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MAGNITUDE OF THE MOON AT LARGE PHASE ANGLES
Magnitude of the Moon
at large phase angles
Wenceslao Segura González
e-mail: wenceslaoseguragonzalez@yahoo.es
Independent Researcher
Abstract. We expose and analyze the proposed models of the visual magnitude of the Moon for
large phase angles (>150º). We devised a method to determine the luminance and illuminance per
unit angular length of the lunar crescent as a function of position and phase angles.
1. Introduction
Photometric measurements of the Moon at large phase angles are difficult since its observation
has to be made at a low altitude above the horizon and therefore, they are 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.
The magnitude and differential illuminance of the Moon at large phase angles is one of the
two parameters needed to determine the time of the first sight of the lunar crescent shortly after its
conjunction with the Sun; the other parameter is the luminance of the twilight sky.
In this research, we analyze the various proposals made for the visual magnitude of the
Moon for phase angles greater than 150º to establish physical criteria for the visibility of the lunar
crescent.
2. Logarithmic law
Allen (1973, p. 144) gives the following formula for the magnitude of the Moon as a function
of the phase angle topocentric
(see appendix)
9 4
12.73 0.026 4 10m
is in degrees. (1) is the visual magnitude at the mean distance from the Moon and measured
before atmospheric absorption. We cannot extend the formula (1) to more than 150º of the phase
angle. (Russell, 1916), (Samaha, Asaad and Mikhail, 1969) and (Segura, 2021a) proposes a law by
which the Moon's magnitude depends on the logarithm of the cube of 180
. Adjusting (1) to a
law of this type *, we obtain
3
2.2543 2.0436log 180m
(2) is applicable for phase angles greater than 135º and is a continuation of formula (1), which is
valid up to a phase angle of 150º (tables 1 and 2 and drawing 5).
Samaha et al. adopting the idea above, found
3
10
4.3456 2.5log 180m
1
* We have adjusted formula (1) between the values of 135º and 150º, since in this interval, formula (1) fits the
logarithmic law. To obtain (2), we find the magnitude m of (1) (entre 135º y 150º) 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.
(1)
(2)
(3)
Wenceslao Segura González
2
applicable for all phase angles (tables 1 and 2)
We assume as a first approximation that the luminance of the Moon is the same at every
point of the crescent; that is, it does not depend on the lunar latitude
, nor on the lunar longitude
,
and only depends on the phase angle. From expression (2), or from (3), we derive the Moon's
luminance. 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 semi-circle and a semi-ellipse (Segura 2018, p.190),
(Segura, 2020). 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 Earth-Moon distance. By the definition of stellar
magnitude
13 .98 2 .5 log
m E
and by (2), we find the integral illuminance
E
of the Moon
expressed in lux at the mean distance between the Earth and the Moon
2.4523 2.4523
7
log 6.4937 log 180 3.2085 10 180 ,
E E
and the luminance in cd/m2 by (A.1) is
2
2.4523
7
2
2
3.2085 10 180 ,
1 cos
E r
BR
this formula is valid on the assumption that the luminance of the lunar crescent is uniform and that
(2) is fulfilled.
B
is the luminance before atmospheric attenuation *.
The above calculations assume that the Moon and Earth are separated by the mean dis-
tance of approximately 384,399 kilometers. However, the Moon-Earth distance is highly variable
due to the variation in the eccentricity of its orbit caused by the action of the Sun. According to
Meeus (1981), the extreme distances between the centers of the Moon and the Earth in the period
between the years 1750 and 2125 are 356,375 and 406,720 kilometers.
By varying the distance between the Moon and the Earth, the apparent size of the Moon,
its illuminance, and therefore its magnitude vary. The relationship between the illuminances ob-
served on Earth when the Moon is at the mean distance
r
, and the distance r is
2 2
2.5log
r r
E E m m
r r
E
and E are the illuminances at the mean distance and the distance r, and
m
and m is the magni-
tude of the Moon at its mean distance from the Earth and at the distance r. From (7), we deduce
that at the smallest distance from the Moon to the Earth, the magnitude increases by -0.16 and
decreases by 0.12 when the Moon is at the greatest distance. The luminance of the Moon at
distance r from the Earth is
2
2
21 cos
2
r r E
E E
B B
R
r
the variation in illuminance caused by the Earth-Moon distance is balanced by the opposite varia-
tion of solid angle of Moon; that is, if the illuminance increases as a result of the Moon being close
to the Earth, it increases, in the same way, the lunar solid angle. Therefore, the luminance of the
Moon is not affected by its distance from Earth.
