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TIME OF THE FIRST SIGHT OF THE LUNAR CRESCENT

Time of the first sight

of the lunar crescent

Wenceslao Segura González

e-mail: wenceslaoseguragonzalez@yahoo.es

Independent Researcher

1

First version December, 21, 2022

Abstract. We calculate, as a function of latitude, the universal time when the visibility of the first

lunar crescent begins. We verified that for the same meridian, the time of the first visibility of the

crescent depends on the latitude and that the atmospheric absorption that attenuates the moonlight

has little influence.

1. Introduction

The Islamic calendar is lunar, meaning that the months begin when observers see the first

crescent of the Moon near the western horizon, shortly after sunset and when some time has

passed since the conjunction of the Sun and the Moon.

Strictly speaking, the Islamic calendar is lunisolar, in the sense that, on average, the duration

of the lunar month is the synodic period of the Moon, which is a time interval that depends on the

movements of the Sun and the Moon (Segura 2022a).

In the Middle Ages, Muslim astronomers devised computational Islamic calendars, that is,

based on mathematical rules and not on physical observation of the Moon (Segura 2020a, 2020b).

On average, these calendars follow the movement of the Moon, since the lunar synodic period

varies very little over time (each century the synodic period expressed in universal time decreases

by 0.031 seconds) (Segura 2018, 172).

However, today the Islamic world follows the tradition of beginning the month when reliable

witnesses see the Moon for the first time after conjunction with the Sun. Although this custom is

faithful to tradition, and some give it religious meaning, it originates numerous problems, which

gives rise to a disparity in the beginning and duration of the months, mainly at the beginning and end

of the holy month of Ramadan.

The lack of centralized religious authority in the Muslim world has prevented single rules

regulating the calendar. This lack of single criteria means that each year there is the painful spectacle

of the same religious holidays are celebrated on different dates in different parts of the world.

Among the problems faced by the Islamic calendar is the influence that geographic latitude

has on the observation of the first lunar crescent, an issue that we partially deal with in this research

and that we have discussed on other occasions (Segura 2022b, 2022c).

The Islamic calendar uses the horizon as a reference because observing the first lunar

crescent is only possible when the Moon is close to the horizon. However, the horizon plane depends

on the latitude of the place of observation; therefore, the lunar calendar depends on the geographical

latitude. Observers located in the same meridian but in different latitudes may have differing views

about the beginning of the month since, in some latitudes, the first lunar crescent will be observable;

however, it will not be visible in other latitudes.

Not only the beginning of the lunar month is dependent on latitude, but also the day, if we

understand that it begins with sunset, a phenomenon also linked to the horizon.

The Gregorian solar calendar does not have these drawbacks since the beginning of the day

in local time is related to the antimeridian of the place, independent of latitude. For all the observers

Wenceslao Segura González

2

of the same meridian, the local solar day begins at the same instant: when the center of the Sun

passes the antimeridian. Furthermore, in the Gregorian solar calendar, there is also no discrepancy

about the beginning of the months because they all have a fixed number of days.

2. Influence of geographic position on the observation of the first crescent Moon

In drawing 1, we show a graph of the topocentric altitude of the Moon's center and its

azimuth difference with the Sun as a function of latitude, for several lunations, null Sun's depression,

and for the longitude of 0º, where we check the effect of latitude in the position of the Moon. The

drawing shows Maunder's lunar visibility criterion (Maunder, 1911). If the position of the Moon is

above the Maunder line, the crescent will be visible, and it will not be otherwise (Segura 2022h).

We verify that for the same meridian, there are latitudes from which the crescent is visible

(because the position of the Moon is above the Maunder line). In contrast, the crescent will not be

visible for other latitudes of the same meridian.

Several similar but different concepts relate the geographical position with the visibility of the

first lunar crescent, which we now specify. We define these concepts for the hypothetical case that

atmospheric conditions remain unchanged.

* The apex is the place on Earth where the lunar crescent is visible in the easternmost

position.The geographic coordinates of the apex vary for each lunation (Segura 2022g).

