Everything hangs on the sun

Islamic prayer times are defined by sunlight, so calculating them is an astronomy problem: given a place and a date, where is the sun at every moment? Once the sun's path is known, each prayer is a rule about that path.

Fajr and Isha — twilight angles

Fajr begins when morning twilight starts: the sun still below the horizon but close enough that its light scatters into the sky. Conventions differ on the angle — ISNA (North America) uses 15° below the horizon for both Fajr and Isha; the Muslim World League uses 18°/17°; Karachi uses 18°/18°; Egypt 19.5°/17.5°. Deeper angle = earlier Fajr and later Isha, which is why method choice matters most at the day's two ends, and why high-latitude summers (where the sun may never reach 18° down) need special rules beyond this site's US scope.

Sunrise and Maghrib — the horizon, corrected

Sunrise and sunset are computed for the moment the sun's upper edge crosses the horizon, which happens when its center is about 0.833° below the geometric horizon — 0.583° because the sun has width, plus ~0.25° because the atmosphere bends its light (refraction), letting you see the sun slightly before it geometrically arrives. Maghrib is that same moment in the evening; Dhuhr begins just after solar noon, the sun's highest point, corrected by the equation of time and your longitude within the time zone.

Asr — the shadow rule

Asr starts when an object's shadow equals a set multiple of its height, plus the small shadow it already casts at noon. The majority schools use a factor of 1 (shadow = height); the Hanafi school uses 2, giving a later Asr. The calculator offers both — it is the single biggest legitimate difference between timetables after the twilight angles.

From angles to clock times

The computation chain runs: date → Julian day → the sun's declination and the equation of time → the hour angle at which the sun reaches each required angle → local clock time via longitude and the time zone (including daylight saving, which your browser's IANA database knows for every US zone). It is the same chain printed timetables use; small differences (a minute or two) come from rounding and slightly different solar constants, which is why this site calls its results estimates and defers to local mosque schedules for congregation.