What is day length?

What is day length?

Day length is the total time any portion of the Sun's disc is above the horizon at a given location on a given date. The U.S. Naval Observatory uses exactly that wording — "duration of daylight" being "the total time that any portion of the Sun is above the horizon" — for the daylight tables it publishes alongside its sunrise and sunset data.¹ Day length is also written as daylength, and called daylight duration, daylight hours, or, in biological contexts, photoperiod. The page treats them as synonyms.

Day length is the duration between two specific events — sunrise and sunset — and inherits its precision from how those events are defined. By the convention the U.S. Naval Observatory and most modern almanacs use, sunrise is the moment the Sun's geometric zenith distance is 90.8333°, equivalently the moment the centre of the Sun is 50 arcminutes (about 0.833°) below the geometric horizon. The 50-arcminute offset is the sum of two physical corrections: 16 arcminutes for the apparent radius of the Sun's disc (so what is rising or setting is the upper limb, not the centre) and 34 arcminutes for standard atmospheric refraction (light bending downward through the increasingly dense lower atmosphere).³ Sunset uses the same threshold mirrored on the other side of the day. Day length is the elapsed time between them.

The same convention is used elsewhere on the site, so the day length reported on a city's sunrise and sunset page is exactly the difference between the sunrise and sunset times shown next to it. Daytime — the colloquial period between sunrise and sunset — and twilight are distinct: the civil, nautical, and astronomical twilight phases extend the period of indirect light past sunset and before sunrise, but day length itself is the strictly above-horizon interval, not the longer twilight-inclusive duration.

Why isn't day length 12 hours at the equinox?

The most widely held misconception about day length is that it equals exactly 12 hours on the day of an equinox. The U.S. Naval Observatory rebuts this directly: "Day and night are not exactly of equal length at the time of the March and September equinoxes," and "the dates on which day and night are each 12 hours occur a few days before and after the equinoxes."⁵ The day on which day and night are exactly equal has been given a separate name — the equilux — a 1980s neologism that has gained currency in the 21st century.⁶

The two physical reasons are the same two corrections that define sunrise and sunset. First, the upper limb of the Sun rather than its centre marks the moment of rise and set, so the Sun is geometrically rising before its centre clears the horizon and still setting after its centre has dipped below. Second, atmospheric refraction lifts the Sun's apparent position by about 34 arcminutes near the horizon, so an observer sees daylight before the geometric Sun reaches the horizon at sunrise and continues to see it for some minutes afterwards at sunset. Both effects extend the day at the expense of the night, and both apply on the equinox as on any other date.³⁶

The numerical bias from these two corrections grows with latitude, because the same vertical offset projects into a longer interval of time when the Sun's path through the sky is shallower. The U.S. Naval Observatory quantifies it directly: at the equinoxes, day is about 7 minutes longer than night at latitudes up to about 25°, "increasing to 10 minutes or more at latitude 50 degrees."⁵ On the equator, where the Sun rises and sets near-vertically, the day-night gap on the equinox itself is closer to 14 minutes — refraction and disc size still apply, and there is no shallow-path attenuation working the other way.⁶ The actual equilux at any given location therefore lands a few days before the March equinox in the Northern Hemisphere (when the Sun is still south of the equator and day length is rising through 12 hours), and a few days after the September equinox; the dates are mirrored in the Southern Hemisphere.

How does day length vary with latitude and season?

Day length depends on three quantities: the observer's latitude (angular distance north or south of the equator), the Sun's declination on the date in question (its angular distance north or south of the celestial equator), and the same horizon-correction angle that defines sunrise and sunset. The first is fixed for a given place. The second varies smoothly through the year between −23.44° at the December solstice and +23.44° at the June solstice, passing through zero at the March and September equinoxes — the figure is the obliquity of Earth's axis, the tilt that gives the year its seasons.³

The qualitative picture follows directly. Near the equator, the Sun is always close to overhead at solar noon and travels a path that crosses the horizon close to vertically, so the time it spends above the horizon is close to half a day regardless of season. Near the poles, the Sun's path is nearly parallel to the horizon, so a small change in declination through the year translates into a large change in how much of that path lies above the horizon. The mid-latitudes sit between the two extremes — Paris, Beijing, Chicago, Cape Town — and see the seasonal swing without the polar-circle extremes.

The three regimes:

• The tropics (between 23.44° south and 23.44° north). Day length stays close to 12 hours all year. At the equator itself, every day is approximately 12 hours and 7 minutes; near the tropic lines, the seasonal range is roughly 11h–13h.² The Sun stands directly overhead at solar noon on two dates each year inside this band — the two zenith passages.
• The mid-latitudes (roughly 30°–60° north or south). Day length swings through a meaningful range. At 50° latitude, the difference between the longest and shortest days of the year is about eight hours: a midsummer day around 16h 20m versus a midwinter day around 8h, with the equinoxes splitting the difference at the 7-minute-longer-than-12h figure quoted above.⁵²
• The polar circles and beyond (above 66.5° latitude in either hemisphere). The Sun fails to rise or fails to set on at least one date each year. At the polar circles themselves, this happens on the solstices only; further toward the poles, the bands of polar day (24h) and polar night (0h) lengthen until, at the poles, each lasts roughly half the year.²

The 23.44° figure governs both ends of the picture. It sets the latitude bands of the tropics — between which the Sun can stand directly overhead — and, by complementary subtraction from 90°, the latitude of the polar circles, beyond which the Sun can fail to rise or set.

How is day length calculated?

