visibility of crescent-1

8/22/2019 VISIBILITY OF CRESCENT-1

1/23

Chapter No. IINTRODUCTION

Slnce rhe anclent ttmes rh. 6preaEn@ ot nw tunar cr6cent marked lhebeginningofa new monlh. With the dvotopment ot ctv zarions org.ntrtng lme tore(ended pe ods lnto weeks, months snd teaE, the tlnar phase cyctes tead to thoevolutio. ot calendaB. lh6refore the pobtem ot dotermining the day of tiFtsEhtingol nw crescent moon arlbcted hlman betngr tt Invotves con stderatio n ora nuhber of astronomlcat es wett as other facto6. On the orher hand th6 probtemol obserylng the new tunar cres.6nt at an eaniesr posstbte momenl ts chalengingfor bolh amateuE and professtonat .5tonome6. Th6 .st,onomtcat paEmetets onwhlch thesolution olthlsprobtem ts bas6d are b eflydtscussed in the ftsl .iicte ot

The circudstarces and parameteE assciated with th problem of sightingnew runar cresce.t Sreatty vrry wirh th v.rytng post o. or obseder on rhe globe.The atfecls ot the geographicat tocation on the probtem ls briefiy discussed in rhenext 6rtlcle. Atempts to determlne crle a tor the delermintngthe flrst ltslbitity otnew runar c@scenr ai..y ptac rpp.ared as any as the Babytonian ers (Faroohieral, 1999, Bruin, 197?, tty.s, 19946). Signncanr adrances were made by Mustimsand AEbs during medlevgt perlod. A brtl nccount ot thes6 ancient aid medievaleffons b discu$ed ln thtd.dtcte ofthts chapter.

Srnce anliqolty changing phas6 o, the Moon and a comptete cyct or rhesevrriallorc has been ured a3 meansot kptnEaccount ofcatendaG. |yas hasgivena delailed accounr of the history of ihe scince of lunar crescent vlstblity and thelslahic Calenda. (rryas, 1994.), Dogger h6 di$ussed lhe history and rhedeveropn. of calenda6 (ooget,1992), Fetngold &orshowitz hav. presenred a(omparEllvesludy or morethan rwnry catendaG or vsrtous (ypes both sncien, a;d


2/23

moden (Reingold & Dschowttz, 2OO1). A numbor otorhor rutnors navo contrtbur.don rcraid tssle. (atr.3hk, 1993, y.s, 1997, Odeh, 2oo4 etc.). A. ,ghr from rhebeginnlng the c6tenda6 rE 6s$ctated wirh th. cycts ot rh heavens, rhe eme.re Evi*ed atongvilh tno a$ociated catendaB in the nen anbb oi thts chapterwlth speclal emphasts on the tst.mic Lunar catendaiDurlng the modrn rims tt wa3 onty in rhe tasl quaner of the 19rh centurylh.l weslem askorom66 stangd erpto,ing the pbblem of ea l6sr sigh trg o, newlunar 'oscent. the tast antcte is a bnot luruoy of lrorature that delcribed andttweropod vaious c,ire.t. o, models for lhe $tulto. ot probtm ot d.t.rmining th

