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FOTHERINGHAM'S GRAPHS OF VISIBILITY OF THE FIRST LUNAR CRESCENT

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Abstract

We call Fotheringham curves of visibility of the first lunar crescent graphs of the altitude of the center of the Moon and its difference in azimuth with the center of the Sun (represented at the moment when the center of the true Sun is on the horizon), which separates the zones lunar visibility and invisibility. These are multiparameter curves, which are dependent on astronomical and atmospheric parameters. In this investigation, we find the Fotheringham graphs deriving them from the Segura (2022b) lunar visibility theory and check their dependence on astronomical and atmospheric parameters.
FOTHERINGHAM'S GRAPHS OF VISIBILITY OF THE FIRST LUNAR CRESCENT
Fotheringham's graphs of visibility
of the first lunar crescent
Wenceslao Segura González
e-mail: wenceslaoseguragonzalez@yahoo.es
Independent Researcher
Abstract. We call Fotheringham curves of visibility of the first lunar crescent graphs of the
altitude of the center of the Moon and its difference in azimuth with the center of the Sun
(represented at the moment when the center of the true Sun is on the horizon), which separates the
zones lunar visibility and invisibility. These are multiparameter curves, which are dependent on
astronomical and atmospheric parameters. In this investigation, we find the Fotheringham graphs
deriving them from the Segura (2022b) lunar visibility theory and check their dependence on
astronomical and atmospheric parameters.
1. Fotheringham's graphs
In 1910 Fotheringham proposed the first modern criterion of visibility of the first lunar crescent
(Fotheringham, 1910). From observations, he drew a graph showing the limiting true altitude of the
center of the Moon and its azimuth difference from the Sun at the moment when the true sun
center is on the horizon. We will call this type of criterion graphs-Fotheringham (drawing 1).
1
Drawing 1.- The blue curves are the Fotheringham criterion, and the red curves are the Maunder criterion.
The remaining modern Fotheringham-type criteria are similar to the Maunder curve. If at the moment when
Sun's center is on the horizon, the position of the Moon (given by apparent or true altitude and difference
in azimuth between the centers of the Moon and the Sun) is above the curves, the crescent will be seen.
With solid lines is the graph for apparent altitudes (with refraction and parallax), and the dashed lines are for
true altitudes.
6
7
8
9
10
11
-5- 10- 15
- 20
- 25 5 10 15 20 25
Azimuth difference between the centers of the Moon and the Sun
Altitude of the
Moon's center
12
Fotheringham
Maunder
First version November 4, 2022
Wenceslao Segura González
There are other Fotheringham-type graphs, which we show in table 1 (Fatoohi, 1998, 94-
144), where we convert the true altitude to apparent altitude (parallax and refraction).
The lunar crescent is not visible for the positions of the Sun and the Moon that appear in the
Fotheringham graphs; over time, the positions of the Sun and the Moon vary as a result of their
daily movement, improving the visibility conditions by increasing the depression of the Sun, until the
time when the crescent is visible.
If we want to follow the movement of the Moon and the Sun from the position shown in the
Fotheringham graphs, it is necessary to know the geographical latitude of the place of observation
and the declination of the Moon, which define the inclination and position of the parallels that in its
movement daily follow the Sun and Moon.
Fotheringham states, «given a clear sky, the problem is almost purely astronomical, not
atmospheric. As the solution considers nothing except the relative positions of the Sun, Moon, and
horizon, it is independent of differences in latitude. It ought, therefore, to apply to any place, subject
to a slight modification for permanent differences in the clearness of the air».
Contrary to the above, Fotheringhan-type visibility curves depend, as we will see later, on the
following parameters: Moon declination, geographic latitude, atmospheric attenuation constant, Earth-
Moon distance, probability of vision, and the atmospheric conditions that determine the luminance
of the twilight sky.
2. Astronomical calculations
The astronomical parameters that characterize the Fotheringham graphs are the Moon's
topocentric declination

, geographic latitude

, and the Earth-Moon distance r. We give arbitrary
values of the topocentric altitude of the center of Moon
0
h
and the difference in azimuth between
centers of the Moon and Sun
0
A
, at the moment when the depression of the center of the true
Sun is zero 0
0
d
and check if there is visibility. In this way, we find a graph with the altitude of the
Moon and its azimuth difference with the Sun, depending on the Moon's declination, geographic
latitude, and Earth-Moon distance (and other non-astronomical factors, as we will see later).
