Equation of time

During the course of the year, the time as read from a sundial can run ahead of clock time by as much as 16 min 33 s (around October 31–November 1) or fall behind by as much as 14 min 6 s (around February 11–12). This difference, known as the equation of time, results from an apparent irregular movement of the Sun caused by a combination of the obliquity of the Earth's rotation axis and the eccentricity of its orbit. The equation of time is visually illustrated by an analemma.

Naturally, other planets will have an equation of time too. On Mars the difference between sundial time and clock time can be as much as 50 minutes, due to its orbit's considerably greater eccentricity.

Apparent time versus mean time

The irregular daily movement of the Sun was known by the Babylonians, and Ptolemy has a whole chapter in the Almagest devoted to its calculation (Book III, chapter 9). However he did not consider the effect relevant for most calculations as the correction was negligible for the slow-moving luminaries. He only applied it for the fastest-moving luminary, the moon.

Until the invention of the pendulum and the development of reliable clocks towards the end of the 17th century, the equation of time as defined by Ptolemy remained a gadget, not important to normal people except astronomers. Only when mechanical clocks started to take over timekeeping from sundials, which had served humanity for centuries, the difference between clock time and solar time became an issue. A new definition of the equation of time was developed, the one which is still in use today.

Apparent solar time (or true or real solar time) is the time indicated by the sun on a sundial, while mean solar time is the average as indicated by the clocks. The equation of time is apparent minus mean solar time.

As the daily movement of the sun is one revolution per day, that is 360° per 24 hours or 1° per 4 minutes, and the sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, clearly the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections due to one's offset from the local time zone meridian and summertime, if any.

The tiny increase of the mean solar day itself due to the slowing down of the earth's rotation, by about 2 ms per day per century, which currently accumulates up to about 1 second every year has nothing to do with the equation of time, and is completely irrelevant at the accuracy given by sundials.

The eccentricity of the Earth's orbit

The Earth revolves around the sun. As such it appears that the Sun moves in one year around the Earth. If the sun moved with a constant speed and along the celestial equator, then it would culminate every day at exactly 12 o'clock, and be a perfect time keeper. But the earth's orbit is an ellipse, and as such the sun seems to move faster at perihelion (currently around 3 January) and slower at aphelion a half year later, according to Kepler's laws of planetary motion. At these extreme instances this effect increases (respectively, decreases) the real solar day by 7.9 sec. This accumulates every day. The final result is that the eccentricity of the earth's orbit contributes a sine wave variation with an amplitude of 7.66 minutes and a period of one year to the equation of time. The zero points are reached on perihelion (at the beginning of January) and aphelion (beginning of June) while the maximum values are at the beginnings of April (negative) and October (positive).

The obliquity of the ecliptic

The sun does not move along the celestial equator but rather along the ecliptic. At the equinoxes part of the yearly movement of the sun goes in a component for the change in declination leaving less for the component in right ascension. The sun slows down by up to 20.3 sec per day. At the solstices on the other hand, all yearly movement is in right ascension only, but at this declination, 23°4, the meridians are closer together, which speeds up the sun by the same amount. The inclination of the ecliptic results in the contribution of another sinewave variation with an amplitude of 9.87 minutes and a period of a half year to the equation of time. The zero points are reached on the equinoxes and solstices, while the maxima are at the beginning of February and August (negative) and the beginning of May and November (positive).

Secular effects

The two above mentioned factors have different wavelength, amplitude and phase, so their combined contribution is an irregular wave. At epoch 2000 these are the values:

minimum	−14:15	11 February
zero	00:00	15 April
maximum	+3:41	14 May
zero	00:00	13 June
minimum	−06:30	26 July
zero	00:00	1 September
maximum	+16:25	3 November
zero	00:00	25 December

E.T = apparent − mean. Positive means: sun runs fast and culminates earlier, or the sundial is ahead of mean time. A slight yearly variation occurs due to presence of leap years, resetting itself every 4 years.

The exact shape of the equation of time curve and the associated analemma slowly changes over the centuries due to secular variations in both eccentricity and obliquity. At this moment both are slowly decreasing, but in reality they vary up and down over a timescale of hundredthousands of years. When the eccentricity, now 0.0167, reaches 0.047 the eccentricity effect may in some circumstances overshadow the obliquity effect leaving the equation of time curve with only one maximum and minimum per year.

On shorter timescale (thousands of years) the shifts in the date of equinox and perihelion will be more important. The former is caused by the precession, and shifts the equinox backwards compared to the stars. But it can be ignored in the current discussion as our Gregorian calendar is constructed in such a way to keep the vernal equinox date at 21 March (at least at sufficient accuracy for our aim here). The shift of the perihelion is forwards, about 1.7 days per century. For example in 1246 the perihelion occurred on 22 December, the day of the solstice. At that time the two contributing waves had common zeropoints, and the resulting equation of time curve was symmetrical. Before that time the February minimum was larger than the November maximum, and the May maximum larger than the July minimum. The secular change is evident when one compares a current graph of the equation of time (see below) with one of about 2000 year ago, for example constructed from the data of Ptolemy.

Practical use

If the gnomon (the shadow casting object) is not an edge but a point (e.g., a hole in a plate), the shadow (or spot of light) will trace out a curve during the course of a day. If the shadow is cast on a plane surface, this curve will (usually) be the conic section of the hyperbola, since the circle of the Sun's motion together with the gnomon point define a cone. At the spring and fall equinoxes, the cone degenerates to a plane and the hyperbola to a line. With a different hyperbola for each day, hour marks can be put on each hyperbola which include any necessary corrections. Unfortunately, each hyperbola corresponds to two different days, one in each half of the year, and these two days will require different corrections. A convenient compromise is to draw the line for the "mean time" and add a curve showing the exact position of the shadow points at noon during the course of the year. This curve will take the form of a figure eight and is known as an "analemma". By comparing the analemma to the mean noon line, the amount of correction to be applied generally on that day can be determined.