Content uploaded by Wenceslao Segura
Author content
All content in this area was uploaded by Wenceslao Segura on Nov 11, 2021
Content may be subject to copyright.
DANJON LIMIT: BRUIN’S METHOD
Danjon Limit: Bruin’s Method
Wenceslao Segura González
email: wenceslaoseguragonzalez@yahoo.es
Independent Researcher
Abstract. Bruin (1977) devised a procedure to find out the visibility of the first crescent Moon. He
applied various simplifications to his theory, not all of them acceptable. We rethink Bruin's method
by making some corrections: we take into account the variation of the luminance of the Moon with
the phase, we use the experimental results of Knoll et al. (1946) on threshold contrast, we apply
Riccò's law, and we consider the atmospheric extinction coefficient to be variable. We use the
theory to derive the Danjon limit.
1. Introduction
The crescent of the Moon is visible for the first time shortly after the conjunction, on the
western horizon and after sunset. We want to know in advance when an observer with good vision,
sees the crescent Moon from a geographical position and with good atmospheric conditions. The
importance of this problem is that the first sight of the lunar crescent marks the beginning of the
lunar month in the Islamic calendar.
This problem has been treated in two different ways: using empirical criteria and by physical
procedures.
Bruin (1977) is considered as the first author who has dealt with the visibility of the crescent
Moon by physical methods, but Samaha, Assad, and Mikhail (1969) previously devised a theory
with physical criteria to find the first vision of the crescent Moon; however, Bruin's research, which
appeared in a prestigious astronomical journal, has been the most widely publicized.
To determine the moment when the first vision of the Moon crescent occurs, we have to
consider the following three data:
a) The luminance of the twilight sky where the Moon is, which depends on the place
of observation; depression of the Sun below the horizon; the altitude of the Moon; azimuth
difference between the Moon and the Sun and other unpredictable factors.
b) The observed luminance of the Moon, which depends on the phase angle; of the
atmospheric extinction coefficient; the altitude of the Moon, and the libration.
c) The limit of human vision or minimum contrast (A.4) is required to see the Moon
against the bright background of the twilight sky. This criterion depends on several factors
such as the age of the observer; shape, orientation, and color of the illuminated object;
prior knowledge of where to look; whether it is mono or binocular vision; whether there is
magnification; whether artificial or natural pupil is used; the duration of the observation;
the number of observers,...
In the critical zone, the visibility of an object is probabilistic (Segura, 2021c), that is to say,
that, under the same conditions, sometimes the observer sees the object and sometimes not, with a
certain probability. Finally, note that the results of the measurements are dependent on the methodology
of the experiment, so the results of the measurements carried out to determine the threshold contrast
are somewhat different from each other.
In this investigation, we consider that the distance from the Earth to the Moon and the Sun is
the average. The altitude of the Moon is apparent, that is, corrected for parallax and refraction; the
Sun's depression is without refraction, and the phase angle is topocentric, that is, it is the selenocentric
angle between the center of the Sun and the position of the observer on the surface of the Earth
(see appendix).
;
1
Wenceslao Segura González
As the observation of the Moon is near the horizon, we identify the parallax of the Moon with
the equatorial horizontal parallax
, to which we give the approximate value of one degree (A.12).
In this investigation, we will expose the Bruin method for the visibility of the crescent Moon;
we will expose its defects and propose a modified method. Specifically, what interests us is the
determination of the Danjon limit (1932 and 1936), the minimum angle between the centers of the
Moon and the Sun at which the crescent can still be visible (Segura, 2021a, 2021b, 2021d).
2. Bruin's method simplifications
As we will see later, Bruin (1977) adopts simplifications in his theory of lunar visibility, some
acceptable and others wrong.
a) He assumes that the luminance of the twilight sky
S
B
is uniform and only depends on
the depression of the Sun below the horizon.
b) Bruin considers the observed luminance of the Moon uniform, that is, the same in all
the illuminated points of the crescent, and assumes that it is 3,600 cd/ m2 at the zenith and
sea level.
c) Assume the observed luminance of the Moon independent of the phase angle
; as we
shall see, this simplification is manifestly erroneous and represents the most severe mistake
of Bruin's method.
d) He gives a fixed value to the atmospheric extinction coefficient k (approximately
0.25).
e) At limit vision, he reduces the center of the crescent to a circle with a diameter equal
to the maximum width of the crescent.
f) Determine contrast and threshold luminance by extrapolating Siedentopf (1941)
measurements.
3. Bruin's method of crescent visibility
To apply Bruin's method and find out if the crescent will be seen at a specific time, we need
to know: the depression of the Sun below the horizon d, the topocentric phase angle
, the apparent
altitude of the Moon h, and the atmospheric extinction coefficient k. With this data, we do the
following:
a) We find the limit altitude
lim
h
, the minimum altitude that the Moon must have at the
moment considered to be visible.
b) If the altitude of the Moon h is greater than the limit altitude, the Moon will be visible,
and it will not be otherwise.
