OFFSET
1,1
COMMENTS
The Islamic calendar is purely lunar. It starts on Friday 0001-Mulharram-1 AH (Anno Hegirae) = AD 662-Jul-16 (Julian calendar) = AD 622-Jul-19 (Gregorian proleptic) = JDN 1948440. Every 12 months is a lunar year containing either 354 days (regular) or 355 days (leap year). Odd-numbered months are 30 days, even-numbered months are 29 days, except month 12 is 30 days in leap years. Each 30-year cycle contains 19 regular years and 11 leap years. Thus, 1 cycle = 30 lunar years = 360 lunar months = 10631 days, and a(n+30*k) = a(n) + k*10631, for all k. Since 10631 is not a multiple of 7, the calendar repeats after 7 cycles = 210 lunar years.
In various locations, the Islamic new moon is chosen to be dated by either (a) the first sighting of the lunar crescent, (b) astronomical new moon tables, or (c) tabular methods. Only the tabular methods are described here. At least five methods exist, differing only in the distribution of leap years. a(n) are calculated here using the most common method (Fazari or West Islam), in which the leap years within each 30-year cycle (first year of cycle is 1, not 0) are years {2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29} = {floor((30*k-1-c) / 11), k = 1..11, c = 3}. Three other tabular methods correspond to other values of c, namely, c = 4 (Kushyar or East Islam), c = 0 (Ismaili), c = -2 (Habash). In a fifth method (Fattuh), the leap years are not spaced evenly enough to fit this algorithm.
In a minority of locations, an epoch date of Thursday AD 662-Jul-15 is used; this subtracts one day from each of the five calculation methods.
The chronological Julian day number (JDN) is the number of days since 4713-Jan-1 BC (Julian proleptic calendar), e.g., 2000-Jan-1 (Gregorian) = JDN 2451545. As used by historians, chronologers and calendarists, it is integer and does not incorporate time or location. The astronomical JDN incorporates both time and location: it equals the chronological JDN at UT (Greenwich) noon, and includes time as a decimal fraction of a day, e.g., JDN 2451545.50 = 2000-Jan-1 24:00 UT.
As of AD 2000, the astronomical synodic month averages 29.530588865 days; the Islamic month averages 10631/360 = 29.5305555555 days, and falls behind the synodic moon by 0.04120 days per century. The astronomical tropical year averages 365.242192 days; the Islamic lunar year averages 12*10631/360 = 354.366666 days, so there are an average of 103.07120 Islamic years per tropical century.
The astronomical new moon of July 622 occurred on July 14 at 05:30 UT = 08:10 Mecca Local Mean Time (MLMT), but the crescent moon was not visible in Mecca until sunset of the next day July 15 (~18:00 MLMT), the start of 0001-Mulharram-1 AH, which is equated with AD 622-Jul-16 (which began 6 hours later at 24:00 MLMT). - Robert B Fowler, Aug 31 2024
REFERENCES
Jean Meeus, Astronomical Algorithms, Willmann-Bell, Richmond, Virginia. Second edition, 1998, chapter 9, pages 73-76.
Edward M. Reingold and Nachum Dershowitz, Calendrical Calculations, Cambridge University, UK. 1st edition, 1997, chapter 6 and appendix B8. 4th edition, 2018. Chapter 7 and Appendix D7.
Edward Graham Richards, Mapping Time, Oxford University, London, 1998. Chapter 15, pages 231-235, 311, 323-324.
Paul Kenneth Seidelmann and Leroy Elsworth Doggett, Explanatory Supplement to the Astronomical Almanac, Mill Valley, 1992. Pages 589-591.
LINKS
Robert Harry van Gent, Islamic Calendar Converter, Mathematical Institute, Utrecht University, December 2019. An excellent source. The "Origin" section clearly describes how the various leap year methods are constructed; the "Islamic calendar converter" section converts Gregorian/Julian dates simultaneously into eight Islamic dates (four leap year methods in two epochs); the "Other online converters" section indicates which of the eight possible Islamic dates are used by ten online calendar converters, most of which do not divulge their method.
Dieter Koch & Alois Treindl, Astrodienst Planetary Positions for AD 622 July (page 272).
E. G. Richards, The identification of days, Figure 22.1 in: Mapping Time: The Calendar and Its History.
P. K. Seidelmann, The Islamic Calendar, Explanatory Supplement 1992.
Wikipedia, Islamic calendar.
Wikipedia, Tabular Islamic calendar.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
FORMULA
a(n) = floor((n*10631+c)/30) + 1948086.
c = 3 is used here; for other calendar methods, see comments section.
The epoch date of July 16 is assumed; for epoch July 15, subtract one from a(n).
Number of days in Islamic year n = a(n+1) - a(n).
Day of week of first day in year n = (a(n) mod 7) = 0 (Monday) to 6 (Sunday).
Julian day number of general Islamic date y,m,d = floor((y*10631+c)/30) + floor(m*59/2) + d + 1948056. Note that this single equation defines the entire Tabular Islamic calendar (for the four tabular methods mentioned in the comments).
EXAMPLE
a(1) = floor((1*10631+3)/30) + 1948086 = 1948440 (JDN).
Year 1 has a(2) - a(1) = 354 days (a regular year).
Year 1 began on weekday (a(1) mod 7) = 4 (Friday).
Year 2 has a(3) - a(2) = 355 days (a leap year).
MATHEMATICA
a[n_] := Floor[(n*10631 + 3)/30 + 1948086];
Array[a, 32] (* Amiram Eldar, Jan 04 2022 *)
IslamicNewYear[n_] := Module[{},
date := DateObject[{n, 1, 1, 12},
CalendarType -> "Islamic",
TimeZone -> "Europe/London"];
jl := CalendarConvert[date, "Julian"];
jd := JulianDate[jl];
MixedFractionParts[jd][[1]]
]; Table[IslamicNewYear[n], {n, 1, 32}] (* Peter Luschny, Feb 13 2022 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Robert B Fowler, Jan 04 2022
STATUS
approved