DEFINITION & CALCULATION of PRAYER TIMINGS

 

Abdurrahman ÖZLEM

 

"It is He Who made the sun to be a shining glory and the moon to be a light (of beauty), and measured out stages for it; that ye might know the number of years and the count of time (10:5)"

"The sun and the moon follow courses (exactly) computed (55:5)"

"He makes... the sun and moon for the reckoning of time... (6:96)"

"And the Sun runs unto a resting place... (36:38)"

"Have you not considered (the work of) your Lord, how He extends the shade? And if He had pleased He would certainly have made it stationary; then We have made the sun an indication of it. Then We take it to Ourselves, taking little by little (25:45-46)

"It is not permitted to the Sun to catch up the Moon, nor can the Night outstrip the Day: each just swims along in its own orbit according to Law (36:40)"

 

 

INDEX

INTRODUCTION

TIMINGS

QIBLA

HIJRI CALENDAR

CONCLUSION

BIBLIOGRAPHY


INTRODUCTION

Our Wise Creator assigned some worships to His esteemed Muslims. Obligatory duties like prayer (salah), fasting and pilgrimage (hajj) are subject to defined timings. "For such prayers are enjoined on believers at stated times (4, 104)" declares that the five daily prayers must be performed within their correct times. The performance of the prayer within meticulously prescribed time periods has always been an integral part of a Muslim’s devotional routine. If an act of the worship is enacted within its prescribed time slot it would be regarded as Ada (complete); but if is performed after its time, it becomes Qada (delayed). Other than the start and end timings of these five prayers, there are also "makrouh timings", when a prayer should be avoided. Whereas prayer and start/end of fasting are subject to daily timings; hajj, fasting and obligatory alms are performed within yearly periods. Since the year is defined as 12 lunar months, it becomes essential to know the beginning day of any month.

In order that all those timings can be easily determined by the people, the daily and monthly timings are bound to the apparent motions of the Sun and the Moon. This has motivated the Muslims to study astronomy and try to calculate these timings accurately.

Our modern era with the highest living tempo necessitates programming even the minutes; so a Muslim should know the definition, calculation and uncertaintities related to prayer timings. This whitepaper aims to help the reader to acquire this basic information and also to "open a conceptual window" in the sky.

Readers who do not have time to examine the whole paper or who are not interested in the theory, may skip to the practical information summarized in the Conclusion section.

 

TIMINGS

In this chapter, we will try to explain begin and end conditions of the five prayer timings, besides the makrouh ones. To clarify, we will state the definition of each timing, expressed in Qur'an and Hadith, and then establish certain rules based on these expressions and demonstrate the calculation of the timings using astronomical methods. Moreover, we will mention some difficulties which are faced when we designate the timings by computation and, in this context, point out to safety margins as well as differences between the prayer calendars. Last, we will study the attitude of prayer timings at extreme latitudes, which are allocations near the poles.

 

PRAYER TIMINGS

These are the start and end times of the daily five obligatory prayers. Begin and finish times for fasting are based on the same timings.

Definitions

The basis for the definition of the pray/fast timings are the Qur'an and Hadiths. Direct or indirect mention of the times of the daily prayers can be found in several parts of the Qur’an (2: 238, 4:103, 11:114, 17:78, 24:36, 30:17-18, 50:39-40, 76:26); the fine designation of the timings are left to the practices of the Holy Prophet (peace be upon whom). Those timings are taught to the Messenger by the angel Gabriel; he first came and they prayed whenever the relevant time has just started; the next day he came again and they prayed just before time had elapsed. So he showed the time slots for each prayer and said “between these times are the times for the prayers”. The Messenger then informed his ashab verbally and practically about these timings. In a Hadith is told that “There is a beginning and an end of the time of each prayer. The time of the Zuhr (noon) prayer is when the sun starts to incline, until the time of the Asr (afternoon) prayer. The time of the Asr prayer is until the yellowing of the sun. The time of the Maghrib (sunset, evening) prayer begins with sunset and lasts as long as twilight. The time of the Isha (night) prayer is when the twilight disappears, up to the middle of the night. And the time of the Fajr (morning) prayer is from the appearance of the dawn until the time of sunrise”. We will now strive to determine the definition of each prayer time by starting out from this and similar Hadiths:

Zuhr Prayer Time: According to the Hadith the Zuhr prayer time starts when the Sun passes the meridian (zenith) and inclines towards west; there is no conflict about the beginning of the Zuhr prayer, as it is also fixed by the Qur'an (17:78). The Zuhr prayer time lasts until the Asr prayer time.

Asr Prayer Time: As can be deduced from the Hadith, the beginning time of the afternoon is not explicitly specified by the Holy Prophet, but shown practically. In several Hadiths, it has been reported that they performed the Asr prayer when "the length of the shadow of something became equal to the length of itself". As to clarify the subject, it has also been quoted that, "at noon time the shadow of a man was as long as a shoe strap, and the Asr was prayed when the man's shadow reached to the length of the strap plus the man's own length". According to that, the first time of the Asr prayer (Asr-I) enters when the shadow of anything grows one exact length of itself from its shadow at noon. However, there is a dissent about the start time of the Asr prayer; Imam Abu Hanifa accepts that the 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 (Asr-II). The end of the afternoon prayer is declared in the Hadith as "the yellowing of the sun". Nevertheless, this should be understood such that it is makrouh to pray the Asr after the yellowing of the sun (Fading); because the time of afternoon prayer continues actually until the sunset, as per Hadith "Who attains to one rakat of Asr prayer before the sun has set, has attained to the Asr prayer".

Maghrib Prayer Time: Starts with the complete sunset in respect of the Hadith. If the west horizon cannot be seen because of any obstruction, the expression "the east hills getting dark" in a Hadith clarifies the beginning of Maghrib prayer. This statement necessitates that the upper limb of the sun must disappear under the horizon, even when observed from the highest site of the district, because during sunset, the sunlight is reflected to the opposite direction, namely to the east. The vanishing of the twilight is defined in the Hadith for the end of Maghrib and beginning of Isha. However, relying on the Hadith where Gabriel came the next day for the sunset prayer at the same time as the previous day, Maliki school infers that the Maghrib prayer timing is a narrow duration starting with sunset. This is also in confirmation with a Hadith which informs that "it is faithful to complete the Maghrib prayer before many stars become visible", so in any case it should be makrouh to delay the Maghrib prayer.

Isha Prayer Time: Begins with the loss of twilight (dusk) and ends at midnight according to Hadith. One narration notifies the end of Isha as "one-third of the night". But a Hadith recounted by Ebu Katada states that "the time of a prayer continues until the time of the following prayer". Considering this, the Isha prayer time should last until the Fajr time, but it is makrouh to pray it after the one-third or one-half of the night.

There is dispute between the schools about the dusk. Imam Abu Hanifa accepts that the twilight is the whiteness (shafaq al-abyad) on the horizon, whereas other madhhab scholars take the redness (shafaq al-ahmar). After sunset, it first appears a red glow on the horizon, which then fades and is replaced by the faint whiteness. In that manner, Isha time of Imam Abu Hanifa enters when the whiteness (any visible light) on the horizon has completely disappeared, which occurs at least 30 minutes after the extinction of the red glow, namely the Isha of the other madhhab scholars. We find out that there was neither a consensus about the Isha time among the ashab; some followed the redness, others the whiteness.

Fajr Praying Time: According to Hadith, it begins with the true dawn and goes on until the sunrise. The true dawn (fajr as-sadiq) is clarified by the Messenger as "the light which spreads horizontally on the horizon", increasing in visibility and clarity. Before the true dawn is observed the false dawn (fajr al-kazib), "a light which appears vertically on the horizon". "The false dawn should not prevent anyone from eating/drinking", therefore it cannot be the time for beginning the fasting nor the Fajr prayer. Accordingly, while the dawn enters, we first observe a vertical faint illumination ascending towards the sky like "the tail of a fox" (or a pillar/pyramid) which is gradually succeeded by a skylight, continuously brightening and horizontally expanding, which is the true dawn, the start of Fajr prayer and also fasting. The characteristic event of false dawn, observed typically before the true dawn, is considered to be the Zodiacal Light. The zodiacal light is originated by meteoric dust particles dropped off from asteroids and comets found in the plane of the solar system which scatter the sunlight in such a way that it is faintly visible from Earth. The zodiacal light forms a faint pyramid (Figure 1), with its floor on the horizon and the tip pointing to the sun orbit (ecliptic). It can be more easily distinguished at low latitudes (near equator).

 

Figure 1.    Zodiacal Light

The sunrise, which finishes the Fajr prayer time, occurs when the upper limb of the sun rises over the apparent horizon and just becomes visible.

Rules

In this section, we will mention how to convert the definitions specified in Qur'an and Hadiths to some rules such that we can obtain the timings by calculation. If we look at the definitions given above, we encounter expressions like sunset, sunrise, sun passing the meridian or yellowing of the Sun. This indicates us that the Sun is the main element to designate the daily prayer timings of Muslims. Similarly, the length of shadows or the occurrence of twilight is related to the position of the Sun on the sky. For this reason, the trajectory of the Sun is the base factor for the establishment of calculation rules. Sample trajectory of the Sun on the sky is sketched below. Figure 2 is prepared for Ankara (latitude of 40º North), the left trajectory at June 21st, the right at December 21st. The orange curve above the plane shows the daytime, the blue one below the plane shows the night part. The circle circumferencing the plane symbolizes the horizon line; north is to the right.

