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APEX OF THE ZONE OF FIRST VISIBILITY OF THE MOON
Apex of the zone of first visibility
of the Moon
Wenceslao Segura González
e-mail: wenceslaoseguragonzalez@yahoo.es
Independent Researcher
Abstract. 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.
1. Terrestrial terminators
We call the terrestrial terminator or isochrone of the Moon (or Sun) the line on the surface of
the Earth of places where the Moon (or Sun) is observed at the same altitude above the horizon and
at the same instant. We can refer to the true or apparent altitude; the terminators will differ in both
cases (Laoucet 2020, 128-137).
In a day a star has twice the same altitude above the horizon, once on the eastern horizon
and once on the western. That is, the line that corresponds to the terrestrial terminator has two
parts, which we call the western terrestrial terminator (when we calculate it for an altitude above
the western horizon) and the eastern terrestrial terminator (when the altitude is above the eastern
horizon).
To calculate the terrestrial terminators of the Moon or the Sun, we choose the date and
universal time, from which we find the mean sidereal time in the prime meridian
m
T
, the declination
of the center of the Moon or the Sun
or
and the right ascension
or
. Arbitrarily chosen
a geographical latitude
we find the hour angle of the Moon H (or the Sun
H
), applying the law
of cosines to the spherical triangle with vertices at the Moon (or the Sun), at the zenith, and the
celestial pole
sin sin sin
cos ,
cos cos
h
H
h is the true altitude for which we compute the terminator. From the inverse of
cos
H
, we find two
values of the argument, H and
H
, which correspond to the hour angles of the west terminator
and the east terminator. After finding the hour angle, we determine the geographic longitude by
,
m
T H
we neglect the equation of the equinoxes. We give new values to the latitude and find the
corresponding geographical longitude, with which we draw the terrestrial terminator. We must
remember that when the star is above the eastern horizon, the hour angle is between 180º and 360º,
while when it is above the western horizon, the hour angle is between 0º and 180º. It must also be
taken into account that the positive geographical longitudes are towards the west and the negative
ones towards the east.
Special terrestrial terminators correspond to the moonrise or moonset (of sunrise o sunset),
which graphically identify the areas of the Earth where these two stars are visible (or invisible).
Other special terminators are those that join the points on Earth where the first crescent of the
1
(1)
(2)
Wenceslao Segura González
Moon is observable at the same time.
In drawing 1, we represent the terminator I for the Moon (drawn in blue) for December 25,
2021, at 12 hours UT and for the true altitude of 4º. II is the solar terminator drawn in red for the
same day and time and a true altitude of 0º.
2.- Terrestrial Terminator Slope
The inclination of the terrestrial terminator with respect to the equator depends on the decli-
nation *. Differentiating (1) with respect to latitude
2
tan
sin ,
cos
dH
Hd
we assume that
0
h
and
sin 0
h
. If we assume that the declination is positive, then for the
western terminator, that is, the one whose altitude is referred to the western horizon, the hour angle
is
0º 180º
H
,
sin 0
H
, therefore
0
dH d
and by (2)
-20 -40 -60 -80 -100 -120 -140 -160
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406080100120140160
20
40
60
80
-20
-40
-60
-80
Drawing 1.- The drawing represents the projection of the Earth's entire surface on a plane. The vertical axis is
the Greenwich meridian, where we mark the geographical latitude. The horizontal line is the equator. To the
right are east longitudes and to the left are west longitudes. We have drawn the plane of the Earth between
longitudes 180º W (far left) and 180º E (far right).
The blue line (I) is the terrestrial terminator of the Moon for December 25, 2021, at 12 hours UT and for a true
altitude of 4º above the horizon. The red line (II) is the terrestrial terminator of the Sun for the same day and
time and at a true altitude of 0º.
The line joining the points on Earth where the Moon is observed at 4º above the western horizon (or western
terrestrial terminator) is the solid blue line. The continuous red line is the one that joins the points on Earth
where the Sun is observed simultaneously with the altitude of 0º above the western horizon. The other halves
of the two curves are the eastern terrestrial terminators, drawn with dashed lines, joining the points on Earth
where the Moon is observed at 4º (or the Sun at 0º) above the eastern horizon simultaneously.
To the east of the western terminator of the Moon (solid blue line) is the area where the altitude of the Moon
above the western horizon
W
h
is less than 4º. To the west of this same terminator is where the Moon is more
than 4º above the western horizon.
