Dr. Abdelhamid Bentchikou is a Biologist Engineer. He has both a Ph. D. in literature and in science. He was one of the founding deans of the University of Constantine, Algeria, from 1970 to 1975.

Dr. Moiz Rasiwala is an astrophysicist who obtained his Ph.D. in 1969 at the Institute of Astrophysics in Paris. He was at the National Centre of Scientific Research (CNRS) from 1963 until 1970. He was head of the physics department at the University of Constantine whilst Dr. Bentchikou was dean of the science faculty.

Mr. Amit Patel is a postgraduate in the field of Information Technology. He is currently working with India based top software services provider company. Amit is an executive engineer for Makkah Calendar Project.

Abstract

This paper wants to shed a new light on the Islamic calendar for Makkah. This very practical problem has impact on the lives of millions of pilgrims across the world because the date of the Hajj, or the great annual pilgrimage, depends on an accurate calendar. Modern astronomical methods make it quite possible to establish an accurate calendar for Makkah decades in advance. This article enlarges upon the traditional Islamic idea of beginning the new month when the first crescent is observed in the evening sky, as reported by two reliable witnesses. The concept of the limiting horizon is introduced as defined by the time interval between sunset and the early morning prayer in Makkah at each birth of the new moon. Any visibility within the limiting horizon, at any intermediate horizon, is considered as visibility at Makkah itself.The concept of the limiting horizon permits the use of plotted visibility curves in order to predict the visibility of the early crescent to a high degree of accuracy. Some recent photographs of the early crescent are published in the article. They validate concretely the accuracy of our hypothesis.

I. Lunar calendars

Lunar calendars, just like solar calendars, are as ancient as the recorded history of civilisations. The only purely lunar calendar in use at present is the Islamic or Hegirian calendar (Hijri Calendar) in which the twelve months of the year follow twelve lunar cycles. Since the lunar year is shorter than the solar year by a period of 11 to 12 days, the Islamic months drift across the seasons in a cycle of approximately 33 lunar years.

Most other lunar calendars, the Hebrew, the Chinese, the Hindu calendars are in fact lunisolar calendars. To keep the months in phase with the seasons, even whilst accounting for the shortness of the lunar year with respect to the solar year, an intercalary 13th month is introduced every three years in general. These calendars are used essentially for religious purposes whereas the Hegirian calendar (Hijri calendar) is also the recognised commercial calendar in some Islamic countries like Saudi Arabia.

II. The Islamic Calendar of Makkah

It is well known that the beginning of the Islamic month is based on the visibility of the earliest crescent moon as reported by two reliable witnesses. Thus, depending on local sighting, the calendar differs from one place to another. Amongst all the possible local Islamic calendars, that of Makkah is specially precious in order to determine the date of the annual pilgrimage. It concerns not only the Arabian peninsula but also the countries to the North, to the South and to the West of the peninsula. On certain months the calendar might also apply to the countries lying to the East. Thus it is essential to create a calendar for Makkah which is both totally scientific in nature and which respects Islamic tradition. In order to be useful, it must be possible to establish such a calendar over a long span of time depending on the predicted visibility of the New Moon.

The calendar of Makkah is not the universal Islamic Calendar / Hegirian calendar. But it can be a decisive step towards a universal calendar for Islam.

III. Islamic tradition : some references to religious texts

The highest authority in Islam is the Koran. Islamic law and tradition (the Sharia) are an expression of the Koran in legislative and social terms. A second authority is contained in the sayings and doings of the Prophet as recorded by a chain of reliable witnesses. These are known as the Hadith. Even in an article of scientific interest it is not irrelevant to reproduce some references to religious texts pertaining to the Islamic calendar. These texts are the ultimate authority in establishing a scientific Islamic calendar. We give the texts as an annex at the end of this article.

IV. Each spot in the world has its own calendar

As remarked before, because of the varying visibility of the new crescent, each place in the world will have its own local calendar. In practice, however, each country will refer the calendar to the capital or some other important metropolitan town. The calendar thus used will be of a conventional nature.Conflict in fact can arise, and has arisen several times, between the conventional calendar and actual local observation.

We thus have the choice between observing the beginning of the Hegirian months from an agreed unique location – which seems rather difficult to admit – or, more broadly, by reference to a unique location.

