Solar Time, Mean Solar Time and Leap Years – How Are These Related?

We all know about leap years, there is one every 4 years except in 2000 – remember that?  Did you know that it is all to do with simplifications we make to the solar day?  The solar day varies over the year depending on where the earth is on its orbit around the sun, so it is simplified to the mean solar day.  It all seems quite complicated, does it not?  Let us start at the beginning and unpick all this.  

The Solar Day – Function Over Accuracy

The solar day is the practical solution to how a day is measured. The solar day is how humans have understood a day even when it was imagined that the sun orbited around the earth. When is was shown that the earth rotated about its axis and orbited the sun there was a desire to accurately define the day. The most accurate measure of a day is the sidereal day, where the reference point is a distant star. This is, unfortunately, not very practical for those of us who like to arise at a regular time and to get as much sunshine as possible. The solar day though has problems too because even though it is convenient to use, it lacks accuracy. Let’s have a look at the details. 

A Quick Review

The sidereal day is defined as the time taken for the earth to complete a single revolution on its axis with reference to the stars.  The solar day, by comparison, is the time it takes the earth to complete a single revolution on its axis with reference to the sun.  Intuitively we understand the solar day more easily, the difference between dawn one day and dawn the following day (or any other reference time). It is when we dissect the details of the solar day and compare it to the sidereal day that the differences become obvious. These differences are small in absolute magnitude but over time they accrue to have a material impact on how we measure time. 

A Quick Bit Of History of The Solar Day

The advent of the solar day can trace its roots back to the oldest time measurement system, the sundial.  The sundial uses the rotation of the earth around the sun (or the sun round the earth as it was envisaged at the time the sundial was invented).  Knowingly or not the sundial works off the solar day which, as we noted above, is the interval between two successive transits of the middle of the sun over the meridian.  

That sounds simple enough, but when we look at the details from afar it becomes a bit more complex. The earth hurtles through space at 67,000 miles per hour orbiting the sun.  Simultaneously we are spinning on our celestial axis at about 1,000 miles per hour at the equator. The earth orbits the sun in a counterclockwise direction and the earth rotates around its axis in a counterclockwise direction when viewed down onto the North Pole.  It is the fact that the earth is spinning on its axis as we orbit the sun that causes the difference between the sidereal day and the solar day.   

The graphic here is an exaggerated example of how the earth transits around the sun. If the earth starts its day at position 1 then as time passes it travels along its orbit around the sun and simultaneously rotates on its axis. When the earth has made it to position 2 of the transit a sidereal day has elapsed. For the earth to complete a full rotation and the earth be in the same position in relation to the sun it must turn a small increment further. This is caused by the relative movement of the sun in the same direction as the rotation of the earth on its axis as the earth progresses on its orbit around the sun. 

Details Details Details

Unfortunately there are further complications using the solar day, it is constantly varying. This arises from two principal causes. The first is that the earth’s orbit around the sun is an ellipse with the sun in one of the focuses (an ellipse has 2 focus, a circle has a single focus). The transit (or its orbit around the sun) of the earth varies depending on its distance from the focus. The earth moves slowest when it is farthest off – i.e. in July and faster when it is close to the sun.  

The other, and much more significant cause, is that the sun moves in the ecliptic and not in the plane of the equator. The consequence of this is that the days are a minute longer in December and half a minute longer in June than they are in April or September.  If you would like to test this, get a digital clock and set it with a sun-dial in November at midday, by February the clock will hit midday 30 minutes before the midday indicated by the sundial.  This explains why the afternoons are so much darker in November in the northern hemisphere than they are the same number of days after the winter solstice.

Ironing Out Imperfections

These variations were well documented and understood to occur at regular intervals so if they were left to accumulate would become material over time. In the 16th century as timekeeping became more and more accurate there was a desire to define the day as a regular period. This defined day should keep in sync with the rotation of the earth. This resulted in the mean solar day.  

The mean solar day is defined as the average length of all solar days in the year, which has no exact number of days. When this definition is first read it can be a little confusing but once unpacked it is easily understood.  The sidereal year is the absolute time taken for the earth to orbit the sun with reference to a fixed point in the solar system (normally distant stars).  Once this period of time is quantified it is divided into the number of mean days, or days of average length, that fit into that period of time.  For the sidereal year, or the time for the earth’s return to the same place among the stars is 365.25634 mean solar days. 

The equinoctial or common year, which is the time taken for the sun to pass from vernal (spring) equinox to vernal equinox is 365.242216 mean solar days, or 365 days 5 hours 48 minutes and 49.5 seconds. This difference is caused by the rotation of the earth on its axis counterclockwise as it orbits the sun counterclockwise and is referred to as precession.  The difference between true solar time and mean solar time for each day is called the equation of time. It is from these small daily differences that the whole concept of the leap year evolves.  

Keeping Precision Makes More Complication

It is quite obvious now how we came to have 365 days in a year, but there are still errors. It is evident that when we celebrate our birthday exactly 365 days from our time of birth we are actually celebrating 5 hours 48 minutes and 49.5 seconds too early as we would not have reached a full cycle around the sun. A quick bit of arithmetic indicates the difference over 4 years to be 23 hours 15 minutes and 18 seconds.  This is how the leap year was introduced on 29 February. 

Those who are particularly quick with the mental arithmetic will notice that having a leap year every 4 years will not quite get us onto the correct schedule either. Our current calendar, the Gregorian Calendar, was adopted in 1582 and by convention the calendar system deals with these issues through adjustments to the number of days in various years. The adjustment allows for a leap year every 4 years, but this has the calendar accruing ahead of the solar year by 44 minutes and 45 seconds every 4 years.  If we round this to 0.75 of an hour we see that in 128 years the count will be a full day behind, thus once every 100 years the leap year is not observed.

But then, after 400 years the day count becomes 26.88 hours ahead, so if the year is divisible by 400 then the leap year is observed.  If you recall there was a Leap Year in 2000 but there will be no leap year in 2100, and this over time keeps the Gregorian calendar in sync (or at least close enough for all intents and purposes) with the position of the earth in relation to the sun.

More Accurate Leap Year System

This is a very intricate process to ensure that we experience regular days that are, more or less, synchronized with dawn and dusk.  There is, though, a more mathematically accurate method for dealing with the errors between solar days, mean solar days and sidereal days.  Edmund Beckett, Lord Grimthorpe initially put forward this proposal in the 19th century in his paper called ‘Astronomy’ and showed conclusively that a more complete correction would be made by simply dropping every 32nd leap-day, or the leap day every 128 years.  But once again, having a regular and convenient process trumped accuracy in the development of our system of managing leap years.

There are a number of further idiosyncrasies that have crept into our time measuring system, you can read about more of these in “A Rudimentary Treatise on Clocks Watches and Bells for Public Purposes” – it is itself an idiosyncratic review of timekeeping of another time.  We are giving it away free, just download it here.