Happy first day of spring! Princeton University Press is celebrating the coming of the crocuses and daffodils with this mathematical post by Oscar Fernandez, author of Everyday Calculus: Discovering the Hidden Math All around Us.

March 20^{th}. Don’t recognize that date? You should, it’s the official start of spring! I won’t blame you for not knowing, because after the unusually cold winter we’ve had it’s easy to forget that higher temperatures are coming. But why March 20^{th}, and not the 21^{st} or the 19^{th}? And while we’re at it, why are there even seasons at all?

The answer has to do with 2 numbers. Don’t worry, they’re simple numbers (not like pi [1]). Stick around and I’ll show you some neat graphs to help you understand where they come from, and hopefully entertain you in the process too.

The first star of this show is the number **92 million**. No, it’s not the current Powerball jackpot; it’s also not the number of times a teenager texts per day. To appreciate its significance, have a look at our first chart:

*Figure 1: The average surface temperature (on the vertical axis) of the planets in our solar system sorted by their distance from our sun (the horizontal axis). From left to right: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto (technically, Pluto lost its planet status in 2006).*

That first planet on the left is Mercury. It’s about 36 million miles away from the sun and has an average surface temperature of 333^{o}. (Bring LOTS of sunscreen.) Fourth down the line is the red planet, Mars. At a distance of about 141 million miles from the sun, Mars’ average temperature is -85^{o}. (Bring LOTS of hot chocolate.) We could keep going, but the general trend is clear: planets farther away from the sun have lower average temperatures.[2]

If neither 333^{o} nor -85^{o} sound inviting, I’ve got just the place for you: Earth! At a cool 59^{o} this planet is … drumroll please … 92 million miles from the sun.

We actually got lucky here. You see, it turns out that a planet’s temperature *T* is related to its distance *r* from the sun by the formula , where *k* is a number that depends on certain properties of the planet. I’ve graphed this curve in Figure 1. Notice that all the planets (except for the pesky Venus) closely follow the curve. But there’s more here than meets the eye. Specifically, the *T* formula predicts that a 1% change in distance will result in a 0.5% change in temperature.[3] For example, were Earth just 3% closer to the sun—about 89 million miles away instead of 92 million—the average temperature would be about 1.5^{o} higher. To put that in perspective, note that at the end of the last ice age average temperatures were only 5^{o} to 9^{o} cooler than today.[4]

So our distance from the sun gets us more reasonable temperatures than Mercury and Mars have, but where do the seasons come from? That’s where our second number comes in: **23.4**.

Imagine yourself in a park sitting in front of a bonfire. You’re standing close enough to feel the heat but not close enough to feel the burn. Now lean in. Your head is now hotter than your toes; this *tilt* has produced a temperature difference between your “northern hemisphere” and your “southern hemisphere.” This “tilt effect” is exactly what happens as Earth orbits the sun. More specifically, our planet is tilted about 23.4^{o} from its vertical axis (Figure 2).

*Figure 2. Earth is tilted about 23.4 ^{o} from the plane of orbit with the sun (called the *ecliptic plane

*).*

Because of its tilt, as the Earth orbits the sun sometimes the Northern Hemisphere tilts toward the sun—roughly March-September—and other times it tilts away from the sun—roughly September-March (Figure 3).[5]

*Figure 3. Earth’s tilt points toward the sun between mid-March and mid-September, and points away from it the remaining months of the year. The four marked dates describe how this “tilt effect” changes the number of daylight hours throughout the year. Assuming you live in the Northern Hemisphere, days are longest during the summer solstice (shorter nights) and shortest during the winter solstice (longer nights). During the equinoxes, daytime and nighttime are about the same length.*

Now that you know how two numbers—92 million and 23.4—explain the seasons, let’s get back to spring in particular. As Figure 3 shows, there are two days each year when Earth’s tilt neither points toward nor away from the sun. Those two days, called the *equinoxes*, divide the warmer months from the colder ones. And that’s exactly what happened on March 20^{th}: we passed the spring equinox.

Before you go, I have a little confession to make. It’s not entirely true that just *two* numbers explain the seasons. Distance to the sun and Earth’s tilt are arguably the most important factors, but other factors—like our atmosphere—are also important. But that would’ve made the title a lot longer. And anyway, I would’ve ended up explaining those factors using *more* numbers. The takeaway: math is powerful, and the more you learn the better you’ll understand just about anything.[6]

[1] The ratio of a circle’s circumference to its diameter, pi is a never-ending, never repeating number. It is approximately 3.14.

[2] Venus is the exception. Its thick atmosphere prevents the planet from cooling.

[3] Here’s the explanation for the mathematically inclined. In calculus, changes in a function are described by the function’s *derivative*; the derivative of *T* is . This tells us that for a small change *dr* in *r* the temperature change *dT* is . *Relative* changes are ratios of small changes in a quantity to its original value. Thus, the relative change in temperature, *dT/T*, is

which is minus 0.5 times the relative change in distance, *dr/r*. The minus sign says that the temperature decreases as *r* increases, confirming the results of Figure 1.

[5] Just like in our thought experiment, the Southern Hemisphere’s seasons are swapped with our own; when one is cold the other is warm and vice versa.

[6] One last thing, I promise. Here are two links that animate Figure 3:

Interactive seasons animation, from McGraw-Hill.

Animation showing how the number of daylight hours change, from Mathisfun.com.