# Sunrise in the west: spoiler

Q: Where and when on Earth can the Sun rise in the west?

A: Near one of the poles in spring, since the earth's motion round the sun is then more significant than its rotation.

The point is that in this situation the sun is not following an arc overhead from one horizon to the other as usual. During the winter the pole is in constant night, with the sun below the horizon. At the end of winter, for an observer near the pole the sun is going round and round the horizon (one circuit per day) but is always just below the horizon so is out of sight. As spring comes the sun's track gradually rises in the sky, so it moves from going round and round the horizon but just below it, to going round and round the horizon but just above it.

"Sunrise" occurs at the point where the sun's rising track moves above the horizon. Consider a ring of observers standing around the pole. They will all see this sunrise happen at the same place on the horizon. However, the direction that they say that it happens will depend on where they're standing relative to the pole. For example, at the north pole an observer who happens to have the pole on their right as they watch the sunrise will call the direction that they're looking "west".

Another way to look at this:

Let T be the angle of tilt of the earth's axis (about 0.4 radians), and let R be its radius (about 6.4e6 m). Suppose for the moment that the earth isn't rotating on its axis, and consider a small region around one of the poles. As the earth moves round the sun then in spring the night-day terminator will move across this region with speed 2 pi R sin T / year, and if we stand in the appropriate part of the region we will call the direction in which the sun appears to be rising west.

Now suppose the earth is rotating on its axis. Then we will be carried from west to east, ie in the same direction as the terminator. However, suppose we are so close to the pole that we are moving slower than the terminator - then the terminator will pass us in the same direction as before, and we will still see the sun rise (slowly) in the west. If our latitude is pi/2 - L then our speed due to the rotation is 2 pi R sin L / day. Thus - assuming that L is small relative to T so that the terminator's speed is constant throughout the region - we must have
2 pi R sin L / day < 2 pi R sin T / year, ie sin L < sin T day/year.

Our distance from the pole is RL. With the above values of T and R we can therefore be up to 6.9 km from the pole.

Non-spoiler

Comments, criticisms and suggestions are welcome. Copyright © Thomas Bending 2022 