The equation of time: why solar noon keeps missing 12:00
The Sun's clock runs fast in November and slow in February, by up to 16 minutes. The two causes are obliquity and orbital eccentricity.
A clock is a machine that counts at a constant rate. The Sun is not. It runs ahead of the clock by up to 16 minutes in early November and behind it by up to 14 minutes in mid-February, and the difference between the two is called the equation of time.
The word is older than algebra in this sense: an “equation” was a correction applied to an observation. The equation of time is the correction you apply to a sundial to get clock time.
| Date | Equation of time | Solar noon vs 12:00 |
|---|---|---|
| February 11 | −14m 15s | 12:14 |
| April 15 | 0 | 12:00 |
| May 14 | +3m 40s | 11:56 |
| July 26 | −6m 30s | 12:07 |
| November 3 | +16m 25s | 11:44 |
| December 25 | 0 | 12:00 |
Four zero crossings a year, two large extremes and two smaller ones — the shape is not a simple sine wave, and that is the clue to its causes.
Cause one: the axis is tilted
The Sun moves along the ecliptic, but time is measured on the equator. Projecting an arc of the ecliptic onto the equator is not linear: near the solstices, a degree of ecliptic longitude corresponds to less than a degree of right ascension, because the ecliptic is at its most oblique there.
The result is a wave with a period of half a year and an amplitude of about ±9.9 minutes. It is zero at the solstices and equinoxes, and at its extremes around November 6 and February 6 — note that this component is in phase with the November maximum, which is a hint.
Cause two: the orbit is an ellipse
Earth’s orbital speed varies by about 6% between perihelion in early January and aphelion in early July. The Sun’s apparent motion along the ecliptic inherits that variation, and the accumulated difference produces a wave with a period of one year and an amplitude of about ±7.7 minutes, peaking around February 11.
Two waves, one annual and one semiannual, added together. That is the entire equation of time, and it explains the asymmetry: the February minimum is the sum of both components at their negative extremes, while the November maximum is the semiannual component’s positive extreme partly fighting the annual one.
Why this is the answer to a common puzzle
“Why is the earliest sunset before the shortest day?” is the equation of time wearing a costume. Day length is set by declination; the clock time of sunset is set by declination and the drift of solar noon. Near the solstice the declination term is nearly flat, so the 16-minute drift dominates and pulls the two ends of the day apart. That page has the numbers.
The analemma
Photograph the Sun from the same spot at the same clock time every clear day for a year and the images trace a figure eight. Its height is the declination cycle, roughly 47° from top to bottom. Its width is the equation of time, about 32 minutes of time. The lobes are different sizes because the two components of the equation are out of phase.
The analemma is on the globe: the figure-eight printed in the Pacific Ocean on older globes is exactly this curve, drawn in a place where nobody would mind.
The practical version
Two corrections decide what a sundial says versus what your phone says, and they are independent:
- Equation of time: up to ±16 minutes, and it depends on the date.
- Longitude within the time zone: up to ±30 minutes, and it depends on where you are in the zone.
Add them and you have the offset of solar noon from 12:00. The sunrise and sunset table prints the resulting solar noon for sixteen cities, and the spread between them — 11:28 in one city, 12:20 in another — is mostly the longitude term, with the equation of time applied to all of them equally on a given day.
Figures are the standard published values for the equation of time; solar noon times are computed from the solar algorithm and agree with the USNO to about a minute.
Frequently asked questions
- What is the equation of time?
- The difference between apparent solar time — what a sundial reads — and mean solar time — what an average clock keeps. It is not an equation in the algebraic sense; the word is used in the older sense of a correction. It ranges from about −14 minutes in mid-February to +16 minutes in early November.
- Why does it have two separate components?
- Because two unrelated things are wrong with the assumption that the Sun moves at a constant rate. Earth's axis is tilted, so the Sun's motion along the ecliptic projects unevenly onto the equator; and Earth's orbit is elliptical, so the Sun's apparent speed along the ecliptic varies. Each produces a wave of different period, and their sum is the equation of time.
- Why is the figure-eight analemma shaped that way?
- The analemma is the Sun's position at the same clock time every day for a year, plotted in the sky. Its north–south extent is the declination cycle; its east–west wobble is the equation of time. The two lobes are unequal because the equation of time is not symmetric — one lobe is the February minimum, the other the November maximum.
- Does the equation of time affect sunrise and sunset times?
- Yes, and it is often the larger of the two corrections near the solstices. A 16-minute shift in solar noon moves both sunrise and sunset by 16 minutes. That is precisely why the earliest sunset is not on the shortest day.
Sources
Each source is a primary or official reference, checked before publication. Links open the original.
- Meeus, Astronomical Algorithms, 2nd edition, chapter 28 — the equation of time, its decomposition into obliquity and eccentricity components, and its annual extremes of about −14 and +16 minutes
- NOAA Global Monitoring Laboratory, Solar Calculator — the implementation of the equation of time behind every solar time on this site (gml.noaa.gov/grad/solcalc/)
- U.S. Naval Observatory — solar transit times, which is what solar noon means and what this page's comparison against 12:00 is built on (aa.usno.navy.mil)
- Royal Observatory Greenwich — the analemma and the historical distinction between apparent and mean solar time that gave the equation its name