To do astronomical calculations, we need to know the Moon's topocentric declination,
geographic latitude, and Earth-Moon distance. With the arbitrary values
0
h
and
0
A
, we find the
values of the Moon's topocentric altitude and its azimuth difference with the Sun for arbitrary
values of the Sun's depression d.
a) Calculation of the azimuth of the center of the Moon when the depression of the Sun is
null
Applying the law of cosines to the spherical triangle with vertices at the pole, zenith, and center of
2
12.0 / 11.1
11.9 / 11.0
11.4 / 10.5
11.0 / 10.1
-
10.0 / 9.1
7.7 / 6.8
0
5
10
15
19
20
23
11.8 / 11.0
11.3 / 10.5
9.7 / 8.9
9.7 / 8.9
9.7 / 8.9
7.3 / 6.5
11.0 / 10.2
10.5 / 9.7
9.5 / 8.7
8.0 / 7.2
-
6.0 / 5.2
-
10.7 / 9.9
10.3 / 9.5
9.4 / 8.6
7.6 / 6.8
6.3 / 5.5
-
-
10.4 / 9.6
10.0 / 9.2
9.3 / 8.5
8.0 / 7.2
6.6 / 5.8
6.2 / 5.4
4.8 / 4.0
10.3 / 9.5
9.9 / 9.1
9.15 / 8.3
7.9 / 7.1
-
6.4 / 5.6
5.6 / 4.8
Fotheringham Maimonides Maunder Schoch Neugebauer Ilyas
Table 1.- Fotheringham-type criteria for visibility of the first lunar crescent. In each column is the minimum
true and apparent altitude of the center of the Moon to see the crescent for each azimuth difference at the
moment when the center of the true Sun is on the horizon.
Minimum true / apparent altitude to see the crescent
FOTHERINGHAM'S GRAPHS OF VISIBILITY OF THE FIRST LUNAR CRESCENT
the Moon, we find the azimuth of the Moon
0
A
when the depression of the Sun is null
0 0
0
0
sin sin sin
cos .
cos cos
h
Ah
We define the azimuth as the angle measured on the horizon from the south in the retrograde
direction.
b) Calculation of the azimuth of the center of the Sun when the depression of the Sun is null
0 0 0
.
A A A
c) Calculation of the topocentric phase angle
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 0 0 0 0 0
cos sin sin cos cos cos cos cos ,
h d h d A h A
we consider depression with the positive sign. Since the Earth-Sun distance is much greater than
the Earth-Moon distance, then
180 ,
L
a
L
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 when its depression is null
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 0 0
sin sin sin cos cos cos cos cos .
d d A A
e) Calculation of the topocentric hour angle of the Sun when the depression of the Sun is
null
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
d
H
f) Calculation of the topocentric hour angle of the Moon when the depression of the Sun is
null
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
0 0
0
0
sin sin sin
cos .
cos cos
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
0
0
sin sin sin
cos .
cos cos
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
.
T H H
j) Calculation of the topocentric hour angle of the center of the Moon when the depression
of the Sun is d
0
.
H H T
k) Calculation of the topocentric altitude of the center of the Moon when the depression of
the Sun is d
0 0
sin sin sin cos cos cos .
h H
l) Calculation of the difference in azimuth between the centers of the Moon and the Sun
when the depression of the Sun is d
3
(1)
(2)
Wenceslao Segura González
4
0
sin sin cos
cos .
cos cos
h d
A
h d
m) Calculation of the topocentric altitude of the center of the crescent
We find the topocentric altitude of the center of the Moon 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 2, there are two right spherical triangles, applying the law of sines
1
1 0 1
0 1
sin
sin sin sin
sin ; sin 180 ,
sin sin sin sin 180
h d h
d
we deduce
0
120
sin sin cos 180
1
sin ; ,
sin sin 180
1
d h
h
when the angles are small, (3) reduces to
1 0
1180
1d 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 2, we find the topocentric altitude of the center of the
lunar crescent
1
1
sin
sin sin ,
sin
c
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
0
1
1 .
180
c
d h
h h R
From drawing 2
1
1cos
cos
cos ; cos
cos cos c
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
c
c
R
A A OD OF A h h
and in the plane trigonometry approximation is
0 1
1
sin ; .
180 tan
c c
d
A A h h
A
B
C
O
h
c
h
d
horizon
Drawing 2.- 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 ; ; ; .
c
OA OC BA R A GD A GF A FD
D
F
G
(3)
FOTHERINGHAM'S GRAPHS OF VISIBILITY OF THE FIRST LUNAR CRESCENT
5
The difference between the results found by spherical and plane trigonometry is negligible for our
problem.
n) Calculation of the apparent altitude of the center of the crescent
With the topocentric altitude
c
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
c ca
ca
ca
h h
hh
with the angles in degrees.