 

Figure 2.    Typical Sun Trajectory

As can be deducted from the figure, the Sun rises from the east side crossing up the horizon and after reaching the peak elevation at zenith, it passes to the west side and it declines back to cross down the horizon and set. What also observed is that the summer and winter trajectories are different; it stays longer above the horizon in summer and crosses the horizon on locations closer to north. On the contrary, it stays shorter above the horizon in winter and crosses the horizon on the locations closer to south. In fact, the sun rises and sets at slightly different points on the horizon each day, following a spiral track. One interpretation of the clause "Lord of the easts and the wests" in Qur'an (55:17) may indicate this feature of the Sun's trajectory.

During the daily turn of the Sun on the sky, its vertical angle to the horizontal plane continuously changes with time. This angle, representing the height of the Sun relative to the horizon, is called as the Sun Elevation Angle (ELEV), which is the base parameter for calculation of prayer timings. Sun Elevation Angle is taken from the center of the Sun. Note that ELEV is maximum at noon, minimum at midnight and zero when the Sun crosses the horizon.

After giving this information, let us continue with the determination of the rules for each timing:

Zuhr Time:

Zuhr time begins when the Sun passes to west, as its definition implies. The Sun is rising on the east side during the first half of the day, whereas it declines on the west side afterwards. At the very middle of the day, it sits on the local meridian (zenith), namely the center of the Sun coincides with the theoretical vertical plane on the north/south direction. At that instant, the Sun points to the exact south on the northern hemisphere (see Figure 2) or to the exact north on the southern hemisphere (except within the equatorial zone). Since it is forbidden to pray at this very moment (zenith/noon time) when the Sun culminates on the sky, we should wait at least until the Sun globe leaves this vertical meridian plane, such that the Sun can be accepted as having declined to the west and the Zuhr time has begun. During this period the meridian line shifts from the center of the Sun to its circular border, the distance of one radius. Since the apparent radius of the Sun on the sky subtends an angle of approx. 0.27° and the Sun moves one degree in four minutes (360° per day = 360° per 1440 minutes), it will be neccessary to pass slightly more than one minute while the Sun moves one radius and the meridian leaves the Sun circle. That's why it should be added 2 minutes to the zenith time in order to find the Zuhr prayer timing.

We mentioned that the Sun reaches its maximum height at zenith time for a particular day. So the Sun Elevation Angle at zenith time is the highest vertical angle for that day. From now on, we will call it as Zenith Elevation Angle and use it later for the calculation of Asr time. When the Sun is positioned in its highest elevation at zenith time, it means that the shadows are shortest at that time. The shortest length of a shadow is called as Zenith Shadow, which takes part in Asr Shadow calculation:

Asr Time:

Asr timing begins when the shadow of any object elongates to its Asr length, as specified in the Hadith. However, there is discrepancy between the schools about the asr shadow length, which was explained in the Definition section. The opinion of Maliki, Shafi, Hanbali schools and also of the Hanafi imams except Imam Abu Hanifa is that the Asr shadow is constituted by adding the length of any object to its Zenith Shadow length. For example, if we assume that a stick of one meter length has a shadow of one half meter at noon time, then the Asr prayer time will enter when its shadow reaches a length of: 0.5 m + 1 m = 1.5 m. This Asr time will be named as the Early Asr or Asr-I. But Imam Abu Hanifa takes the Asr shadow as twice the length of any object plus its Zenith Shadow length; i.e. the shadow of our stick in the former example has to attain a length of: 0.5 m + 2 m = 2.5 m. This Asr time of Abu Hanifa will be called as Late Asr or Asr-II. The formation of the Zenith, Asr-I and Asr-II shadows is shown in Figure 3.

 

Figure 3.    Determination of Asr Time

Taking into consideration these different opinions between the schools, it would be safer to perform the Zuhr prayer before the Early Asr and the Asr prayer after the Late Asr thus conforming to all schools. Nevertheless, in a case of difficulty, a missed Zuhr prayer may be performed after the Early Asr until the Late Asr without Qada, obeying Imam Abu Hanifa, and performing the Asr prayer after the Late Asr.
 

Sunset & Sunrise Times:

We learn from the Hadith that the Sun must be completely set in order that the Maghrib time enters. This will be valid when the upper limb of the Sun sinks below the apparent horizon line. The apparent Sun Elevation Angle is 0° if the center of the Sun is just crossing the horizon. To touch the upper limb the horizon, we have to wait that the Sun moves down one radius, which is 0.27°. However, we must consider additionally that the atmosphere refracts the sunlight:

The atmosphere makes the entering Sun rays change their direction. Since under the refraction, the rays are directed to the denser layer, the light coming from the space will be diverted steadily towards the Earth as it moves through the still denser atmosphere, increasing the apparent Sun elevation angle. This causes the Sun to be seen higher than its actual position. So the astronomical height (true Sun Elevation Angle) is smaller than its height observed from Earth (apparent Sun Elevation Angle). The difference in-between is known as the Refraction Angle:

 

Refraction Angle = apparent Sun Elevation Angle - true Sun Elevation Angle

 

As ascending from Earth into the space (increasing altitude) the density of the atmosphere gradually decreases and diminishes practically after approx. 85 kilometers (mesopause). In fact, its pressure vanishes to 1% at already 30 km. altitude (upper stratosphere), which can be assumed as its physical equivalent width. Under this assumption, one may think that the refraction ratio is constant and independent of the incidence of the light; but it is not true:

Since the Earth is spherical, so is its atmosphere. As the Sun declines towards the horizon and the Sun Elevation Angle decreases, the distance traveled by the Sun rays increases (Figure 4).

 

Figure 4.    Optical Atmosphere Width depending on Sun Elevation

This "optical" width of the atmosphere is equal to its physical width when the lights come vertical to the Earth; but as the Sun declines, the optical width increases gradually. Especially during sunset and sunrise when the sun is very close to the horizon, the optical width grows several times. That's why; the refraction of the Sun rays by the atmosphere is not constant but a function of the angle between the Sun and the horizon (Sun Elevation Angle). The change of the Refraction Angle as a function of the apparent Sun Elevation Angle is depicted in Figure 5.

 

Figure 5.    Refraction Angle vs. Sun Elevation Angle

It should be mentioned that the values represented in the graphics are valid for standard atmospheric conditions (STP); the real refraction angle may deviate up to 10% depending on several meteorological conditions such as air temperature, air pressure (altitude), humidity or particle quantity. For example the refraction will be most in a cold moist salty morning when the Sun rises above the sea horizon. The general refraction equation is given below, where the pressure is in millibars and the temperature in °C:

Refraction Angle = 0.017° / tan((ELEV + 10.3° / (ELEV + 5.11°)) * p / 180°) * Pressure / 1010 * 283.15 / (273.15 + Temperature) 

In respect of the information so far we can deduce that the Sun is in fact completely under the horizon when we observe it is setting. This fact, visualized in Figure 6, must be considered when produce the rule for Maghrib time:

 

Figure 6.    Observation of the Sun Below the Horizon

The necessary elevation for the Sun to set fully, which is the condition for the Maghrib time to enter, is the radius angle of the Sun on the sky, namely -0,27° (negative value stands for below horizon). But this is the apparent angle, as observed from the Earth. True Elevation angle will be found by subtracting the Refraction Angle from this value as per the former equation. Figure 5 shows the Refraction Angle as 0.57° for an apparent Elevation Angle of zero. Therefore the true Sun Elevation Angle will be -0.27° - 0.57° = -0.84° at the time the Sun is just observed to set completely. Again this is a theoretical value, which is employed in most prayer calculators; the real angle however can be different since the refraction is influenced by temperature or humidity. In a research done in year 2003, the mean Refraction Angle was observed as 0.714° at sunrise and 0.579° at sunset (refraction is larger at sunrise because of lower temperature and higher humidity). Note that the sunset is described in one Hadith narration as the "disappearance of the Sun's eyebrow", which was interpreted as that the red belt right over the Sun circle must also slide down below the horizon after the sunset. To account for this requirement, we prefer to take the true Sun Elevation Angle for sunset as -1° instead of -0.84°.

At sunrise on the other hand, which ends the Fajr time, the upper limb of the sun just appears on the horizon when the apparent elevation is again -0.27°. Considering the former research, the proper ELEV for sunrise should be: -0.27° - 0.714° = -0.984°. Accordingly, we will take the Sun Elevation Angle at sunrise also as -1° instead of -0.84°.

The standard deviations of the refraction values obtained in this research are 0.108° for sunset and 0.184° for sunrise, so the uncertainty for sunset/sunrise timings will generally be less than 1 or 2 minutes, respectively, except the extreme latitudes.

Another parameter affecting the timing is the height. In case the sight location is at higher altitude than the viewable horizon in the sunset/sunrise direction, the look angle changes due to the roundness of the Earth. If we stand for example on the hill of a mountain and the Sun is setting over the sea, we will observe slant below the horizontal plane and we can see the Sun for a longer time, causing the sunset to occur later and the sunrise earlier. The look angle down the horizon in degrees is roughly the square-root of the altitude difference in kilometers. For example when watching the sunset over the sea (in winter) from the Çamlıca hill in İstanbul (267 meters altitude), the view angle will be approx. 0.5°, corresponding to nearly 3 minutes delay. The exact equation for the slant view angle is given below, where H is the altitude difference and R is the radius of the Earth.

Slant Angle = arccos(H / (H + R)) 

So the sunset/sunrise calculations should include a provision for slant angle compensation, considering the altitude difference between the highest location in the neighborhood and the point over which the sun sets/rises. This is generally done by adding a sufficient safety margin to the computed timing.

