### Islamic Calendar

# Personal website of Wenceslao Segura

# - - - - - PAPERS - - - - -

We describe the arithmetic or computational Islamic calendar of medieval Muslim astronomers. We classify the different calendars of this type, also called tabular, finding the posible intercalation crieteria.

With the chronological Julian day, we obtain precise rules to convert this tabular calendar to the Julian or Gregorian calendar and vice versa.

The Islamic tabular calendar is, on average, very close to astronomical reality; however, there is an error that is accumulative and that we determine precisely.

This paper analyzes the Islamic era and its relationship with the pre-Islamic calendar that existed in Arabia before the arrival of Islam.

We expose the techniques to find computational lunar calendars. We distinguish between regular and semi-regular calendars. We study the Islamic calendar proposed by Rashed, Moklof, and Hamza, and we use the chronological Julian day to do the conversion to other calendars.

We calculate the Danjon limit or the smallest angular distance between the Moon and the Sun with which we can see the lunar crescent, using the model developed by astronomers at the Helwan Observatory. We found that the Moon could seen with the naked eye at 5.6º away from the Sun in exceptional conditions. With a more realistic calculation, we find 7.1º for the Danjon limit. We show that this limit angle es highly dependent on atmospheric absorption, varyng significatly when the extinction coefficient is modified. We find a altitude above the horizonat which it is easier to observe the Moon crescent, which depends exclusivly on the extinction coefficient. Finally, we show that Helwan method has unsatisfactory foundations, although the results we derive are in agreement with what has been found in other theories of lunar visibility.

We show that the criteria of lunar crescent visibility of al-Kwarizmi (9th century) and al-Qallas (10 century) is not the Indian criterion, according to which the Moon will be visible if bwteen the moonset and sunset there are more than 48 minutes. Therefore, we distinguished two new visibility criteria: al Kwarizmi and al-Qallas, which we analyze and generalize.

We calculate in detail the maximum width of the illuminated part of the Moon and the phase, or proportion of the illuminated area to the total surface. We do the calculations from the geocentric anda topocentric points of wiew.

The Arab astronomer and mathematician al-Battani (858-929) developed a theory to determine when the Moon would first be visible it was new. In this paper, we study this criterion with current astronomical knowloge.

We show that the extent of the illuminated area of the Moon is greater then 50% of its surface; that is, a part of the lunar hemisphere opposite the Sun is illuminated, whichi occurs because the Sun is largen than the Moon.

There is widespread confusion about the concept of tropical year. Although it is correctly defined from the mean longitude of the Sun, at the same time it is identified with the movement of the Sun respect to the seasonal points. In this study, we distinguish between the tropical year, as it is commonly defined, and the seasonal year, or time between two consecutive passages of the Sun by a particular seasonal point. We found that the mean value of the possible four years seasonal averages (Spring, Summer, Autumn and Winter) coincides with the tropical year. We evaluated the variation of the seasonal years for an interval of time of a few thousands of years. And finally we estimated the durations of the various types of years in universal time units, adapting them for use in the calculation of the calendarists’ errors.

When observing the first Moon crescent, it is necessary to gather information about the Moon: site in the sky where it is at sunset, luminosity, width, and orientation of its horns. We calculate the angles that the midpoint of the crescent forms with the vertical and the hour circle, data that allows us to know the orientation of the Moon's horns.

Conferencia divulgativa sobre el calendario gregoriano, donde se destacan algunos aspectos astronómicos, curiosidades del calendario y los intentos para reformarlo.

By the Sultan method (2007b), we determine the Danjon limit: the minimum angle between the Moon and the Sun at which we can see the Moon. We corrected several errors made by Sultan; in particular, we calculated by a different method the magnitude of the Moon at large phase angles. We verify that the Danjon limit is dependent on the atmospheric extinction coefficient. We found that Sultan's method is not satisfactory since it makes the visibility dependent on the luminance of the Moon when the factor that determines whether or not it will be seen is the illuminance.

We show techniques for finding the arc-light, or angle between the centers of the Sun and the Moon. We describe the periodicity of the Moon's ecliptic latitude and its effect on the arc-light. We verify that the arc-light at the New Moon time has a periodicity of approximately 173.5 days. We define the topocentric New Moon, which occurs when there is a relative minimum of the topocentric arc-light.

We analyze some of the periodic parameters that characterize the Moon: latitude, inclination of the orbit, tropic velocity, synodic velocity, lunation, distance from Earth, as well as the periodicities of other phenomena that have some relationship with calendars: lunar day, the interval between consecutive moonsets, synodic and ecliptic movement, effect of the variation of the Earth's eccentricity, half lunations, difference between mean and true lunations and lunations depending on the phase.

This letter is a brief report on how to deal with predicting the first visibility of the Moon crescent. We warn against the erroneous interpretation of Blackwell's threshold visibility experiment and that calculations must be made for a lunar width less than the resolving power of the human eye.

Schaefer (1991) determined the Danjon limit or minimum angle between the Sun and the Moon from which the Moon can be seen shortly after the conjunction. Schaefer's method uses Hapke's (1984) lunar photometric theory and considers a fixed value for the threshold illuminance.

We show Schaefer's method and its shortcomings, and we expose a modified theory, where the threshold illuminance to see the lunar crescent depends on several factors, mainly atmospheric absorption. We consider that vision is a probabilistic phenomenon; that is, when we use the experimental data of Blackwell (1946), we cannot be sure whether or not the Moon will be seen.

Finally, we conclude that «perhaps» Hapke's theory overestimates the shielding of the sun's rays by the irregularities of the lunar surface at large phase angles.