3. Lunar photometry
To find the visibility criterion based on the altitude of the Moon and its difference in azimuth
with the Sun (or Fotheringham graphs), we use Segura's theory (2022b), by which we divide the
lunar crescent into portions of one minute of angular length. Table 2 shows the stellar magnitude of
each of these portions (Segura, 2022a), expressed as a function of the position angle of the considered
lunar portion and the topocentric phase angle.
To find the magnitude increase caused by atmospheric absorption, we use the formula of
Kasten and Young (1989) and the atmospheric attenuation constant, which is another parameter of
the Fotheringham graphs.
The central part of the lunar crescent (which has a null position angle) has the greatest
illumination (or smaller magnitude) due to its greater width, and the Sun's rays fall with less obliquity,
having less shielding by the irregularities of the lunar surface. Therefore, the problem is to know if
this central part will be seen (Segura, 2021a).
The portions into which we divide the lunar crescent are smaller than the resolving power of
the human eye (which we assume is 1 minute of arc); therefore, it will be the illuminance that
reaches the observer (and not the luminance) that determines whether you will see the image
(Segura, 2021b). The size and shape of the portions into which we have divided the lunar crescent
is insignificant in the viewing process since we observe them as punctual images.
4. Threshold magnitude
The process of seeing the lunar crescent consists of seeing a luminous object (the crescent)
0
10
20
30
40
50
60
70
80
90
-2.34
-2.23
-2.05
-1.77
-1.39
-0.87
-0.13
1.10
3.25
-1.86
-1.75
-1.57
-1.29
-0.90
-0.35
0.44
1.64
3.79
-1.28
-1.17
-0.97
-0.67
-0.27
0.26
1.03
2.28
4.44
-0.52
-0.42
-0.21
0.09
0.05
1.04
1.82
3.07
5.24
0.55
0.66
0.87
1.17
1.59
2.13
2.92
4.17
6.34
150 155 160 165 170
Topocentric phase angle
Position
angle
Table 2.- Illuminance in magnitudes of the portions of 1' of angular length in which we divide the Moon
crescent, depending on the angles of phase and position (Segura 2022a).
Differential magnitude of the crescent
2.40
2.50
2.72
3.02
3.44
3.99
4.78
6.04
8.20
175
Wenceslao Segura González
against a bright background (the twilight sky). The luminance or bright of the sky after sunset
depends on the altitude in the sky and its difference in azimuth with the center of the Sun. We take
the data from Koomen et al. (1952) (table 3); we do a triple interpolation by Lagrange polynomials
to find the intermediate values.
We call threshold illuminance the minimum illuminance of the image (the crescent Moon) to
see it against a bright background (the twilight sky). We take the data from Knoll, Tousey, and
Hulburt (1946) applicable to non-resolvable images. From these data, we find the relationship
between the threshold illuminance
th
E
and the background luminance
s
B
(Segura, 2021c)
2
log 6.72548 0.50020167 log 0.05862778 log ,
th S S
E B B
the formula (4) is independent of image size as long as it is below the resolving power of the human
eye. With (4), we find the threshold magnitude by
13.98 2.5log .
th th
m E
For example, we calculate table 4, showing the threshold magnitude with the measurements of
Koomen et al. and formula (4) for
0
A
.
To determine if we will see the crescent, we need to know the depression of the Sun d, the
apparent altitude of the central part of the crescent
ca
h
, and its azimuth difference
c
A
with the
center of the Sun. We find the magnitude of the central portion of the Moon by interpolating table
1 and then find the apparent magnitude; then, we find the threshold magnitude by triply interpolating
table 3 and applying formula (4). Finally, we calculate the visibility coefficient
th Moon
m m
where
Moon
m is the apparent magnitude of the Moon (i.e., corrected for atmospheric absorption
and calculated for the apparent position of the center of the crescent). If
0
, there is visibility,,
and otherwise, there is not.
5. Correction for Earth-Moon distance and for viewing probability
As the Earth-Moon distance varies, the lunar magnitude varies. The relationship between
the illuminances observed on Earth when the Moon is at the mean distance
r
, and distance r is
2 2
2.5log
r r
E E m m
r r
E and
E
(m and
m
) are the illuminances (magnitudes) at the distance r and mean distance
r
.
6
2.01
0.51
-1.11
-2.19
-2.93
2.26
0.94
-0.42
-1.74
-2.52
2.21
0.88
-0.49
-1.79
-2.65
3º
6º
9º
12º
15º
2
log (cd m )
s
B
Altitude
0º 10º 30º
Table 3.- The logarithm of the twilight sky luminance as a function of the Sun's depression, altitude y
diferencia de acimut, measured in Maryland (39.0º N, 76.7º W, and 30 m height) by Koomen et al. The
luminance of the sky
s
B
is in cd/m2. We find the intermediate values with a triple interpolation with the
Lagrange polynomials.
