June 29, 2022, was the shortest day ever on record, at 1.59 milliseconds less than 24 hours. You probably didnâ€™t notice, since 1.59 milliseconds goes by pretty fast, but it raises some interesting theoretical questions, like: Since it fell on a Wednesday, should you subtract those missing 1.59 milliseconds from your workday or from your sleep? And: Could Earthâ€™s days get even *shorter*?

That may sound like a wild idea, but thereâ€™s a reason to think it could happen: climate change. Burning fossil fuels adds carbon dioxide to the atmosphere, which melts glaciersâ€”and how that affects the Earthâ€™s motion might have some bearing on the length of a day. Iâ€™ll take you through it step by step.

What Is a Day?

First, letâ€™s go over some basics about how we determine what a â€śdayâ€ť is. Weâ€™ll start with the sun. For the sake of this calculation, we can assume that the sun just stays in place. Of course, this isn't actually trueâ€”in fact, the sun orbits the center of the galaxy. But this motion has no measurable effect on the length of a day.

The Earth has two motions that change the length of a day. First, the planet orbits the sun, with one orbit taking a year. Second, it rotates around an axis passing from its north pole to the south pole. Each rotation takes about a day. The Earth's rotational axis isn't exactly perpendicular to the orbital plane. Instead, it's tilted about 23.4 degrees.

There are several definitions of â€śdayâ€ťâ€”probably more than you realize. There is, of course, the common sense of day versus night, in which day means the part of the solar cycle when your location on the Earth faces the sun and it is light outside. Using that definition, it's fairly obvious that the length of a day isn't constantâ€”surely you have noticed that daylight is shorter in the winter and longer in the summer. This difference is caused by the tilt of the Earth's axis. During the summer, the northern hemisphere is tilted toward the sun. This means that the sun rises higher in the sky and takes a longer path from sunrise to sunset, so there is more daylight time. For the northern hemisphere, the reverse happens in the winter. (The seasons are the opposite in the southern hemisphere.)

But this isn't the kind of day we want to talk about. There are two other definitions we can use. One is the time from noon on one day to noon on the nextâ€”defining â€śnoonâ€ť as when the sun is at its highest point in the sky. This is called a *solar day*, and it has a value of 24 hours, or 86,400 seconds.

Does that mean it takes the Earth 24 hours to make a complete rotation? Actually, no. The Earth makes one rotation in 23 hours, 56 minutes and about 4 seconds. This shorter time frame is called a *stellar day*. You could define it as the time it takes a star you observe in the sky to get back to the same apparent position from your vantage point on Earth. Why is this different than a solar day? It's because the Earth is both rotating on its axis *and* orbiting the sun at the same time.

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Let's consider an example with an imaginary planet. In this solar system, the planet completes one orbit around its sun in 8.6 solar days, instead of 365 days, as the Earth does. (I'm using a shorter year because it magnifies the difference between solar and stellar days, so you can see it more easily.)

Here is an animation showing the difference between solar and stellar days for this planet. The arrow shows when a certain spot on the planet points at a distant star (which would be *way* outside the frame) or at its sun. The instant when it points at the sun is when the sun would be at the highest point in the sky for an observer on that spot.

Notice that for a stellar day, the planet does indeed make one complete revolutionâ€”with a time of 0.648 â€śtime units.â€ť (I also made up imaginary units of time for this example.) However, at this point in the motion, the sun isn't back to the same spot in the planet's sky, because during that stellar day the planet moved. It takes 0.726 â€śtime unitsâ€ť before the arrow points back to the sun. So, in this case, the solar day is a little bit longer than a stellar day, just like on Earth.

Is it possible for the solar day to be *shorter* than the stellar day? Yup. If the planet rotates in a direction opposite to its orbital rotation, then this backward rotation will get the sun back to the highest point sooner. Here's what that looks like:

However, because of the way solar systems form, planets usually rotate in the same direction as their orbital motion. In our solar system, only Venus rotates backward. (OK, Uranus rotates on its sideâ€”Iâ€™m not sure if that counts as backward.) But still, the point is that a solar day is different than a stellar day.

Changes in a Solar Day

For our make-believe planet, the length of each solar day was the same as the previous solar day. On Earth, this isn't true. The difference is that our imaginary planet had a circular orbit, and the Earth's orbit isn't perfectly circularâ€”it's close, but not exact.

Here's what the imaginary planet would look like with an elliptical orbit. Note: I'm not showing the rotation of the planet on its axis. Instead, I have a red vector arrow to represent the planet's velocityâ€”the longer the arrow, the faster the planet is moving.

