Supposing I stand at 89 degrees latitude?

Question

(either north or south, and let's assume I get there somehow). It is the spring or autumnal equinox. How far apart will time zones be?

Answer 1

The time zones have nothing to do with the sun's declination and therefore the time of the year.

Although time zones vary to accommodate with the different country boundaries, they are near the poles divided in 24 sections of 360 degrees of longitude, i.e. 15 degrees of longitude apart.
One degree of longitude equal 60 nautical miles time the cosine of the latitude. At latitude 89 N (or S) one degree of longitude is then 1.047 NM long. (one nautical mile is 1852 meters).

One time zone of 15 degrees would then be 15.7 NM wide at latitude 89 N (or S).

Answer 2

Time zones are a human invention, therefore humans get to decide where the zones are. If you are at 89 degrees latitude, then you are within 60 miles of the pole.

If it is the south pole, then you should be in radio communication with the South Pole station. The time zone they use is New Zealand time, because that is where their supply base is. Therefore, you should also use the NZ time zone.

If you are near the north pole, then the time zone to use depends on where your supply base is, and what time zone the people there are using. It is common to use UTC (time zone Zulu), because that is understood by the Russian / Canadian / Norwegian / American military and scientists who are in radio contact with you -- and perhaps trying to rescue you!

That way, you don't have to change time zones ever time you move 15.7 miles.

Trying to use the time zones from further away from the pole is nonsense, and just asking for trouble. You might arrive at your pickup point 2 hours late, and find that the rescue plane has already left. Or you might try dangerous rushing over the ice, and then find you are 2 hours early, and could have taken the safe way (and not break your leg).

Answer 3

One degree of latitude is about 69 statute miles, which is the radius of the 89° latitude circle with the pole at the center. The circumference of that circle would be 2*π*69=433.53 miles, so each time zone would be 433.5 / 24 = 18 miles apart.