Depression
A
2.01
0.66
-0.71
-2.06
-2.89
2.03
0.58
-0.79
-2.14
-2.89
3º
6º
9º
12º
15º
2
log (cd m )
s
B
Altitude
0º 10º
Depression
1.92
0.51
-1.12
-2.27
-2.93
2.16
0.81
-0.51
-1.89
-2.69
2.15
0.84
-0.57
-1.93
-2.74
3º
6º
9º
12º
15º
2
log (cd m )
s
B
Altitude
0º 10º 30º
Depression
22.5º
A
45º
A
1.80
0.33
-1.27
-2.4
-2.98
20º
(4)
(5)
FOTHERINGHAM'S GRAPHS OF VISIBILITY OF THE FIRST LUNAR CRESCENT
7
From the previous formula, it follows that the magnitude of the Moon can vary by 0.287 between its
extreme positions (Meeus, 1981).
Table 2 calculates the magnitude for the mean distance between the Earth and the Moon;
for another distance, it must be corrected by (5).
Blackwell’s experiment (1946) shows that threshold vision is a probabilistic process. We
distinguish three zones of visibility of a luminous object on a bright background. The first area is
when the contrast between the image and the background is very high, the probability of vision is
100%: we always see the object.
Another is the zone of zero visibility; due to the small contrast, the observer never sees the
luminous object. The third is the critical zone of visibility, with an intermediate contrast, where
there is a probability of seeing the object. Under the same conditions, the same observer sometimes
sees the object and sometimes not, with a certain probability given by Blackwell’s results.
Blackwell’s experiment showed that the probability of vision in the critical zone does not
depend on the luminance of the background and is almost independent of the size of the object,
depending exclusively on the threshold contrast. Blackwell gave the threshold contrast of his
experiment for a probability of vision of 50%; that is, the observer sees the object half of the times
he observes it. With a contrast lower than the threshold, the probability of vision is lower and vice
versa.
Blackwell's experiment gives the factor
p
(drawing 3 and table 5), by which we must
-1.3579
-0.5533
0.2422
1.0100
1.7344
2.4026
3.0048
3.5340
3.9861
4.3600
4.6575
4.8833
5.0449
5.1530
-1.4015
-0.5850
0.2117
0.9749
1.6929
2.3558
2.9558
3.4870
3.9453
4.3287
4.6372
4.8725
5.0385
5.1408
-1.4416
-0.6146
0.1842
0.9441
1.6562
2.3155
2.9137
3.4465
3.9101
4.3016
4.6193
4.8626
5.0318
5.1286
-1.4782
-0.6421
0.1577
0.9150
1.6242
2.2799
2.8774
3.4123
3.8806
4.2788
4.6039
4.8632
5.0246
5.1163
-1.5674
-0.7117
0.0946
0.8512
1.5570
2.2103
2.8084
3.3482
3.8255
4.2354
4.5725
4.8304
5.0019
5.0793
-1.5411
-0.6906
0.1132
0.8690
1.5746
2.2276
2.8250
3.3632
3.8383
4.2457
4.5805
4.8372
5.0097
5.0917
-1.6097
-0.7474
0.0645
0.8260
1.5362
2.1934
2.7950
3.3375
3.8167
4.2276
4.5640
4.8192
4.9856
5.0546
-1.5903
-0.7307
0.0784
0.8369
1.5442
2.1989
2.7983
3.3396
3.8183
4.2294
4.5670
4.8244
4.9939
5.0669
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-1.5114
-0.6674
0.1343
0.8903
1.5970
2.2508
2.8479
3.3845
3.8566
4.2602
4.5910
4.8449
5.0173
5.1040
Apparent altitude and
0
A
Depression
02 4 6 8 10 12 14 16
Table 4.- The threshold magnitude of a luminous object with an angular diameter equal to or less than 1' to
be seen at twilight as a function of the apparent altitude and the true depression of the Sun, and for
0
A
.
We find the data by a double interpolation of the first table 3 with the polynomials of Lagrange and the
results of Knoll et al. The threshold magnitude for depression of has been obtained by extrapolation
using the Lagrange polynomials. From the depression of 15º, we assume that the level of night illumination
has been reached; therefore, the threshold magnitude remains constant for depressions greater than 15º.
We found similar tables for the azimuth differences of 22.5º and 45º.