The following procedure calculates the limit altitude:
a) We determine the luminance of the sky, knowing the depression of the Sun.
b) We find the maximum width of the lunar crescent
max
w
knowing the topocentric phase
angle (A.11).
c) From the experimental data, we find the threshold contrast and the threshold luminance
th
B
for viewing the Moon.
d) From the law of atmospheric absorption (A.6) and (A.7), we find the limit altitude or
altitude at which the Moon has a luminance equal to the threshold luminance.
Bruin developed a graphical method and plotted lim
h d
as a function of d for various phase
angles. He used a single value for the atmospheric extinction coefficient k; a more thorough
investigation requires that k be variable.
For each value of the phase angle, Bruin gives values to the depression d, finding by the
method indicated above the limit altitude
lim
h
. Thus he obtained several curves, each of them
characterized by the value of the phase angle.
To determine if the Moon would be visible, he determined for the time of sunset of the day of
interest, the apparent altitude of the Moon h, the geocentric depression of the Sun d, and the
topocentric phase angle
. Then draw the horizontal line
h d
on the graph. If this line intersects
the curve characterized by the phase angle
, the Moon will be visible; the cutoff points will
indicate the beginning and end of the Moon's visibility period. We will not see the Moon if the
horizontal line
h d
is below the graph (drawing 2).
2
DANJON LIMIT: BRUIN’S METHOD
4. Riccò's law for non resolvable images
Riccò's law proposed in 1877 applies to images of unresolvable size, that is to say, smaller in
size than the resolving of the human eye (A.14), which we estimate to be 1 arc minute; then the
image will be seen if its illuminance exceeds a threshold limit
th
E
that depends exclusively on the
background luminance
S
B
(Segura, 2021c). Let us consider a circular image, with uniform luminance,
diameter
0
, and solid angle
0
, smaller than the resolution of the human eye; then the threshold
contrast when the image is on a background of luminance
S
B
is
0
0
, ,
th S
th
th S
S S
E B
B
C B B B
for another image of size
, solid angle and not resolvable, its threshold contrast will be
2 2
0 0
, , ,
th S
th S th S th S
S
E B
C B C B C B
B
we have applied (A.3) and (A.9); then Riccò's law for not resolvable images is
2
,
th S S
C B B
the coefficient
only depends on
S
B
2
0 0
,
S th S
B C B
In table 1, we have calculated the coefficient
S
B
according to Knoll, Tousey, and Hulburt
(1946) for a probability of vision of 100%. Table 2 represents coefficient
S
B
according to
Blackwell (1946), which corresponds to a probability of vision of 50%. Finally, in table 3 are the
results of Seidentopf (1941).
We have obtained Riccó's law assuming that for images smaller than the resolving of the
3
7.1671
6.9609
6.7255
6.4607
6.1667
5.8433
5.4906
5.1086
4.6972
4.2566
44.99
29.01
21.55
10.24
5.21
2.83
1.65
1.03
0.68
0.49
0.37
0.30
0.26
Table 1. From the data of Knoll et al. (1946), we have calculated the threshold illuminance
th
E
for the
background luminance
S
B
.
th
B
is the threshold luminance obtained by
0
th th
B E
,
0
is the solid angle of
a circle with an angular diameter of 1 minute. In the last column is the coefficient
S
B
that appears in
Riccó's law (1) and that we calculate by (2). The asterisks signify extrapolated values when fitting the curve
with a sixthorder polynomial curve.
2
log
cd m
s
B
log
lux
th s
E B
S
B
8
0 0
1 6.64572 10 sr
1.0241
1.6465
2.8311
5.2090
10.251
21.585
48.625
117.18
302.18
833.41
2*
1.7*
1.5*
1
0.5
0
0.5
1
1.5
2
2.5
3
3.5
0
,
th S
B B
(1)
(2)
Wenceslao Segura González
human eye, the illuminance that reaches the observer is the factor that determines the visibility of
the image on a bright background (Segura, 2021c), that is, that there is a threshold illuminance and
if the illuminance of the image is greater than the threshold, the image will be seen.
The opposite argument is also true. If we assume that Riccó's law is valid, that is, that the
threshold contrast is inversely proportional to the square of the angular diameter of the image
(supposedly circular), then there is a threshold illuminance, showing that for nonresolvable images,
the factor that determines its visibility is the illuminance and not luminance.
Riccó's law is exact for nonresolvable images and approximately true for small images
although larger than the resolving of the human eye,
as confirmed by the experiment of Blackwell
(1946).
4
Table 2. Riccò's law coefficient
derived from Blackwell's data for an image of 0.595 arc minutes and a
probability of vision of 50%. For a probability of 100%, we must multiply
by 2.