V. The official calendar used by the Saudi Arabian government

Known as the Umm al-Qura calendar the Saudi Arabian government uses the following criteria for determining the beginning of the Islamic month. These criteria apply since the Islamic year 1423 (current Islamic year: 1432):

If on the 29th day of the lunar month the two following conditions are satisfied, then the next day is the first day of the new lunar month:

1. The geocentric conjunction occurs before sunset
2. The moon sets after the sun

Otherwise, the current lunar month will last 30 days.

It is obvious that the actual sighting of the young crescent is not taken into account. Many discrepancies have been reported about the beginning of the lunar month under these circumstances. Even if the conjunction is shortly before sunset, with no possibility of viewing the crescent on that particular evening, the month will be declared as over, contrary to Islamic tradition.

VI. The criteria for observing the new moon

Low on the horizon, faint and thin, the young crescent is indeed difficult to observe. The following scientific criteria determine the actual observation of the crescent:

1. The arc of light (elongation, meaning separation of the moon from the sun),
2. The arc of vision (the altitude the of moon from the sun),
3. The altitude of the moon above the local horizon,
4. The width of the crescent,
5. The distance of the moon from earth,
6. The distance of the earth from the sun.

We will exploit these terms in greater detail in the course of this article. In practice, at least ten hours must elapse after conjunction before the angular separation between sun and moon will be sufficient to give a chance to observe the crescent. Local atmospheric conditions will also obviously play a role.

VII. What to do in practice

Obviously, the traditional method would be to try to observe the crescent in Makkah itself. Normally, if the crescent is not visible on the 29th lunar day, the month would be extended to 30 days. The revolutionary new idea we have developed is the following: if, after conjunction, the new moon is sighted to the west of Makkah before the morning prayer, the sighting can be considered as having taken place in Makkah itself. The month will be ended at 29 days in this case. Otherwise it will be prolonged to 30 days. Instead of the narrow interval of time around sunset in order to observe the crescent, we now dispose of the whole night (until the morning prayer) to observe the crescent anywhere west of Makkah. This leads to the concept of the limiting horizon and of an intermediate horizon developed in the next paragraph. We might add, that the time of the morning prayer in Islam, called fajr in Arabic, is of solar nature, defined as the moment of early dawn.

VIII. The method used for elaborating the Makkah Islamic Calendar

The earth rotates around its axis once every 24 hours. This means that every point on any given longitude describes a circle of 360° around the axis of rotation defined by the straight line which traverses the North and the South poles. We can thus affirm that 24 hours (or 24 x 60 = 1440 minutes) correspond to 360°.Knowing the time of sunset in Makkah on the evening after the birth of the new moon, as well as the time of the fajr prayer which follows – meaning the fajr of the day which follows that of the birth of the new moon – it is very simple to calculate the time difference between the two. In fact, this interval of time varies between roughly 12 hours when the nights are the longest and 9 hours when they are the shortest. The simple rule of three tells us that if 24 hours correspond to 360°, then 12 hours correspond to 180° and 9 hours to 135°. For each day of birth of the new moon, this elementary rule of proportionality thus allows us to convert into a space interval – or, more precisely, into an interval of longitudinal degrees – the time interval between sunset at Makkah and the fajr of the next day, both referred to the new moon.

The calculation of the Limiting Horizon (LH) is now very easy : it is the visibility to the extreme West of Makkah, circumscribed by fajr in this town.

Now Makkah is situated at a longitude of 39.8° East of the Greenwich meridian. Let us simplify this figure to 40° East and let us give a concrete example: suppose that the duration between sunset in Makkah on the day of the new moon birth and the prayer of fajr on the next day is 12 hours, which corresponds to 180° of longitude. Towards the West of Makkah, we will have to cover 40° in order to reach the Greenwich meridian and again cover a further 140° in order to reach our limiting horizon. In this case, LH = 140° West.

We postulate that any visibility of the crescent until 140° West will be considered as visibility referred to Makkah. A corollary immediately follows: if the visibility is achieved before the limiting horizon, at an Intermediate Horizon (IH), it will be already considered as visibility referred to Makkah.

In practice, we have used the visibility curves provided by Syed Khalid Shaukat . Some examples of recent visibility curves are given below, whereas the method of plotting the visibility curves is given in the last section of this article. [More such visibility curves are also available at Crescent Visibility Predictions Section.]