Isha & Fajr Prayers:

The definition of Isha according to Hadith is that the twilight (dusk) disappears whereas the Fajr is designated as the appearance of the true dawn. But twilight is a phenomenon directly stemming from the existence of the atmosphere and its scattering property. If the Earth would not be wrapped up with an aerial cortex, the whole sky will be completely dark as soon as the Sun sets. So there would be no twilight and neither the Fajr/Isha timings. Well, how is the twilight caused by scattering?

Since the dimension of the molecules constituting the atmosphere is much smaller than the wavelength of the sunlight, the rays will be scattered while passing through the atmosphere. This phenomenon, known as Rayleigh scattering in literature, varies depending on the wavelength (i.e. color) of the light. Due to the inverse relation between scatering and wavelength, the blue rays with relatively small wavelength are more heavily scattered than the red rays. Nearly 5% of the red rays from the Sun are scattered whereas this ratio is 20% for blue rays. The scattered blue rays paint the sky in blue and since the sunlight has now less blue, it seems a little yellowish. In fact, the Sun color observed from the space is dull white, the color of a fluorescent bulb. Yes, we give thanks to our Lord Who gives a scattering property to the atmosphere, thereby "warming" the Sun and coloring the pitch-dark sky (Figure 7).

 

Figure 7.    View of the Sky from outside the Atmosphere

When the property of wavelength-dependent-scattering combines with effect optical-thickness-grow, the panorama becomes delightful: As the Sun declines towards horizon, the scattering of blue rays increases and the Sun gets more yellow. Near sunset, it turns into a golden ball and its neighborhood becomes orange, whereas blue content increases as we move away from the sun, forming a color palette on the sky, from orange to yellow, then from purple to turquoise (Figure 8). After sunset, the twilight covers the west sky like a strip ascending from red to blue. This strip descends as the time passes and first the redness fades away, followed by the vanishing of the remaining dim whiteness.

 

Figure 8.    Effect of Scattering

Up to now, we have touched a bit on the causes of occurrence of red and white twilight regarding Isha & Fajr timings, as described in the Definitions section. Next, we will mention the difficulties in calculating the dawn/dusk times.

Rules defined for Zuhr, Asr and Maghrib (and Sunrise) prayer times are directly referred to the Sun's position on the sky, which can be calculated today astronomically with a very high accuracy. So it is possible to establish the values for the Zuhr & Asr timings within an uncertainty of several seconds, as long as the geographic location is precisely known. The sunset/sunrise times are a bit more indefinite because of the variance in the refraction by the atmosphere, within 1-2 minutes. However, the situation is somewhat different for Isha/Fajr; the determination of those timings are connected to the events of (true) dawn and (red/white) dusk, respectively, of which the instant of appearance cannot be fixed such definitely due to the following reasons:

In the light of the above-mentioned clarifications, we will remind again that the basis for the designation of the prayer timings is the observation. Especially for Isha and Fajr timings, the horizon must be observed whenever possible. Otherwise, when we are bound to rely on calculations, then we have to select safer timings, based on the uncertainties described above. As such, Isha & Fajr prayers should be delayed somewhat, hastening the Maghrib prayer on the contrary, which conforms to sunnah.

Now let's try to establish rules for calculation of Isha/Fajr timings… A vertical angle must be assigned for these timings to execute computations. Ancient Muslim scientists, especially around the 15th century when the astronomical prayer calculations became popular, adopted the value of -19° as the proper vertical angle for the twilight boundary, which was also used in Turkey until 1982 for Fajr. It was then also accepted that the vertical angle difference between the red and white dusk is 2°, such that the vertical angle for Isha regarding Ebu Hanifa (red dusk) became -17°, which is still used by many countries. But the more recent observatory studies in the western world (HM Nautical Almanac Office, US Naval Observatory and others) on the twilight stages show that the astronomical twilight occurs between -12° and -18° of vertical angle and that sky illumination after -18° is so faint that it is practically imperceptible. This limiting value of -18° has then been accepted by some countries as criteria for Fajr.

It has been recently debated, especially in England and U.S.A., whether the true dawn is valid at -18°. It has been asserted that the value of -18° is the boundary for astronomical twilight and it is not conforming to the definition of the true dawn which requires the horizontal spread. The observation of the zodiacal light by the Belgian astronomer Marcel Minnaert was around a depression angle of -17°, and he could distinguish the onset of twilight at -16/17° using photometric instruments. Dr. Mohammad Ilyas found similar results and he could only detect small changes in sky illumination after -16° with optical tools. A study in 2005 at ESO-Paranal Observatory (Chile) has akin outcomes (Figure 9). The graph shows the zenith brightness in  mag arcsec-2. X-axis is the Sun's zenith angle in degrees, so it must be subtracted from 90° to convert it into elevation angle. The measured data shows that the twilight brightness diminishes down to the night-sky illumination and then stays flat after around -16°. The solid line represents the simple-scatering model, which clearly knees at -10°, implying the end of red dusk. Following the results of such observations, some scholars accepted a vertical angle of -15° both for Fajr and Isha.

Figure 9.    Sky Brightness vs. Sun Elevation

To overcome this difficulty about the Fajr/Isha timings, some researchers attempt to perform long-term observations. One of the most comprehensive observations was started in 1987 September in Blackburn/Lancashire (England) and lasted near one year with the attendance of the religious scholars. Some interesting findings were obtained at the end of this observation:

  1. Although dawn and dusk are two symmetrical phenomena astronomically, it has been perceived that the relevant depression angles are not always identical, which is commented as the atmospheric conditions are not the same for dawn and dusk. In fact, the detection of the first illumination on the sky by an eye adapted to darkness is easier than the detection of the last illumination by an eye adapted to brightness. Besides, the humidity, fog and especially temperature levels are rather different at dawn compared to dusk. Yet another factor is that the dawn and dusk occur at opposite locations on the horizon with unlike air and surface layers.
  2. ELEV values for Fajr/Isha are not constant throughout the year but are seasonally changing. Values increase a bit in winter for example, which can be explained as the higher refraction of the light by the colder (and denser) atmosphere. So it was deducted that the angle values will vary in different locations with distinct climate.
  3. The angles were altering within one year between -12°/-16° for Fajr (tabayyun) and -9°/-11° for Isha (red dusk).

Regarding all these researches, we conclude that it is not possible to assign a constant and reliable vertical angle for Fajr/Isha, they change in time and location. Although observation is essential, it is not generally possible especially under city illumination. So we suggest to allow a gray zone of angle for Fajr as well as Isha, say -18°/-15° for the onset of true dawn, -16°/-12° for tabayyun, -9°/-12° for the red dusk and -15°/-17° for white dusk. Ideally we should refrain from these zones of uncertainty as much as possible (for example to start fasting at -18° but perform Fajr prayer at -12° or perform Maghrib prayer before -9° but perform Isha prayer after -17°, whereby conforming to sunnah). Nevertheless, this ambiguity should be considered as a blessing not a burden; in any case of difficulty, we believe that we can use these zones, similar to the case of Early/Late Asr.

Below are tabled reference ELEV values for the calculation of Fajr/Isha timings in some countries; differences stem from the inherent vagueness both for the definition and the detection (Table 1).

 

CountryFajrIsha
Europe, Far East, USA (partly)- 18- 17
Pakistan, Bangladesh, India, Afghanistan- 18- 18
Africa, Syria, Iraq, Lebanon, Malaysia- 19.5- 17.5
Canada, USA and UK (partly)- 15- 15
Arabian Peninsula- 19*)

Table 1.

*) The Isha timing is calculated in the Arabian Peninsula by adding 90 minutes to the Maghrib time (corresponds to -19°/-21°); 120 minutes added during Ramadhan.

In locations with higher latitudes, the twilight does not disappear throughout the night, especially in summer, since the Sun does not depress sufficiently below horizon. In that situation it is not possible to compute the Fajr/Isha timings astronomically; whether or when to perform these prayers will be clarified in section Extreme Latitudes.

Astronomical Calculation

Spherical trigonometry is utilized to astronomically compute the prayer timings, of which the definitions and rules in terms of Sharia are given above. All the timings are determined in reference to noon (zenith), so it is essential to find the noon time of the relevant day first. Perhaps there may be a sign to this astronomical reality in the Hadith declaring the prayer timings, where Zuhr was mentioned first. Thus we will start with the Zuhr prayer and continue in the order as in the Hadith to express the calculation methods of each timing in a lucid way.

How can we find the noon time of a day?

To clarify the case, let's examine first the civil time system we are using. Here, the mean noon time, which is the center of the day, is taken as 12:00 for 0º longitude (Greenwich). So when the Greenwich Mean Time (GMT) is 12:00, in any location on this main meridian, the Sun will coincide with the meridian, in average. Before explaining why "in average", we will first expound the local time.

Local time is determined according to the Standard Time Zone of a location. As an example, Turkey uses the time zone of +02:00, such that the local time of Turkey is 2 hours in advance relative to Greenwich (when it is average noon in Greenwich, the local time in Turkey will be 14:00). Thus we should add the time zone value of a location to the GMT in order to find the Local Time and vice versa. Horizontally wide nations like USA or Russia employ multiple time zones. In case the Daylight Saving Time (DST) is applied, one extra hour must be added to find the local time. Hence we will use the following formula for the local time:

 

Local Time = GMT + Time Zone + DST

 

This formula gives the average noon time at the main meridian in local time, if we set GMT=12. Well, what about the noon time on a specific meridian?

Since one mean day is taken as 24 hours and our planet rotates meanwhile 360° around itself with respect to Sun, Earth will axially move 360 / 24 = 15 degrees of angle within one hour. Therefore in a district on the 15° East meridian, the mean noon time will occur one hour earlier, namely at 11:00 GMT.