3

-5

6

9

12

15

18

21

-10-15-20 5 10 15 20

Azimuth difference between the centers of the Moon and the Sun

Topocentric altitude

Moon’s center

0º

0º

0º

0º

0º

I

II

III

IV

V

Maunder

South

North

50º

80º

80º

60º

Drawing 1.- We represent the topocentric altitude of the Moon and its difference in azimuth with the Sun at

the moment when the depression of the Sun is null, the meridian of longitude 0º and for the following dates

of the year 2025: September 23 (I), March 30 (II), January 30 (III), July 25 (IV) and February 28 (V). The blue part

of the curves (on the left) corresponds to positive latitudes, and the red one (on the right) to negative

latitudes.The union between the positive and negative parts corresponds to latitude 0º. The red dots are the

positions of the Moon at positive latitudes every 10º, and the blue squares are the positions of the Moon for

negative latitudes every 10º. The graphs do not have the same extension because at latitudes above the polar

circles (arctic or antarctic), the Sun may be permanently above the horizon, and it is impossible to see the

crescent.We do not draw the positions of the Moon when it is below the horizon because it does not

correspond to the visibility situation. The black curve is the Maunder visibility criterion for topocentric

altitudes. If the position of the Moon is above this curve, we will see the crescent, and we will not see it if it

is below the curve. We verify that for the same meridian, there are latitudes in which the Moon is visible and

other latitudes (preferably extreme) in which it is not visible, which shows that latitude is one of the main

factors to know if we will see the crescent or not.

Horizon

Sun

TIME OF THE FIRST SIGHT OF THE LUNAR CRESCENT

3

* Place of the first sight of the crescent is the place on the Earth's surface where

we see the crescent before anywhere else. Its position is close to the apex, but they

are different places.

* Isochrones are the places on Earth where the lunar crescent is visible at the same

time (Segura 2022b).

* The line of change of month is a line drawn on the Earth's surface that separates

the places where there is visibility of the lunar crescent from those others where its

visibility is not possible; that is, it is the line that divides the Earth's surface into two

zones that have different months. The lines of change of month are different for

each lunation; these lines have the shape of a horizontal quasi-parabola, and its vertex

is the apex (Segura 2022b).

* The moment of best vision of the crescent is when the observer in the same

place sees the crescent more easily. The evening when we see the lunar crescent for

the first time, begins with the Moon hidden by the Sun's rays when the solar depression

is small. As the Sun moves below the horizon, the luminance of the twilight sky

decreases, causing the crescent to become visible. The more the Sun descends below

the horizon, the less luminosity of the sky and the easier it is to see the crescent.

However, when the Moon approaches the horizon, the atmospheric absorption increases,

decreasing the illuminance of the Moon and therefore decreasing the ease of vision

until reaching a moment in which the Moon is no longer visible, even before crossing

the horizon. There is a moment between the two situations, characterized by the Sun's

depression, in which the visibility of the crescent is optimum.

* The latitude of optimum viewing is the place of a meridian where the visibility of

the crescent is more accessible than in the other places of the same meridian, that is

to say, the place where we observe the crescent at the highest altitude above the

horizon for the same depression of the Sun. The latitude of optimum viewing depends

on the depression of the Sun.

* The line of optimum viewing is a line that joins the places of optimum viewing of

each meridian for a certain lunation.

* The latitude of first and last crescent visibility it is the place where the crescent

is visible for the first or last time in the same meridian. These places are points of the

line of change of month since it separates viewing from non-viewing in the meridian.

The previous concepts show the importance of geographical position in observing the lunar

crescent. In this investigation, we found the moment the crescent's visibility begins for latitudes of

the same meridian.

3. The universal time when the visibility of the crescent begins

Chosen a meridian of longitude

we want to calculate for the day in which the lunar month

begins, the moment in universal time in which the observation of the crescent begins in each

latitude of that meridian. Although we can make rigorous astronomical calculations, we use an

approximate algorithm, with which we will make an error of a few minutes, which is not important

since it will be less than the error made in the choice of other parameters, especially the atmospheric

ones.

In the first part of the procedure, we choose an initial reference position for the center of the

Moon, finding for the date considered, latitude 0

0

and depression of the Sun 0

0

d

, the

geocentric coordinates of the Moon: geocentric altitude of the center of the Moon

0

h

, azimuth

difference between the centers of the Moon and the Sun

0

A

, the geocentric declination of the

Moon

0

and the moment of the true sunset

0

t

(when the center of the true Sun, that is, without

refraction, is on the horizon). With these data, we find for latitude

and depression d, the topocentric

coordinates of the Moon: topocentric altitude

T

h

, azimuth difference between the Moon and the

Sun

A

, the topocentric declination of the Moon

T

and the declination of the Sun

(see

appendix 1 ).

