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LENGTH OF ISLAMIC MONTHS
Length of Islamic Months
Wenceslao Segura González
email: wenceslaoseguragonzalez@yahoo.es
Independent Researcher
Abstract. For the central zone of the Earth (approximately 50ºN50ºS), 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.
1. Introduction
The month of the Islamic calendar begins when the Moon is observed for the first time after
being in conjunction with the Sun. The Islamic month is regulated by the synodic period of the
Moon, or the time between two consecutive conjunctions of Moon and Sun. The mean duration of
the synodic month or lunation is 29.53 days, with a small oscillation (Segura, 2021a).
The observation of the first crescent lunar depends on the observer's geographical position.
The month begins in some positions on Earth on a day, and in other places, it begins on another day.
For geographical positions in the central zone of the Earth, Islamic months can only have
lengths of 29 and 30 days. On average, the duration of the Islamic month must coincide approximately
with the synodic period; there are almost the same months of 29 as 30 days. For higher latitudes,
Islamic months may have 28 y 31 days.
A particular situation occurs when we observe in very high latitudes (either north or south)
because, in these regions, it may happen that there is no moonset, either because the Moon is
permanently below the horizon or above it. Therefore, the observation criterion of the first lunar
crescent used to know the beginning of the month is not applicable.
2. Maunder Criterion
There are several empirical criteria to determine the moment of visibility of the first lunar
crescent. Among them are the criteria that use the altitude of the Moon's center and the difference
in azimuth between the Moon and Sun. The Maunder (1911) criterion that we use for simplicity,
belongs to this group.
These criteria are not satisfactory, mainly because they consider that atmospheric conditions
are the same everywhere. However, atmospheric factors affect the observation of the first lunar
crescent, mainly atmospheric absorption, which attenuates the light intensity coming from the Moon
and has substantial variability, even in the same place and at intervals of only a few hours.
Maunder's criterion (table 1) gives the true altitude of the center of the Moon h to see the
crescent according to the azimuth difference
A
between the Moon and the Sun at the time of the
true sunset
0
h
.
In drawing 1, we show the Maunder criterion and divide the plane
h A
into four zones. If
the altitude of the Moon is greater than 11º (zone A), the Moon will be visible. Zone B is limited by
the Maunder curve (continuous blue line) and the altitudes 11º and 6º. If the Moon is in zone B, it
will be visible, regardless of the azimuth difference. Zone C is between the Maunder curve and the
azimuth differences +20º and 20º. If the Moon is in C it will be invisible.
1
Wenceslao Segura González
2
0º
5º / 5º
10º / 10º
15º / 15º
20º / 20º
11.0º
10.5º
9.5º
8.0º
6.0º
Difference of
azimuth between
the centers of the
Moon and the Sun
Minimum true
altitude
center of the Moon
to be visible when
the true altitude
of the Sun is zero
Table 1. Maunder lunar visibility criterion. The second column is the minimum true or geocentric altitude of
the Moon to be visible with the azimuth difference of the first column when the true altitude of the center of
the Sun is 0º.
Drawing 1. Interpretation of the Maunder criterion. With a continuous blue line, we represent the curve
corresponding to table 1. The vertical axis is the true altitude of the center of the Moon. The horizontal axis
is the difference in azimuth between the centers of the Moon and the Sun, calculated at the time of true
sunset. That is when the true altitude of the center of the Sun is zero. The crescent will be seen if the Moon
is in zone A, regardless of the azimuth difference.The Moon will be visible if its true altitude is greater than 11º.
The solid blue curve bounds zone B, and if the coordinates of the Moon are in that zone, the Moon will be
visible. Zone C corresponds to the invisibility of the Moon. Finally, zone D is undetermined since the
Maunder criterion gives values up to
20º
of azimuth difference. Therefore, we do not know the extrapolation
of the Maunder curve (blue line) in zone D.
A
A
B
B
C
C
D
D
Azimuth difference between the Moon and the Sun
True altitude of the Moon
When
A
increases, the arclight (or arc of separation between Moon and Sun) and phase
angle (or selenocentric angle between the centers of the Earth and the Sun) also increase; that is,
the phase of the light increases. Moon, therefore, increases its illuminance, increasing the chances
of seeing the crescent. In addition, the higher the
A
, the lower the luminance of the twilight sky,,
which favors the vision of the first crescent. In zone D, the crescent can be seen at a lower
altitude; therefore, the extrapolation of the Maunder curve in zone D does not exceed 6º. However,
