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Abstract

We show that in high geographic latitudes (approximately > 50º north or south), the lunar months of 28 and 31 days are possible.
Abstract.- We show that in high geographic latitudes (approximately > 50º north
or south), the lunar months of 28 and 31 days are possible.
1.- Lunar calendar
The lunar calendar is regulated by the conjunction of the Moon and the Sun; more specifically,
by the synodic period of the Moon, of 29.53 days of average duration. This calendar is not strictly
lunar, in the sense that it is not regulated exclusively by the Moon but also by the movement of the
Sun (Segura, 2022a).
The Islamic calendar is an observational lunar calendar; that is to say, the month begins
when on the western horizon, shortly after sunset, the lunar crescent is first seen after the conjunction
of the Moon with the Sun.
In the central zone of the Earth, between latitudes
50º
approximately (Segura, 2022b),
the duration of the lunar months can only be 29 or 30 days (Schaefer, 1992); this is because, in this
central area of the Earth, the crescent is seen on two consecutive days; that is to say, that there is
one day between the first and last vision of the crescent. For each lunation, in some areas of the
Earth, the month has 29 days and in others 30 days (Segura, 2022b)
Neither the Babylonian astronomers nor the Arab astronomers of the Middle Ages, nor
even the research on the vision of the lunar crescent developed since the beginning of the last
century, paid attention to the observation of the lunar crescent in high latitudes (Ilyas, 1988). With
the spread of Islam, the month's duration at latitudes above 50º acquires an interest; this is the
issue we deal with below.
2.- The inclination of the Moon's path with the horizon
In the short time between sunset and observing the lunar crescent, the daily motions of the
Moon and Sun follow approximately celestial parallels. The angle formed by the equator with the
horizon is 90
(

is the geographic 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.
The azimuth
A
of the intersection of the parallel of declination

with the western horizon
at latitude

is (Segura, 2022b)
sin
cos .
A
The angle

of the parallel with the horizon is
Lunar months of 28 and 31 days
Wenceslao Segura
wenceslaotarifa@gmail.com
1
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sin
tan .
tan
A
therefore, the higher the latitude, the smaller angle between the parallel (or path of the Moon) and
the horizon. Let
A
be the azimuth difference between the Moon and the Sun at sunset;

is the
declination of the Moon and
the declination of the Sun at sunset. Then in the northern hemisphere,
A
increases with latitude, as long as
0
, and the opposite for the southern hemisphere
(Segura, 2022c).
For the same age of the Moon (or what is equivalent, for the same arc-light
L
a
or angular
separation between the Moon and the Sun), the altitude h of the Moon above the horizon at sunset
will be smaller by how much greater is
A
cos
cos
cos
L
a
h
A
.
Due to atmospheric absorption, to see the crescent, the altitude of the Moon must reach a
specific limit, which depends on
A
. For significant azimuth differences (which can occur at high
latitudes), the crescent will take longer to see because the Moon will take longer to pass the
altitude limit.
In short, at high latitudes, the difference in azimuth between the center of the Moon and the
place of the sunset may be significant (more than 20º), and we have to wait longer to see the
crescent. So observation of the crescent at high latitudes may take two or more days from its first
sight elsewhere on Earth.
If the observation of the crescent at high latitudes is delayed at the beginning of the month,
its duration can be 28 days. However, if the delay occurs at the end of the month, its duration can
reach 31 days.
The above reasoning is valid up to latitudes close to the polar circles (exactly up to
61.5º
)
(Segura, 2022b), since in these areas, it may happen that there is no moonset for one or several
days. For these places, redefining the criteria to start the lunar month is necessary.
3.- Extended Maunder Criterion
There are numerous criteria to determine when the crescent will be seen for the first time.
One set of these criteria uses the altitude of the Moon and the azimuth difference from the Sun at
sunset as parameters.
Although deficient for several reasons, we will use Maunder's (1911) criterion for
simplification. However, the original criterion only applies up to
20º
A
; therefore, it is necessary
to make extrapolation to apply it to more considerable azimuth differences. In drawing 1, we
represent the extended Maunder criterion (Segura, 2022b) (Ilyas, 1988) that we will use to predict
the vision of the first lunar crescent for observers located at high latitudes.
We note that to find the extended Maunder criterion, we accept Ilyas's assumption that the
minimum altitude of the Moon to see the crescent is 4º, a value that may be excessively small
(Fatoohi, 1988, 111-114). Therefore, visions of the Moon with altitudes between 4º and 5º must be
considered doubtful.
4.- Length of months observed at high latitudes
Tables 1, 2, 3, and 4 show the months of 31 and 28 days predicted for the 21st century for
various geographical positions and using the extended Maunder criterion.
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2
4
6
8
10
12
-10-20-30-40-50-60 10 20 30 40 50 60
True altitude
of the Moon
Azimuth difference between the Moon and the Sun
Drawing 1.- 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. 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.
Zul-Qida
1507 AH
05-07-2084
05-08-2084
Beginning
End
7.81º
-17.6
7.13º
-34.3
-4.38º
-16.2
04-07-2084
03-08-2084
120º W
22º S
20º E
31
Zul-Qida
1508 AH
24-06-2085
25-07-2085
Beginning
End
8.39º
-3.60º
7.76º
-33.1
-3.60º
-14.7
23-06-2085
23-07-2085
N
129º W
20º S
12º E
31
Month First
vision Apex Length
Table 1.- Months of 31 days in the 21st century from 50º N and 0º longitude. In this table and following are
the months with durations of 28 and 31 days, according to the extended Maunder criterion. The crescent is
visible in the evening of the day before it appears on the tables. The altitude h of the Moon is the altitude
true in true sunset (that is, when the true altitude of the center of the Sun is zero). The first vision is the day
when the crescent Moon is seen for the first time in the month considered. In the Apex column are the
geographical coordinates of the most easterly point at which the lunar crescent is seen in the considered
lunation. In the last column is the length of the month.
In the table, we verify that the difference between the declinations of the Moon