In the second part of the calculation, we start with the topocentric position of the Moon at

latitude

and depression

0

d

, previously calculated. Using Segura's theory of lunar visibility

Wenceslao Segura González

4

10

20

30

40

- 10

- 20

- 30

15.5 16 16.5 17 17.5 18 18.5 19 19.5

Universal Time of the first visibility of the crescent

Latitude

Drawing 2.- The lines represent the universal time in which the lunar crescent begins to be visible for the

indicated longitude and latitude, on January 30, 2025. In this lunation, we find that the first visibilities occur

in the northernmost positions, the opposite occurring for other lunations. The points correspond to the

Moon's positions every 10º of latitude. The three lines on the left are shorter because from latitudes 40º N and

30º S, we cannot see the crescent for the indicated longitudes. To calculate the graphs, we have used the lunar

visibility theory of Segura (2022d) (see appendix 3), considering an atmospheric absorption coefficient of

0.25. Similar graphs are found with other lunar visibility criteria.

The visibility of the Moon begins in positions more to the east in the same parallel. However, there are

positions to the west that see the crescent before other positions to the east. For example, in position A

(longitude 30º E), the crescent is visible at 16h 2m. In position B (longitude 40º E, that is, more to the east than

A), the visibility of the crescent is somewhat later, at 16h 29m.

0º

18.0 18.5 19.0 19.5 20.0 20.5

10

20

30

40

50

- 10

- 20

- 30

Latitude

Universal Time of the first visibility of the crescent

2025 / 1/ 30

0º

2025 / 7 / 25

0º

Drawing 3.- The lines show the moment of visibility of the crescent at 0º longitude for two dates: January 30,

2025 (negative Sun declination) and July 25, 2025 (positive Sun declination). When the declination of the Sun

is negative, the visibility of the crescent begins in the most northerly positions. However, when the declination

of the Sun is positive, the visibility of the crescent begins in the most southern positions. We use Segura's

lunar visibility theory (2022d), with an atmospheric absorption coefficient of 0.25.

30º

E

40º E

20º E

A

B

10º E

TIME OF THE FIRST SIGHT OF THE LUNAR CRESCENT

(2022d) we find the depression of the Sun for which the crescent begins to be visible. Varying the

latitude, we determine the depression in which there is visibility in other meridian places.

To simplify the calculation, we assume that the geocentric declinations of the Sun and the

Moon remain invariable in the observations in the same meridian; however, we assume that the

right ascensions

and

and the phase angle vary uniformly..

Knowing the true sunset time for 0

0

and 0

0

d

we can approximately find the true

sunset time for any other latitude for the same day and meridian by the techniques presented in

appendix 4.

Using Segura's theory of lunar visibility, we find the depression of the Sun in which the

visibility of the Moon begins, knowing the topocentric coordinates of the Moon at the latitude

(appendix 2).

Finally, by an approximate calculation (appendix 4), we find the moment in universal time

when the visibility of the lunar crescent begins.

4 Results

In drawing 2, we show the results of the techniques described in the previous section. We

draw some lines that are the moments in which the visibility of the lunar crescent begins on January

30, 2025, at the indicated latitudes and longitudes.

We verified that the moment of observation of the crescent strongly depends on the latitude

of the place. For example, for latitudes in 0º longitude, the difference in moments of the first

visibility exceeds an hour and a half.

We also see that the lines represented have a negative inclination concerning the time axis

(horizontal axis); this means that the visibility of the crescent begins at the northernmost position,

and the time of the last observation is at the southernmost latitude. However, for other lunations,

the inclination of the lines is positive, and the first observation of the crescent occurs at the

southernmost latitude.

In drawing 3, we show two lines of the first observation of the crescent for two different

dates and the same longitude, verifying that their inclinations differ. The cause of this diversity is

the sign of the declination of the Sun. On January 30, 2025, the Sun's declination is approximately

-17.4º, and the slope is negative. On July 25, 2025, the Sun's declination is approximately 23.3º, so

the slope of the line is positive (appendix 4).

We have shown (Segura 2020d) the significant influence that atmospheric absorption has on

the visibility of the first lunar crescent. We now show the influence of atmospheric attenuation at

5

30

20

10

0

-10

-20

-30

5.54º

5.18

5.01

4.99

5.10

5.43

6.62

17h-19m

17-30

17-42

17-54

18-08

18-24

18-48

4.92º

4.64

4.58

4.53

4.58

4.77

5.25

6.41º

5.79

5.54

5.52

5.70

6.28

Not visible

17h-16m

17-28

17-40

17-52

18-06

18-21

18-41

17h-23m

17-33

17-44

17-57

18-10

18-28

Not visible

Depression in which it begins

the visibility of the lunar crescent

First visibility of the crescent

(hours and minutes)

Latitude

Table 1.- We calculate the universal time at which the visibility of the crescent of the Moon begins at -10º

longitude on January 30, 2025, for various latitudes and various atmospheric attenuation coefficients. We

found that the lower the atmospheric absorption, the visibility of the crescent is earlier; however, the differences

between the first visibilities of the crescent are only a few minutes. Although atmospheric absorption is an

essential factor in determining the visibility of the crescent, it hardly influences the moment at which the

visibility of the lunar crescent begins.

