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DANJON LIMIT: SCHAEFER’S METHOD
1
Danjon Limit: Schaefer’s Method
Abstract. Schaefer (1991) determined the Danjon limit or minimum angle between the Sun and the
Moon from which the Moon can be seen shortly after the conjunction. Schaefer's method uses
Hapke's (1984) lunar photometric theory and considers a fixed value for the threshold illuminance.
We show Schaefer's method and its shortcomings, and we expose a modified theory, where the
threshold illuminance to see the lunar crescent depends on several factors, mainly atmospheric
absorption. We consider that vision is a probabilistic phenomenon; that is, when we use the
experimental data of Blackwell (1946), we cannot be sure whether or not the Moon will be seen.
Finally, we conclude that «perhaps» Hapke's theory overestimates the shielding of the sun's rays
by the irregularities of the lunar surface at large phase angles.
1. Introduction
The conjunction of the Sun and the Moon occurs when the centers of both stars have the
same apparent ecliptic longitude; that is, the longitudes are corrected for precession, nutation and
aberration. Unless there is a total solar eclipse, the Moon has an ecliptic latitude and therefore an
angle of separation with the Sun at the time of conjunction, therefore even at the time of conjunction,
an area although tiny of the Moon is illuminated. At the conjunction, the Moon is not visible from the
Earth since the scattering of the solar rays in the atmosphere very close to the Sun produces a very
intense brightness, hiding the weak lunar rays.
The Moon has an apparent movement faster than the Sun; both stars are moving apart, then
the Moon occupies places in the sky where the luminosity produced by the scattering of the solar
rays is small. Simultaneously, the illuminated area of the Moon is increases and therefore is more
luminous; these two circumstances increase the chances of seeing the Moon shortly after the
conjunction. When there is sufficient separation between the two stars, the crescent Moon is
visible in the twilight sky shortly after sunset and near the western horizon.
Our problem is to find the minimum angular distance between the Sun and the Moon from
which we can observe the crescent Moon. This extreme angle is called the Danjon limit.
The Danjon limit is essential in the Islamic calendar since it is impossible to see the crescent
at a smaller angular separation; however, that the Moon is at a greater angle than the Danjon limit
does not ensure that we see the crescent (table 1).
In this research, we expose the method devised by Schaefer (1991) to find the Danjon limit,
using the integration of the equation of the lunar photometric theory of Hapke (1984). We expose
the shortcomings of Schaefer's method and propose a modification similarly based on Hapke's
theory.
In the appendix, we define the basic concepts and photometric units.
2. Illuminance per unit of angular length
Consider the flat image of the crescent Moon seen from Earth (drawing 1). The position
angle
identifies a portion of the crescent. R is the radius of the Moon, and r is the distance Moon
Earth, both in units of length.
dE is the illuminance (see appendix) of the portion ABCD of the crescent measurement on
Earth; we call illuminance per unit angular length to
Wenceslao Segura González
email: wenceslaoseguragonzalez@yahoo.es
Independent Researcher
Wenceslao Segura González
Fotheringham (1910)
Maunder (1911)
Schoch (1928)
Neugebauer (1929)
Danjon (1932)
Danjon (1936)
Koomen et al. (1967)
Samaha et al. (1969)
McNally (1983)
Ilyas (1983)
Ilyas (1984)
Ilyas (1987)
Yallop (1988)
Schaefer (1988a)
Schaefer (1991)
Fatoohi at al. (1998)
Odeh (2004)
Segura (2006, p. 124133)
Sultan (2007a, 2007b)
Hasanzadeh (2012)
Ahmed (2016)
Segura (2021a)
Segura (2021b)
12º
11º
10.7º
10.4º
6.7º
7º
2º
8º
5º
10.5º
10.5º
4º
8º
8.8º
7º
7.5º
7.6º
7.6º11.4º
7.5º
0.25º
5º
8.5º 0.1º
5.6º y 7.1º
10º
Author Danjon
limit Comments
Geocentric angle at sunset. Very pure sky. It is obtained
from observations.
0
Z
.
Idem.
Idem.
Idem.
Topocentric angle. Extrapolated from observations.
Idem.
Moon photographed by a rocket at a altitude of 175 km.
Deduced from Helwan model.
Deduced for atmospheric turbulence.
Topocentric angle. Extrapolation of observations.
Idem.
Minimum topocentric altitude at sunset. Extrapolation of
observations.
Geocentric angle. Deduced from observations.
Geocentric angle. Obtained from observations and
corroborated by Schaefer's theory (1988b).
Topocentric angle. Deduced from the integration of
Hapke's lunar photometric theory (Bowell et al., 1989).
Topocentric angle. Obtained from observations.
Topocentric angle. Obtained from observations without
optical aid.
0.6º
Z
.
Topocentric angle, which is dependent on the extinction
coefficient,
0.1 0.3
k
.
Topocentric angle. Found from the lunar visibility theory
of (Sultan, 2007a) and (Sultan, 2007b).
Topocentric angle. Deduced from observations and from
McNally's theory (1983).
Topocentric angle. Deduced from observations.
5.6º for ideal conditions and 7.1º for good atmospheric
conditions. Application theory of Samaha et al. (1967).
Topocentric angle. Good atmos pheric conditions.
Application theory of Sultan (2007a, 2007b).
Table 1. Proposed values of the Danjon limit. Z is the azimuth difference between the centers of the Sun and
the Moon at the time of observation, and k is the atmospheric extinction coefficient.
2
DANJON LIMIT: SCHAEFER’S METHOD
, ,
dE
G r
Rd r
is the geocentric phase angle (see appendix), and
Rd r
is the angle in radians of the arc BC in
drawing 1 measured from Earth.
The illuminance per unit angular length G depends on the phase angle, the position angle, and
the distance from the Earth to the Moon. We do not take into account the variation of the Sun
Moon distance. The unit of G is lux/rad. Unless otherwise stated, we will assume that G is the
illuminance per unit angular length at the mean distance between the Moon and the Earth.
The illuminance of a finite portion of the crescent Moon between position angles
1
and
2
is
2
1
1 2
, ,
R
E G d
r
then the total illuminance of the crescent is
2
0
2 , ,
R
E G d
r
factor 2 is to take into account the illuminance of the two cusps of the Moon.
The magnitude of the crescent as a function of the phase angle is (A.5)
13.98 2.5 logm E
E
is the illuminance calculated by (2). Illuminance decreases with the square of the distance;
if
E
is the illuminance of the Moon at the mean distance from Earth
r
, then the illuminance E at
a distance r is
2
2
E r
E r
therefore the relationship between the magnitude
m
when the Moon is at the mean distance and
the magnitude m at the distance r is
5log .
r
m m
r
Drawing 1. One of the portions in which we divide the image of the crescent Moon projected on the zy
plane; the xaxis is directed towards the Earth; the Sun is in the xy plane, and
is the angle of position of the
portion ABCD. The length of the arc BC is Rd
, where 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 a point inside the lunar portion.
z
y
O
l