day or lh fl6t vlslblltty ot tun.r crescont at any ptac on th6 Elobe du nE 2olh

Ll PARAMETERS FOR VISIlIILITY OF NEW LUNARCRf,SCDNT

The Moon. the ody natunt sateuirc oflhe Eanh. is eoiog ound lhe [anh h anorbit thot is a hiShly iftgrlar ellipe Thc i resutariries in lhe pd$ of rhe M@n a duc lollrc facl Et its norion is govemed by not onty tbe gEln ional pu oflhe Eanh. bur isalso aaf.clcd by toan) of rhe ncishholiing cetestial objecN (Danby, 1992). \\,nh rHadh s varying dhr&ce fron lhc Sun, thc e,Iecioa $e Sun,s gnvnodonat pull v,ricssrbstmti ly snh rheElarile posnionsofthe Eanh and the Moon. Moreov.!,lhe afcorsorthe confisur.lion oflheshole Sotrsrslem (losirions ofa| lhe major ptMcrs a.d $cmajor ar.rcid9 on rhc morion oflhe Moon is oor nesligibte, Thus, accoding ro rhc''Eplremirids Lun,irs Prisiemcs,, popularly tnoh d Chapont,s tuM tho,y ELp,2000/82 (Chapronl-Touze & Chapronr. t 983, Cluprcnr-Touzc & Chaprcnt, t99l), tor lheb.sl possible plecision iD te lonSirude, hdnde md lhe disrance (bclsen lhc Mon andthe Eonh), dere aE equired as nany as 15i227 priodic rrms. On lbe othf hand, fordcteminhg 6e position ofrhe Sun b r simile d.8E of acc@cy one rqdB 2,425periodic rems in vicw olrhe ,Variarions S.culairs des Orbnes plandaiet, rhe Fienchpr.netary thory lnoM as VSOP87 (B&raSnon & Fl!Bo!, t988, Meus' 1998). The


3/23

accurat dterni.alio. of lhc posiiiom of the Su dd tl'e Moon is tbe fisl slep towardsexplorins |he circunsr.ee of rlE vGibihy of ihc !a lud .esnt, Handling sdh alaqe hnmber of pdiodi. terms mak6 this fiEt slep very ciucial. The lest of the sludydep.nds on th. Elativc pcitio$ of lhe Ss and thc M@n ed rhe hod&i at md anerthc aunsel for a.y place on the &nh.

These theoris detemio. lhe c.l$i.l ecliptic coodiiates of the sold sy$enobj*|' fid dc b6ed on sphencal polar c@rdinats. The fig Ll.l sho\s the spheiicalpol.r coordiMle sys&n, The origin of lh. syslen O is nher lhe ccntrt of lh E nh otlhat ofthe Sun. The xy'plane k the plane olsliptic, the plane i. shich the E horbi6iound the Su, The x-dis poin6 in the diEtion ol the vcmal Equimr l which is thepoinr of intersection of the Ecliptic (palh of rhe Eanh arcund rhe Sun) and lhd celesrialequaror (rvhos plde coimides wnh rhe pl.ne oa rhc lereslrial equalor). P is $e objslllhc Moon in gcocemric sysem or thc Earth in llelioccn(ric sy(em) $hosc stheicalporq @dimrs tp. 0. 0) e p = lo4. 0 = 4oP 6d e-zzo?.\i\ee P ist\.prcjccrion ollhe position P ollhe objccl on ro the xy-phne. In the celestial eclipticcmdinaB the cetesial loicitudc I = o dd rhe cel.$ial l6(ludeP= 9f - el= zPoP ).whcn fie hcli@enilic eclipric coordinates ofthe E nh aa elaluared usins VSoP ft.yaE $en tdslomed inlo lh gc@nric eliplic coodimles of lhe Sun The corjuncdonor $e Bidh ofNe\ Men occu6 wben }{1 = Is.

Fig. No I I I


4/23

SuDpoe ('-.r",/.)!d (.,,r",t) E thc p6isc dist nc6, etipri. tonSnld...d Oc hnud. of lhc Moon .nd rt Sltr Bperiv.ty, Ff.Gd ro fi. nsr cquircx olrh. d.y of @njucrion. For .ny t@tor on th. Esnh wnh Gcri.l eotdimrcs. I rh.rcGrdal lonsiud. ot rh. d@ .d t is rh. Lcid.l tlrirud., thc ti6t scD i. ded.rcmimtion of dc visibihy ot ft. nd t!@ cE$.ni is ro d.r.diF rh. aclutdyMicrl tin (TD oi rD T., of rlE @njwtion. N.xt, om Eqdc coBidcrine rh.la.!l tima ofslrdns ofrt. Su {d rlE M@n. IfT, lrd Tm (CoordiDr.d Urivcdt Tin.TUC) bc lhe rinB of rh. lcat sunsa ud rhc hooEt, ihcn d. i* G$cnr h.y b.visiblc only ilT, < To, This La4 to lh. p.md.r LAG - Tr - Tr.

