Hegira Calendar

http://www.numericana.com/answer/calendar.htm#moon 

(2002-12-28)   Hegira Calendar   [AH = Anno Hegirae]
The Islamic calendar is called  Hijri  (or Hijrah calendar).

The origin of the Muslim calendar is "1 Muharram 1 AH"  (i.e., Friday, July 16, 622 CE) and predates by a few weeks the "flight from Mecca" (Hijra, Latin: Hegira) which, according to Muslim tradition, took place in September 622 CE.
The numbering of years from the date of the Hegira was introduced in AD 639  (17 AH) by the second Caliph, 'Umar ibn Al-KHaTTab (592-644).  The monthly Islamic calendar itself had already been in use since  AD 631  (10 AH) as the Quran prescribes a  lunar  calendar  without  embolismic months  (9:36-37).

Before  10 AH, a long forgotten "Arabian calendar" was probably used, which was similar to the Jewish calendar and had an intercalary month, now and then, in order to compensate for the steady drift of the lunar cycle with respect to the solar seasons.

Since an Islamic year (12 lunar months) falls shorts of a tropical year by almost 11 days, the Islamic calendar isn't related to the seasons.  Muslim festivals simply drift backwards and return roughly to the same seasonal point after a period of 33 Islamic years (which is about a week longer than 32 tropical years).
Traditionally, the beginning of a new Islamic month is defined locally from the time when the thin crescent of the young moon actually becomes visible again at dusk, a day or so after the new moon.  If the moon can't be observed for any reason, the new month is said to begin 30 days after the last one did.

Tabular Islamic Calendars :

Printed Islamic calendars are most commonly based on standard  arithmetic  predictions of moon sightings.  We present the most common of 8 extant variants.

Such schemes were devised by Muslim asronomers  after the eighth century CE.  Historians use this routinely to convert an Islamic date to a Gregorian one  (unless a knowledge of the day of the week allows a precise synchronization with the relevant local observational calendar).

A regular cycle of 30 years is used, which includes 19 years of 354 days and 11 years of 355 days (modulo 30, the long years are: 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, and 29).  The average Islamic month is thus equal to 29.53055555... days, which is about 2.9 s shorter than the actual mean synodic lunar month of 29.530588853 days (it takes about 2428 tropical years to build up a discrepancy of a whole day). The standard Islamic year is tabulated below:

Number  Month Name Days 1 MuHarram 30 2 Safar 29 3 Raby' al-awal 30 4 Raby' al-THaany 29 5 Jumaada al-awal 30 6 Jumaada al-THaany 29 7 Rajab 30 8 SHa'baan 29 9 RamaDHaan 30 10 SHawwal 29 11 Thw al-Qi'dah 30 12 Thw al-Hijjah 29 or 30

30 Islamic years (10631  days) 0 10 20 1 11 21 2 12 22 3 13 23 4 14 24 5 15 25 6 16 26 7 17 27 8 18 28 9 19 29

Important Islamic Celebrations  

Date	Arabic	English
10 MuHarram	Ashura	Remembrance of Muharram
12 Raby' al-awal	Maulidun-Nabi	Birth of the Prophet
27 Rajab	Laylatul-Mi'raj	Night of Ascension
15 SHa'baan	Laylatul-Bara'ah (Shabi-Baraat)	Night of Record
1 RamaDHaan	RamaDHaan	Fast of Ramadan (first day)
27 RamaDHaan	Laylatul-Qadr	Night of Power
1 SHawwal	Eid al-Fitar	Breaking of the Fast
10 Thw al-Hijjah	Eid al-Adha	Sacrifice Festival  [3½ days]

Hijri Calendrical Formulas :   (2007-06-22)

The average Islamic year (12 months) is  10631 / 30 = 354.36666... days.  If day 0 (zero) is the first day of the above cycle of 30 Islamic years, then the number of the year to which day N belongs equals  floor ((30N+k)/10631)  provided  k  is between  26  (included)  and  27  (excluded).

Such a choice of  k  ensures that two critical inequalities are satisfied:  For year 16 to be longer than year 15, day 5669 must belong to year 16 and not 15.  This requires  k  to be  at least  16*10631-30*5669 = 26.  On the other hand, for year 26 to be longer than year 27, day 9567 must be in year 26 rather than 27, which implies that  k  must be strictly less than  27*10631-30*9567 = 27.  These two inequalities are sufficient to satisfy the 60 constraints imposed by the entire 30-year pattern.

