Muslim Prayer (Namaaz) Time Calculator

In His Name, the Most High

Preamble

Muslims perform five prayers a day. Each prayer is given a certain prescribed time during which it must be performed. This document briefly describes these times and explains how they can be calculated mathematically, with appropriate formulea provided. This document forms the basis of how the library used by this website calculates the prayer times.

Definitions

To determine the exact period for each prayer (and also for fasting), we need to determine ten points of time per a day. These times are defined in the following table:

What follows is information on how to calculate the above times mathematically for any location if the coordinates of the site are known. This forms the basis of his this library/API works.

Astronomical Measures

There are two astronomical measures that are essential for computing prayer times. These two measures are the equation of time and the declination of the Sun.

The equation of time is the difference between time as read from a sundial and a clock. It results from an apparent irregular movement of the Sun caused by a combination of the obliquity of the Earth's rotation axis and the eccentricity of its orbit. The sundial can be ahead (fast) by as much as 16min 33s (around November 3) or fall behind by as much as 14min 6s (around February 12), as shown in the following graph:

The declination of the Sun is the angle between the rays of the sun and the plane of the Earth's equator. The declination of the Sun changes continuously throughout the year. This is a consequence of the Earth's tilt (that is the difference in its rotational and revolutionary axes).

The above two astronomical measures can be obtained accurately from The Star Almanac, or can be calculated approximately. The following algorithm from U.S. Naval Observatory (Alternative Link) computes the Sun's angular coordinates to an accuracy of about 1 arc-minute within two centuries of 2000.

d = jd - 2451545.0;  // jd is the given Julian date 

g = 357.529 + 0.98560028* d;
q = 280.459 + 0.98564736* d;
L = q + 1.915* sin(g) + 0.020* sin(2*g);

R = 1.00014 - 0.01671* cos(g) - 0.00014* cos(2*g);
e = 23.439 - 0.00000036* d;
RA = arctan2(cos(e)* sin(L), cos(L))/ 15;

D = arcsin(sin(e)* sin(L));  // declination of the Sun
EqT = q/15 - RA;  // equation of time

Calculating Prayer Times

To calculate the prayer times for a given location, we need to know the latitude (Lat) and the longitude (Lng) of the location, along with the local Time zone for that location.

We also obtain the equation of time (EqT) and the declination of the Sun (D) for a given date using the algorithm mentioned above.

Dhuhr

Dhuhr can be calculated easily using the following formula:

Dhuhr = 12 + TimeZone - Lng/15 - EqT.

The above formula indeed calculates the midday time, when the Sun reaches its highest point in the sky. A slight margin is usually considered for Dhuhr as explained below:

A Note on Dhuhr

Dhuhr has been defined in several ways in the fiqh (Islamic Judicial Law) literature:

1. When the Sun begins to decline (Zawaal) after reaching its highest point in the sky.
2. When the shadow of an indicator (a vertical stick) reaches its minimum length and starts to increase.
3. When the Sun's disk comes out of its zenith line, which is a line between the observer and the center of the Sun when it is at the highest point.

The first and the second definitions are equivalent, as the shadow length has a direct correlation to the Sun's elevation in the sky via the following formula:

ShadowLength = Object Height × cot(SunAngle).

The Sun's angle is a continuous function over time and has only one peak point (the point at which the tangent to its curve has zero slope) which is realized exactly at midday. Therefore, according to the first two definitions, Dhuhr begins immediately after the midday.

The third definition is slightly different from the previous two definitions. According to this definition, Sun must passes its zenith line before Dhuhr starts. We need the following information to calculate this time:

• The radius of the Sun (r): 695,500 km
• Distance between the Sun and the Earth (d): 147,098,074 km to 152,097,701 km

Having r and d, the time t needed for Sun to pass its zenith line can be computed using the following formula:

t = arctan(r/d) / 2π × 24 × 60 × 60

The maximum value obtained by the above formula (which corresponds to the minimum Sun-Earth distance) is 65.0 seconds. Therefore, it takes approximately 1 minute until Sun's disk comes out of its zenith that should be considered into consideration for calculating Dhuhr, if the third definition is used.

There are three definitions for the Dhuhr time as described above. According to the first two definitions, Dhuhr = midday, and according to the third definition, Dhuhr = midday + 65 sec.

Sunrise and Sunset

The time difference between the mid-day and the time at which sun reaches an angle α below the horizon can be computed using the following formula:

Astronomical sunrise and sunset occurs at α=0.

However, due to the refraction of light by terrestrial atmosphere, actual sunrise appears slightly before astronomical sunrise, and actual sunset occurs after astronomical sunset.

