. Tricky if you don't have a sea level horizon.
4. Stay warm for a few hours.
5. Measure all the altitudes again.
6. Repeat Steps 3 through 5 for at least 12 hours.

If all the altitudes remain unchanged, you're at the pole. Congratulations!

If not, all the stars will wiggle up and down the same amount. A $2^\circ$ wiggle (diameter of 4 full moons) means you're $60$ nautical miles from the pole. Pick a star that has just reached its highest point and move the appropriate distance away from that star. Go back to Step 1 above and repeat...

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Thank you! I have read how Amundsen measured the Sun's altitude while searching for the South Pole, so obviously there exists a daytime method as well. However, I believe that he know which day of the year he was in, so he would know the expected changes in altitude due to the Earth's orbit around the sun. –  dotancohen Nov 21 '13 at 6:48

You can actually figure out if you are at a pole without ever going outside.

Set a pendulum swinging, the bigger the better. If it takes 23hours and 56 minutes (a sidereal day) to complete one full precession clockwise, you are at the north pole. If it takes that long and goes counterclockwise then you are at the south pole.

If it doesn't precess at all, then break out your summer shorts because you are on the equator.

You can actually figure out what your latitude is this way in general, it will be the arcsin of how many degrees it precesses in one day.

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Yours and others methods suggested here, would take some time on Venus, though. –  LocalFluff Jun 29 at 5:20