Bismillah hir-Rahman nir-Rahim
(In the Name of ALLAH, THE MOST BENEFICENT, THE MOST MERCIFUL)
THE MUSLIM STUDENTS' ASSOCIATION OF COLUMBIA UNIVERSITY
102 Earl Hall, Columbia University, NEW YORK, NY 10027.
THE POSSIBLE ORIENTATIONS (SHAPES) OF A HILAL (CRESCENT MOON)
*************************************************************
2 Shawwal 1417
10 February 1997
Dr.Mohib.N.Durrani.
E-Mail: mnd0@columbia.edu
Home Page: http://Hilal-Sighting.Com
Copyright (c) 1997.
Permission for free distribution is granted to all.
Please give reference and credit.
As Salam alikum.
We will explore the different orientations (shapes) of possible
Hilals (Crescent Moons), and the reasons for them.
There have been many questions on acceptable Hilal shapes.
The questions that need to be answered are:
Are ALL reported Hilal shapes valid ?
If not, what are the valid Hilal shapes, and why ?
These postings are to educate ourselves, so that, when Hilal sighting
reports are made, we can be our own evaluators, and accept only those
Hilal shapes that are possible, and do NOT accept those Hilal shapes
that are not possible.
This posting has seven (7) sections.
This posting will be followed with at least one more, on:
"The Hilal of the First of the Month, and the variations in
the Altitude, Thickness, and Time between Sunset and Moonset"
inshallah.
Your brother in Islam,
Mohib.
1. INTRODUCTION
2. A SPHERE ILLUMINATED BY LIGHT
3. PHASES OF THE MOON
4. VALID HILAL SHAPES ON THE HORIZON
5. POSITIONAL ASTRONOMY
6. MEASUREMENTS IN THE SKY
7. CONCLUSION
1. INTRODUCTION
***************
We will start with an explantion of the different apparent shapes
of the illuminated portion of a sphere, that is illuminated from a
distant light source, when observed from different points around the
sphere. This is then compared to the Phases of the Moon.
The explanations are based on two dimensional (2-D) geometry.
The next section on Valid Hilal (Crescent Moon) Shapes on the Horizon
is based on three dimensional (3-D) geometry. This is because of the
inclination of the path of the Moon around the Earth to the path of
the Earth around the Sun.
The next two sections deal with the Accuracy of Positional Astronomy
and Approximate Measurements of Angles in the Sky. The last section
is the Conclusion.
In order to keep the explanations concise, some of the statements are
approximated. A more precise explanation would involve additional
background in sciences and the greater precision not necessary to
understand most of the required phenomena.
2. A SPHERE ILLUMINATED BY LIGHT
********************************
The explanations given here are based on elementary physics of
light-and-shadow around spheres (the Moon) that are illuminated from a
light source of parallel light (the Sun). The changes to the
apparent shape of the lighted sphere is dependent on the location
of the observer (on earth).
The underlying assumption is that light travels in "straight" lines.
Hence, the following explanations could also be understood with
principles of elementary Geometry.
Consider parallel light coming from the left, from a far away source
(Sun), and striking a sphere (Moon). Only the half of the sphere,
towards the light, would be flodded with "Light" while the other half
of the sphere, which is away from the light, would be "Dark". Hence
the illuminated part of the shpere would be from equivalent points 6, 7, 8,
9, 10, 11, and 12 on the sphere, and the equivalent dark points (XXXX)
on the sphere are 12, 1, 2, 3, 4, 5, and 6. The points 6 and 12 are
the transition points and hence are included in both groups.
The point "0" is located at the center of the sphere.
(-------|-------)
LIGHTED | DARK
HALF | HALF
12
PARALLEL LIGHT 11 1
SPHERE
-----------------) 10 _____ 2
-----------------) / |XXX\
-----------------) 9 ( 0 XXX) 3
-----------------) \ |XXX/
-----------------) 8 ----- 4
(MOON)
PARALLEL LIGHT 7 5
6
We will now shift our origin to different points (relative positions of
the Earth). That is, the locations from where we are going to look at
the Moon will be from different places. These places are the relative
places of the Earth.
a. When WE (the EARTH) are at a distant point (away from the Lunar surface)
on a line joining the center of the sphere, 0, to the point 3, the Moon
appears as an Astronomical New Moon, which is completely dark, and hence
is invisible.