3.- Magnitudes of Mercury and the Moon
We compare the stellar magnitude of Mercury with the lunar magnitude since they are
stars without atmosphere; Mercury has a surface structure similar to the Moon, although with a
smoother orography, and its magnitude has been measured up to a 170º phase angle, which does
* The phase angle varies with the distance from the Earth to the Moon. But for the extreme distances from the
Moon to the Earth, the differences in phase angle are minimal and can be neglected.
(4)
(5)
(6)
(7)
MAGNITUDE OF THE MOON AT LARGE PHASE ANGLES
3
not it has been possible to do with the Moon. The general formula for the magnitude of the planets
when the distance between the Earth, Sun and planet is one astronomical unit is
0
V V V
m m m
V
m
is the visual magnitude,
0
V
m is the magnitude when the phase angle is zero, and
V
m
is the phase coefficient (Harris, 1961, p. 272-342), (Hilton, 2005), (Mallama, Wang and Howard,
2002). According to Mallman and Hilton (2018) Mercury's visual magnitude fits the following sixth-
order polynomial
2 3 2 5 3
7 4 9 5 12 6
0.613 6.3280 10 1.6336 10 3.3644 10
3.4265 10 1.6893 10 3.0334 10 ,
V
m
is the topocentric phase angle in degrees. We adapt formula (8) to the Moon (Victor Reijs private
communication). We assume that
m
is the same function for the Moon, and to fit the independent
coefficient, we assume that there is continuity with (1) from 150º (Rijs suggests continuity from
125º)
2 3 2 5 3
7 4 9 5 12 6
13.3518 6.3280 10 1.6336 10 3.3644 10
3.4265 10 1.6893 10 3.0334 10 ,
m
the magnitude (9) corresponds to the mean distance between the Earth and the Moon. (9) is
applicable from the phase angle 150º; for smaller angles, we continue using (1) (tables 1 and 2, and
drawing 4).
4.- Extrapolation of Allen's formula
When the phase angle is 180º, we find by (1) that the magnitude of the Moon is -3.97, when
in reality, it must be infinite, indicating that its luminance is null, as long as we do not take into
account the earthshine *; this means that (1) is not applicable at large phase angles.
Samah et al. suggest extrapolating Allen's formula (1) but imposing the condition that the
luminance is null when the phase angle is 180º. The previous authors suggest a linear extrapolation
of the luminance of the Moon deduced by (1) from a particular value of the phase angle (which we
take as 150º) to the phase angle of 180º.
Assuming that the luminance is uniform throughout the crescent, we can determine
B
from the magnitude as a function of the phase angle. By (A.1), (A.2), and (4), we find
13.98
1.5
2
2
10
.
1 cos
2
m
E
BR
r
We do the calculations at the mean distance between the Sun and the Moon. Samah et al. observed
that for large phase angles, the luminance obtained from formula (1) is approximately linear with
the phase angle, which suggests the aforementioned linear extrapolation.
By (1) and (10), we calculate that the luminance of the Moon at 150º is 313.78 cd/m2. So
extrapolating linearly to
180º 0
B
, we find the luminance of the Moon at phase angles greater
than 150º. The results are in drawing 5 and tables 1 and 2.
5.- Lunar magnitude derived from Hapke's theory
Consider the flat image of the lunar crescent seen from Earth (drawing 1). The position
angle
is the one that identifies a portion of the crescent. R is the radius of the Moon, and r is the
distance from the Moon to Earth, both in units of length.
If dE is the illuminance of the infinitesimal portion ABCD of the crescent measured on Earth
* The magnitude of earthshine depends on whether the light reflected by the Earth is from seas or clouds, being between
-1.30 and -3.69, with an average value of -2.71 (Agrawall, 2016) (Jackson, 1943). For phase angles that are not exces-
sively large (less than 165º), the magnitude of the illuminated part of the Moon is much greater than the magnitude of the
earthshine, so (1) does not take it into account. When the phase angle is close to 180º, earthshine is of the same order as
the magnitude of the crescent; therefore, it is necessary to distinguish between the magnitude of the entire Moon and the
magnitude of the part illuminated by the Sun.
(8)
(9)
(10)
Wenceslao Segura González
4
(drawing 1), we call illuminance per unit angular length
, ,
dE
G r
Rd r
is the topocentric phase angle,
R r d
is the angle in radians of arc BC (drawing 1) measured
from Earth, and d
is in radians.
We note that the illuminance per unit angular length G depends on the phase angle, the
position angle, and the distance from the Earth to the Moon. The unit of G is lx/rad. Unless stated
otherwise, we will assume that G is the illuminance per unit angular length at the mean distance
between the Moon and Earth.
