Origin of the equation of time

Shadows is using the French convention for the equation of time: it is defined as the difference between the Mean Time and Solar Time (see below).

The Equation of Time provides the time difference between Solar Time and Mean Time. Solar Time is given by the Sun's Hour Angle; Mean Time is given by the division of a day by 24 hours of 3600 seconds (the second is a primary quantity defined from the light speed). Therefore the Solar Time is not synchronized with the Mean Time, because of variations of astronomical origin.

From a day to the following, the Earth has progressed on its orbit and the direction of the Sun has slightly changed (see the figure below). This variation is not constant over the year as the orbital speed varies with the distance to the Sun (the orbit is elliptic).

In addition, the Earth's axis is inclined on the ecliptic plane so the apparent orbit of the Sun. The projection of its position on the celestial equator introduces another periodical variation.

The ecliptic plane (in red) is inclined on the celestial equator (in bleu) by around 23. The A-B arc, travelled by the apparent Sun is projected in A'-B'. When the Sun is close to the vernal point (g) the A'-B' arc is shorter than the A-B arc (left above figure). It is the case around the equinoxes. When the Sun is closed to its maximum declination, around the solstices (right above figure), the A'-B' arc is longer than the A-B arc.

This two phenomenon are at the origin of the equation of time. They are superimposed and give the double-sinusoide shape of the curve.

Convention used for the equation of time

Two conventions are used for the equation of time: Mean Time - Solar Time (as in Shadows) or Solar Time - Mean Time (as in most English books). This convention reverses the curve but the values are to be used in an opposite manner, so that the result is the same.

This convention may be modified in the preferences Equation of time.

The following table explains how to use the value of the equation of time (EoT) given by the program depending on the convention:

Convention for the EoT

MT - ST*

ST - MT

To get the Solar Time from the Mean Time

subtract the EoT

add the EoT

To get the Mean Time from the Solar Time

add the EoT

subtract the EoT

* default value used by Shadows.

Note: adding a negative number is equivalent to subtracting this umber without its sign (12 plus (-5) equals: 12 minus 5 = 7). Subtracting a negative number is equivalent to adding this number without its sign (12 minus (-5) equals: 12 plus 5 = 17.

Using the equation of time

The following explanation is given for the convention used in Shadows (MT-ST) and in parenthesis for the opposite convention (ST-MT). When the equation of time is positive (negative), the length of the day (time difference between two crossings of the meridian) is greater to 24 hours. In this case the True Sun (defining the Solar Time) is late compared to the Mean Sun. The delay reached +/- 15 minutes depending on the date, which is not neglectable at all! It sometimes puzzles people looking at a sundial because the difference with the clock time is important (a lot of people think in Europe that they just need to add 1 hour during the winter and 2 hours during the summer to the Solar Time to get the clock time).

Values of the equation of time are slightly different from one year to the next. That is why it is calculated for the reference year, which can be changed in the Options menu. After four years, the values become the same again. Therefore the equation of time can be calculated for a given year or as a mean of four years.