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The question of time reckoning has always played a central role in human history in general, and has ever since the advent of Islam preoccupied the best minds among Muslim astronomers for its relevance to religious practice. While the timings of the five daily canonical prayers are determined by the relative position of the Sun with respect to a local horizon, the determination of the new Lunar Hijri month relies on the visibility of the first thin lunar crescent following a Luni-Solar conjunction. To date, the scientific prediction of the visibility of the first crescent remains difficult and challenging for various technical reasons. To be able to predict its visibility, scientists have to first resolve the celestial mechanics problems: a daunting task that can be performed using high performance computing. The next challenge is not as trivial as the first one for it involves the reflection of light off the Moon and its transmission through the Earth’s atmosphere, a highly variable and turbulent medium that affects the propagation and absorption of light. The final limitation is directly connected to human physiology since one requires naked-eye observation. If the dynamics of the Moon-Earth system is now known and understood with a remarkable precision, predicting the physical state of the atmospheric medium over any given time of interest is obviously a complex and difficult task for anyone familiar with the field of fluid dynamics and atmospheric physics. This in turn leads to the logical conlcusion that a crescent visibility-based calendar is unfeasible since it requires real time observations. By combining celestial mechanics computations with atmospheric radiative transfer conditions and naked-eye observations, one can clearly get a feel for the amount of uncertainty involved in predicting first crescent visibility.

Our purpose in the present article is not to develop a critical scientific assessment of the way the crescent visibility question has been addressed in the academic literature over the past decades; this is neither the place nor the proper format for a technical report. We, however, examine how the scientific results are employed and manipulated by individuals, basically amateur astronomers, who have little or no science practice that would enable the grasp of subtleties embedded in the academic research on the various  relevant topic. With the virtual world of the Internet, irreducible complex and sophisticated topics have been reduced to fit the “popular” demand, and presented with fundamental loopholes that we would precisely like to address here. Moreover, this very critique raises some fundamental points scattered in the literature that in no way contribute towards elucidating the crescent-visibility problem and its solution while at the same time add confusion where none is needed. What follows is in no way a comprehensive review of the field or for that matter a full critique, and we do not pretend to hold the master key to all the problems raised below. However, we are fully aware that some of these problems need to be addressed thoroughly and scholarly. Moreover, we believe in full academic transparency, and our short critique is meant to bring the whole field related to crescent visibility to an academic level.

Our critique is focused solely on the scientific component of the crescent visibility challenge; we are in no way qualified to comment on the legal aspect. The various schools of jurisprudence have over the past centuries clearly identified their respective positions. However, we should point out that for this problem to be resolved, modern jurisconsults have to familiarize themselves with the tools of modern science: the theoretical and the experimental ones.



The determination and observation of the first lunar crescent is of primordial importance in Islam. Ever since the birth of Islam, Muslims of all walks of life, on the eve of religious events, would carefully scan the western horizon at sunset hoping to catch a glimpse of the thin crescent that would signal the end of a lunar month and the beginning of a new one. In the absence of observation, the beginning of the next lunar month is postponed by a full day. It is absolutely clear that the setup of a lunar calendar cannot be based on visibility reports of the first thin crescent; this would mean waiting for the 29th day of each lunar month before deciding about the start of the next one. This approach is obviously problematic since it does not have any predictive potential: a necessary condition for the setup of any calendar. This state of affairs did not prevent scholars from attempting to tackle the problem and propose solutions with very limited predictive potential for reasons to be elucidated. The proposed solutions available to the general public have not been able to resolve the unfortunate controversy that arises year after year on the eve of each and every Muslim holiday. This in turn has put government officials under a great deal of pressure given the fact that in our modern times we are lost in the zoo of information looking for the Phoenix of knowledge. This “état de fait” is due to a number of factors that we barely detail below. But before embarking on the scientific study of the problem at hand, which we have to address eventually in a longer publication, we would like to bring forth one important source of confusion, namely the ‘reproductive flourishing’ of the number of  individual contributors to the question related to crescent visibility. These contributors have in general very few scientific credentials if none, and seem to have become internet stars. However, we have no doubt that in most cases the intention is noble, but we ought to remind those contributors that this matter necessitates scholarship for it is of utmost importance: it deals with the religious affairs of Muslims all around the world. The use of the available technical literature requires some mastering of various fields of research; indeed, we have observed that the web is littered with information that is clearly misleading since it does not reflect the scientific consistency and expectancy of most of the refereed literature, and in most cases fails to convey the consequences and the validity limits of the studies reported. Another source of confusion, which will not be addressed in this short paper, is related to reports by trustworthy witnesses of observation of the lunar crescent under unfavourable atmospheric physical conditions. The polemic raised by such reports, when diffused by mass media and when reaching the general public, can only take the level of confusion to yet another height.