we cannot determine the extrapolation exactly with the criterion data (table 1).
2
4
6
8
10
12
14
16
5101520253035 5 10 15 20 25 30 35
LENGTH OF ISLAMIC MONTHS
3. Limit of the Maunder criterion
The trajectories that the Sun and the Moon follow in their daily movement are approximately
a celestial parallel. The variation of the declination of the Moon is on average approximately
5'/ hour. The time that interests us in the moonset is around half an hour, that is to say, that in that
time the declination of the Moon varies approximately 2.5 minutes of arc. Since we don't need
much precision, we consider that the Moon follows a parallel while approaching the horizon.
For high latitudes (north or south), the inclination of the celestial parallels with the horizon
decreases. Therefore, for high latitudes, the difference in azimuth between the Moon and the Sun
increases at the time of observation of the crescent, so the position of the Moon may be in zone D
of drawing 1, where we cannot apply the Maunder criterion, which requires generalization.
Next we calculate the geographical latitude at which we can still apply the Maunder criterion.
The angle formed by the equator with the horizon is 90
(
is the geographical latitude),
with a negative slope in the northern hemisphere and positive in the southern hemisphere, as seen
by the observer located on the surface of the Earth. Nevertheless, the angle that the parallels form
with the horizon depends on the latitude of the place and the declination of the parallel.
From the position triangle with vertices at the zenith, celestial pole and sky point we deduce
by the cosine theorem
sin sin sin
cos cos cos
sin sin sin cos cos cos
sin cos
sin cos
h
Ah
h H
H
Ah
A is the azimuth (measured from south retrograde), h is the true altitude,
is the declination, and H
is the hour angle.
0
h
corresponds to the cutoff point of the parallel (of declination
) with the
west horizon, the azimuth of this point is
sin
cos .
cos
A
If the declination is positive,
A
is greater than 90º; that is, the cutoff point is in the quadrant WN;
if the declination is negative, then
A
is less than 90º, and the cutoff point is in the quadrant WS.
For a point of altitude close to zero and declination
,
A A
, then we develop A in series of
powers, and we only take the first two terms of the expansion; that is, we consider the linear
approximation
cos cos sin ,
A A A A A
since h is small then sin
h h
and
cos 1
h
, then by the first equation (1)
sin
cos tan
cos
A h
from (2), (3) and (4) we find in the linear approximation
sin sin
tan
sin tan tan
A A
A h A h A A
A
that corresponds to a straight line, with a negative or positive slope depending on whether the
latitude of the place of observation is north or south, respectively. The angle
of the parallel with
the horizon is
sin
tan .
tan
A
The parallel is steeper when the observation is further north or further south.
Applying (5) to the Moon and the Sun at the true sunset
0
h
we find (Ayari, 2020, 221
224)
tan
.
sin
A A A A A h
A
To calculate the declinations of the Moon
(or the Sun
), we apply the law of sines to the
3
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Wenceslao Segura González
90 180 270 360
4.9
5.0
5.1
4.2
5.3
Ecliptic longitude at the conjunction
Drawing 2. Maximum differences in the absolute value of the declinations of the Sun and the Moon at the
time of conjunction, calculated when the lunar ecliptic latitude has the highest absolute value. When the
lunar latitude is positive, the declination of the Moon is greater than that of the Sun, and the opposite when
the latitude of the Moon has the most considerable negative value. The greatest differences occur at the
solstices and the smallest differences at the equinoxes. For other ecliptic latitudes of the Moon
it will
be less than shown in the drawing. The lowest value of
corresponds to the lowest absolute value of
the latitude of the Moon, that is when
0
, then
0
.
spherical triangle with vertices at the ecliptic pole, the celestial pole, and the Moon (or the Sun). We
calculate for the conjunction (when Moon and Sun have the same ecliptic longitude) and at the true
sunset of the Sun, finding
sin cos sin sin cos sin
sin sin sin
L
L
is the obliquity of the ecliptic (approximately 23º 26');
is the ecliptic latitude of the Moon
(extreme values
5º 7
), and L is the ecliptic longitude, the same for the Sun and the Moon because
they are in conjunction.
From the formulas (8), we find drawing 2, where we verify that the maximum difference
between the declinations of the Moon and the Sun at the conjunction is approximately 5º when the
lunar latitude is maximum. For smaller values of
, the difference of the declinations is smaller.
Approximately, the declination of the Moon increases by 2º every day; that is to say, one day
after the conjunction (when it is already possible to see the lunar crescent), the maximum value of
the difference of the declinations is approximately 7º.
For the following calculation, we must bear in mind that when
by formula (7)
A