The illuminance of a finite portion of the lunar crescent between position angles
1
and
2
is
2
1
1 2
, ,
R
E G d
r
and the total illuminance of the crescent is
2
0
2 , ,
R
E G d
r
the factor 2 is to take into account the two horns of the Moon.
In Hapke's (1963, 1966, 1981, 1984) photometric model for smooth surfaces, the bidirectional
reflectance distribution function
r
f
or ratio between the emitted luminance and the irradiance
reaching the surface (see appendix) is
, , 1 1
4
i
r i
i
w
f i P H H
i is the angle of incidence of the light coming from the Sun with respect to the normal to the surface
of the Moon
is the angle of emergence of the ray that is directed towards the observer on Earth,
cos
i
i
, and
cos
. The factor
i i
is the Lommel-Seeliger law; w is the average
single-scattering albedo;
P
is the opposition effect function, which measures the sudden increase
in luminance of the Moon at phase angles close to zero;
is the average single-particle scattering
Drawing 1.- ABCD is one of the infinitesimal portions into which we divide the image of the lunar crescent
projected onto the z-y plane; the x-axis is directed towards the Earth; the Sun is in the x-y plane, and
is the
position angle of portion ABCD. The length of arc BC is
Rd
, R is the radius of the Moon in units of length,
d
is in radians. l is the distance from the center of the Moon to an interior point of the lunar portion.
z
y
O
l
Rd
A
B
C
D
(11)
(12)
(13)
MAGNITUDE OF THE MOON AT LARGE PHASE ANGLES
5
function (see appendix) and H is the Chandrasekhar function for isotropic dispersions that explains
the effect of multiple dispersions. The
1 4
coefficient appears by normalizing the scattering function.
For phase angles greater than 90º, the opposition effect is null; therefore,
0
P
, which is
applicable to large phase angles as in our case. The scattering function is expressed by the Legendre
polynomials
2
1
1 3 1 ....,
2
b c
it is enough with the first terms. b and c are constants. The Chandrasekhar function is
1 2
1 2
H
1
w
.
When the surface is rough, as it happens on the Moon, there are two effects: the shadows
that are projected on the part of the surface because other parts block the arrival of light, and
effective surface tilt, caused by the inclinations of the irregularities of the lunar surface that have
different orientations, making some areas more or less illuminated and more or less seen by the
observer.
For a rough surface, the bidirectional reflectance distribution function is (Hapke, 1984)
, , 1 1 , ,
4
i
r i
i
w
f i P H H S i
the effect of the shadows is described by the function
, ,
S i
and the effect of the micro-
inclinations of the surface requires replacing cosi and cos
by the effective cosines
i
and
.
Equation (14) needs six parameters: the albedo w of a single scattering; h and S(0) to evaluate
the opposition effect function; the coefficients b and c of the Legendre polynomials; and
, which
is the topographic slope that gives a measure of roughness on a small scale.
Helfenstein and Veverka (1987), adjusting (14) to the measurements of Rusell (1916), Rougier
(1933), Shorthill et al. (1969), and Lane and Irvine (1972), obtained the value of the six average
parameters for the Moon. They only used photometric values of the Moon up to the phase angle of
150º; that is, we cannot ensure that for higher phase angles, which occur in the first visibility of the
lunar crescent, the Helfenstein and Veverka parameters coincide with the empirical data.
Suppose the coordinate system with the origin at the Moon's center and the x-axis directed
towards the center of the Earth, with the Sun in the x-y plane. The spherical coordinates referred
to this system are the selenocentric coordinates of the Moon: latitude
and longitude
(drawing
2). Note that this coordinate system is not fixed on the Moon's surface like Earth coordinates but
depends on the Moon's position relative to Earth.
The angles of incidence and reflection are related to the selenographic coordinates by (Lester,
McCall and Tatum, 1979)
cos cos cos ; cos cos cos ,
i
therefore the luminance found by (13) and (14) is put as a function of
and
.
The distance l (drawing 1) from the center O of the Moon to a point P located inside the
portion ABCD of the lunar crescent as observed from Earth is related to the coordinates z and y
sin ; cos sin
cos
sin ,
sin sin sin
z R y R
z z dz
l dl R d
l
the relation between the angle of position and the latitude of the point P is
2
tan
tan tan tan sin tan cos
sin cos
z d
d
y
then
3 2
2 2
cos
.
cos 1 sin tan
R
dl d
The surface of an element of a portion into which we divide the Moon (ABCD drawing 1)
(14)
Wenceslao Segura González
with sides d
and dl is
2
Rd dl
dS Rd dl d
r
r is the distance from the Moon to Earth, R is the radius of the Moon, and d
is in radians. The
illuminance of an infinitesimal portion having the position angle
is
2
3 22 2 2
2
1 cos
, , , .
cos 1 sin tan
Rd Rd R
E Bd Bdl B d
r r r
Using the luminance calculated by Hapke's theory (1984) with the parameters proposed by
Helfenstein and Veverka, Schaefer (1991) performed the integration of (15), calculated the
illuminance per unit angular length G, and expressed it as a magnitude that is, in logarithmic form:
2.5log ,
G
, the results of the integration are represented in drawing 3, where it is verified
that when we approach the end of the lunar horns, the illuminance decreases sharply. This decrease
in luminosity is mainly because the width of the Moon decreases when we approach the end of the
horns and to the increase in shielding produced by irregularities of the lunar surface.