To the west of the eastern terminator of the Moon (dashed blue line) is the area where the Moon is observed
at an altitude
E
h
of less than 4º above the eastern horizon. While to the east of this terminator are the places
where the altitude of the Moon is greater than 4º above the eastern horizon. Similar reasoning applies to the
solar terminator.
II (Sun)
I (Moon)
Greenwich's meridian
Equator
* In our graphic representation, the east or negative longitudes are to the right; therefore, the slope of the
terminator is the trigonometric tangent of the angle between the terminator and the right part of the horizontal
axis, that is, where the longitudes are negative.
4º
W
h
4º
W
h
4º
E
h
4º
E
h
2
(3)
APEX OF THE ZONE OF FIRST VISIBILITY OF THE MOON
0,
d dH
d d
so for the west terminator, when the declination is positive, the longitude decreases when the
latitude increases; that is, the slope that the terminator forms with the equator is positive. For the
east terminator (and in the case of positive declination), we will have
360º 180º
H
and
sin 0
H
,
then
0
dH d
, and by (4)
0
d d
, therefore if latitude increases longitude also increases,
i.e., the slope respect the equator is negative. The opposite situation occurs when the declination is
negative.
In drawing 1, we have drawn with continuous lines the western terrestrial terminators of the
Sun and the Moon for 12 hours UT on December 25, 2021; that is, we have calculated them
knowing the altitude above the western horizon (0º for the Sun and 4º for the Moon, in our example).
The west terminator of the Sun has a negative slope with respect to the equator since its declination
for that day is negative; for the Moon, the western terminator has a positive slope since its declination
for the day considered is positive. We have represented the eastern terminators (with dashed
3
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40
60
80
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-40
-60
-80
20 -20 -40 -60 -80 -100 -120 -140 -160 -180
Drawing 2.- In the flat representation of the terrestrial surface, we draw two western terrestrial terminators of
the Sun. The red line (I-II) corresponds to January 25, 2022, 10h UT and
0
h
; the blue line (IV-III) is the
terminator of August 10, 2022, 13h UT, and
0
h
.
We check that the inclination of the western terminator with respect to the equator depends on the sign of the
declination. The red terminator corresponds to the declination -18º 53' 34'' (negative slope with respect to the
right part of the horizontal axis), and the blue terminator corresponds to the declination 15º 27' 51'' (positive
slope).
The geographical longitude extension of each terminator is 180°, the same extension as the eastern terminators
(not shown in the drawing), which have opposite inclinations with respect to the equator.
To the west of the terminators are areas where the altitude of the Sun above the western horizon is greater
than 0º and have lower altitudes in positions to the east of the terminators. Terminators have inflection points
at both ends. In the northernmost area of the north inflection point of the red terminator (zone I), the altitude
of the Sun will be less than 0º for some days (depending on the latitude of the observation site). Zone II, the
altitude of the Sun is greater than 0º for some days (which depends on the latitude of the observation site).
To the north of the north inflection point of the blue terminator (zone III) are the places where the Sun is seen
with an altitude greater than 0º for some days. In zone IV, the Moon will be observed with
0
h
for a few days.
The Sun will be permanently above the horizon in zones II and III, and in zones I and IV, the Sun will be
permanently below the horizon.
Greenwich Meridian
Equator
I
II
IV
III
(4)
Wenceslao Segura González
lines), those calculated for altitudes measured above the eastern horizon (0º for the Sun and 4º for
the Moon). Finally, add that the eastern terminators have opposite slopes to the western terminators.
From (1), we verify that
dH d
calculated by (3) is a function that increases when the
declination increases, a result that is independent of the value of altitude and latitude. For positive
declination and west terminator, by (4) we verify that when the declination increases,
d d
becomes more negative; the greater the latitude, the greater the variation in longitude; that is, the
slope decreases. The opposite occurs when the declination is negative.
Therefore we verify that the slope of the west terminator depends on the declination, being
greater when the declination is more smaller and vice versa. The opposite occurs for the east
terminator.
3.- Terrestrial Terminator Movement
As we show in drawing 1, the complete terminator extends over the Earth's entire surface;
that is, its extension is 360º in longitude. Each of the two terminators (western and eastern) extends
180º in longitude.