Thus we can calculate the noon time on any meridian using the following formulae:

 

Mean Zenith Time (GMT) = 12 - Longitude/15

Local Mean Zenith Time = 12 + Time Zone + DST - Longitude/15

 

Times in the formulae are in hours; whereas longitude is in degrees, positive for east of Greenwich. According to that, for example, the mean noon time in summer for İstanbul at 29° East will be 13:04, and the Zuhr time 13:06.

Now we return back to what mean time is...

The worldwide civil time system takes a day as 24 hours, which is in fact only an average value per day. However, the real duration of a day is changing periodically throughout the year. So each day, one complete rotation of the Earth takes several seconds more or less than 24 hours and the real time advances or delays against the mean time. As these daily tiny drifts accumulate, the total drift can reach 16 minutes. Figure 10 displays this time drift within a year (drift in minutes vs. day of year). To find the real noon time for a specific day, the relevant drift value should be subtracted from 12:00. For example, the Greenwich zenith time for the 72nd day of the year will be 12:10. Drift is known in literature as "Equation of Time" (EoT). On the picture below (right), you can also observe the outcome of this drift. The picture is formed by the superposition of the Sun's position on the sky at the same time (12:00) of every day; the compound effect of eccentricity and obliquity cause the Sun to deflect horizontally and vertically, constituting an "eight" shape, or "analemma". You can take a look at the animation or at a series of beautiful analemma photographs have been taken by the astrophotographer Antony Ayiomamits.

 

Figure 10.    Annual Drift of the Noon Time (EoT) and the Analemma

But why does the real duration of a day change throughout the year?

Our basic day, which also forms a basis to the daily prayer timings, is referenced to the Sun and it is called in the astronomical literature as tropical day. It is one complete rotation of the Earth as seen from the Sun.  This is also the duration of one complete turn of the Sun (from zenith to zenith) on the sky as seen from Earth. The trajectory tracked by the Sun on the sky stems from two rotations:

  1. rotation of the Earth around its own axis
  2. rotation of the Earth around the Sun   

The rotation of the Earth around itself is normally referenced to a very far star and it is named as the sidereal day. One sidereal day is 23 hours, 56 minutes and 4 seconds, i.e. near 4 minutes shorter than our known day. This self-rotation has a constant speed and does not change through the year (except that it is slowing down rather gently, several milliseconds each year, which is already corrected as leap seconds).One complete turn of the Earth around the Sun has a period of 365.2422 days (24 hours each). The tropical day is the composition of these two rotations. Both rotations are in the same direction (counter-clockwise when looked from the north), so the Earth must rotate slightly more than one sidereal turn to complete the tropical turn. The difference is the angle which the Earth moves in one day around the Sun, namely 360 degrees per 365.2422 days or 0.9856474º in average. Hence one mean tropical day occurs at each 360.9856474º rotation of the Earth around itself thus it is longer than one sidereal day. The exact angle value and consequently the duration is different for each tropical day because of the non-uniform speed of the Earth around the Sun.  The speed is not constant because of the elliptical nature of the trajectory of the Earth (eccentricity of the orbit is 0.0167 for epoch J2000). The square of the speed is proportional to the cube of the distance from the Sun, as to the 3rd law of Keppler. The difference of the true tropical day and the mean tropical day is symbolized with the EoT variable.

If we add this "beautiful" EoT feature which the Earth exhibits while gyrating with a celestial love, we end up with (1):

 

Local Zenith Time = 12 + Time Zone + DST - Longitude/15 - EoT

 

Zuhr time is obtained by adding 2 minutes to this local zenith time. All the other timings are acquired by adding/subtracting the time for the Sun to reach the specific vertical angle, using spherical trigonometry, as below (2):

 

HA = ± arccos(sin(ELEV)-sin(DECL)*sin(LAT))/(cos(DECL)*cos(LAT)))

Time = Local Zenith Time + HA/15

 

In this equation, HA represents the angle on the trajectory of the Sun between the zenith and the point when the Sun has the height of ELEV. This angle is named as Hour Angle; it is negative for a.m. (Fajr and Sunrise) and positive for p.m. (Asr, Maghrib and Isha). By dividing the Hour Angle into 15, we convert it into hours. Near the polar zones with high latitudes, if the Sun cannot reach ELEV, the parameter of arccos will become greater than one such that the arccos function is undefined and the timing does not occur astronomically. ELEV stands for the Sun Elevation Angle for that timing, LAT for the latitude of the location (positive for northern hemisphere) and DECL for the Declination Angle. DECL is defined as the angle between the Earth axis and the plane vertical to the ecliptic. In other words, DECL is the angle between the rays of the sun and the equatorial plane (the latitude crossing the Earth-Sun line). This angle is zero at equinoxes, +23,4º at summer solstice and -23,4º at winter solstice. Declination Angle is calculated as follows:

 

DECL = arcsin(sin(OBL) * sin(L))

 

OBL stands for the obliquity of the Earth axis (23.439º for J2000), whereas L represents the ecliptic angle, which is literally known as Sun's True Longitude. Ecliptic angle is zero at vernal (spring) equinox. The ecliptic angle has been measured 280.46º at J2000 (at 12:00 GMT of January 1st, 2000). Each day it increases 0.9856474º in average, as explained above. Following equation is then valid, where ML shows the mean ecliptic angle, or Sun's Mean Longitude, and D is the symbol for the number of days from J2000:

ML = 280.46º + 0.9856474º * D

The real ecliptic angle under the influence of eccentricity, will be obtained by the following equations. M represents the Mean Anomaly, which is the angle to perihelion (where Earth is closest to the Sun) and it is measured 357.53º at J2000. Note that the daily angular motions of M and the ecliptic angle are not exactly the same (0.9856474º vs. 0.9856003º), which tells us that the perihelion/aphelion drifts very slowly (1º every 58 years) because of the precession of the Earth's axial motion. At J2000, M is zero on January 3rd; but 10,000 years before, M was zero in summer, i.e. Earth was most-distant to the Sun (aphelion). In fact, not only the axial precession, but also the eccentricity and obliquity make oscillations in long-term. Moreover, other factors like apsidal precession and orbital inclination have an influence on the Earth's position (see Milankowitch Cycles).

M = 357.53º + 0.9856003º * D

L = ML + 1.914º * sin(M) + 0.02º * sin(2 * M)

RA = arctan(cos(OBL) * sin(L) / cos(L))

EoT = (ML - RA) / 15
 

RA is the Sun's Right Ascension. EoT is in hours. Above equations are accurate to about 1 arc-minute (1') in short-term, namely within two centuries of 2000. Since DECL and EoT keep changing within the same day, the instantaneous values should be used in the equation (2) for high accuracy. This in turn necessitates an iterative method, where DECL and EoT is updated with respect to the time value obtained by (2) and the time value is recalculated at next iteration.

All the prayer timings except the Asr can be directly calculated by the time equation (2). For Asr we need an additional computation to find the relevant ELEV, because the rule specifies shadow lengths:

We defined the Asr the time when the shadow length of any object reaches its shadow length at noon plus one (Asr-I) or two (Asr-II) times its own length. The shadow length of any object depends on the vertical angle of the Sun at that time. We can write the following equation obeying the rules of trigonometry:

 

cot(ELEV) = shadow length / object length

 

We can define now the Asr Vertical Angle for the Asr shadow:

 

cot(Asr Vertical Angle) = Asr shadow length / object length

 

Regarding the rules for Asr-I and Asr-II, we now write:

 

cot(Asr-I  Vertical Angle) = 1 + zenith shadow length / object length

cot(Asr-II Vertical Angle) = 2 + zenith shadow length / object length

 

And the zenith shadow length depends on the Zenith Vertical Angle as follows:

 

cot(Zenith Vertical Angle) = zenith shadow length / object length

 

Therefore we deduce:

 

cot(Asr-I  Vertical Angle) = 1 + cot(Zenith Vertical Angle)

cot(Asr-II Vertical Angle) = 2 + cot(Zenith Vertical Angle)

 

As can be seen from Figure 2, the Zenith Vertical Angle is different in summer than in winter. Zenith Vertical Angle, which is also a function of latitude, can be calculated by using the following formula:

 

Zenith Vertical Angle = 90º - |LAT - DECL|

 

Again, the latitude is positive for northern hemisphere. The absolute value is necessary for the equatorial zone. So the Asr equations (3) become:

 

cot(Asr-I  Vertical Angle) = 1 + |tan(LAT - DECL)|

cot(Asr-II Vertical Angle) = 2 + |tan(LAT - DECL)|

 

The variation of refraction angle, which was depicted in Figure 5, also influences the Asr shadow lengths. Both the zenith and Asr shadows appear slightly shorter because of refraction. Since the refraction increases as the Sun approaches the horizon, the shortening effect is smallest for the zenith shadow and greatest for Asr-II shadow. To compensate for this shortening effect, the Asr vertical angles computed by the former equations (3) should be corrected as follows (detailed information about this correction is available in the relevant paper):

 

Corrected Asr-I  Vertical Angle = Asr-I  Vertical Angle * 1.00065 – 0.0439º

Corrected Asr-II Vertical Angle = Asr-II Vertical Angle * 1.00191 - 0.0817º

 

MAKROUH TIMINGS

Definitions

The time intervals when praying is prohibited were defined in Hadith as such:

“There are three time periods the Holy Prophet forbid us to perform a ritual prayer as well as to bury our corpses:

Based on this Hadith, the makrouh timings are gathered into three, namely Ishraq, Mid-day and Fading:

Ishraq: It is proscribed to perform prayer between the Sunset and Ishraq time. In one Hadith was told that the Qada of a missed Fajr prayer was performed "when the Sun had ascended and whitened", in another Hadith was ordered that "prayer must be abandoned until the Sun reaches a spears' height and its (weak, yellow) rays disappear".

