Why doesn't the sun rise in the arctic winter?

Because The Sun Heat Is So Far Away

It is because when the northern hemisphere is tilted away from the sun at the December Solstice, the sun's rays can't reach any area north of the Arctic Circle, so it stays dark all day.

The Arctic Circle marks the southernmost extent where the sun doesn't rise for a few days in winter and also where the sun doesn't set for a period in the summer.