What an almanac actually returns for sunrise and sunset
An almanac calculates sunrise and sunset by deriving the Sun's apparent position from a date, latitude, longitude, and the civil UTC offset, then reporting the local clock times when the Sun crosses an apparent zenith of 90.833° in the morning and evening. The 90.833° value is not a geometric horizon — it bundles atmospheric refraction at the horizon together with the Sun's visible angular radius so that the reported minute matches when an unobstructed disk would appear or disappear for a sea-level observer. This is the same convention used by NOAA's Global Monitoring Laboratory and by the compact implementations of Jean Meeus's astronomical algorithms, which is why both almanacs and online tools built on them agree to within a minute across most populated latitudes. The Sunrise Sunset Calculator applies exactly this chain of steps: it converts the chosen date to a Julian day, computes the Sun's geometric longitude and mean anomaly, corrects for orbital eccentricity, derives the apparent longitude and corrected obliquity of the ecliptic, then calculates solar declination and the equation of time before combining those with latitude and the time-zone offset to produce local civil times.
Almanac-style outputs are rounded to the nearest minute and reported in the civil time zone you specify, not in UTC. The clock times are anchored to the longitude and equation of time at the date's Julian noon, so a result for a given date belongs to a 24-hour window centered on that Julian day. When the longitude offset and a manually selected civil UTC offset push an event across midnight, the calculator marks it with a parenthetical previous- or next-day flag rather than silently wrapping the hour. This is one reason the almanac approach asks you to enter the UTC offset yourself: civil time zones are political boundaries, and the model has no way to know whether your location is on standard time, daylight saving time, or a historical exception without being told.
The same chain of intermediate values — solar declination and the equation of time — is exposed by the tool as a diagnostic display, so you can compare what the calculator is doing with what another implementation reports. That transparency is useful because two almanacs can disagree on the zenith assumption, the exact list of correction terms, or how the hour angle is converted into civil minutes, even though both are technically "NOAA-style" approximations.
The four inputs an almanac method requires
Every almanac sunrise and sunset calculation, whether it lives in a printed nautical almanac or in a browser tool, starts from the same four pieces of information: a calendar date, a latitude, a longitude, and the civil UTC offset that applies on that date in that location. The almanac does not need your street address, your elevation, your local weather, or a permission prompt to read your location — it only needs the numeric values, which is why these calculators can run entirely on the page without sending anything to a remote service.
| Input | Format | Accepted range | Notes |
|---|---|---|---|
| Date | Gregorian calendar | 1901-01-01 to 2099-12-31 | Outside this window the compact Meeus equations lose their intended accuracy |
| Latitude | Decimal degrees, north positive | -89.8° to +89.8° | Northern hemisphere is positive, southern hemisphere is negative |
| Longitude | Decimal degrees, east positive | -180° to +180° | New York uses a negative value; Sydney uses a positive value |
| Civil UTC offset | Hours and minutes from UTC | Any signed offset | Must reflect daylight saving if the chosen date falls inside DST |
The sign convention matters more than it looks. Western longitudes (the Americas) are negative; eastern longitudes (Europe, Africa, Asia, Oceania) are positive. North latitudes are positive; south latitudes are negative. Mixing the signs is the single most common reason an almanac result looks like it is off by several hours.
The civil UTC offset is intentionally separate from the longitude. Two locations on the same meridian can have different offsets because they belong to different countries, and one location can have different offsets on different dates because of daylight saving rules. The calculator does not guess a time-zone name or apply a historical DST rule — you enter the offset that applies to the chosen date, and that becomes the anchor for the displayed clock time.
How to calculate sunrise and sunset using almanac formulas
The almanac approach is a fixed sequence of intermediate values, and once the inputs are in place the arithmetic is mechanical. The Sunrise Sunset Calculator carries out this sequence in the browser and reports the local clock times at the end. Use it as follows.
- Choose a Gregorian date between 1901 and 2099 — the lightweight Meeus equations used here are documented to that accuracy scope.
- Enter the latitude in decimal degrees, with north positive and south negative, between -89.8° and +89.8°.
- Enter the longitude in decimal degrees, with east positive and west negative, between -180° and +180°.
- Enter the civil UTC offset that is in force on that date for that location, including daylight saving time if applicable — for example, New York is UTC-5 in January and UTC-4 in July.