Notice that when the planet gets closer to the sun, it speeds up. Then it slows down when it gets farther away. There are a couple of ways to explain this phenomenon, but I'm going to use the idea of angular momentum.

To be honest, the math needed to fully understand angular momentum can get a little ugly. So, instead, I'm just going to explain this with a nice demonstration.

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Imagine that you are sitting in one of those office chairs that can swivel around. You give yourself a spin by pushing on the floor with your foot. After that, you lift your feet off the ground and spin like your life depends on it. You feel like you are the ruler of the world in your spinning chair, so you hold out your arms like Leo Dicaprio on the bow of the *Titanic*. Then you start to feel dizzy, so you pull your arms in. Guess what happens?

When you pull your arms in, you *increase* your rotational speed. But your angular momentum doesnâ€™t change. In this example, the angular momentum (L) is a product of your rotational velocity (Ď‰) and something called the moment of inertia (I).

OK, a couple of comments. First, physicists use the nonobvious letter L for angular momentum because there are only so many letters to choose from and â€śaâ€ť was already being used for acceleration. That means we can write the angular momentum as:

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Also, the term â€śmoment of inertiaâ€ť isn't very descriptive, and it can be the most difficult part to understand. It's a quantity that depends not only on the mass of an object but also on how that mass is distributed around the axis of rotation.

If you hold your arms outstretched from your body, you have two masses that are far from the axis of rotation (which runs through the center of your body). When you pull your arms in close, the centerâ€™s distance to these masses decreases, which causes your moment of inertia to also decrease. With a decrease in the moment of inertia, the only way your angular momentum can remain constant is if your angular velocity increases. It is possible to have a change in angular momentumâ€”such as when you push with your foot on the floor to start the chairâ€™s spin. But once your foot is off the ground, your angular momentum is constant.

What does this have to do with a solar day for a planet orbiting a star? As the planet moves closer to the star, its moment of inertia decreases, just like when you pull your arms close to your body while sitting on that spinning chair. The angular momentum of this system has a constant value, so this decrease in the moment of inertia means that the planet has to move faster.

Hereâ€™s a model of that imaginary rotating planet with a non-circular orbit. Every time the sun gets back to the same position in the sky, I'm going to draw a line to the sun, so you can see when the solar day happens.

When the planet moves closer to the sun, it travels at a higher speed. This means that as it rotates on its own axis, it travels through a greater angle, such that it must rotate an extra amount to get the sun back to the same position. As a result, the solar days will be longer when the planet is closer to the sun.

Just for fun, here is a plot of the length of the solar day during three years, or three orbits:

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For the Earth's orbit, the difference in the length of a day isn't quite as noticeable. The variation in the solar day is about 7.9 seconds longer or shorter, depending on where the Earth is in its orbit.

The Stellar Day Can Also Change

OK, so due to the change in the planetâ€™s orbital motion, the solar day isn't constant. But the actual rotation rate of the Earth can also changeâ€”which means the stellar day is not constant either.

There are actually three ways to change the rotational speed of the Earth (and thus the length of a stellar day), and they all have to do with angular momentum. The first way is to exert a torque on it. Think back to the example of the spinning office chair. When you put your foot on the floor and push, your foot twists the chair, providing a torque. The magnitude of a torque depends both on how hard you push and how far the force is from the rotation point. (We call that distance a â€śtorque arm.â€ť) A longer torque arm means you get a larger torque. That's why door handles are as far away from the hinges as possibleâ€”so that you can apply a greater torque with a smaller force.

A torque changes the angular momentum of an object. Your foot on the floor made the chair go from zero angular momentum (or not rotating) to some nonzero value. You could also use your foot to stop the chair from spinning. In both of these cases, the torque causes a change in angular momentum. If the Earth has a change in angular momentum, that could mean a change in angular velocity, which in turn would change the length of a stellar day.

Is there ever a torque on the Earth? Yes. The most noticeable torque on the Earth comes from the moon. There is a gravitational interaction between parts of the moon and parts of the Earth, since both bodies have mass. If both had uniform mass distribution, the net force from the gravitational interaction would create a total torque of zero.

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However, neither are perfectly even. They have parts with more mass and parts with less mass. This means that as the moon moves around the Earth, it produces a nonzero torque and decreases its angular momentum. Over time, the angular momentum of the Earth will keep decreasing until only one side of the planet ever faces the moonâ€”just like only one side of the moon faces the Earth. This will at some point make the length of a stellar day about 28 days long. (Don't worry, it won't happen anytime soon.)