In drawing 2, we have represented as examples the terminators of the Sun for an altitude of
0º at two different moments, when the declination is negative (red line) and for positive declination
(blue line), where we verify that the inclination with respect to the equator is different, for the
reasons deduced in 1.
The terminator is a wavy line bounded by the north and south. Further north or south of these
inflection zones are places where for some time (depending on latitude), the Moon or Sun is
permanently seen at a higher or lower altitude than required to calculate the terminator, such as we
explain in drawing 2.
The terminator moves towards the west (see drawing 3). For the terminator of the Sun, the
average speed is 360º every 24 hours. For the Moon, it is a slightly slower speed due to the Moon's
faster apparent motion.
To calculate the speed of the Moon with respect to the Earth, that is, with respect to the
meridian, we take into account the length of the sidereal day or interval between two consecutive
passages of the mean equinox through the meridian h m s
( 23 56 4 .09)
and which corresponds to
the speed
4
20
406080100120140160180 -20 -40 -60 -80 -100
20
20
20
20
-20
-40
-60
-80
Drawing 3.- We show the hourly displacement of the western terrestrial terminator of the Sun. The green line
on the right is the western terminator of the Sun on November 9, 2022, at 16h UT for an altitude of -5º. The red
line is the terminator for the same day and altitude, and at 20h UT, the blue line (on the left) corresponds to 24h
UT. We observe that the Sun's terminator moves west (left) at a rate of 360º every 24 hours. The Moon's
terminator is also moving west, but with an average speed of 14.49º/h.
Equator
Greenwich
Meridian
APEX OF THE ZONE OF FIRST VISIBILITY OF THE MOON
h m s
360º
15.041º ,
23 56 4 .09
h
and the tropic month of the Moon or interval between two consecutive passages of the Moon with
respect to the mean equinox d h m s
( 27 7 43 4 .68)
, that corresponds to the speed
d h m s
360º
0.54 9 º .
27 7 43 4 .68
h
Since the movement of the Moon and the equinox have opposite directions, the average speed of
the Moon with respect to the Earth is
15.041 0.549 14.492º ,
h
therefore, on average, the terminator of the Moon makes a complete revolution with respect to the
Earth in 24h 50m.
The extreme north and south latitudes of the terminator depend on the declination. The
smaller the declination (in absolute value), the greater the slope and, therefore, the lower the extreme
latitudes of the inflection zone, as we see in drawing 4, where we represent two terminators for
declinations of 23º and 5º. In this drawing, the terminator corresponding to declination 5º has a
greater slope than that of 23º and has a greater extension in latitude.
4.- Maunder criterion
Islamic month begins on the evening when looking at the western horizon sometime after
sunset, the Moon is seen for the first time after being in conjunction with the Sun and therefore
invisible from Earth.
A fundamental problem in the Islamic calendar is theoretically anticipating the evening when
we see the first lunar crescent by observing the sky with the naked eye. Four data sets are needed
to solve this problem:
a) The astronomical coordinates of the Moon and the Sun: that is, ecliptic longitude, ecliptic latitude,
and distances Earth-Sun and Earth-Moon.
b) Geographical coordinates of the place of observation: longitude, latitude, and height.
b) Atmospheric conditions that affect light scattering in the twilight sky and atmospheric extinction.
c) The visual acuity of the observer and the observation conditions: monocular or binocular vision,
with the artificial or natural pupil; with or without optical aid; experience in observing the crescent;
5
Drawing 4- Terrestrial terminators for declinations of 23º (blue curve) and 5º (red curve). We observe that the
slope of the terminators is greater when the declination is smaller. Consequently, the inflection zones reach
greater latitude (north or south) the lower the declination.
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80120160200 -40 -80 -120 -160 -200
20
40
60
80
-20
-40
-80
-100
5º
5º
23º
23º
Equator
Greenwich
Meridian
Wenceslao Segura González
knowledge of how and where the Moon will be observed; the age of the observer; the number of
observers,...
There is another factor on which the observation of the first lunar crescent depends. The
vision is probabilistic in the critical zone of visibility, that is, at the limit of human vision. Therefore,
under the same conditions, sometimes the crescent is seen and sometimes not, with a certain
probability that we know from experiments (Blackwell, 1946).
We are interested in how the geographic position affects the vision of the crescent. Therefore,
we consider that the atmospheric conditions and the observation techniques are unchanged. In
addition, we will neglect the effect of the variation of the distances from the Earth to the Sun and
the Moon, and the height of the observer above the Earth's surface; finally, we do not consider the
probabilistic nature of the critical vision; so only four astronomical variables and two geographic
coordinates will be involved.
