Prairie Mountain Sunrise, Part 2

The Science

After we returned to Calgary from our December 29 Prairie Mountain sunrise hike, I started to wonder about the small anomaly that led us to pick that day. Specifically, the fact that the day with the latest sunrise does not align with the winter solstice in the northern hemisphere. Before I get into the science, here’s another photo from that great morning.

I did some research and learned the interesting astronomical reasons for this phenomenon. I first looked up the sunrise/sunset tables for southern Alberta. The solstice occurred at 8:27pm on December 21, but the latest sunrise occurred more than a week later. Not only that, the date of our earliest sunset was well before the solstice… way back on December 12, to be exact. 

What’s going on here? After all, these differences are not small. 

While we think of the solstice as the shortest day based on sunlight hours, it isn’t the shortest “solar day”, defined as the measured time from noon on one day to noon on the next. In fact, solar days are the longest in December.

A recent article in Scientific American explains that there are two reasons for this counterintuitive result. With all credit to the excellent minutephysics video by Henry Reich (“Why December Has the Longest Days”) referenced in the article, I’ve reproduced the two reasons in the following chart.

First, the shape of the Earth’s orbit is not a circle but an oval, an ellipse. The difference between the earth’s nearest and farthest points from the sun is small, about 3% of its average orbital distance of 150 million kilometres. This matters because as the earth reaches its closest point (perihelion), it moves faster through space. The faster movement lengthens the time needed for a given line of longitude to come around the next day to align with the sun. This effect adds about eight seconds to the solar day.

Besides the eccentricity of the earth’s orbit, the tilt of the earth’s axis also contributes to the disparity between solar day and clock day. This effect, called the obliquity effect, lengthens the days by about 20 seconds around the solstices, and shortens them by about 20 seconds around the equinoxes.

The impact of these two factors on the solar day is well known. There is even an “equation of time” to relate solar time to clock time. Mathematically, the effect is represented by two sine functions. The frequency of the eccentricity curve matches the earth’s annual rotation, and the tilt curve goes through two cycles each year. The figure below was generated on PlanetCalc. It shows how adding the curves results in solar days that are shortest in February and longest in December. There is also a smaller peak in the spring and a dip in the summer.

Because perihelion occurs close to the winter solstice (on January 2), the two day-lengthening effects are additive, totalling about 30 seconds a day at the peak in November. These “extra” seconds are pushed forward to subsequent days, making solar noon later and later at that time of the year. And because sunrises and sunsets are symmetrical around solar noon, we get the observed result: the earliest sunset gets shifted backward (before the solstice) and the latest sunrise gets pushed forward (after the solstice).

Whew!

I expected the answer to this question to be simple, but it’s taken me a few tries and a lot of soak time to understand.

Somehow it seems appropriate, having just started a new year (and passed the perihelion), to recognize that the universe is full of mystery.