By (12), we calculate the total illuminance of the Moon and the lunar magnitude by (A.2). In
drawing 3, we represent the illuminance per unit angular length as a function of the phase angle for
the mean distance between the Earth and the Moon.
6.- Illumination per unit angular length
From the magnitude of the Moon as a function of the phase angle
, we find the luminance
of the Moon if we assume that the luminance is the same at all the crescents (10). Then the
luminance only depends on the phase angle and does not depend on the position angle. In drawing
6, we represent the luminance of the Moon for several models of the lunar magnitude in the
assumption that the luminance is uniform. For the Schaefer model, we draw the luminance of the
center of the Moon as a dashed line.
To determine the visibility of the first lunar crescent, we need to calculate the illumination
per unit angle (11). If
d
is the solid angle of an infinitesimal portion of the lunar crescent (portion
ABCD of drawing 1) of position angle
and
is the width of the crescent in that portion in radians
6
Earth
Sun
z
x
y
R
P
cos sin
R
Drawing 2.- Definition of selenographic coordinates.
is the lunar latitude, and
is the longitude, whose
signs are defined in the drawing.
Moon’s
center
O
(15)
MAGNITUDE OF THE MOON AT LARGE PHASE ANGLES
as seen from the Earth's surface, then the illumination per unit of angular longitude in the case of
uniform luminance is
, , .
,
dE dE
B G B
R
dd
r
By (A.6)
2 2 2
cos
, 1 .
sin cos cos
R
G B
r
We find by (17) the illuminance of each portion into which we divide the Moon crescent. Suppose
we divide the crescent into portions like ABCD (drawing 1), of angular length
R r d
, then the
illumination measured at the Earth's surface for each of those portions is
2
22 2 2
cos
, , 1 ,
sin cos cos
R R
E G B
r r
as an example of the application of (18), we divide the lunar crescent into portions of length 1' of
arc *; using table 2 we find
B
in cd/m2. We represent the result in drawing 7.
7.- Lunar crescent luminance as a function of position angle
From the magnitude of the lunar crescent, we cannot find the luminance of each lunar
portion as a function of the angle of position. Assuming, as we have done before, that the luminance
is uniform is unsatisfactory because the luminance depends on the angle of inclination with which
the solar rays reach the surface of the Moon, and this inclination depends on the selenographic
position. The luminance is a function of the selenographic geographic coordinates and the phase
angle.
Since we are interested in very thin crescents, we can simplify and assume that the lumi-
nance depends on the phase angle and the position angle according to the law
7
Drawing 3.-Integration of the Hapke equation for large phase angles. On the vertical axis is the illuminance per
unit length
2.5log ,
G
(G in lx/rad), and the horizontal axis is the position angle in degrees.
2
4
6
8
10
10 20 30 40 50 60 70 80 90
Illuminance per unit angular length
Position angle
150º
155º
160º
165º
167.5º
170º
172.5º
175º
* We choose 1 minute of arc because it is approximately the resolving power of the human eye.
(16)
(17)
(18)
Wenceslao Segura González
, , .
B B S
In
,
S
, we include the factors that depend on the inclination with which the sun's rays reach
the Moon, among which are the macro and micro shielding of the lunar surface.
By applying (19) to the center of the crescent
0
0 0
, 0 , 0
,
, , ,
, 0
B B B S
S
B B B f
S
8
100
110
120
130
140
150
155
160
165
170
175
-9.73
-9.28
-8.78
-8.21
-7.55
-6.81
-6.39
-5.95
-5.48
-4.97
-4.43
-9.73
-9.28
-8.78
-8.21
-7.55
-6.81
-6.32
-5.72
-4.96
-3.88
-2.03
-9.93
-9.49
-8.99
-8.4
-7.67
-6.73
-6.14
-5.41
-4.48
-3.16
-0.90
-9.73
-9.28
-8.78
-8.21
-7.55
-6.81
-6.22
-5.50
-4.57
-3.25
-0.99
-10.12
-9.71
-9.22
-8.60
-7.81
-6.80
-6.21
-5.56
-4.84
-4.06
-3.23
-
-
-
-
-
-6.81
-6.31
-5.17
-3.69
-1.57
2.00
Allen
Angle
of phase Logarithmic
law.