Mid-day: The Sun to "stand on top" should mean the transit (noon) time. But in one Hadith was narrated that “Holy Messenger deemed it makrouh to pray at mid-day, except Friday.”. Some scholars perceived the mid-day as the half of the canonical day, which is the time period of fasting, that is between the Fajr and Maghrib.

Fading: This is the symmetric of the Ishraq, when it is makrouh to pray from this time until sunset.

Rules

We already explained in the preceding sections that the yellowing of the Sun owes to the refraction of the atmosphere such that its optical thickness increases as the Sun approaches the horizon and its blue rays are scattered more heavily, causing the Sun to become pale. Although this change does not occur in a sudden and sharp manner, it has been admitted that the Sun must decline below 5° in vertical in order to become clearly yellow. Therefore, it is a general assent to take the Sun's vertical angle as 5° for the computation of Ishraq and Fading timings. Note that Ishraq is also the time for Eid salah.

Regarding the literal definition of Mid-day, it would be banned to pray only during the period the Sun (circle) is on the zenith, as explained for Zuhr timing. This, in turn, will cover two symmetrical intervals of 2 minutes, on either side of the noon-time; thus only 4 minutes before Zuhr will be makrouh. On the contrary, considering the broader definition, Mid-day becomes the center between Fajr and Maghrib. In this case, it will be makrouh to perform prayer for an interval equal to the half of the duration of Fajr prayer (between Fajr and Sunrise). If, for example, Fajr is at 04:00, Sunrise at 06:00, Zuhr at 12:02 and Maghrib at 18:00, Mid-day will start at 11:00 and prayer will be forbidden until 12:02 (see Figure 11). Sunset and Sunrise are shown in purple, Isha in green, Fajr in blue and Zuhr in yellow. Mid-day is in orange.

 

Figure 11.    Half of Canonical Day

Astronomical Calculation

Ishraq and Fading are computed by entering ELEV=5° into the equation (2); arccos will be taken as negative for Ishraq and positive for Fading. For Mid-day, we prefer to take the average of Fajr and Maghrib, considering the safer definition.

 

SAFETY MARGINS

We stated before that there are some inherent uncertainties, more or less, for each timing obtained by calculation. They may either arise by their definitions or originate from the geographical/meteorological conditions. Besides that, it may be appropriate to add some margin of safety to be convinced that the time has really entered. It was especially important in the past, when a single timetable was prepared for a broad city and the time variations within the area were neglected; or when the clocks used by the inhabitants generally drift from the true astronomical clock. That's why it has been a common practice to add/subtract a small amount of time, called Safety Margin. It is negative for Fajr and Sunset (also for Mid-day and Fading, if applied), positive for all others. Note that for locations with high latitudes, where the Sun trajectory is aslope, the vertical angle of the Sun changes more slowly and the uncertainty increases, necessitating a greater safety margin..

The default values in the Alperen software, in minutes, together with the relevant conditions given in the Rules section, are listed in Table 2.

 

TimeConditionSafety
FajrELEV = -18°

-2

SunsetELEV = -1°-7
IshraqELEV = 5° 10
Mid-dayZenith - (Sunset - Fajr) / 20
Zuhr Zenith + 2 min.+5
Asr Asr-I Vertical Angle+5
Fading ELEV  = 5°0
Maghrib ELEV  = -1°+7
Isha ELEV = -17°+2

Table 2.

Note that a larger value is acquired for Sunset/Maghrib, when the uncertainties are higher due to refraction and viewing height differences. Fajr/Isha does not need high margins, since already safer angle values are selected for them. Margins can be decreased if accurate coordinates are known and the clock is accurate (using GPS), particularly at low latitude. Here we must be aware of that the real values of the prayer timings will be found after extracting the Safety Margin value from the tabulated timings. For example, we should be conscious of that the Asr time already ends at 16:47 if it is displayed as 16:52.

 

EXTREME LATITUDES

We can divide the Earth into three zones, regarding the formation of the seasons and the variation of day/night durations.

The first one is the equatorial zone between ± 23.4° latitudes around the equator, where the Sun tracks a near-vertical trajectory; therefore the lengthening/shortening of the daytime during summer/winter is not prominent so the four seasons cannot be distinguished and a tropical climate exists all the year (Figure 12).

 

Figure 12.    Summer and Winter Trajectories of the Sun in Mecca (21.43° latitude)

On the contrary, the Sun will follow a rather horizontal trajectory within the polar zone with latitude greater than ±66°. The day/night durations in winter and summer exhibit great difference (Figure 13). In some days near solstice the Sun will not even rise nor set. Thus, on polar locations some of the daily prayer timings or even all of them will not occur astronomically.

 

Figure 13.    Summer and Winter Solstice Trajectory of the Sun in Oslo (59.93° latitude)

In the region between these extreme zones, which house the most of the population, the daytime prolongs/shrinks in summer/winter period depending on the latitude. At higher latitude, the sun moves more horizontally, so the night duration in summer will be shorter, which causes the Isha and Fajr timings to approach each other. This poses a difficulty to perform the Isha prayer, since it is squeezed into a late and small time.

Figure 14 displays the World Prayer Map for June, where the squeeze of the prayer durations is visualized. As moving to north, daytime (yellow/orange region) broadens and the night squashes. Moreover, the twilight period expands (blue region), increasing the time for Fajr (dawn to sunrise) and Maghrib (sunset to dusk) prayers, but the time for Isha (dusk to dawn) diminishes (gray region). After a certain latitude it disappears completely, when dusk and dawn coincide, so the twilight is continuous; the sky darkens up to the midnight but before reaching full darkness, it begins to lighten again, causing the absence of Isha/Fajr. The upper half of Asia and North America (continuous blue region and no gray region) suffer from this attitude in summer, as shown in Figure 14. In the polar zone even the blue region disappears, bringing about the absence of night (sunset/sunrise) such that Sunset and Maghrib timings cannot be established either.

 


Figure 14.    Change of Prayer Timings with respect to Latitude

The smallest vertical angle of the Sun at midnight can be calculated by using the following formula:

 

Dip Vertical Angle = |LAT + DECL| - 90°

 

If we take the Fajr/Isha vertical angle (twilight limit angle) as -17.5°, twilight will be continuous in several summer nights for latitudes greater than 49°, such that Fajr/Isha times will not occur astronomically.

Shortening of Isha/Fajr Interval

As described above, in the summer season and on the locations with high latitude nearer the poles, the Isha timing either does not appear, or even if it appears, it starts in a late time with a short duration, inconvenient especially in Ramadhan. This condition exist notably at latitudes above 45º. Since the preceding scholars didn't manifest any judgment on this matter, the I. European Islamic Seminar was organized in 1980 in Brussels. It was then decreed to determine the Isha timing of every day by "takdir" for latitudes greater than 45º. According to this application, the Isha/Fajr interval for these extreme latitudes is to be postulated as the interval at Mecca; so the Isha time is to be accepted as 80 minutes after sunset for every day of the year. The problem with this judgment is that in seasons other than summer, although Isha time occurs astronomically and its interval is acceptably wide, it is pulled earlier, namely before dusk. For example on May 1st when twilight disappears at 22:26 in Hamburg, the takdir method initiates Isha already at 21:19.

There are other takdir methods also, such as "One-Seventh of the Night", where Isha starts at the 1/7th of the astronomical night period. But this or any other process is to be applied to all of the Isha timings of the year, regardless whether the time occurs or not, thus the aforementioned problem persists.

What we recommend for the solution of this difficulty is that to consider the Hadiths related to the prayer times as a whole and interpret accordingly: As we attempted to explain in the Definitions section, there are two statements about Isha; it starts with the onset of the dusk and continues until the midnight (or one-third of the night). Accordingly, the time will exit before it enters at higher latitudes where the twilight disappears after one-third of the night or even persists until midnight. So our advice is to take the either the first one-third or the first half of the canonical night (between Sunset and Fajr) as the latest Isha if twilight continues at that time (Figure 15). By limiting the Isha timing as such, the real (astronomical) time will be used for Isha whenever it occurs before this boundary. Figure 15 (left) shows the canonical night and the two alternative latest Isha timings (1/3 in dotted and 1/2 in solid line). Sunset and sunrise are shown in purple, Isha in green, Fajr in blue and noon in yellow. Asr is in orange.

 

Figure 15.     Recommended Latest Isha / Shortest Night Timings (left), Shortest Day Timings (right)

Concerning the Fajr time, we evaluate that the astronomical timing is to be regarded as long as the emergence of the true dawn is observable. For latitudes over 49°, where the twilight persists and onset of the dawn cannot be distinguished, it is a common practice to take the astronomical midnight (center of sunset/sunrise) as the start of Fajr timing, when the Sun dips, i.e. the sky twilight brightness is minimum. However, since the Sun follows a rather horizontal trajectory for those higher latitudes (Figure 13), the vertical Sun angle hardly changes at midnight and it is not possible to discriminate that the sky illumination propagates. As an example, the Sun elevation angle at 49° latitude is -17.5° at June 21st midnight, and it changes only 0.3° and becomes -17.2° even after 30 minutes. For locations nearer to the poles, the angular difference will be smaller. So we advice to allow the Fajr timing to approach the midnight half an hour at most, whenever the dip vertical angle does not reach to -18°. Thus we will limit the earliest Fajr time to 30 minutes after astronomical midnight, as we did for Isha.

