What is the equation of time?

What is the equation of time?

The equation of time is the difference, in minutes, between the time a sundial reads at a particular moment and the time a clock running on uniform time reads at the same moment. The U.S. Naval Observatory states the definition exactly: "the Equation of Time is the difference apparent solar time minus mean solar time."¹ Apparent solar time is what the actual Sun in the sky shows — the time told directly by the Sun's position relative to the local meridian. Mean solar time is the steady time scale a clock keeps, defined by an imaginary "mean Sun" that moves at a perfectly uniform rate along the celestial equator and completes one circuit in exactly 24 hours.¹

The sign convention is fixed. The equation of time is positive when the true Sun is ahead of the mean Sun in the sky — equivalently, when a sundial reads "fast" relative to a mean-solar-time clock at the same location. A value of +10 minutes on a given date means a sundial reads ten minutes later than local mean solar time; a value of −10 minutes means it reads ten minutes earlier.¹ An equivalent framing comes from solar noon: NOAA describes the equation of time as "an astronomical term accounting for changes in the time of solar noon for a given location over the course of a year."⁶ Solar noon arrives ahead of or behind 12:00 mean solar time at any location by exactly the number of minutes that the equation of time gives that day.

The word "equation" in the term carries its older meaning of "reconciliation of a difference," from the medieval Latin aequātiō diērum, "equation of days." It is not an algebraic identity; it is a correction table.³

What does its annual curve look like?

The equation of time traces a smooth, irregular curve through the year. Its cumulative magnitude is small — USNO notes that "the fractional change in the rate of the Sun's apparent daily motion is tiny (about 0.03%)" — but the day-by-day offsets accumulate into a peak-to-peak swing of about 30 minutes.¹ Britannica gives the same upper bound: "a correction, never exceeding 16 minutes."²

The two annual extremes are unequal. The sundial runs fastest at +16 min 33 s around 3 November, and slowest at −14 min 6 s around 11 February.³ Between those extremes the curve has a smaller positive hump in mid-May (peaking near +3.7 minutes) and a smaller negative trough in late July (bottoming near −6.5 minutes). The curve crosses zero — apparent and mean solar time agree exactly — four times a year, near 15 April, 13 June, 1 September, and 25 December.³ Those four zero-crossings are why a sundial in a public square is a passable timepiece on those specific dates and a poor one in between.

What causes it?

Two independent physical effects combine algebraically into the equation of time. They are roughly comparable in size but have different periods, so they slip in and out of phase through the year. The composite curve has its zero-crossings where the two terms briefly cancel, and its extremes where they add constructively.¹

The first effect is the obliquity of the Earth's axis, tilted about 23.44° from the perpendicular to the orbital plane. The Sun's apparent path on the sky is the ecliptic — a great circle inclined to the celestial equator at the same angle. Mean solar time, by contrast, is referenced to a fictitious mean Sun moving uniformly along the celestial equator. Even if the Earth's orbit were perfectly circular, the projection of the actual Sun's motion onto the equator (the quantity that defines apparent solar time at the meridian) would not be uniform: it runs faster near the equinoxes and slower near the solstices. This produces a sinusoidal contribution that completes two cycles per year. USNO: "The part of the Equation of Time due just to this effect is zero at the equinoxes and solstices, and can reach ±10 minutes at other times of the year."¹

The second effect is the eccentricity of the Earth's orbit, about 0.0167. The Earth moves faster near perihelion in early January and slower near aphelion in early July, by Kepler's second law. Faster orbital motion means the Sun appears to advance slightly faster eastward against the stars, which slows its rate of return to the meridian; slower orbital motion does the reverse. The contribution is a sinusoidal term that completes one cycle per year. USNO: "the cumulative effect of these changes in orbital speed, with the largest offsets in early April (–7.5 minutes) and early October (+7.5 minutes)."¹

The two terms have very different shapes — semi-annual versus annual — and their phases do not line up neatly, because perihelion and aphelion fall a few weeks after the December and June solstices respectively, not at them.⁴ The result is the asymmetric four-extreme curve described in the previous section: the early-November maximum is the largest because both terms are pulling positive at the same time; the mid-February minimum is the deepest negative excursion for the same reason; and the smaller mid-May and late-July features are where the two contributions partially cancel.

How is it calculated?