Formula (2)
Logarithmic
law.
Formula (3)
Allen's linear
extrapolation
Polynomial
law.
Formula (9)
Hapke's
theory
Visual magnitude of the Moon
Table 1.- Visual magnitude of the Moon for large phase angles according to various models.
150
155
160
165
170
175
315.2
306.5
317.2
362.5
510.6
1237.8
315.2
286.0
257.1
224.7
186.5
136.0
292.8
243.0
192.9
144.3
96.4
48.0
315.2
261.8
209.5
157.5
104.7
52.2
313.7
259.9
220.7
201.6
220.7
408.9
313.8
284.4
154.6
70.2
22.3
3.3
Allen
Angle
of phase Logarithmic
law.
Formula (2)
Logarithmic
law.
Formula (3)
Allen's linear
extrapolation
Polynomial
law.
Formula (9)
Hapke's
theory
Moon luminance (cd/m2)
Table 2.- Luminance of the Moon for large phase angles assuming that the luminance is uniform throughout
the crescent. For Allen's law and the polynomial law (9), we observe a considerable increase in the luminance
for the largest values of the phase angle, which shows that both formulas do not apply to very high phase
angles since the luminance it must decrease with phase angle because light from the Sun is incident at a
greater angle of inclination.
(20)
(19)
MAGNITUDE OF THE MOON AT LARGE PHASE ANGLES
Table 4.- Moon magnitude for various models. We observe differences from the 150º phase angle.
Table 5.- Moon magnitude for various models. From the 150º phase angle there is a marked darkening of the
Moon.
- 2
- 4
- 6
- 8
- 10
- 12
- 14
- 2
- 4
- 6
- 8
- 10
- 12
- 14
20 40 60 80 100 120 140 160 180
20
2
4
40 60 80 100 120 140 160 180
MagnitudeMagnitude
Phase angle
Phase angle
9
Hapke’s theory
Allen
Formula
(9)
Allen
Formula
(2)
Allen’s
linear extrapolation
Wenceslao Segura González
10
Drawing 6.- Luminance of the lunar crescent assuming that the luminance is uniform throughout the crescent.
The dashed black line is the luminance of the center of the crescent according to the Schaefer integration of
the Hapke equation. We note that the luminance deduced from (9) (red line) increases with phase angle, which
is impossible, meaning that this model is not acceptable for large phase angles. We also check that the
luminance derived from Hapke's theory for phase angles greater than 170° is approximately the same as the
luminance derived from the magnitude assuming uniform luminance.
Luminance (cd/m
2
)
100
200
300
400
500
Phase angle
155 160 165 170 175
Luminance
deducted from (9)
Hapke's theory. luminance
in the center of the crescent
Hapke's theory.
Uniform luminance
Luminance
deducted from (2)
Extrapolation
Allen's
10 20 30 40 50 60 70 80 90-10-20-30- 40-50- 60-70- 80-90
Illuminance
of a portion of 1'
(in lux)
Position angle
Drawing 7.- The vertical axis is the illuminance of a portion of the Moon one minute of arc in angular length,
measured in lux. The horizontal axis is the position angle,
;
0
corresponds to the center of the crescent.
The curves correspond to phase angles of 160º, 165º, and 170º. The calculations have been made with the
lunar magnitude (2) and applying (18). We verify that the lunar portion that corresponds to the center of the
crescent has greater illumination due to being wider and because the solar rays arrive with less inclination.
The illuminance decreases when the position angle increases, becoming zero at the crescent poles.
160º
165º
170º
6
25 10
6
15 10
6
10 10
6
5 10
6
20 10
MAGNITUDE OF THE MOON AT LARGE PHASE ANGLES
to find the function
,
f
, we apply (20) to the integration of the Hapke equation (drawing 3).
We check that
,
f
depends mainly on
, so we neglect its dependence on
. Averaging, the
result fits approximately a second order polynomial
5 2
, 0.9982 0.00634 5.217 10 .
f
We caution that this reasoning is highly speculative. The errors we make by using (21) are masked
by the decrease in the width of the crescent, which is the main cause of the decrease in illuminance
per unit angular length of the crescent.
With the function
,
f
, we found the luminance
,
B
, knowing the lunar magni-
tude
m
. It is first necessary to find the relationship between
0
B
and
m
. From (20)
finds the total illumination of the crescent
2
0 0
0
, , 2 , , ,
R R
dE B f d E B f d
r r
with (A.2), we find
E
as a function of
m
, with (22), we find
0
B
, and by (20), we find
the luminance of the Moon as a function of the angle of position
13.98
2.5
02
0
13.98
2.5
2
0
110
2, ,
,
, 10
2, ,
m
m
r
BRf d
f
r
BRf d
finally, by (16), we find the illuminance per unit angular length as a function of the phase and
position angles
13.98
2.5
2
0
, ,
, 10 .
