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) |
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) |
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.
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 |
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.