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## 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.