gives higher values than when
for the northern hemisphere, and the opposite occurs in the
southern hemisphere. Therefore, 7º is approximately the maximum difference between the
declinations of the Moon and Sun at crescent observation, and we do the calculation for the northern
hemisphere.
Maunder's criterion is applicable for any value of h and
A
except for zone D of drawing 1.
This zone is characterized by meeting
6º
h
and
20º
A . With the formula (7), and estimation
of the maximum value of the difference of the declinations calculated above, we verify that the
maximum value of the geographical latitude for which the Moon cannot be found in zone D is
4
(8)
LENGTH OF ISLAMIC MONTHS
approximately 50º (Reynold, 1939).
We found the previous value for the extreme conditions of
23.44º
and
7 16.44º
12.08 1.32 .
A h
In drawing 3, we show the Maunder criterion and the lines
h A
of the Moon for various
latitudes. With a continuous red line, we draw the line (9). We check that it corresponds to the
maximum geographical latitude for which the path of the Moon towards the horizon does not pass
through zone D. The other lines that appear correspond to the latitudes of 60º (dashed line) and 30º
(dotted line) and with the same declinations.
The above calculation is approximate, applicable in extreme situations, and assumes a rectilinear
path of the Moon near the horizon; the Maunder criterion can usually be applied in most cases to
latitudes greater than 50º. However, we cannot ensure that there are positions of the Moon in zone
D (drawing 1), where we do not know the extrapolation of the Maunder curve.
4. Length of the months according to their position concerning the Month Change Line
To find the length of the Islamic months, we divide the Earth's surface into three zones: the
central zone, which is approximately between latitudes 50º N and 50º S; two intermediate zones
that are between 50º north or south to 61.5º north or south and the polar zones from 61.5º north or
south to the poles.
For latitudes higher than 61.5º (north or south), there may be no moonset (see appendix). We
omit the polar zone because knowing the length of the months observed from these positions
requires a modification of the criterion to establish the beginning of the month.
Limiting ourselves to the central zone, we show in table 2, as an example, that the same
month of the Islamic calendar can have 29 or 30 days depending on the geographical position of the
place of observation. In table 2, we give the length of the months of the year 1442 Hijra for the
positions of Greenwich, Jakarta, and Mexico. In the same table, we see that the year's duration
also depends on the geographical position; Greenwich and Jakarta have 355 days, and it is 354 days
for Mexico. We have found table 2 with Maunder's criterion.
Schaefer (1992), using the physical method developed in the LunarCal software (Schaefer,
1990), reaches the same conclusion: the months are 29 or 30 days for observers located in what we
have called the central zone.
Drawing 3. We represent the Maunder criterion with a continuous blue line. The straight lines are the paths
followed by the Moon in its movement towards the moonset. We verify that latitude 50º N (or 50º S) is the
maximum for the Moon not to pass through zone D. The other lines correspond to other latitudes. We have
calculated all the paths of the Moon for the extreme declinations
23.44º
and
16.44º
.
5
True altitude of the Moon
Azimuth difference between the Moon and the Sun
40º
50º
50º
60º
5
10
15
510152025303540 5 10 15 20 25 30
D
D
(9)
Wenceslao Segura González
20
50
100
40
60
Equator
Greenwich’s
meridian
Drawing 3. The red curve is the Month Change Line for January 3, 2022, drawn on a flat projection of the
Earth's surface.The vertical axis is the Greenwich meridian, with the scale in latitudes and the horizontal axis
is the equator, with the scale in longitudes. The places on Earth inside the Month Change Line see the first
lunar crescent on the evening of January 3, 2022; therefore, January 4, 2022, will be the first day of the new
Islamic month. Places east of the Month Change Line fails to see the crescent Moon on the evening of
January 3, 2022, therefore January 4th will still belong to the previous month.The Line of Change of Month
extends from the apex in the east to the antimeridian of Greenwich (180º longitude) in the west.
For the day examined, the apex is in the southern hemisphere. We verify that the two branches of the Line of
Change of the Month (in the northern and southern hemispheres) are not symmetrical.
6
Muharram
Safar
Rabi AlAwal
Rabi AlThani
Jumad AlUla
Jumad AlThani
Rajab
Shaban
Ramadhan
Shawwal
ZulQida
ZulHijja
Greenwich Jakarta Mexico
30
30
29
30
29
29
30
29
30
29
30
30
30
29
30
29
30
29
30
29
30
30
29
30
29
30
29
30
29
30
29
30
29
30
29
30
Table 2. The months of the Islamic year 1442 for Greenwich (coordinates 51.5º N, 0º), Jakarta (6.2º S, 106.8º E),
and Mexico (19.5º N, 99.1º W). We verify that the duration of the months depends on the observer's geographical