7.5881
7.5061
7.2237
6.9202
6.5735
6.1820
5.7706
5.3689
4.9781
4.5060
2
log
cd m
s
B
S
B
1.0974
1.3254
2.5391
5.1080
11.3480
27.9545
72.08
181.7677
447
1325.63
th
B
1
0.5
0
0.5
1
1.5
2
2.5
3
3.5
8
0 0
0.595 2.35275 10 sr
4.0023
1.5995
1.0246
0.5864
0.4408
0.3170
0.2579
0.2038
0.1752
0.1484
log
lux
th s
E B
7.4239
7.1198
6.7963
6.5189
6.2033
5.9468
5.6092
5.3027
4.9908
4.6867
5.67
3.61
2.41
1.44
0.94
0.54
0.37
0.24
0.15
0.10
2
log
cd m
s
B
log
lux
th s
E B
S
B
8
0 0
1 6.64572 10 sr
0.57
1.14
2.41
4.56
9.42
17.01
37.01
74.95
153.71
309.60
1*
0.5*
0
0.5
1
1.5
2
2.5
3*
3.5*
0
,
th S
B B
Table 3. Riccò's law coefficient
derived from Seidentopf (1941). The asterisk means extrapolations
DANJON LIMIT: BRUIN’S METHOD
5. The magnitude of the Moon
Photometric measurements of the Moon at large values of the phase angle are difficult since
its observation has to be made at a low altitude above the horizon and therefore, is highly affected
by atmospheric attenuation; also, the observation has to be done with twilight light, therefore the
Moon's own illumination is added to the illumination of the sky, and finally, it must be added that the
Moon is rarely observed, and for a very short time, with a phase angle greater than 170º because
the brightness twilight sky masks the light emitted by the Moon.
Allen (1973, p. 144) gives the following formula for the magnitude of the Moon out of the
atmosphere as a function of the geocentric phase angle
9 4
12.73 0.026 4 10m
is in degrees. We cannot extend the formula (3) to more than 150º of the phase angle, since the
luminance of the Moon at large phase angles is affected by the shielding and microshielding of the
lunar surface caused by the inclination of the solar rays close to the horizon of the Moon. We check
that (3) is unsuitable for use at large phase angles. In summary, (3) gives an illuminance greater
than the real one for large phase angles.
(Samaha, Asaad and Mikhail, 1969) and (Russell, 1916) proposes a law by which the Moon's
magnitude depends on the logarithm of the cube of 180
. Adjusting (3) to a law of this type we
obtain *
3 2.8026
3.62548 2.33551log 180 3.62548 2.5log 180m
,
(4) can be extended for high phase angle. We assume that (4) is the geocentric magnitude at the
mean distance from Earth
r
, hence
is the geocentric phase angle. From formula (4), we derive
the Moon's luminance, assuming it is the same throughout the crescent. The surface of the lunar
crescent of phase angle
seen in the direction of the Earth is
2
1
1 cos
2
S R
which is the subtraction between the areas of a semicircle and a semiellipse (Segura 2018, p.190),
(Segura 2020b). Its solid angle is
2
2 2 1 cos
2
S R
r r
R is the radius of the Moon, and
r
is the mean distance from the Moon to the Earth. We define the
magnitude by (A.5), then by (4), we find the geocentric illuminance at the mean distance from the
Moon
E
expressed in lux
2.8026 2.8026
8
log 7.0422 log 180 9.0742 10 180E E
and the luminance in cd/m2 is by (A.3)
2
2.8026
8
2
2
9.0742 10 180 ,
1 cos
E r
BR
this formula is valid on the assumption that the luminance of the lunar crescent is uniform and
before atmospheric absorption. Let us indicate that luminance is independent of the distance from
the Moon since it is an intrinsic magnitude of the luminous object.
As we see the crescent near the horizon, the relationship between the topocentric
and
geocentric
phase angles is
1º.
In table 4, we have applied (5), and we find the luminance of the Moon as a function of the
topocentric phase angle, assuming that the luminance of the Moon is uniform. In Table 4, we see
that the luminance of the Moon is highly dependent on the phase angle; therefore, it is unacceptable
5
* We have adjusted formula (3) between the values of 90 and 150 degrees, thus avoid considering the
opposition effect of the Moon. To obtain (4), we have first put the magnitude m of (3) as a function of
3
log 180
, and we fit it to the straight line
3
log 180m a b
, determining the parameters a and b.
Sahama et al. (1969) adopt the formula
3
4.5245 2.5log 180m
instead of (4).
(3)
(4)
(5)
Wenceslao Segura González
to assume, as Bruin does, that the luminance is the same regardless of the phase angle.
The luminance of the Moon is not uniform; it depends on the selenocentric geographic
coordinates, especially on the lunar longitude. However, since the Moon crescent has a minimal
width, it is acceptable to take the uniform luminance.
Schaefer (1991) used Hapke's lunar photometric theory to find the luminance of the Moon at
large phase angles. However, as we have shown (Segura, 2021d), Hapke's theory gives an excessive
darkening of the Moon when the phase angle is large.
