Visibility of Luna and Lunar Crescent

1582 (rdday) ii the Crcgorian alendd. Ther.by aU dare conversioo algoilbms have to
kecp accoul of thh skipping of days in r]te solar calendar. Difercnl counrries, culluics
and Fligiou omhMiiics adaprd to rhis DodilicElior ar diff@.1 !inc. Ir is lheEtorc a
high hrk for hisrorifls ro tccp racl of r]E apprepriare dalcs.
,n every solr qlendr thft @ iwctle mo.lbs. For simple dithmcric Ea$ns
theE colld hdve been seven months of 30 dars and fi ve of I I days (in a nomal ye.r and
six ofl0 days plw 3ix or 3l days in a lcap re&). How;ver in pncrice sincc rh€ ime of
CEek it w6s knoM lhar d€ wi.r6 h.lfofa nodat ,€u h6 l8l days whee6 rhc
summ.rhalf@ ains I84 days. The it&n for ih. sue is lie faler molio. of thc Eznh
rhc Frihelion. thc Eanh clossr ro $; Su dar oeun on aound January 4,i. Srdnins
with January cvery ancinale nonlh is of3t days tiu Juty. rcbru.ry h of28 days (o.29
days for a le.p yctu) sd lhe Br G ofto days.&h. The.ltdar€ nonrLs ofjl days
dd l0 dals omintB $.d frcm AuSul ro DMnber
ln a Strictly Lunar cat€ndd, lhar is ba$d on $e luharion pcriod in one year lhcle
arc eilher 354 days or 155 days. Tbe clrcnr ave!.8. of$e lunation period h 29.510589
days or 29 days, 12 hou6. 44 ninui4 dd 2.9 $con.l$(A$nrcmical Atnaiac. 2007).
HoEEr ihis aEase is.hrging, Aaodin8 ro rh. tun5. lltery of Chaprcnl-Tou&, dd
chapont lh.e vdiadons !E eoukd for by the tolowing cxpEssion (Chdpon!
'lbur'ind Chapro , 1988):
20.5305888511 + 0.o0oo0o2t62l, a_ I.o4! IO.
o ! a. Da)5 ( r.4 r)
whce T n givcn b, ( L4.2). An, rrEdicutu phe cycl. my v.ry fmn te nes hv uD to
seven hom. Tnus rhis priod vtuj.s froh !romd 29.2 days ro mor lnan 29.3 days fom
nontn lo nonlh. Tnerefore in aU Iutu calend&s rhe numbe! ofdays in a nomh is eithcr
29 or 10. l. Arithmetic Lunar calcndq rheE $ lllenate hontns of29 and 30 davs. tn
obiwrional lue Rt.ndd $.8 6 bc s mey 4 rhE. rcns{ulirc mon$s o, 2,
days qch od d mdy s folr conKud@ momh5 of 30 days e.ch { oys, I 994). In an
l5