position. To determine the month's first day, we use the Maunder criterion. We note that the length of the
Islamic year for Greenwich and Jakarta is 355 days but 354 days for Mexico; therefore, the length of the year
of the Islamic calendar depends on the place of observation. We have obtained the results from the MoonCal
software (Ahmed, 2001).
20
40
60
150
180 50 100
Greenwich’s
antimeridian
Apex
LENGTH OF ISLAMIC MONTHS
1
1
2
2
29 day month 29 day month
30 day month
30 day month 30 day month
Drawing 4. We represent two consecutive Month Change Lines in red. Number 1 is for the previous month,
and number 2 is for the month after. The two Month Change Lines divide the Earth's surface into three zones,
where observers will have months of the indicated durations. We verify that for all Islamic months observed
from the central zone of the Earth, there are places where the duration is 29 days and others where the
duration is 30 days.
We warn that this rule may not be fulfilled for places at high latitudes; this will occur if the observer far to the
north or far south sees the crescent two or more days after the earliest sight.
7
C
C’ B’ A’
To anticipate the duration of an Islamic month observed from a particular geographical position,
we use of the concept of the Month Change Line (Segura, 2022), which is the curve drawn on the
surface of the Earth for the places where the first lunar crescent is seen before any other place on
the same terrestrial parallel. The Month Change Line has a quasiparabolic shape, whose vertex
we call the apex. In drawing 3, we represent the Month Change Line for January 3, 2022.
The area of the Earth inside the Month Change Line is the zone of first lunar visibility. There
are the places on Earth where the evening of the considered day, the crescent of the Moon is seen
for the first time and therefore it is the terrestrial area where the new Islamic month begins.
However, places east of the Month Change Line will continue with the previous month.
The zone of first lunar visibility extends from the apex in the east to the Greenwich antimeridian
in the west. The Month Change Line is different each month, changing the extension in geographical
longitude of the zone of first lunar visibility.
In each lunation, the apex (and the entire Month Change Line) moves on average 191º to the
west
0.53 360
, although there is a large dispersion (Segura, 2022). The same relative positions
of the Moon and the Sun occur every 29.53 days approximately. During this time, the Earth rotates
29 times on its axis, plus the arc covered in the remaining 0.53 days.
As a result of the shift of the Month Change Line to the west, on some occasions, it is to the
west of the previous Line and on other occasions, it is to the east, as shown in drawing 4. The
BA
29 day month
Wenceslao Segura González
8
Month Change Line 1 is the one with which the month begins, and Line 2 is the one with which it
ends. In table 3, we deduce the duration of the months according to the observer's geographical
position in drawing 4.
The two Month Change Lines of consecutive lunations divide the Earth's surface into three
zones as far as the month's duration is concerned, as explained in drawing 4. We verify that every
Islamic month has two durations, 29 and 30 days, depending on the place of observation.
The duration of the months according to their position concerning the two Month Change
Lines are deduced bearing in mind that the months begin with a difference of one day for observers
in geographical positions located on both sides of the Month Change Line; and that the months end
on the same day if they are in the same side of the Month Change Line.
We conclude that for the terrestrial zone between latitudes 50º N50º S (what we have called
the central zone of the Earth), the months can only have durations of 29 and 30 days.
5. Extended Maunder Criterion
In the plane
h A
(drawing 1) are zones D, where we do not know how to apply the
Maunder criterion, which according to table 1, gives the limit of altitude of the Moon up to 20º of
separation from the Sun.
We show that at high latitudes, the inclination of the parallels decreases, which causes the
difference in azimuth between the Moon and the Sun to increase. We show that from 50º latitude
(north or south), the difference in azimuth may be greater than 20º, and the altitude of the Moon is
less than 6º; that is, the Moon would be in zone D of drawing 1. Therefore, it is necessary to extend
the Maunder criterion to apply it to latitudes greater than 50º, which we have called the intermediate
zone of the Earth's surface.
When the azimuth difference increases, the phase angle and, therefore, the illuminance of
the Moon increases; furthermore, increasing A decreases the luminance of the twilight sky;
therefore, with the increase in A, the altitude of the Moon in which it is possible to see its first
crescent decreases.
Another factor in observing the first lunar crescent is atmospheric attenuation, which increases
Begins of the month
A B C