The different colours represent extended zones of different patterns of visibility on earth between the latitudes 60° North and 60° South. The curves are plotted for the day of birth of the new moon (except when visibility is impossible on earth on this day) and the two following days.

We have retained the most convenient visibility, represented by the green and the blue fields. We have avoided the exclusive use of the green fields, because this leads to an impossible result: in 1430, for instance, we would end up with too many months of 30 days and a Hegirian year of close to 360 days.

One also has to emphasize that the blue fields represent vast regions of the globe and perfect conditions of visibility will certainly be achieved at some place or another. We further point out that the use of an optical instrument in case of necessity is hardly contrary to Islamic law.

IX. Calculation of the Q factor or the “Ease of visibility” of the early crescent

We have already remarked that it is no easy feat to observe the early crescent with the naked eye. However, if astronomy tells us that the crescent is going to be visible and gives us its ease of visibility, then we can redouble our efforts of sighting and even try to photograph the crescent, atmospheric conditions permitting. The following paragraphs explore this “ease of visibility”.

IX.1 The notion of “Best time of visibility” or Tb

One might think that the best time for observing the new crescent moon is just before it sets in the night sky. This is however not so. If the crescent moon is observed too early after sunset, the sky might be still too bright to obtain visibility of the faint object. If we wait too much, then the intrinsic visibility of the crescent will diminish, and, again, it will not be visible. In his paper, Yallop (B. D. Yallop,A Method for Predicting the First Sighting of the New Crescent Moon, HM Nautical Almanac Office, NAO Technical Note N° 69, June 1997, Updated April 1998) uses earlier results to give an empirical formula for the “Best Time” of visibility. According to him:

Tb = Ts + 4/9 * Lag

In this equation Ts is the time of sunset and Lag is the time difference between sunset and moonset (in minutes, for example). We will use the example of observing the new moon of November 2009 in Makkah and at an Intermediate Horizon IH. The new moon is born on the 16th at 19 H 15 M (universal time). This is too late to give any visibility on the 16th in Makkah. On the next day, 17th November, there is still no visibility in Makkah, but the Intermediate Horizon at 30° W 30° S is green. As an example, we will calculate the Q factor for 17th November 2009, both for Makkah and for IH as above.

Best time of observation in Makkah (17/11/2009 ; 21°25’0” N 39°49’0” E, seconds neglected:
Ts = 14:38
Moonset = 15:01
Lag = 0:23
Tb (Makkah) = 14:38 + 4/9(23 minutes) = 14:48

Best time of observation at IH 30°W 30° S (17/11/2009, Using Sunset Moonset computing in MICA)
Ts = 20:36
Moonset = 21:39
Lag = 01:03
Tb (IH) = 20:36 + 4/9(63 minutes) = 21:04

IX.2 Coordinate systems

Angles will be measured either in geocentric, topocentric or celestial coordinates.Geocentric means from the centre of the earth. A position measured in geocentric coordinates does not depend on latitude of longitude since observations are from a fixed point.

Topocentric means from each local horizon. Topocentric coordinates from Makkah will use the horizon at Makkah.

Celestial coordinates are used to fix position of heavenly bodies on the celestial sphere. The celestial sphere is the huge sphere which seems to surround us day and night. It is an indefinite projection of the spherical earth. The projection of the earth’s equator on the celestial sphere is the celestial equator. The apparent path described by the sun on the celestial sphere is known as the ecliptic. Since the axis of the earth is tilted by 23.5° with respect to the plane of its orbit, the planes of the ecliptic and of the celestial equator are also tilted at the same angle. They intersect in two points called the vernal and the autumnal equinox.

Celestial coordinates are like longitude and latitude on earth. Celestial longitude is also called Right Ascension. It is measured along the celestial equator starting from the vernal equinox and counted in degrees or, more traditionally, in hours, minutes and seconds (24 hours = 360 degrees). Celestial latitude is also called Declination. It is always measured in degrees, positive if the celestial body is north of the celestial equator, negative if it is south of the celestial equator.

Finally, azimuth is the clockwise angle starting from the North until the direction of the heavenly body is reached.

IX.3 Four definitions

1X.3.1 Arc of Sight or ARCS

It is the angular distance in degrees between the Moon’s centre and the horizon at the time of local sunset. It is equal to the topocentric altitude of the Moon at local sunset. The angle is to be measured at the “best time”.

