Sunrise, sunset, and the winter solstice

Where I live in Michigan, sunrise this time of year is about 8:10 AM, while sunset is around 5:20 PM. Centered on the winter solstice, a graph of sunrise and sunset through the year looks like this (sans Daylight Savings Time):

A close look suggests a couple oddities. First, even though Earth's orbit is more or less circular (eccentricity 0.017), the curves don't look quite like sine waves; the sunset curve looks flattened on the right (winter and spring), while sunrise looks flattened on the left (summer and fall). Second, the earliest sunset seems to happen before the latest sunrise. Let's check:

Sure enough, the latest sunrise is more than three weeks after the earliest sunset, and neither falls on the winter solstice.

This asynchrony is due to the rotation of the solar terminus. Even though days are getting shorter as Earth's axis tilts away from the sun, at its furthest extreme the day-to-day change in hours of daylight is small. At some point (December 9th), sunset time is impacted less by Earth's tilt than by the solar terminus moving away from us, so sunset gets later even though we haven't reached the winter solstice.

The delay in latest sunrise is caused by the same phenomenon. Even though we've already passed the shortest day, hours of daylight is still changing so little from axial tilt that movement of the solar terminus dominates. Eventually (on January 3rd), the lengthening daylight caused by tipping back toward the sun overcomes spin of the solar terminus, and sunrise begins to get earlier again.

Though most of the year the rotation of the solar terminus has less influence than the changing tilt of the Earth, you can still see its impact in the skewed shapes of the curves. The sunset curve is pushed left, flattening it on the right, while the sunrise curve is pushed right, flattened on the left.

Naturally, the exact details differ by latitude, which we'll explore in the next post.