6. Atmospheric absorption
The atmospheric absorption is characterized by two factors: the extinction coefficient k and
the airmass X, a measure of the distance traveled by light through the atmosphere. The extinction
coefficient is highly variable, and we cannot know its value in advance. As we will see, this coefficient
greatly affects the visibility of the crescent and is, therefore, the most uncertain factor in the theory
of the visibility of the lunar crescent.
We are interested in calculating the minimum altitude at which the Moon must be to be
visible; that is, by (A.6), we have to calculate the airmass
2.5 log M
th S
B
X
k B B
M
B
is the luminance of the Moon according to the phase angle and outside the atmosphere,
and
th
B
is the threshold luminance that depends exclusively on the luminance of the sky
S
B
(table
4). Finally, from (A.7) or table 6, we determine the apparent altitude of the Moon or limit altitude to
see it.
7. Modified Bruin's method
Taking the ideas of Bruin, we develop a modified method, where we admit the following
simplifications:
a) The sky luminance
S
B
only depends on the depression of the Sun below the horizon
and we use Bruin's measurements (table 5).
b) The crescent has a uniform luminance.
c) We assume that at the limit of visibility, the visible zone of the Moon is a circular disk
of diameter the maximum width of the crescent
max
w
(A.11).
d) Riccò's law (1) is valid.
6
90
100
110
120
130
140
150
160
170
175
860.1
748.6
647.2
553.7
465.9
381.9
299.6
216.3
127.7
78.3
(degrees)
S
B
(cd/m2)
Table 4. Average luminance of the Moon according to formula (5) as a function of the topocentric phase
angle.
(6)
DANJON LIMIT: BRUIN’S METHOD
7
e) We take the coefficient of Riccò's law (2) from the Knoll et al. experiment (table 1)
from which we deduce the threshold contrast and the threshold luminance
th
B
.
f) The luminance of the Moon
M
B
depends on the phase angle by equation (5) and table
4.
g) We deduce the airmass X of the limit altitude of the Moon from (6).
h) We calculate by (A.7) (table 6) the limit altitude.
The procedure to build the graphs that will allow us to know when the Moon will be visible is as
follows:
a) For an extinction coefficient k and a topocentric phase angle
, we give values to the
depression of the Sun d.
b) Once the depression is known, we interpolate table 5 and calculate the luminance of
the sky
S
B
.
c) Knowing the topocentric phase angle, we determine the maximum width of the crescent
by (A.11).
d) Knowing the luminance of the sky, we determine the coefficient
from table 1.
e) By Riccò's law (1), we calculate the threshold contrast and the threshold luminance of
the Moon
th
B
, where we take
max
w
.
f) By (6), we calculate the airmass
lim
X h
of the limit altitude of to see the Moon.
g) By (A.7) find the apparent altitude limit
lim
h
.
h) We represent in the diagram the point
lim
,
d h d
.
i) We take a new value for the depression and obtain a new point, and so on until we can
draw the curve corresponding to the extinction coefficient k and the phase angle
.
j) We obtain new curves with other values of the phase angle and the extinction coefficient.
In table 7, we show an example of the calculations for a topocentric angle of 167º and
atmospheric extinction coefficient 0.25. Until the depression of the Sun of 5º, the Moon is not
observable since the luminance of the sky is very intense. For depression of 5º, there is a limit
0
1
2
3
4
5
6
7
8
9
10
11
12
13*
14*
3.102
2.932
2.659
2.296
1.773
1.227
0.705
0.182
0.341
0.818
1.251
1.6125
1.9375
2.2042
2.4296
Table 5. Luminance of the twilight sky as a function of the Sun depression without refraction, according to
Bruin measurements. Asterisks mean extrapolated values.
2
log
cd m
S
B
degrees
d
Wenceslao Segura González
altitude of the Moon, although this does not mean that it will be visible; for this, the altitude of the
Moon for the corresponding depression of the Sun must be greater than the limit altitude.
In drawing 1, we have drawn some graphs that help us determine if the Moon will be visible.
The graphs are characterized by the topocentric phase angle and the atmospheric extinction
coefficient. In drawing 1, we show the effect of extinction. The right end of the graphs, which we
represent with a broken line, has been obtained by linear extrapolation.
We have found drawing 1 with the measurements made by Knoll et al. that give the minimum
of vision with a probability of 100%. However, we can modify the results for another probability.
Let
th
C p
the threshold contrast to see an image with a probability of vision p and
50%
th
Cthe threshold contrast to see the same image under the same conditions with a probability
8
0
1
2
3
4
5
6
7
8
9
10
37.9196
26.3106
19.4332
15.1477
12.302
10.3058
8.8415
7.7281
6.8565
6.1577
5.5860
h (º) KastenYoung
Airmasss
Table 6. Airmass as a function of apparent altitude, according to the KastenYoung formula (A.7). h is the
apparent altitude of the Moon, that is, corrected for parallax and refraction.
67 8 9 10 11 12 13 14 15
15
25
35
45
55
Drawing 1. Some of the curves obtained by the modified Bruin’s method. The curves with solid lines
correspond to an extinction constant of 0.25. The dotted curve is for
0.2
k. The dashed lines are the linear
extrapolations.