The apparent enthusiasm sparked by the question of the crescent visibility during the last few decades is not synonymous of significant progress. Visibility criteria were developed and put forward in order to bring some beneficial solutions. These criteria, as they can be used in practice, are all geometric in nature except the one introduced by Schaefer, i.e. based on the relative position of the Moon with respect to the Sun. One has to determine a critical angular separation between the two luminaries that allows a sufficient contrast therefore enabling the observation of  the new-born crescent in the western sky. For example, the Danjon[1],[2] limit poses that the lunar crescent cannot be seen with a naked-eye if the angular distance between the Moon and the Sun (ARCL, Arc of Light) is less than ~ 7 degrees. Although Danjon’s limit remains a solid empirical threshold (violations of Danjon’s limit are rare, and in general should be taken with caution), yet it does constitute a sufficient condition for visibility. In general, the geometry-based criteria can be illustrated in what we call the DAZ-ARCVvisibility curve discussed below. Other criteria that appear to be not directly related to angular distance, such as the time lag (moonset and sunset time difference) or the Moon age, are of the same nature nevertheless. The present report is by no means a comprehensice assessment of the scientific progress on the question. However, we focus our critique on the popularized geometrical argument and criterion, which carries potential impacts on insitutions throughout the Islamic world. Through our analysis, we raise at the same time the main limitation of the criterion that is occulted by the amateur astronomers.

The DAZ-ARCL visibility curves

The crescent visibility curve is designed to help separate between regions characterized by a set of variables (altitude, azimuth, …) where the crescent is potentially visible, and regions where it cannot be seen. This approach was used  by Muslim astronomers during the classical era to make visibility predictions. The most simple of these visibility curves is that proposed by Al-Khawarizmi[3], and which relies on a single variable. Ever since Fotheringham[4] published his pioneering study in 1910 on lunar crescent visibility, the standard was set for the generation and production of visibility curves. Indeed, all visibility curves that appear in the modern literature use the relative altitude of the Moon with respect to the Sun (ARCV) and the relative azimuth (DAZ). In practice, both DAZ and ARCV are calculated for a given time at sunset, and at a given location. A different marker is then assigned to each (DAZ-ARCV) vector based on whether the crescent is visible or not. To be visible, a small (large) DAZ requires a high (low) altitude ARCV. After collecting a large and significant number of observations, a scatter plot is produced in the DAZ-ARCV plane. A visual inspection of the scatter plot is then relied upon to draw/insert a line, a hypothetical line ARCV=f(DAZ), separating the data into a set of points reflecting crescent visibility and a second set reflecting the non-visibility. The line is monotonically regular, and one should expect that from this subjective insertion one is bound to end up with some discordant observations. Moreover, this hypothetical line has no physical justification, and the points that lie on it do not necessarily satisfy the geometrical constraints of the problem. The important point to take home is the fact that the visibility on the scatter plot is based solely on a visual inspection and therefore becomes highly subjective. Nowhere in the literature has this point been addressed.

In order to increase the visibility potential, the DAZ-ARCV curves are extended to include aided-observations using binoculars and telescopes. The resulting visibility curve is nothing but the naked-eye visibility curve shifted in the y-direction (ordinate direction). Again, this artifice is nowhere justified. It implicitly assumes that the used instruments operate like a naked-eyed with an enhanced optical power only. The actual effects, which are inherent to the instrument characteristics, are rarely questioned or addressed properly. Worse, the data collection used to make the visibility curves is built from observations made with various instruments a priori having different optical characteristics. In the case of binocular equipment, issues related to calibration are completely ignored or not documented properly. It is clear, from the few arguments mentioned above, that a solid  scientific experimental procedure need to be followed in order to justify the use of the “aided visibility curve”.

For a marginal Earth-Moon-Sun configuration, where uncertainties are the highest, the DAZ-ARCV curves are of a minor interest and sometimes even useless. For this critical configuration, the criteria of geometrical nature reach their own limit. For this reason, in particular, there exist in the literature several visibility criteria with various degrees of reliability, which basically constitute a source of disagreement in the prediction, although often minor. In reality, and at this level of magnitude (relative), weather and atmospheric conditions predominate; the Celestial Mechanics problem becomes secondary and the atmospheric  radiative transfer problem becomes dominant. The refraction, absorption and diffraction of light reflected by the Moon and propagating through the Earth’s atmosphere becomes the main problem to resolve. This kind of treatment was begun by B. Schaefer whose work is not available in the literature for an assessment. We have here identified the heart of the problem, which constitutes the main scientific obstacle to the elaboration of the Hijri calendar. The problem is accentuated because that the marginal configuration described above occurs on a regular basis.


