Equation of Time

Two suns, two clocks

Imagine two suns crossing your sky. One is the real Sun — call it the apparent Sun. It rises, climbs, and reaches its highest point (true noon) at a moment that drifts a little earlier or later from one week to the next. A sundial tracks this Sun exactly; the shadow tells apparent solar time.

The other is a bookkeeping fiction astronomers invented to make timekeeping sane: the mean Sun. It travels along the celestial equator at a perfectly constant rate, completing one circuit in exactly one tropical year. Its crossings of the meridian define mean solar time, and from it we derive the uniform 24-hour day, the second, and ultimately the clock on your wall. The mean Sun never speeds up or slows down — that is the whole point of inventing it.

The equation of time is simply the running difference between these two:

EoT = (apparent solar time) − (mean solar time)

When EoT is positive, the real Sun is ahead of the clock — true noon arrives before clock noon, so the sundial reads "fast." When EoT is negative, the Sun is running behind and the sundial reads "slow." The word "equation" here is archaic: it means the correction you must add to apparent time to equate it with mean time — a usage that predates algebra-class equations entirely.

The first wave: eccentricity

Earth's orbit is not a circle but an ellipse, with eccentricity e ≈ 0.0167. By Kepler's second law, a planet sweeps equal areas in equal times, so Earth moves fastest at perihelion (closest approach, around January 3, ~147.1 million km) and slowest at aphelion (around July 4, ~152.1 million km). The difference in orbital speed is only about 3.4%, but it accumulates into a noticeable timing error.

Why does orbital speed touch the Sun's apparent motion? Because the Sun's eastward drift against the stars each day mirrors Earth's progress around its orbit. Near perihelion, Earth covers more orbital angle per day, so the Sun appears to slide eastward faster, the solar day stretches slightly past 24 hours, and apparent noon creeps later. Near aphelion the reverse happens. This produces a smooth annual sine with one peak and one trough per year — amplitude roughly ±7.7 minutes, with zeros at perihelion and aphelion.

The second wave: obliquity

The larger contributor is the obliquity — Earth's axial tilt of ε = 23.44°. The Sun travels along the ecliptic, the plane of Earth's orbit, but clocks are keyed to the celestial equator, which is tilted by ε relative to it. Even if the Sun moved at a perfectly uniform speed along the ecliptic, its projection onto the equator — the coordinate that actually governs the length of the solar day — would not be uniform.

Near the equinoxes the ecliptic crosses the equator at a steep angle, so a fixed step along the ecliptic projects to a shorter step in right ascension; near the solstices the ecliptic runs parallel to the equator, so the same step projects to a longer one. Geometrically this is exactly the spherical-trigonometry factor of cos ε at the solstices versus 1/cos ε at the equinoxes. The result is a semiannual sine — two peaks and two troughs a year, with zeros at the solstices and equinoxes — and an amplitude of about ±9.9 minutes.

Adding the two waves

The full equation of time is the sum of the annual eccentricity wave and the semiannual obliquity wave, slightly offset in phase. Because they have different periods and peaks, they reinforce in some seasons and cancel in others, producing an asymmetric curve with two unequal maxima and minima:

Equation of time through the year (apparent − mean, minutes; values rounded)

Date	EoT (min)	Sun vs. clock	What dominates
Feb 11	−14.2	Sun ~14 min slow	both terms negative
Apr 15	0	agree	terms cancel
May 14	+3.7	Sun ~4 min fast	obliquity peak
Jun 13	0	agree	terms cancel
Jul 26	−6.5	Sun ~7 min slow	obliquity trough
Sep 1	0	agree	terms cancel
Nov 3	+16.4	Sun ~16 min fast	both terms positive
Dec 25	0	agree	terms cancel

A useful closed-form approximation, accurate to better than a minute, is:

EoT ≈ 9.87·sin(2B) − 7.53·cos(B) − 1.5·sin(B) minutes, where B = 360°·(N − 81)/365 and N is the day of the year.

The first term is the semiannual obliquity wave; the cosine and sine in B together encode the annual eccentricity wave and its phase. Modern almanacs use far more terms, but this captures the shape — including the steep dive to about −14 minutes in February and the broad +16-minute hump in late October and early November.

Seeing it in the sky: the analemma

Photograph the Sun from a fixed spot at the same clock time — say, exactly noon — every few days for a year, and the images trace a slender figure-eight called the analemma. Two effects shape it:

• Vertical extent — the Sun's declination swinging ±23.44° between solstices, the direct fingerprint of obliquity on the seasons.
• Horizontal width — the equation of time, the Sun pulled east or west of the mean position by up to ~31 minutes of arc-time end to end.

The figure-eight is lopsided — the lower (winter) loop is fatter than the upper (summer) loop — because the broad November +16-minute excursion is larger than the February −14-minute one. That asymmetry is the eccentricity term tilting the scales. On Mars, with eccentricity nearly 0.094 and a different tilt, the analemma is a teardrop rather than a figure-eight; the shape is a planet's signature.

Everyday consequences

Where the equation of time shows up

Phenomenon	Effect of EoT
Earliest sunset	Falls ~2 weeks before the winter solstice (~Dec 7 at 40°N), not on it
Latest sunrise	Falls ~2 weeks after the solstice (~Jan 4)
Sundial reading	Off from clock time by up to +16 / −14 min before longitude correction
Solar-panel / heliostat aiming	Solar noon offset must be applied to track the true Sun
Celestial navigation	Almanac EoT converts observed Sun to GMT for longitude fixes

The earliest-sunset puzzle is the everyday face of the equation of time: in early December the Sun is running "fast" and gaining on the clock, so solar noon drifts later each day; sunset, measured against a fixed clock, bottoms out before the solstice even though daylight keeps shrinking until the solstice. Most people notice the symptom — "the evenings start getting lighter before Christmas" — without knowing it traces back to an ellipse and a tilt.

Common misconceptions

• It's about the seasons. No — seasons are the Sun's declination (height). The equation of time is the Sun's timing (east-west), a different axis of the analemma.
• It's caused by daylight saving time. No — it's astronomical and DST-independent; clocks were already mean-time before DST existed.
• It's a small, fixed offset. It changes daily, spanning about 30 minutes peak to peak across the year.
• Only eccentricity matters. The obliquity term is actually the larger of the two (±9.9 vs ±7.7 minutes).
• A sundial is "wrong." A sundial is perfectly right about the real Sun; it's your uniform clock that runs on an invented average.
• It applies only on Earth. Every planet has one, set by its own e and ε — Mars's is a teardrop.