0.2

k

0.25

k

0.3

k

0.3

k

0.25

k

0.2

k

Wenceslao Segura González

the moment the crescent visibility begins. Table 1 shows the depressions and the universal time at

which the visibility of the first lunar crescent begins for various latitudes and atmospheric attenuation

constants. We verified that the depression in which the observation of the crescent begins is affected

by atmospheric absorption; however, it hardly influences the moment in which the visibility of the

crescent begins.

5. Referencias

* Kasten, F. and Young, A. T. (1989). «Revised optical air mass tables and approximation formula»,

Applied Optics 28, 4735-4738.

* Knoll, H. A.; Tousey, R. and Hulburt, E.O. (1946). «Visual Thresholds of Steady Sources of Light

in Fields of Brightness from Dark to Daylight», Journal of the Optical Society of America 36

(8), 480-482.

* Koomen, M. J.; Lock, C.; Packer, D. M.; Scolnik, R.; Tousey, R. and Hulburt, E. O. (1952).

«Measurements of the Brightness of the Twilight Sky», Journal of the Optical Society of America

42-5, 353-356.

* Maunder, E. W. (1911): «On the Smallest Visible Phase of the Moon», The Journal of theBritish

Astronomical Association 21, 355-362.

* Segura, W. (2018). Movimiento de la Luna y el Sol, eWT Ediciones, 2018.

* Segura W. (2020a). «Two Lunar Crescent Visibility Critera: Al Khwarizmi and al-Qallas». https:/

/www.researchgate.net/publication/341977445_Two_lunar_crescent_visibility_criteria_al-

Khwarizmi_and_al-Qallas.

* Segu ra, W. (2020b). «Luna r Crescent Visi bil i ty Crite rion of al-Ba ttani». https://

www.researchgate.net/publication/342589407_Lunar_Crescent_Visibility_Criterion_of_al-Battani

* Segura, W. (2022a). «The Islammic Calendar is Lunisolar». https://www.researchgate.net/

publication/360370578_The_Islamic_calendar_is_lunisolar.

* Segura, W. (2022b). «Apex of the Zone of Fir st Visibility of the Moon». https://

www.researchgate.net/publication 360370828_Apex_of_the_zone_of_first_visibility_of_the_Moon

* Segu ra, W. (20 22c). «Islamic Calendar a nd Geogr aphic Coordinates». https: //

www.researchgate.net/publication/361283442_Islamic_calendar_and_geographic_coordinates.

* Segura, W. (2022d). «Fotheringham Graphs of Visibility of the First Lunar Crescent».

https://www.researchgate.net/publication/365125238_FOTHERINGHAM%27S

_GRAPHS_OF_VISIBILITY_OF_THE_FIRST_LUNAR_CRESCENT.

* Segura, W. (2022e). «Magnitude of the Moon at large phase angles». https://www.researchgate.net/

publication/362491492_Magnitude_of_the_Moon_at_large_phase_angles

* Segura, W. (2022f). «Visibility Window of the First Lunar Crescent». https://www.researchgate.net/

publication/363785232_Visibility_Window_of_the_First_Lunar_Crescent.

* Segura, W. (2022g). «Isl amic Ca lendar a nd Geo graph ic Coor dina tes». https://

www.researchgate.net/publication/361283442_Islamic_calendar_and_geographic_coordinates.

* Segura, W. (2022h). « Optimal Latitude to See the Lunar Crescent». https://www.researchgate.net/

publication/365893281_Optimal_latitude_to_see_the_lunar_crescent

-----------------------------------------------------------------------------------------------------------------

A P P E N D I C E S

-----------------------------------------------------------------------------------------------------------------

Appendix 1: Algorithm to find the topocentric coordinates of the Moon

We show the approximate calculation to find the topocentric altitude of the center of the

Moon, the azimuth difference, and the topocentric declination of the Moon, calculated based on

latitude, longitude and depression of the Sun.