Rd
A
B
C
D
3
P
(1)
(2)
(3)
(4)
Wenceslao Segura González
The extreme distances of the Moon are 356,375 km and 406,720 km (Meeus, 1981); applying (4),
we check that the difference of the extreme magnitudes of the Moon as consequence of the
distance is 0.29.
3. 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.
If we observe a finite image but with an angular size smaller than that Airy disk, the size of
the image on the retina after passing through the pupil will remain the same as for the point image.
Therefore any image that has an angular size smaller than that of the Airy disk, we consider it a
point since the image it produces on the retina has the same size. If the angular size of an image is
smaller than that of the Airy disk, we say that the image is not resolvable, in the sense that we
cannot distinguish any aspect of its interior.
Suppose the human eye sees a circular image of uniform luminance (see appendix) of an
angular size larger than the resolving of the human eye. The image that the crystalline projects on
the retina is a disk that has an area A.
is the angle in radians of the image, which is approximately
the same angle of the retinal image, f is the focal length of the lens, and D is the diameter of the
circular image projected on the retina, then
2 2
,
4
D f A f
the solid angle of the image at the observer's position is by (A.10)
2
,
4
comparing the previous equations, we find that the area of the retinal image is
2
.
A f
is the luminous flux or light power that reaches the retina from the image, S the area of the pupil,
B the luminance of the image, and
r
E
the illuminance of the image on the retina
2 2
.
r
ES B S S
E B
A A f f
The vision is determined by the number of photons that reach the retina per unit area and per
unit time. That is, it depends on the retinal illuminance
r
E
, which is proportional to the luminance of
image B, as long as we neglect the variation in the size of the pupil with luminance. So for objects
of resolvable size, the viewing threshold depends on the threshold luminance of the image. If the
luminance of the image is greater than the threshold luminance, then the image will be viewed.
The situation changes when the image has an angular size
smaller than the resolving of the
human eye, which we can estimate at one arc minute. So the image projected on the retina has the
same size as Airy's disk regardless of the image size.
The image area on the retina of an unresolvable image is
2 2
4
r
A f
and the retinal illuminance is
2 2
2 2
4
4
r
r
r
ES S
E E
A f
f
that is, the illuminance
r
E
is proportional to the illuminance E that reaches the pupil. The detection
of an image requires a minimum value of
r
E
, therefore a minimum value or threshold of E. In the
4
DANJON LIMIT: SCHAEFER’S METHOD
5
case of nonresolvable images, the size of the image does not matter, but what determines whether
it will be seen or not is the flux luminous that reaches the unit of surface, that is to say, its illuminance.
The threshold illuminance
th
E
is the same for all nonresolvable images regardless of size
and only depends on the luminance of the background.
3. Threshold illuminance as a function of background luminance
In Blackwell's experiment (1946), a luminous image of 0.595 arc minutes of angular diameter
was projected onto a lighted screen; the experience was repeated with varying levels of contrasts
(see appendix) and with various background luminances. The observers were young women with
good vision who knew where to look to see the image.
The observers were not given a time limit in each test, but in the course of the experiment, it
was found that, in general, 15 seconds was enough time to detect the vision threshold.
The precision of this test was lower than in experiments where the image was projected in
several different positions. The cause was attributed to the fatigue of the observers when looking
exclusively in one direction.
Blackwell made 90,000 observations and gave the results for a probability of vision of 50%,
that is, for the same conditions, the observer would see the image half of the times he looks at it.
As a secondary conclusion of Blackwell's experiment is that the threshold contrast, when
the observer knows where the image will appear and with more time of observation, is less than
when the image is projected in several positions, and the observer has a limited time to say if have
seen the image or not. From Blackwell's data, we have prepared table 2, where we compare the
contrasts for the two types of experience carried out by Blackwell.
Instead of the threshold contrast, we express the results in threshold illuminance. For a
circular image of small angular diameter
(in radians), the solid angle is (A.10)
2
,
2
for a uniformly illuminated image
E B
, then if
th
C
is the threshold contrast, the threshold
illuminance of a luminous object that is also illuminated by the background light of luminance
S
B
is
2
,
2
th th S
E C B
equation that we have used to obtain table 3.
We calculate the threshold illuminance per unit angular length by
0
th
th
E
G
we assume circular image with an angular diameter 0
1'
. In table 3, we calculate
log
th
E
and
2.5log
th
G
as a function of the background luminance,
th
G
is expressed in lux/rad.
4. Hapke equation
In Hapke's photometric model (1963, 1966, 1981), for smooth surfaces, the bidirectional
reflectance distribution function
r
f
(see appendix) or quotient between the emitted luminance
, ,
B i
and the luminous flux density F of a collimated beam (see appendix) is
, , 1 1
4
i
r i
i
w
f i P H H
i is the angle with which the sunlight falls with respect to the normal to the surface of the Moon,
is the angle of emergence of the ray directed towards the observer on Earth,
cos
i
i
,
cos
and
is the geocentric phase angle The factor
i i
is the LommelSeeliger
law (Fairbairn, 2005); w is the average singlescattering 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 singleparticle scattering function (see appendix) and H is the
Chandrasekhar function for isotropic dispersions that explains the effect of multiple dispersions.
The coefficient
1 4
appears by the normalization of the dispersion function.
(6)
(5)
Wenceslao Segura González
6
3.5
3
2.5
2
1.5
1
0.5
0
4.5067
4.9392
5.3699
5.7759
6.1795
6.5515
6.9181
7.2126
0.5
1
1.5
2
2.5
3
3.5
4
7.4992
7.6797
7.8500
7.8835
7.9274
8.1104
8.2951
8.5031
Table 3. The logarithm of the threshold illuminance for images smaller than the human eye's resolving as a
function of the logarithm of the background luminance, derived from Blackwell (1946).
th
G
is the threshold
illuminance per unit angular length for viewing a circular image of one minute of angular diameter. For a
background luminance of 5.60 cd/ m2,
2.5log
th
G
takes the value 8, which is the threshold required by
Schaefer's theory (see below).
2.4260
3.5074
4.5841
5.5991
6.6081
7.5381
8.4546
9.1908
9.9073
10.3586
10.7843
10.8681
10.9778
11.4353
11.8971
12.4170
lux rad
th
G
2
cd m
S
B
2
cd m
S
B
lux rad
th
G
lux
th
E
lux
th
E
121
55.2
18.2
9.68
3.60
2.132
2.062
1.889
1.730