To predict the observation of the crescent, we need to know the luminance of the Moon
(which depends on the topocentric phase angle) and the luminance of the twilight sky, which
depends on the depression of the Sun and the horizontal coordinates of the Moon (apparent altitude
and azimuth difference with the Sun) (Segura, 2021a). From the spherical triangle with vertices at
the Sun, Moon, and zenith, we find a relationship between these four variables, so only three are
independent: the apparent altitude of the Moon's center, the difference in azimuth between the
centers of the Sun and the Moon, and the depression of the Sun below the horizon.
Then, an applicable visibility criterion in our simplified model consists of giving the extreme
values of the altitude and azimuth difference that the Moon must have for a certain depression of
the Sun for the Moon to be visible. Such a criterion was proposed by Fotheringham (1910) and
others (Fatoohi, 1998, p. 94-144). These authors gave the true altitude of the center of the Moon
for azimuth difference when the true altitude of the center of the Sun is zero. It should be noted
that the correct criterion must use the apparent altitude, not the true one.
Although this type of criteria is very simplified, we adopt it in our study to investigate the
influence of geographic coordinates on the vision of the crescent. We will use the Maunder criterion
(1911) as a function of the true altitude of the center of the Moon and the difference in azimuth
between the centers of the Moon and the Sun at the moment of geocentric sunset, that is, when
the true altitude of its center is 0º.
To know if we will see the crescent Moon, we find for the day considered the moment of
geocentric sunset, and for that time, we calculate the true altitude of the Moon and the difference
in azimuth between the Moon and the Sun, for which it is necessary to know the geographical
coordinates of the observation site. If when drawing the calculated position in drawing 5, it is
above the curve of the Maunder criterion, the crescent will be visible.
5.- Zone of first visibility of the Moon
To apply the Maunder criterion, we use drawing 5, which corresponds to January 3, 2022,
where a curve is drawn for each geographical longitude. On the vertical axis is the true altitude of
the Moon and on the horizontal axis is the difference in azimuth between the Moon and the Sun at
the time of geocentric sunset. The curves have been calculated for various latitudes (from 65º S to
65º N).
The intersection points of the curve of the Maunder criterion with each of the curves associated
with the various longitudes give the minimum altitude of the Moon in which it is still visible for the
considered longitude. In drawing 6, we represent a curve for each geographical longitude, which
relates the true altitude of the Moon and the geographical latitude. To calculate drawing 6 we use
the same procedure as for drawing 5, except that now the variables are altitude and latitude. From
this drawing 6, we find the latitude corresponding to the minimum altitude to see the crescent,
which we have previously calculated.
Knowing the latitude and longitude, we represent the line on the Earth's surface that is the
limit of vision of the lunar crescent. In table 1 are the coordinates of this line for January 3, 2022
and in drawing 7, we represent this line in red for the same day in a flat projection of the Earth's
surface.
The geographical points placed to the east of the line are the places where the crescent will
6
APEX OF THE ZONE OF FIRST VISIBILITY OF THE MOON
7
4
8
12
16
20
48 12 16 20-4-12-16
-20º -30º -40º
-50º
-60º
-65º
-10º
0º
10º
20º
30º
40º
50º
60º
65º
True altitude
of the Moon
Azimuth difference between the Moon and the Sun
Drawing 5.- The red lines are the altitude and difference of azimuth (between the Moon and the Sun) at the
moment of the true sunset, that is, when the true altitude of the center of the Sun is zero. We have calculated
the curves for January 3, 2022. Each curve corresponds to a geographical longitude. The geographical
longitude of the curves goes from 100º W, the most external, to 120º E, the most internal, in intervals of 20º.
The points are the geographic latitude for which we have calculated the altitude and azimuth difference,
ranging from 65º N (to the left) to 65º S (to the right).
4
8
12
16
10 20 30 40 50 60-10-20-30-40-50-60
Geographic latitude
True altitude
of the Moon
Drawing 6.- From the curves, we find the geographical latitude of the place of observation, knowing the true
altitude of the Moon and the geographical longitude for the moment in which the true altitude of the Sun is
0 degrees. Each curve corresponds to a geographic longitude. The outermost curve corresponds to 100º W
and the innermost to 120º E; the rest have a difference in longitude of 20º between them. We have calculated
the curves for January 3, 2022. We verify that the drawing is not symmetric concerning latitude 0º. For the
chosen date, the maximum altitudes of the Moon are in the southern latitudes.