2, ,
m
f
r
GRf d
As an example of this procedure, we apply (23) and (24) to the lunar magnitude given by
(2). Tables 3 and 4 show the luminance and the illuminance per unit angular length as a function of
the position angle and the phase angle. In drawing 8, we represent the illuminance of each lunar
portion of 1 minute of arc.
8.- Conclusions
There are no reliable measurements of the Moon's magnitude at phase angles greater than
150°. However, knowing the illuminance per unit angular length (11) for large phase angles is
necessary to predict the vision of the Moon's first crescent.
We have discussed various methods for determining the magnitude of the Moon at large
phase angles. The polynomial law (9) is not acceptable since, for this model, the luminance of the
Moon increases for large phase angles, which is impossible.
The Danjon limit found with Hapke's theory is greater than that found by other procedures
(Segura, 2021b), which makes us think that Hapke's theory, as integrated by Schaefer, gives the
Moon excessive dimming at large phase angles.
Therefore, we point out that the logarithmic law model (2) and (3) is the most realistic to
represent the magnitude of the Moon
m
at large phase angles.
What interests us is not the integral illuminance of the Moon but its illuminance per unit
length as a function of the position angle
,
G
. However, finding
,
G
from
m
is not
possible, it is necessary to establish some hypotheses. We impose the condition (19) and (20), then
the luminance of a zone of the Moon characterized by its angle of position is the luminance of the
center of the crescent multiplied by the factor
,
f
, in which the effects caused by the inclina-
tion of the sun's rays are included.
We have approximately found the function
,
f
from the integration of the Hpake
equation for large phase angles. Once the function
,
f
is known, it is possible to find the
11
(21)
(22)
(23)
(24)
Wenceslao Segura González
luminance in each place of the lunar crescent if we know the magnitude of the entire crescent, and
it is also possible to find the illuminance per unit of angular length.
The theory that we have developed to calculate the luminance as a function of the position
angle is very uncertain, but we trust that it will be helpful to find the moment of the vision of the first
lunar crescent.
9.- References
1.- Agrawal, D. C. (2016). Apparent Magnitude of Earthshine: a Simple Calcualtion, European
Journal of Physics 37, 035601.
2.- Allen, C. W. (1973). Astrophysical Qantities, University of London, 1973.
12
0
10
20
30
40
50
60
70
80
90
391.7
367.1
331.4
293.8
255.4
218.2
174.3
117.1
61.6
0
361.1
338.3
305.5
270.8
235.4
201.1
160.7
108.0
56.8
0
327.0
306.4
276.6
245.2
213.2
182.1
145.5
97.8
51.4
0
288.8
270.6
244.3
216.6
188.3
160.8
128.5
86.3
45.4
0
240.8
225.7
203.8
180.6
157.0
134.2
107.2
72.0
37.9
0
175.7
164.6
148.6
131.8
114.5
97.9
78.2
52.5
27.6
0
150 155 160 165 170 175
Phase angle
Position
angle
Table 3.- Luminance in cd/m2 of the lunar crescent as a function of the phase angle and the position angle for
the lunar magnitude (2). The luminance decreases slightly with phase angle but has a very pronounced
decrease with increasing position angle. When the position angle is 90º, the luminance must be zero.
0
10
20
30
40
50
60
70
80
90
1.56
1.66
1.85
2.13
2.51
3.03
3.77
5.00
7.15
2.04
2.15
2.33
2.61
3.00
3.55
4.34
5.54
7.69
2.62
2.73
2.93
3.23
3.63
4.16
4.93
6.18
8.34
3.38
3.48
3.69
3.99
4.40
4.94
5.72
6.97
9.14
4.45
4.56
4.77
5.07
5.49
6.03
6.82
8.07
10.24
6.30
6.40
6.62
6.92
7.34
7.89
8.68
9.94
12.10
150 155 160 165 170 175
Phase angle
Position
angle
Table 4.- Illuminance per angular length unit as a function of phase angle and position angle expressed in
magnitudes:
2.5log ,
G
and
,
G
in lx/rad.
MAGNITUDE OF THE MOON AT LARGE PHASE ANGLES
3 Hapke, B. (1963). A Theorical Photometric Function for the Lunar Surface, Journal of
Geophysical Research 68(15), 4571-4586.
4 Hapke, B. (1966). An Improved Theoretical Lunar Photometric Function, The Astronomical
Journal 71(5), 333-339.