Another issue is to assign a more convenient elevation angle for the calculation of Fajr and Isha timings. The Isha/Fajr interval will broaden (or will exist at least, if not) if a lower depression angle value is selected. If we accept, for example, -15° instead of the safer value of -17° for Isha, the interval will expand from 99 to 170 minutes for Munich. If we consider the red dusk and take the angle as -10°, the duration even increases to 290 minutes, which is a significant gain especially in Ramadhan.

Non-existence of the Timings

We mentioned above that at higher latitudes, the day/night becomes very short, or even absent (in the polar region) for some days of the year. Although a relatively small proportion of the World population live there, it has been always an important issue, whether or how to perform the prayer near the poles. The solution of this problem is referred to the following Hadith of the Holy Messenger: “Dajjal will stay 40 days on Earth. One day of this will be like a year, one (other) day will be like a month, one (other) day will be like a week and the other days will be like your normal days”. As the ashab asks whether it is enough to perform a single prayer in the long days, he replied that it would not be enough and the daily prayer timings should be determined by "takdir". Accordingly, appropriate prayer timings should be adopted for "long" days, i.e. whenever sunset/sunrise does not occur.

Up-to now, different opinions were declared by the scholars about how the "takdir" should be realized in that case. Early judgments such as Nearest Day or Nearest Location refer to some times/places when/where the timings are existing and convenient, and apply their durations to those locations with extreme latitudes. Nevertheless, this application may imply a constant interval between the timings throughout the year, similar to the case for Isha, causing some timings, which actually exist astronomically, to become non-conforming. Our recommendation for those timings, namely for Maghrib (sunset) and Sunrise, is to limit the shortest day and the shortest night to three hours, thereby leaving a sufficient interval to perform the prayers. According to this, the shortest day in winter will begin (latest Sunrise) at 90 minutes before noon (zenith) time and it will end (earliest Maghrib) 90 minutes after the noon. Asr time may be determined as the middle of noon and sunset, so we will limit the earliest Asr time as 45 minutes after the noon, allowing 45 minutes each for Zuhr and Asr prayers. Likewise, latest Fajr / earliest Isha may be selected as 60 minutes before sunrise / after sunset (see Figure 15 right). On the contrary, the shortest night in summer will start at 90 minutes before the astronomical midnight and finish at 90 minutes after. In that case, 60 minutes will be available for each of Maghrib, Isha and Fajr prayers, if we take the latest Isha time as one-half of the canonical night and the earliest Fajr time as half an hour after the astronomical midnight, as advised before (Figure 15 left). Latest Asr in summer may be predicated as 60 minutes before sunset, as appropriate. Zenith time is always when the Sun has highest elevation, although under horizon and Zuhr is calculated accordingly. Astronomical midnight occurs always 12 hours after zenith when the Sun reaches deepest elevation, even above horizon, as calculated by Equation (2).

The limiting values for each timing at extreme latitudes may be expressed in hour-angles for convenience. The cosine of this angle may then be taken as the limiting value for the parameter to the arccos function in Equation (2). Limit hour-angle values for each timing are given in Table 3, suitable for the application of the takdir method recommended above. Note that the hour-angle values for summer (shortest night) are 180° shifted (cosine values are negative) such that the timings will be limited around the midnight, not around zenith as will be in winter. If our takdir method of limiting is not to be applied and the pure astronomical timings are to be calculated, then hour-angle values of 0°/180° (cosine values +1/ -1) should be used for winter/summer respectively, in order the arccos function does not return an invalid number. In this case, the non-existent timings will appear either at noon or at midnight.

 

TimeUpper Limit
(Winter)
Lower Limit
(Summer)

Fajr

37.5°

172.5°

Sunset

22.5°

157.5°

Zuhr

-

-

Asr

11.25°

142.5°

Maghrib

22.5°

157.5°

Isha

37.5°

172.5°

Table 3.

 

QIBLA

Qibla is one of the conditions which makes the prayer valid. It is the direction which the Muslims face during their daily worships. Qibla, which is the direction of the Kaaba is apodictic by the verse “Turn then thy face in the direction of the Sacred Mosque: Wherever ye are, turn your faces in that direction” (2:144). In the era of the Holy Messenger, when the Muslims were living near to Mecca, it was relatively easy to determine the Qibla direction; however after the expansion to Europe, Africa and the Far East, the Muslim astronomers had prepared various tables for assigning the Qibla direction using spherical trigonometry.

 

QIBLA ANGLE

Qibla is the horizontal direction of Kaaba, which we must face with chest during the prayer. So the Qibla Angle is defined as the horizontal angle between the Kaaba direction and the geographic north (pole). By means of this angle we may easily fix our Qibla direction; we first face to north and then turn clockwise an angle equal to Qibla Angle.

Formerly, some people assumed the Earth surface as plain and they tried to estimate the Qibla direction by inspecting the prevalent Earth maps. Kaaba direction was then interpreted as the route of the travel to Kaaba on the map. The line connecting America and Kaaba is positioned as directing to southeast on the maps based on Mercator projection. A small group of Muslims living in America still pray facing this southeast direction. On the other hand, the vast majority adopt the Qibla Angle found by spherical trigonometry. The basis for that is the assumption that Qibla is not only the square building, but rather an endless pivot through the Kaaba on both sides, up and down the Earth, which also the angels are believed to circumambulate of. This theoretical axis will then be visible from anywhere on Earth, even from space, so we will "turn our faces in the direction of the Sacred Mosque, wherever we are".

As to this method, Qibla Angle is defined to be the horizontal angle of the line connecting this theoretical axis and our location to the geographic north (our meridian). Figure 16 shows that this line points to northeast (with respect to the meridian) for America.

 

Figure 16.    Qibla Direction in America

The formula (4) below is used for calculating the Qibla Angle anywhere on Earth. LAT and LON are our location's latitude and longitude, respectively. Clockwise is positive. The values 21.4225º and 39.8262º are the latitude and longitude of Kaaba, respectively.

 

Qibla Angle = -arctan(sin(LON-39.8262º)/(cos(LAT)*tan(21.4225º)-sin(LAT)*cos(LON-39.8262º))

 

The World Qibla Map showing the global Qibla Angles is given in Figure 17. Angles are differentiated in color slices of 15° each. You can find your Qibla direction wherever on the Earth, by just having this map.

 

Figure 17.    World Qibla Map

 

DIRECTION FINDING

Having determined the Qibla Angle this way, we should decide how to estimate the direction. For the designation of the direction, we can make use of a compass, the Sun, the Moon or stars. Each method will now be explained briefly:

Compass:

This is the angle indication method most frequently applied. It is based on the visualizing the magnetic field of the Earth by means of a magnet needle which can freely move on the horizontal plane. One obvious disadvantage of this method is that the magnetic poles of the Earth do not coincide with the geographic poles. That's why the direction pointed by the compass needle (magnetic north) deviates from the true north (Figure 18). The angle drift between the magnetic and the geographic north is named as Magnetic Declination.

 

Figure 18.    Compass Drift (Magnetic Declination)

The Earth magnetic field is originated by the induction of the solid and liquid iron present in the Earth's core. Magnetic poles are points where the field is perpendicular to the surface. On these locations, the horizontal component of the magnetic flux is zero thus the compass will indicate a random direction. Although the magnetic poles are generally positioned near the geographic poles, they continuously move in time. The south geomagnetic pole is located at 64° latitude, whereas the north pole shifts from 81° latitude in 2001 to 86° as of 2015. The north geomagnetic pole has moved a total of 1100 kilometers during the 20th century and nowadays it drifts 50 kilometers northwest per year. The disturbance of the charged particles from the Sun may also cause to shift the pole up to 80 kilometers within a day.

As can be deduced from this general information, the compass drift angle is not constant but changing with time and place. Earth Magnetic Declination Map is a chart which shows the declination on anywhere over the world at a fixed time (Figure 19). The compass drift for England for example is inferred as near 0° from the map; however it was -15° one century before (each color represents 5°).

 

Figure 19.     Earth Magnetic Declination Map (2017)

In order to produce an earth declination map for the future, it is necessary to foresee the annual change of the geomagnetic field for all latitudes and longitudes. Since the modeling of the field is rather complex, only short-term (5 year) estimations can give results with sufficient accuracy. One of the most detailed works on this subject is known as the International Geomagnetic Reference Field (IGRF). This model with its latest (12th) revision, also used by the Alperen software, enables worldwide field calculations between years 1900~2019 with an accuracy of 0.1 nT. The 2015 revision of another model, WMM (World Magnetic Model) on the other hand, produces successful results for years 2015~2019.

The horizontal component of the geomagnetic field varies between 10~40 µT, except near geomagnetic poles. In the vicinity of the poles, the horizontal flux decreases considerably, Qibla determination with compass will no longer be reliable. Moreover, it should be kept in mind that environmental influences and the quality of the compass may affect the correctness of the angle obtained. Our advice is to use the compass only if the alternative methods described below are not applicable. Even then, we recommend measuring the angle on multiple places, especially when indoor.