Wenceslao Segura González
not be visible on the evening of January 3, 2022, therefore in the places of this area, on January 4,
2022, will continue the month Jumada al-Awwal.
In the points of the Earth located to the west of the red line, the crescent will be seen on
January 3, 2022; that is, for those places, January 4, 2022, will be the 1 of the new lunar month
Jumada al-Thani.
The geographic area inside the red line in drawing 7 is called the zone of first lunar visibility.
The red line that limits it is called International Lunar Date Line by Ilyas (1997, p. 109-155);
however, it seems more accurate to call it Month Change Line because it is the line that, on the
day the crescent begins to be seen for the first time, separates points that have different months.
The Month Change Line has an approximately parabolic shape; its vertex P is called the
apex, which is the position furthest to the east (right in the drawing) in which the crescent is seen
on the day in question.
6.- Apex movement
The position and shape of the Month Change Line is different for each lunation. Sometimes
the apex is in the northern hemisphere and sometimes in the southern. The latitude of the apex
depends on the difference between the declinations of Moon
and Sun
. The larger
, the
more north the apex will be, and the smaller
, the more south the apex will be.
There is proportional correspondence approximate between the geographical latitude of the
apex and
. The extreme values of the latitude of the apex depend on the lunar visibility
criteria chosen and can exceed 50º north and south, which correspond to
10º
. In drawing
9 we represent the latitude of the apex as a function of the difference between declinations of the
100
80
60
40
20
0
-20
-40
-60
-80
-100
47.2
44.6
41.9
38.8
35.4
31.1
26.5
20.9
13.8
4.1
-21.6
-65.0
-64.4
-63.4
-62.2
-60.8
-58.7
-56.2
-53.6
-49.5
-43.4
-26.0
Longitude Latitude
North South
Table 1.- We show the geographical coordinates of the Month Change Line for January 3, 2022, which joins
the points on Earth that have the geographical longitude limit for the visibility of the lunar crescent for each
latitude. The Month Change Line has two branches, one in the northern hemisphere, which corresponds to
the second column of latitude, and the third column corresponds to the southern hemisphere. The coordinates
of the table are the intersection points of the line that corresponds to the Maunder criterion, with the altitude-
azimuth difference curves that we have found in drawing 5. From this drawing, we obtain the longitude and
altitude of the Moon; with them, we deduce from drawing 6 the latitude that corresponds to each geographical
longitude for the extreme visibility of the lunar crescent. We observe that for a longitude of -100º, the
Maunder criterion curve intersects the altitude-azimuth difference curve at two points located in the southern
hemisphere, which means that the apex of the Month Change Line for the considered day is in the southern
hemisphere.
8
APEX OF THE ZONE OF FIRST VISIBILITY OF THE MOON
Drawing 7.- In red is the Month Change Line for January 3, 2022. The horizontal line is the equator, and the
vertical line is the Greenwich meridian. The blue lines are the isochrones or western terrestrial terminators of
the Sun when its altitude is -5º and for the hours of universal time indicated in the drawing. In our approximation,
the blue lines are the places on Earth where the first lunar crescent is simultaneously observed. Point P is the
apex or end of the Month Change Line, the point furthest to the east where the first crescent Moon is visible
on the date in question.
P
is the point of first lunar visibility, the place on Earth where the lunar crescent is
observed earliest. The terminator that passes through the point
P
and is tangent to the Month Change Line
is the one corresponding to the moment in which the first lunar crescent is visible before anywhere else on
Earth, which occurs at 12h 19m UT. From the drawing, we deduce that the geographic coordinates of
P
are
13º S and 94.0º E. The apex coordinates deduced from the graph are 23.8º S and 100.0ºE.
24
h
22
h
20
h
18
h
14
h
16
h
P
P
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40
-20
-40
-60
Drawing 8.- Month Change Line in the graphical representation of the MoonCal software for January 3, 2022.