Bruin (1977) devised a procedure to find out the visibility of the first crescent Moon. He applied various simplifications to his theory, not all of them acceptable. We rethink Bruin's method by making some corrections: we take into account the variation of the luminance of the Moon with the phase, we use the experimental results of Knoll et al. (1946) on threshold contrast, we apply Riccò's law, and we consider the atmospheric extinction coefficient to be variable. We use the theory to derive the Danjon limit.

We verify that the Islamic calendar is not exclusively lunar but is also related to the movement of the Sun; for this reason, we say that the Islamic calendar has some lunisolar aspects.

The month of the Islamic calendar begins with the first observation of the crescent of the Moon. This phenomenon is highly dependent on the geographical position of the observation site. We expose the dependency of the first sighting of the Moon on latitude and longitude. We define the concepts: terrestrial terminator, Month Change Line, zone of first lunar visibility, apex, point of the first vision of the crescent, and isochrones. We check the dependence of these concepts on the equatorial coordinates of the Sun and the Moon.

We describe the global view of the Moon's crescent and show the movement of the apex and the point of first visibility of the crescent.

We show that in high geographic latitudes (approximately > 50º north or south), the lunar months of 28 and 31 days are possible.

We expose and analyze the proposed models of the visual magnitude of the Moon for large phase angles (>150º). We devised a method to determine the luminance and illuminance per unit angular length of the lunar crescent as a function of position and phase angles.

For the central zone of the Earth (approximately 50ºN-50ºS), Islamic months have lengths of 29 and 30 days depending on the place of Earth from where we observe the first lunar crescent. We verify that all the lunar months have two durations for the central zone, one of 29 days and the other of 30 days. For higher latitudes (50º N or S to 61.5º N or S), we find that months can have 28 and 31 days lengths. We determine the length of the lunar months using the Month Change Line concept, applying the extended Maunder criterion.

We call Fotheringham curves of visibility of the first lunar crescent graphs of the altitude of the center of the Moon and its difference in azimuth with the center of the Sun (represented at the moment when the center of the true Sun is on the horizon), which separates the zones lunar visibility and invisibility. These are multiparameter curves, which are dependent on astronomical and atmospheric parameters. In this investigation, we find the Fotheringham graphs deriving them from the Segura (2022b) lunar visibility theory and check their dependence on astronomical and atmospheric parameters.

The latitude of optimum viewing of the lunar crescent is the latitude for a specific meridian where it is easiest to see the lunar crescent. We show an algorithm to determine the optimum latitude, which depends on the meridian and the depression of the Sun. We draw the line of optimum viewing or line that joins the places of optimum viewing of each meridian, which is different for each lunation and depends on the depression of the Sun.

We calculate, as a function of latitude, the universal time when the visibility of the first lunar crescent begins. We verified that for the same meridian, the time of the first visibility of the crescent depends on the latitude and that the atmospheric absorption that attenuates the moonlight has little influence.

We verify that the Islamic calendar is not exclusively lunar but is also related to the movement of the Sun; for this reason, we say that the Islamic calendar has some lunisolar aspects.

We define the width of the window of visibility of the first lunar crescent as the interval of altitudes of the Moon between which we can see the crescent. We define the duration of the visibility window or time during which we see the crescent; and we also define the altitude of optimal vision of the crescent. We check the parameters on which the visibility window depends. We determine the variation of the visibility window with the phase angle, the atmospheric attenuation constant, the latitude of the observation site, and the declinations of the Sun and the Moon.

In the evening when we see the crescent of the Moon for the first time, sometime after its conjunction with the Sun, there is a time when we see the crescent more easily, which we call the best time for visibility of the crescent of the Moon. We present software that determines this best time, knowing the date, the geographic coordinates, and the atmospheric conditions.

We explain in depth the Julian and Gregorian lunisolar calendars, that is, the calendar that allow us to determine the day of Easter. We define and use the golden number, the epact, the dominical letter, the concurrent, the regular,... In the end, we expose the algorithms to find Easter Sunday,

We have developed a software called "Crescent Moon Visibility" which provides information on the visibility of the first lunar crescent under certain conditions. This software takes into account the physical laws and the capacity of human visual perception to determine whether the crescent can be seen or not. It calculates the probability of observing the crescent based on the atmospheric attenuation.

We have developed a software called "Crescent Moon Visibility" which provides information on the visibility of the first lunar crescent under certain conditions. This software takes into account the physical laws and the capacity of human visual perception to determine whether the crescent can be seen or not. It calculates the probability of observing the crescent based on the atmospheric attenuation.

Among the bi-parametric empirical criteria designed to determine when we will see the lunar crescent for the first time are multi-functional criteria, characterized by using several functions or visibility curves, which provide information about the difficulty of seeing the crescent. We show in detail six of these criteria and compare them with the criterion deduced from the Crescent Moon Visibility (from now on CMV) software, based on physical principles, which defines several visibility functions characterized by the probability of vision of the crescent. We analyze the proposals to determine the best time to view the crescent. We conclude that the best known of them, the one presented by Yallop, has an erroneous deduction. All six empirical criteria considered are compatible with CMV, and the various visibility curves of these criteria are crescent vision probability curves. From these empirical criteria, we deduce that the probability of 20% is the lowest to see the crescent. The empirical criteria analyzed are simplifications of the complex problem of visibility of the crescent of the Moon, in particular, because they do not consider the variable atmospheric attenuation.

Medieval Arab astronomers must have calculated the Julian calendar date of the first day of the Islamic calendar. We repeat this calculation using the Julian day and find the desired date is July 16, 622. We use the Islamic computational calendar developed in the Middle Ages to find this result.