The standard implementation in modern solar-position software is a five-term truncated series, derived from Jean Meeus' Astronomical Algorithms and shipped, in effectively the same form, in NOAA's solar calculator.⁷⁸ With y = tan²(ε/2) (ε is the corrected obliquity of the ecliptic), e the orbital eccentricity, L₀ the Sun's geometric mean longitude, and M the Sun's geometric mean anomaly:⁵

E = y · sin 2L₀ − 2e · sin M + 4ey · sin M · cos 2L₀ − ½y² · sin 4L₀ − ⁵⁄₄ · e² · sin 2M

The output is in radians; multiply by 180/π and again by 4 minutes per degree to convert to minutes of time. The structure of the expansion can be read directly: each term is a sinusoid in either L₀ or M, or a product of both. The leading term y · sin 2L₀ is the obliquity contribution at twice-per-year frequency; the second term −2e · sin M is the eccentricity contribution at once-per-year frequency; the remaining three are smaller cross-terms.⁵

The U.S. Naval Observatory publishes a parallel approximate solar-coordinates algorithm, "essentially the same as that found on page C5 of The Astronomical Almanac," which arrives at the same numerical answer through a different route. It computes the Sun's right ascension RA directly from the geometric mean longitude q, then recovers the equation of time as EqT = q/15 − RA, with q in degrees, RA in hours, and the division by 15 converting degrees to hours.⁹ Both algorithms reference the J2000.0 epoch (Julian date 2451545.0, corresponding to 2000 January 1.5 UT) as their time origin.⁹

The textbook implementation is more accurate than most readers expect. NOAA's solar-noon results are "theoretically accurate to within a minute for locations between +/- 72° latitude"; USNO claims approximately one arcminute of solar position accuracy within two centuries of the year 2000.⁷⁹ NOAA's web calculator is valid from −2000 to +3000 in calendar date; its spreadsheet version is restricted to 1901–2099 by an approximation in its Julian-day handling.⁷ The solar noon calculator on this site uses the same formula for any city and date.

Why does it matter?

The most direct application is sundial calibration. A sundial reads apparent solar time by construction; converting that reading to civil clock time requires three corrections — the longitude offset within the time zone, any active daylight saving time shift, and the equation of time at that date. The longitude term is constant for a given location, the daylight-saving term is constant within each season, but the equation of time changes day by day. It is therefore the only correction that needs a year-round table to be read off the sundial. Many fine sundials are inscribed with a small graph of the equation of time along the plinth or pedestal so the reader can compute the correction without a separate reference.¹

The equation of time is also embedded in every modern algorithm for solar noon, sunrise, and sunset. NOAA's solar-noon formula is the canonical example:⁵

solNoonLocal = 720 − 4·λ − E + 60·Z

The 720 is twelve hours expressed in minutes; 4·λ is the longitude correction (four minutes per degree east, since the Earth turns 15° per hour); E is the equation of time in minutes at that day's noon; and 60·Z converts the location's UTC offset back to local clock time. Sunrise and sunset are then computed as offsets from solar noon, so the equation of time propagates into them as well. At a given location, the gap between local clock noon and sunrise (or sunset) varies through the year by the same amount the equation of time varies — typically minutes, occasionally tens of minutes.⁵

The equation of time is also the resolution of a small puzzle that catches most readers off guard: at mid-northern latitudes, the earliest sunset of the year does not fall on the December solstice when day length is shortest. It typically falls a week or two earlier; the latest sunrise falls a week or two after the solstice. The asymmetry is the equation of time at work — its curve is sliding rapidly through its early-November maximum during late autumn, which lifts the entire daily clock pattern earlier and shifts the earliest sunset away from the actual solstice. The southern hemisphere sees the analogous offset around the June solstice. The sunrise & sunset calculator shows the effect for any specific location.

Mean solar time replaced apparent solar time as the basis for civil clocks gradually through the 18th and 19th centuries. Maskelyne's Nautical Almanac of 1767 was the first widely used reference to publish daily equation-of-time values; mean time "did not supplant apparent time in national almanacs and ephemerides until the early 19th century."³ Until that transition, public clocks were periodically reset against the local sundial; afterwards, the sundial was reset against the clock, with the equation of time as the bridge.

What is the analemma?

The most direct way to visualise the equation of time is the analemma — "a diagram showing the position of the Sun in the sky as seen from a fixed location on Earth at the same mean solar time over the course of a year."⁴ If the Sun is photographed at the same civil-clock instant on many dates throughout a year, the resulting trace is a slim figure-eight in the sky, not a single point.

The two coordinates of the figure-eight have direct astronomical meaning. The east–west extent is the equation of time itself: the angular displacement of the Sun east or west of where the mean Sun would be at the same instant, in degrees of arc, with one degree corresponding to four minutes of time. The vertical extent is the Sun's declination — its angular distance north or south of the celestial equator, which varies smoothly between −23.44° at the December solstice and +23.44° at the June solstice.⁶ The figure-eight's long axis is therefore about 47° from end to end, regardless of latitude; only the orientation of the figure on the sky changes with where on Earth the camera is placed.⁴

The two lobes of the figure-eight are unequal because perihelion and aphelion fall a few weeks after the December and June solstices respectively, which puts the once-per-year eccentricity term and the twice-per-year obliquity term slightly out of phase.⁴ The northern (boreal-summer) lobe is the smaller one in the conventional Earth-up orientation. The asymmetry of the analemma's two lobes is therefore the same asymmetry that produces the +16 min 33 s peak in November and the only-slightly-shallower −14 min 6 s trough in February.³

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