2.561
2.547
2.460
2.328
1.721
17 %
19 %
23 %
27 %

2.130
2.060
1.857
1.658
1.111
2.553
2.538
2.428
2.241
1.569
17 %
19 %
24 %
26 %
29 %
121
55.2
18.2
9.68
3.60
2.114
2.020
1.738
1.437
0.812
2.464
2.428
2.276
2.051
1.322
14 %
17 %
24 %
30 %
39 %
1.959
1.857
1.421
1.070
0.385
2.212
2.131
1.914
1.672
0.924
11 %
13 %
26 %
36 %
58 %
Angular
image size
(minutes) % increment % increment
Angular
image size
(minutes)
% increment % increment
Table 2. Variation of the logarithm of the threshold contrast Cth when the exposure time increases and the
image always appear in the same position, according to the Blackwell experiment (1946). We show the results
for four background luminances. The percentage is the increase in the logarithm of the threshold contrast
when the experiment is modified. When the exposure is 6 seconds instead of 15 seconds, the threshold
contrast increases, making the image more difficult to see.
Exposure 6 s Exposure 15 s Exposure 15 sExposure 6 s
Exposure 6 s
Exposure 15 s
Exposure 6 s Exposure 15 s
log
th
C
log
th
C
log
th
C
log
th
C
2
342.6 cd m
S
B
2
34.26 cd m
S
B
2
3.426 cd m
S
B
2
0.3426 cd m
S
B
log
S
B
log
th
E
2.5log
th
G
log
S
B
log
th
E
2.5log
th
G
DANJON LIMIT: SCHAEFER’S METHOD
For phase angles greater than 90º, the opposition effect is null, therefore
0
P
, which
applies to large phase angles. The Legendre polynomials express the scattering function
2
1
1 3 1 ....
2
b c
it is enough to take the first terms. b and c are constant. Chandrasekhar's function is
1 2
1 2
H
1
w
.
When the surface is rough, like on the Moon, there are two effects: shadows projected on
the surface because other parts block the arrival of light, and effective surface tilt, caused by the
inclinations of the irregularities of the surface lunar that have different orientations, which means
that some areas are more or less illuminated and that are more or less seen by the observer.
When we deal with a rough surface, the bidirectional reflectance distribution function (6) 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
cos
i
and
cos
by the effective cosines
i
and
.
Equation (7) requires six parameters: the albedo w of a single scattering; two parameters to
evaluate the function of the opposition effect; the coefficients b and c of the Legendre polynomials;
and
which is the topographic slope that gives a smallscale roughness measure.
Helfenstein and Veverka (1987), adjusting (7) 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. The photometric data of the Moon are applicable up to the phase angle
of 150º, that is to say, that we cannot assure that for higher phase angles, which are those that
occur in the first visibility of the lunar crescent, Helvenstein and Veverka parameters are correct.
Suppose the coordinate system with the origin at the center of the Moon and with the xaxis
directed towards the center of the Earth; the Sun is in the xy plane (drawing 2). The spherical
coordinates in this system are the selenocentric coordinates of the Moon: latitude
and longitude
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
7
(7)
Wenceslao Segura González
. The coordinate system is not fixed on the surface of the Moon like the terrestrial coordinates but
depends on the position of the Moon with respect to the Earth.
The angles of incidence and reflection are related to the selenographic coordinates by (Lester,
McCall & Tatum, 1979)
cos cos cos ; cos cos cos ,
i
then we put the luminance found from (6) and (7) in terms of
,
and
.
5. Integration of Hapke's equation
The distance l (drawing 1) from the center O of the Moon to the point P located inside one
of the portions of the lunar crescent as observed in the flat lunar image seen from Earth, is related
to the coordinates z, y
sin ; cos sin
cos
sin ,
sin sin sin
z R y R
z z dz
l dl R d
l
the relationship between the angle of position and the latitude of 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 area of an element of one of the parts into which we divide the Moon with sides d
and
dl is
2
Rd dl
dS Rd dl d
r
r is the distance from the Moon to the Earth, R is the radius of the Moon, and d
is in radians. The
illuminance of a 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 (1987), Schaefer (1991) integrated (8); he calculated the illuminance per
unit of angular length (1) of a lunar portion and he expressed it as a magnitude, that is to say, in
logarithmic form
2.5log ,
G
, the integration result is in drawing 3; we verify that when we
approach the end of the lunar cusp, the illumination decreases sharply. This decrease in luminosity
is mainly because the width of the Moon decreases as we approach the end of the cusps.
We calculate the total illuminance of the Moon by (2) and the lunar magnitude by (3). In
drawing 4, we represent lunar magnitude as a function of phase angle for the mean distance
between the Earth and the Moon. Until
150º
, we have used the formula (A.8), and the remaining
values we deduce from the Schaefer integration, that is, from the integration of the curves in
drawing 3.
6. Schaefer's method of lunar visibility
Schaefer (1991) assumes, without a solid basis, that the threshold of illuminance per unit
angular length (in magnitudes) must be a fixed quantity. The value chosen by Schaefer is 8 because
it conforms to the experimental results of Danjon (1936). Then the portion of the Moon with an
illuminance per unit angular length (in magnitudes) greater than 8 will be visible and otherwise
invisible.
From drawing 3, we find that the Moon will cease to be seen when the phase angle is 172.1º
because then the illuminance per unit of angular length of the central part of the Moon
0
is
less than 8, which is considered the vision threshold. Therefore the Danjon limit or minimum arc
light (see appendix) between the Sun and the Moon in which the Moon can still be seen is 7.9º
8
(8)
DANJON LIMIT: SCHAEFER’S METHOD
according to Schaefer's theory. A value close to that found by other authors (table 1).
The previous value is geocentric since, in the integration of Hapke's equation (drawing 3),
the phase angle is geocentric. As the azimuth difference between Sun and Moon is zero and both
stars are close to the horizon, then the topocentric arclight is (see appendix)
LT T L
a h h h h a
LT
a
and
L
a
are the topocentric and geocentric arclight,
T
h
and h the altitudes of the topocentric
and geocentric Moon and
h
the altitude of the Sun that, because its parallax is small, we identify
the topocentric and geocentric value, finally
is the equatorial horizontal parallax of the Moon
which at the mean distance is approximately 1º. By (9), the topocentric Danjon limit derived from
Schefer's theory is approximately
7.9 1 7º
.
Schaefer's theory also explains the phenomenon of the shortening of the lunar cusp discovered
by Danjon (1932), according to which as the phase angle increases, the angle between the ends of
the Moon decreases because the lunar areas furthest from the center is no longer visible. According
Drawing 3. Integration of the lunar photometric equation of Hapke (1984) by Schaefer (1991). On the horizontal
axis is the position angle in degrees. The vertical axis is the illuminance per angular unit of the crescent of the
Moon
,
G
(in lux/rad) expressed in magnitudes:
2.5log ,
G
. The horizontal broken line is the
threshold illuminance per unit length in magnitudes according to Schaefer. Above this line, there is visibility,
and there is not if the value of
2.5log ,
G
is below it.
10 20 30 40 50 60 70 80 90
11
10
9
8
7
6
5
4
3
2
1
Angle of position in degrees
Illuminance per unit of angular length (in magnitudes)
