Danjon noticed that the length (cusp to
cusp) of the new crescent moon was less than 180 degrees and suggested that the
cause of the shortening is the shadows of the lunar mountains. McNally, however,
attributed the crescent shortening to atmospheric seeing, while Schaefer
suggests that length shortening is due to sharp falling off of the brightness
towards the cusps.
We
attribute length shortening to the Blackwell contrast threshold; we consider
the thin crescent as a group of discs of varying angular size, and each has its
equivalent Blackwell disc, the largest being at the centre of the crescent. The
discs become smaller in the direction of the cusps, therefore the Blackwell
thresholds become higher. According to this model, if we know the apparent
diameter of the Moon and the width of the crescent, we can calculate the
approximate visible length of the crescent.
Introduction
In 1931 August,
Danjon at the Strasbourg Observatory, France, noticed that the length (cusp to
cusp) of the crescent Moon, which was only 16.6 hours before conjunction,
extended only 70-80 degrees instead of 180 degrees. He suggested that the cause
of the shortening is the shadows of the lunar mountains. However, McNally did
not accept this interpretation and showed that the height of the mountainous
lunar terrain compared to the lunar radius is not sufficient to be the cause of
shortening; he attributed the crescent shortening to atmospheric seeing.
Schaefer, in turn, rejected McNally’s explanation, showing that the resolution
of the human eye is larger than the size of the crescent disc so that seeing
has no effect on the perceived width. He suggested that length shortening is
due to sharp falling off of the brightness towards the cusps, emphasizing that
detection threshold does not depend on the surface brightness of the Moon but on
the total brightness integrated across the crescent. He points out that as the
extreme parts of the crescent are narrower than the resolution of the eye "the
detection threshold does not depend on the surface brightness of the moon, but
on the total brightness integrated across the crescent".
Discussion
We disagree with
Schaefer because we believe that human vision is more sophisticated than he
suggests. Briefly, human vision involves the simultaneous interaction of the
eye and the brain through a network of neurons, receptors, and other
specialized cells. The retina contains neural circuitry that converts light
energy to action potentials that travel along the optic nerve to the brain. If
the retinal area that is illuminated is small enough then the photons will fall
entirely within the centre of the receptive field. If enough photons fall into
the receptive field, the ganglion cell will respond by firing. According to
Ricco's Law of spatial summation, if we increase the area of the stimulus so
that it is still within the centre of the receptive field, then more photons
would be collected over this larger area and so a lower intensity of light
would be required. Ricco's Law of spatial summation has been completely
disregardedby Schaefer: "the
experiments (see also Blackwell 1946) show thatunresolved circles of light have a threshold that is independent of the
source size, yet which depends on the total brightness within the circle. The
basic idea is that all that matters is how much light is received by any
resolution element of the eye and not how the light is spread over the
‘pixel’".
As far as I know
Schaefer is the most prolific contributor to the subject that we are
discussing, but his paper implies a measure of confusion with some photometric definitions
such as brightness, surface brightness, integrated brightness, and total integrated
brightness.In spite of this (i.e., if
we neglect Schaefer’s length-shortening interpretation), he got very good
empirical, results represented by his Fig. 2.
Clark describes Blackwell’s 1946 data in
his excellent book entitled Visual Astronomy of the Deep Sky; Clark
has added additional comments since the book's publication (1990), at
http://clarkvision.com/visastro/omva1/index.html. At this site, Clark illustrates Blackwell’s 1946 data in a diagram (his
Fig.2.6) which he explained as follows: "Here we notice that for objects
with small angular sizes, the smallest detectable contrast times the surface
area is a constant. As an object becomes larger, this product is no longer
constant. The angle at which the change occurs is called the critical visual
angle. An object, smaller than this angle, is a point source as far as the eye
is concerned. (A point can be considered the angular size smaller than which no
detail can be seen.)"
In effect, Clark
specifies the domain of utilization of Ricco’s Law, which should be less than
the critical visual angle. For stimuli smaller than the critical visual angle, i.e.,
on the left side of Clark’s figure, we conclude two things: the first was
mentioned by Clark himself while the second was mentioned neither by Schaefer
nor by Clark.(i) When the contrast between surface brightness of the stimulus and
surface brightness of the background is higher than Blackwell’s contrast
threshold, the object is seen as a point source; and (ii) when it is less than
Blackwell’s threshold, the object couldn’t be seen at all. Therefore, the
visibility of small stimuli__smaller than the critical visual angle is
characterized by the left-side curves of Clark’s
Figure. Fromthese curves, we find thatthe stimulus visibility alwaysdepends on its luminance, itsdiameter, and the background luminance even
if it is smaller than the critical visual angle.