The apex coordinates are 24º S and 100º E, coinciding with drawing 7. The point of first sighting is 11º S, 96º
E, something different from what we have calculated. We found in the excellent MoonCal software errors in
calculating the coordinates of the point of first vision of the first lunar crescent
9
12 19
h m
Wenceslao Segura González
Drawing 9.- The horizontal axis is the difference between the declinations of the Moon and the Sun, and the
vertical axis is the latitude of the apex of the zone of first visibility. Points represent latitude and difference of
declinations of the Moon and Sun for the apex of each lunar month of the years 2023-2025. The drawing
shows that when the Moon's declination is greater than that of the Sun
0
, the apex is in the
northern hemisphere, and the greater the difference, the further north it is. Suppose the declination of the Sun
is greater than that of the Moon
0
, then the apex is in the southern hemisphere (this rule is
sometimes broken when
is small). The straight blue line is the linear regression, roughly fitting the
points. The apex's latitudes further north or south are approximately 40º N and 40º S. However, in exceptional
conditions, they exceed 50º north or south. We have found the latitude of the apex by MoonCal software.
-1
10
1 2 3 4 5 6-2-3-4-5-6-7
20
30
40
-10
-20
-30
-40
Latitude of apex
Difference declinations of the Moon and the Sun
Moon and the Sun, and we verify that it adjusts approximately to a linear correspondence.
has a period of 27.3 days (tropical month), and
has a period of one tropical year
(approximately 365.25 days). The difference between these two periodic functions is another
periodic function with a period of roughly 354.4 days, that is, a lunar year or twelve synodic
months.
The geographical latitude of the apex varies periodically, with a period of one lunar year or
354.4 days (that is, 12 lunations), which means a sequence of six consecutive lunations with the
apex in the northern and another six consecutive lunations with the apex in the southern hemisphere.
In drawing 10, we calculate the difference between the declinations of the Moon and the Sun,
taking as reference the new Moon on January 2, 2022. We notice two periods of the variation of
the declination difference: the short period has a duration of 27.3 and corresponds to the tropic
period of the Moon, which coincides with the declination period of the Moon (Segura, 2021b), and
a long period formed by thirteen tropical months, which coincides with twelve lunations or synodic
periods.
In drawing 11 we see the two variations of the latitude of the apex. The horizontal axis of
this drawing is elapsed time. In drawing 12, we have drawn the variation of the difference in
declinations of the Sun and the Moon as a function of time, verifying that it has the same periodicity
as the latitude of the apex.
A period long of 19 solar years modulates the short periods of 354.4 days. In some lunar
years, the maximum latitudes (north or south) exceed 50º, and in others, they are barely 4º. 19
solar years is also a period of the difference in declinations of the Moon and the Sun. The 19 year
period is the Meton cycle which contains 235 lunations.
In the Meton cycle of 19 solar years (which lasts 6,939 days, that is, 16 normal years of
365 days and 4 years of 366 days because they are leap years), there are 254 tropical months of
10
APEX OF THE ZONE OF FIRST VISIBILITY OF THE MOON
40 80 120 160 200 240 280 320 360 400
-40
-20
20
40
Drawing 10.- In red we represent the variation of the difference in declinations of the Moon and the Sun. The
horizontal axis is the days counted from the moment of the new Moon on January 2, 2022, to which we give
the value 2. The vertical axis is the difference between the declinations. We observe two periods. The short
period has a duration of 27.3 days, that is, the tropic period of the Moon, and it is the period of variation of the
Moon's declination. There is another period of 354.4 days or a lunar year of thirteen tropical periods or twelve
synodic periods, whose ends we have marked with a square dots. The blue line joins the points corresponding
to 24 hours after the new Moon, which approximately coincides with the moment of observation of the first
lunar crescent. The blue line has a period of one lunar year (whose ends we have marked with an arrow), and
we verify that in six consecutive lunations, the declination difference is positive (which means that the apex
will be in the northern hemisphere), and in six consecutive lunations, the declination difference is negative;
therefore, the apex is in the southern hemisphere.
10
20
30
40
50
60
-10
-20
-30
-40
-50
-60
Drawing 11.- Variation of the latitude of the apex or vertex of the zone of first lunar visibility. On the vertical
axis is the latitude of the apex, and on the horizontal axis each mark corresponds to the null ecliptic longitude
of the Sun; therefore, there is a year between consecutive marks. The short period of apex corresponds to
twelve lunations or 354.4 days. Another period modulates the short periods and has a duration of 19 solar
years. In the drawing, we verify that there is great dispersion of the maximum latitudes of the apex in each
short period. The drawing's highest maximum latitudes in a short period are
54º
, and the lowest maximum
latitudes are
4º
. The periodicity of the drawing is a reflection of the same periodicity of the difference in
declinations of the Moon and Sun. To make the graph, we calculated the ecliptic longitude of the Sun's center
in the geographical position of the apex at the time of the apparent setting of the Sun. To determine the
geographical position of the apex, we have used the Moon Calculator software (Ahmed, 2001).
