Equation of Time

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Most of us forget that our modern 24 hour day, which has exactly 86,400 seconds, doesn't actually exist except in our system of timekeeping. Two successive sunrises or sunsets are almost never exactly 24 hours to the second, and few of us live where the sun is exactly overhead at noon. There are always differences and errors, and astronomers and physicists have developed a very accurate time keeping system that we use to reference what we see in the world around us. The variation between the real movements of the sun and our standard 24 hour clock is called the Equation of Time.

There are two specific main reasons for the Equation of Time: 1. The eccentricity of the Earth’s orbit causes us to speed up and slow down in different parts of our annual orbit. 2. The Earth|Earth axis is tilted to our orbit, and so the Sun’s apparent motion along the (tilted) elliptic has a varying effect when viewed along the equatorial plane.

We will show the correct reasoning and then the mathematics to be able to calculate this effect. There are a couple of other reasons why a sundial might not be precisely accurate. One is a necessary correction between the actual Longitude of the sundial and the central Longitude of the time zone there, where four minutes correction is necessary for each degree difference. The other is related to the fact that our year is close to one-fourth day different from an integer number of days, and so there is a cycling over each four years regarding a slight shift of this Equation of Time pattern.

There are very slow changes in two of the quantities upon which the Equation of Time depends. We include an extremely accurate clock which determines the difference between the modified clock formula based on the Equation of Time and the sidereal time over a number of years. The clock is accurate to around 0.01 second and can be simulated to the number of years input by the user (up to 10³⁰⁰ years).

History

Prior to around 500 years ago, there were not yet any mechanical clocks and so daily life was generally based on the apparent motions and positions of the Sun and Moon in the sky. An interesting detail is that they decided to define an interval which we now call an hour as being 1/12 of the daylight period on that day. The result was that an hour in the summer was much longer than an hour in the winter! But intervals of time were not yet that important to anyone! Ancient Greeks had invented crude water clocks, which therefore obviously had some major problems of their own, but they also discovered that there were some other variations, such as where the Sun seemed to cross the Meridian Circle ahead of or behind where they might have expected it to have been at noon, over a range of what we would now call over half an hour. (about February 12 compared to November 3).

When clocks were invented around 500 years ago, and then greatly improved in accuracy around 250 years ago, the design of those mechanisms were all based on processes that occurred at precisely accurate and repeatable intervals. And so it was natural that the length of a day or hour or year was defined as a specific interval of time. This was a wonderful advance but it caused some minor errors when it was used to try to accurately calculate where the Sun or Moon might be at any moment, specifically at noon! In fact, around November 3 of each year, the two clocks are different by more than 16 minutes! Around February 12, they are different by about 14 minutes in the opposite direction!

The fact that Mariners needed to accurately know the time in order to determine their Longitude and Latitude, made this a serious issue to deal with! Since that was the only use of really accurate timekeeping, the number that we now call the Equation of Time was initially defined as always being in one direction, based on the operation of ships. Much later, when accurate time became useful for business and commerce, the current definition was adopted, where the True Sun is sometimes ahead of and sometimes behind clock time.

We will call Mean Solar Time as being what we call clock time (with Mean essentially meaning Average), and we will call Apparent Solar Time as being the time which should be true based on the apparent location of the Sun at that instant. In a more technical way, we define the first as being the Hour Angle of the Mean Sun and the second as the Hour Angle of the True Sun. The Mean Sun is fictitious, and is defined as moving with uniform angular speed along the Celestial Equator.

These can equally be defined as the Right Ascensions of the Mean Sun and the True Sun.

The Equation of Time continuously varies throughout the year, from the combined effect of two primary causes. (There are many very tiny effects which also exist, mostly due to gravitational perturbation effects of other planets, and some interesting consequences of the Moon's motion, but they are generally very tiny effects and mostly considered to simply be perturbations of the two primary effects: (1) Eccentricity of the Earth’s orbit and (2) Obliquity of the Earth’s Elliptic orbit.)

A couple centuries ago, the definition was opposite the sign of what is currently defined, which causes some confusion regarding the correct sign. According to modern definitions, if the Equation of Time is a negative number, that is an indication that the true Sun has moved ahead of the Mean Sun, and therefore a sundial would give a time that was fast, and so the correction needs to be negative to get to Clock time.

The two primary causes of variation are the varying speed of the Earth in its orbit due to the eccentricity of that orbit (and Kepler and Newton's analysis of the speed variations which result); and due to the fact that the Earth's rotational axis is tilted to the plane of its orbit, which is called the Obliquity of the Ecliptic, which is essentially a geometric factor.