1


28
29
1
2


29
30
1
2


29

End of the month
A’ B’ C’

1


29
30

1


29

1
2


30

Table 3. The first table corresponds to the first of the graphs in drawing 4. The points on the Earth located in
zones B and C began the month on the same day since they are in the same zone concerning the Month
Change Line 1. However, zone A begins he month one day later. In other words, day 1 of zone A coincides with
day 2 of zones B and C, as shown in the table. Zones A and B end the month on the same day since the two
zones are east of the Month Change Line 2. The month in B must necessarily have a duration of 30 days;
otherwise, in zone A, the length of the month would have been 28 days, which is impossible for geographical
latitudes of the central terrestrial zone defined above. Therefore, in zone A, the month has a duration of 29
days. Zone C ends the month one day before zones A and B because it is to the west of the Month Change
Line 2, and its duration is 29 days.
The second table corresponds to the second graph of drawing 4. The reasoning is similar to the first table,
deducing the month's duration according to the Earth's area divided by the two Monthly Change Lines.
Day of the month


LENGTH OF ISLAMIC MONTHS
considerably when the Moon is near the horizon. There is an altitude limit from an azimuth difference
for which it is impossible to see the Moon due to high atmospheric absorption.
Using the results of Ilyas (1988), we assume that from
35º
A
the minimum altitude for
viewing the crescent is 4º. Joining this assumption with the Maunder criterion of table 1, we find the
curve of drawing 5, which corresponds to the extended Maunder criterion, which is somewhat
different from the one deduced by Ilyas *. The curve is found by interpolating with a 1D cubic
spline. Table 5 shows the numerical values of the extended Maunder criterion.
5. Length of the Islamic months in the intermediate zone of the Earth
The Month Change Line continues beyond the Greenwich antimeridian, extending for another
360º of longitude. We call its interior the second zone of first lunar visibility because in it are the
places that will see the crescent of the Moon one day after the first vision, thus beginning the new
month. Places on Earth that are outside this second zone of first visibility will continue into the
previous month and see the crescent two or more days after the first vision, as a result of which the
months could have 28 or 31 days.
We limit ourselves now to the terrestrial areas between latitudes 50º61.5º (north or south),
which we have called the intermediate zone, where the vision of the lunar crescent may be two or
more days after the first vision of the Moon.
If the delay is at the end of the month, then the length of that month may be 31 days.
However, if the delay occurs at the beginning of the month, the month may have 28 days.
In table 6, we have put, as an example, the duration of two months, one of 28 and the other
of 31 days, observed at the latitude of 55º N and calculated by the extended Maunder criterion.
6. Conclusions
For the central zone of the Earth, we find that the duration of the Islamic months, those that
begin with the observation of the first lunar crescent, have 29 or 30 days. The length of the months
depends on the geographical position of the place of observation. Also, all months have 29 days for
some places and 30 days for others. We find the months according to the geographical position
9
2
4
6
8
10
12
102030405060 10 20 30 40 50 60
True height
of the Moon
Azimuth difference between the Moon and the Sun
Drawing 5. Extended Maunder Criterion. Maunder's original criterion applies up to azimuth differences
20º
(dashed vertical lines). Following Ilyas, we assume that from
35º
A
, the minimum altitude for viewing the
Moon is 4º. If the Moon is above the curve of the drawing, it will be visible and invisible if it is below the
curve.
* To find his extrapolation Ilyas used the physical theory of Bruin (1977) which is excessively simplified
(Segura, 2021b), so his results are doubtful. Even so, we use it for simplicity since we are interested in
knowing the behavior of the Islamic calendar at high altitudes rather than the exact results. Fatoohi (1988, 111
114) considers that the minimum altitude of the Moon to see the crescent must be greater than the 4º that Ilyas
assumes.
Wenceslao Segura González
Begins of the month 27July2006 26July2006
Table 6. Begins and ends of Rajab 1427 AH and Safar 1505 AH for an observer located at 55º N and 0º
longitude. The crescent observation is the previous day's evening that appears on the table. In the fourth
column is the true altitude of the Moon and its difference in azimuth with the Sun at true sunset. To determine
the beginning of the month of Rajab and the end of the month of Safar, we apply the Maunder criterion.
Moreover, to find the end of Rajab and the beginning of the month of Safar, we apply the extended Maunder
criterion. In the last column is the day the month begins and ends earliest; for its calculation, we use the
Maunder criterion.
The month of Rajab ends two days after the first sight of the crescent; as a result, this month at latitude 55ºN
has a duration of 31 days. The month of Safar begins three days after the first vision of the crescent, and thus
