Equation Of Time

by Dominic Ford, Editor
Equation Of Time

The Earth, as seen by Apollo 17

The equation of time is the offset between mean time and solar time, which varies through the year.

Mean time is the time that is shown by clocks. The length of the second is defined by some physical process – for example, the oscillations of a crystal or the swinging of a pendulum – which remains constant at all times.

Solar time is judged by the position of the Sun in the sky, where midday is the moment when the Sun appears highest in the sky. In solar time, the length of each second is one 86,400th of the time it takes the Sun to complete a circuit around the sky.

Over the course of the year, these two measurements of time deviate by up to 15 minutes, in a fixed pattern which repeats every year.

Sidereal time

The Earth's daily cycle of day and night is usually explained by saying that the Earth spins about its polar axis once every 24 hours. As it does so, the Sun revolves around the sky. Once in each revolution, the Sun is highest in the sky, at noon. Twelve hours later, at midnight, it is lowest in the sky – usually some distance below the horizon.

The truth is a little more complicated. The first problem with this simple picture is that the Earth doesn't rotate once every 24 hours, but a little faster than that. It turns once every 23 hours and 56 minutes.

This is the period with which stars of the night sky appear to revolve around the celestial poles, called the sidereal day. Any given star will rise at exactly the same point on an observer's horizon every 23 hours and 56 minutes.

The time of day, however, is defined not by the positions of stars in the sky, but by the position of the Sun.

Unlike the stars, the Sun moves appreciably across the celestial sphere from one day to the next – by around a degree, or twice its diameter. This motion is in the opposite direction to the sky's apparent daily rotation. After each sidereal day, the stars return to the same positions, but the Sun is a full two Sun-widths behind where it appeared the previous day. It takes the sky's rotation just under four minutes to travel this extra distance.

As a result, stars traverse the sky once every 23 hours and 56 minutes, but the Sun does so once every 24 hours.

Seasonal changes

In practice, there are additional complications. This four-minute difference between the sidereal day and the solar day is not the same every day.

The chart below shows the Sun's annual path through the constellations – along the ecliptic – projected onto a rectangular grid of right ascension and declination.

The Sun moves leftward across the map by an average of one degree (four minutes in right ascension) each day. But its path is not a straight line.

At the solstices, the Sun's movement in declination comes to a standstill at the tropics. The Sun is moving horizontally across the map above, with no up-down motion.

As the Sun passes the celestial equator at the equinoxes, it moves along a diagonal path with considerable up-down motion. At these times, the Sun's rate of motion in declination is at its greatest.

There is a corresponding variation in the Sun's rate of movement in right ascension. At the equinoxes, when it traces a diagonal line, it moves comparatively slowly in right ascension. Conversely, at the solstices, when the Sun moves along a horizontal line, its right ascension changes comparatively quickly.

The Earth's elliptical orbit

There is a further complication. The Earth's orbit is not quite circular, but oval shaped (an ellipse). It makes its closest approach to the Sun around January 3, and is furthest from the Sun around July 4.

According to Kepler's Laws of planetary motion, planets move more quickly when they are closer to the Sun, and more slowly when they are further from the Sun. The Earth is no exception, and so the Sun's apparent rate of motion along the ecliptic is not completely steady over the course of the year. It moves fastest in January, and slowest in July.

Combined, these effects cause the length of each day to vary by up to 20 seconds through the year. In June and December, it takes the Sun less time to get from one midday to the next as compared to March and September.

As a result, in local solar time the length of the second changes slightly depending on the time of year. Seconds pass more quickly in June and December than in March or September.

With the advent of accurate mechanical clocks, it has become preferably to find a definition of the length of the second that is constant throughout the year. For clockmakers, this means their mechanisms do not need to run at different rates on different days of the year. For scientists, it means that experiments which involve making precise measurements of how long processes take do not suffer inaccuracies due to the units of time themselves varying in length.

Mean time

Mean time is just such a system. The length of the second is constant throughout the year, defined to be one 86,400th of the average time it takes the Sun to move from noon (when the Sun is highest in the sky) on one day to noon on the following day.

On any given day of the year, the length of the solar day may differ by up to 20 seconds from this, and since these errors accumulate day by day, the time at which noon occurs according to a clock showing mean time may drift at a rate of up to ten minutes per month relative to solar time.

However, the length of the mean day is chosen such that there is an exact balance between the times of year when noon drifts earlier in the day, and those when it drifts later in the day. After one whole year has elapsed, noon has returned to exactly the same mean time.

The equation of time

In a mean time system, the time interval between noon (when the Sun is highest) and 12:00 is called the equation of time, where the word equation is used in a historical sense, not to mean an algebraic expression but rather a correction.

Although the equation of time varies by up to 20 minutes through the course of the year, it repeats almost exactly the same pattern each year, since the Earth follows almost exactly the same path around the Sun from one year to the next.

The figure below shows difference, in minutes, between solar time and mean time over the course of the year (click to enlarge it). It is also available in pdf format.

The Sun's analemma

Suppose that a photograph is taken at exactly 12:00 (mean time) each day, showing the Sun's position in the sky.

As the seasons change, the Sun's altitude at midday changes: it culminates at a higher altitude in the summer than in the winter.

However, the Sun's east-west position also varies through the year because of the equation of time. At some times of year, it appear further to the east, approaching its highest point in the sky. At other times it appears further to the west, having already passed its highest point in the sky.

The result is a figure-of-eight pattern called an analemma. The diagram below shows he analemma that would be recorded by a northern-hemisphere observer, with circles indicating the position of the Sun. The same pattern would also be seen in the southern hemisphere, viewed upside down and with the summer and winter solstices swapped.




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