Geographical lat itude of the apex
11
Wenceslao Segura González
the Moon (each of 27.322 days). The Moon's declination period is a tropical month (Segura, 2021b),
while the Sun's declination period is a solar year (specifically a tropical year). In the 19 solar years,
there is an integer number of declination periods for the Sun (19 periods) and another integer
number of declination periods for the Moon (254 periods). The difference in the declinations of
these stars determines the geographic latitude of the apex; we verify that the latitude of the apex
has a period of 19 years because, after this time interval, the declinations of the Moon and the Sun
return to have the same values. The same happens with the difference between the two declinations.
In each lunation, the apex moves on average 191º to the west. The movement of the apex is
caused by the duration of the synodic period, which has a mean duration of 29.53 days. Therefore,
the apex will move 0.53 days in each lunation, equivalent to 191º (=0.53x360).
The displacement towards the west of the apex in each lunation, although it has an average
value of 191º, has a great dispersion, having an approximately periodic variation, of a period of just
over four hundred days (a few days longer than the 412-day period of the lunation) and dispersion
of more than two hundred degrees (Segura, 2021b).
In drawing 13, we represent the increase in longitude of the apex in two consecutive lunations.
The average value found in the 7 years analyzed is 195º, very close to 191º, which increases
longitude on average. We verify the great dispersion of the increase in the longitude of the apex,
which goes from 96º to 303º.
In drawing 14, we show the geographical longitude of the apex from the year 2023 to 2036,
which are values scattered throughout the extension of the Earth's surface; we do not observe a
correlation between the data.
7.- Isochrones
In the drawing 7, we show some blue lines, which we call isochrones, are lines on the
Earth's surface in which are the places where the first lunar crescent will be seen simultaneously.
The isochrones of 24, 22, 20, 18, 16 and 14 hours UT on January 3, 2022, are drawn. The Month
Change Line limits the isochrones since the crescent has not yet been seen outside this line.
Maunder's criterion that we are applying establishes the true altitude of the Moon at true
sunset for the crescent to be visible. However, the first lunar crescent cannot be seen at sunset; it
is necessary to wait some time for the depression of the Sun to increase and the luminance of the
twilight sky to decrease, increasing the chances of sighting the crescent of the Moon.
To simplify our calculations in the drawing of the isochrones, we assume that the depression
of the Sun is 5º then the crescent will be seen for the first time, as long as it meets the Maunder
2
4
6
8
-2
-4
-6
-8
-10
Drawing 12.- Difference in declinations of the Moon and the Sun at the time of the apparent sunset at the
apex. The vertical axis is the difference between the declinations, and the marks on the horizontal axis
correspond to the null ecliptic longitude of the Sun. We verify that there are two periods. The short period is
one lunar year and the long period is 19 solar years.
Declination difference
12
APEX OF THE ZONE OF FIRST VISIBILITY OF THE MOON
13
120
160
200
240
280
0180 0 180 0 180 0 180 0 180 0 180 0 180
Ecliptic longitude of the Sun
Apex longitude increase in
two consecutive lunati ons
Drawing 13.- On the vertical axis is the westward displacement of the longitude of the apex in each lunation.
On the horizontal axis is the ecliptic longitude of the Sun since the beginning of the year 2024. In the drawing,
we see a period of approximately 417 days and a dispersion of 207º. The average value of the displacement
towards the west is 191º. The horizontal line corresponds to the mean value of the geographical longitudes of
the apex, which for the period considered is 195º, which corresponds to a mean lunation of 29.54 days.
50
100
150
200
-50
-100
-150
-200
25 50 75 100 125 150
Geographic longi tude o f the apex
Drawing 14.- Geographic longitude of the apex of the zone of first lunar visibility. On the horizontal axis, we
number the lunations from 2023 to 2036. We observe an uncorrelated dispersion of the lunar longitude of the
apex between the limits 180º W and 180º E; that is, the apex can have any geographical longitude and with the
same probability.
criterion. Although this is not entirely exact, at least it allows us to know the influence of the
geographical coordinates in observing the first crescent of the Moon.