We choose a geocentric reference position of the center of the Moon for a given date and

for longitude

, corresponding to latitude 0

0

and the Sun's depression 0

0

d

, where the Moon

has geocentric altitude

0

h

, azimuth difference with Sun

0

A

and geocentric declination

0

. The

usual techniques of spherical astronomy calculate these data. With them, we find approximately

the topocentric position of the Moon for a depression of the Sun d and a latitude

. In the following

6

TIME OF THE FIRST SIGHT OF THE LUNAR CRESCENT

reasoning, we neglect the parallax in azimuth and in the coordinates of the Sun.

a) Moon azimuth

Applying the law of cosines to the spherical triangle with vertices at the pole, zenith and center of

the Moon, we find the azimuth of the center of the Moon

0 0 0 0

0

0 0 0

sin sin sin sin

cos .

cos cos cos

h

A

h h

We define the azimuth as the angle measured on the horizon from the south in the retrograde

direction.

b) Sun azimuth

0 0 0

.

A A A

c) Declination of the Sun

0 0 0 0 0 0 0

sin sin sin cos cos cos cos .

d d A A

Since the time interval between the observation of the crescent at latitude

0

and latitude

is

small, we assume that the declinations of the Sun and Moon remain unchanged.

d) Phase angle

0 0 0 0 0 0 0 0

cos sin sin cos cos cos cos cos .

h d h d A h A

e) Geocentric hour angle of the center of the Sun

Applying the law of cosines to the spherical triangle with vertices at the Sun, zenith and pole we

find

0 0 0

0

0 0

sin sin sin

cos 0.

cos cos

d

H

f) Geocentric hour angle of the center of the Moon

0 0 0 0

0

0 0 0

sin sin sin sin

cos .

cos cos cos

h h

H

g) Right ascension and phase angle variations

We assume that the right ascensions of the Moon

and the Sun

and the phase angle

vary

uniformly in the time interval

T

0 0 0

; ; .

T T T

The mean variations are

360º 27.3 0.0366º º; 360º 365 0.00274º º; 360º 29

.53 0.0339º º .

d d d

h) Calculation of the geocentric hour angle of the Sun for latitude

y depresión d

0

0

sin sin sin

cos ,

cos cos

d

H

we consider depression with the positive sign.

i) Sidereal time interval elapsed

0

,

1 1

H H

H

T H H T T

we identify the intervals of sidereal time and universal time.

j) Geocentric hour angle for Moon when latitude is

0

1 .

T H H T H H T

k) Geocentric altitude of the center of the Moon for latitude

0 0

sin sin sin cos cos cos .

h H

l) Azimuth difference between the Moon and the Sun for latitude

y depresión d

0

sin sin cos

sin sin cos

cos .

cos cos cos cos

h d T

h d

A

h d h d

m) Topocentric altitude of the Moon

We assume that the parallax does not affect the azimuth, and since the calculations do not have to

7

Wenceslao Segura González

8

be very precise, we will use the simplified formula to find the topocentric altitude (Segura 2018, 31-

35)

sin sin

tan ,

cos

T

h

h

h

is the equatorial horizontal parallax of the Moon.

n) Topocentric hour coordinates

0 0 0

0

0 0 0 0

0 0 0

0 0

0 0 0 0

cos sin cos sin

tan ; tan

cos cos sin cos cos cos sin cos

sin sin sin sin sin sin

tan cos ; tan cos .

cos cos sin sin cos cos sin sin

T T

T T T T

H H

H H

H H

H H

H H

Then for latitude

we find the Moon's topocentric altitude

T

h

, the Moon-Sun azimuth difference

A

and the Moon's topocentric declination

T

. After this calculation, we choose a new latitude,

obtaining a new position on the Moon, until completing all the latitudes that interest us in the chosen

meridian.

The process continues by choosing a new meridian of longitude

and repeating the previous

calculations, with which we will obtain the graph of the positions of the Moon for the new meridian

as a function of latitude (drawing 1).

Appendix 2: Algorithm to find the apparent altitude and azimuth difference of the crescent

center of the Moon

Knowing the topocentric position of the center of the Moon

, ,

T T

h DA

when the depression

of the Sun is null, and the latitude is

, we want to find approximately the depression for which the

visibility of the lunar crescent begins.

, ,

T T

h DA

we find it by the algorithm developed in appendix

1.

a) Calculation of the azimuth of the center of the Moon for 0

0

d

Applying the law of cosines to the spherical triangle with vertices at the pole, zenith, and center of

the Moon, we find the azimuth of the Moon

0

A

when the depression of the Sun is null

sin sin sin

cos .

cos cos

T T

T

h

Ah

b) Calculation of the azimuth of the center of the Sun for 0

0

d

.