5.- Hapke, B. (1981). Bidirectional Reflectance Spectroscopy. Theory, Journal of Goephysical
Reesearch 86(B4), 3039-3054.
6.- Hapke, B. (1984). Bidirectional Reflectance Spectroscopy. Correction for Macroscopic
Roughness, Icarus 59, 41-59.
7.- Harris, L. (1961). Photometry and Colorimetry of Planets and Satellites, in Planets and Satellites,
The University Press.
8.- Helfenstein, P., Veverka, J. (1987). Photometric Propierties of Lunar Terrain Derived from
Hapke’s Equation, Lunar and Planetary Institute 18, 415-416.
9.- Hilton, J. L. (2005). Improving the Visual Magnitudes of the Planet in The Astronomical Almanac.
I. Mercury and Venus, The Astronomical Journal 29, 2902-2906.
10.- Jackson, J. (1943). The Brightness of Earthshine on the Moon, Monthly Notes of the
Astronomical Society of South Africa 2, 24.
13
10 20 30 40 50 60 70 80 90
100
200
300
400
Luminance (cd/m
2
)
Position angle
150º
175º
Drawing 8.- Crescent luminance as a function of phase angle for lunar magnitude (2), for phase angles 150,
155, 160, 170, and 175 degrees.
150
155
160
165
170
175
0.0003827
0.0002644
0.0001680
0.00009447
0.00004189
0.00001045
Phase
angle
I
Table 5.-
2
0
, ,
I f d
.
Wenceslao Segura González
11.- Lane, A. P. and Irvine, W. M. (1972). Monochromatic plane curves and albedos for the lunar
disk, American Astronomy Society 78, 267-277.
12.- Lester T. P.; McCall, M. L. and Tatum J. B. (1979). Theory of Planetary Photometry, The
Journal of the Royal Astronomical Society of Canada 73(5), 233-257.
13.- Mallama, A.and Hilton, J. L. (2018). Computing Apparent Planetary Magnitudes for The
Astronomical Almanac, https://arxiv.org/abs/1808.01973.
14.- Mallama, A.; Wang, D. and Howard, R. (2002). Photometry of Mercury from SOHO/LASCO
and Earth: The Phase Function from 2 to 170º, Icarus 155-2, 253-264.
15.- Meeus, J. (1981). Extreme perigées and apogées of the Moon, Sky and Telescope 62, 110-
111.
16.- Rougier, G. (1933). Photométrie Photoélectrique Globale de la Lune, Annales de l’Observatoire
de Strasbourg 2, 205-339.
17.- Russell, H. N. (1916). The Setellar Magnitudes of the Sun, Moon and Planets, Astrophysical
Journal 43, 103-129.
18.- Samaha, A.E.; Asaad, A. S. and Mikhail, J. S. (1969). Visibility of the New Moon, Bulletin of
Observatory Helwan, 84.
19.- Schaefer, B. E. (1991): Length of the Lunar Crescent, Quarterly Journal of the Royal
Astronomical Society 32, 265-277.
20.- Shorthill, R. W.; Saari, J. M.; Baird, F. E. and LeCompete, J. R. (1969). Photometric Properties
of Selected Lunar Features, NASA Contractor Report CR-1429.
21.- Segura González, W. (2018). Movimientos de la Luna y el Sol. Una introducción, eWT
Ediciones.
22.- Segura González, W. (2020). Moon's crescent width, https://www.researchgate.net/publication/
343219170_Moon's_crescent_width.
23.- Segura González, W. (2021a). Danjon Limit: Bruin’s Method, https://www.researchgate.net/
publication/356128660_Danjon_Limit_Bruin's_Method.
24.- Segura González, W. (2021b). Danjon Limit: Schaefer’s Method, https://www.researchgate.net/
publication/355188668_Danjon_Limit_Schaefer's_Method.
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
14
- 20- 30- 40- 50- 60- 70- 80-90 - 10 10 20 30 40 50 60 70 80 90
160º
165º
170º
6
25 10
6
20 10
6
15 10
6
10 10
6
5 10
Drawing 7.- The vertical axis is the illuminance of a portion of the Moon one minute of arc in angular length,
measured in lux. The curves correspond to phase angles of 160º, 165º, and 170º. The calculations have been
made with the lunar magnitude (2) and applying (18). We find that there is a very sharp maximum near phase
angle 0.
Position angle
Iluminance
MAGNITUDE OF THE MOON AT LARGE PHASE ANGLES
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 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
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 (1) and (2) 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 (3)
reduces to
.