175º
172.5º
170º
167.5º
160º
165º
155º
150º
170
P
167.5
P
165
P
160
P
9
(9)
Wenceslao Segura González
to Schaefer's theory, the interpretation of this phenomenon is that the illuminance of the cusp of the
Moon decreases when the phase angle increases, becoming lower than the threshold value of
vision.
From drawing 3, we calculate the shortening of the horns of the Moon according to the
phase angle. For example, point
170
P
marks the visibility for the Moon with a phase angle of 170º.
The position angle of
170
P
is 44.4º; therefore, the areas of the Moon with a greater position angle
are not visible because their illuminance per unit length is less than the threshold; therefore, the
150
155
160
165
167.5
170
172.1
174.2
168.5
161.1
143.0
123.5
88.8
0
Phase angle
(geocentric) Crescent length
Table 4. Length of the cusp of the Moon as a function of the phase angle, obtained by the Schaefer method.
To obtain the topocentric phase angle, we calculate the parallax correction. Due to the positions occupied by
the Sun and the Moon at the moment of sighting the crescent, we have assumed an approximate parallax of
1º. The length of the crescent is found by drawing 3
Phase angle
(topocentric)
Arclight
(topocentric)
151
156
161
166
168.5
171
173.1
29
24
19
14
11.5
9
6.9
10
100 110 120 130 140 150 160 170 180
2
2
4
6
8
10
Magnitude of the Moon
Geocentric phase angle in degrees
Drawing 4. The magnitude of the Moon as a function of the geocentric phase angle. We have found the
magnitude until the phase angle 150º from Allen's formula (A.8). The following magnitudes we found from the
Schaefer integration of Hapke's equation (7). We observe that from 160º, there is a gradual darkening of
crescent Moon caused by a decrease in the apparent size of the illuminated part of the Moon and by the
shielding suffered by the solar rays that arrive almost parallel to the lunar horizon.
DANJON LIMIT: SCHAEFER’S METHOD
angle between the two visible ends of the Moon is 88.8º. We do the same calculation for the
remaining points
167.5
P
,
165
P
,
160
P
, .... The result is in table 4.
7. Schaefer method shortcomings
For our research, the angular width of the Moon (see appendix) is less than the resolving
power of the human eye; therefore, the illuminance is the factor that determines visibility; there is,
therefore, a threshold illuminance. However, the threshold illuminance cannot be constant as proposed
by Schaefer, since it must depend on the background luminance (table 2), which in twilight is a
function of the depression of the Sun, of the difference in azimuth between the Sun and Moon and
the altitude of the Moon above the horizon.
We can assume that the limit 8 proposed by Schaefer corresponds to a fixed depression of
the Sun and neglect the effect of the azimuth difference, but then as the phase angle varies, the
altitude of the Moon will also vary, causing a variation of the luminance of the sky where the Moon
is and therefore of the threshold illuminance. But the significant variability of the atmospheric
extinction coefficient also significantly affects the illuminance of astronomical objects that are
close to the horizon, as occurs with the Moon in the first observation.
In summary, although there is a threshold illuminance to see the lunar crescent, it cannot be
constant but will depend on the positions of the Sun and the Moon and the extinction coefficient of
the atmospheric
8. Modification of the Schaefer method
We modify Schaefer's method in the sense that the threshold illuminance per unit of angular
length varies according to the positions of the Sun and the Moon and the atmospheric extinction
constant. The input data to determine the visibility of Moon are: the Moon's apparent altitude above
the horizon ha; the geocentric phase angle
; the atmospheric extinction constant k; and the sky's
illuminance as a function of the apparent altitude, the difference in azimuth, and the Sun's depression
d, which depends on the place of observation. Since we want to find the Danjon limit, we assume
that the azimuth difference between the centers of the Sun and the Moon is zero because this is the
best position so that the arclight or angular distance between the two stars is the smallest possible.
As we have said, increasing the phase angle shortens the lunar cusp. There will be a phase
angle for which it is visible only a portion of the Moon's center, for having greater illuminance. We
assume that this portion has a size smaller than the resolving human eye; therefore, it will have the
Drawing 5. Arc length shortening of the Moon as a function of the topocentric arclight or angle between the
centers of the Sun and the Moon. The line formed with segments corresponds to Danjon's (1936) results, and
the square points are the results obtained by the method of Schaefer (1991).
Topocentric arclight (degrees)
Lenght of crescent (degrees)
30
60
90
120
150
180
10 15 20 25 30 40 50
11
Wenceslao Segura González
width
(see appendix), always less than one arcminute, according to the Moon phase angle, and
we assume that its angular length is one minute of arc; that is, the lunar portion is approximately a
rectangle with sides 1’ and
, to ensure that its angular size is smaller than the resolving of the
human eye.
Suppose we know the depression of the Sun d (which we take as a positive number) and the
geocentric phase angle
. If we assume that the distance MoonSun is much greater than the
distance from the Earth to the Moon, then by (A.13) and (A.14)
180 180 1º
L T
a h d h d h h h
h and
T
h
are the geocentric and topocentric altitudes of the Moon. Later we do the correction for
refraction (A.16), and we find the apparent altitude
a
h
of the Moon. We determine the illuminance
of the sky
, , Z
S a
B h d
where the Moon is interpolating in table 5, obtained from the data of
Samaha, Asaad, and Mikhail (1969) measured in Helwan (Egypt).
Interpolating in table 3, we find the threshold illuminance that the Moon must have to be
visible with the background luminance found in table 5. The threshold illuminance per unit length
obtained from the integration of Hapke's equation (drawing 3) is before that the atmospheric
absorption. Therefore if
th
m
is the threshold magnitude of a portion of the Moon smaller than that
of resolving human eye, k is the extinction constant, X is the airmass (A.7); we calculate
th th
m m kX
(A.6), which is the magnitude of the lunar portion outside the atmosphere so that
the threshold magnitude at the Earth's surface is
th
m
. Finally,, from
th
m
we deduce the threshold
illuminance of the Moon outside the atmosphere, and dividing it by 1 minutearc expressed in
radians, we find
2.5log
th
G
.
In table 7, we have done the calculations for a geocentric phase angle of 168.5 degrees and
extinction constants of 0.15 and 0.17, representing excellent visibility conditions. Interpolating table
6, the minimum value of
2.5log
G
to view the Moon with a geocentric phase angle of 168.5º is
6.068. From table 7, we deduce that for
0.15
k
, the Moon would be visible with a topocentric
arclight of 10.6º (that is, the depression and 1.6º of the topocentric altitude of the Moon without
refraction), a value that is the Danjon limit of the Schaefer’s modified method. For more realistic
extinction constants, the Danjon limit will have a higher value.
It is important to note that we adopted Blackwell's visibility criteria, whose results are calculated
for a 50% probability of vision; so if the Moon has a geocentric phase angle of 168.5º and an
12
30
21.7
15
9.3
3.3
336
504
668
840
1008
22.3
16.7
7.8
1.17

