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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.

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 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.

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.

We expose and analyze the proposed models of the visual magnitude of the Moon for large phase angles (>150Âº). 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.

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

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.

For the central zone of the Earth (approximately 50ÂºN-50ÂºS), 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 (50Âº N or S to 61.5Âº 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 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.

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 RiccÃ²'s law, and we consider the atmospheric extinction coefficient to be variable. We use the theory to derive the Danjon limit.

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 Â«perhapsÂ» Hapke's theory overestimates the shielding of the sun's rays by the irregularities of the lunar surface at large phase angles.

Brief report on how to calculate the first visibility of the Moon crescent. We warn against the misinterpretation of Blackwell's threshold visibility experiment. We state that the width of the first lunar crescent is less than the resolving power of the human eye, so the determining factor for visibility is the illuminance of the Moon and not its brightness.

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.

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.

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.

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 be seen with the naked eye at 5.6Âº away from the Sun in exceptional conditions. With a more realistic calculation, we find 7.1Âº for the Danjon limit. We show that this limit angle is highly dependent on atmospheric absorption, varying significantly when the extinction coefficient is modified. We find a height above the horizon at which it is easier to observe the Moon crescent, which depends exclusively 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 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 and topocentric points of view.

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

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

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 possible intercalation criteria. 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.