◆ How Zmanim Are Calculated

This page documents the exact algorithms and halachic opinions used by ZmanimPro. All calculations are performed by the open-source PhpZmanim library (LGPL-2.1), a PHP port of Eliyahu Hershfeld's KosherJava API.

1. Solar Position (Sunrise & Sunset)

Sunrise and sunset are computed using the NOAA Solar Calculator, based on Jean Meeus' Astronomical Algorithms (2nd edition, Willmann-Bell, 1998). The key formula:

cos(H) = (sin(h₀) − sin(φ)·sin(δ)) / (cos(φ)·cos(δ))

Where:

• H = hour angle of the sun
• h₀ = altitude of the sun's center = −0.833° (for sunrise/sunset)
• φ = observer's latitude
• δ = solar declination

The value −0.833° accounts for:

• 34′ atmospheric refraction (light bends as it passes through the atmosphere, so the sun appears higher than it actually is)
• 16′ solar radius (we measure when the upper limb of the sun appears, not its center)

Accuracy: ±0.01° in solar position, corresponding to roughly ±1 minute of time at mid-latitudes.

2. Halachic Hours (Sha'ot Zmaniyot)

Jewish law divides daylight into 12 equal "halachic hours." There are two primary opinions on what counts as "daylight":

A halachic hour under the GRA = (sunset − sunrise) ÷ 12. We display GRA times by default; Magen Avraham times are available on request via the API.

3. Key Zmanim Formulas

4. Candle Lighting Time

Candle lighting marks the acceptance of Shabbat. The time is computed as sunset − candle offset:

5. Havdalah (End of Shabbat)

Havdalah is recited at nightfall (tzait haKochavim), when three medium-sized stars are visible. We display the 72-minute opinion (Rabbeinu Tam simplified) by default, and also show 8.5° (GRA) (~36 min at equinox, the standard Tzait HaKochavim) and 16.1° (Rabbeinu Tam degrees) (~72 min at equinox) times.

Rabbeinu Tam (a more stringent opinion) defines nightfall as 4 mil after sunset. This is calculated either as fixed 72 minutes (4 × 18 min/mil) or astronomically as 16.1° below the horizon. Many observant individuals wait for this time before performing melacha (work forbidden on Shabbat), as a stringency (lechumra).

6. Hebrew Calendar Algorithm

The Hebrew calendar is computed by the algorithm described by Maimonides in Hilchot Kiddush HaChodesh (12th century) and refined by Gauss (1802). Key rules:

• Months alternate 29 and 30 days (defective vs. full months).
• Leap years (with Adar II added) occur in years 3, 6, 8, 11, 14, 17, 19 of a 19-year cycle.
• Molad (mean lunar conjunction) is computed; Rosh Hashana is postponed by 4 rules (Dehiyot) to prevent Yom Kippur from falling adjacent to Shabbat, etc.

This is implemented in PhpZmanim's JewishCalendar class, which we use for all Hebrew-date conversions on this site.

7. Sources of Variation

Minor differences (1–2 minutes) can occur between Zmanim calculators due to several methodological factors. Understanding these helps explain why times may differ slightly from one source to another:

• Solar calculator: Different implementations use NOAA, Jean Meeus, or the higher-precision Solar Position Algorithm (SPA). We use the NOAA calculator by default.
• Elevation data: Altitude affects sunrise/sunset by roughly 1 minute per 1,500 meters. Sources using different elevation models will produce slightly different times.
• Rounding conventions: Some sources round to the nearest minute; others truncate. We round to the nearest minute.
• Halachic opinions: The choice between 8.5° and 72 minutes for nightfall, or between GRA and Magen Avraham for halachic hours, produces different Zmanim. We display both opinions where relevant.

PhpZmanim supports multiple calculators (NoaaCalculator, SunTimesCalculator, ZmanimCalculator), allowing users to select the precision level they need. Our default configuration uses the NOAA calculator, which balances accuracy and computational efficiency.

Frequently Asked Questions