The angular velocity of the annual motion of the Sun along the ecliptic is
ω = (360 / 365.25) = 0.986 ° / day.
Near the vernal equinox, when the Sun rises at the North Pole, this velocity will have a vertical component equal to
ωv = ω sin ε = 0.393 ° / day.
Here ε is the angle of inclination of the equator to the ecliptic, equal to 23.5 °. Considering that the angular diameter of the Sun at the vernal equinox is 32 ′ or 0.53 °, it turns out that the sunrise at the North Pole will stretch for 1.357 days, or 1 day and 8.6 hours.
The moon will rise much faster. Its average angular velocity is
Ω = (360 / 27.32) = 13.177 ° / day,
Ωv = Ω sin ε = 5.254 ° / day,
and with an average angular diameter of 31 ‘, the sunrise will last 2.36 hours, which is very little by “polar standards”
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