252
336
503
671

29.2
21.3
16
7.8
3
84
126
167
252
336
27.8
22
17.2
13.8
6.5
33.6
50.4
67.3
84
134.4
Altitude
(degrees)
Altitude
(degrees)
Altitude
(degrees)
Altitude
(degrees)
30.5
21.7
12.8
3.8

8.4
10.8
33.6
67.2

26
17.8
8.4
3.8

4.2
8.4
16.8
25.2

27.3
22.7
15
6

0.84
1.68
3.36
6.72

29.7
22.8
16.6
11.8
5.3
0.25
0.50
0.84
1.26
2.52
Altitude
(degrees)
Altitude
(degrees)
Altitude
(degrees)
Altitude
(degrees)
Table 5.Twilight sky luminance
S
B
at Helwan for Sun's depression d and zero azimuth difference (Samaha,
Asaad and Mikhail, 1969). The unit of sky luminance
S
B
is cd/m2 . The altitude is apparent.
1º
d
2º
d
3º
d
4º
d
5º
d
6º
d
7º
d
8º
d
S
B
S
B
S
B
S
B
S
B
S
B
S
B
S
B
(10)
DANJON LIMIT: SCHAEFER’S METHOD
13
5
6
7
8
9
10
6.5
5.5
4.5
3.5
2.5
1.5
5.71
4.73
3.73
2.81
1.89
1.02
1.765
1.365
0.900
0.536
0.0013
0.386
5.966
6.280
6.625
6.892
7.213
7.434
0.935
1.720
2.583
3.250
4.053
4.605
9.224
10.782
12.969
15.826
20.034
26.136
0.449
0.103
0.638
0.876
1.048
0.685
0.633
0.113
0.378
0.560
0.647
0.162
4.691
5.242
5.777
6.015
6.187
5.824
4.506
5.026
5.518
5.699
5.787
5.301
dh ha
X
0.15
k
0.17
k
Table 7. Determination of the Danjon limit by the Schaefer’s modified method for extinction constants of 0.15
and 0.17. We do the calculations for a geocentric phase angle of 168.5º. d is the depression of the Sun, h is the
geocentric altitude of the Moon, and
a
h
is the apparent altitude, that is to say, topocentric and with refraction,
expressed in degrees. Knowing d and
a
h
, we interpolate in table 5 the logarithm of the luminance of the sky
S
B
in the position occupied by the Moon. From table 3, we interpolate the value of the logarithm of the
threshold illumination to observe an object smaller than the resolving one. By formula (A.5), we calculate the
threshold magnitude
th
m
of the portion of the Moon. X is the air mass calculated by (A.7).
For the atmospheric extinction constant 0.15, we calculate the magnitude
th
m
that the Moon must have
outside the atmosphere for its light to have the threshold magnitude
th
m
(th th
m m kX
) when passing
through it (A.6). From
th
m
, we calculate the illuminance per unit angular length
th
G
, assuming the Moon
formed by portions of 1 minutearc longitude.
From table 6 we find that the maximum value of
2.5log
G
when the phase angle is 168.5º is 6.070. From table
7 we find that
2.5log
th
G
exceeds this value when
0.15
k (that is to say, , max
th
G G
), but for
0.17
k the
Moon is no longer visible ( max
th
G G
). That is, the extreme topocentric arclight or Danjon limit is
9 1.55 10.6º
LT T
a d h ,
T
h
is the topocentric altitude of the center of the Moon without refraction.
The result shows that in exceptional atmospheric conditions, there is a 50% probability of seeing the Moon
at 10.6º from the Sun. But it is possible to see it with a smaller angle, or it can be invisible when the angle is
greater than 10.6º.
150
155
160
165
167.5
170
172.5
175
1.28
2.00
3.20
4.65
5.60
6.77
8.25
10.39
Geocentric
phase angle
(degrees)
Table 6. Maximum illuminance of the Moon per unit of angular length (that is, for
0
) as a function of the
geocentric phase angle. Data were taken from drawing 3 derived from the integration of Hapke's equation.
0º
max
2.5log
G
Wenceslao Segura González
extinction constant of 0.15, there is a 50% probability of seeing the Moon, which does does not
guarantee if it will be seen or not
It follows from our reasoning that the shortening of the lunar cusp depend on the altitude
above the horizon of the Moon, the atmospheric extinction constant, and the phase angle; and that
the Danjon limit depends on k.
9. Determination of the visibility of the crescent
To determine if the crescent will be seen using Schaefer's modified method, we need to
know: the geocentric phase angle
, the extinction constant k and the azimuth difference Interpolating
table 6, we find the maxim illuminance per unit angular length
max
G
for the phase angle
that
is, the one corresponding to the lunar portion
0
. The threshold illuminance per unit of angular
length
, , ,
th a
G d h k p
is found from Blackwell's results. In table 7, we have analyzed the lunar
visibility for a geocentric phase angle of 165º, an extinction constant of 0.3, and a null difference
between the azimuths of the Sun and the Moon.
We have done table 8 just like we did table 7. We compare the value of the threshold
illumination per unit of angular length
, , ,
th a
G d h k p
with the maxim illuminance per unit of length
of the Moon
max
, 0
G G
deduced by interpolating table 6. If max
th
G G
, the crescent
14
1
2
3
4
5
6
7
8
9
10
11
12
13
14
14
13
12
11
10
9
8
7
6
5
4
3
2
1
13.1433
12.1454
11.1486
10.1532
9.1596
8.1682
7.1798
6.1953
5.2163
4.2454
3.2870
2.3495
1.4486
0.6134
2.8603
2.6183
2.3275
2.0298
1.6504
