Thread by @AlexKontorovich

With summer coming soon, I was asked to explain the mathematics of the seasons.

"The Mathematics of the Seasons, an Ode to Trigonometry"

Many creatures on Earth experience seasons, here transitioning from spring towards summer. Why does this actually happen? And how do we know?

(2/n) The first giant leap in our understanding of this phenomenon came once humanity accepted heliocentrism, that the Earth revolves around the Sun, and not, as one might naturally guess, the other way around. Today we could find this out by sending rockets into space...

(3/n) and having them report back their locations/images over time. But actually these ideas were understood long, long before such technology existed, from basic mathematics (trigonometry)!

Though early attempts were made already by the ancient Greeks...

(4/n) heliocentrism was not fully accepted until the 16th and 17th centuries, when Copernicus, Galileo, and others made fantastically precise (for their day) calculations of the positions of the Sun, Moon, and planets, as viewed from Earth.

(5/n) These paths over time look quite bizarre; for example, about once every two years, Mars stops its west-to-east motion, turns around, and moves "backwards" (east-to-west) for a few months, before resuming its west-to-east motion.

(6/n) If the Earth is still (as seems to us to be the case), then the gods play some astronomical game of billiards to get the heavens to move as they do. But if one has the brilliant idea to try putting the Sun at the center of the system,

(7/n) then after some non-trivial spherical trigonometry, all of the perceived motion is consistent with almost circular motion about the Sun!

Kepler soon refined "almost circular" to "elliptic," that is, the path any planet takes around the Sun is an ellipse,

(8/n) with the Sun at one focus. It is not until Newton that Kepler's elliptic motion theory is itself explained in simpler terms: as a direct consequence of Newton's universal law of gravitation. (The possibly apocryphal story of the apple dropping on Newton's head is *not*

(9/n) about gravity pulling the *apple* towards Earth, as is commonly told, but about Newton realizing that the same law pulled both the apple *and the Moon* to the Earth, resulting in the latter being our biggest satellite.)

(10/n) This elliptic motion, which puts us sometimes closer and sometimes farther from the Sun, gives us our first guess at the seasons: perhaps this distance is responsible for making the planet hotter or colder?

(11/n) This is a theory that takes little effort to dismiss, for the seasons are experienced antipodally on Earth (our summer is winter in Australia); but both have the same distance to the Sun at the same time. In fact, we are closest to the Sun in January, and farthest in July!

(12/n) This dismissal also suggests the next theory underlying the seasons, this time the correct one: the axis of Earth's spin (which in the northern hemisphere points directly to Polaris, the North Star) is not perpendicular to its plane around the Sun!

(13/n) Around June 21st, this northern axis leans exactly "in" towards the Sun, while on December 21st, it leans the farthest away from the Sun (and the opposite happens in the southern hemisphere)

(14/n) Where did we get this tilt in the first place, by the way? An interesting guess is that early Earth was hit by a hypothetical ancient planet, Theia, and the impact tilted our spin axis, sending an enormous amount of debris into space, which eventually formed into our Moon!

(15/15) So next time you're at the beach enjoying some summer rays, tip your hat to Theia, mythical mother to Selene, goddess of the moon. And to mathematics, in particular trigonometry, for revealing her to us.