Actual sunrise and sunset can be computed using the following formulae:

Sunrise = Dhuhr - T(0.833)
Sunset = Dhuhr + T(0.833)

If the observer's location is higher than the surrounding terrain, we can consider this elevation into consideration by increasing the above constant 0.833 by 0.0347 × sqrt(h), where h is the observer's height in meters.

This would make the fomulae:

Sunrise = Dhuhr - T(0.8333 + (0.0347 * sqrt(h)))
Sunrise = Dhuhr + T(0.8333 + (0.0347 * sqrt(h)))

Fajr and Isha

There are differing opinions on what angle to be used for calculating Fajr and Isha.

The following table shows several conventions currently in use in various countries:

As an example, according the Shia Ithna Ashari, Leva Research Institute, Qum:

Fajr = Dhuhr - T(16) and Isha = Dhuhr + T(14)

Asr

There are two main opinions on how to calculate Asr time. The majority of schools (including Shafi'i, Maliki, Ja'fari, and Hanbali) say it is at the time when the length of any object's shadow equals the length of the object itself plus the length of that object's shadow at noon.

The dominant opinion in the Hanafi school says that Asr begins when the length of any object's shadow is twice the length of the object plus the length of that object's shadow at noon.

The following formula computes the time difference between the mid-day and the time at which the object's shadow equals t times the length of the object itself plus the length of that object's shadow at noon:

Therefore, in the first four schools of thought the formula would be:

Asr = Dhuhr + A(1)

Whereas in the Hanafi school the formula would be:

Asr = Dhuhr + A(2)

Maghreb

In the opinion of the Sunni schools of Islamic Judicial Law, the time for Maghreb prayer begins once the Sun has completely set beneath the horizon, that is:

Maghreb = Sunset

It should be noted that within the Sunni Judicial Law, some calculators suggest 1 to 3 minutes after Sunset on the basis of a precaution.

Shia Judicial Law suggests that, as long as the redness in the eastern sky appearing after sunset has not passed overhead, Maghreb prayer should not be performed. It is usually taken into consideration by assuming a twilight angle (as shown in the tables above):

Maghreb = Dhuhr + T(4).

Midnight

Midnight is generally calculated as the mean time from Sunset to Sunrise:

Midnight = 1/2(Sunrise - Sunset)

However, the Shia Jurists opine that the juridical midnight (the ending time for performing Isha prayer) is the mean time from Sunset to Fajr:

Midnight = 1/2(Fajr - Sunset)

Higher Latitudes

In locations at higher latitude, twilight may persist throughout the night during some months of the year.

In these abnormal periods, the determination of Fajr and Isha is not possible using the usual formulas as described above. In order to overcome this problem, several solutions have been proposed, three of which are described below:

Middle of the Night

In this method, the period from sunset to sunrise is divided into two halves. The first half is considered to be the "night" and the other half as "day break". Fajr and Isha in this method are assumed to be at mid-night during the abnormal periods.

One-Seventh of the Night

In this method, the period between sunset and sunrise is divided into seven parts. Isha begins after the first one-seventh part, and Fajr is at the beginning of the seventh part.

Angle-Based Method

This is an intermediate solution, used by some recent prayer time calculators.

var α == TheTwilightAngleForIsha, 
var t = α/60.

The period between sunset and sunrise is divided into t parts. Isha begins after the first part.

For example, if the twilight angle for Isha is 15, then Isha begins at the end of the first quarter (15/60) of the night.

Time for Fajr is calculated similarly.

In case Maghreb is not equal to Sunset, we can apply the above rules to Maghreb as well to make sure that Maghreb always falls between Sunset and Isha during the abnormal periods.

Acknowledgments

This project is built on the excellent work PrayTimes.org by Hamid Zarrabi-Zadeh.

Development Considerations

I have taken the formulae and calculations, sanity checked them as applicable, and examined the source code that various other contributors have written - found at Prayer Times Code Snippets. However, given that some of the code is extremely out-dated, and in some cases doesn't work as it should, I have decided to re-write the code, fixing coding issues and ensuring the algorithms required are correctly implemented. The code has been built using Microsoft dotnet, and is available as a nuget package, as an api (which can be self-hosted as a docker container and for which the source code is also available), and off course the source code for this site (also available as a docker container).

The source code for the Library is available on Github.

Disclaimer

It should be noted that while every effort has been taken to ensure the veracity of the calculations provided by the library, and a great deal of testing has been placed, this is provided AS-IS, with no warranty or claim that it is completely accurate and without flaws.

Related Projects

Licensing

The full source code (library, website, api, mobile apps) are made available (under the GNU Lesser General Public License).

Credits

The following projects have been very useful in building the prayer times library that is used by this project, and deserve to be acknowledged:

• https://github.com/stankovski/prayer-times
• https://github.com/mohamedmansour/prayer-times-extension
• https://github.com/zulfahmi93/prayer-times/