A A A

c) Calculation of the topocentric phase angle for 0

0

d

Applying the law of cosines to the spherical triangle with vertices at the centers of the Sun and

Moon and the zenith, we find the topocentric phase angle

0 0 0

cos sin sin cos cos cos cos cos .

T T T T

h d h d A h A

Since the Earth-Sun distance is much greater than the Earth-Moon distance, then

180 ,

T LT

a

LT

a

is the arc-light or topocentric angle between the centers of the Sun and the Moon.

d) Calculation of the topocentric declination of the Sun for 0

0

d

Applying the cosine theorem to the spherical triangle with vertices at the Sun, the zenith, and the

pole, we find the topocentric declination of the center of the Sun

0

0 0 0

sin sin sin cos cos cos cos cos .

d d A A

e) Calculation of the topocentric hour angle of the Sun for 0

0

d

Applying the cosine theorem to the spherical triangle with vertices at the Sun, zenith and pole, we

find the topocentric hour angle of the Sun when its depression is zero

0 0

0 0

0

sin sin sin

cos tan tan .

cos cos

T

d

H

f) Calculation of the topocentric hour angle of the Moon for 0

0

d

Applying the law of cosines to the spherical triangle with vertices the Moon, the zenith and the pole,

we find the topocentric hour angle of the Moon when the depression of the Sun is zero

TIME OF THE FIRST SIGHT OF THE LUNAR CRESCENT

0

sin sin sin

cos .

cos cos

T T

T

T

h

H

g) Next, we give arbitrary values to the depression d, gradually increasing it. We assume that the

declinations and right ascensions of the Sun and the Moon and the phase angle remain unchanged

during the short time it takes for the Moon to reach the horizon.

h) Calculation of the topocentric hour angle of the Sun for depression d and latitude

0

0

sin sin sin

cos .

cos cos

T

d

H

i) Calculation of elapsed sidereal time

The sidereal time

T

elapsed since the Sun had zero depression to the new position with depression

d is

0

.

1

T T

H H

T

j) Calculation of the topocentric hour angle of the center of the Moon when the depression

of the Sun is d

0

1

.

1

T T

H H T

k) Calculation of the topocentric altitude of the center of the Moon when the depression of

the Sun is d

sin sin sin cos cos cos .

T T T T

h H

l) Calculation of the phase angle when the depression of the Sun is d

0

.

T T

T

m) Calculation of the difference in azimuth between the centers of the Moon and the Sun

when the depression of the Sun is d

sin sin cos

cos .

cos cos

T T

T

h d

Ah d

n) Calculation of the topocentric altitude of the center of the crescent

Previously, we found the topocentric altitude of the center of the Moon

T

h

and the azimuth difference

A

between the Moon's and the Sun's centers. However, we are interested in the altitude and the

azimuth difference of the central part of the crescent, which will allow us to find the luminance of

the twilight sky around the center of the crescent.

In drawing 4, there are two right spherical triangles, applying the law of sines

1

1 1

1

sin sin sin

sin

sin ; sin 180 ,

sin sin sin sin 180

T T T

T T T

T T

h h

d

d

we deduce

12

sin sin cos 180

1

sin ; ,

sin sin 180

1

T T

T

T T

d h

h

9

B

C

O

cT

h

d

Horizon

Drawing 4.- Point A is the center of the Moon, B is the center of the crescent and C the center of the Sun. The

horizontal line is the horizon.

1 0 1

; 180 ; ; ; .

T T T c

OA OC BA R A GD A GF A FD

G

F

D

T

h

A

Wenceslao Segura González

when the angles are small, then

1

1

180 ,

1

T T

T

d h

which is what we find when we use plane trigonometry instead of spherical. Again using the sine

theorem on the spherical triangle in drawing 4, we find the topocentric altitude of the center of the

lunar crescent

1

1

sin

sin sin ,

sin

T

cT T

T

R

h h

R is the apparent radius of the Moon, which is 15.5 minutes in the Earth-Moon mean distance. For

small angles, we can use the plane trigonometric, finding

1

1 .

180

T

cT T

T

d h

h h R

From drawing 4

1

1cos

cos

cos ; cos ,

cos cos

T

T

T cT

R

OD OF =

h h

then the difference in azimuth between the center of the crescent and the center of the Sun is

1

1 1

1cos

cos

cos cos ,

cos cos

T

T

c

T cT

R

A A OD OF A h h

and in the plane trigonometry approximation is

1

1

sin ; .