Threshold magnitude
-1.3109
-0.5194
0.2751
1.0486
1.7807
2.4553
3.0602
3.5873
4.0324
4.3954
4.6803
4.8949
5.0513
5.1653
18
Wenceslao Segura González
8
98
90
80
70
60
50
40
30
20
10
2.0
1.62
1.40
1.23
1.11
1.00
0.86
0.74
0.58
0.39
p
p
Table 5.- Relative contrast based on the probability of vision, deduced from drawing 3.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10
20
30
40
50
60
70
80
90
100
Relative contrast,
p
Probability (%)
Drawing 3.- Average probability curve in Blackwell's experience. Relative contrast 1 corresponds to a probability
of 50%. If another probability is desired, the curve determines the coefficient
p
by which the threshold
contrast must be multiplied. For example, a probability of 90% corresponds to the relative contrast 1.62, which
is the factor by which the contrast must be multiplied. The maximum of the curve corresponds to 98%
probability. Curve reproduced from Blackwell's work.
multiply the threshold contrast (
th th s
C B B
) for a probability of 50% to find the threshold contrast
for probability p
50% ,
th th
C p p C
for a uniform luminance image
50%
th th
E p p E
FOTHERINGHAM'S GRAPHS OF VISIBILITY OF THE FIRST LUNAR CRESCENT
and the relationship between the magnitudes is
100% 2.5 log 100% 2.5 log 2.5 log 2.
100%
th th th
p
m p m m p
In our model, we use a vision probability of 100%, so if we want to do the calculation for a
probability p, we have to apply (6).
6. Shape of the visibility curve
In drawing 4, we represent a Fotheringham graph of visibility of the first lunar crescent,
which depends on latitude
30º
, atmospheric attenuation constant
0.2
k, probability of
vision
100%
p, and Earth-Moon distance
384,400
r km
.
In drawing 4, we see a bell-shaped curve, almost symmetrical to the vertical axis, with a
maximum in its central position. In addition, we represent two almost straight lines, which are the
limits of impossibility; that is, they limit the areas of the graph where the Moon cannot be because
then the declination of the Sun would exceed its maximum values
23.5º
.
The points in drawing 4 correspond to the apparent altitude of the center of the Moon and its
difference in azimuth with the Sun at the moment when the true Sun's center is above the horizon.
We divide drawing into three zones: visibility (when the position of the center of the Moon is above
the curve and between the lines of impossibility), the invisibility of the crescent (when the position
of the center of the Moon is below the curve and between the lines of impossibility) and the area
where the Moon cannot be (the areas to the right and left of the drawing).
The limits of impossibility (like those shown in drawing 4) depend exclusively on the Moon's
geographical latitude and topocentric declination. To determine these limits, we use formulas (1)
and (2), finding the limit line
0 0
h f A
for the moment when the center of the true Sun is on the
horizon
1 1
0 0 0
0 0 0
0
sin sin sin sin
cos cos .
cos cos cos
h
A A A h
2
4
12
6
8
10
14
- 5
- 10- 15- 20- 25- 30- 35- 40 403530252015105
Topocentric altitude of the Moon
Azimuth difference between the centers of the Sun and the Moon
Visibility Visibility
Invisibility Invisibility
Impossibility Impossibility
Impossibility
Impossibility
Drawing 4.- Fotheringham graph of lunar crescent visibility, calculated for 30º latitude, atmospheric extinction
coefficient 0.2, mean Earth-Moon distance, and 100% viewing probability. The graph has the shape of a
quasi-symmetric bell with a central maximum. On the sides, we have drawn two very approximately straight
lines, which correspond to the limit of impossibility; that is, it separates the areas where the Moon cannot be
found because then the declination of the Sun would exceed its extreme values. Between the two lines are the
zone of visibility (upper part of the curve) and the zone of invisibility (lower part of the curve).
1
P
2
P
9
(6)
Wenceslao Segura González
In drawing 5, we represent several lines of impossibility depending on the latitude and
declination of the Moon, verifying that the higher the latitude, the lines are more oblique, and the
higher the declination, the lines are farther from the Sun. In some cases (see below), the impossibility
lines do not intersect the Fotheringham curve, but they intersect a critical part of the curve in other
cases.
Fotheringham graphs must be bell-shaped. Let P be a point on the Fotheringham curve, that
is, the limiting position of the Moon at true sunset to be visible. Taking P as the center, we represent
four zones, as shown in drawing 6. If the Moon were in zone A, it would be visible, and therefore
the visibility curve would not pass through this zone; in effect, if the Moon were at A, it would have,
for the same depression of the Sun, higher altitude than if it had been at P, which means that it
would have less atmospheric absorption; furthermore, if the Moon were in zone A, it would have a
greater arc-light than if it were at point P, decreasing the phase angle and increasing the luminosity
of the Moon, which would favor its vision.