The Lunar Date Line: A Gross Approximation

The Malaysian astronomer Mohamed Ilays[5] is one of few Muslims, in the modern era,  who has made a genuine technical effort to address the crescent visibility problem. In his seminal book, “Islamic Calendar, Times & Qibla,” he developed a modern formulation in terms of mathematics of reckonings related to Islamic rituals. Among Muslim scholars, Ilyas is considered as one of the pioneers in the field of crescent visibility. Ilyas[6],[7] introduced in 1982 the Lunar Date Line (LDL), a line that separates regions on the Earth’s surface where the lunar crescent could be seen from those regions where it remains invisible. Similar to the change date line (CDL), the Lunar Date Line (LDL) concept could be used as a reference for the determination of the monthly sequence in the Hijri calendar: In regions located West of the LDL a new Lunar month starts while in those located East of the LDL one more day has to pass before a new Hijri month starts.  However, a visibility criterion has to be selected first for this LDL insertion to become feasible, and as a consequence, there are as many LDLs as criteria. The LDL is rather a didactic visual tool which is not meant to address the fundamental question of the crescent visibility. In fact, under critical conditions the LDL is useless and even misleading. The Doggett-Schaefer Moonwatch campaign[8] showed that the uncertainty of the LDL could reach 105o in longitude, almost 1/3 of the planet! Still, this has not prevented the wide use of the LDL by neophytes who are quick to the release button for the instant display on the net. Globally, the LDL reflects the eastward motion of the Moon with respect to the horizon, which renders western regions more favorable than the eastern parts for crescent observation.


Erudition Exposed

The establishment of an Islamic Lunar calendar has always been a focal point of interest for Muslims, and this has in turn led many Muslim scientists and respected scholars to address this need and propose practical and reliable solutions. In this section, we present what we believe is a fair and critical assessment of a sample of papers and works published by Muslims on various platforms related to the net. The purpose of this exercise is to demonstrate that the treatments and the solutions proposed and found in the literature often reflect the lack of erudition and depth with which the question of the crescent visibility has been presented. We believe that this very shallow handling of this important question constitutes part of the problem. In no way do we intend to alter the integrity of the people named below, our concern is strictly academic, and our criticism is aimed at elevating the standard to a true academic one. We have approached this exercise as a refereeing task for a professional academic journal.

1.     Khalid Shaukat

For years, Khalid Shaukat has been running the widely popular website, which provides very useful information on prayer times, moon visibility and other matters related to Astronomy and Islam. Data on crescent visibility are based on Shaukat’s own criterion.  What makes the case of Shaukat astonishing is that his criterion has not been published in the peer review literature, and is nowhere to be found, not even on his own website, and we are not the first ones to report this (see for example Leong Wen Xin[9]). We find ourselves in a situation where no scientific assessment of Shaukat’s work can be carried out, and we have no way to find out how Khalid Shaukat derived his criterion. We have no doubt about the sincerity of Khalid Shaukat. However, we find this situation completely incongruous since this information should be available to the public, and a full assessment of the criterion has to be carried before adopting it in one form or another or rejecting it.

2.     Mohamed Odeh

Odeh is the vice-president of the International Crescent Observatory Project (ICOP), an active group whose  main objective is the collection of lunar data. Odeh is quite active on the web, and many public institutions from around the Muslim and Arab countries frequently solicit his expertise on the crescent visibility, and other astronomy related matters. He even set up a criterion for crescent visibility labeled with his name (Odeh’s criterion). The criterion was derived from work published by Odeh in 2004[10]. Odeh’s criterion lies within the  DAZ-ARCV family of curves,  and is based on a collection of 737 observations of which about half were undertaken by ICOP. In his study, Odeh used topocentric values for the Moon Age (Age), the arc of vision (ARCV), the arc of Light (ARCL) as well as the width of the crescent moon (W). Odeh never explains in his 2004 publication10 why he used topocentric variables rather than the universal geocentric ones. The choice of a topocentric framework results in a meridian-dependent Age of the Moon (for example), which renders the context completely confusing, and one wonders about the underlying physical interest for this unnecessary transformation. Moreover, the positions of the Moon (such as ARCV) used to fit the data are calculated (Celestial Mechanics) NOT observed. So, Odeh for some obscure reasons replaced the calculated geocentric variables with the topocentric ones, which are also calculated. Nowhere in his 2004 paper does Odeh discuss how topocentric variables provide a better fit to the observations. The use of topocentric variables ought to be justified considering that no similar previous studies since Fotheringham (1910) have utilized them. Moreover, there is another striking ingredient in Odeh’s paper. To carry his study, the author followed the footstep of Yallop’s note published in 1998[11] . To determine the critical visibility curve, a conventional procedure is used since Fotheringham (1910) in which a table (ARCV, W) or equivalently (ARCV, DAZ) is introduced to separate positive from negative observations. The numerical values given on the table are afterwards fitted using a third degree polynomial in lunar width W (or DAZ). Yallop’s fit provides the following model:

q = (ARCV – (−0.1018W3 + 0.7319W2 – 6.3226W + 11.8371))/10

while Odeh’s model is given by:

V = ARCV – (−0.1018W3 + 0.7319W2 – 6.3226W + 7.1651)

Except for the y-intercept, it appears that the numerical values of the coefficients in Odeh’s model are quite striking: Odehs’ findings are, to the fifth digit, EXACTLY 10 times higher than Yallop’s (we should point out that Yallop’s division by 10 was introduced for convenience to confine the values of q between -1 and 1). Assuming that there is no error of any kind, Odeh’s critical visibility curve is just a translation (along ARCV−axis) of Yallop’s one. This situation is quite suspicious since both Yallop and Odeh used different sets of observations and therefore one would expect the fits not to be quasi identical, not to the fifth digit. Worse, Odeh used topocentric ARCV while Yallop did not, i.e. the y-variable is not exactly the same. It appears to us that Odeh’s fit is bogus or at least needs a very serious clarification. Finally, the third degree polynomial fit in W is not consistent with Table V of the paper and on which Odeh constructed his new criterion. Odeh also claims that his criterion is precise, meaning that there is an improvement when compared to previous similar studies. However, Odeh does not provide any quantitative argument to support his claim.

3.     Nidhal Guessoum’s bi-zonal calendar

To remedy the question of the crescent visibility, Guessoum[12] went further by calling for a bi-zonal calendar, which separates the ancient world (Asia-Europe+Africa) from the American continent. Guessoum criterion can be formulated as follows:

The new month begins in both areas if the conjunction occurs before dawn at Mecca. If the conjunction occurs between the dawn of Mecca and noon GM, the new month begins in the American zone to be postponed for the day after in the ancient world”.

Guessoum labels his solution “Kepler solution” echoing the Copernican revolution, which dismissed the complicated epicycles used in the middle age to model the planetary motion around Earth to replace it by a heliocentric system. In reality, Guessoum’s criterion remains solely based on a geometric argument with no relevance to the problem he set himself to dismiss. Guessoum does not elaborate on his rule and we are consequently left unable to make any technical assessment of his work. What follows is a critique mainly based on what Guessoum published on the net and made available on various websites. We have combed the professional academic literature to no avail.

First, switching from Mecca to Greenwich meridian seems a bit injudicious or even heteroclite. In addition, using Fajr as a reference instant is ambiguous. Moreover, Guessoum claims that 73% of the observations comply with the criterion. Is it assumed that a visibility criterion was adopted? Also, what about the 27% of cases left (basically 1 out of 4), a score that dismisses any claim for an authentic calendar. As presented, Guessoum’s solution appears completely inconsistent since it relates his bi-zonal criterion with the observations without any physical base. For example, he does not explain why using the Mecca meridian, although symbolically understandable though not legally justified, leads to a better result. This inconsistency could be the source of serious problems. For example, the length of the same month could be different whether you are in Zone 1 or Zone 2. A reliable calendar cannot be built only on observations whose validity is questionable, and it is far more preferable to rely on sound and understood physics.

Guessoum’s criterion is just an artifice to predict the beginning of a month and by no means forms a basis for a calendar. In fact, any other meridian reference will provide similar outcomes. An authentic calendar possesses resources to include deviant or eccentric situations. Testing the reliability of a calendar in terms of statistics is not pertinent for as far as academic standard requirements are concerned.