(Horizontal line = local horizon)

1X.3.2 Arc of Sight or ARCS

ARCL is the angle subtended at the centre of the Earth by the centre of the Sun and the centre of the Moon. ARCL allows the calculation of the width of the lunar crescent (W or WOC) according to a formula given later. ARCL is frequently called the lunar elongation.

[The following two diagrams have been taken from the article of Ilias M. Fernini, Yallop’s criterion as a test for the earliest crescent visibility, College of Science, Department of physics, U.A.E. University, Al-Ain, P.O. Box, 17550 U.A.E. Date of publication not available.]

1X.3.3 Delta azimuth

Delta Azimuth or DAZ is the difference in azimuth between the Sun and the Moon at a given latitude and longitude, the difference is in the sense azimuth of the Sun minus azimuth of the Moon.

1X.3.4 The Arc of Vision or ARCV

Delta Azimuth or DAZ is the difference in azimuth between the Sun and the Moon at a given latitude and longitude, the difference is in the sense azimuth of the Sun minus azimuth of the Moon.

ARCV is the geocentric difference in altitude between the centre of the Sun and the centre of the Moon for a given latitude and longitude, ignoring the effects of refraction.

Angles ARCL, ARCV and DAZ satisfy the equation
Cos ARCL = Cos ARCV * Cos DAZ
so only two of the angles are independent variables. ARCL and ARCV are not directly observable and have to be computed from the celestial longitudes and latitudes of the Sun and the Moon.

IX.4 Example of Makkah

First of all we calculate ARCL and ARCV. For this we need from MICA L_S, the geocentric celestial longitude (or Right Ascension) of the Sun and L_M, the geocentric celestial longitude of the Moon. We need D_M, the celestial latitude (or declination) of the Moon. We further need DAZ as defined above. Values for Makkah on 17th November 2009 at 14:48 (Best Time).

L_S = 235.39°
L_M = 245.01°
D_M = – 3.76° (the moon is south of the celestial equator)
Azimuth of Sun = 250.62°
Azimuth of Moon = 241.97°
DAZ = 8.65°
RP = Radius vector of the moon (earth – moon geocentric distance = 392 357.996 km = 61.52 (in terms of the equatorial radius of the earth or 6378.1370 km)
Finally, ARCS or AltM (topocentric altitude of moon) = 1.75°

Now:
ARCL = Cos-1 (Cos(L_M – L_S) * Cos D_M)
= 10.32°
ARCV = Cos-1 (Cos ARCL/Cos DAZ)
= 5.6484°

We now calculate the semi diameter SD of the Moon in seconds of arc with the following formula:
SD = 56204.92/RP * (1 + Sin AltM/RP)
= 0.2539°

Yallop gives for the width of the lunar crescent W the following equation:
W = SD * (1 – Cos ARCL)
= 0.2468’ (minutes of arc)
= 0.0041°

Finally, with ARCV in degrees and W in minutes of arc,
Q = (ARCV – 11.8371 + 6.3226 W – 0.7319 W2 + 0.1018 W3)/10
= – 0.467

According to Yallop:

Easily visible or green means                             Q > +0.216
Visible under perfect conditions or blue means     +0.216 ≥ Q > – 0.014
Optical aid probably needed or grey                   – 0.014 ≥ Q > – 0.160
Optical aid certainly needed or red                    – 0.160 ≥ Q > – 0.232
Not visible or black                                         Q < -0.232

So, on 17th November 09 at 14:48, visibility is not there at Makkah (black).

IX.5 Example of intermediate horizon 30°W, 30°S at 21:04

We have:
L_S = 235.65°
L_M = 248.32°
D_M = – 3.55°
Azimuth of Sun = 243.47°
DAZ = 243.47 – 244.69 = 1.22 (always +)
Azimuth of Moon = 244.69°
Zenith distance of Moon = 84.26°
AltM = 90° – 84.26° = 5.74°
Distance earth –moon (geocentric) = 393315 km so RP = 393315/6378 = 61.67
SD according to MICA (illumination of disc) = 15’ 11.14” = 911.14”
ARCL = Cos-1 (Cos(L_M – L_S) * Cos D_M)
= 13.15°
ARCV = Cos-1 (Cos ARCL/Cos DAZ)
= 13.10°
SD = 56204.92/RP * (1 + Sin AltM/RP)
= 0.2536°
W = SD * (1 – Cos ARCL)
= 0.3986’ (minutes of arc)
= 0.0066°
Finally, with ARCV in degrees and W in minutes of arc,
Q = (ARCV – 11.8371 + 6.3226 W – 0.7319 W2 + 0.1018 W3)/10
= + 0.367
According to the Yallop criteria, Q indicates easy visibility in the green.