Crescent-length
calculations
We consider the thin lunar crescent as a
group of discs of varying angular size, and each has its equivalent Blackwell
disc. The largest disc is at the centre of the crescent. Discs become smaller
in the direction of the cusps, therefore Blackwell thresholds become higher. To
obtain Blackwell’s contrast threshold, Cth, for discs of diameters less than
0.6 minute of arc, we extrapolate the data in table VIII of Blackwell4. Fig.1
is an example of our extrapolations. We found that the smallest width to be
seen depends mainly on three local factors: site elevation, sky luminance, and
the zenith distance of the Moon at the moment of observation. It is also
depends on whether the Moon is near apogee or perigee. For our site at Mouneef
(1990 metres, 44 E, 13 N), the two smallest widths to be seen are about 0.16
arc minute when the Moon is near apogee and about 0.18 arc minute when the Moon
is near perigee.
From standard software and Fig.2, we can
calculate the length of the visible crescent as follows: on the vertical diameter
of the Moon’s disc (Fig. 2), the length of the group of discs starting with the
disc at the centre and ending at the end of one cusp is equal to r + W/2,
then
r + W/2 = W + W1 + W2 +…
= W + W [1W/(r + W/2)] + W [1W/(r + W/2)]2 + …(1)
From Danco5, the right side of Eqn.1
represents a decreasing geometric progression whose sum is equal to W/W/(r +
W/2), which is clearly equal to the left side of Eqn.1.
In Fig. 2, let L/2 be the distance starting
at the centre of the Moon’s disc and endingat the beginning of the largest invisible width w(for
our site w = 0.14 arc minute when the Moon is near apogee and w = 0.16 arc
minute when the Moon is near perigee)
L /2 = r – w/W
/(r + W/2)
= r – w (r + W/2)/W
then,
L = 2r – 2w
(r + W/2)/W,
where Lrepresents the vertical component of the
visible crescent length in arc minutes. Then the visible crescent length in
degrees will be: L /2r X 180.
If we
take w = 0.15 arc minute as an approximate value for the biggest invisible
width, we get a simple formula for calculating crescent length. It depends only
on two factors: the apparent diameter of the Moon, D, and the width of the
crescent, W.
Then
L = D – 0.3 (D + W)/2W.(2)
Putting the model to
the test
In the following, we test our model
calculations using two well-documented observations. We ask the reader to
compare our results with those given by the only existing model Schaefer’s
model.
Danjon’s observation
of 1932 August 13: (reported length =70
80ｰ)
D=
32.88 arc minute
W=
0.27 arc minute
L= D –
0.3 (D + W)/2W
= 14.46 arc minutes
The crescent length in degrees will be
L/D X 180 =
79.2
By using MOONC60 software
(http://www.starlight.demon.co.uk/mooncalc), which adopts Schaefer's model, we
get L/DX 180= 93
Stamm’s observation of 1996 January 20:
(reported length = 45)
D = 33.32 arc minute
W = 0.19 arc minute
L = D – 0.3 (D + W)/2W
= 6.9 arc minute
The crescent length in degrees will be
L/D X 180 =
37.1
By using Schaefer's mode l we get L/DX 180= 56
Conclusions
Our model gives a new explanation for the
cause of length shortening of the new crescent Moon and it introduces a method for
calculating the approximate visible length of the thin crescent Moon. Further, our
photometric model gives the same results obtained by Schaefer’s empirical model
but with a different interpretation concerning the cause of length shortening.
References
(1) A. Danjon,L’Astronomie, 46, 57, 1932.
(2) D. McNally,
QJRAS, 24, 417, 1983.
(3) B. E. Schaefer,
QJRAS, 32, 265, 1991.
(4) H. R. Blackwell,
JOSA, 36, 624, 1946.
(5) P. E. Danco,
Higher Mathematics in Problems and Exercises (Mir Publishers,Mosc ow), 1983.
(6) M. B. Pepin, S&T, 92, 104, 1996.
Figure Captions
FIG. 1: An example of
our extrapolation of the data in table VIII of Blackwell4, the crescent width
is equal to 0.1 arc minute.
FIG.2. Starting from
the centre of the crescent and going in the direction of the cusps, the
crescent contains two groups of discs; each group represents a decreasing geometric
progression. The visible crescent length in degrees will be: L /2r X 180, whereLrepresents the vertical
component of the visible crescent length in arc minutes and is given byL = 2r – 2w (r+W/2)/W, and where r is the
apparent semi-diameter of the Moon, W is the width of the crescent, andwisthelargest invisible width. For our site w =
0.14 arc minute when the Moon is near apogee and w = 0.16 arc minute when the
Moon is near perigee.
