Introduction
The Moon orbits the Earth with average orbital velocity 1.022 Km/sec at average distance of 385,000 kilometers. Due to its elliptical orbit, the distance between moon and earth varies from 356,400 kilometers (at perigee) to 406,700 Kilometers (at apogee). The mean orbital inclination of the lunar orbit is 5.14o to the ecliptic plane. Both the rotation of the moon and its revolution around earth takes 27 days, 7 hours, and 43 minutes. This synchronous rotation is caused by an unsymmetrical distribution of mass in the moon, which has allowed earth's gravity to keep one lunar hemisphere permanently turned towards earth.
A lunar conjunction, the new moon, is the event when moon and the sun have same ecliptic longitude. The average period from one conjunction to the next conjunction is 29.53 days. It is a well-established fact and accurate to the level of minutes. Eclipses and new moon can be calculated for number of years with precision.
The Earth Moon System (Credit: NASA http://visibleearth.nasa.gov/)
The Phases of the Moon
The sunlight always illuminates half of the moon. Due to cycle of the moon, amount of reflected light towards Earth varies with its phases. The astronomical definition of “New Moon” is different from traditional concept. The traditional concept of the new moon is the thin crescent where some amount of light is reflected from the moon’s surface. At new moon no part of light reflected and elongation (the angular separation between the Sun and Moon, with Earth as the reference point) is equal to zero degree.
Phases of the Moon (Credit: www.farmersalmanac.com)
Science of New Crescent Moon Visibility
The sighting of new crescent moon is significant in Islam. The Islamic calendar or Hijri calendar is based on moon’s cycle. The hijri month starts on the day following the observation of a new crescent, soon after a conjunction.
Since the time of Babylonians, the research of visibility new crescent is still going on. The Babylonians criterion for sighting the new crescent moon was based on the age of the Moon and the time difference between sunset and moonset (lag). The Babylonians came to know that the visibility of new crescent with naked eye is possible when the age of the Moon is greater than 24 hours and lag is more than 48 minutes (LJ, Fatoohi, 1999). The Muslims astronomers of medieval era like Yaqub Ibn Tariq, Habash, Al-Khwarzmi, Al-Farghani, Al-Tabari, Al-Battani and Al-Biruni addressed the issue of new crescent visibility and developed their own criteria (M.Ilyas, 1994).
In the beginning of twentieth century, the models developed by J.K Fotheringham (1910), E.K Maunder (1911), A. Danjon (1932) laid the foundation for predicting early visibility of the new crescent. Later, the different visibility criteria were proposed by F. Bruin (1977), B.E Schaefer (1988) and M. Ilyas (1994). Bruin developed visibility curves for early visibility of lunar crescent using parameters like crescent width (illuminated width along the moon’s diameter) and sky brightness at the time of sunset. Schaefer extended the bruin’s work and included the atmospheric parameters in his model. In order to calculate visibility of new crescent moon, following variables should be considered:
- Moon’s Age (Age): The time interval between conjunction and the time of observation.
- Moon’s Lag Time (Lag): The time interval between sunset and moonset or moonrise and sunrise.
- Moon’s Altitude: The angular distance of the Moon above the horizon.
- Arc of Light (ARCL): The angular separation (elongation) between the Sun and the Moon.
- Arc of Vision (ARCV): The angular difference in altitude between the Sun and the Moon.
- Relative Azimuth (DAZ): The angular difference in azimuth between the Sun and the Moon.
- Crescent Width (W): The width of the lit area of the Moon measured along the Moon’s diameter.
- Danjon Limit: It is the limiting value of topocentric elongation at which new crescent cannot be seen. Danjon proposed this value around 7°.
Basic geometric parameters for crescent visibility prediction (Credit: M. Odeh, 2004)
Dr. Barnard Yallop (1997) developed single parameter empirical model based on 295 observations of lunar crescent. By applying q-test to 295 observations carried out from 1859 to 1996, he divided the visibility types and codes into six regions. The model also deduced the best time for the observation of new crescent by adding factor of 4/9 lag to the sunset time. Following are the visibility regions with codes.
A: Easily visible with the naked eye
B: Visible with naked eye under perfect atmospheric conditions
C: May need optical aid to find/locate the thin crescent moon before it can be seen with the naked eye.
D: Can only be seen with binoculars or a telescope
E: Below the normal limit for detection with a telescope
F: Not visible, below the Danjon limit.
Later, M. Odeh (2004) proposed a new criterion for lunar crescent visibility using 737 observations. Half of them were observed by the Islamic Crescent Observation Project (ICOP). This criterion is based on two variables, the topocentric ARCV and the crescent width. His model divides the visibility into four zones.
Zone A: Crescent is visible by naked eyes.
Zone B: Crescent is visible by optical aid, and it could be seen by naked eyes.
Zone C: Crescent is visible by optical aid only.
Zone D: Crescent is not visible even by optical aid and sets the Danjon limit 6.4o.
Visibility Criterion Used for 5-Year Lunar Calendar for Pakistan
Based on empirical models of Yallop and Odeh, the following criterion has been used for the prediction of visibility new crescent moon at the time of sunset.
- Moon Altitude ≥ 6.5°
- W (Crescent Width) ≥ 0.17 arcmin
- Illumination ≥ 0.8% or Elongation ≥ 9°
- Lag ≥ 38 minutes
On the basis above criterion for the visibility is used for making of 5-year lunar calendar for Pakistan. The new crescent data for Gilgit, Peshawar, Islamabad, Lahore, Muzaffarabad, Quetta, Karachi and Jiwani has been processed. Our minimum criterion lies well within the region C of Yallop and Zone C of Odeh mentioned above. However, the prediction criterion does not take atmospheric conditions into account at the observing site because it is a random phenomenon.
The present lunar calendar has been prepared purely on scientific perspective with certain limitations. Most of the new crescents are predicted to be sighted by naked eye under favorable atmospheric conditions. However, few may require optical aid to find/locate under perfect atmospheric conditions. Among eight cities mention above, one location should fulfill the set minimum criterion to declare 1st of lunar month on the next day.
This calendar comprises of information about lunar conjunctions, month-to-month & day-to-day Hijri years in comparison with Gregorian from 2019-24.
References:
1) J. K. Fotheringham, MANRA, 70, 527, 1910.
2) E. W. Maunder, JBAA, 21, 355, 1911.
3) A. Danjon, L’Astronomie, 46, 57, 1932.
4) F. Bruin, Vistas Astron., 21, 331, 1977.
5) B. E.Schaefer, QJRAS, 29, 511, 1988.
6) M. Ilyas, QJRAS, 35, 425, 1994.
7) B. D. Yallop, RGO NAO Tech. Note 69, 1997.
8) L. J. Fatoohi et al, JHA, 1999.
9) M. Odeh, Experimental Astronomy, 18, 39, 2004
10) Islamic Crescent Observation Project (ICOP) Database, www.icop.org
11) United States Naval Observatory (UNSO) Crescent Moon Database, www.usno.navy.mil/USNO
12) Her Majesty’s Nautical Almanac Office, New Crescent Database
13) Accurate times Software
14) MoonCal 6.0 Software