Notes and definitions
Fajr
Fajr begins with the break of true dawn — that is, twilight which appears in the East and continues to expand both horizontally and vertically.
The Shafiis, Hanafis and Hanbalis say that fajr ends at sunrise. The Malikis say that fajr lasts until the brightness is such that people's faces are clearly distinguishable by someone with average sight in a place not covered by a roof, and when the stars cease to be visible; and that its extended time (for those with a legal excuse) lasts until sunrise.
Note True dawn is normally preceded by the false dawn — a column of light flanked on either side by darkness which extends up along the ecliptic. It resembles the tail of a black wolf, and corresponds to the phenomenon known as the zodiacal light. True dawn is followed by the red dawn. The altitude of the sun at the time of true dawn is by the determination of many Muslim astronomers, -18° (fajr altitude). This value was widely used throughout Islamic history, although slight variations (between -17° and -20°) have also been mentioned and are still in use today. Angles as small as -15° have no pre-modern precedent and are a result of observations affected by artificial anthropogenic skyglow (Spitschan et al 2016, Falchi et al 2016). When the sun does not reach the fajr altitude (and hence twilight is perpetual), the sun anti-transit time (also known as solar midnight) is used. This is the point at which the signs of fajr establish, i.e. twilight moves from West to East, stops decreasing and begins to increase filling the horizon until the sun appears. This time is also equivalent to both nisf ul-layl (half magrib to sunrise) and aqrab ul-bilad (proportion of night at the nearest place where the fajr altitude is reached). (! Notes on when to perform esha and fajr during persistent twilight: Patel 2017, Patel 2016.)

Duha/ishraq
The time of duha/ishraq begins from the moment the sun is elevated one spear's length, and lasts until istiwa. This is also the period of the Eid prayer for the Hanafis, Hanbalis and Malikis. The Shafiis say the time for Eid prayer extends from sunrise to istiwa.
Note A spear (rumh) is 12 spans and as a measure in astronomy according to astronomers and manuscripts, 4.5° (karaha altitude), although slight variations (notably 5°) have also been mentioned.

Zohr
Zohr commences immediately after istiwa. Istiwa is the time when the sun is at its zenith and prayer is forbidden. It is a momentary period in which the sun transits the celestial meridian; technically calculated here as the period beginning with the crossing of the sun's leading limb and ending with the exit of its trailing limb, at which time zohr commences.
The Shafiis and Hanbalis say that zohr ends when the shadow of an object is the length of the object increased by the length of its meridian shadow (asr mithl). The Malikis hold the same but say its extended time lasts to sundown. They also maintain that there is a shared time period between zohr and asr the length of which is the time required to complete four rakahs of prayer (or two for a traveller) but differ as to if this period is at the end of zohr or the beginning of asr. The Hanafis say that zohr ends when the shadow of an object is twice the length of the object increased by the length of its meridian shadow (asr mithlain), although the Sahibain hold that zohr ends when the shadow of an object is the length of the object increased by the length of its meridian shadow, in agreement with the other three schools.

Asr
The Shafiis, Hanbalis and Malikis maintain that asr begins when the shadow of an object becomes longer than the length of the object increased by the length of its meridian shadow (asr mithl). However, the Malikis maintain that there is a shared time period between zohr and asr the length of which is the time required to complete four rakahs of prayer (or two for a traveller) but differ as to if this period is at the end of zohr or the beginning of asr. The Hanafis say that asr begins when the shadow of an object becomes longer than twice the length of the object increased by the length of its meridian shadow (asr mithlain), although the Sahibain hold that asr begins when the shadow of an object becomes longer than the length of the object increased by the length of its meridian shadow, in agreement with the other three schools.
The Hanafis and Shafis maintain that asr ends at sundown, but the Hanafis consider it reprehensible to delay the asr prayer until changes becomes visible in the appearance of the sun. The Hanbalis say that asr ends when the shadow of an object becomes twice the length of the object increased by the length of its meridian shadow (asr mithlain). The Malaikis say that asr lasts until sunlight reflected off the ground and walls takes on a yellowish hue. Both the Hanbalis and Malikis say the extended time of asr lasts to sundown.
Note The sun's disc is said to change visibly is when it reaches the height of a spear above the horizon, or according to some when looking at the sun's disc no longer causes discomfort to the eyes. The former is 12 spans and as a measure in astronomy, 4.5° (karaha altitude), although slight variations (notably 5°) have also been mentioned.

