Daylight Deltas

We have seasons on Earth because of our planet’s axial tilt. In relation to the ecliptic – the orbital path Earth takes around the Sun – we’re slanted just over 23 degrees from an “upright” position, based on the planet’s spinning axis. As we go around our star, this tilt causes one hemisphere to receive more sunlight than the other for half a year, and vice versa. The points of maximum and minimum tilting occur at the solstices. When a hemisphere points toward the Sun, it’s summer; when that hemisphere points away, it’s winter.

Halfway between the solstices lie the equinoxes. These moments occur when the Earth appears, from a cosmic observer, to be “standing straight up” in relation to the Sun. These points produce nearly identical lengths of day and night because neither hemisphere is tilted closer to the Sun. The equinoxes mark the beginnings of spring and autumn.

This interplay between orbit and tilt leads to a constantly changing amount of daylight for most points on the planet (the exception is the equator, which experiences approximately 12 hours of daylight and 12 hours of night year-round).

During a hemisphere’s summer solstice, days are longest, while the winter solstice brings the shortest day.

How quickly does the amount of daylight change as we orbit the Sun? Does the amount we gain or lose on a given day vary, or is it an equal fraction?

It’s certainly easy to see the difference in day lengths between the solstices, and I’m sure most people notice equinox dates feature more sun than the winter solstice and less than the summer solstice. But, if you’ve never dived into the minutiae of day length over a year, these changes on a smaller scale can be effortlessly overlooked.

How the amount of sunlight a place receives changes day-to-day is not constant and requires a little math to illustrate. Don’t worry, there won’t be a quiz at the end of the article.

Let’s start with an illustration of how the Sun appears to move through the sky from a spot near 45 degrees north latitude, in Traverse City, Michigan.

The amount of sunlight on a date depends on one’s location and the declination of the Sun. This term refers to the angle above the equator at which a body appears. For the Sun, the value can be positive (between the spring and autumn equinoxes) or negative (between the autumn and spring equinoxes).

Using astronomer Bob Moler’s fantastic program, we can visualize how this fact plays out in Traverse City. During the winter solstice, the Sun rises far in the south and takes a very low angle through the sky (maximum of 22 degrees). Days are short.

On the equinoxes, the Sun rises due east and sets due west, while reaching 45 degrees above the horizon. Day and night are roughly equal.

On the summer solstice, our star rises far north of due east and traces quite an angle in the sky, reaching 69 degrees altitude at noon. This combo produces the longest day of the year.

The Sun’s maximum point in the sky changes every day.

If you mapped this point each day for a year, an interesting shape emerges.

It’s a sine wave!

Circular motion spread out over time and mapped to a flat surface presents as a sine wave. As it turns out, the lengths of daylight for locations not within the Arctic or Antarctic Circles form a nice sinusoidal graph.

The solstices mark the high and low points of the graph above, while the equinoxes occur where the Sun intersects with the x-axis, which represents the equator.

If you remember calculus, you can infer a few things from this graph about our initial questions. How quickly does the amount of sunlight change daily, and is this change constant?

We can think of the Sun’s path on the wave as its north-and-south movement, its change in declination. As it moves along the sine wave, the north-to-south or south-to-north movement is not equal. Derivatives display the mathematics behind the phenomenon, but we can visualize it by adding a tangent to the sine wave.

The red line in the graphic above represents the slope of the curve at any point. If the wave represents the Sun’s motion, we can think of the tangent as the amount it moves north-south on a given day.

At the maximum and minimum, note that the line is flat, indicating zero movement. This fact matches what we observe on the solstices. The amount of added or subtracted daylight on the solstices, depending on the season, is essentially zero. If we traced the Sun’s path through a year, it would appear to “stand still” in the sky at the solstice (i.e, it would neither move southward nor northward).

Now consider the slope of the tangent where it intersects the x-axis. These points, representing the equinoxes, display the steepest slopes, implying maximum north-south movement. Once again, this attribute matches observation. The biggest amount of change in daylight from one date to the next transpires on the equinoxes!

If all this math sounds like gobbledygook, simpler charts might help illuminate the trend.

These tables display the times of sunrise and sunset for TMAC HQ in Columbus, as well as the length of the day and the delta (change) in that length from the previous day. Highlighted in red are the solstices and equinoxes.

As you can see, on the summer solstice, the difference between June 20 and June 21 was less than a second. This happening is mirrored in the third image, as just a minute delta exists between December 20 and the solstice a day later.

Conversely, on the equinoxes of images two and four, the change from day to day is more than two-and-a-half minutes!

So, if you feel like the advent of autumn is stealing bigger chunks of daylight than a month or two earlier, you’re right. If you’re reading this article on a fall equinox week, each day is about 150 seconds shorter than the previous one. Of course, the reverse is true near the spring equinox, where we get more than two minutes more sunlight each day.

If you’re into mathematics or have a keen eye for data, you might have noticed something else, which answers the other part of our initial query. Is the rate of change the same day-to-day? In other words, the biggest changes happen near the equinoxes, but is the acceleration from zero to max constant?

Nope!

The second derivative of a position or speed function yields acceleration. The slope of the second derivative essentially shows us how fast the rate changes at any point. For sin(x), the second derivative equals -sin(x). Applying this to sunlight lengths, the biggest point of acceleration occurs at the solstices. This fact makes sense. If we start from a point of “standing still” and move north or south, we’re kicking the Sun into high gear. Likewise, though the greatest amount of change from one day to the next happens at the equinoxes, the rate of acceleration is at its minimum. You can see this idea in the graphs above. For the days around an equinox, the daylight delta is nearly the same for a couple of weeks.

A great metaphor to illustrate the larger point is one of a car on a road. If you start from zero, you’re at a solstice. You push the gas pedal to get the vehicle rolling, accelerating from zero toward a maximum cruising speed. As you near your top-end speed, you ease off on the gas just a bit, accelerating just a bit to nab those last few miles per hour. When you hit that top velocity, cruise control keeps you steady for just a bit (you’re at the equinox!) before you notice a stop sign in the far distance. You ease onto the brake, as there’s no need to stop early, starting to slow. As you reach the stop sign, you apply the final, firm touch to the brake to bring you to a halt. You’ve gone from one solstice to an equinox to another solstice, and you’re ready to start the process anew. During this drive, your speed matches the amount of sunlight you gain or lose each day – highest at the equinox – while the rate at which you’re speeding or slowing is equal to how much that delta changes from day to day.

Looking at the data, autumn gains a depressing attribute. The amount of change each day is so high that from September 8 to October 1, a period of 23 days, we lose a full hour of sunlight in Columbus. Further, I use the lengthening days after the winter solstice to boost my outlook, but gaining two seconds, six seconds, and 10 seconds on the first three days doesn’t really inspire much positivity.

BONUS FACT: Since the amount of daylight depends on the declination of the Sun and its maximum angle at a given latitude, the yearly graph mimics the sine wave of the Sun’s path through the sky. This correlation breaks down above the Arctic Circle and below the Antarctic Circle. The sunlight length graph in those places would not look like a sine wave but rather like a sawtooth wave, either 24 hours long or 0 hours long. In the Arctic, between March and September, the Sun is always up; between September and March, the Antarctic is bathed in constant light. The path of the Sun during this constant daytime is fascinating!