Sun:

We may prefer to make use of the Sun for determining Qibla direction, which is more reliable than the compass:

Sun Horizontal (Azimuth) Angle is the angle of its projection on the ground relative to north, while it moves in its trajectory on the sky. The shadow of any object points to the opposite direction of the Sun, so it is very easy to find the horizontal projection of the Sun. The Sun Horizontal Angle at any time can be very definitely calculated by the following formula (5):

 

HA = 15 * (Local Time - Local Zenith Time)

Sun Horizontal Angle = -arctan(sin(HA)/(cos(LAT)*tan(DECL)-sin(LAT)*cos(HA)))

Sun Vertical Angle = ELEV = arcsin(cos(HA) * cos(LAT)*cos(DECL) + sin(LAT)*sin(DECL)))



To designate the Qibla using the Sun Horizontal Angle, we first face to the Sun. Then we subtract the Sun Horizontal Angle (5) from the Qibla Angle (4) and turn this angle right (clockwise). For example, if the Sun Horizontal Angle at that time is 250° and the Qibla Angle is 160° then we should turn 90° left (counterclockwise) such that the Sun is at our right in order to face to Qibla.

Qibla Time, which is displayed on some calendars for each day, is the moment when the Sun Horizontal Angle (5) is equal to the Qibla Angle (4); that is when the directions of the Sun and Qibla coincide such that anyone facing to the Sun is facing to Qibla at the same time. It is useful when the Qibla Time is in the daytime (the Sun is observable), which is the case for most of the European and Asian countries. Qibla Time can be calculated as follows:

m = arctan(cos(Qibla Angle)*cot(LAT))

ELEV = arcsin(cos(m)*sin(DECL)/sin(LAT)) - m

HA = -arcsin(cos(ELEV)*sin(Qibla Angle)/cos(DECL))

Qibla Time = Local Zenith Time + HA/15

If Qibla points to north on the Nothern hemisphere or to south on the Southern hemisphere, Qibla Time occurs in night and Qibla direction cannot be identified. In that case, 180° are added to Qibla Angle to find the Anti-Qibla Angle, such that the shadow of an object points to Kaaba at Qibla Time. For high accuracy, Qibla Time should be recalculated using the iterated values of DECL and Local Zenith Time.

On the World Qibla Day (approx. May 28th and July 16th) the Sun Declination Angle becomes equal to the latitude of Kaaba, so that the Sun trajectory moves over the Kaaba (Figure 12 left). On these days at the Zenith Time of Kaaba, the Sun is exactly on top of the Kaaba (ELEV = 90°) and the Kaaba axis points directly to the Sun. This means that at this time (Qibla Time of the World Qibla Day) anyone on Earth heading to the Sun will be pointing to Qibla. This special "World Qibla Time" is 09:18 GMT for May 28th and 09:27 GMT for July 16th. We can obtain the local time for any place by adding the relevant time-zone to this World Qibla Time.

Note that World Qibla Day is only useable two times per year and the Qibla Time one per day, whereas Sun Horizontal Angle can be used anytime during the day as long as the Sun is visible.

Moon & Stars:

In the night when the Sun is absent, we may benefit from the Moon or the stars. The verse: "It is God who created the stars so that you could find your way thereby in the darkness of the land and sea" (6:97) should indicate this reality. The Moon, tracks a similar trajectory as of the Sun and completes one turn nearly in 25 hours. However, the visibility of the Moon in the night depends upon its phase; when near full moon, it may be observed on the sky for a greater portion of the night. Just like the Sun, an azimuth angle (Moon Horizontal Angle) can be determined for the Moon, which is the angle between its projection on the ground and the north. Again, to designate the Qibla using the Moon Horizontal Angle, we first face to the Moon and then we subtract the Moon Horizontal Angle from the Qibla Angle (4) and turn this angle right (clockwise). But we will omit here the computation of the Moon Horizontal Angle, since it is much more complex than that of the Sun; interested readers may refer to our Prayer Timings Excel spreadsheet, or Alperen source codes or to the relevant works in the Bibliography section. You can also use particular software such as Moontool or Moon Calculator to calculate the moon trajectory. Instead, you may prefer to obtain the instantaneous azimuth angles online from the Australian Geoscience's WEB page.

We can use the stars in case the Moon is invisible. North Star (Polaris) is the most convenient tool for the inhabitants of the northern hemisphere. It can be easily fixed on the sky by the asterism Big Dipper (Plough) consisting of the seven brightest stars (shown in blue in Figure 20) of the constellation Ursa Major. Since Polaris has an angle of 89.3° to the equatorial plane, it points to the north with an accuracy of better than ±1°. It will be enough to face to Polaris first and then to turn right the Qibla Angle to find the Qibla direction. Note that the position of the North Star on the sky gradually changes over time because of the precession of the Earth's rotational axis with a period of approx. 26,000 years. So after 2100, Polaris will start to deviate from the north.

 

Figure 20     Ursa Major & Polaris

 

HIJRI CALENDAR

Besides the daily prayers, Muslims are obligated to fulfill some yearly worships, such as fasting, alms, pilgrimage, eid prayer and sacrifice. Worships with yearly period are based on the Hijri calendar. This calendar takes the month as one complete lunation of the Moon. So the Hijri (lunar) year consists of 12 lunar months, apodictic, as defined by the verse “According to the Book of God, from the day He created the heavens and the earth, the number of months are twelve” (9:36). Muharram is accepted as the first month of the year. A lunar month has a length of either 29 or 30 days; but it may appear four consecutive 30-day months or three consecutive 29-day months. The duration of a lunar year is meanly 354.36 days, so it is 11 days shorter than an average solar year.

 

START of LUNAR MONTH

The lunar month starts on the evening when the crescent is first observed on the sky, as expressed in the Hadith: “Do not fast until you see the Hilaal and do not give up fasting until you see the Hilaal, but if you cannot see it then act on estimation”. So the definition of the beginning of an Islamic month is the actual observation of the new crescent on the west horizon after sunset, upon which the 1st day of the new lunar month begins (religiously, the day starts at sunset). If the crescent cannot be detected at the end of the 29th day because of the clouds, the new month is delayed for another day and the following day is accepted as the 30th of the previous month, predicated on the Hadith "... and if it is hidden then regard the month of Ramadhan as of 30 days". Nevertheless it that case, it is possible to see the new crescent of Shawwal on the 28th evening if actually both months had 29 days length. Contrary, if the new crescent is not present on the 30th evening on a clear sky, it will be the definite evidence that we started the preceding month one day earlier. The real observation of the waxing crescent by multiple testifiers is the condition for the new Islamic month. However, the need to establish a calendar beforehand and (hopefully) to enable the Islamic World to begin/end fasting or celebrate Eids on the same days, has given rise to determine the new month by calculation. This can be only possible by defining some "crescent visibility criteria", based on which we can ascertain that the crescent will be observable on the sky to predict the date of a new month. The history of attempts for establishing criteria to compute the moment of earliest visibility has started even before 5,000 years, namely in Babylonian era.

 

MOON's MOTION and CONJUNCTION

After expounding the definitions about the start of an Islamic month, we now aim to clarify the formation of the crescent and the phenomenon of visibility. So we will first explain the motion of the Moon:

The Moon moves around the Earth in counter-clockwise direction and completes one turn in 27.3216 days. This period, referenced again to a very far star, is named as the sidereal month. Since the Moon rotates around itself with the same speed as around the Earth, only its same side is visible from the Earth. But the Moon also rotates around the Sun together with the Earth. Since the Moon is not a light source and it only reflects the rays of the Sun, its shape or visibility will depend upon the position relative to the Sun. Therefore one full lunation, as seen from Earth, will be equal to the period of one complete turn around the Earth when observed from the Sun. This period from one conjunction to the next, is named as synodic month and has an average duration of  29.5306 days (Figure 21). It is somewhat longer than the sidereal month, since it moves in the same direction around the Earth as the Earth rotates around the Sun. Since the Moon's orbit is rather elliptic (eccentricity is 0.0549 and its geocentric distance varies between 356 and 407 megameters), its rotational speed changes in time and one synodic month may alter between 29.27 and 29.84 days. That's why a lunar month can be either 29 or 30 days. In the presentation about the calculation of the speed of light by Muhammad Zuhdi, based on the work of Prof. M.H. Elnaby, you can watch the animation about the motion of the Moon. The angle between the Sun-Earth line (dotted in Figure 21) and the projection of the Moon on the ecliptic plane is defined as the Moon Phase Angle. It is 0º at conjunction, 90º  at first quarter, 180º at full-moon and 270º at last quarter.

 

Figure 21.    Sidereal and Synodic Months

The orbit of the Moon around the Earth has a small angle of 5.1454º with the ecliptic (Figure 22). Because of this angle, the Moon makes vertical oscillations of ±5° around the ecliptic plane with a period of one sidereal month. The instantaneous value of the vertical angle of the Moon with respect to the ecliptic is called as the Moon Declination Angle. If this angle is sufficiently small (< 1º) during conjunction, the Moon enters between the Sun and the Earth such that a Sun Eclipse occurs. On the contrary, if Moon Declination Angle is small enough while full-moon, the Earth positions itself between the Sun and the Moon, causing a Moon Eclipse. If Moon's orbit were not declined, we would see an eclipse every 15 days. Our Creator, Who assigns specials prayers for the eclipse times, attracts our attentions to the stupendous and fabulous celestial composition He established.

 

Figure 22.    Moon Declination

 

VISIBILITY CRITERIA of NEW MOON

A lunar month starts astronomically when the Sun-Earth line coincides with the Earth-Moon line projected onto the ecliptic plane, which is named as conjunction. At this moment, all the Sun, the Moon and the Earth are on the same line so the Moon's lit hemisphere is just on the opposite to the Earth therefore it is not possible to see any trace of it. Since the religious condition for the lunar month to begin is that the new crescent must be observed, the Moon has to separate a certain distance from the point of conjunction such that the sunlight reflected from its edge (crescent) can reach the Earth (Figure 23). So the time of conjunction which can be computed very precisely can never be used as the earliest visibility.

