London Unified Prayer Timetable
London Unified Prayer Timetable

Technical

The London Unified Prayer Timetable is calculated using standard astronomical formulas from Astronomical Algorithms by Jean Meeus (second edition 1998). The equations used are reproduced below.

Coordinates for calculations

Distances to London have traditionally been measured to Charing Cross.

This is a good average point for the mosques in inner London. It also lies exactly between the most eastern and western edges of the M25, which is the boundary of the region covered by this prayer timetable.

Latitude and longitude for Charing Cross: 51.5073°N, 0.12755°W.

Fajr and ‘Ishā

The times for Fajr and ‘Ishā are based on the UK observations of Hizbul Ulama; they determined the times between Fajr and sunrise, and between sunset and ‘Ishā.

Accuracy and geographic coverage

The timetable has been calculated using Microsoft Excel, then checked using JavaScript. The times in Excel and JavaScript when rounded to the nearest minute match exactly.

The National Oceanic & Atmospheric Administration makes the following observation about calculations: “although the sunrise and sunset results are theoretically accurate to within a minute, due to variations in atmospheric composition, temperature, pressure and conditions, observed values may vary from calculations.”

The intention is for these prayer times to be applicable anywhere in the London region contained within the M25. The time difference from chosen centre, Charing Cross, to the farthest points east and west is about 100 seconds. To allow for this, also for elevations above sea level and for variations in atmospheric refraction, the following adjustments are made:

  • Sunrise:3 minutes earlier
  • ‘Asr:2 minutes later
  • Sunset:3 minutes later

An allowance is made as follows to avoid Zawal (when the Sun is at its highest point):

  • Zuhr:5 minutes later

These allowances ensure the calculated prayer times can be used throughout the London region. There will therefore be a 3-minute difference between the prayer timetable and, for example, the sunrise and sunset times on the BBC Weather website and app.

Islamic dates

The Umm al-Qura calendar is used for Islamic dates. As conversions for the Umm al-Qura calendar using either ICU Islamic calendar or Microsoft Excel occasionally return an incorrect date, this website uses official Umm al-Qura calendar data from the King Abdulaziz City for Science and Technology (KACST).

Note that for actual Islamic dates, such as the start and end of Ramadān, and the start of Hajj, the sighting of the new crescent moon takes precedence.

Formulas used in calculations

There follows the sequence of calculations used. Unless otherwise stated, references are from Jean Meeus, ‘Astronomical Algorithms’, second English edition published in 1998.

Note: equations that are wider than the screen can be scrolled horizontally.

The Sun’s mean longitude in degrees in range 0° to 360° (Meeus, page 183):

L0=280.4664567+360007.6982779 r+0.03032028 r2+r349931−r415300−r52000000L_0 = 280.4664567 + 360007.6982779\,r + 0.03032028\,r^2 + \frac{r^3}{49931} - \frac{r^4}{15300} - \frac{r^5}{2000000}

where r is the time measured in Julian millennia. The last three terms can be omitted without affecting desired accuracy.

The Sun’s mean anomaly in degrees (Meeus, page 338):

M=357.5291092+35999.0502909 T−0.0001536 T2−T324490000M = 357.5291092 + 35999.0502909\,T - 0.0001536\,T^2 - \frac{T^3}{24490000}

where T, the time measured in Julian centuries of 36525 ephemeris days from the epoch J2000.0 (2000 January 1.5 TD), is given by

T=JD−245154536525T = \frac{JD - 2451545}{36525}

where JD is the Julian Day (Meeus, page 143). Note that the Julian Ephemeris Day is not introduced, as the difference in accuracy is only a fraction of a second.

The eccentricity of the Earth's orbit (Meeus, page 151):

e=0.016708634−0.000042037 T−0.0000001267 T2e = 0.016708634 - 0.000042037\,T - 0.0000001267\,T^2

Sun’s equation of centre (Meeus, page 164):

C=(1.914602∘−0.004817∘ T−0.000014∘ T2)sin⁡M+(0.019993∘−0.000101∘ T)sin⁡2M+0.000289sin⁡3MC = (1.914602^{\circ} - 0.004817^{\circ}\,T - 0.000014^{\circ}\,T^2)\sin M + (0.019993^{\circ} - 0.000101^{\circ}\,T)\sin 2M + 0.000289\sin 3M

Sun's true longitude ⊙ and true anomaly v in degrees (Meeus, page 164):

⊙=L0+C\odot = L_0 + C
v=M+Cv = M + C

Sun’s radius vector, or the distance between the centres of the Sun and the Earth, expressed in astronomical units (Meeus, page 164):