180 tan

c T cT

T T

dA A h h

The difference between the results found by spherical and plane trigonometry is negligible for our

problem.

o) Calculation of the apparent altitude of the center of the crescent

With the topocentric altitude

cT

h

we find the apparent altitude

ca

h

of the center of the crescent by

Bennet's formula (Segura 2018, 35-44)

1 1

60 7.31

tan 4.4

cT ca

ca

ca

h h

hh

with the angles in degrees.

Appendix 3: Lunar visibility theory

a) Calculation of the apparent magnitude of the center of the crescent

We assume that the lunar crescent is divided into portions with an angular length of 1 minute, that

is, approximately the size of the resolving power of the human eye. The magnitude of the central

portion as a function of the phase angle is (Segura 2022e)

d

Sun

Horizon

Parallel

Drawing 5.- We represent the Sun below the horizon with a depression d. The inclined line is the parallel of

declination

that the Sun travels in its daily movement, which forms an angle

with the horizon. D is the

length of arc AB.

A

B

D

10

TIME OF THE FIRST SIGHT OF THE LUNAR CRESCENT

The central part of the crescent will be the first to be seen after the conjunction with the Sun.

Therefore, the problem of determining the visibility of the lunar crescent is limited to checking if the

central part of the crescent will be visible, which has a width that, for the cases of interest, are

smaller than the resolution of the human eye. To find the magnitude increase caused by atmospheric

absorption, we use the formula of Kasten and Young (1989) and the atmospheric attenuation constant.

p) Twilight sky luminance

The luminance of the twilight sky depends on the Sun's depression, the altitude, and the azimuth

difference from the Sun. In our theory, we use the luminance measured by Koomen, Lock, Packer,

Scolnik, Tousey and Hulburt (1952). To find the luminance at any position, we extrapolate Koomen's

results using the Lagrange polynomials.

q) Threshold illuminance and threshold magnitude of a point object on a bright background

We have to detect a luminous object (the central part of the lunar crescent) of a size smaller than

the eye's resolution, which is on a bright background (the twilight sky). We will see the crescent

when it exceeds a certain level of contrast with the background. To find the threshold illuminance,

we use the measurements of Knoll, Tousey and Hulburt (1946), which we adjust to the following

polynomial expression (Segura 2022f)

2

log 6.72548 0.50020167 log 0.05862778 log ; 13.

98 2.5log ,

th S S th th

E B B m E

th

E

is the threshold illuminance to see an unresolvable object against a background of luminance

S

B

, and

th

m

is the corresponding threshold magnitude.

r) Calculation of the visibility coefficient

We define the coefficient of visibility by

th M

m m

M

m

is the apparent magnitude of the central part of the Moon. If the visibility coefficient is negative

the crescent is not visible and will be seen when

is greater than zero. Naturally, the greater the

value of

, the easier it will be to see the increasing

Appendix 4: Calculation of the moment of the first observation of the crescent

Knowing the time of true sunset for 0

0

and 0

0

d

for the considered day and longitude

, we can approximately find the time of true sunset for any other latitude for the same day and

meridian.

The hour angle of the Sun at the reference position is 0

90º

H

. For another latitude

and

0

d

the hour angle is

0

cos tan tan .

H

The sidereal time on the prime meridian at sunset is

T H

and the universal time of sunset for latitude

is

0

1.002737

h

T T UT

t

0h

T UT

is the sidereal time in the prime meridian at 0h UT of the day considered. The denominator

is a factor to convert sidereal time into universal (Segura 2018, 15-18).

If

0

t

is the universal time of the sunset for

0

and

0

d

then neglecting the variation of

the right ascension of the Sun and identifying the sidereal and universal time intervals, we find

1

0 0 0 0

24

90º cos tan tan 90º ,

360

t t t H H H H t t

expressed in units of time.

Knowing the depression d at which the crescent begins to be seen at a latitude

, we find

11

-2.34 -1.86 -1.28 -0.52 0.55 2.40

165

150 155 160 170 175

Magnitude

Phase

angle

Wenceslao Segura González

12

approximately the moment of universal time of that instant. Using plane trigonometry in the drawing

5, we find

sin

d

D

D is the arc AB, and

is the angle formed by the horizon and the parallel of the Sun corresponding

to the declination

0

. The angle measured from the center of the celestial sphere and covered by

the Sun in

t

is

vt

, where v is the daily speed of the Sun, approximately 15º/day;

t

is the time

between sunset

t

and the moment t of visibility of the crescent. The arc D is

0

cos ,

D vt

therefore, the time when the first lunar crescent is observed is

1

0 0

0 0

24

cos tan tan 90º .