If the Moon were in zone B, we would not see it; therefore, the visibility curve cannot pass
through this zone. Indeed, if the Moon were in B, for the same depression, it would have a lower
altitude than if it were in P, having greater atmospheric absorption; In addition, since the Moon is in
B, it would have less arc-light and therefore a greater phase angle, which means that it would have
less luminosity. Due to these circumstances, the Moon would be more difficult to see than if it had
been in P, but since P corresponds to the limit position, the crescent could not be seen at B.
Therefore, the visibility curve can only go through zone C (drawing 6); it must be bell-shaped
and therefore have a maximum.
The slope of the visibility curve increases sharply when the altitude is small. In drawing 4, we
show the points
1
P
and
2
P
with the lowest and highest slopes. Suppose the Moon is in position
2
P
at the true sunset of the Sun. In that case, we will see it when A is very large, and the depression
of the Sun is small, but from table 3, we verify that at large A, the twilight luminosity decreases
10
2
4
6
8
10
12
14
16
18
20
22
- 10
- 20- 30- 40- 50- 60 605040302010 70 80 90
Apparent altitude of center of the Moon
Azimuth difference between the centers of the Sun and the Moon
Drawing 5.- Impossibility limits for -23.5º declination of the Sun. The continuous lines correspond to the
latitude of 30º and the dashed lines to the latitude of 50º. The red lines correspond to topocentric declination
of the Moon of 28º. The blue lines correspond to declination of 0º, and the black lines are for declination of -
28º. We found that the higher the latitude, the steeper the visibility lines, and the higher the declination of the
Moon, the more the impossibility line is further north of the Sun. The impossibility lines for the 23.5º declination
are located further to the south. All lines have a negative slope and the opposite slope they would have if the
latitudes were negative. The line for latitude 50º and declination 28º, corresponding to an extreme situation, is
different from the previous ones, which are approximately straight. In addition, we cannot extend this curve
to a greater altitude.
30º
28º
50º
28º
30º
50º
30º
28º
50º
28º
FOTHERINGHAM'S GRAPHS OF VISIBILITY OF THE FIRST LUNAR CRESCENT
11
Drawing 6.- P is a point on the visibility curve. As shown in the text, the curve cannot continue through zones
A and B, so it must continue its descent through zone C. Therefore, the visibility curve is inclined and
directed downwards.
A
P
Visibility curve
C
considerably; therefore, the crescent will need less luminosity to be seen, that is, the phase angle
does not need to be very large, and therefore the Moon does not have to be excessively far from
the Sun, this means that the visibility curve approaches the Sun, causing its slope to increase.
7. Factors on which the visibility curves depend
In the following drawings, we represent the visibility curves without the impossibility lines
(except drawing 11) for simplification. All the curves are similar; they are bell-shaped, with a
central maximum and an increase in slope at their ends.
In drawing 7, we show two Fotheringham-type visibility curves for latitudes 0º and 45º. We
found that they are little affected by latitude. However, this does not mean that the vision of the
lunar crescent is independent of latitude.
Higher the latitude, the steeper the path the Moon follows as it approaches the moonset; this
means that at the same depression of the Sun, the altitude of the Moon is less than that observed in
a place at a lower latitude. However, the lower the altitude, the greater the atmospheric absorption
and, therefore, the more difficult it is to see.
In drawing 8, we show three Fotheringham-type visibility curves for various atmospheric
extinction coefficients (0.15, 0.20, and 0.30); we verify that the curves highly depend on atmospheric
absorption. When k is smaller, the lunar crescent is visible at a lower altitude; its luminosity is
greater, and we can see it in a brighter sky. That is, there are situations where we do not see the
crescent with one atmospheric absorption constant but might see it with another smaller constant.
In drawing 9, we check the effect that the vision probability has. The smaller p is, we see the
crescent at a lower altitude; this does not mean that if the position of the Moon is above the
Fotheringham graph for probability p, we will see the crescent, but rather that there is a probability
p of seeing the crescent, that is, some observers will see the crescent and others will not (as long
as all the observers have the same visual acuity and make the observation from the same place).
However, if the position of the Moon is above the graph corresponding to the probability of 100%,
all observers will see the crescent.
In drawing 10, we draw visibility curves for the extreme Earth-Moon distances, verifying
that at a shorter distance, we observe the crescent at a lower altitude due to its greater illumination
observed from Earth.
Finally, in drawing 11, we represent the effect that the topocentric declination of the Moon
has on the Fotheringham visibility graphs. In this drawing, we have represented the limits of
impossibility. Only an impossibility line cuts the curve for declinations of 28º and -28º.
8. Visibility curve for extreme conditions. Danjon limit
We verify that the Fotheringham graphs are multiparametric and very different depending on
the atmospheric absorption, the probability of seeing, or the declination of the Moon. Therefore, for
these curves to be helpful, they must be prepared for each specific situation.
