On Clocks and Sundials, Sunsets and Solstices.

If you put a vertical stake in the ground, and observe every day, the time when its shadow points due south, you may come up with a surprising result.  If your clock is accurate enough you will find that the interval between the so-called "solar noon" from one day to the next is not exactly twenty-four hours.  In fact you will find that it varies from day to day.  

What this means is that "solar time" (i.e. time measured on an sun dial) and time as measured by a terrestrial clock - say a pendulum clock or the digital watch on your wrist are not exactly the same.  We'll come to the reason for this in a moment.  But first let me show you the magnitude of the difference.  It varies throughout the year.  The following shows the difference between the two.

The horizontal axis is the day of the year.  The vertical axis is the difference (in minutes) between clock time and solar time.  Above the horizontal axis the clock is ahead of the sundial and below the axis it is the other way around.  So now in mid-December the clock is slightly ahead of the sundial, but soon it will change (on December 25 to be precise) and then will be behind (until 15th April).  The relationship between solar time and clock time captured in the graph is known as the Equation of Time.  

Now for the explanation.  Those with a mathematical background might suspect that the above graph is the sum of two sine functions with different phases, amplitudes and periods, and indeed that is what they are.  That is because the phenomenon is due to two causes.  

The first is that the earth's orbit is elliptic and not circular.  This means that the velocity of the earth varies throughout the year as it orbits the sun.  It is fastest at the two "pointed ends" of the ellipse which occur in mid-winter (3rd January) and mid-summer (3rd. July) and slowest on the "flat ends" in Spring and Fall.   The consequence of the variation in earth's speed is that the time from solar noon to solar noon varies. This is because when the earth is travelling faster it will take more time to complete a rotation to face the sun (because the earth has moved around the sun - from position 1 to position 3 in diagram below) than it would when it was travelling slower i.e the distance from 1 to 3 is greater in January and July than in March and September, so the angle to the sun is from 1 and 3 is greater in the former than the latter. 

This implies that in mid-summer and mid-winter, the interval between solar noons  should be larger than in the Fall and Spring. This yields the larger amplitude sinusoid (period one year).  

The other reason for the difference relates to the tilt of the earth (which gives rise to the seasons).  I won't go into details, which can be found in Wikipaedia at 

https://en.wikipedia.org/wiki/Equation_of_time

This is the reason behind the second, shorter period (six month), sinusoid.  

When these two sinusoids are summed one gets the equation of time graph above.

Apparently the variations in solar time were known to the ancient Babylonians (see Wiki article)  and Ptolemy tabulated the equation of time.  Ptolemy lived in Roman Egypt and was born about 100 CE.  In the Middle Ages various Islamic astronomers made improvements to Ptolemy's tables, especially the contribution due to the earth's tilt. 

One consequence of the difference between solar time and clock time is a fact which the very observant may have noted at this time of year. It is that the time of sunset has been gradually getting later since December 8 here in Victoria, BC (since Dec. 11th in London) even though the shortest day is not until 21st. of the month.  On the other hand, the time of sunrise will continue to get later until early in the New Year.  

In the December, up to the 25th, solar time is earlier than clock time, but with the difference diminishing as the month progresses.  This means that sunrises and sunsets are earlier than they would be if the earth's orbit were circular.  This pushes the earliest sunset to occur before the solstice (December 21st.) and the latest sunrise to occur after the solstice.

I am filled with admiration for the astronomers and mathematicians, both ancient and more modern, who worked all of this out.  The motion of the celestial spheres has been a continuing area of inquiry for some of the greatest mathematical minds of the ages. Apparently since time of Kepler (around 1600) many luminaries have studied planetary orbits and their  stability including

Laplace, Lagrange, Gauss, Poincaré, Kolmogorov, Vladimir Arnold, and Jürgen Moser.  

A heavenly galaxy of stars indeed!