Sun depression (degrees)
65
168º
168º
170º
lim
degrees
h d
DANJON LIMIT: BRUIN’S METHOD
9
0
1
2
3
4
5
6
7
8
9
10
11
12
3.102
2.932
2.659
2.296
1.773
1.227
0.705
0.182
0.341
0.818
1.251
1.613
1.938
1264.70
855.07
456.04
197.70
59.29
16.87
5.07
1.52
0.456
0.152
0.0561
0.0244
0.0115
0.2918
0.3095
0.3477
0.4190
0.5763
0.8711
1.3958
2.4005
4.4532
8.4091
15.918
25.508
41.200





92.77
44.68
23.03
12.82
8.07
5.64
3.93
2.99





1.942
5.115
7.992
10.537
12.548
14.105
15.673
16.858





30.94
10.99
6.76
4.88
3.91
3.37
2.88
2.60





35.94
16.99
13.76
12.88
12.91
13.37
13.88
14.60
d
log
S
B
S
B
S
B
th
B
X
lim
h
lim
d h
Table 7. Calculation of the limit altitude for a topocentric phase angle of 167º and an atmospheric extinction
coeffecient of 0.25. We determine
S
B
by table 5. We calculate
S
B
by table 2. We find
th
B
by Riccó's law
(1). We calculate the air mass by (6) and
lim
h
by table 6.
30
25
20
15
10
6 7 8 9 10 11 12
Sun depression (degrees)
lim
h d
168º 31
0.25k
164º 49 '
0.25k
8º 31
h d
12º 54
h d
Drawing 2. Prediction of lunar visibility by the modified Bruin method. The curve with a solid line corresponds
to the time of sunset on March 14, 2021. The continuous horizontal line is the altitude of the Moon plus the
depression of the Sun (which we take as a positive value) for the same moment. Since the horizontal line does
not intersect the curve, the Moon will not be visible.
The discontinuous curve is the one that corresponds to the time of sunset on February 12, 2021. The
discontinuous horizontal line is the altitude of the Moon plus the depression of the Sun for the same moment.
As the horizontal line intersects the curve, the Moon will be seen. The vision of the crescent begins
approximately when the Sun is 5º 48‘ below the horizon (point A) and will end when the solar depression is 10º
40' (point B). The two curves correspond to observations at the geographical position 36º 1' N, 5º 22' W, and
an atmospheric extinction constant of 0.25.
AB
max
167º 0.25 145.08 0.398 Probability 100%
M
k B
Wenceslao Segura González
of 50%, according to Blackwell
50% 50% .
th th th th
C p p C B p p B
In drawing 3, we show the function
p
according to Blackwell's results. The horizontal axis is
the function
, and the vertical axis is the probability of vision p expressed in units.
To find the threshold luminance for a probability p from the luminance found in Knoll et al.,
that is, with a probability of 100%, we apply
1
100% 100%
100% 2
th th th
p
B p B p B
From drawing 3 we find
100% 2
and deduce the value of
p
.
In drawing 4, we compare the curves for the probabilities 100%, 50%, and 20% for the
topocentric phase angle 168º and the atmospheric extinction constant 0.2. The graph corresponding
to the lowest probability of vision is the one with the lowest limit altitude, which indicates that we
can see the Moon at the most inferior distance from the Sun; we can see it with more anticipation.
8. Visibility of the crescent Moon by Bruin's modified theory
To find out if the lunar crescent will be visible we calculate the topocentric phase angle and
the apparent altitude of the Moon h when the depression of the Sun is 0. We assume that during the
time between sunset and vision of the Moon, the angle
h d
does not vary..
In drawing 2, we analyze the visibility of the Moon on March 14, 2021, and February 12,
2021, at a location with geographic coordinates of 36º 1' N and 5º 22' W and assuming that the
atmospheric extinction constant is 0.25. We determine the phase angle of the Moon at sunset of the
days considered and represent the visibility curves for the two moments considered and the straight
lines
h d
for the two days.
In drawing 2, we see that for March 14, 2021, the horizontal line
h d
does not cut the
10
00.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Relative contrast,
Probability, p
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
by which the threshold
contrast for a probability of 50% must be multiplied. For example, a probability of 90% corresponds to a
threshold contrast of 1.62, which is the factor by which to multiply the contrast for a probability of 50%, to
find the threshold contrast for the 90% probability. The maximum of the curve corresponds to a 98% probability.
Curve reproduced from Blackwell's work.
DANJON LIMIT: BRUIN’S METHOD
curve, indicating that the Moon will not be seen that afternoon. However, for February 12, 2021,
the horizontal line
h d
cuts the visibility curve; therefore, the Moon will be seen that day. Cutting
points of the
h d
line with curve indicate the depression of the Sun at the beginning and end of
the vision of the crescent.
