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An Islamic Calendar for Makkah - Page 3

IX. Calculation of the Q factor or the “Ease of visibility” of the early crescent

We have already remarked that it is no easy feat to observe the early crescent with the naked eye. However, if astronomy tells us that the crescent is going to be visible and gives us its ease of visibility, then we can redouble our efforts of sighting and even try to photograph the crescent, atmospheric conditions permitting. The following paragraphs explore this “ease of visibility”.

IX.1 The notion of “Best time of visibility” or Tb

One might think that the best time for observing the new crescent moon is just before it sets in the night sky. This is however not so. If the crescent moon is observed too early after sunset, the sky might be still too bright to obtain visibility of the faint object. If we wait too much, then the intrinsic visibility of the crescent will diminish, and, again, it will not be visible. In his paper, Yallop (B. D. Yallop,A Method for Predicting the First Sighting of the New Crescent Moon, HM Nautical Almanac Office, NAO Technical Note N° 69, June 1997, Updated April 1998) uses earlier results to give an empirical formula for the “Best Time” of visibility. According to him:

    Tb = Ts + 4/9 * Lag

In this equation Ts is the time of sunset and Lag is the time difference between sunset and moonset (in minutes, for example). We will use the example of observing the new moon of November 2009 in Makkah and at an Intermediate Horizon IH. The new moon is born on the 16th at 19 H 15 M (universal time). This is too late to give any visibility on the 16th in Makkah. On the next day, 17th November, there is still no visibility in Makkah, but the Intermediate Horizon at 30° W 30° S is green. As an example, we will calculate the Q factor for 17th November 2009, both for Makkah and for IH as above.

Best time of observation in Makkah (17/11/2009 ; 21°25’0” N 39°49’0” E, seconds neglected:
Ts = 14:38
Moonset = 15:01
Lag = 0:23
Tb (Makkah) = 14:38 + 4/9(23 minutes) = 14:48

Best time of observation at IH 30°W 30° S (17/11/2009, Using Sunset Moonset computing in MICA)
Ts = 20:36
Moonset = 21:39
Lag = 01:03
Tb (IH) = 20:36 + 4/9(63 minutes) = 21:04

IX.2 Coordinate systems

Angles will be measured either in geocentric, topocentric or celestial coordinates.Geocentric means from the centre of the earth. A position measured in geocentric coordinates does not depend on latitude of longitude since observations are from a fixed point.

Topocentric means from each local horizon. Topocentric coordinates from Makkah will use the horizon at Makkah.

Celestial coordinates are used to fix position of heavenly bodies on the celestial sphere. The celestial sphere is the huge sphere which seems to surround us day and night. It is an indefinite projection of the spherical earth. The projection of the earth’s equator on the celestial sphere is the celestial equator. The apparent path described by the sun on the celestial sphere is known as the ecliptic. Since the axis of the earth is tilted by 23.5° with respect to the plane of its orbit, the planes of the ecliptic and of the celestial equator are also tilted at the same angle. They intersect in two points called the vernal and the autumnal equinox.

Celestial coordinates are like longitude and latitude on earth. Celestial longitude is also called Right Ascension. It is measured along the celestial equator starting from the vernal equinox and counted in degrees or, more traditionally, in hours, minutes and seconds (24 hours = 360 degrees). Celestial latitude is also called Declination. It is always measured in degrees, positive if the celestial body is north of the celestial equator, negative if it is south of the celestial equator.

Finally, azimuth is the clockwise angle starting from the North until the direction of the heavenly body is reached.

IX.3 Four definitions

1X.3.1 Arc of Sight or ARCS

It is the angular distance in degrees between the Moon’s centre and the horizon at the time of local sunset. It is equal to the topocentric altitude of the Moon at local sunset. The angle is to be measured at the “best time”.

(Horizontal line = local horizon)

1X.3.2 Arc of Sight or ARCS

ARCL is the angle subtended at the centre of the Earth by the centre of the Sun and the centre of the Moon. ARCL allows the calculation of the width of the lunar crescent (W or WOC) according to a formula given later. ARCL is frequently called the lunar elongation.

[The following two diagrams have been taken from the article of Ilias M. Fernini, Yallop’s criterion as a test for the earliest crescent visibility, College of Science, Department of physics, U.A.E. University, Al-Ain, P.O. Box, 17550 U.A.E. Date of publication not available.]

1X.3.3 Delta azimuth

Delta Azimuth or DAZ is the difference in azimuth between the Sun and the Moon at a given latitude and longitude, the difference is in the sense azimuth of the Sun minus azimuth of the Moon.

1X.3.4 The Arc of Vision or ARCV

Delta Azimuth or DAZ is the difference in azimuth between the Sun and the Moon at a given latitude and longitude, the difference is in the sense azimuth of the Sun minus azimuth of the Moon.

ARCV is the geocentric difference in altitude between the centre of the Sun and the centre of the Moon for a given latitude and longitude, ignoring the effects of refraction.

Angles ARCL, ARCV and DAZ satisfy the equation
Cos ARCL = Cos ARCV * Cos DAZ
so only two of the angles are independent variables. ARCL and ARCV are not directly observable and have to be computed from the celestial longitudes and latitudes of the Sun and the Moon.