1.2338
0.7889
0.3493
0.1115
0.4914
0.8181
1.2562
1.6318
1.9800
5.0595
5.2680
5.5100
5.7517
6.0581
6.3776
6.7063
7.0069
7.2765
7.4943
7.6140
7.7670
7.8588
7.8822
1.3313
0.8100
0.2050
0.3993
1.1653
1.9640
2.7858
3.5373
4.2113
4.7558
5.0550
5.4375
5.6670
5.7255
4.3216
4.6562
5.0467
5.5076
6.0589
6.7283
7.5560
8.6007
9.9518
11.7490
14.2168
17.7162
22.7951
30.0853
2.6278
2.2069
1.7190
1.2530
0.6524
0.0545
0.5190
0.9571
1.2258
1.2311
0.7900
0.1226
1.1715
3.3000
2.5116
2.9325
3.4204
3.8863
4.4870
5.0849
5.6584
6.0965
6.3652
6.3705
5.9294
5.2620
3.9679
1.8394
I
I
I
I
I
V
V
V
V
V
V
V
I
I
Table 8. Determination of the visibility of the Moon for a geocentric phase angle of 165º and for an extinction
coefficient of 0.3. From table 6, we find that the maximum value of
2.5log
G
for
165º
is 4.65. The
columns are geocentric depression of the Sun, geocentric altitude of the center of the Moon, apparent
altitude of the center of the Moon, the logarithm of the luminance of the sky in the place where the Moon is
located, the logarithm of the threshold illuminance to see the Moon, threshold magnitude of a lunar portion
of one minute of arc located in the center of the lunar crescent, airmass, threshold magnitude outside the
atmosphere and the threshold illuminance per unit length expressed in magnitudes. In the last column, it is
indicated whether or not the Moon will be visible. If max max
2.5log 2.5log
th th
G G G G
then the
Moon will be visible with a 50% probability. I indicate invisibility and V visibility. p is the probability of
vision.
dhX
log
S
B
log
th
E
th
m
th
m
2.5log
th
G
a
h
max
165º 0.3 50% 165º 4.65
k p G
DANJON LIMIT: SCHAEFER’S METHOD
will be seen, and otherwise, it will not be. With the data in table 8, we have constructed drawing 6.
The horizontal broken line corresponds to the illuminance per unit of angular length, expressed in
magnitudes, that has a portion of the Moon of 1 minute of arc located in the center of the crescent,
as it follows of table 6, this horizontal line is the limit between visibility or nonvisibility
of the Moon.
When the altitude of the Moon is considerable (right part of the graph), the depression of the
Sun is small. Therefore the sky is very illuminated, preventing the Moon from being seen, as shown
in the right part of drawing 6, in which the curve is below the horizontal line. On the contrary, when
the altitude is small (left part of drawing 6), the sky's illumination is low because the depression of
the Sun is great. Then the Moon is no longer visible because the light coming from the Moon is
strongly absorbed by the atmosphere. The lunar visibility zone is located between the two previous
zones. The curve has a maximum, which corresponds to the altitude of the Moon where the crescent
can best be seen.
We define the visibility coefficient by
max
log
log
th
G
P
G
th
G
and
max
G
are the threshold illuminances per unit of angular length and the illuminance per unit
of angular length of the center of the Moon calculated by drawing 3. If P is greater than 1, the
Moon will be visible and otherwise invisible. In drawing 9, we represent the visibility criterion
according to the Schaefer’s modified method for various geocentric phase angles and atmospheric
extinction constants.
15
Drawing 6. Graph obtained from table 8. The horizontal broken line is the limit to see the Moon for a phase
angle of 165º deduced from table 6. In the upper part of this line are the positions in which the crescent is
visible, and in the bottom are the positions in which the crescent of the Moon is not visible. When the altitude
of the Moon is large (right of the curve), the depression of the Sun is small, and the sky is very illuminated,
being impossible to see the crescent Moon. When the altitude of the Moon is small (left of the curve), the sky
is poorly illuminated, but the light from the Moon that is close to the horizon is strongly absorbed by the
atmosphere and does not reach the limit required for vision. Between both extremes are the positions where
the Moon is visible. The maximum of the curve represents the moment of the best visibility of the crescent.
We find that the Moon is visible between the apparent altitudes of 8º 53 ’to 1º 50’. The best time for visibility
is at altitude 4º 41’. These values will be different for other phase angles and other extinction coefficients.
Apparent altitude of the Moon
Illuminance per unit of angular length in magnitudes
1 3 5 7 9 11 13
2
3
4
5
6
(11)
2.5log , , ,
th a
G d h k p
max
2.5logG
Wenceslao Segura González
9. Vision probability
We determine the sensitivity of the human eye to see a luminous object in a lit background by
the experiment of Blackwell (1946). As we have stated, it follows from this experiment that the