9. Danjon limit
We call the Danjon limit the minimum topocentric angular distance (or arclight) between the
centers of the Moon and the Sun in which it is still possible to see the Moon; that is, at a smaller
angular distance, it is impossible to see the Moon. The smallest arclight is found when the azimuth
difference between the Moon and the Sun is zero. Then
LT
a h d
h
is the topocentric altitude of the Moon without refraction. By (A.12)
180 180 180
LT R
a h d h d R
R
R
is the angle of refraction calculated by (A.10) and h is the apparent altitude (with refraction
and parallax) of the Moon.
To find the Danjon limit, we have to determine the topocentric phase angle that has a visibility
curve such that its minimum or lower point is tangent to the horizontal line 180
R
R
.
R
R
is the
angle of refraction for the limiting height of the minimum of the visibility curve. Drawing 5 shows
the conditions that the phase angle corresponding to the Danjon limit must meet.
It is important to note that the Danjon limit is a topocentric angle. It depends on the atmospheric
extinction coefficient and the probability of vision; therefore, it is not a fixed value, not even for the
same place of observation.
To set conditions for calculating the Danjon limit, we assume exceptional atmospheric
conditions and consider an extinction coefficient of 0.15 and a probability of vision of 20%. Making
calculations like those in table 7 for various phase angles, we find that the conditions required in
drawing 5 occur when the topocentric phase angle is 169.9º, that is, the Danjon limit in Bruin's
theory according to the requirements above is 10.1º, a little higher than that found by other authors.
If we assume other atmospheric conditions and another probability of vision, the obtained
11
11
13
15
17
19
67 8 9 10 11 12
100%
50%
20%
Drawing 4. Lunar visibility curves for a 168º topocentric phase angle, extinction constant 0.2 and various
vision probabilities. When the probability of vision is small, the Moon can be seen at a smaller angular
distance from the Sun; therefore, we can see it with more anticipation.
Sun depression (degrees)
lim
degrees
h d
(7)
Wenceslao Segura González
Danjon limit will be different.
10. Conclusions
In 1977 Bruin proposed a method to determine the visibility of the crescent Moon by comparing
the luminance of the twilight sky with the luminance of the Moon. Bruin proposed some simplifications,
such as assuming that the Moon always has the same luminance; we have corrected it in our
modified method proposal. Bruin did not consider the variation of the atmospheric extinction
coefficient nor the probability of vision.
Like Bruin, we associate a curve to each phase angle which depends on the coefficient of
vision and the probability of vision. Using a graphical method, we determine from these graphs
whether the Moon will be visible or not.
Finally, applying the modified Bruin method, we have determined a value for the Danjon limit,
which we have calculated for an extinction coefficient of 0.15 and a vision probability of 20%. The
topocentric arclight of the Danjon limit that we have found is approximately 10.1º, a little higher
than the value found by other authors.
11. Appendix
* Luminance. Luminance B is the luminous flux (or luminous power) emitted per unit area
perpendicular to the direction of emission and per unit solid angle
2
cos
d
B
dS d
its unit is
1 2
lm sr m
or
2
cd m
;
is the angle between the direction in which the luminance is
measured and the normal to the luminous surface element dS.
* Iluminance. Illuminance E is the luminous flux that reaches the unit area normal to the direction
of incidence and has the unit
2
lm m
or lux
cos
d
E
dS
dS
is the surface on which the light falls and
is the angle between the normal to the surface
dS
and the incidence direction.
* Relationship between luminance and illuminance. There is a relationship between luminance
and illuminance. If d is the solid angle of the surface
dS
on which the light falls observed from
the light source, r is the distance between emitting and receiving surfaces, then it is satisfied
12
lim
h d
Sun depression
Drawing 5. The phase angle of the Danjon limit for an extinction coefficient k and a probability P is the
topocentric phase angle
D
for which the minimum of the visibility curve is tangent to the horizontal line
180
D R
R
. The Danjon limit is the topocentric arclight 180
LT D
a
.
, ,
D
k P
180
D R
R
(A.1)
(A.2)
DANJON LIMIT: BRUIN’S METHOD
2
cos
,
dS
d
r
the solid angle
d
of the emitting surface element dS seen from the surface on which the light
falls is
2
cos
dS
d
r
by (A.1) we find
2 2 2
2
cos
cos cos
cos
d d d dE
B dE Bd
dS
dS d d dS d
dS r
d
is the solid angle of an element of the emitting surface as measured by the observer. In the
special case that the surface has uniform luminance (the same over the entire surface),
then
.
E B
* Contrast. We define the contrast of a image of luminance
B
that is on a background of luminance
BS as
S
S
B B
CB
it is a dimensionless quantity that, for our purposes, is always a positive number. The observed
luminance B of the Moon is the sum of the luminance of the Moon
M
B
, after going through the
atmosphere and the luminance of the twilight sky
S
B
.