the month has 28 days.
Vision of the
crescent
Coordinates
lunars to the
sunse t
On the first day the
Moon is visible
10
7º 52
17º 6
h
A
End of the month 27August2006 5º 40
32º 3
h
A
25August2006
Rajab 1427 AH
Safar 1505 AH
Begins of the month 5November2081 2November2081
4º 15
25º 53
h
A
End of the month 3December2081 2December2081
5º 21
46º 31
h
A
0º
5º / 5º
10º / 10º
15º / 15º
20º / 20º
25º / 25º
30º / 30º
35º / 35º
>35º / <35º
11.0º
10.5º
9.5º
8.0º
6.0º
5.2º
4.5º
4º
4º
Difference of
azimuth between the
centers of the Sun
and the Moon
Minimum true
altitude center of
the Moon to be
visible when the
true altitude of the
Sun is zero
Table 5. Extended Maunder visibility criteria. Up to the difference in azimuth
20º
, the values are the same
as in table 1. The following values correspond to the extension of the Maunder criterion, constructed by
assuming that from the difference in azimuth
35º
, the minimum altitude of the Moon to be visible is 4º. The
latest values are found by interpolating the data from the Maunder criterion (table 1) with the condition that
from
35º
of azimuth difference, the minimum altitude of the Moon to see the crescent is 4º.
LENGTH OF ISLAMIC MONTHS
concerning the Lines of Change of the Month.
We show that the Maunder criterion, which we have chosen to simplify our investigation,
applies up to approximately
50º
latitude. At higher latitudes, the azimuth differences between the
Moon and the Sun can be greater than 20º, and extending the Maunder criterion is necessary.
Following some conclusions of Ilyas, we consider that from the azimuth difference of
35º
,
the minimum altitude to see the crescent is 4º. With this result, we obtain the extended Maunder
criterion applicable for high latitudes.
When the geographic latitude is greater than 50º, the day the month begins or ends, maybe
two or more days behind the day the crescent was first observed. As a result, lunar months at high
latitudes can have 28 or 31 days (Huber, 1982).
Our investigation extends up to
61.5º
because from here, there may be no moonset, a
circumstance that would require changing the criteria to determine when the beginning of the new
month occurs.
7. Appendix: Declinations for which there are no sunsets and moonsets
There are latitudes for which sometimes there is no sunset. From the spherical triangle of
vertices the celestial pole, the Sun, and the zenith, we find
sin sin sin
cos cos cos
h
H
H
is the hour angle,
h
is the true altitude above the horizon,
is latitude, and
is declination. If
H
is greater than 1 or less than 1 then there is no sunset. We distinguish four possibilities,
according to the sign of the declination and if we observe in the northern or southern hemisphere.
a) Positive declination and observation in the northern hemisphere. There are latitudes for
which the Sun is always above the horizon, at least for one day; therefore, there is no sunset, this
occurs when
180º
H
and
cos 1
H
, then by (10)
sin sin sin
1 cos sin 50 89º 10
cos cos
h
50
is the true altitude of the upper limb of the Sun at apparent sunset (Segura, 2018, 78).
b) Positive declination and observation in the southern hemisphere. There are latitudes for
which the Sun is always below the horizon, at least for one day, and therefore there is no sunset,
this occurs when
0
H
and
cos 1
H
, then by (10 )
sin sin sin
1 cos sin 50 90º 50
cos cos
h
of the two signs of the inverse of the cosine we have taken the negative.
c) Negative declination and observation in the northern hemisphere. There are latitudes for
which the Sun is always below the horizon. The latitude limit occurs when
0
H
then
sin sin sin
1 cos sin 50 90º 50 .
cos cos
h
d) Negative declination and observation in the southern hemisphere. There are latitudes for
which the Sun is permanently above the horizon, which occurs when
180º
H
, therefore
sin sin sin
1 cos sin 50 89º 10
cos cos
h
we have taken the negative sign of the inverse of the cosine.
The limits of latitudes when declination is positive for which there is no sunset
are:
89º 10 0 65º 43 48
(the Sun above the horizon and
is the obliquity of the ecliptic or
maximum value of the declination of the Sun) and
90º 50 67º 23 48
(the Sun is below
the horizon). And the limits when declination is negative are:
90º 50 67º 23 48
(the Sun is
below the horizon) y
89º 10 0 65º 43 48
(the Sun above the horizon).
We apply the same reasoning to the Moon (Segura, 2018, p. 211212), in which we find that
there will be no moorise or moonset from the latitude
when
sin 0, 7275 0º 34 sin sin
1cos cos
11
(10)
Wenceslao Segura González
is the equatorial horizontal parallax and
0,7275 0º 34
is the true altitude of the upper limb of
the Moon at moonset; from the above formula, we deduce the limits of the latitude for which there
will be no moonsets. If the declination is positive the limits are: 90º 7 30
(Moon over the
horizon) and 89º 52 30
(Moon below the horizon). If the declination is negative the extremes
are 89º 52 30
(Moon below the horizon) y 90º 7 30
(Moon above the horizon).
The extreme of these limits in the assumption of mean distance between the Earth and the
Moon are:
61º 24 24
y
61º 9 24
for maximum positive declination and
61º 9 24
y