For all the above, we can find an extreme curve, that is, prepared with the extreme values of
the parameters: atmospheric extinction coefficient 0.11, probability of vision 10%, Earth-Moon
distance 356,000 km and dependent on the declination of the Moon, considering the graph independent
Wenceslao Segura González
12
Drawing 7.- Fotheringham graphs for latitudes of 0º (red curve) and 45º (black curve). We verify that the
visibility curve is almost independent of geographic latitude. The other parameters of the curve are declination
0º, atmospheric extinction coefficient 0.2, probability of vision 100%, and mean distance Earth-Moon. We
omit the limits of impossibility for clarity.
Topocentric altitude
Moon’s center
Azimuth difference between the centers of the Sun and the Moon
- 5- 10- 15- 20- 25- 30- 35- 40 5 10 15 20 25 30 35 40 45- 45
4
5
6
7
8
9
10
11
12
13
45º
Drawing 8.- Fotheringham graphs for various atmospheric extinction coefficients:
0.3
k (black curve),
0.2
k (blue curve) and
0.11
k (red curve). We found that atmospheric absorption considerably modifies
the lunar visibility graphs. The lower the extinction constant we can see the lunar crescent at a lower altitude.
The remaining parameters of the curves are declination 0º, latitude 30º, probability of vision 100%, and mean
distance Earth-Moon.
Topocentric altitude
Moon’s center
Azimuth difference between the centers of the Sun and the Moon
- 5- 10- 15- 20- 25- 30- 35- 40 5 10 15 20 25 30 35 40 45- 45
4
5
6
7
8
9
10
11
12
13
0.2
k
0.3
k
0.11
k
FOTHERINGHAM'S GRAPHS OF VISIBILITY OF THE FIRST LUNAR CRESCENT
13
Topocentric altitude
Moon’s center
Azimuth difference between the centers of the Sun and the Moon
Drawing 9.- Fotheringham graphs for various vision probabilities: 100% (blue curve), 50% (black curve), and
10% (red curve). We note that the viewing probability significantly affects the lunar visibility curves. The
lower the probability, the easier it is to see the crescent.The other parameters are Moon declination 0º, latitude
30º, and mean Earth-Moon distance.
- 5- 10- 15- 20- 25- 30- 35- 40 5 10 15 20 25 30 35 40
4
5
6
7
8
9
10
11
12
4
5
6
7
8
9
10
11
12
5 10 15 20 25 30 35 40- 5
- 10
- 15- 20- 25- 30- 35- 40
Topocentric altitude
Moon’s center
Drawing 10.-Fotheringham graphs for extreme Earth-Moon distances. The red curve corresponds to the
maximum distance (406,720 km), and the blue curve is to the minimum distance (356,375 km). At a smaller
distance, the Moon is more luminous as seen from the Earth, which facilitates observation; that is, we can
observe it at a lower altitude. The other parameters are Moon declination 0º, latitude 30º and viewing probability
100%.
Azimuth difference between the centers of the Sun and the Moon
Maximum distance
Minimal distance
Wenceslao Segura González
14
of geographic latitude.
The result appears in drawing 12, where we represent the visibility curves for various
topocentric declinations (0º, 28º, and 28º). If the position of the Moon in the diagrams is below the
curves, we understand that it is impossible to see the crescent; however, there is no guarantee that
the crescent can be seen if the position of the Moon is above the curves.
The Danjon limit is the minimum topocentric arc light with which we can see the lunar
crescent. From drawing 12, we verify that the Danjon topocentric limit is approximately 7.8º, a
value consistent with that given by other observers (Segura, 2021d). We add that a modification of
the atmospheric conditions of the sky caused, for example, by the elevation of the observation site,
could decrease the previously calculated Danjon limit.
9. Conclusions
In 1910 Fotheringham devised a criterion for the visibility of the first lunar crescent, using a
true altitude-azimuth difference diagram between the centers of the Sun and the Moon when the
center of the true Sun is on the horizon. Fotheringham warned that the graph was independent of
geographic latitude and subjected exclusively to some variation due to atmospheric causes.
Other Fotheringham-type criteria have been proposed, all using observational data. In this
investigation, we find Fotheringham graphs but using the physical theory developed by Segura.
We conclude that the Fotheringham graphs are multiparametric; they depend on several
parameters, and there is not a single graph, as has been assumed so far. We verify that the
geographical latitude barely modifies the graph, although the observation of the crescent is very
dependent on the geographical position of observation. However, the graphs are very dependent on
atmospheric absorption and viewing probability and also depend on the Moon's declination.