M S S
SM
S S S
B B B
B B
B
C
B B B
* Stellar magnitude. We define stellar magnitude so that an increase of 5 of its units corresponds
to an increase of 100 times its illuminance. We take as reference that an illuminance of 1 lux has a
magnitude of 13.98; therefore, the visual magnitude m is determined by (Allen 1973, p.201)
13.98 2.5 log .
m E
the unit of E is lux.
* Atmospheric extinction. When light passes through the atmosphere, it undergoes a weakening
called extinction, caused by three factors: Rayleigh scattering by molecules, scattering by aerosols,
and molecular absorption, mainly ozone. The attenuation of light rays entering the atmosphere
follows the BeerLambert law
0.4 0.4 2.5
10 10 log
kX kX
B
E E B B X
k B
E and B are the illuminance and luminance observed,
E
and
B
illuminance and luminance outside
the atmosphere, X es el airmass, a measure of the distance traveled by light in the atmosphere, k
is a constant called the extinction coefficient expressed in magnitudes per airmass.
As a consequence of atmospheric extinction, the stellar magnitude m after passing through
the atmosphere is
13.98 2.5 log 13.98 2.5log
m E E kX m kX
m
is the magnitude of the Moon before atmospheric attenuation.
There are several formulas for airmass X that conform to realistic atmospheric models;
among them is the formula of Kasten and Young (1989)
1.6364
1
cos 0.50572 6.07995º 90º
Xz z
z is the apparent zenith distance in degrees (that is, topocentric and with refraction).
* Phase angle. The geocentric phase angle
is the selenocentric angle between the centers of the
Sun and the Earth. Note that when the Moon is in conjunction is not
180º
, because there is
ecliptic latitude of the Moon. The topocentric phase angle
T
is the selenocentric angle between
13
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
Wenceslao Segura González
the observer's position on the Earth's surface and the center of the Sun.
* Arcligth. The geocentric arclight
L
a
is the angle measured from the center of the Earth between
the Sun and the Moon centers. The topocentric arclight
LT
a
is the angle measured from the
observation point on the Earth's surface between the Moon and Sun's centers.
* Allen's formula. The bolometric magnitude of the Moon, that is, measured for all wavelengths, in
the absence of atmosphere and at the mean distance from the Earth, is calculated by (Allen 1973,
p.144)
9 4
12.73 0.026 4 10m
this formula cannot be extrapolated to large phase angles because they do not consider the attenuation
of the illuminance of the Moon due to the inclination with which the solar rays reach its surface.
Following a suggestion from Russell (1916) we obtain a formula for the magnitude of the
Moon that depends on the cube of 180
(Segura, 2021b)
3 2.8026
3.62548 2.33551log 180 3.62548 2.5log 180 .
m
* Solid angle of a circular surface. When projecting a circular image of angular diameter
onto
a sphere of radius r, it forms a spherical shell, whose area is
2
2 1 cos
2
S r
then the solid angle is
2
2 1 cos .
2
S
r
If
is very small
2
1
cos 1
2 2 2
expressed
in radians, therefore
14
Drawing A.1. In the celestial sphere, we have drawn the horizon, which is the horizontal circle. In the center
of the sphere is the observer. S and M is the Sun and the Moon, Z is the zenith of the observation site. We
measure azimuth in a retrograde direction from the meridian to the point where the horizon intersects the star's
vertical. The arc of the great circle between the Sun and the Moon is the arclight.We measure azimuth from
the south. From triangle SZM, we calculate the arclight by the cosine theorem. The angle at vertex
is the
azimuth difference between the Sun and the Moon.
horizon
Z
S
M
North
cardinal point South
cardinal point
meridian
LT
a
(A.8)
DANJON LIMIT: BRUIN’S METHOD
15
2 2
.
2 2
E B
As the total luminance of the image is the sum of the luminance of the image B and the
background luminance
S
B
, then using (A.2) and (A.3), the illuminance of the image is
2 2
,
2 2 S
E B B C
E is the illuminance caused exclusively by the image,
S
B
is the background luminance, and C is the
contrast,
is in radians.
If
is not very small, then the illuminance as a function of luminance for a circular image is
2 1 cos 2 .
S
E B C
* Refraction. The angle of refraction is
0
R
R z z
z is the geometric zenith distance (without refraction), and
0
z
is the apparent zenith distance (with
refraction),
R
R
is measured in arc minutes. Bennett (1982) (Meeus, 1991, p. 102) obtained an
empirical formula that gives with a very good approximation the angle of refraction for all the
values of the apparent altitude of a star above the horizon
0
0
1
7.31
tan
4.4
R
R
hh
valid for normal atmospheric pressure and 10 ºC,
0
h
being expressed in degrees. In (A.16), the
angle of refraction is in minutes and the apparent altitude in degrees.
When we know the geometric altitude h (not including refraction) and not the apparent altitude
0
h
, we continue to use (A.10) to calculate the angle of refraction, using successive approximations.