61º 24 24
for maximum negative declination.
8. Bibliography
1. Ahmed, M. (2001). Moon Calculator (MoonCalc). Version 6.0.
2. Ayari, L. (2020). Sighting the Month.
3. Bruin, F. (1977): The First Visibility of the Lunar Crescent, Vistas in Astronomy 21, 331358.
4. Fatoohi, L. J. (1998). First visibility of the lunar crescent and other problems in historical astronomy.
Doctoral theses, Durham University.
5. Huber, P. J. 1982. Astronomical dating of Babylon I and Ur III. Occasional papers of the
Near East 1, 107199.
6. Ilyas, M. (1988). Limiting altitude separation in the new Moon’s first visibility criterion, Astronomy
and Astrophysics 206, 133135.
7. Maunder, E. W. (1911). On the Smallest Visible Phase of the Moon. The Journal of the British
Astronomical Association 21, 355362.
8. Reynold, J. H. 1939. The Mohammedan calendar and the first visibility of the new moon in
Egypt. Royal Astronomical Society. Occasional Notes 1(7), 100101.
9. Schaefer, B. (1987). An Algorithm for Predicting the Visibility of the Lunar Crescent in Pro
ceedings of the IIIT Lunar Calendar Conference, The International Institute of Islamic Thourght,
10:110:10.
10. Schaefer, B. (1990). LunarCal. Western Research.
11. Schaefer, B. (1992). The Length of the Lunar Month. Archaeoastronomy 17, 3242.
12. Segura, W. (2018). Movimientos de la Luna y el Sol. EWT Ediciones.
13. Segura, W. (2021a). Periods of the Moon. https://www.academia.edu/5077 9982/
Periods_of_the_Moon.
14. Segura, W. (2021b). Danjon Limit: Bruin’s Method, https://www.researchgate.net/publica
tion/356128660_Danjon_Limit_Bruin's_Method
15. Segura, W. (2022). Apex of the zone of first visibilidty of the Moon. https://www.academia.edu/
78415798/Apex_of_the_zone_of_first_visibility_of_the_Moon.
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