# How the length of a day changes over the year

Revised 10 September 2019

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 longitudeSee 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”.

Notes

Strictly speaking the terms right ascension and declination should be used rather than latitude and longitude.

## 35 thoughts on “How the length of a day changes over the year”

1. There are two typographical errors in the article. 1) 7 lines below the 3-dimensional picture of the sun on the ecliptic, at the end of the sentence starting with “Therefore”, the “.” should be replaced with a “,”, and then “the sun …”. 2) Below the 2-dimensional picture of the sun around the earth, line 5 “23 hours 40 seconds” should be “23 hours 59 minutes 40 seconds”.
I also have a request. How can one explain the “tilt” effect using a diagram where the sun is at the center, and the earth revolves on a circular orbit with a tilted axis around it? If explained this way, it should be much easier to understand and discussed with students.

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1. Thank you for pointing out the typos. I have corrected them.
With regard to your second question it is harder to visualise why the the tilt of the Earth’s axis causes a variation in the length of a solar day, which is actually larger than that due to the ellipticity of the Earth’s orbit. Interestingly, a recent BBC website article about the variation of the length of the solar day throughout the year ignored this cause altogether.
https://www.bbc.co.uk/news/magazine-30549149

This may have been because the author didn’t understand themselves and thus was unable to explain it! Although of course. I am only speculating here.
I find it is easier certainly easier to understand this effect (which many people struggle to understand) by considering the variation in the speed of the apparent motion of the Sun around the ecliptic. However, at some point I should explaining it in a frame in which the Earth orbits the Sun. If/when I do so I will add this explanation as an addendum to this post.

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2. […] The combination of these two factors gives the equation of time shown in the picture above. For more details on how these two factors work together to vary the length of the day, see my post How the length of the day varies throughout the year. […]

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3. […] The combination of these two factors gives the equation of time shown in the picture above. For more details on how these two factors work together to vary the length of the day, see my post How the length of the day varies throughout the year. […]

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4. […] The combination of these two factors gives the equation of time shown in the picture above. For more details on how these two factors work together to vary the length of the day, see my post How the length of the day varies throughout the year. […]

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5. […] Interestingly, the length of the solar day isn’t constant. It varies throughout the year in a predictable way between a minimum of 23 hours 59 minutes and 39 seconds in mid-September and a maximum of 24 hours 30 seconds just before Christmas day. The reasons for this variation are twofold: firstly, the Earth moves in an uneven speed in its elliptical orbit around the Sun and secondly the Earth’s axis is tilted. They were discussed in detail in my post  How the length of a day changes over the year  […]

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6. […] Interestingly, the length of the solar day isn’t constant. It varies throughout the year in a predictable way between a minimum of 23 hours 59 minutes and 39 seconds in mid-September and a maximum of 24 hours 30 seconds just before Christmas day. The reasons for this variation are twofold: firstly, the Earth moves in an uneven speed in its elliptical orbit around the Sun and secondly the Earth’s axis is tilted. They were discussed in detail in my post  How the length of a day changes over the year  […]

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7. […] Curiously, the size of the photo voltaic day isn’t fixed. It varies all year long in a predictable approach between a minimal of 23 hours 59 minutes and 39 seconds in mid-September and a most of 24 hours 30 seconds simply earlier than Christmas day. The explanations for this variation are twofold: firstly, the Earth strikes in an uneven velocity in its elliptical orbit across the Solar and secondly the Earth’s axis is tilted. They had been mentioned intimately in my publish  How the length of a day changes over the year  […]

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8. Thank you for explaining it so well. Years ago I made a table top sundial with a gauged adjustable dial to compensate for the rough differences from month to month, just rotating it a few degrees left or right according to the season. Fun geeky stuff!

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1. Thank you! !

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9. […] Well, now hold on, is the day actually 24 hours?  No, as it turns out.  The varying speed of Earth in its orbit, along with the tilt of the Earth’s rotational axis with respect to the path of its orbit around the Sun make the length of an actual day, a full rotation, vary over the course of a year.  In fact, earlier this week, we had our shortest full day of the year – at only 23 hours, 59 minutes, and 40 seconds.  (This variation is very nicely explained by fellow blogger, The Science Geek, here.) […]

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10. I wish I knew this 50 years ago. I could have taught my geography teacher something new!

