1. Introduction
The early sighting of a new crescent moon is one of the earliest astronomical activities performed by human civilizations. This sighting marked the start of the lunar month in many such civilizations. The first evidence of these activities came to us from the Babylonians (6th Century B.C.), who lived in the fertile lands of Mesopotamia (modern-day Iraq) [
1]. They relied on the crescent moon to indicate the start of their calendar months and years. They used a standard criterion to predict the moon’s sight, which was sufficient for that purpose. Other civilizations, such as Indians and Chinese, also used lunar sight and still do to this day, although use different criteria to determine the start of a lunar month [
2]. Jewish people also still use the moon cycle to indicate the start of the months in their lunisolar calendar, although they use arithmetic calculation rather than moon sighting.
Centuries after the Babylonians, in the 7th Century C.E., a new religion, Islam, had emerged from the deserts of the Arabian Peninsula. Three of the five main pillars of Islam depend on the start of the lunar months. They are Fasting-Breakfasting “
Siam-Fitr”, Pilgrimage “
Hajj”, and Alms “
Zakat”. The first two pillars are performed in a specific lunar month in a year, while the third is performed once in the first month of each lunar year [
2]. Reckoning of the first crescent moon after the conjunction is an essential indication for Muslims to begin their lunar months and perform their religious duties.
Through the Islamic golden age (between the 9th and 11th centuries C.E.), many famous Muslim astronomers laid down specific criteria for moon sighting that depended on the age of the moon (the time passed since the conjunction) and the lag of the moon (the time between the sunset and the moonset) [
3]. Some considered the arc length of celestial degrees between the sun and the moon (elongation) at sunset. However, they could not predict the sighting perfectly because they did not consider the local conditions of the horizon at the time of observation, and they omitted the effect of changing the apparent width of the crescent moon and its relation with the brightness of the sky [
4].
Since the start of the previous century, various theories have been proposed to design a criterion for moon visibility suitable to predict the start of lunar months. These theories relied on drawing a borderline between positive (visible) and negative (invisible) observations or on defining a visibility region using interpolations. All these methods were empirical and depended on the subjective viewpoint of the observer. Many observation records yielded incorrect predictions [
5]. There is also interference between the two mentioned regions, which was difficult separate empirically. This situation could be resolved by drawing a barrier region between them. Therefore, an objective-oriented method that depends on the presented datasets themselves should make the visibility regions more realistic and reliable.
The development of machine learning (ML) techniques in recent years resulted in many considerable advancements in modern technology. ML algorithms have proven their ability in numerous fields, such as pattern recognition.
Artificial neural networks (ANN’s) are of the most widely used technology in pattern recognition due to their ability to analyze and learn patterns hidden inside the data that they consider, as well as their ability to train themselves on new data to rectify and improve their performance. They can predict the output of any input not presented in the training data, making them flexible in handling lost and misleading information. Many studies have been performed related to ML with ANNs in the field of pattern recognition and classification [
6,
7,
8,
9].
In this study, a two-layered ANN was used to learn moon sighting data tables provided by Schaefer, Yallop, and Odeh to generate a moon sighting region recognizer to predict the possible condition of the first sighting of a new crescent moon. This ANN was used to build a calendar for four Islamic Hijri years (1440–1443 A.H.) based on the local horizon of Baghdad, the capital of Iraq, as a part of a larger project to build a unified Islamic Hijri Calendar for Iraq. The results of the ANN recognizer were compared with the current official Hijri Calendar in Iraq, which relies on astronomical calculations rather than moon sighting.
The remainder of this study is organized as follows. In
Section 2, we outline a literature review of related works. In
Section 3, we illustrate the methods used to calculate the essential parameters for moon sighting and demonstrate the observational dataset and the design of the proposed pattern recognizer ANN. In
Section 4, we present the results of ANN training and the newly constructed calendar. In
Section 5, we discuss the obtained results.
2. Related Works
Many astronomers and scientists developed arithmetic criteria for the first visibility of the crescent moon at the beginning of the previous century. Many of their criteria were based on the minimum elongation of the moon suitable for visibility.