observation in the critical visibility zone is probabilistic. The numerical values that we have used
from Blackwell's investigation apply to a 50% probability of vision.
We adapt the results of the experiment to use a different 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 of 50%, according to Blackwell
50% .
th th
C p p C
In drawing 10, 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.
By definition of contrast (A.3)
50%
; 50% ,
th th
th th
S S
B p B
C p C
B B
th
B p
is the threshold luminance to see the Moon with a probability of vision p, and
50%
th
B is
the threshold luminance for the probability of 50%. The sky luminance
S
B
is the same in the two
formulas. From (12), we deduce the relationship between the threshold illuminances per unit of
angular length for the probabilities p and 50%
2.5log 2.5log 50 % 2.5log .
th th
G p G p
Table 9 shows the value of
as a function of the probability of vision p deduced from
drawing 10. In the third column of table 9, we calculate the term that we have to apply in (13) to
find the visibility of the Moon with another probability of vision.
The results that we have found in previous calculations must be modified if we want a
probability of vision different from 50%. For example, with a probability of 20%, the Danjon limit
16
0.4
Visibility coefficient
0.6
0.8
1.2
1.4
1.6
1 3 5 7 8 10 11
Apparent altitude
of the Moon
Drawing 9. Visibility of the Moon according to Schaefer's modified method for various geocentric phase
angles, for an atmospheric extinction coefficient of 0.3 and zero difference between the azimuths of the Sun
and the Moon. The horizontal line corresponds to
1
P
. There will be visibility for the positions of the Moon
located above this line, and the Moon will be invisible for the positions that are below the horizontal line. For
a phase angle of 164º, the Moon will be visible between the apparent altitudes of 10º 41' and 1º 35'; however,
for the phase angle of 167º, the Moon will be invisible for an extinction constant of 0.3. The maximums of the
curves or points of optimal vision of the Moon are between the apparent altitudes of 5º 23' and 3º 40'. The
probability of vision is 50%.
(12)
(13)
DANJON LIMIT: SCHAEFER’S METHOD
17
that we find in table 7 will be lower because the contrast is smaller. Repeating table 7, we find that
the topocentric Danjon limit for a probability of 20% and
0.15
k
is approximately 10º, only a few
tenths less than that corresponding to the probability of 50%, which shows us that the effect of the
probability of vision it is not a factor that significantly affects the calculation of the Moon's visibility.
10. Danjon limit and shortening of the lunar horns
Schaefer's method explains the Danjon limit, that is, the existence of a minimum arclight for
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 10. 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.
10
20
30
40
50
60
70
80
90
0.38
0.59
0.72
0.87
1
1.11
1.24
1.40
1.62
Table 9. Coefficient
for a probability of vision
p.
is the factor by which we must multiply the
threshold contrast for a 50% probability of
vision to find the threshold contrast for a
probability p. The third column is the term
that must be added to
2.5log 50 %
th
G to
determine
2.5log th
G p
(13).
2.5log
1.05
0.61
0.36
0.15
0
 0.11
 0.23
 0.37
 0.52
p
(%)
Wenceslao Segura González
the vision of the crescent Moon. However, as we have shown, the Danjon limit depends on the
phase angle, but it is also highly dependent on the atmospheric extinction coefficient, the altitude of
the Moon, and the azimuth difference between the Moon and the Sun. Therefore, the Danjon limit
does not depend exclusively on the phase angle, and we cannot compare the arclight limits measured
at different times and in other places.
Schaefer's method also explains the shortening of the horns of the Moon. But this shortening
depends, in addition, on the phase angle, the atmospheric attenuation, and the altitude of the Moon.
In table 9, we calculated the angular length of the Moon as a function of the phase angle, the
extinction constant, and the apparent altitude of the center of the Moon. We verify that all three
factors affect the length of the crescent Moon, showing that the phase angle is not the only parameter
to consider. The length of the lunar horns is small shortly after the first vision of the crescent and
when the Moon is near the horizon. There is an altitude of the Moon at which the length of the
crescent is maximum. In table 9, we verify that when the apparent altitude of the Moon is between
4 and 5 degrees above the horizon, the crescent reaches its maximum angular length.
11. Conclusions
We have exposed Schaefer's method to determine the Danjon limit, which uses the illuminance
18
12
11
10
9
8
7
6
5
4
3
2
1