We determine the characteristics of the Fotheringham graphs: they are bell-shaped, almost
symmetric, with a central maximum, and at the extremes, it has an increase in slope.
We check that the Moon cannot occupy any position in the
h A
diagram. There are
zones, which we call impossibility, in which if the Moon were there, the Sun would have a declination
that would exceed its limit values. These zones are limited by approximately straight lines, with an
4
5
6
7
8
9
10
11
12
13
- 5
- 10- 15- 20- 25- 30- 35- 40 5 10 15 20 25 30 35 40
Topocentric altitude
Moon’s center
Azimuth difference between the centers of the Sun and the Moon
Drawing 11.- Fotheringham graphs for various topocentric declinations of the Moon:
(solid blue line),
28º
(dotted red line), and
28º
(dashed black line). We draw the limits of impossibility, then the zone
of visibility of the lunar crescent is above the curve, and if the position of the Moon were below the curve, the
crescent would be invisible. We find that declination radically changes the visibility curve. The other parameters
are latitude 0º, mean Earth-Moon distance and 100% chance of seeing.
28º
28º
FOTHERINGHAM'S GRAPHS OF VISIBILITY OF THE FIRST LUNAR CRESCENT
15
5
6
7
8
510 15 20 25 30 35- 30 - 25 - 20 - 15 - 10 - 5
Azimuth difference between the centers of the Sun and the Moon
Topocentric altitude
of center of the Moon
Drawing 12.- Fotheringham limit graphs for various topocentric declinations of the Moon. We understand
that if the position of the Moon is below these curves, it is impossible to see it; however, there is a possibility
of seeing the Moon if its position is above the curves.
28º
28º
inclination depending on the terrestrial hemisphere.
We have drawn the Fotheringham-type curves for an extreme situation when the best visibility
conditions exist. From these graphs, we deduce the value of the topocentric Danjon limit, which for
the optimal conditions chosen is 7.8º, very close to the values given by other authors, which confirms
the hypotheses on which we base our theory.
In the algorithm used, we assume that the right ascension and declination of the Sun and the
Moon do not vary in the time it takes for the Moon to reach the moonset. We consider the observer's
visual acuity excellent, and we have not considered the variation of atmospheric absorption with
the elevation of the place of observation nor the changes in the luminance of the sky with the site
and season (Mikhail et al., 1995) (Arumaningtyas and Raharto, 2008).
10. Bibliography
* Arumaningtyas, E. P.; Raharto, M. (2008). A Study of the Impact of Sky Brightness on Hilal
Visibitility, Proceedings of the 2nd International conference on Mathematics and Ntural Sicences,
p. 1311-1314.
* Blackwell, H. R. (1946): Contrast Thresholds of the Human Eye, Journal of the Optical Society
of America 36 (1), 624-643.
* Fatoohi, L. J. (1998): First Visibility of the Lunar Crescent and other Problems in Historical
Astronomy, Durham University, Thesis.
* Fotheringham, J. K. (1910): On the smallest visible phase of the Moon, Monthly Notices of the
Royal Astronomical Society 70, 527-531.
* 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.
* Meeus, J. (1981): Extreme perigées and apogées of the Moon, Sky and Telescope 62, 110-111.
* Mikhail, J. S.; Assad, A. S.; Nawar, S. and Hassanin, N. Y. (1995): Improving the Crescent
Visibility Limits Due to Factors Causing in the Sky Twilight Brightness, Earth, Moon and Planets
28º
28º
Wenceslao Segura González
16
70, 109-121.
* Segura, W. (2018): Movimiento de la Luna y el Sol, eWT Ediciones.
* Segura, W. (2021a): Predicting the First Visibility of the Lunar Crescent, Academia Letters,
Article 2878.
* Segura, W. (2021b): Danjon Limit: Bruin's Method, https://www.researchgate.net/publication/
356128660_Danjon_Limit_Bruin%27s_Method.
* Segura, W. (2021c): Danjon Limit: Sultan's Method, https://www.researchgate.net/publication/
350609317_Danjon_Limit_Sultan%27s_Method.
* Segura, W. (2021d): Danjon Limit: Schaefer’s Method, https://www.researchgate.net/publication/
355188668_Danjon_Limit_Schaefer%27s_Method.
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* Segura, W. (2022b): Visibility Window of the First Lunar Crescent, https://www.researchgate.net/
publication/363785232_Visibility_Window_of_the_First_Lunar_Crescent.
ResearchGate has not been able to resolve any citations for this publication.
Magnitude of the Moon at large phase angles
  • W Segura
* Segura, W. (2022a): Magnitude of the Moon at large phase angles, https://www.researchgate.net/ publication/362491492_Magnitude_of_the_Moon_at_large_phase_angles.