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1. Me too 😉

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11. Reblogged this on The Science Geek and commented:

Not many people know this, but next Tuesday 18 September, is the shortest solar day of the year. I’ve decided to re-blog my post from 2015 on this interesting fact.

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12. […] If you want to know more about how these  factors work together to vary the length of the solar day, see my post September 18 the Shortest Day. […]

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13. I loved the simplicity of your explanation of how earth’s elliptical orbit changes length of earth’s solar day. But I just can’t get how the earth’s tilt changes our solar day length. Could you perhaps explain it more briefly focusing on earth’s movement rather than sun’s apparent movement? Thanks.

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1. Thank you, glad you enjoyed the post.

Of the two causes of the variation of the length of the solar day during the year.

The factor due to the ellipiticity of the Earth’s orbit causing its speed to vary around the Sun is relatively easy to grasp, even though it is the smaller of the two effects.

The second factor due to the Earth’s tilt is harder for many people to understand. It is certainly not intuitive why this should happen, so I am not surprised that you struggled with it 🙂

This factor is best explained by considering the Sun to take a year to make an orbit around the Earth, which is what it appears to do as seen from the Earth and that , as explained in the post, its speed varies in this “apparent orbit” varies throughout the year.

I will look at putting some extra wording in a future version of this post to explain this effect in more detail.

The Science Geek

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14. […] 1)The average length of day when calculated over a entire year is just over 24 hours. However, over the course of a year the the actual length of a solar day varies. This variation throughout the year is due to entirely different effects than the slowing of the Earth’s rotation. It is at its longest – 24 hours and 30 seconds – around Christmas day and it is shortest at around 23 hours 59 minutes and 38 seconds in mid September. This is variation is described in more detail in my post  September 18 the shortest day. […]

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15. […] The combination of these two factors gives the equation of time shown in the picture above. For more details on how these two factors work together to vary the length of the day, see my post September 18 the Shortest Day. […]

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16. “He’s blinding me… with science!”
Awesomeness!

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17. Reblogged this on LexaLovesFUNFASHIONFOOD and commented:
“He’s blinding me ….with science “!!!

I love learning and this article and the graphs as well as diagrams is awesome. And I never use the word lightly.
Thanks Science Geek Guy

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18. Very cool!

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19. Wonderful article on a subject of great interest to us. You have a wonderful way of explaining scientific everyday occurrences!

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20. That did skirt the boundary of what I expect to deal with on a post–but how much nicer it is to be expected to think more, rather than less. Scientifical-ness is like kinky sex–no matter how out there, you can still be pretty sure there are others who are into the same thing…

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1. Thank for your interesting comment 😉

I am glad you enjoyed the post and hope my explanation was clear enough
The Science Geek

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21. Being a person that loves science, your blog is really fun to read and love the way you put things up ♡

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1. Thank you very much. To be honest I thought when I was writing this post that the subject matter might appear to a number of my readers as being a bit dull and I nearly didn’t complete it

Liked by 1 person

1. I loved it! Keep up the good stuff😊. And everyone has there own preferences there is always a set of people made for every kind of writing.

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22. […] September 18 The Shortest Day. […]

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23. Thanks for an intriguing post. Lock-step modern education overlooks the excitement of knowing that you do not know!

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24. I feel like I just got off of a playground merry-go-round. It’s interesting how all these motions, angles, and elipticals blend together. Thanks.

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1. Thanks for your comment. To be honest when I was writing this I thought that perhaps because this was such a narrow subject it might not be interesting to non scientists and I nearly didn’t complete the post.

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25. It is amazing how the various motions can be tracked and plotted. For most people, they don’t have any idea it is going on. For us geeky types, it is cool stuff.

I got intrigued with motions of the Moon a couple of years ago. Perhaps you and your readers would enjoy seeing it. http://bit.ly/1NQ3k8a

Thanks for the interesting post.

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