R=1.000001018 (1−e2)1+ecos⁡vR = \frac{1.000001018\,(1 - e^2)}{1 + e\cos v}

Sun’s apparent longitude in degrees (Meeus, page 164):

λ=⊙−0.00569∘−0.00478∘sin⁡(125.04∘−1934.136∘ T)\lambda = \odot - 0.00569^{\circ} - 0.00478^{\circ}\sin(125.04^{\circ} - 1934.136^{\circ}\,T)

Mean obliquity of the ecliptic in degrees (Meeus, page 147):

ε0=23∘26′21.448′′−46.8150′′ T−0.00059′′ T2+0.001813′′ T3\varepsilon_0 = 23^{\circ}26'21.448'' - 46.8150''\,T - 0.00059''\,T^2 + 0.001813''\,T^3

Corrected obliquity of the ecliptic in degrees (Meeus, page 165):

ε=ε0+0.00256∘cos⁡(125.04∘−1934.136∘ T)\varepsilon = \varepsilon_0 + 0.00256^{\circ}\cos(125.04^{\circ} - 1934.136^{\circ}\,T)

Sun’s right ascension α and declination δ in degrees (Meeus, page 165):

tan⁡α=cos⁡ε sin⁡λcos⁡λ\tan \alpha = \frac{\cos \varepsilon \, \sin \lambda}{\cos \lambda}
sin⁡δ=sin⁡ε sin⁡λ\sin \delta = \sin \varepsilon \, \sin \lambda

The equation of time in degrees (Meeus, pages 184–5):

E=ysin⁡2L0−2esin⁡M+4eysin⁡Mcos⁡2L0−12y2sin⁡4L0−54e2sin⁡2ME = y \sin 2L_0 - 2e \sin M + 4ey \sin M \cos 2L_0 - \tfrac{1}{2} y^2 \sin 4L_0 - \tfrac{5}{4} e^2 \sin 2M

where y=tan⁡2 ⁣(ε2)y = \tan^2\!\left(\tfrac{\varepsilon}{2}\right). Multiply E by 4 to convert from degrees to minutes of time.

Thus, the time, in hours, of solar noon:

m0=12−L15−E15m_0 = 12 - \frac{L}{15} - \frac{E}{15}

where L is the geographic longitude in degrees, measured positively east from Greenwich, negatively to the west (note that if comparing with Jean Meeus, he goes against convention by denoting longitude positively to the west).

The hour angle at sunrise in degrees (Meeus, page 102):

H=cos⁡−1 ⁣(sin⁡(−0.833333∘)−sin⁡φ sin⁡δcos⁡φ cos⁡δ)H = \cos^{-1}\!\left( \frac{\sin(-0.833333^{\circ}) - \sin \varphi \, \sin \delta}{\cos \varphi \, \cos \delta} \right)

where φ is the latitude. The value -0.833333° comprises 16’ for the semidiameter of the Sun and 34’ for atmospheric refraction.

Thus, the time, in hours, of sunrise Sr and sunset Ss respectively:

Sr=m0−H15S_r = m_0 - \frac{H}{15}
Ss=m0+H15S_s = m_0 + \frac{H}{15}

Time, in hours, when the length of an object’s shadow equals the length of the object itself plus the length of that object’s shadow at noon:

A1=m0+cos⁡−1 ⁣(sin⁡(cot⁡−1(1+tan⁡(φ−δ)))−sin⁡φ sin⁡δcos⁡φ cos⁡δ)⋅115A_1 = m_0 + \cos^{-1}\!\left( \frac{\sin\left( \cot^{-1}(1 + \tan(\varphi - \delta)) \right) - \sin \varphi \, \sin \delta}{\cos \varphi \, \cos \delta} \right) \cdot \frac{1}{15}

where the division by 15 converts from degrees to hours.

In Excel, where the calculation is in radians, and the result is required as a fraction of one day, the division is instead by 2π (first multiply by 360/2π to convert to degrees, then divide by 15 to convert to hours, then divide by 24 to convert to a fraction of one day).

Similarly, when the shadow equals twice the length:

A2=m0+cos⁡−1 ⁣(sin⁡(cot⁡−1(2+tan⁡(φ−δ)))−sin⁡φ sin⁡δcos⁡φ cos⁡δ)⋅115A_2 = m_0 + \cos^{-1}\!\left( \frac{\sin\left( \cot^{-1}(2 + \tan(\varphi - \delta)) \right) - \sin \varphi \, \sin \delta}{\cos \varphi \, \cos \delta} \right) \cdot \frac{1}{15}

(Mohamoud A. Mohamoud, ‘Tracing the Shadow: Mathematical Calculation of Prayer Times Using Spherical Trigonometry’, Middle-East Journal of Scientific Research 25 (8), published in 2017, page 1656)