and the Sun
is always
negative for the northern hemisphere. Therefore the difference in azimuth between the center of the Moon
and the Sun at sunset is negative; that is, the Sun is further north than the Moon. Note that we measure the
azimuth from the south in the retrograde direction. From the table, we check that
is greater at the time
of the vision of the crescent with which the lunar month ends.
We note that in the northern hemisphere, the 31-day months occur close to the summer solstice. Finally, we
verify that the absolute value of the azimuth difference is greater when the month ends, which is why the
crescent is delayed by two days.
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h
A
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Ramadhan
1449 AH
29-01-2028
29-02-2028
Beginning
End
5.85º
5.09º
5.29º
36.57º
9.87º
18.94º
28-01-2028
27-02-2028
14º N
37º E
35º N
12E
31
Rabi
al-Awwal
1469 AH
29-12-2046
29-01-2047
Beginning
End
8.72º
15.09º
7.37
31.25º
1.99º
13.95º
29-12-2046
27-01-2047
S
158º W
16º N
90º W
31
Month First
vision Apex Length
Jumad
al-Ula
1511 AH
27-11-2087
28-12-2087
Beginning
End
9.59º
14.14º
10.74º
29.73º
0.98º
9.76º
27-11-2087
26-12-2087
S
17E
16º N
91º W
31
Jumad
al-Thani
1512 AH
15-12-2088
15-01-2089
Beginning
End
7.24º
18.82º
8.94º
31.74º
5.44º
12.72º
14-12-2088
13-01-2089
13º N
81º W
26º N
11º W
31
Zul-Qida
1522 AH
23-01-2099
23-02-2099
Beginning
End
9.53º
13.35º
8.53º
27.11º
0.74º
10.72º
23-01-2099
21-02-2099
16º S
16E
N
151º W
31
Table 2.- Months of 31 days in the 21st century for an observer at 50º S and longitude 0º. The difference in
azimuth of the centers of the Moon and the Sun at sunset is always positive. We check that
is
positive, the opposite of what happens if the observation is made in the northern hemisphere. The 31-day
months occur on dates close to the winter solstice when observed from the southern hemisphere. We check
that the apex is further north at the beginning of the lunation with which the month ends.
The conclusions obtained from the tables are the following:
a) The theoretically found tables clearly show that there are months of 31 and 28 days in
the area between 50º-61.5º, north or south, approximately.
b) In months of 31 days, the lunation with which the month ends is two or more days
behind the first vision.
c) In months of 28 days, the lunation with which the month begins is delayed by two or
more days.
d) Months with 31 days are much more frequent than those with 28.
e) In the northern hemisphere, in the months of 31 and 28 days,
0
A
for the lunations
at the beginning and end of the month. In the southern hemisphere, the opposite occurs.
f) 31-day months occur near the summer solstice in the northern hemisphere and the
opposite in the southern hemisphere.
g) 28-day months occur near the winter solstice in the northern hemisphere and the
h
A
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Rajab
1426 AH
07-08-2005
07-09-2005
Beginning
End
7.33º
-17.1
6.14º
-32.9
-2.88º
-13.4
06-08-2005
05-09-2005
10º N
111º W
S
6º W
31
Rajab
1427 AH
27-07-2006
27-08-2006
Beginning
End
7.85º
-17.1
5.71º
-32.0
-2.51º
-13.4
26-07-2006
25-08-2006
12º N
125º W
S
20º W
31
Month First
vision Apex Length
Safar
1452 AH
03-06-2030
04-07-2030
Beginning
End
8.80º
-14.9
7.72º
-33.1
-0.06º
-11.0
03-06-2030
02-07-2030
12º N
17E
11º S
39º W
31
Safar
1453 AH
23-05-2031
23-06-2031
Beginning
End
9.55º
-14.2
9.13º
-30.9
0.63º
-14.2
23-05-2031
21-06-2031
12º N
17E
S
56º W
31
Shawwal
1464AH
17-09-2042
18-10-2042
Beginning
End
4.87º
-25.5
4.02º
-39.2
-10.3
-14.14
16-09-2042
15-10-2042
16º S
13E
