Suddenly Springtime: the Nonlinearity of Seasons

Why does a change in the seasons always seem to creep up on us? Winter has a way of seeming like it’ll never end, like every day closer to springtime brings only another minute of sunlight—and then, nearly all at once, you’re enjoying a sunset at 7 PM in nothing more than a light jacket.

The official move from one season to another is marked by the solstices—the longest and shortest days of the year—and the equinoxes, days of roughly equal light and darkness…but the way that things change between these points is more complicated than you might think.

Our planet’s axis of rotation is tilted, relative to the plane of its orbit around the sun. While practically everyone knows that this is what gives rise to the seasons, the geometry of the situation isn’t something that most people think about very often after learning it, so it’s worth a quick refresher.

As Earth revolves around the sun, the axis of rotation stays fixed. While we’re on one side of the sun, the northern hemisphere is tilted toward the sun and gets more direct sunlight. Once we swing around to the other side, the northern hemisphere is tilted away.

While the tilt DOES change over time, a process called axial precession, it takes something like 25,000 years to complete one revolution.
Image Credit: Blueshade, via Wikimedia Commons (CC BY-SA 2.0)

When one hemisphere is tilted away, the same rays of light spread out over a wider area—like a flashlight pointed at a sharp angle to the ground, rather than straight down.

This changes both the overall intensity of the light and the amount of air it has to travel through, which affects its color.
Image Credit: Peter Halasz (CC BY-SA 3.0)
Now, we know that the winter solstice is the shortest day of the year, the summer solstice is the longest, and the equinoxes are right in between. On a graph, those four points of information look something like this:

The y-axis’s values will obviously vary depending on your latitude—these are for Maryland, where the American Physical Society’s headquarters is located.
Image Credit: Stephen Skolnick (CC BY-SA 3.0)
The human brain is an unparalleled pattern-finding machine, and we often do it without even realizing. It’s easy to interpolate, to draw lines connecting those four points, and even if we don’t think about it, we kind of end up drawing those lines in our heads regardless, to describe how the length of the day and the temperature will change with the seasons—but this is where it’s easy to go wrong.

Before someone pointed out the error of my ways, my initial assumption about how the day’s length changes over the course of the year—the curve I had drawn in my head without even realizing it—looked like this:

Image Credit: Stephen Skolnick (CC BY-SA 3.0)

Seems fair, right? If we’re halfway from summer to winter, i.e. at the autumn equinox, the day’s length is halfway between the values at the solstices.  If we’re halfway between the summer solstice and the autumn equinox, the intuitively obvious conclusion is that the length of the day should be halfway between those two values, as on the curve above. It’s the simplest way to connect those four data points, and it makes perfect sense if those four are all you’re working with.

But hopefully, seeing it on the graph, you can spot what’s wrong with that thinking—relatively few things in nature move in a sawtooth-wave kind of way. Physical phenomena tend to vary smoothly, and the seasons are no exception. You’ll encounter the word “smooth” a lot in physics and mathematics, and you’ll often find it accompanied by another phrase: “continuously differentiable“.
While that sounds like heavy math jargon, it just means that there are no sharp corners to a curve. Differentiable means that you can find the rate of change in a function, like the slope of a line (the derivative, in calculus terms)—and continuously differentiable just means the slope won’t abruptly change from one value to a very different one, the way it does in the graph above. With this in mind, what would a smooth, differentiable curve to connect those four points look like?

If you guessed “a sine wave”, you’re right!
(But really, if you’re not sure about something in physics, “sine wave!” is usually a solid guess.)
Image Credit: Stephen Skolnick (CC BY-SA 3.0)

It’s not quite fair to say that the graph takes that shape because “nature is smooth”, of course—the reason the real curve looks like this is because we live on a sphere, and the geometry of circles and sine waves are intimately related.

As demonstrated spectacularly in this .gif.
Image Credit: 1ucasvb
What this all means, though, is that the rate at which the seasons change also changes. At the solstice, the rate of change in the day’s length slows to a crawl, picking up steam again until it’s practically barreling through the equinoxes—not only the balance points between day and night, but also the times of greatest change. If you look halfway between the spring equinox and the summer solstice, you’ll see that the day length is already most of the way to its maximum value! This is what is meant by the title of this post: the seasons change in a nonlinear way, staying at their extremes longer than you might expect, and then changing more rapidly.

There’s a deeper insight into some core mathematics here, too: the relationship between sine and cosine. In the .gif above, you can see that the points where the red curve reaches its maximum are the points where the blue curve crosses the midline—the “zero”—and vice versa. This is no coincidence—if you’ve taken calculus, you know that the cosine function is the derivative of the sine function, and vice versa: each describes the rate of change in the other.

The day I properly understood that was a minor formative moment for me—I still remember feeling the “click” of it in my mind, stopping in my tracks on the way home from class and going “Ohhhhh” aloud. But even if you’re not such a geek, you can hopefully still appreciate that we’re past the worst of winter, and heading for the steepest part of the sine curve—meaning it’s going to get warm here in a hurry!

—Stephen Skolnick

P.S. Apologies to any of our readers in the southern hemisphere—but presumably you’re just as sick of the heat by now as we are of the cold!

For more on seasons and sunlight, check out our previous post: Physics in the Autumn Sunrise.

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