 

Figure 23.     Conjunction and Earliest Visibility

In order an observer's eye to distinguish the very first crescent on the sky, it has to attain a certain thickness. When the young crescent is very thin, its tips cannot be discriminated by the eye such that the crescent is observed shorter. The amount of the necessary thickness depends on the contrast, i.e. the ratio between the brightness of the Moon and the surrounding sky brightness. Because of this, it cannot be detected in daytime on fully bright sky by the naked eye, as long as its thickness (of its center) subtends an angle of approx. 2.75' (0.8 mrad). But we may see a thinner crescent on a darker sky, which is the case during sunset. Nevertheless, in the waxing crescent phase, the Moon follows the Sun just from behind (Figure 23) such that the Moon sets soon after the Sun. This means that the earliest crescent can be visible within a rather short period (several minutes). There is a large database of visibility observation over the centuries. The smallest separation angle, namely the angle between the Sun-Earth and the Earth-Moon connection lines, which is recorded in 2002 for a positive visibility by naked eye is 7.6º at an altitude of 2,200 m. Danjon, in 1932, asserted a limit angle of 7º, known as Danjon limit. Note that this angle of 7º is topocentric as seen from the observer, which corresponds to 8º geocentric (from the Earth center) because of parallax. At this limit, the crescent thickness is about 0.25'. The "best time" for the first visibility comes out to occur when the Moon is at 4~5º geocentric elevation, just before its pale light is absorbed by the optically thick atmosphere, and the Sun is 4º below the horizon in order the sky brightness dims sufficiently. The horizontal position of the Moon is nearly the set point of the Sun. Danjon limit is valid only for naked eye; the existence of the crescent can be verified at smaller separation angles with optical aid. For example, on June 15th 2007, Martin Elsässer could detect the crescent when the separation angle was only 4.7º, i.e. 2.7 hours after conjunction, even before sunset. This observation was made at an altitude of 1,800 meters, using a telescope, imaging infrared (IIR) detector and image processing software for contrast enhancement. In space, where the sky brightness is almost zero, the crescent is distinguishable at smaller separation angles; the vestige of a baby crescent was sensed as a glint on the coronagraph mounted on a rocket in an experiment in 1966, when the separation angle was only 2º.

Following relation exists between the Separation Angle and the Crescent Width:

 

Crescent Width = Moon Diameter / Moon Distance * Moon Illumination ≈ 15.5' * (1-cos(Separation))

 

The Moon Phase Angle is the horizontal and the Moon Declination Angle is the vertical component of the Separation Angle, so the following equation is valid under spherical trigonometry (6):

 

cos(Separation) = cos(Phase) * cos(Declination)

 

Note that the minimum Separation Angle at conjunction is equal to the Declination Angle, which can be between 0º and 5º. So without the atmosphere created or without its scattering property, there would be no sky brightness, consecutively we could be able to see the crescent below 8º separation angle with naked eye, such that we can see the Moon even at conjunction for many lunations and the determination of a new month will be impossible. For the Separation Angle to reach the value of 8º, the Phase Angle must progress to 6.1º ~ 8º, depending on the Declination Angle, and correspondingly a time interval  of 11~18 hours must pass from the conjunction on.

Once again; although the conjunction is objectively definable and accurately computable, the first visibility, being the condition for the new month, is subjective and depends on the observer as well as the atmospheric conditions, like in the dawn/dusk case.  So it is not possible to determine the occurrence instant by astronomical calculation, and observation is essential. But on the information given above, we can assert that it is not possible to see the crescent by naked eye unless some conditions exists. So we can then calculate the earliest time of visibility on the assumption of some criteria. The common parameters introduced by these criteria are Sunset-Moonset time interval, Moon elevation at sunset. More detailed information is available on the paper, where a simplified visibility criterion has been proposed, which is also used in the Alperen software.

The visibility criteria is useful to determine whether the crescent is visible at a specific time and location. Crescent Visibility Maps prepared by means of these criteria, estimate where on Earth the first crescent will be visible. A visibility map has the shape of a parabola, which starts at the point of first visible location (apex) and widens towards west. Figure 24 is a Crescent Visibility Map produced by the Ehille software for the Hijri Month of Rajab 1437.

 

Figure 24.     Crescent Visibility Map

During the crescent phase of the Moon, the remaining dark part of it also radiates a hardly visible light (Figure 25). As the crescent grows, the luminosity of this dark part diminishes. This radiation, named after Leonardo da Vinci since discovered first by him, is caused by the sunlight reflected by the Earth to the Moon and then back to the Earth (Figure 26). Note that the view of the Earth from the Moon is almost the opposite of the view from the Earth; that is the Earth seems nearly full when the Moon Phase Angle is small; and as a crescent during the full-moon phase. So during the new-moon period, the "full-earth" reflects maximum sunlight to the Moon and this Vinci effect is at most. As the crescent of the Moon thickens, the Vinci illumination weakens, proportional to the dark area percent of the Moon.

 

Figure 25.     View of the Moon under Da Vinci Effect

 

Figure 26.     Formation of Da Vinci Effect by Double Reflection

Some Shafi scholars claim that for cities which are apart from each other more than 134 km, the month may start on different days if the new Moon has not been observed on the same evening (ikhtilaful-matali'). Their evidence is the verse "Those of you who witness this month shall fast therein" (2:185). For this opinion, a Crescent Visibility Map using earliest visibility criteria will estimate the start of a lunar month for any local position. But according the other schools, if authentic moonsighting news comes from other areas, then local moonsighting is not necessary. The wide interpretation of this thought is if the New Moon has been attested in any location of the world, then all the others accept the start of the month. Under this admission, an International Hijri Calendar will be necessary. This is especially important for the Hijri months of Ramadhan, Shawwal and Dhil-Hijja for the unification of fasting and pilgrimage.

Muslim countries have adopted different calendars, Saudi Arabia for example relies on its Umm-ul Qura calendar. To resolve the problem of different start dates for lunar months, experts from various domains come together from time to time. Some examples are the Quwait conference in 1973 and Istanbul Crescent Visibility Conference in 1978. This conference has some important outcomes, which we consider to be valuable for the formation of an International Hijri Calendar. There has been decided that; it is essential the crescent is visible (1), geocentric Moon Separation Angle should be at least 8º (2) and Moon Elevation should be at least 5º (3) in the calculations for the astronomical estimation of new moon visibility. All countries over the world will use the same Unified Hijri Calendar, so the new month will start on the same date at the local sunset (4). The first day of a lunar month will be the Gregorian day following the day on which the new moon becomes visible, i.e. the Separation Angle reaches 8º (5). This means that if the Moon Separation Angle attains 8º until 23:59 GMT, the crescent will (most probably) be visible on the very west border of the world (west coasts of the continents North/South America), and since the Sun has already set in the rest of the world, this evening will be the beginning of the first day (see Figure 24, where the crescent is first visible at 23:47 GMT of April 7th and April 8th is the 1st of Rajab). If the Moon Separation Angle becomes 8º after 23:59 GMT, then it will (most probably) be visible only when the Sun sets at the very east coasts of Aisa. That means if the visibility parabola lies on the Great Ocean without touching America, the next month will just begin in the following evening. For the cases when the start of visibility is near 00:00 GMT, the decisiveness of the calculation vanishes and observation from the American west coast (vicinity of the apex of the parabola) becomes indispensable. Using this rational rule, which is also implemented in the Alperen software, we prepared a Hijri Calendar Table for years 1901-2099 for the convenience of the reader.

 

CONCLUSION

Here we will summarize some of the information given above as to help for the daily practices of the reader:

 


BIBLIOGRAPHY

  1. Moon Calculator v6.0, Dr. Monzur Ahmed, Birmingham, 2001
  2. Questions on Moonsighting, Khalid Shaukat, 1999
  3. Questions on Prayer Times, Khalid Shaukat, 1998
  4. Questions on Qibla Direction, Khalid Shaukat, 1999
  5. The Correct Qibla, S.Kamal Abdali, 1997
  6. Fajar and Isha, Yaqub Ahmed Miftahi, 2005
  7. How to Calculate Prayer Times?, Türkiye Calendar
  8. National Geomagnetism Program
  9. International Geomagnetic Reference Field
  10. Blue Twilight in a Simple Atmosphere, Phil Ekstrom, 2002
  11. Atmospheric Optics, Ord.Prof.Dr. Craig F. Bohren, Pennsylvania State University
  12. UBVRI Twilight Sky Brightness at ESO-Paranal, F. Patat, European Southern Observatory, 2006
  13. Variability in the Astronomical Refraction of the Rising and Setting Sun, Russell D. Sampson, 2003
  14. Computational Astronomy and the Earliest Visibility Of Lunar Crescent, Muhammad S. Qureshi, 2005
  15. Rational Design of Lunar-Visibility Criteria, Roy E. Hoffman
  16. US Naval Observatory
  17. World Magnetic Model
  18. Islamic Research Foundation International
  19. http://moonsighting.com
  20. Islamic Crescent's Observation Project, International Astronomical Center
  21. Atmospheric Refraction Applet
  22. Astronomical Determinations for the Beginning Prayer Time of Isha’, University of Malaya, 2012
  23. Monte Carlo Studies of the Sky Radiation at Twilight, 1974
  24. How accurate are the computed timings for sunrise and sunset?, Akbar Ali S.F.A. Saifee, 2016
  25. Variation in the equation of time, Kevin Karney
  26. Prayer Times Calculation, Hamid Zarrabi-Zadeh
  27. Solar Position Algorithm for Solar Radiation Applications, Ibrahim Reda and Afshin Andreas, 2008