49.0
83.4
105.0
118.8
125.6
126.4
119.7
104.4





51.0
83.9
98.7
106.4
103.2
82.6
47.3







43.8
77.7
81.9
63.5


7.6
38.7
79.7
105.4
123.5
135.0
142.3
144.7
143.4
139.8
125.2
83.3



52.2
87.4
109.4
123.3
130.7
132.6
127.7
114.9
61.8





58.4
89.2
113.2
121.7
119.5
104.4

Crescent Moon longitude (degrees)
Table 10. Angular length of the lunar crescent as a function of the phase angle, the extinction coefficient, and
the apparent altitude of the Moon. The lines mean that the Moon is not visible for those conditions. We
verify that Schaefer's modified method explains the shortening of the horns of the Moon, but this shortening
is not exclusively a function of the phase angle but is affected by the extinction constant and the altitude at
which the Moon is. The shortening increases with the extinction constant. The table also shows that the
shortening of the horns is more significant when the Moon becomes visible (high values of the altitude) and
when the Moon is soon to disappear from vision because it is very close to the horizon. There is an intermediate
altitude where the Moon is seen at its maximum length. To find the data in the previous table, we have
calculated
th
G
for each of the situations, and then we have found in drawing 3 the value of
for which the
illuminance per unit of angular length coincides with
th
G
for the phase angle considered.
a
h
164º
165º
166º
0.3
k
0.3
k
0.3
k
0.2
k
0.2
k
0.2
k
DANJON LIMIT: SCHAEFER’S METHOD
the Moon deduced from Hapke's equation (1984). Schaefer proposed that the threshold illuminance
per unit of angular length for the vision of a luminous object is a value constant.
We show that the threshold illuminance per unit of angular length depends on the luminance
of the twilight sky where the Moon is; therefore, it cannot have a constant value. We found that the
Danjon limit depends on atmospheric absorption, a factor that Schaefer did not take into account.
No obstante, Schaefer evalúa el error cometido cuando se toma 2.5logGth=8, estimándolo en 1.5
magnitudes, es decir que el valor umbral va de 6.5 a 10.5, una dispersión excesivamente grande,
por esto es mejor considerar desde un principio la variabilidad del valor umbral de la iluminancia por
unidad de longitud como hemos hecho nosotros.
We propose a modification of Schaefer's method, continue using Hapke's equations to find
the illuminance of the Moon and we find the threshold illuminance from the results of Blackwell's
experiment calculated for a 50% probability. We have exposed the method to express the results on
lunar visibility for another different probability.
The illuminance decreases at the ends of the Moon mainly because its width becomes small;
when it is less than the threshold illuminance, it is no longer visible, which explains the shortening of
the lunar horns. However, this phenomenon depends not only on the phase angle but on the
atmospheric extinction constant and the altitude of the Moon.
In short, we have stated that the problem of the visibility of the crescent of the Moon and in
particular the determination of the Danjon limit, requires a more complex solution; in particular, the
atmospheric extinction constant, the probability of vision and the relative position of the Moon with
respect to the Sun must be taken into account.
We have verified that there is an altitude of the Moon in which its observation is optimal
(drawing 9). In this position, the visibility coefficient P takes the highest possible value for the
conditions considered.
The value found for the Danjon limit by Schaefer's method, despite its uncertainty, is higher
than that found by other authors (table 1); this leads us to think that perhaps Hapke's photometric
theory gives an excessive darkening of the Moon at large phase angles, assuming an excessive
shielding of the solar rays.
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
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
19
(A.1)
Wenceslao Segura González
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
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
* 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
,
i i
y
,
r r
are the spherical coordinates of the direction of incidence and reflection. The
BRDF is measured in the inverse of stereoradian.
* 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, McCall and
Tatum 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.
* 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)
20
(A.2)
(A.3)
(A.4)
DANJON LIMIT: SCHAEFER’S METHOD
13.98 2.5 log .
m E
la unidad de E es 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
10 10
kX kX
E E B 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 2.5log 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
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.
* Elongation. Elongation is the difference between the apparent ecliptic longitudes (corrected for
precession, nutation and aberration) 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 arclight
L
a
and elongation
M S
L L
is (Segura 2018, p. 191193)
cos cos cos .
L M M S
a L L
There is a relationship between the arclight 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
the above equations are applicable to topocentric values (Segura 2020b).
* 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
which is valid until
150º
. 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
21
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
Wenceslao Segura González
22
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
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
* Geocentric to topocentric horizontal coordinate conversion. From the geocentric horizontal
coordinates, altura y acimut, h, A of the Sun or the Moon, we calculate the topocentric horizontal
coordinates
T
h
,
T
A
by the equations (Segura 2018, p. 3135)
cos cos cos cos sin sin
cos sin cos sin
sin sin sin cos ,
T T
T T
T
s h A h A
s h A h A
s h h
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 Z is the
azimuth difference between the Sun and the Moon.
horizon
Z
S
M
North
cardinal point South
cardinal point
meridian
LT
a
(A.10)
(A.11)
DANJON LIMIT: SCHAEFER’S METHOD
T
s r r
, r and
T
r
are the geocentric and topocentric distances from the center of the Sun or
Moon.
is the equatorial horizontal parallax, defined by
sin rad,
E E
R R
r r
RE is the equatorial radius of the Earth. In equation (A.11),
and
are the geographic and
geocentric latitudes of the observation point on the Earth's surface that are related by
2
2
tan tan ,
E
b
R
6356.77
b
km is the semiminor axis of the Earth's ellipsoid of revolution,
6378.16
E
R km,
and
is the quotient
E
R R
, where
R
is the distance from the center of the ellipsoid of
revolution of the Earth to the point of observation. The parameter
is calculated by
2 2 2
.
1 1 cos
E
E
b R
b 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.
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’’.
* Refraction. The angle of refraction is
23
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.12)
(A.13)
(A.14)
(A.15)
Wenceslao Segura González
24
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.16) 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
r
R is the radius of the Moon and r the EarthMoon distance,
is expressed in radians.
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