Developing accurate criteria for moon sighting in the modern era began with Fotheringham [
10] in 1910 and Maunder [
11] in 1911, who categorized moon sighting regions as visible/invisible using moon observation data collected by Julius Schmidt in Athens, Greece, between 1859 and 1867. They used one parameter for crescent visibility, i.e., elongation, which is the distance in arc degrees between the sun and the moon. They claimed that the shortest elongation for naked-eye visibility should be 12° (Fotheringham) or 11° (Maunder). In 1932, Danjon [
12] claimed that the moon could not be seen if its elongation is less than 7°, thereafter referred to in the literature as “Danjon’s limit”. Many moonsighting campaigns led by McNally [
13] and Schaefer [
14] attempted to break that limit [
15]; the results were recorded in tables and used by many others, such as Yallop [
16] and Odeh [
17].
McNally claimed that the moon could reflect the light from the sun to the earth if its elongation was 5.5° at least. However, Schaefer criticized McNally’s claim and confirmed Danjon’s limit [
18]. Ilyas [
19] agreed with Maunder’s limit and modified it to 10.5° for naked-eye visibility. Yallop [
16] agreed with Ilyas’ limit, although decreased it to 10°. Fatoohi [
1] modified Danjon’s limit to 7.5° after considering many data records which belong to the Babylonian era. Finally, Odeh [
17] claimed that one of the recorded observations by the
Islamic Crescent Observation Project (ICOP) broke the Danjon’s limit. In that observation, the crescent was seen by a telescope at elongation of 6.4°. However, this observation is unique and unreliable without further investigations and similar observational conditions [
20].
Table 1 illustrates the minimum elongation for visibility according to several authors:
The criteria above are simple because they give one parameter only to judge whether the crescent would be seen or not. However, ancient and contemporary astronomers used many complicated criteria. A comprehensive list of these criteria and their parameters is shown in [
3].
Many previous authors attempted to add other parameters to moon visibility criteria. They considered the physical aspects and their relation with early visibility, such as sky brightness and eye perception [
21]. However, other astronomers omitted the effect of the atmosphere for theoretical visibility [
13].
Fotheringham-Maunder proposed the first mathematical criterion, which extracted it from observational data [
10,
11]. It has two parameters. These parameters are the
azimuthal difference between the sun and the moon (which is abbreviated as
) and the altitude difference (which is referred to as
the arc of vision and abbreviated as
) between the sun and the moon. These parameters are computed at the time of sunset at the observation location. This criterion draws a line between the positive observations (visible by the naked eye V) and negative observations (invisible by the naked eye I). For instance, if the moon’s position was under that line, it would not be seen, and vice versa.
In 1977, Bruin [
4] modified Fotheringham-Maunder’s criterion. He considered the western sky, the moon’s surface brightness, and the solar depression (the altitude of the sun below the horizon). Bruin presented a new parameter, (
the crescent width, abbreviated as
). The crescent width is the arclength of the crescent, which is proportional to the radius and the illuminated fraction of the moon. Bruin used the arc of vision as a function of the crescent width and found that the moon will not be visible if its width is less than 0.5′ (30 arcseconds).
Ilyas [
22], on the other hand, reformed Bruin’s criterion and found that the minimum limit of visible width of the moon was 0.25′; (15 arcseconds). In [
19], he developed a new criterion and claimed that the moon could not be visible if it had been found on the horizon when the depression of the sun was 4° (
). He verified the Maunder and Bruin visibility criteria and extended their curves using extrapolation to cover visibility conditions in higher geographic latitudes. Furthermore, Ilyas found that Danjon’s limit was just an extrapolation, therefore he increased the limit to 10.5° for naked-eye visibility. This issue was due to observation in the lower-tropical geographic latitudes rather than the middle latitudes when Danjon recorded his observations in Strasbourg Observatory, France.
Schaefer [
21] claimed that some of Bruin’s data were misleading, and some of his assumptions were incorrect regarding twilight sky brightness, lunar surface brightness, and physiological data of lunar vision. Schaefer added other parameters, such as the visual extinction coefficient and atmospheric clarity. In his study above, he demonstrated the condition of the first moon visibility and claimed that his result was better than the results of Danjon and Bruin.
Yallop [
16] used the results of Maunder and Bruin and the observational data collected by Schaefer to derive a new
q-test criterion based on a published formula in the
Indian Astronomical Almanac in 1979. That formula was useful for heliacal rising and setting. He presented the concept of the
best time, which is the time of the optimal observational condition. He claimed that the best time
is the time which comes after the sunset
in exactly four-ninths of the moon lag, i.e.,:
where
is the time period between the sunset and the moonset.