Magrib
Magrib begins when the sun has disappeared entirely under the horizon. (! Notes on uncertainty: Schaefer & Liller 1990, Young 2004, Sampson et al 2003.)
The Shafiis and Hanbalis say that magrib ends with the disappearance of the red twilight glow. The Hanafis say that magrib ends with the disappearance of the white twilight glow, although the Sahibain hold that magrib ends with the disappearance of the red twilight glow. The Malikis say that magrib lasts only as long as is required to perform magrib prayer and fulfill the requisite conditions for its validity as well as the azan and iqamah.

Esha
The Shafiis, Hanbalis and Malikis say that esha begins from the time the red twilight glow disappears. The Hanafis say that esha begins from the time the white twilight glow disappears, although the Sahibain hold that esha begins from the time the red twilight glow disappears, in agreement with the other three schools.
The Shafiis and Hanafis say that esha lasts until true dawn. The Hanbalis and Malikis say that esha lasts until the first third of the shari night (1/3 shari night (magrib to fajr)), and that only its extended time lasts until true dawn.
Note The altitude of the sun at the time of the disappearance of white twilight is by the determination of many Muslim astronomers, -18° (esha altitude). This value was widely used throughout Islamic history, although slight variations have also been mentioned and are still in use today. When the sun does not reach the esha altitude (and hence the signs of esha do not appear), no time is given. (! Notes on esha when its signs are absent: Radd al-Muhtar ala al-Dur al-Mukhtar of Ibn Abidin 1855, Furber 2015, Patel 2016)

Technical Notes
Astronomical coordinates are calculated using VSOP87D and have a precision better than one arc-second. However, ranges are used for rise/set (rise/set uncertainty) and fajr/esha (fajr/esha uncertainty) because of unpredictable day-to-day variations in atmospheric refraction, uncertainties in terrain, and because apparent solar altitudes for fajr/esha are generally cited to 1⁄2 degree. Rise/set/transit times have been validated against those published by the United States Naval Observatory (USNO), with agreement to within 1⁄2 second on average. Note: Fajr/esha/karaha input altitudes are apparent solar altitudes; temperature/pressure inputs are at the observer; refraction coefficient is the ratio of ray curvature to earth curvature; night fractions use today's magrib and tomorrow's fajr; and night seasonal/temporal hour length uses today's magrib and tomorrow's sunrise. Moon visibility is at the observer and best time, calculated using Odeh 2005. Qibla magnetic bearing is calculated using the International Geomagnetic Reference Field (IGRF) model. Day/night overlay uses Georgiev's Nite. Night sky brightness uses Falchi et al 2016. Jurisprudential differences are mainly referenced from al-Fiqh ala al-Madhahib al-Arbah of Al-Jaziri. Full version: www.muwaqqit.com. Mobile version: www.muwaqqit.com/mobile.

Endorsements
Mufti Sajid Patel (specialist in the times of prayer, UK); Shaykh Haitham al-Haddad (Islamic Sharia Council, UK)

REST API
The REST API gives developers the ability to use Muwaqqit.com salah times and astronomical data in their own applications. Data can be requested in JSON or XML using the appropriate endpoint: https://api.muwaqqit.com/api.json or https://api.muwaqqit.com/api.xml. The following query string parametres can be used: d date as yyyy-mm-dd, eg. "d=2016-09-06"; diptype set "diptype=apparent" to use horizon dip for fajr/esha/ishtibak; ea esha angle in degrees, eg. "ea=-18.0"; eh elevation of the horizon in metres, eg. "eh=5.0"; eo elevation of the observer in metres, eg. "eo=5.0"; fa fajr angle in degrees, eg. "fa=-18.0"; fea fajr/esha uncertainty in degrees, eg. "fea=0.5"; ia karakah angle in degrees, eg. "ia=4.5"; isn istibak un-nujum angle in degrees, eg. "isn=-10.0"; isna istibak un-nujum uncertainty angle in degrees, eg. "isna=0.5"; k refraction coefficient, eg. "k=0.155"; ln longitude as a decimal, eg. "ln=0.0458383"; lt latitude as a decimal, eg. "lt=0.0458383"; p pressure at the observer in mb, eg "p=1010.0"; t temperature at the observer in celsius, eg "t=15.0"; rsa sun rise/set uncertainty in degrees, eg. "rsa=0.5"; tz time zone name according to the tz database where any special characters are escaped, eg. "tz=Europe%2FLondon"; vc hilal sighting model – set "vc=5.65" for naked-eye clear, "vc=3.53" for naked-eye difficult, "vc=2.00" for naked-eye very difficult, "vc=-0.96" for optical aid. The API returns data for one month at a time. Non-registered requests are rate limited to 1 request per 5 seconds. Register your project by emailing the address below. Data is licensed under a Creative Commons Attribution License. This allows derivative use on the condition of appropriate attribution and linking back to Muwaqqit.com.