E 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
* Number of 10th magnitude stars per square degree. With S10 (v), we represent the number of
tenth magnitude stars per square degree of the sky. We define this unit as the luminance of a one-
degree square portion of the sky with a uniform illuminance equal to that of a tenth-magnitude star.
For a star of
10
m
we find by (A.2)
10 13.98
10 2
2.5
10
10 2.558 586 10 lm m .
E
The square degree is a solid angle unit (Guthrie 1947), since 1 degree is
180
radians, then
2 2
2 2
1deg rad sr.
180 180
Note that the square degree is not the area of a spherical square with a side 1º. The spherical
square has for its sides arcs of great circles of equal length and the four equal angles. The area of
the spherical square is
1 2 2
4sin tan 2
S
is the side of the square in radians. When
is small, S is very close to
2
if we have a spherical
square with side 1º then
2
1 deg
S.
Since the luminance of the square degree is uniform, then by (A.1)
15
(A.1)
(A.2)
Wenceslao Segura González
10 2
7 2
10
2
2
2.558 586 10 lm m
1 10 8.399 342 10 cd m
1deg 180 sr
E
E
B S v
therefore
2 6
1cd m 1.190 569 10 S10 v .
* 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 Beer-Lambert law
0.4
010
kX
E E
E is the observed illuminance,
0
E
illuminance outside the atmosphere, X es el air-mass, a measure
of the distance traveled by light in the atmosphere, k is a constant called the extinction coefficient
expressed in magnitudes per air-mass.
As a consequence of atmospheric extinction, the stellar magnitude m after passing through
the atmosphere is
0 0
13.98 2.5 log 13.98 2.5log
m E E kX m kX
0
m
is the magnitude of the Moon before atmospheric attenuation.
There are several formulas for air-mass 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.
* 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
0º
, because there is
an ecliptic latitude of the Moon. The topocentric phase angle
is the selenocentric angle between
the observer's position on the Earth's surface and the center of the Sun.
* Arc-ligth. The geocentric arc-light
L
a
is the angle measured from the center of the Earth between
the Sun and the Moon centers. The topocentric arc-light
L
a
is the angle measured from the
observation point on the Earth's surface between the Moon and Sun's centers.
* Elongation. Elongation is the difference between the apparent ecliptic longitudes of the Moon
M
L
and the Sun
S
L
. The elongation and the arc-light are different; they are only equal if the
latitude of the Moon
M
is zero. The relationship between geocentric arc-light
L
a
and elongation
M S
L L
is (Segura, 2018, p. 191-193)
cos cos cos .
L M M S
a L L
There is a relationship between the arc-light and the phase angle
sin 180
sin
L
a
r r
r
is the distance from Earth to Sun, and r is the distance from Earth to Moon. If we neglect r
against
r
, then we find
180 ,
L
a
(A.4) and (A.5) are applicable to topocentric values (Segura, 2020). From (A.3), we find the
geocentric arc-light, and by making the parallax correction, we determine the topocentric arc-light.
* Irradiance. It is the luminous flux that falls per unit area (not necessarily perpendicular to the
direction of propagation); it is measured in lux and mathematically defined by
.
d
R
dS
* Bidirectional Reflectance Distribution Function (BRDF). Suppose a surface with an irradiance
,
i i
R
and that as a result has a luminance
,
r r
B
, the bidirectional reflectance distribution
function is defined by
,
,
r r
r
i i
B
fR
16
(A.3)
(A.4)
(A.5)
MAGNITUDE OF THE MOON AT LARGE PHASE ANGLES
17
* Luminous flux density. It is the luminous flux through the unit surface oriented at right angles to
the direction of propagation. It applies exclusively to plane wavefront radiation incident on a surface.
The unit is lux, and we represent it by the letter F.
* Scattering function We assume a surface that scatters the light that reaches it. The scattering
function tells us how light is scattered according to the direction of emission, Lester et al. (1976).
The scattering function is defined as the luminous flux reflected per unit solid angle divided by the
luminous flux that falls on the surface and is subsequently scattered.
2
2
.
r
r i
i
d d
d d d
d
d
is the solid angle at which the radiation is emitted. The situation that interests us is when a
collimated beam of light, or a plane wavefront, arrives on the dispersive surface, with a luminous
flux density i
F d dS
2i
r r
S
d
d dSd FdSd d FdSd
dS
r
d
is the luminous flux reflected by the entire illuminated surface at a solid angle
d
, assuming
that F is the luminous flux density that will be scattered; that is, it does not include the luminous flux
that is absorbed by the surface and is therefore not scattered. dS es la superficie «eficaz» en la
dirección de la densidad de flujo luminoso.
* 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
r
R is the radius of the Moon and r the Earth-Moon distance,
is expressed in radians and measured
from Earth.
(A.6)
