Prayer Times in Livonia, MI Today

1. Inputs

2. Method parameters

3. Formulas

1. D = JD − 2451545.0
2. g = 357.529 + 0.98560028 D ; q = 280.459 + 0.98564736 D
3. L = q + 1.915 sin g + 0.020 sin 2g ; e = 23.439 − 0.00000036 D
4. RA = atan2(cos e · sin L, cos L) / 15 ; δ = asin(sin e · sin L)
5. EqT = q/15 − RA
6. Dhuhr (local mean solar time) = 12 − EqT ; clock time = Dhuhr − λ/15 + UTC offset
7. T(a) = (1/15) · acos[(sin a − sin φ · sin δ) / (cos φ · cos δ)]
8. Sunrise = Dhuhr − T(−0.833°) ; Maghrib = Dhuhr + T(−0.833°)
9. Fajr = Dhuhr − T(−Fajr angle) ; Isha = Dhuhr + T(−Isha angle)
10. Asr = Dhuhr + T(a) with a = arccot(s + tan|φ − δ(noon)|), s = 1 or 2
11. 0.833° = 0.567° refraction + 0.2667° solar semi-diameter

φ is latitude, λ longitude (east positive), δ the Sun's declination. The Sun's position is evaluated at each event's own time and the calculation is iterated twice. Clock times come from the IANA time-zone rules, so daylight saving time is always applied correctly.

4. Worked example for 2026-10-13

Sun at solar noon:

5. Measured accuracy

The engine is tested against an independent implementation (adhan-js 4.4.6) on 377 cases across 15 US cities and 13 dates, including daylight-saving transition days and Fairbanks, Alaska. The largest difference measured is 59.9 seconds.

Exception: in 5 Alaskan winter cases the Sun barely climbs above the Asr altitude, so Asr becomes very sensitive and differed by up to 4.2 minutes. Pages flag such dates.

6. Limits

• These are astronomical calculations, not a religious ruling. Mosques publish their own iqama times and may use another method or add a few minutes of caution; follow your local mosque.
• Elevation and local horizon (mountains, buildings) are not taken into account.

7. Method reference

8. Sources

• Jean Meeus, Astronomical Algorithms, 2nd ed., Willmann-Bell, 1998 (Julian Day, chapter 7)
• U.S. Naval Observatory, Approximate Solar Coordinates (low-precision formulas for the Sun)
• The Astronomical Almanac, Section C: low-precision formulas for the Sun's coordinates
• NOAA Global Monitoring Laboratory, Solar Calculator
• PrayTimes.org, Prayer Times Calculation (Dhuhr, sunrise/sunset with 0.833°, Asr shadow rule, method table, high-latitude methods)
• Islamic Society of North America (ISNA); angles 15°/15° as listed in the PrayTimes.org method table
• Muslim World League: Fajr 18°, Isha 17°
• Egyptian General Authority of Survey: Fajr 19.5°, Isha 17.5°
• University of Islamic Sciences, Karachi: Fajr 18°, Isha 18°
• Umm al-Qura University, Makkah: Fajr 18.5°, Isha 90 minutes after Maghrib (120 in Ramadan)
• IANA Time Zone Database
• GeoNames (geonames.org), cities15000, CC BY 4.0 — city coordinates and population
• timezone-boundary-builder (ODbL) — offline timezone polygon lookup
• adhan-js 4.4.6 (MIT) — independent reference implementation used for the accuracy tests

Methodology v1.0, reviewed 2026-10 · engine 1.0. Methodology and changelog