34º S
123º W
31
Jumad
al-Ula
1491 AH
02-07-2068
02-08-2068
Beginning
End
5.25º
-25.9
5.16º
-40.7
-8.97º
-18.4
01-07-2068
30-07-2068
15º S
82º E
31º S
113º W
31
Zul-Qida
1506 AH
17-07-2083
17-08-2083
Beginning
End
5.92º
-20.6
4.27º
-40.7
-5.64º
-19.4
16-07-2083
15-08-2083
73º W
22º S
11E
31
Shawwal
1507 AH
5-06-2084
06-07-2084
Beginning
End
10.02º
-12.0
8.37º
-29.3
2.58º
-8.45º
05-06-2084
04-07-2084
42º N
13E
120º W
31
Table 3.- Months of 31 days in the 21st century for an observer located at 55º N and 0º longitude. We find the
same guidelines as in the previous tables. However, the month of Shawwal 1464 AH does not coincide with
dates close to the summer solstice. The extended Maunder criterion that we are using considers that the
minimum altitude of the Moon to see the crescent is 4º, a perhaps small value, so the cases with an altitude
between 4º and 5º must be questioned.
h
A
Muharram
1452AH
04-05-2030
04-06-2030
Beginning
End
10.45º
-8.07º
10.64º
-26.72
5.30º
-1.77º
04-05-2030
03-06-2084
34º N
68º E
12º N
17E
31
Safar
1453AH
23-05-2031
23-06-2031
Beginning
End
7.98º
-15.4
5.29º
-32.0
0.62º
-8.65º
23-05-2031
21-06-2031
12º N
170 º E
S
56º W
31
Month First
vision Apex Length
Ramadhan
1471AH
03-06-2049
04-07-2049
Beginning
End
9.22º
-23.8
4.40º
-40.1
-1.69º
-13.5
02-06-2049
01-07-2049
N
70º E
13º S
154º W
31
Zul-Qida
1473AH
11-06-2051
12-07-2051
Beginning
End
6.09º
-27.0
4.82º
-44.6
-5.66º
-15.8
10-06-2051
09-07-2051
S
66º E
23º S
84º W
31
Rajab
1482 AH
09-12-2059
06-01-2060
Beginning
End
4.93º
-43.61
7.76º
-24.7
-2.24º
0.02º
06-12-2059
05-01-2060
31º S
64º W
26º S
11E
28
Shaban
1489AH
23-10-2066
20-11-2066
Beginning
End
4.17º
-47.7
4.49º
-28.4
-12.7
-4.10º
20-10-2066
19-11-2066
51º S
29º W
32º S
15E
28
Rabi
al-Thani
1491AH
02-06-2068
03-07-2068
Beginning
End
7.32º
-20.0
4.02º
-37.7
-1.59º
-12.7
01-06-2068
01-07-2068
N
102º W
15º S
82º E
31
Ramadhan
1506 AH
18-05-2083
18-06-2083
Beginning
End
10.64º
-6.99º
10.65º
-25.7
8.67º
-0.93º
18-05-2083
17-06-2083
44º N
42º E
24º N
14E
31
Shawwal
1507 AH
05-06-2084
06-07-2084
Beginning
End
8.62º
-13.4
4.81º
-30.3
2.54º
-8.56º
05-06-2084
04-07-2084
24º N
13E
120º W
31
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h
A
7
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Rabi
al-Thani
1508 AH
01-12-2084
29-12-2084
Beginning
End
7.95º
-36.5
7.92º
-15.6
-0.08º
3.41º
29-11-2084
28-12-2084
28º S
27º E
S
148º W
28
Shaban
1508 AH
25-05-2085
25-06-2085
Beginning
End
8.96º
-12.1
6.06º
-30.2
3.13
-7.09º
25-05-2085
23-06-2085
26º N
93º E
N
129º W
31
Month First
vision Apex Length
Table 4.- Months of 28 and 31 days during the 21st century for an observer located at 60º N and 0º longitude.
We find three 28-day months close to the winter solstice. In these short months, the delay in seeing the
crescent occurs on the lunation with which the month begins.
opposite in the southern hemisphere.
h) If
1
A
is the azimuth difference at the time of seeing the crescent when there is a
delay of several days and
2
A
is the azimuth difference in seeing the crescent when
there is no delay of a few days, so for the northern hemisphere
1 2
A A
and
1 2
A A
for the southern hemisphere. 1 may correspond to the crescent with
which the month begins (months of 28 days) or which it ends (months of 31 days).
i) The differences in declination of the Moon and the Sun for the months of 31 and 28
days are generally negative for the northern hemisphere and positive for the southern.
However, it always happens that this difference in absolute value is greater on the lunation
in which the delay in seeing the crescent occurs.