Due to the quasi-parabolic shape of the Month Change Line, the length of the isochrones is
smaller when closer to the apex. There is an isochrone of null longitude; therefore, it will be
tangent to the Month Change Line. The point of tangency is
P
in drawing 7, where the isochrone
corresponding to 12h 19m UT passes, which is when the crescent is seen for the first time.
Wenceslao Segura González
14
We assume that the zone of first visibility ends at the international date line or antimeridian
of Greenwich. From here begins a new zone of visibility of the crescent Moon corresponding to a
later day.
8.- Conclusions
We have defined the Month Change Line as the curve on the Earth's surface, which at the
beginning of the Islamic month, separates places that belong to different months: to the east are
the sites that continue with the previous month and to the west are the places where the new
month begins.
We call the zone of first visibility of the lunar crescent the places on Earth inside the Month
Change Line Line since in it are the places where the first crescent of the Moon is seen on the
considered date. Outside this area and for the day in question, it is impossible to see the first lunar
crescent.
In the main conclusion, we show that the line separating the places where the new Islamic
month begins from those other places where they continue with the previous month is a complex
line that varies each lunation and depends on the equatorial coordinates of the Moon and the Sun.
The Month Change Line has a quasi-parabolic shape, with its end directed towards the east,
where the apex is located, which is the most easterly place where the first crescent of the Moon is
seen on the considered date. There is a new apex at each lunation, which shifts in both latitude and
longitude with respect to the apex of the previous month. While the displacement of the apex in
latitude is relatively small, the variation in longitude of the apex from two consecutive lunations is
considerable.
It is interesting to point out that even though the Islamic calendar is lunar, the latitude of the
apex has a period of 19 years, corresponding to the Meton cycle, which is characteristic of lunisolar
calendars. The Islamic calendar is not purely lunar because the Sun also regulates it.
There is a point on Earth's surface where the first lunar crescent is seen before any other
place on Earth and does not coincide with the apex, although both points are close and are located
at the eastern end of the Month Change Line. We call this place on Earth, which is different for
each lunation, the point of first visibility of the first crescent of the Moon. Its position with respect
to the apex varies, mainly depending on the difference in declination between the Moon and the
Sun. Sometimes the point of first visibility of the first lunar crescent is to the north of the apex and,
at other times to the south.
We have calculated the terrestrial lines where the crescent is observed simultaneously and
that are parts of the terrestrial terminator of the Sun in the simplified theory that we have developed.
They are complex lines whose slope depends on the declination of the Sun. We call these lines
isochrones, and each of them corresponds to a particular moment.
For all that we have said, we verify that the observer's geographical position determines the
moment in which he will see the first crescent of the Moon; in addition, this geographical dependency
changes in a complex way, depending on the equatorial coordinates of the Sun and the Moon.
An untreated issue is the observation of the first lunar crescent at high latitudes. In these
places, we cannot apply the traditional rule of the beginning of the Islamic month, coinciding with
the first lunar crescent observation.
9.- Bibliography
1.- Ahmed, M. (2001). Moon Calculator (MoonCalc). Version 6.0.
2.- Blackwell, H. R. (1946). Contrast Thresholds of the Human Eye. Journal of the Optical
Society of America, 36 (1), 624-643.
3.- Fatoohi, Louay J. (1998). First visibility of the lunar crescent and other problems in historical
astronomy. Doctoral theses, Durham University.
4.- Fotheringham, J. K. (1910). On the smallest visible phase of the Moon. Monthly Notices of the
Royal Astronomical Society, 70, 527-531.
5.- Ilyas, M. (1997). Astronomy of Islamic Calendar. A. S. Noordeen.
6.- Laoucet Ayari, M. (2020). Sighting the Month.
APEX OF THE ZONE OF FIRST VISIBILITY OF THE MOON
15
6.- Maunder, E. W. (1911). On the Smallest Visible Phase of the Moon. The Journal of the British
Astronomical Association, 21, 355-362.
7.- Segura, W. (2021a). Predicting the First Visibility of the Lunar Crescent. Academia Letters,
Article 2828.
8.- Segura, W. (2 021b) . Periods of the Moon. https://www.a cademia.edu/50779982 /
Periods_of_the_Moon.