Accuracy and limitations of calculations
by Dr. Monzur Ahmed

Bismillaahir Rahmaanir Raheem

The moon orbits around the earth in a predictable way and the earth orbits the sun in a predictable way. Therefore there are certain aspects of the moon (and sun) that can be calculated with a very high degree of accuracy. These aspects include:

1. The date/time of conjunction (astronomical new moon).

2. The altitude and azimuth of the moon and sun from any place on earth at a particular time.

3. The moonrise, moonset, sunrise and sunset times at a particular location.

4. The elongation of the moon.

5. The phase, age and width of the crescent. etc. etc.

Any observatory in the world should be able to supply the above data. Many computer programs such as MoonCalc (http://www.ummah.net/ildl/mooncalc.html) can also provide the data (and more). The above information can be calculated with a level of uncertainty which is negligible. For example, conjunction time can be calculated to an accuracy of better than one second of time. Altitude and azimuth may be calculated to an accuracy of 1/100th of a degree or better. [As a practical, "observable" example, consider solar eclipses which by definition occur at times of conjunction. Using calculations, it is possible to predict that the next total solar eclipse is on 11th Aug 1999, 11:04 TD; it is also possible to calculate the path of totality on a world map. The eclipse will appear total from South England and the path will cross through Europe towards Pakistan (covering Karachi) and India. NB NEVER LOOK DIRECTLY AT THE SUN DURING A SOLAR ECLIPSE - RISK OF RETINAL BURN AND BLINDNESS].

Just knowing the above list of crescent information (which can always be calculated very accurately) can be useful in determining if a crescent is visible. For example...

- A new crescent (hilal) cannot be seen before conjunction because it does not exist. In fact, many hours have to elapse after conjunction before the hilal is visible from anywhere in the world. Hence any claims of hilal sightings before conjunction must be wrong.

- A new crescent cannot be seen after the moon has set because it is not above the horizon. Hence any reports of sightings after moonset must be rejected also.

- Although it is possible to see a several days old crescent before sunset, a thin new crescent can only be seen *after* sunset. This is because the sky is too bright before sunset and for a short while after sunset due to solar glare. Hence any sightings of young crescents before sunset must be rejected.

Therefore it is possible to reject some claims off sighting as "obvious honest mistakes". However, it is sometimes not possible to say with absolute certainty whether a crescent will be visible from a certain location on a particular evening. The crescent may be above the horizon at sunset but this fact alone is not enough to guarantee visibility. Lunar visibility depend on many factors some of which are not easy to predict.

Predictable factors:

Altitude and azimuth of moon relative to sun
Elongation of moon (angular separation from sun)
Crescent width
Age of moon etc

Unpredictable factors:

Weather and atmospheric conditions
Quality of eyesight of observers
Experience of observer etc

Since ancient times, astronomers and civilisations have attempted to predict crescent visibility using mathematical models of varying levels of sophistication. Fore example, the ancient Babylonians had a simple moonset lag 48mins rule. The Muslims also did much work in this area, eg:

767-778 ..... Yaqub Ibn Tariq

740-840 ..... Habash

-830 ........... Al-Khwarizmi

850-929 ..... Al-Battani

................... Al-Farghani

826-901 ...... Thabet b. Qurra

-986 ........... Al-Sufi

973-1048 ..... Al-Biruni

...................... Ibn Sina

1135-1204 .... Maimonides (actually a Jew, aka Moses Ibn Momen)

1258-1274 .... Nasir Al-Din

....................... Al-Tusi

After the 13th century CE there was little new work (coinciding with the sack of Baghdad) until 1910 when Fotheringham published his famous paper. Since then there has been much work by both Muslims and non-Muslims in the area of predicting crescent visibility. For example, (one of) Ilyas' criterion depends on altitude of the moon (alt) at sunset and the difference in azimuth (daz) between the sun and moon at sunset. A graph is drawn of alt versus daz - if the moon's parameters are above certain threshold values then the crescent should be visible pending favourable weather conditiond and vice versa if the parameters are below the threshold values.

Modern visibility criteria can correctly predict crescent visibility in about 85% of cases. Most of the mistakes will be in cases of "borderline" crescents. However, despite this level of uncertainty, remember that using calculations we can still reject certain sightings with absolute confidence (eg "sightings" before conjunction, after moonset and before sunset).

Wasalaam. -- Monzur Ahmed
http://www.ummah.net/astronomy