Predicting the First Visibility of the Lunar Crescent

Abstract

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.

Full text

ACADEMIA Letters
Predicting the First Visibility of the Lunar Crescent
Wenceslao Segura
We observe the rst visibility of the lunar crescent on the western horizon shortly after sunset,
and it is the beginning of the month in the Islamic calendar. There are numerous techniques
for predicting the day of the rst sight of the Moon, but our interest is in the physical method,
which began with the investigations of Samaha, Assad, and Mikhail (1969) and Bruin (1977).
To predict when we will observe the rst crescent Moon, we need:
1. The topocentric altitude of the Moon; azimuth dierence between the centers of the Sun
and the Moon (DZ); Sun depression (d); Earth-Moon distance and topocentric phase
angle (or selenocentric angle between the observer’s position and the center of the Sun).
2. The luminance of the Moon without atmospheric absorption for the phase angle; (we
understand luminance as the luminous ux per unit of the luminous area perpendicular
to the observation direction and per unit of solid angle).
3. The atmospheric extinction coecient for the place of observation, which determines
the luminance of the Moon Bm at the surface of the Earth.
4. The twilight sky luminance Bs, as a function of dand DZ.
5. The threshold contrast or the threshold illuminance for viewing the Moon in the twilight
sky; (we dene the contrast by C=Bm/Bs, and the illuminance is the luminous ux that
reaches the observer per unit area normal to the observation direction).
Photometric measurements of the Moon at large phase angle are dicult since its observa-
tion has to be made at a low altitude above the horizon and therefore, is highly aected by
atmospheric attenuation; also, the observation has to be done with twilight light, therefore the
Moon’s own illumination is added to the illumination of the sky, and nally, it must be added
Academia Letters, August 2021
Corresponding Author: Wenceslao Segura, wenceslaotarifa@gmail.com
Citation: Segura, W. (2021). Predicting the First Visibility of the Lunar Crescent. Academia Letters, Article
2878.
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©2021 by the author — Open Access — Distributed under CC BY 4.0

that the Moon is rarely observed (and for a very short time) with a phase angle greater than
170º, because the brightness twilight sky masks the light emitted by the Moon.
To nd the luminance of the Moon some researchers (Sultan, 2006) (Segura, 2021) have
extrapolated the empirical formula of the magnitude of the Moon as a function of the phase
angle (Allen, 1973, p. 144). Others (Schaefer, 1991) have applied Hapke’s (1984) lunar
photometric theory. However, we do not have experimental data to conrm that these methods
give the true luminance of the Moon, which at large phase angles is highly inuenced by macro
and micro-shields and by the libration.
The extinction coecient and the luminance of the sky are easy to measure, but it is im-
possible to anticipate their value, not even one day in advance. The luminance of the sky does
not critically aect the view of the crescent, but a small variation in the extinction coecient
signicantly aects the prediction of when the Moon we will see for the rst time.
Blackwell (1946) did extensive research on the human eye’s sensitivity to see a luminous
object against a lit background. His results have been applied to the visibility of the crescent
Moon but without being correctly understood.
Blackwell’s experiment shows that threshold vision is a probabilistic process. We distin-
guish three zones of visibility of a luminous object on a bright background. The rst area is
when the contrast is very high, the probability of vision is 100%; we always see the object.
Another is the zone of zero visibility; due to the small contrast, the observer never sees the lu-
minous object. The third is the critical zone of visibility, with an intermediate contrast, where
there is a probability of seeing the object. Under the same conditions, the same observer
sometimes sees the object and sometimes not, with a certain probability given by Blackwell’s
results.
Blackwell’s experiment showed that the probability of vision in the critical zone does not
depend on the luminance of the background and is almost independent of the size of the object,
depending exclusively on the threshold contrast. Blackwell gave the threshold contrast of his
experiment for a probability of vision of 50%; that is, the observer sees the object half of the
times he observes it. With a contrast lower than the threshold, the probability of vision is
lower and vice versa.
Researchers have not noticed that Blackwell’s experiment gives: 1) the threshold contrast
as a function of image size and background luminance, and 2) the probability of vision for
the ratio C/Cth;Cis the contrast of the object and Cth the threshold contrast according to
Blackwell’s tables.
Other researchers (Knoll, Tousey & Hulburt, 1946) devised experiments to nd the thresh-
old contrast for 100% probability. Comparison of the results of the various visual sensitivity
experiments give similar results, but they are highly dependent on the experimental procedure
Academia Letters, August 2021
Corresponding Author: Wenceslao Segura, wenceslaotarifa@gmail.com
Citation: Segura, W. (2021). Predicting the First Visibility of the Lunar Crescent. Academia Letters, Article
2878.
2
©2021 by the author — Open Access — Distributed under CC BY 4.0

