Sunrise & Sunset Calculator
Enter latitude, longitude, date and timezone offset; the tool runs the NOAA / Jean Meeus solar-position algorithm (*Astronomical Algorithms*, 2nd ed.) and returns sunrise, sunset, solar noon and day length, plus the day's solar declination (δ) and equation of time (E). A 90.833° zenith accounts for atmospheric refraction (~34′) and the apparent radius of the solar disk (~16′). Accuracy is ~1 minute for latitudes below 72° and years 1800–2200. Useful for photographers, hikers, solar-panel alignment, gardeners, astrologers and amateur astronomers.
Enter a valid monthly income (> 0), monthly savings (0 ≤ savings ≤ income), current balance, real return rate (0–50 %) and safe-withdrawal rate (2–10 %).
Sunrise
05:39
Solar noon
12:24
Sunset
19:10
Day length
13h 31m
Solar declination
23.4°
Equation of time E
−1.5 min
Formula
JD = Gregorian Julian Date; T = (JD − 2451545) ⁄ 36525 λ_sun ≈ L₀ + C − 0.00569 − 0.00478·sin Ω δ = arcsin(sin ε · sin λ_sun) E (min) = 4·rad2deg(y·sin 2L₀ − 2e·sin M + 4ey·sin M·cos 2L₀ − ½y²·sin 4L₀ − 1.25e²·sin 2M), y = tan²(ε ⁄ 2) Hour angle H = arccos((cos 90.833° − sin φ·sin δ) ⁄ (cos φ·cos δ)) Solar noon (UTC, min) = 720 − 4·longitude − E; sunrise / sunset = noon ∓ 4H
- · The 90.833° zenith includes 0.833°: ~34′ to compensate for atmospheric refraction (which lifts the sun visually by ~34 arcminutes near the horizon) plus the ~16′ angular radius of the solar disk. Sunrise is defined as the moment the *upper* limb first touches the horizon, not the centre.
- · The equation of time (apparent solar time vs. mean solar time) swings between about −14 min in mid-February and +16 min in early November. It is the sum of the eccentricity term (~7.7 min amplitude) and the obliquity term (~9.9 min amplitude) and traces out the familiar "figure-of-eight" analemma in the sky.
- · Two extremes: inside the polar circles you get midnight sun (sun never sets) near the summer solstice and polar night (sun never rises) near the winter solstice. The latitude at which each first occurs depends on date; the outer limits are ±66.56° (the Arctic / Antarctic Circles).
- · References: NOAA Earth System Research Laboratory Solar Calculator (https://gml.noaa.gov/grad/solcalc/); Jean Meeus, *Astronomical Algorithms*, 2nd ed. (1998), chapters 22, 25, 28; USNO Astronomical Applications Department.
- · Limitations: (1) the 0.833° refraction term is a global average — real refraction depends on temperature, pressure and humidity and can vary by 5–10 arcminutes; (2) elevated observers (mountaintops, tall buildings) see the sun set slightly later — the full NOAA SPA adds an altitude term; (3) at latitudes ≥ 72° near the start or end of midnight-sun / polar-night the equations are very sensitive and accuracy drops to ~5 minutes.
- · Golden hour is roughly the hour after sunrise / before sunset (sun 0–10° above horizon); blue hour is the ~30 minutes before sunrise / after sunset when the sun is between −6° and −4° (also called civil twilight).
Frequently asked
Why do two cities at the same latitude have sunrise / sunset times hours apart?
Two reasons. First, longitude directly maps to time: every 15° of longitude is one hour of solar time. London (0°E) and Moscow (37.6°E) are both near 51°N, but Moscow runs about 2.5 hours ahead by clock; expressed in UTC their sunrise times are roughly 2.5 hours apart — exactly what 37.6° / 15° gives you. Second, civil time zones do not have to follow geography. China spans roughly 60° of longitude but uses a single UTC+8 (Beijing time), so Ürümqi in Xinjiang (87°E) can have a clock-time sunrise as late as 10:00. Spain sits near the prime meridian yet observes UTC+1 (Central European Time), pushing Madrid sunsets to ~21:30 in midsummer — a political-time-zone effect, not an astronomical one. This tool takes longitude and timezone offset as separate inputs so you can isolate the two contributions.
Why are sunrise earliest, sunset latest and the day longest on the summer solstice?
You might expect them all on the same day — but the earliest sunrise and the latest sunset do *not* both fall on the solstice. The longest day does coincide with the solstice (around 21 June in the Northern Hemisphere), because the solar declination δ reaches its maximum (~+23.44°) on that date and so does the geometric hour angle H. But the clock time of sunrise also includes the equation of time E, and E is about −2 to −3 minutes around the June solstice, rising to about +6 minutes by late July. So at northern mid-latitudes (e.g. New York at 41°N) the earliest sunrise lands around 14 June, the latest sunset around 28 June, with the solstice itself in the middle. The longest day is still the solstice. This left-right asymmetry comes from the asymmetric "figure-of-eight" shape of E in the analemma.
Is sunrise defined as the centre of the disk crossing the horizon, or the upper limb crossing?
The standard adopted by the USNO, IAU, NOAA, KMA, HKO and almost every modern almanac is: sunrise / sunset is the moment the *upper limb* of the sun touches the visible horizon. To express this in terms of the sun's centre, add the sun's angular radius (~16′) and the atmospheric refraction near the horizon (~34′) — together ~50′ ≈ 0.833°. So when the geometric centre of the sun is still 0.833° below the horizon, the upper limb is just touching it visually. This tool, like all NOAA / Meeus-based calculators, uses a 90.833° zenith (i.e. a −0.833° solar altitude) as the cut-off. If instead you want the moment the sun's centre crosses the horizon (sometimes called geometric sunrise), use 90°. Civil twilight uses 96° (−6°), nautical twilight 102° (−12°), and astronomical twilight 108° (−18°).
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