b. When WE (the EARTH) are at a distant point (away from the Lunar surface)
on a line joining the center of the sphere, 0, to the point 12, the Moon
appears as a First Quarter Moon, that is half illuminated.
c. When WE (the EARTH) are at a distant point (away from the Lunar surface)
on a line joining the center of the sphere, 0, to the point 9, the Moon
appears as a Full Moon, which is fully illuminated.
d. When WE (the EARTH) are at a distant point (away from the Lunar surface)
on a line joining the center of the sphere, 0, to the point 6, the Moon
appears as a Last Quarter Moon, where the other half is illuminated.
e. When WE (the EARTH) are at a distant point (away from the Lunar surface)
on a line joining the center of the sphere, 0, to the point 2, the Moon
appears as a HILAL (Crescent Moon).
*** IT IS IMPORTANT TO UNDERSTAND WHY THE ILLUMINATED PORTION OF ***
*** THE SPHERE APPEARS AS A CRESCENT !! ***
(Try to use a large sphere, 1 foot or 30 centimeters in diameter,
in a dark room. Use a single light source. Keep the center of the
sphere and the light source at eye level. You can walk around the
sphere and see the "phases" of the sphere.)
The border of the illuminated portion of the sphere with the dark
portion is a great circle (with center of circle as the center of
the sphere). When this border circle is viewed from an angle (a point
not in the plane of the circle) it appears elliptical (flattened).
The illuminated portion of the sphere appears like a crescent because
we are able to see only a small portion of the illuminated sphere
(between the circular illuminated edge of the sphere and the elliptical
boundary of the illuminated portion ) and a major portion of the dark
(not illuminated) sphere.
NOTE: For the Sphere, we have the center of the Sphere, the source
of Light (center of the Sun) and our Eye, on the same horizontal
level. This condition would give the following shape of the
Crescent Sphere.
)\ |\ /| /(
) \ | \ / | /(
) \ | \ / | / (
) ) | ) ( | ( (
) ) | ) ( | ( (
) ) | ) ( | ( (
) / | / \ | \ (
) / | / \ | \(
)/ |/ \| \(
CRESCENT FIRST THIRD CRESCENT
WAXING QUARTER QUARTER WANING
(Increasing) (Decreasing)
*** THE ENDS OF THE CRESCENT ("Horns") ALWAYS POINT AWAY ***
*** FROM THE SOURCE OF LIGHT !!! ***
3. PHASES OF THE MOON
*********************
The Sun-Earth-Moon form a three-body system such that the combined
Earth-Moon revolves around the sun in 1 year (365.25 days), and the
Moon revolves around the Earth in 1 Lunar month (29.53 days).
Consider the Sun to be at the left with Sunlight falling on the
Earth-Moon system, as shown below. Consider the Earth to be at "0"
and the Moon to revolve conterclockwise around the Earth. Hence the
Moon can take the positions: 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1.
12
PARALLEL LIGHT 11 1
C
-----------------) 10 _____ 2
-----------------) / \
-----------------) 9 D ( 0 ) B 3
-----------------) \ EARTH /
-----------------) 8 ----- 4
A
PARALLEL LIGHT 7 5
6
As mentioned earlier:
The Earth (Origin at Earth) is at position 0.
The Astronomical New Moon is when the Moon is at position 9,
the First Quarter is when the Moon is at position 6,
the Full Moon is when the Moon is at position 3, and
The Third Quarter is when the Moon is at position 12.
Consider the Earth to rotate in a counter-clockwise direction.
The locations A, B, C, and D are "stationary" in space, while the
Earth rotates.
An observer at location "A" on the Earth would just have had Sunset,
an observer at location "B" on the Earth would have midnight,
an observer at location "C" on the Earth would just have had Sunrise, and
an observer at location "D" on the Earth would have noon.
*** IT IS IMPORTANT TO UNDERSTAND WHY THE PARTICULAR LOCATIONS ***
*** INDICATE SUNSET, MIDNIGHT, SUNRISE, AND NOON !! ***
The following observations can also be made about the Cycle of the
phases of the Moon:
a. Astronomical New Moon (Invisible)
b. Crescent Moon (Visible AFTER sunset, for some time)
c. First Quarter (Visible in early night, until about midnight)
d. Full Moon (Visible ALL night)
e. Third Quarter (Visible after midnight, until sunrise)
f. Waning Crescent Moon (Visible BEFORE sunrise, for some time)
g. Astronomical New Moon (Invisible)
4. VALID HILAL SHAPES ON THE HORIZON
************************************
The Hilals that are visible on the western horizon are always crescents.