The standard implementation of day length is the one shipped in NOAA's solar calculator, derived from Jean Meeus's Astronomical Algorithms — the canonical reference textbook for the field — and used in effectively the same form by most modern sun-position software.⁴⁷ The algorithm computes the hour angle of sunrise H₀: the angle, measured westward along the celestial equator, between the meridian (the imaginary north–south line through the local zenith) and the Sun's position at the moment of sunrise. The Sun's hour angle increases at 15° per hour as the Earth rotates, so the day from sunrise to sunset spans an arc of 2H₀ in hour-angle terms, equivalent to a duration of 2H₀ / 15° hours.

The exact formula NOAA uses, pulled directly from its public JavaScript source:⁸

cos H₀ = cos 90.833° / (cos φ · cos δ) − tan φ · tan δ

The three inputs are the standard horizon-correction angle of 90.833° (the same 90° + 50 arcminutes that defines sunrise and sunset), the observer's latitude φ, and the Sun's declination δ on the date in question. The output H₀ is in radians; multiplying by the conversion factor and doubling gives day length:

day length (hours) = 2 · H₀ · 180° / π / 15°

NOAA's implementation iterates the calculation: it computes solar noon for the date first, then evaluates δ at the instant of solar noon, then derives the sunrise hour angle from that δ. A second pass refines the calculation if the equation of time is changing rapidly across the day. The complete chain — declination, equation of time, longitude, time-zone offset — produces the local clock times of sunrise and sunset for any city; their difference is the day length.⁸

Two edge cases fall out of the formula. When the right-hand side is greater than 1, no real H₀ exists — the Sun never rises that day, and day length is zero (polar night). When it is less than −1, again no real H₀ exists — the Sun never sets, and day length is 24 hours (polar day, the midnight Sun). The combination cos H₀ > 1 occurs at high latitudes when latitude and declination have opposite signs (winter pole); cos H₀ < −1 occurs when they share a sign at sufficient magnitude (summer pole).⁸

The textbook algorithm is more accurate than most readers expect. NOAA states that its results are "theoretically accurate to within a minute for locations between +/- 72° latitude, and within 10 minutes outside of those latitudes."⁴ The published Astronomical Almanac, jointly produced by the U.S. Naval Observatory and HM Nautical Almanac Office, tabulates the same quantities to one-minute precision for a worldwide grid every year.⁹ The day length calculator on this site uses the same formula for any city and date.

Why doesn't day length change at a constant rate?

Day length changes fastest near the equinoxes and slowest near the solstices, by an amount that depends on latitude. At mid-latitudes, the day shortens or lengthens by about three minutes per day around the equinoxes; near the solstices, the rate falls to under a minute per day.⁶ The variation is geometric rather than astronomical surprise: the rate of change of day length is proportional to the rate of change of solar declination, and that rate is itself a sinusoid of the year — steepest where the declination passes through zero (the equinoxes) and flattest where it reaches its extremes (the solstices).

The slowdown near the solstice is the etymological content of the word itself. Solstice comes from Latin sol ("sun") plus sistere ("to stand still"); for several days either side of the solstice, day length is changing so slowly that the Sun's height at noon — and the time of sunrise and sunset — appears to hold steady. The same effect makes the solstice difficult to pin down by direct observation: the Sun's noon altitude and the time of sunrise change by less than the noise floor of casual measurement for several days running. The equinox, by contrast, falls at the moment of maximum day-length change, which is part of why the dating of the equinoxes is precise to within a few minutes while pre-modern fixings of the solstice were accurate only to within a day or two.

One subtle consequence in clock time: the earliest sunset of the year at mid-northern latitudes does not fall on the December solstice, even though that is the shortest day; it typically falls a week or two earlier, with the latest sunrise a week or two after the solstice. The asymmetry is the equation of time at work — the difference between apparent solar time and mean solar time is sliding through its early-November maximum during late autumn, biasing the entire daily clock pattern earlier. The day length itself still bottoms at the solstice; what shifts is the clock-time framing of the rise and set events that bracket it.

What happens at the polar circles?

Inside the polar circles — above latitude 66°33′ in either hemisphere — there are dates each year on which day length is 24 hours (the Sun never sets) and dates on which it is zero (the Sun never rises). At the polar circle itself, this happens on the local summer and winter solstices respectively; further poleward, the bands of permanent day (the midnight Sun) and permanent night (polar night) lengthen, until at the pole each lasts roughly half the year.²

The 66°33′ figure is a direct geometric consequence of the 23.44° axial tilt: 90° − 23.44° = 66.56°. It is the highest latitude at which the Sun's declination at the solstice can still bring it down to or up to the horizon. North of this line, on the date of the December solstice, the Sun's declination of −23.44° puts it permanently below the local horizon for at least 24 hours. The same line in the south hemisphere — 66°33′ south, the Antarctic Circle — gives the equivalent on the opposite solstice.

Day length also has a refraction-driven softening near the polar circles. The standard 34-arcminute refraction allowance lifts the Sun's apparent horizon-crossing slightly, so the real-world band of permanent day or night is shifted poleward of the geometric polar circles by a few tenths of a degree. Locations at exactly 66°33′ therefore see one or two days of midnight Sun around the June solstice each year rather than zero, on the strength of refraction alone. The effect is small at the polar circles themselves, but in the high Arctic the same correction reshapes the boundary dates of polar day and polar night by a few days at each end.³

The polar regions are also where the assumptions baked into casual day-length computation start to fray. Atmospheric refraction varies with temperature, pressure, and season — the standard 34-arcminute figure is a long-run average — and the variation is largest precisely where the Sun is grazing the horizon for hours at a time. Almanacs caution that rise/set times within the Arctic and Antarctic Circles can be reliably computed only to within several minutes, with the error climbing toward the poles.⁴

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