# Equation of time

The calculator plots the equation of time - the difference between apparent solar time and the mean solar time. It also draws the two components: the orbit eccentricity effect and the obliquity effect.

This calculator displays the equation of time. The equation of time is the difference between apparent solar time and mean solar time in minutes. Also, the individual components of the equation of time are displayed. If the graph is above zero, then apparent solar time is ahead of mean solar time. If below zero, it lags mean solar time.

To display the graph of the equation of time, we used the approximate formula given by Reingold and Dershowitz in the book Calendrical Calculations. ^{1}

## Apparent Solar Time

If you measure the length of the solar day with the help of an accurate clock, you will find the duration of a solar day is not equal to 24 hours.

This duration either increases or decreases from season to season. Besides the maximal difference of about 30 seconds, during several days, these differences accumulate and become noticeable. The accumulated difference between the time on a sundial and an ordinary clock can reach 16 minutes. Thus, the ** apparent solar time** displayed by the sundial runs unevenly and cannot be used to measure equal intervals of time with adequate precision.

## The reasons for the unevenness of the apparent solar time

Ptolemy found two main reasons for the irregularity of solar time:

An anomalistic solar day is the period comprising the passage of the 360 time-degrees of one revolution of the equator plus that stretch of the equator which rises with, or crosses the meridian with, the anomalistic motion of the sun [in that period].

This additional stretch of the equator, beyond the 360 time-degrees, which crosses [the horizon or meridian] cannot be a constant, for two reasons: firstly, because of the sun’s apparent anomaly; and secondly, because equal sections of the ecliptic do not cross either the horizon or the meridian in equal times. Neither of these effects causes a perceptible difference between the mean and the anomalistic return for a single solar day, but the accumulated difference

over a number of solar days is quite noticeable.^{2}

We see in this text of the 2nd century AD that the ancient astronomers correctly understood the two main reasons that affect the unevenness of the solar day. They are the tilt of the Earth's axis and the uneven motion of the Sun (read the Earth) relative to the stars.

### The effect of obliquity

During the solstices, the sun moves almost parallel to the celestial equator. Its speed of movement is deducted from the celestial sphere diurnal rotation speed to a greater extent. Therefore, near the solstices, the solar day duration is maximum. The sun moves at the maximum angle to the celestial equator during the equinoxes. The speed of its movement is deducted from the celestial sphere diurnal rotation speed to the smallest extent. This shortens the length of the solar day. The obliquity effect curve has a period of half a year. It passes at zero close to the times of the solstices and equinoxes.

### The effect of eccentricity

The Earth moves around the Sun by the ellipsoidal orbit. The Sun is at one of the focuses of the ellipsoid. According to Kepler's second law, the Earth speed at the closest point to the Sun (perihelion) is maximum. At the opposite place (aphelion), the Earth's velocity is minimal. Accordingly, at perihelion, solar days are lengthened the most, whereas at aphelion they are shortened. The eccentricity effect graph points close to zero correspond to the aphelion and perihelion. The period of this curve is one year.

## Mean Solar Time, Historical Reference

Despite the impossibility of direct measurement, the need to introduce the mean solar time arose among ancient astronomers.

Quoting Ptolemy again:

Now the [maximum] subtractive result from both effects

occurs over the interval from the middle of Aquarius to [the end of] Libra, and

the [maximum] additive one over the interval from [the beginning of] Scorpio to the middle of Aquarius. Both of these intervals produce a maximum additive or subtractive result which is composed of about 3⅔° due to the effect of the solar anomaly, and about 4⅔° due to the [variation in the time of] meridiancrossing.

Thus the maximum difference arising from the combination of both

the above effects is 8⅓ time-degrees, or 5/9ths of an hour, between the [true] solar days over either of these intervals and the [corresponding] mean solar days, and twice as much, 16⅔ time-degrees, or 1⅑ hours, between the [true] solar days of one such interval and those of the other. Neglect of a difference of this order would, perhaps, produce no perceptible error in the computation of the phenomena associated with the sun or the other [planets]; but in the case of the

moon, since its speed is so great, the resulting error could no longer be overlooked, since it could amount to 3/5 of a degree.^{3}

Ancient astronomers had to introduce the mean solar time to measure the exact movement of the Moon across the celestial sphere. The Moon, in turn, acted as a reference point for finding stars by the stellar catalog, so the accuracy of its movement was vital.

Mean solar time is convenient. Unlike apparent solar time, it flows evenly. Nowadays, any electronic or mechanical watch can measure it with high accuracy.

### Equation of time

There were no accurate clocks in ancient times. For many centuries it was necessary to be content with a sundial, which shows apparent solar time. The apparent solar time, as we found out above, goes unevenly. So, ancient astronomers used the equation of time math to evaluate the mean solar time by sundial readings. The mean solar time was convenient for calculating the motion of celestial bodies observed in the sky.

On the contrary, nowadays, watches that measure the mean solar time are more common. So knowing the equation of time, we can calculate the readings of a sundial.

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