# Wandering Autumn

Exploring change and the life that comes with it

# Day After Day

January 20, 2017

Did you know that the length of the day depends on the length of the year?

It’s a bit of a surprising fact if you haven’t been exposed to it before, especially since it’s not something we particularly think about. We carry on our day-by-day without thinking about it.

But, when you start contemplating worldbuilding, it becomes a little more important.

Let’s dive in.

You probably learned in school that a day is the amount of time it takes for the Earth to spin once on its axis. This is wrong.

Kind of.

Turns out, there are two different kinds of “days”, that have two different lengths.

The time it takes for the Earth to spin once on its axis is known as a sidereal day. You may remember “sidereal” from my earlier post on orbital period. It’s derived ultimately from the Latin word for “star”: sidus, and in this instance, means “the amount of time relative to the fixed stars”.1 Sidereal year, as it turns out, means the same thing: one revolution of the Earth around the Sun relative to the fixed stars.

Here’s the thing: if we actually define a day by the amount of time it takes for the Earth to spin on its axis, we run into a problem: it’s relative to the fixed stars. Think about it. If right now the sun is directly overhead, then six months later—when the Earth is on the other side of the Sun—it will be directly behind us. So right now it’s noon, and half a year from now, it’s midnight.

This is absurd.

Though I’m hinting at the other kind of day here, a solar day, which is basically the amount of time from high noon to high noon.

But the Earth is going around the Sun, and in the time it takes for the Earth to make one spin on its axis, it’s moved a little on its path around the Sun, so a solar day is a little longer than a sidereal day.

And it’s a solar day that depends on the length of the year. The logic behind this isn’t all that difficult: the shorter the year, the farther along—as an angle—the planet is, so the farther it has to spin from a sidereal day to make a solar day.

So a day depends on the length of the year.

A bit of an aside: there’s no way to predict an arbitrary planet’s sidereal day. It depends entirely on the angular momentum when the solar system was formed—and the interactions of all of the other bodies in the solar system since then.

For example, it’s thought that the gravitational effect of the moon slowed Earth’s days down a very long time ago. Based on some science, we know the Earth days were much shorter a long time ago—and we’re pretty sure Earth has kept the same year.

I’m munging the two kinds of days here, but I think you can figure out I mean solar days, since that’s what we actually tend to mean in common usage.

We can do the math to relate these three quantities. Let’s call them $T_Y$ for the sidereal year, $T_O$ for the solar day, and $T_I$ for the sidereal day. For simplicity, we’re going to assume a perfectly circular orbit; obviously any extremely eccentric orbit will break this, but it’s going to be close enough for orbits such as Earth’s that have a very low eccentricity.

Let’s start with assuming that at time $t = 0$, we are at solar noon with the sun, and the planet’s angle relative to the sun is $\theta = 0$.

After one solar day, the planet will have moved along its orbit. We don’t care that much about distance, we only care about angle, and it will have moved an angle of:

Because we know the number of degrees in a circle, and and we know the time it takes to go around that circle, we know the angular velocity of the planet. Our angle is just the standard time times velocity to get distance.

We are also making the assumption that the planet is rotating in the same direction—that is, clockwise or counter-clockwise—as its orbit. This means that the solar day is longer than the sidereal day.2 So we know that the planet spins once and then a little more.

Because of the magic of geometry, we know that the angle of that “little more” is the same angle that the planet has gone around its orbit. Which means:

Again, we know the angular velocity of the planet’s spin, and so we can find out how much angle it’s spun in a length of time. And we know in that time it’s traced out one circle and a little more.

With a little algebra, then, we can determine the sidereal day for the planet based on its solar day and year length—both of which are easier to measure:

I leave deriving the other two formulae as an exercise for the reader

To double-check, let’s input Earth values, in hours:

And that is about 23 hours, 56 minutes, and 4 seconds, which is what Wikipedia says.

Hooray.

Ultimately, the math behind this isn’t quite so complicated. And now you know that a day is not, in fact, the amount of time it takes for the Earth to spin around its axis.

1. That is, all of the stars other than the sun. However, they’re not actually “fixed”, as it were, because everything in the universe is moving. But they move so slowly relative to the timeframe we’re working with, they may as well be fixed. And, it’s the best we have to work with.

2. In instances where the planet rotates a different direction, the math is a little harder, but not by much. Those situations would simply be rarer because of angular momentum and solar system creation dynamics.

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