A short story to meditate on

In 1572, following a recommendation of the Council of Trent (1545-1563), the newly elected Pope Gregory XIII (1502-1585) had the task to reform the Julian calendar, which was enforced back then. The first steps taken by the Pope were to designate Aloysius Lilius, an astronomer from Naples, and later the German Jesuit multidisciplinary scholar Christopher Clavius, to lead a scientific group (committee of experts) to address the question of the Julian calendar and its shortcomings. For pedagogical and instructive reasons, we briefly remind the reader what the reform was all about. Although the Julian calendar accounts for the leap year, which was applied every 4 years, a gap (due to a forward drift with respect to the tropical year) of 0,0078 day still occurs after every year. Ignoring this “error” would lead after 1000 years (for example) to an accumulation of a drift of 7.8 days or 3.12 days after 400 years. To overcome this problem, Gregorian reform suggested skipping the leap year for secular years that are not divisible by 400 (such 1700, 1800). This is not the end of the story. After this judicious adjustment, the Gregorian calendar still leaves a (forward) gap of 0.0004 day after each year, which gives an accumulation of one full day after 3333 years at which time the current calendar will end up with one more extra day.  Since the current calendar began in 1582, the month of February of year 4915 (1582+3333) has been set to include 27 days only. This story illustrates how handling a calendar is a very serious issue and cannot be left to “apprentice sorcerers”. Moreover, although Pop Gregory XIII was an expert in canon law, it did not prevent him to carry one the most important reform in human history, one that is widely used today. The Pope was certain that the issue of the calendar raised then by the council could be resolved by science only. Although astronomers such as Tycho Brahe and Johannes Kepler and the Catholic princes of Europe hailed the reform, many Protestants see in it the work of the Antichrist and refuse to accept it. More than a historical event, the council of Trent is considered as a bifurcation point from the medieval to the classical Church, and one of the most important ecumenical council in Catholic history. Something to meditate at some point !

In conclusion, we have stressed above some science-related elements that prevent the elaboration of a Hijri calendar based solely on geometric arugments. We did not evoke the current attitude of religious scholars including the jurisconsults, vis-a-vis this very question. This obviously cannot be ignored given their influence on Islamic institutions throughout the Muslim world. We do not consider ourselves experts in the field of jurisprudence, but we, however, believe that any current legal opinion should not ignore the scientific progress realized on the question of crescent visibility. One cannot use sophisticated communication devices on real time, on a global scale, and deny at the same time that science has its claim when it comes time reckoning. If vehicule-size spacecrafts can be monitored while located several Astronomical Units from Earth, it means that an underlying reality exists and it cannot simply be refuted using legal arguments. However, we have noticed that over the past few years, Islamic institutions from around the world have started to reject reports of crescent observation “occurring” before the luni-solar conjunction; a sign of a significant progress in the attitude of many religious scholars.

[1] Danjon, A., l’Astronomie, 46, 57, 932.

[2] Danjon, A., Bull. Soc. Astron. France, 50, 57, 1936

[3] Kennedy, E. S., and M. Janjanian, The crescent visibility table in Al-Khawarizmi’s Zij. Centaurus 11, 73-78, 1965.

[4] Fotheringham, J. K., On the smallest visible phase of the Moon, Mon. Not. Roy. Ast. Soc., 70, 527, 1910.

[5] Mohamed Ilyas, Islamic Calendar. Times & Qibla, Berita, Kuala Lumpur, 1984.

[6] Mohamed Ilyas, Earliest global visibility of the new Moon 1981-85, J. R. Ast. Soc. Canada, 76, 371, 1982a.

[7] Mohamed Ilyas, New Moon’s first visibility: review of Astronomy and current Islamic calendrical practices, Islamic Culture, 56, 43, 1982b.

[8] Doggett and Schaefer, Lunar visibility, Icarus, 107, p 388-403, 1994

[9] Leong Wen Xin, Lunar Visibility and the Islamic Calendar, 2001. On page 23 of his document, L. W. Xin writes: “We do not know the exact criteria that led to the figures used above. Nevertheless, Khalid Shaukat is a research scientist since 1967 and has been a consultant on moon sighting for the Islamic Society of North America and had developed a website on moon sighting ( His criterion

has been one of those more reliable one”.

[10] M. Odeh, New Criterion for Lunar Crescent Visibility, Experimental Astronomy, 18, p39-64, 2004.

[11] Yallop, A Method for Predicting the First Sighting of the New Crescent Moon, NAO Technical note, No69, 1998.

[12] Guessoum, Two-Zone solution


Karim Meziane Senior Research Scientists, Physics Department, University of New Brunswick, Canada

Abdelhaq M Hamza Professor, Physics Department, University of New Brunswick, Canada