The equation of time describes the discrepancy between two kinds of solar time. The two times that differ are the apparent solar time, which directly tracks the diurnal motion of the Sun, and mean solar time, which tracks a theoretical mean Sun with uniform motion along the celestial equator. Apparent solar time can be obtained by measurement of the current position (hour angle) of the Sun, as indicated (with limited accuracy) by a sundial. Mean solar time, for the same place, would be the time indicated by a steady clock set so that over the year its differences from apparent solar time would have a mean of zero.
The equation of time is the east or west component of the analemma, a curve representing the angular offset of the Sun from its mean position on the celestial sphere as viewed from Earth. The equation of time values for each day of the year, compiled by astronomical observatories, were widely listed in almanacs and ephemerides.
The equation of time can be approximated by a sum of two sine waves:
[minutes]
where:
where represents the number of days since 1 January of the current year, .
Concept
During a year the equation of time varies as shown on the graph; its change from one year to the next is slight. Apparent time, and the sundial, can be ahead (fast) by as much as 16 min 33 s (around 3 November), or behind (slow) by as much as 14 min 6 s (around 11 February). The equation of time has zeros near 15 April, 13 June, 1 September, and 25 December. Ignoring very slow changes in the Earth's orbit and rotation, these events are repeated at the same times every tropical year. However, due to the non-integral number of days in a year, these dates can vary by a day or so from year to year. As an example of the inexactness of the dates, according to the U.S. Naval Observatory's Multiyear Interactive Computer Almanac the equation of time was zero at 02:00 UT1 on 16 April 2011.
The graph of the equation of time is closely approximated by the sum of two sine curves, one with a period of a year and one with a period of half a year. The curves reflect two astronomical effects, each causing a different non-uniformity in the apparent daily motion of the Sun relative to the stars:
the obliquity of the ecliptic (the plane of the Earth's annual orbital motion around the Sun), which is inclined by about 23.44 degrees relative to the plane of the Earth's equator; and the eccentricity of the Earth's orbit around the Sun, which is about 0.0167.
The equation of time vanishes only for a planet with zero axial tilt and zero orbital eccentricity. Two examples of planets with large equations of time are Mars and Uranus. On Mars the difference between sundial time and clock time can be as much as 50 minutes, due to the considerably greater eccentricity of its orbit. The planet Uranus, which has an extremely large axial tilt, has an equation of time that makes its days start and finish several hours earlier or later depending on where it is in its orbit.
Notation
The United States Naval Observatory states "the Equation of Time is the difference apparent solar time minus mean solar time", i.e. if the sun is ahead of the clock the sign is positive, and if the clock is ahead of the sun the sign is negative. The equation of time is shown in the upper graph above for a period of slightly more than a year. The lower graph (which covers exactly one calendar year) has the same absolute values but the sign is reversed as it shows how far the clock is ahead of the sun. Publications may use either format: in the English-speaking world, the former usage is the more common, but is not always followed. Anyone who makes use of a published table or graph should first check its sign usage. Often, there is a note or caption which explains it. Otherwise, the usage can be determined by knowing that, during the first three months of each year, the clock is ahead of the sundial. The mnemonic "NYSS" (pronounced "nice"), for "new year, sundial slow", can be useful. Some published tables avoid the ambiguity by not using signs, but by showing phrases such as "sundial fast" or "sundial slow" instead.
History
The phrase "equation of time" is derived from the medieval Latin aequātiō diērum, meaning "equation of days" or "difference of days". The word equation is used in the medieval sense of "reconciliation of a difference". The word aequātiō (and Middle English equation) was used in medieval astronomy to tabulate the difference between an observed value and the expected value (as in the equation of the centre, the equation of the equinoxes, the equation of the epicycle). Gerald J. Toomer uses the medieval term "equation", from the Latin aequātiō (equalization or adjustment), for Ptolemy's difference between the mean solar time and the apparent solar time. Johannes Kepler's definition of the equation is "the difference between the number of degrees and minutes of the mean anomaly and the degrees and minutes of the corrected anomaly."
The difference between apparent solar time and mean time was recognized by astronomers since antiquity, but prior to the invention of accurate mechanical clocks in the mid-17th century, sundials were the only reliable timepieces, and apparent solar time was the generally accepted standard. Mean time did not supplant apparent time in national almanacs and ephemerides until the early 19th century.
Early astronomy The irregular daily movement of the Sun was known to the Babylonians.
