# How long will the Sun rise at the North Pole? moon rise?

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,

vertical projection

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”