Yallop generated the observational data provided by Schaefer at his best time, then he used the arc of vision
(in degrees) versus crescent width
(in arcminutes) to develop his criterion as follows:
Table 2 illustrates the crescent conditions according to Yallop:
In 2001, Caldwell and Laney [
23], a team of two astronomers in the
South African Astronomical Observatory (SAAO), developed the
SAAO Criterion, another criterion for crescent moon visibility. This criterion depends on
the arc of light (abbreviated as
) which is slightly less than the elongation, and the arc of vision
(for the bright limb of the moon rather than its center) at the time of the sunset. This criterion was simple; it does not need a complicated polynomial to calculate. The SAAO q-test criterion states that:
Table 3 illustrates the crescent conditions according to the SAAO team:
Caldwell [
24] investigated the correlation of moon lag with the arc of light. Meanwhile, Qureshi [
25] proposed a modification to Yallop’s criterion depending on recently collected observation data.
In 2005, Odeh [
17] proposed the most recent moon sight criterion. He used the dataset of moon sightings provided by Schaefer and Yallop and added other collected records from many crescent watchers and the ICOP. Odeh used more than 700 records in his study and built his criterion based on the Indian formula by altering Yallop’s criterion offset and region control values. Furthermore, he used Yallop’s best time for data collection and used
and
to develop it as follows:
Table 4 illustrates the crescent conditions according to Odeh:
Other researchers attempted to formulate visibility criteria for the crescent moon. Some of them proposed a line referred to as the International Date Line (IDL), a line that divides the globe into two lunar month-start regions. They claimed that the line would resolve the problem of differences in lunar months towards constructing a universal Hijri calendar [
26,
27]. Al-Mostafa [
28] proposed a criterion for moon visibility and reckoning the first day of the lunar month in KSA. A recently collected dataset by Alrefay et al. [
20] of moon sighting observations is useful for a new robust criterion.
All the previous proposed criteria relied on empirical techniques such as interpolating data of the moon at the observation time. This study aims to divide the moon visibility regions according to a prediction algorithm based on ML using the concept of Pattern Recognition built in the ANN. Before getting to the design procedure, it is necessary to demonstrate the essential parameters of the moon sighting and how to compute them.
5. Discussion
Figure 5 gives a complete comprehensive analysis of the original observation dataset. First, the crescent moon would not be visible by any means available (except the CCD imaging) if the crescent width was less or equal to 0.12′ (
is about 6.86° or elongation of 7.12°), which proved Danjon’s limit and Schaefer investigations. In the region of the lower limit of naked eye visibility, the moon would be certainly visible for any condition of the sky if the crescent width was 0.3′ (
is about 11° or elongation of 11.25° provided that
), which confirmed the results of Maunder and Ilyas as the minimum elongation for naked-eye visibility. The crescent width might be greater than 0.3′ and could not be seen by naked eye if its
was very large. For larger values of the crescent width
′ (
), it is obvious that the moon could not be seen by any means available if the
, which confirms the finding of Ilyas. Furthermore, the minimum value of arc of vision for the crescent moon visibility is when
.
Secondly, it is enough to divide the moon sight area into three regions rather than four as in Odeh’s criterion or five as in Yallop’s criterion. The invisible region is contiguous, and so is the visible one. The middle area was only a thin line between the two others.
Thirdly, there was a red spot inside the blue region. The reason was performing the old observations using the naked eye only. This spot should be green (probably visible) rather than red. Schmidt recorded these sightings as “invisible” rather than “probably visible” because he used the unaided sight only.
The crescent moon may be visible by the unaided eye in the green region, but it remains unlikely to be seen compared with the blue one. Furthermore, a visible area under an invisible area implies an issue related with recording sight results. The green region might contain blue spots, which indicates that the crescent could be visible by the unaided eye if the conditions were appropriate. Whenever the moon sight data approaches the red region, moon sighting will be less likely.
Figure 5 shows that the official Hijri Calendar in Iraq started the new lunar month on the following day of the moon sight, regardless of the crescent moon sight result (invisible or probably visible), yet most of the results were invisible. So one should consider that issue and reform the official Hijri calendar to use observational results rather than astronomical calculations, which most countries in the Middle East have used.