Book III of Ptolemy's Almagest (2nd century) is primarily concerned with the Sun's anomaly, and he tabulated the equation of time in his Handy Tables. Ptolemy discusses the correction needed to convert the meridian crossing of the Sun to mean solar time and takes into consideration the nonuniform motion of the Sun along the ecliptic and the meridian correction for the Sun's ecliptic longitude. He states the maximum correction is time-degrees or of an hour (Book III, chapter 9). However he did not consider the effect to be relevant for most calculations since it was negligible for the slow-moving luminaries and only applied it for the fastest-moving luminary, the Moon.
Based on Ptolemy's discussion in the Almagest, values for the equation of time (Arabic taʿdīl al-ayyām bi layālayhā) were standard for the tables (zij) in the works of medieval Islamic astronomy.Reprinted in:
Early modern period
A description of apparent and mean time was given by Nevil Maskelyne in the Nautical Almanac for 1767: "Apparent Time is that deduced immediately from the Sun, whether from the Observation of his passing the Meridian, or from his observed Rising or Setting. This Time is different from that shewn by Clocks and Watches well regulated at Land, which is called equated or mean Time." He went on to say that, at sea, the apparent time found from observation of the Sun must be corrected by the equation of time, if the observer requires the mean time.
The right time was originally considered to be that which was shown by a sundial. When good mechanical clocks were introduced, they agreed with sundials only near four dates each year, so the equation of time was used to "correct" their readings to obtain sundial time. Some clocks, called equation clocks, included an internal mechanism to perform this "correction". Later, as clocks became the dominant good timepieces, uncorrected clock time, i.e., "mean time", became the accepted standard. The readings of sundials, when they were used, were then, and often still are, corrected with the equation of time, used in the reverse direction from previously, to obtain clock time. Many sundials, therefore, have tables or graphs of the equation of time engraved on them to allow the user to make this correction.
The equation of time was used historically to set clocks. Between the invention of accurate clocks in 1656 and the advent of commercial time distribution services around 1900, there were several common land-based ways to set clocks. A sundial was read and corrected with the table or graph of the equation of time.
If a transit instrument was available or accuracy was important, the sun's transit across the meridian (the moment the sun appears to be due south or north of the observer, known as its culmination) was noted; the clock was then set to noon and offset by the number of minutes given by the equation of time for that date. A third method did not use the equation of time; instead, it used stellar observations to give sidereal time, exploiting the relationship between sidereal time and mean solar time. The more accurate methods were also precursors to finding the observer's longitude in relation to a prime meridian, such as in geodesy on land and celestial navigation on the sea.
The first tables to give the equation of time in an essentially correct way were published in 1665 by Christiaan Huygens. Huygens, following the tradition of Ptolemy and medieval astronomers in general, set his values for the equation of time so as to make all values positive throughout the year. This meant that any clock being set to mean time by Huygens's tables was consistently about 15 minutes slow compared to today's mean time.
Another set of tables was published in 1672–73 by John Flamsteed, who later became the first Astronomer Royal of the new Royal Greenwich Observatory. These appear to have been the first essentially correct tables that gave today's meaning of Mean Time (previously, as noted above, the sign of the equation was always positive and it was set at zero when the apparent time of sunrise was earliest relative to the clock time of sunrise). Flamsteed adopted the convention of tabulating and naming the correction in the sense that it was to be applied to the apparent time to give mean time.
The equation of time, correctly based on the two major components of the Sun's irregularity of apparent motion, was not generally adopted until after Flamsteed's tables of 1672–73, published with the posthumous edition of the works of Jeremiah Horrocks.
Robert Hooke (1635–1703), who mathematically analyzed the universal joint, was the first to note that the geometry and mathematical description of the (non-secular) equation of time and the universal joint were identical, and proposed the use of a universal joint in the construction of a "mechanical sundial".
18th and early 19th centuries
The corrections in Flamsteed's tables of 1672–1673 and 1680 gave mean time computed essentially correctly and without need for further offset. But the numerical values in tables of the equation of time have somewhat changed since then, owing to three factors:
General improvements in accuracy that came from refinements in astronomical measurement techniques, Slow intrinsic changes in the equation of time, occurring as a result of small long-term changes in the Earth's obliquity and eccentricity (affecting, for instance, the distance and dates of perihelion), and The inclusion of small sources of additional variation in the apparent motion of the Sun, unknown in the 17th century but discovered from the 18th century onwards, including the effects of the Moon (See barycentre), Venus and Jupiter.