Using  k = 26,  we obtain a formula  (valid for an  indefinite  number of 30-year Islamic cycles)  which gives the Islamic year Y corresponding to day N :

Y   =   floor ( [ 30 N + k ] / 10631 )

Conversely, the number N_Y corresponding to the first day of Islamic year Y is:

N_Y   =   ceiling ( [ 10631 Y - k ] / 30 )

Subtracting this quantity from the original day number (N) we obtain a number N' from 0 to 354 within the Islamic year.  From this number N', the Islamic month is not difficult to obtain.  Conversely, we may also get the number of the first day of the month.  The number within the month is obtained by subtracting that from N'.
All this can be embodied into two computer routines which convert a day number to an Islamic date (hdate) and vice-versa (hday).  For compatibility with the similar routines for the Gregorian and Julian calendars, we count days from the MJDN origin, with an offset of 451915 days.  The following implementations are for the TI-92, TI-89 and  Voyage 200  handheld calculators.  

hday({y2,m2,d2}) - hday({y1,m1,d1}) is the number of days between two Hijri dates.
hdate ( hday ( {yyyy,mm,dd} )) puts a "generalized" Hijri date in standard form.
date ( hday ( {yyyy,mm,dd} )) obtains a Gregorian date from a Hijri date.
hdate ( day ( {yyyy,mm,dd} )) obtains a Hijri date from a Gregorian one.

Competing Variants of the Tabular Islamic Calendar :

The aforementioned date of  Friday  July 16, 622 CE  is by far the most common starting point of the Hegira calendar, but it may also be reckoned from  Thursday  July 15, 622 CE.  This so-called "Thursday" calendar would be obtained with an offset of 451916 days (instead of 451915) in both of the above routines.
There are no fewer than  30  regular  intercalatory patterns which would be based on the same 30-year period (of 10631 days) as above.  Apparently, only four of those have ever been advocated (as tabulated below).
The four extant intercalatory patterns agree that years 2, 5, 13, 21 and 24  (modulo 30)  are "long" years of 355 days, but disagree on some of the remaining 6 long years in the 30-year cycle of 10631 days.  This amounts to different values of  k  for the calendrical formulas introduced above  (with k = 26) :

The  8  Extant Variants of the Tabular Islamic Calendar

Long Years (modulo 30)	kmin	kmax	Friday	Thursday
2, 5, 7, 10, 13, 15, 18, 21, 24, 26, 29	25	< 26	Ic	Ia
2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29	26	< 27	IIc	IIa
2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29	29	< 30	IIIc	IIIa
2, 5, 8, 11, 13, 16, 19, 21, 24, 27, 0	1	< 2	IVc	IVa

To make the above calendrical functions match your favorite variation:

• Change the constant 26 (which appears 3 times) into 25, 26, 29 or 1.
• Use 451915 for a Friday calendar, or 451916 for a Thursday calendar.

The above numbering of the 8 extant variants of the Tabular Islamic Calendar follows the classification given by Robert Harry van Gent, who calls "civil" (c) the tabular calendar based on the usual starting point of  July 16, 622 CE  and "astronomical" (a) the "Thursday" calendar based on a July 15 starting point.

Ic	- Kushyar ibn Labban  (AD 971-1029). - Ulugh Beg  (AD 1393-1449). - Convertisseur de date,  Ministère des Habous et des Affaires Islamiques. - Gregorian-Hijri Dates Converter, by Waleed Muhanna.
Ia	- Microsoft's algorithm  (misleadingly called the Kuwaiti algorithm). - Islam Online's Date Converter. - Al-Islam's Agenda-Date Converter  (Ministry of Islamic Affairs, Saudi Arabia).
IIc	- Gnu Emacs editor  (courtesy of Dershowitz and Reingold). - Calendrica, by Edward M. Reingold and Nachum Dershowitz. - Numericana, by Gérard P. Michon. - Calendar Magic, by Alex Balfour. - Today's Date, by Doug Zongker. - Java Calendar Conversions, by Mark E. Shoulson. - Fourmilab's Calendar Converter, by John Walker. - Conversion of Islamic and Christian dates, by Johannes Thomann. - Muslim Holidays.  Dates of Religious and Civil Holidays Around the World. - Hijri/Gregorian/Julian Converter, by Tarek Maani. - The Islamic Calendar, by Claus Tøndering.
IIIa	- Fatimid Calendar.  Misri calendar.  Bohra calendar. - Date Exchange, by Sualeh Fatehi. - Hijri Calendar  (Dawoodi Bohra Version).
IVa	- Ahmad ibn 'Abdallah Habash al-Hasib al-Marwazi  (d. ca. AD 870). - Abu Arrayhan Muhammad ibn Ahmad al-Biruni  (AD 973-1048). - Elias of Nisibis or Elias bar Senaya,  Patriarch Elias I of Tirhan  (1028-1049).

Many of the above authors do point out that such arithmetic approximations are not a substitute for the actual observational Islamic calendar sanctioned by religious authorities.  Reingold and Dershowitz  (Calendrica)  also provide an "observational" [sic] Islamic calendar, based on more precise astronomical computations to better predict the religious beginning of each Islamic month.

Crescent Moon Visibility and the Islamic Calendar (USNO)   |   Islamic Calendar(s)
The Umm al-Qura Calendar of Saudi Arabia   |   July 16, 622   |   Islamic Calendar