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## Why do we have leap seconds?

At midnight on the night of Monday the 30th of June, an extra second was added to our clocks. A so-called leap second. Did you enjoy it? Me too đź™‚ I got so much more done….. But, why do we have leap seconds?

In this blog here, I explained the difference between how long the Earth takes to rotate $360^{\circ}$ (the sidereal day) and how long it takes for the Sun to appear to go once around the Earth (the mean solar day). We set the length of our day, 24 hours, by the solar day. If there are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, then there should be $24 \times 60 \times 60 = 86,400 \text{ seconds}$ in a solar day. But, there aren’t! The Earth’s rotation is not consistent, that is if we measure the length of a mean solar day, it is not consistently 86,400 seconds. This difference is why we need leap seconds.

A leap second was added at midnight on the 30th of June. It was the first leap second to be added since 2012.

But, how do we accurately measure the mean solar day (the average time the Sun appears to take to go once around the sky) , and what is causing the length of the mean solar day to change?

## How do we define a second?

When the second was first defined, it was defined so that there were 86,400 seconds in a mean solar day. But, since the 1950s, we have a very accurate method qof measuring time, atomic clocks. Using these incredibly accurate time pieces (the most accurate atomic clocks will be correct to 1 second over some tens of thousands of years) we have been able to see that the mean solar day varies. It varies in two ways, there is a gradual lengthening, but there are also random changes which can be either the Earth speeding up or slowing down its rotation.

## How do we measure the Earth’s rotation so accurately

In order to measure the Earth’s rotation accurately we use the sidereal day, which is roughly four minutes shorter than the mean solar day. By definition, the sidereal day is the time it takes for a star to cross through a local meridian a second time. But, actually, stars in our Galaxy are not good for this as they are moving relative to our Sun. So, in fact, we use quasars, which are active galactic nuclei in the very distant Universe; and use radio telescopes to pinpoint their position.

## The gradual slowing down of the Earth’s rotation

There is a gradual and unrelenting slowing down of the Earth’s rotation, which may or may not be greater than the random changes I am going to discuss below. This gradual slowing down is due to the Moon, or more specifically to the Moon’s tidal effects on the Earth. As you know, the Moon produces two high tides a day, and this bulge rotates as the Earth rotates. But, the Moon moves around the Earth much more slowly (a month), so the Moon pulls back on the bulge of the Earth, slowing it down. To conserve angular momentum, the Earth slowing down means the Moon moves further away from the Earth, about 3cm further away each year.

## The random fluctuations in the Earth’s rotation

In addition to the unrelenting slowing down of the Earth’s rotation due to the Moon, there are also random changes in the Earth’s rotation. These can be due to all manner of things, including volcanoes and atmospheric pressure. These random fluctuations can either speed up or slow down the Earth’s rotation.

We have been having leap seconds since the 1970s when atomic clocks became accurate enough to measure the tiny changes in our planet’s rotation. Since them we have added a leap second when it is decided that we need it, typically but not quite once a year. However, having that extra second at the end of June can cause glitches with computers, and so there are discussions to remove the leap second and replace it with something larger on a less frequent basis.

## How long is a day?

How long is a day? It seems like a stupid question. As everyone knows, there are 24 hours in a day. The Earth rotates on its axis once every 24 hours. Or does it?

## The difference between a ‘solar day’ and a ‘sidereal day’

In fact, there is a slight difference between how long the Earth actually takes to rotate $360^{\circ}$ (the ‘sidereal day’, the day as measured by the motion of stars in the sky) and how long it takes for the Sun to appear to go once around the Earth (the ‘solar day’). This is because, during the course of a day, the Earth has moved a little bit in its orbit about the Sun, and so the Earth has to rotate a little bit more than $360^{\circ}$ to bring the Sun back over the local meridian. We measure our day by the solar day, as otherwise the time of local noon would drift away from midday more and more, which we clearly do not want. (You may notice that this is related to the difference between a sidereal month and a synodic month, which I discussed here.)

The difference between a solar day and a sidereal day, which comes about because of the Earth’s motion about the Sun.

## Kepler’s 2nd law

This difference is easy to measure, with a sidereal day being, on average, 4 minutes shorter than a solar day. This means that stars rise about 4 minutes earlier from day to day, or over the course of a month about 2 hours earlier. But, this 4 minute difference is not constant. It changes because the Earth is orbiting the Sun in an ellipse, not a circle. This means that the Earth’s speed in orbit changes, it travels faster when it is closer to the Sun (in January), and slower when it is further from the Sun (in July). This fact, which was first noticed by Kepler, is now known as Kepler’s 2nd law of planetary motion. It is illustrated below.

Kepler’s 2nd law of planetary motion states that a planet will sweep out an equal area in equal time, so that in the same period of e.g. 1 month, the three areas A will be equal. This means that a planet travels quicker when it is near the Sun, and slower when it is further away.

When the Earth is travelling quicker it has to rotate a little bit more to complete a solar day, and when it is travelling slower it has to rotate a little bit less. So, the length of the actual solar day changes in the course of a year, but in a cyclical fashion (this is known as the equation of time, something I will explain more in a future blog). The equation of time is the reason for the East-West motion of the Sun as shown in the analemma, which I discuss here.

Because of these changes in the difference between a sidereal day and a solar day at any given time of the year, we define something called the mean solar day, and it is the mean solar day which should be 24 hours, or 86,400 seconds long. But, it isn’t! In a blog next week, I will explain how the Earth’s period of rotation is not consistent, and this is why we had a leap second at midnight on the 30th of June this year.