As we have seen, THE ENDS OF THE HILAL ("HORNS") ALWAYS POINT AWAY
FROM THE SUN !!!
The Hilal, on the horizon, is located such that it is sometimes to the
left, or above, or to the right of the Sun. We know beforehand, from
calculations, where the Hilal is located for any particular date and
place on the Earth. Three dimensional (3-D) geometry gives a good
indication of the shape of the Hilal.
To report the orientation of the Hilal to another person, one of the
easiest ways is to imagine a clock in the sky with 12'O Clock at the
Top, 3'O Clock to the Right, 6'O Clock at the Bottom, and 9'O Clock
to the Left. Then the Crescent can be easily described as, for example,
from 4'O Clock to 5, 6, 7, and 8'O Clock orientation.
The shape of the Hilal on the horizon could take any of the follwoing
orientations, depending on whether the Hilal is to the LEFT of the Sun,
ABOVE the Sun, or to the RIGHT of the Sun:
SHAPE "A" SHAPE "B" SHAPE "C"
********* ********* *********
\ /
\ /
\\ //
|) (|
\ // \ / \\ /
\\_ _/ / \\__ __// \ \_ _//
\ \___/ / \ \___/ / \ \___/ /
\----/ \-----/ \----/
HILAL TO LEFT HILAL ABOVE HILAL TO RIGHT
OF SUN SUN OF SUN
From 2, 3, to From 4, 5, to From 5, 6, to
.. 6, 7'O Clock .. 7, 8'O Clock .. 9, 10'O Clock
Positions Positions Positions
IT IS IMPOSSIBLE FOR A HILAL TO THE LEFT OF THE SUN
TO LOOK LIKE SHAPE "B or C" !!!
IT IS IMPOSSIBLE FOR A HILAL TO THE RIGHT OF THE SUN
TO LOOK LIKE SHAPE "A or B" !!!
IT IS IMPOSSIBLE FOR A HILAL TO LOOK LIKE AN INVERTED CRESCENT !!!
i.e. INVERTED (horizontally) SHAPES of "A, B, or C" !!!
It should also be noted that the Hilal will be near the Sun, with the
path of the setting Hilal parallel to the path of the setting Sun.
The Hilal will be a maximum of +/- 5 degrees above or below the path
of the Sun. The Hilal would be in the Westerly direction, not in the
east, nor in the north nor south.
The Hilal should not "rise" nor drift "left or right", as a cloud could,
but it should set at an angle nearly equal to the compliment of the
latitude of the place (90 minus latitude).
5. POSITIONAL ASTRONOMY
***********************
Science has progressed to such an extent that Positional Astronomy is
now very precise (less than one arc-second in angular measure) for the
Celestial objects, as seen from any place on the surface of the Earth.
The location of the Hilal in the sky varies with the latitudes and
longitudes of the observing locations. Please see the next posting
for the variations of "First Day Hilals".
6. MEASUREMENTS IN THE SKY
**************************
It is best to be able to use a sextant to measure angles in the sky,
especially the altitude (height above the horizon) of the Hilal.
An approximation is to use our finger-widths (each finger at a distance
of an arm from the eye is 1.5 degrees), fist-width (a fist at an arm's
distance from the eye is 10 degrees), and out-stretched-fingers, from
end of thumb to end of little finger (20 degrees).
The imaginary clock, with 12'O Clock position on top, 3'O Clock position
to the right, etc. is also very useful for describing the orientation of
the Hilal.
7. CONCLUSION
*************
The above information would make self evaluation of Hilal Sighting easy
and shapes in the sky that cannot represent a Hilal would automatically
be eliminated by us, thus avoiding "honest mistakes", inshallah.
The Hints on observing Hilal, given in my Homepage may also be of use.
http://Hilal-Sighting.Com
Some references are:
References: 1. Textbook in Spherical Astronomy, W.M.Smart,
Cambridge University Press, 1977
2. The Dictionary of Astronomy, Ed.V.Illingworth
Facts on File Inc., 1994