Another way to change the length of a stellar day is with a transfer of angular momentum. Imagine that you are an astronaut on a space walk, and for some reason you are holding a spinning bicycle wheel. Now you use your hand to slow down the rotation of the wheel. With this decrease in angular velocity of the wheel, the angular momentum of the system consisting of the astronaut (you) and the wheel would also decrease.

But wait! You are in space and there isn't a torque to go along with that decrease in angular momentum. So, you know what happens? *You* start spinning as the wheel slows down. With the increase in the rotational speed of the person, the total angular momentum stays constant. This is what we mean by "transfer of angular momentum"â€”in a sense, the angular momentum from the bicycle wheel is transferred into the angular momentum of you, the spinning astronaut.

The Earth is sort of like an astronaut with a spinning wheel. It has two parts that can rotate with different speedsâ€”the crust (the hard part that you live on) and the outer core, which is made of liquid iron-nickel metals. With this geology, it's possible that there could be a differential rotation between the core and the crust. Any changes to the rotation of the liquid core would produce a corresponding change in the rotation of the crust. Since we measure a stellar day based on the rotation of the crust, this transfer of angular momentum would change the length of a day.

The final way to change the length of a stellar day is to change the moment of inertia for a spinning Earth. This is like what happens when a spinning ice skater pulls their arms in closer to their body. With the arms tucked in, their moment of inertia will decrease. However, with zero torque on the system, a decrease in the moment of inertia would mean an increase in the angular velocity.

The Earth, of course, doesn't have arms to pull in closer to its surface. However, it does have glaciers, which are usually in mountains. When a mountain glacier melts, the resulting water doesn't just sit on the mountain. It flows downhill closer to the surface of the Earthâ€”which is a bit like an ice skater pulling in their arms. And as you know, melting glaciers can be a consequence of climate change.

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Letâ€™s see what effect that might have on the length of a day.

Estimating the Change Due to Glaciers

Because there are many factors that contribute to the angular velocity of the Earth, it's difficult to fully account for all the factors that go into making the â€śshortest day ever.â€ť

So Iâ€™m going to look at the impact of just one thingâ€”the melting of glaciers. Let's start with the Earth rotating once every 86,400 seconds, which is one solar day, and assume that there are glaciers with frozen ice at the top of some mountains. I need to estimate the moment of inertia of the Earth (IE) with glaciers of mass mg at the tops of mountainsâ€”a distance of rg1 from the axis of rotation. Then I need to find the moment of inertia when the water from the melted glaciers is at a shorter distance of rg2â€”the distance of this water from the axis of rotation when it is at sea level.

Notice that this value of r is the distance from the glacier to the axis of rotation (an imaginary line running from the north to the south pole). If a glacier was at the equator, this r value would be the radius of the Earth plus the height of the mountain. However, if you move to higher latitudes, the r value is less than the radius of the Earth. Maybe this diagram will help:

If I know the mass of the glaciers, along with the initial and final distances, then the conservation of angular momentum equation would look like this:

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Yes, that looks bad, but don't worry. The important thing is that if I estimate some stuff, I can calculate a value for the final angular velocity.

First, although there is an equation for the moment of inertia for a uniform solid sphere, the Earth is neither uniform nor completely solid. For this exercise, Iâ€™ll estimate the moment of inertia for the Earth as â…–MR2 (where M and R are the mass and radius of the Earth) and then add a correction factor, which should be good enough.

The mass of the melting glaciers is a bit more complicated. According to this paper, there are around 215,000 glaciers on the Earth with a total volume of about 159 x 103 cubic kilometers. Of course, if we know the volume, then we can get the mass using an ice density of 917 kilograms per cubic meter.

There's one other thing to consider: Melting glaciers at the north and south poles (and in the Arctic region) wouldn't really affect the moment of inertia. Since they are already very close to the axis of rotation, their water melting and moving closer to the center of the Earth won't change their r values. In effect, only about 10.3 percent of this estimated glacier volume would contribute significantly to a change in the moment of inertia.

But just for fun, let's say all of this ice is at a latitude of 45 degrees (that's midway between the equator and the poles), at a height of 5,000 meters above sea level. With all of that, I have the following calculations. (You can see all the details in this Python code.)

There you have it: Melting these glaciers would make the day shorter by 0.67 microseconds, or 6.67 x 10-7 seconds. That's not a big difference, but it *is* a difference. And while losing a tiny fraction of a second in the length of a day is insignificant next to the environmental devastation caused by melting glaciers, I hope that it doesn't happen.