* Width of the crescent. The width
of a zone of the crescent Moon of position angle
when the
phase angle is
is
2 2 2
cos
, 1 ,
sin cos cos
R
wr
R is the radius of the Moon, r the EarthMoon distance,
is expressed in radians and
is the angle
of position. The maximum width of the Moon corresponds to
0
max
1 cos .
R
w
r
* Parallax: To calculate the topocentric arclight, we apply the cosine theorem to the spherical
triangle of drawing A.1
cos cos cos cos sin sin .
LT T T T T T T
a A A h h + h h
,
T T
A h
is azimuth and altitude of the Sun and
,
T T
A h
those of the Moon, all of them topocentric.
We neglect the parallax of the Sun and identify its topocentric altitude with the geocentric one;
furthermore, since the parallax in azimuth is minimal, we also neglect it; in other words, we will only
correct for parallax the altitude of the Moon and its distance from the place of observation.
E
S
r
LT
a
r
T
Drawing A.2. Positions of the observer on the Earth E, Moon M, and Sun S.
T
is the phase angle and
LT
a
the lightarc, both topocentric,
r
(A.9)
(A.10)
(A.11)
Wenceslao Segura González
Applying the sine theorem to the triangle in drawing A.2
1
sin sin sin
sin ,
LT T LT
T
a r a
r r r
r
is the distance from the Sun to the observation point, and
r
the distance between the Sun and
the Moon's centers. Since
r r
, then by (A.13) 180
T LT
a
.
Applying the cosine theorem to the triangle in drawing A.2
2 2 2 cos
T T TL
r r r r r a
T
r
is the topocentric distance from the center of the Moon.
If the azimuth difference between the Sun and the Moon is zero, then (A.12) reduces to
cos cos 180 .
LT T LT T T
a h h a h h
When the Moon is very close to the horizon then
sin ; cos 1; sin
T T T
h h h
and from equations (A.11) it follows that
T
h h
is the equatorial horizontal parallax of the Moon, which when it is at the mean distance from the
Earth is 57’ 2.6’’.
* Resolving power: Suppose a point image observed at a great distance through a circular diaphragm,
which could be the pupil of the eye. When light passes through the diaphragm, the diffraction
phenomenon occurs, the image observed through the diaphragm is a central circle surrounded by
circular rings. The central image is called the Airy disk and has the angular diameter
2.44 ,
r
is the wavelength and
is the diameter of the diaphragm or pupil.
12. Bibliography
* Allen, C. W. (1973): Astrophysical Qantities, University of London.
* Bennett, G. G. (1982): «The Calculation of Astronomical Refraction in Marine Navigation», Journal
of the Institute for Navigation 35, 255259.
* Blackwell, H. R. (1946): «Contrast Thresholds of the Human Eye», Journal of the Optical
Society of America 36(1), 624643.
* Bruin, F. (1977): «The First Visibility of the Lunar Crescent», Vistas in Astronomy 21, 331358.
* Danjon, A. (1932): «Jeunes et vieilles lunes», L'Astronomie 46, 5766.
* Danjon, A. (1936): «Le croissant lunaire», L'Astronomie 50, 5765.
* Kasten, F. and Young, A. T. (1989): «Revised optical air mass tables and approximation formula»,
Applied Optics 28, 47354738.
* 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),
480482.
* Meeus J. (1991): Astronomical Algorithms, WillmannBell.
* Russell, H. N. (1916): «The Setellar Magnitudes of the Sun, Moon and Planets», Astrophysical
Journal 43, 103129.
* Samaha, Abd ElHamid; Asaad, Adly Salama and Mikhail, Joseph Sidky (1969): «Visibility of the
New Moon», Helwan Observatory Bulletin 84, 137.
* Schaefer, Bradley E. (1991): «Length of the Lunar Crescent», Quarterly Journal of the Royal
Astronomical Society 32, 265277
* Siedentopf, H. (1941): «Neue Messungen der visuellen Kontrastchwelle», Astronomische
Nachrichten 271(5), 193203.
* Segura González, Wenceslao (2018): Movimientos de la Luna y el Sol, eWT Ediciones.
* Segura González, Wencesl ao (2020b): «Position of the bright limb of the Moon»,
www.researchgate.net/publication/349109783_Position_of_the_bright_limb_of_the_Moon.
* Segura González, Wenceslao (2021a): «Danjon Limit: Helwan’s Method», www.researchgate.net/
16
(A.12)
(A.13)
(A.14)
DANJON LIMIT: BRUIN’S METHOD
publication/348622116_Danjon_Limit_Helwan_Method.
* Segura González, Wenceslao (2021b): «Danjon Limit: Sultan’s Method», www.researchgate.net/
publication/350609317_Danjon_Limit_Sultan%27s_Method.
* Segura, W. (2021c): «Predicting the First Visibility of the Lunar Crescent», Academia Letters,
Article 2878.
* Segura González, Wenceslao (2021d): «Danjon Limit: Schaefer’s Method», www.researchgate.net/
publication/355188668_Danjon_Limit_Schaefer's_Method.
17