15sin cos 360 15 sin cos

d d

t t t t t

The angle

between the parallel where the Sun is and the horizon, we calculate it by

2 2

0

cos sin

tan .

sin

When the declination of the Sun is negative, then

0 0

if 0 and if 0 ,

t t t t

which explains that the crescent is first seen in the northern latitudes. The opposite occurs when

the declination of the Sun is positive.

Appendix 5: Software LATITUDE1 and LATITUDE2

To find approximately the depression of the Sun when the lunar crescent begins to be visible

for a specific geographical position, we use two software with the following characteristics:

LATITUDE1

Determine the topocentric position of the Moon based on the latitude and the depression

of the Sun (applying the algorithm of appendix 1).

Initial data: Geocentric position of the Moon

0 0 0

, ,

h A

for a reference position, defined by

latitude 0

0

, depression of the Sun 0

0

d

, geographic longitude

, for the date considered.

Input: Latitude

.

Output: Moon topocentric position

, ,

T T

h A

.

LATITUDE2

Determines the depression of the Sun in which the lunar crescent begins to be visible

(applying the algorithms of appendixes 2 and 3).

Initial data: Topocentric position of the Moon

, ,

T T

h A

for latitude

, depression of the

Sun 0

0

d

, geographic longitude

for the date considered.

Output: Depression d, in which the crescent of the Moon begins to be visible.

Appendix 6: Tables

For example, we put below two of the tables used to make drawings 2 and 3, calculated by

the approximate method previously exposed and therefore have an error of a few minutes.

TIME OF THE FIRST SIGHT OF THE LUNAR CRESCENT

Latitude Moon

topocentric

altitude

T

h

Difference

of azimuth

A

Moon's

topocentric

declination

T

True

sun se t

t

(hours and

minutes)

First

visibility of

the crescent

t

(hours and

minutes)

70º

60

50

40

30

20

10

0

-10

-20

-30

-40

-50

-60

-70

3.44º

7.10

9.77

11.91

13.56

14.69

15.29

15.32

14.81

13.74

12.17

10.13

7.64

4.66

0.53

-13.65º

-12.87

-11.41

-9.33

-6.85

-4.07

-1.07

1.89

4.92

7.80

10.45

12.80

14.83

16.57

18.31

-15.86º

-15.80

-15.71

-15.23

-15.47

-15.31

-15.15

-14.98

-14.80

-14.63

-14.47

-14.32

-14.19

-14.08

-13.99

Not visible

Not visible

Not visible

6.06º

5.30

4.98

4.83

4.82

4.91

5.19

6.09

Not visible

Not visible

Not visible

Not visible

14h-15m

16-02

16-45

17-12

17-32

17-47

18-01

18-13

18-26

18-40

18-55

19-14

19-41

20-25

22-12

Not visible

Not visible

Not visible

17h-46m

17-58

18-09

18-21

18-34

18-47

19-03

19-25

Not visible

Not visible

Not visible

Not visible

Depression = 0º Depression

when the

visibility of t

he crescent

begins

d

0 0 0 0

0 0 0

16º14 25 1º53 25 14º 54 33 0º

18 13 22 0º 0º 17º 26 4

h m s

h A d

t UT

January 30, 2025

Wenceslao Segura González

April 28, 2025

Latitude Moon

topocentric

altitude

T

h

Difference

of azimuth

A

Moon's

topocentric

declination

T

True

sun se t

t

(hours and

minutes)

First

visibility of

the crescent

t

(hours and

minutes)

60º

50

40

30

20

10

0

-10

-20

-30

-40

-50

-60

8.92º

12.38

14.71

16.28

17.15

17.31

16.80

15.63

13.85

11.52

8.67

5.30

1.189

-17.17º

-13.88

-10.59

-7.13

-3.58

-0.78

3.75

7.03

10.00

12.53

14.53

15.88

16.31

24.86º

24.98

25.14

25.30

25.47

25.65

25.83

26.01

26.18

26.33

14.53

15.88

26.63

Not visible

5.52º

4.36

4.09

3.98

3.99

4.08

4.26

4.62

5.41

Not visible

Not visible

Not visible

21h-16m

20-07

19-28

19-01

18-39

18-20

18-03

17-46

17-27

17-05

16-38

15-59

14-50

Not visible

20h-47m

19-54

19-22

18-58

18-38

18-21

18-05

17-49

17-33

Not visible

Not visible

Not visible

Depression = 0º Depression

when the

visibility of

the crescent

begins

d

0 0 0 0

0 0 0

11º12 15 8º 26 1 22º 22 30 0º

17 57 27 0º 0º 23º19 34

h m s

h A d

t UT