j) In the northern hemisphere, the lunation in which there is a delay (two or more days)
in seeing the crescent has the apex further south and the opposite in the southern
hemisphere.
k) The greater the geographical latitude, the more frequent the months of 31 and 28
days.
l) Exceptionally, there are months of 31 days in which the crescent with which it begins
and with which it ends is only one day behind the first vision, (for example, the month of
Muharram 1452 AH from table 4).
m) We have found the possibility that the month has 32 days; this would happen in
Shawwal 1646 (table 3) if we assume that the lowest altitude to see the crescent is
somewhat greater than 4º.
4.- Conclusions
Using a simplified criterion to predict the vision of the first lunar crescent, we verify that at
high geographical latitudes (greater than 50º approximately), there are months of 31 and 28 days;
this happens because the crescent with which the month begins (or with which it ends) is delayed
two or more days to the first vision in the considered month.
Analyzing the months of 31 and 28 days in various latitudes (50º, 55º and 60º), we find the
h
A
circumstances under which it occurs; we verify that in the northern hemisphere, the months of 31
days occur near the summer solstice and at the winter solstice in the hemisphere south; and 28-
day months occur near the winter solstice and the opposite in the southern hemisphere.
5.- References
1.- Fatoohi, L. J. (1998). First visibility of the lunar crescent and other problems in historical
astronomy. Doctoral theses, Durham University.
2.- Ilyas, M. (1988). Limiting altitude separation in the new Moon’s first visibility criterion, Astronomy
and Astropysics 206, 133-135.
3.- Maunder, E. W. (1911). On the Smallest Visible Phase of the Moon. The Journal of the British
Astronomical Association 21, 355-362.
4.- Schaefer, B. (1992). The Length of the Lunar Month. Archaeoastronomy 17, 32-42.
5.- Segura, W. (2022a). The Islamic calendar is lunisolar. https://www.researchgate.net/publication/
360370578_The_Islamic_calendar_is_lunisolar.
6.- Segura, W. (2022b). Length of Islamic Months. https://www.researchgate.net/publication/
361192704_Length_of_Islamic_Months.
7.- Segura, W. (2022c). Apex of the zone of first visibility of the Moon. https://www.researchgate.net/
publication/360370828_Apex_of_the_zone_of_first_visibility_of_the_Moon.
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ResearchGate has not been able to resolve any citations for this publication.
Article
The minimum altitude separation between the moon and the setting sun for new moon's earliest visibility is found to be close to 4°. This result together with an earlier study on limiting elongation makes the new Moon's earliest visibility criterion most complete and comprehensive.
On the Smallest Visible Phase of the Moon
  • E W Maunder
Maunder, E. W. (1911). On the Smallest Visible Phase of the Moon. The Journal of the British Astronomical Association 21, 355-362.
The Islamic calendar is lunisolar
  • W -Segura
-Segura, W. (2022a). The Islamic calendar is lunisolar. https://www.researchgate.net/publication/ 360370578_The_Islamic_calendar_is_lunisolar.
Length of Islamic Months
  • W Segura
Segura, W. (2022b). Length of Islamic Months. https://www.researchgate.net/publication/ 361192704_Length_of_Islamic_Months.
Apex of the zone of first visibility of the Moon
  • W -Segura
-Segura, W. (2022c). Apex of the zone of first visibility of the Moon. https://www.researchgate.net/ publication/360370828_Apex_of_the_zone_of_first_visibility_of_the_Moon.