Three twilights, three thresholds, three different jobs
Twilight is scattered sunlight reaching you from a part of the atmosphere that is still lit while your own patch of ground is not. It fades continuously, so any boundary drawn across it is a convention — but the three conventions in use are old, precise and internationally consistent, each tied to a practical task.
Civil twilight ends when the Sun is 6° below the horizon. Under a clear sky that is roughly the point at which ordinary outdoor activity stops being possible without artificial light, and it is the threshold written into many national road-lighting and aviation regulations. The brightest planets and a handful of first-magnitude stars are visible by then.
Nautical twilight ends at −12°. The name is literal: this is the limit at which a navigator can still make out the sea horizon while enough stars are visible to shoot altitudes against it. Outside that window a sextant sight is either starless or horizonless, which is why the two nautical twilight periods are the working windows for celestial navigation — see the sight reduction calculator for what happens next.
Astronomical twilight ends at −18°, the point beyond which scattered sunlight no longer measurably brightens the sky background. Only after that does the sky reach its natural darkness, which is what deep-sky observing and long-exposure astrophotography need. Above roughly 48.5° latitude, there are weeks around midsummer when the Sun never gets that low and astronomical night simply does not occur.
The formula, one variable at a time
Every twilight time comes from the same expression that gives sunrise, evaluated at a different altitude: cos ω = (sin h − sin φ sin δ) / (cos φ cos δ). The only thing that changes between civil, nautical and astronomical twilight is h — −6°, −12° or −18°.
The hour angle ω is degrees of Earth rotation from solar noon, so the morning crossing is at noon − 4ω minutes and the evening crossing at noon + 4ω. Solar noon itself is 720 − 4λ − EoT + 60·tz, correcting for your longitude within the zone and for the equation of time.
Because ω grows monotonically as h falls, the crossings always come in order: astronomical dawn, then nautical, then civil, then sunrise. The duration of each twilight band is the difference of two hour angles times four — civil twilight lasts 4(ω₋₆ − ω₋₀.₈₃₃) minutes — and is symmetric between morning and evening on a given date because the same two angles serve both ends.
Golden hour and blue hour are photographic rather than official terms, and this calculator uses the common definitions: golden hour runs from the Sun at +6° down to −4°, and blue hour from −4° to −6°. Neither is standardised by any body, so a source quoting different limits is not wrong, only using a different convention. Both are computed with exactly the same equation.
Refraction is handled less carefully at these depths than at sunrise, and deliberately so. At −6° and below, the Sun is well clear of the horizon and refraction changes its apparent altitude by only a few arcminutes, whereas at the horizon it amounts to 34. The −0.833° sunrise value in the table includes the refraction and semidiameter allowance; the twilight thresholds are geometric, which is how the standard definitions are written.
Worked example: civil twilight at the equator and at 45°N
Start at the equator on an equinox, where φ = 0 and δ ≈ 0. The sin φ sin δ term vanishes and cos φ cos δ = 1, so the expression collapses to cos ω = sin h.
- Sunrise threshold. cos ω = sin(−0.833°) = −0.014538, so ω = 90.833°.
- Civil threshold. cos ω = sin(−6°) = −0.104528, so ω = 96.000° exactly — arccos of −sin(6°) is 90° + 6°.
- Duration. 4 × (96.000 − 90.833) = 4 × 5.167 = 20.67 minutes. Tropical twilight really is that short.
- Astronomical threshold. cos ω = sin(−18°) = −0.309017, so ω = 108.000°, meaning the Sun is above −18° for 2 × 108 / 15 = 14.4 h.
- Darkness. 24 − 14.4 = 9.600 hours of genuinely dark sky.
Now move to 45°N on the June solstice, where δ = +23.44°. The denominator is cos 45° cos 23.44° = 0.648754 and sin 45° sin 23.44° = 0.281279.
- Sunrise. cos ω = (−0.014538 − 0.281279)/0.648754 = −0.455977, so ω = 117.128°.
- Civil. cos ω = (−0.104528 − 0.281279)/0.648754 = −0.594689, so ω = 126.491°.
- Duration. 4 × (126.491 − 117.128) = 37.45 minutes, nearly twice the tropical figure.
- Astronomical. cos ω = (−0.309017 − 0.281279)/0.648754 = −0.909893, so ω = 155.490° and darkness is 24 − 2 × 155.490/15 = 3.268 hours.
Push on to 55°N on the same date and the astronomical bracket becomes (−0.309017 − 0.325831)/0.526245 = −1.206, which is outside the range of the arccosine. There is no solution, the Sun never sinks to −18°, and darkness is zero for the night.