- Read the sunrise, solar noon, and sunset clock times from the result, or read the polar-day or polar-night state if the Sun does not cross the modeled apparent horizon that day.
- Treat the reported minute values as apparent-horizon estimates at sea level, and compare them with what you actually see from your local terrain.
Behind those six steps, the calculator converts the date to a Julian day, computes the Sun's geometric longitude and mean anomaly, applies the equation of the center to account for orbital eccentricity, derives the apparent longitude, corrects the obliquity of the ecliptic, then calculates solar declination and the equation of time. Solar noon follows from the longitude and the equation of time; sunrise hour angle comes from a cosine formula evaluated at the apparent zenith of 90.833°; the hour angle is then added to and subtracted from solar noon to produce the two crossings, which are converted from UTC minutes to local civil minutes, rounded, and flagged with a day offset if necessary.
Why the standard zenith is 90.833° and not 90°
The most distinctive number in an almanac sunrise calculation is the 90.833° zenith. It is not the geometric horizon and it is not the distance to the center of the Sun — it is a bundled approximation for two physical effects that are nearly constant at sea level.
The first effect is atmospheric refraction. Near the horizon, the Earth's atmosphere bends the Sun's image upward by roughly 34 arcminutes on average, so the Sun's disk becomes visible while its true geometric center is still below the flat horizon. The second effect is the Sun's own visible radius, about 16 arcminutes, which means the disk first appears when the upper limb is at the horizon, not when the center is. Adding the two together and converting to degrees gives the 90.833° convention.
NOAA documents this convention in its solar calculation details, and notes that refraction is modeled rather than measured — actual weather, air density, and temperature can move the visible event by several minutes. If your goal is a precise horizon observation for photography, navigation, or astronomy, treat the calculator's minute as the average case and allow for your local conditions on top of it.
When the answer is a polar state instead of a clock time
Near the polar circles and beyond, the Sun may not cross the apparent horizon at all on a given date. The almanac catches this case through the cosine of the sunrise hour angle: if the cosine comes out above 1, no real angle exists and the Sun is below the apparent horizon for the full local date (polar night); if it comes out below -1, the Sun never drops below the apparent horizon (polar day). Rather than return an impossible clock time, the calculator reports the polar state.
The thresholds and the resulting behavior are summarized below.
| Hour-angle cosine | What it means | What the calculator reports |
|---|---|---|
| Between -1 and +1 | A real hour angle exists for that date and location | Local clock times for sunrise and sunset, plus solar noon |
| Above +1 | The Sun stays below the apparent horizon all day | Polar night state for the selected date |
| Below -1 | The Sun stays above the apparent horizon all day | Polar day state for the selected date |
Tromsø in Norway is a useful reference case: at the June solstice the Sun is above the apparent horizon continuously and the tool reports polar day, while at the December solstice it is below the apparent horizon continuously and the tool reports polar night. If you are scheduling an activity in such a location, the polar state is a feature rather than an error message — the almanac is telling you that the event you asked about does not occur on that date.
How the almanac result compares with what you observe
According to NOAA, its own calculator is theoretically accurate to about a minute within ±72° latitude, with possible errors up to ten minutes beyond that range. Independent checks against the U.S. Naval Observatory rise and set data allow a roughly three-minute difference to absorb model and rounding choices, and most city-level coordinates agree to within that band across ordinary locations and seasons.
What still moves the visible event away from the modeled minute is everything that NOAA explicitly does not measure: local weather and cloud cover, air pressure and temperature, elevation above sea level, the terrain around the observer, and the height of buildings or trees near the horizon. A city-center coordinate is adequate for broad planning, but if your horizon is shifted by a ridge, a building, or a valley wall, the visible first light or last light will arrive several minutes earlier or later than the almanac says. For navigation, religious rulings, legal determinations, or telescope work at high latitudes, a full ephemeris that includes elevation, pressure, temperature, and a specific event definition is the appropriate reference.
If you are using these times at sea, the same almanac chain applies, but the conventions around civil time, watch-keeping, and twilight definitions differ from a land observer's. A separate guide to calculating sunrise and sunset on a ship walks through those practical differences.
Related reading: Calculate Sunrise and Sunset from Latitude and Longitude.