Note March 25 2021
This topic is also available on YouTube
Most people are probably unaware of this but the length of a solar day, which is the natural day measured by the rising and setting of the Sun isn’t always 24 hours. It varies slightly throughout the course of the year and the days in mid September are the shortest solar days in the year. This post discusses this curiosity, which is not widely known.
Background- the variation in the length of the day.
Although a day for practical timekeeping purposes is always 24 hours, the actual length of a solar day, which is the time difference between two successive occasions when the Sun is at its highest in the sky, varies throughout the year. As shown in the graph below, it is at its longest, 24 hours 30 seconds, around Christmas Day and is at its shortest, 23 hours 59 minutes 38 seconds, in mid-September.
How the length of a solar day differs from the average value of 24 hours.
The y-axis shows the difference in seconds between the length of a solar day and 24 hours on a given date measured in seconds. So, for example:
- 10 means 24 hours 10 seconds
- 20 means 24 hours 20 seconds
- -10 means 23 hours 59 minutes 50 seconds.
If we look in detail around the middle of September then we get the following
September 18 is the shortest day of the year, although the difference in day length between September 18 and the days either side is extremely small!
Why does the length of the solar day vary?
The variation in the length of the solar day is not due to the change in the rotation speed of the Earth around its axis. Although the Earth’s rotation speed does vary and is gradually slowing down, as described in my previous post on leap seconds, the effects are very small and unpredictable. In fact the time for the Earth to turn once on its axis will vary by only 0.005 seconds during a year. Whereas, the variation in the length of a solar day is both predictable and is much larger.
There are actually two different causes of this variation. Firstly, the Earth moves in an elliptical (oval shaped) orbit around the Sun and secondly the Earth is tilted on its axis. I’ll now talk about each of these effects in turn.
Effects due to the elliptical orbit
The Earth takes roughly 23 hours and 56 minutes to make a complete rotation on its axis. However a day is clearly not 23 hours and 56 minutes long! This is because during the time it has performed one rotation the Earth has moved around the Sun a little. So if we take the point in time when the Sun is at its highest in the sky the Earth needs to make slightly more than one complete turn for the Sun to be at the highest point in the sky on the following day (0.00274 of a turn to be precise). It takes an extra 4 minutes to make this extra small fraction of a turn, which is why a day is, on average, 24 hours long. This is shown in the diagram below.
If we take a point on the Earth when the Sun is highest in the sky, then after 1 rotation the Sun won’t be at the highest in the sky again. The Earth has to make slightly more than one rotation for this to happen.
In fact it is a little more complicated than this because the Earth’s orbit is oval shaped its distance from the Sun varies throughout the year. It is at its closest in early January and its furthest away in early July. When the Earth is closest to the Sun it moves more rapidly in its orbit and, when it furthest away, it moves more slowly. This is shown in the diagram below.
- In January, when the Earth is moving faster in its orbit, if we take a point in time when the Sun is at its highest in the sky the Earth needs to make slightly more than 1.00274 of a turn for the Sun to be at the highest point in the sky on the following day. So a solar day is slightly longer than 24 hours.
- In July, when the Earth is moving more slowly in its orbit, the Earth needs to make less than 1.00274 of a turn for the Sun to be at the highest point in the sky on the following day. This would make a day shorter than 24 hours.
If the ovalness of the Earth’s orbit were the only effect then the length of a solar day would be the longest on Jan 2 at 24 hours and 10 seconds and the shortest at 23 hours 50 seconds on July 4. However, this isn’t the only effect, the tilt of the Earth’s axis causes a larger variation in the length of a solar day than the ovalness of the Earth’s orbit and I’ll talk about that now.
Why should the tilt of the Earth’s axis cause the length of day to vary?
Most people will remember from high school that the tilt of the Earth’s axis causes the seasons. At first sight it is not clear why the tilt of the Earth’s axis should also cause the length of a solar day to vary and this cause is harder to understand than the variation due to the ovalness of the Earth’s orbit. I will explain this below. I hope my explanation is clear.
As seen from the Earth, the Sun appears to orbit the Earth once a year and the Earth spins on its axis every 23 hours 56 mins. This is shown in the diagram below. This is a perfectly valid thing to do even though of course, in reality, the Earth orbits the Sun. Interestingly, astronomers use a coordinate system to give the position of the Sun and the planets assuming that they are in orbit around the Earth.
As mentioned above, the Earth takes roughly 23 hours and 56 minutes to make a complete rotation on its axis. During this time, the Sun appears to have moved around the Earth a little. If we take a point in time when the Sun is at its highest in the sky, once again the Earth needs to make an extra 0.00274 of a turn for the Sun to be at the highest point in the sky on the following day.
However because of the tilt of the Earth’s axis, when we look at the picture in three dimension the Sun appears to follow the path shown below. This path is called the ecliptic and is tilted compared to the Earth’s equator.
The Sun appears to move at the same speed around the ecliptic throughout the year, taking 1 year to do a complete circuit. However, as you can see from the diagram:
- At the equinoxes, in March and September, the Sun is moving steeply in latitude and thus changes more slowly in longitude. See notes.
- At the solstices, in June and December, the Sun’s latitude doesn’t change very much and the Sun moves more rapidly in longitude.
Therefore, when we look at the picture in two dimensions, the Sun appears to move at an uneven speed in longitude in its imaginary orbit around the Earth.
It is this uneven speed at which the Sun moves in longitude which causes the variation in the length of a solar day.
If this were the only effect then the length of a solar day would be the longest at the June and December solstices, at around 24 hours and 20 seconds, and its shortest at 23 hours 59 minutes 40 seconds at the equinoxes.
Putting the two effects together
The combination of the two effect is shown in the graph below.
In the graph:
- The horizontal axis gives the months of the year and the vertical axis gives the difference in the length of a solar day from 24 hours in seconds. So, for example, a value of 10 means 24 hours 10 seconds, 20 means 24 hours 20 seconds, -10 means 23 hours 59 minutes 50 seconds.
- The blue line gives the difference due to the tilt of the Earth’s axis.
- The red line gives the difference due to the ovalness of the Earth’s orbit.
- The black line gives the overall effect which is the combination of the blue and red lines.
What effects does this have?
Because the length of a solar day varies throughout the year, the natural time measured by a sundial drifts up to 15 minutes ahead or behind the time measured by an accurate clock. Astronomers call this difference the “equation of time”.
Strictly speaking the terms right ascension and declination should be used rather than latitude and longitude.