Twilight thresholds and what each one is for
| Name | Solar altitude | What it marks | Duration at the equator, equinox |
|---|---|---|---|
| Golden hour | +6° to −4° | Warm, low-contrast light; long soft shadows | about 40 min |
| Sunrise / sunset | −0.833° | Upper limb at the refracted horizon | instant |
| Blue hour | −4° to −6° | Deep blue sky balancing artificial light | about 8 min |
| Civil twilight | 0 to −6° | Outdoor activity possible without lights; brightest planets visible | 20.7 min |
| Nautical twilight | −6° to −12° | Sea horizon and navigational stars both visible | 24.0 min |
| Astronomical twilight | −12° to −18° | Residual sky glow still detectable by instruments | 24.0 min |
| Night | below −18° | Sky at its natural darkness | 9.60 h |
At the equator on an equinox each band is simply 4·(ω₂ − ω₁) minutes with ω = 90° + |h|, which is why the nautical and astronomical bands are exactly 24 minutes each: 6° of altitude at 4 minutes per degree.
How to read the result
Read the darkness figure first if you are planning an observing session. Zero hours means the sky never reaches astronomical night, which happens every summer above about 48.5° latitude and is the reason deep-sky imaging effectively pauses in northern Europe and Canada from May to July. A small positive number tells you how narrow the window is.
Twilight duration grows with latitude and with the season, and it grows for a geometric reason: near the poles the Sun's daily path is shallow, so it takes far longer to descend the same six degrees. At the equator civil twilight is about twenty minutes; at 45°N in midsummer it is nearly forty; above the Arctic Circle it can last all night.
Morning and evening twilight are the same length on a given date, because both come from the same pair of hour angles. They are not centred on the same clock times, though — solar noon is displaced from 12:00 by longitude and by the equation of time, so dawn and dusk are symmetric about solar noon, not about midday on your watch.
Where a threshold is never crossed the calculator says so in words rather than printing a time. “Never this low” means the Sun stays above that altitude all night; “Not reached” means it never climbs that high all day, which is what deep polar night looks like. Both are correct answers, and inventing a time for either would be worse than useless.
Assumptions and limits
- The twilight thresholds are geometric. Only the −0.833° sunrise line carries the refraction and semidiameter allowance; −6°, −12° and −18° are the altitudes of the Sun's centre, as the standard definitions specify.
- A clear, unobstructed sky is assumed. Cloud, haze and light pollution all change what you can actually see, sometimes by more than a whole twilight band.
- Moonlight is ignored. A gibbous Moon can brighten the sky well beyond anything twilight contributes; check the moon phase calculator before planning around the darkness figure.
- Golden hour and blue hour are conventions, not standards. The +6° to −4° and −4° to −6° limits used here are common but not universal.
- No daylight saving is applied. Enter the UTC offset actually in force on the date.
- Terrain is not modelled. A mountain to the west delays the useful end of golden hour without changing the Sun's altitude at all.
Where the −6°, −12° and −18° definitions come from
The three twilight thresholds are the definitions published by the United States Naval Observatory's Astronomical Applications Department and used identically by other national almanac offices. They are stated in terms of the geometric altitude of the centre of the Sun below the horizon, which is why they do not carry the refraction allowance that sunrise and sunset do. The underlying solar position here follows the NOAA implementation of Meeus's low-precision solar algorithm, accurate to well under a minute of time at the altitudes involved — far better than the visual judgement any of these definitions is standing in for.
Related calculations and when to use them
Use this page for the low-light windows either side of the day. For the day itself, the sunrise and sunset calculator gives the horizon crossings and the compass bearings, and the daylight hours calculator gives the length of the day and how fast it is changing.
Astrophotographers pairing this with a session plan will want the moon phase calculator for sky brightness and the star trail length calculator for the maximum exposure a given focal length allows. Navigators taking star sights during nautical twilight need the sidereal time calculator to know which stars are up.
Landscape photographers should treat the golden hour figure as a starting point rather than a schedule. The quality of the light depends on the Sun's altitude, which this calculates exactly, but also on atmospheric clarity and on what the terrain does to your actual horizon. In a valley the Sun can disappear behind a ridge while still at +8°, ending the light half an hour before the arithmetic says it should.
Key terms
- Civil twilight
- The period when the Sun's centre is between the horizon and 6° below it. Many lighting and aviation rules are written against this boundary.
- Nautical twilight
- Between 6° and 12° below the horizon, when both the sea horizon and navigational stars are visible — the working window for a sextant.
- Astronomical twilight
- Between 12° and 18° below, the last stage in which scattered sunlight still measurably brightens the sky.
- Golden hour
- A photographic convention for the Sun between about +6° and −4°, when light is warm and shadows are long. Not an official definition.
- Blue hour
- The window from about −4° to −6°, when the sky is a deep saturated blue and balances well against artificial lighting.