IX.6 Plotting visibility curves and taking photographs

Once the value of the Q factor is known it is in principle easy – with a good computer programme – to plot visibility curves by calculating this value for a closely knit grid of points on the globe.

It is to be noted that the visibility curves on the world map are not for an instant of time. They are a composite of local sunset time at every point on earth showing the possibility of sighting the moon on one specific day (date of Gregorian calendar) shown on top left. That specific day begins at the International Dateline (IDL) at 180º from the prime meridian of the Greenwich. The day continues towards west of IDL, and ends at the IDL traversing the 24 hour period of a specific day.

In March 2010, the new moon was born on the 15th at 12:02 UTC. The visibility curves showed a green belt in Canada on the next day. On our behalf, two photographs were taken of the early crescent on March 16th:

Photo1

Above photograph was taken by Roy Bishop at Grand Pre, Nova Scotia at approximately 45°07’ latitude and 64°18’ longitude, the moon being 26 hours past new.

Photo2

Above Photograph was taken by Steve Irvine, also in Canada, at 44.46’ latitude and 80.57’ longitude, the moon being 27 hours and 9 minutes past new.

Both photographs are brilliant and a conclusive proof of the applicability of our calendar. They both are also published on our dedicated Website in Photo Gallery. We have mobilised several observatories and amateur astronomers to provide us early crescent photographs at places determined by the visibility curves and situated within the limiting horizon for Makkah. All these photographs will be published immediately on our Website.

X. A dedicated Website

In order to make the new Makkah calendar available to the general public, we have created a dedicated Website with the following address:
http://www.makkahcalendar.org
Following are some important links of website.
Islamic Calendar |  Islamic Calendar Photo Gallery |  New moon visibility predictions |  Islamic Calendar Download Section
At present the reader will find the calendars from Islamic years 1431 to 1450 (current year 1431). We will keep on adding additional years in order to reach Islamic year 1500.

Conclusion

In his days, Galileo established, against all current thought, that it is the earth that revolves around the sun, and not the contrary. In our days, science – celestial mechanics in particular – clearly establishes that every point on the globe has its own calendar so that, ipso facto, a national calendar is neither justified nor legitimate. It is then remarkable that astronomy can still be used in order to define an accurate calendar for any given point of the globe for decades in advance. A grid of such calendars for various metropolitan cities of the world can contribute towards the establishing of a universal Islamic calendar, or at least calendars that rely on accurate scientific data.

ANNEX TO THE ARTICLE : ISLAMIC TEXTS PERTINENT FOR THE MAKKAH CALENDAR (SEE § III)

“If you are asked about the phases of the moon, tell them: ‘They are marks in time, intended for men and for fixing the pilgrimage.’” (Koran, Surat 2, verse 189).
As stipulated by the Prophet in several Hadiths, the Hegirian month begins with the visibility of the new moon. A few are quoted below:

First Hadith

“We are an illiterate community. We neither write nor count. The months are sometimes like this and sometimes like that, i.e. sometimes with 29 days and sometimes with 30.” (Reported by Al Bukhari, vol. 5, p. 2485, according to the narration of the son of Omar, transmitted by Said Ibn Awz.)

Second Hadith

“Fast when you see the crescent and finish fasting when you see the crescent. If you do not perceive it, complete the month of Sha’ban with 30 days.” (Reported by Al Bukhari, vol. 5, p. 3476, according to the narration of Abu Hurayra, transmitted by Mohammed Ibn Ziyad.)

Thus the need for vision, otherwise instruction for completing the month in 30 days.

Third Hadith

“If you see it, fast, and if you see it (again), stop fasting. If you do not see it, determine it by calculation.” (Reported by Al Bukhari, vol. 5, p. 2467, according to the narration by the son of Gurat.)

Calculation is thus necessary !