(pupil size, if there is an articial pupil; exposure time of the luminous object; its shape and
orientation; the proportion of its dimensions; chromaticity; visual acuity of the observer;…)
The human eye’s retina responds to the number of photons that reach it per unit area and
unit of time, that is, the retinal illuminance (Er). When the luminous object is greater than the
resolving power of the eye, which we estimate to be one arc minute, the retinal illuminance is
proportional to the object’s luminance. However, when the object has an angular size smaller
than that of the resolution, Er is proportional to the illuminance that reaches the eye’s pupil
(Segura, 2021).
For the situations of interest, the Moon has a smaller width than that of optical resolution,
which means that we must adapt the experimental results of Blackwell and other researchers
and nd the threshold illuminance, which only depends on the luminance of the background
and not on the size of the Moon.
To solve the visibility of the crescent, we use the phenomenon discovered by Danjon (1932
and 1936), according to which there is a shortening of the lunar horns that increases with the
phase angle. For this reason, we only have to analyze the visibility of the central part of
the Moon, which is the last that will be visible, and we assume to be circular and therefore
applicable the Blackwell’s results.
First we nd the luminance of the central part of the Moon according to the phase angle,
and then we nd the illuminance of the Moon (E). From Blackwell’s results, we nd the
threshold illuminance Eth knowing the luminance of the sky. Using the ratio E/Eth and the
Blackwell probability distribution, we nd the probability of seeing the crescent Moon. If
E=Eth, there is a 50% probability; if E>Eth, the probability will be higher, and lower if
E<Eth. Except for extreme situations, we will always have a probability dierent from 100%
of observing the crescent Moon, which will increase if there is more than one observer.
References
Allen, C. W. (1973). Astrophysical Quantities. University of London.
Blackwell, H. (1946). Contrast Thresholds of the Human Eye. Journal of the Optical Society
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Bruin, F. (1977). The First Visibility of the Lunar Crescent. Vistas in Astronomy,21, 331-
358.
Danjon, A. (1932). Jeunes et vieilles lunes. L’Astronomie, 46, 57-66.
Danjon, A. (1936). Le croissant lunaire. L’Astronomie, 50, 57-65.
Academia Letters, August 2021
Corresponding Author: Wenceslao Segura, wenceslaotarifa@gmail.com
Citation: Segura, W. (2021). Predicting the First Visibility of the Lunar Crescent. Academia Letters, Article
2878.
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©2021 by the author — Open Access — Distributed under CC BY 4.0

Hapke, B. (1984). Bidirectional reectance spectroscopy 3. Correction for macroscopic
roughness. Icarus 59 (1), 41-59.
Knoll, H., Tousey, R. & Hulburt, E. (1946). Visual Thresholds of Steady Point Sources of
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Segura, W. (2021). Danjon Limit: Sultan’s Method. https://www.researchgate.net/publication/
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Academia Letters, August 2021
Corresponding Author: Wenceslao Segura, wenceslaotarifa@gmail.com
Citation: Segura, W. (2021). Predicting the First Visibility of the Lunar Crescent. Academia Letters, Article
2878.
4
©2021 by the author — Open Access — Distributed under CC BY 4.0