Using th. cclipric c@rdiNl6 ot |he sun 6d thc Mon drt.ut.l.d for rh. 1.. o,dy orho noD.ni of tiD., Oc cqq|oi.t @rdin c! of rh. |m bodiB (a.,4)!nd(a..t,) cd bc oh.ircd. Vhcdd. rhc nght aeBion k $c djspt@mcnr of$. objerfbm wdal .qdnox .long lh. cctcsli.t cqu|or (in $. q. qudtut s I ). md d js $cdelin.rion or rhe displ.ccftnl of$. obj.cl fbm dc pt.E or ue.qullor t c.l lourAigl. tf is rh.i oblaircd fon Oc dilT.Ene of thc leEt Sid.c.t Tin. (r_SD and rhishr e.6iotr. This liMuy gies ihc leat horianr.t c@diiat s, dimuft (,r). rhcdispl&.md ol rhe objer fbi rhc diEcdon of lh. Nonh rowrds E 5r ,tong $.hodenlal in l@l sty .nd rlE ald|ud. (r), rh. hciehr or rh. objer lbov hori&o. Aftcr.djustus for $. cf@rioi ed $. h.ishr of rh. obencr's to..rion .bovc s.. trycl rhcropo..nric c@rdiiaks (,4., rr ).nd (,{, .}. ) of lhe M@n and de sun, 6!..rivty aE

lo almosr all rhc nod.ls fo..&lid hoo!-nddng, rh. mj.nt s rcll .s lh.mod.m, th. difieE@ ofeinurlrs (D,,tz = l!r. - /r.l. can.d rlodvc szimurh) .id thrt otrltiluda (ARcv - ,- -/'., c.llcd e of vision) $ rhoM in rhc fig 1t.2, at th. rin. oflc.l su@t T, Th. e of vkion is ale l.m.d s F of d.fft$ion. TIF tig, |.t.2 deshow rhc *pa{ion bctwn ln. Sun lnd rh. M@ that is tsM !s 0E .rc of tilnlabbrtviacd s ARCL .rd is .ls tioM $ .tong$ion.


5/23

Ap&1 frh rh. Elni* &idrh, rlE e ofvision !d tn. e oflighq dE dirc.itlor nnien visibility of luE d.gnr rauircr ro aat it|ro @mid.6iion a @dba ofother pm$rd. On. e.h p@ntq t ihe As. ofthe Moon (AcE) = Tt To d.ffneds th tine elaps.d sine th la.t onjuncli@ rill rh. doc of(rlE sn!.r or tlE any orhdrclevut tioe) obsvltion Arbrhe inponDt lir.ror is dE Widft of Cres.r (w) thtidepIds on rhc digre of$e Moor. As dE din@ of 6. Moon frm rhe 2trt veies6on &@.d 0.14 nillion km to a@nd 0.4 nillid tD ln. Fni{i.rr.d of rhe luwdisc vri6 liom I t e ninutF io | 65 e minur6.

Fig I L2Ihun if rhc M@n ir dGc$ to in anh.r th. rirc of obs{tion rne c|ts@i

would b wid.g lnd th6 bndttrd Winh of tlE qtsfl i! di.6tly prcFrlo@t lo rhcPh.F (P) of th. M@, thll is r tunqio of dE ARCL. r\ @mpla. ltul of all rh.$

22.3

T.T.

t-T-. T.


6/23

BdTimeofvhibilily TrAge oflhe M@netTr ACArc'ofvision ARCVRelative AziDurh DAZArcorlight(Elo!8afion) ARCLPbseofcqc.n PWidlhofcrcs@nl w

A conputarioml stnlegy may be lo stad b, deternining $e tim. of Bidh of lheNew Moon or conjlnctio. or th. Moon *ith the snn. For .ny plee on the slobederemirc rh. dm ot suner |har follos rhe timc of bidn of ncw M@n. For this monenrconpute thc e@entic .cliplic coorditulcs of both lte Sun aod $. Moon. Tlmfomthe* coodinat.s to lhe lGal horizonbl c@rdinates. Tbereby coFPute tnc LAC,lhe Age,ARCL, ARCV. DAZ and the width W. The ddoih of all $6e computations shau bc

5.6,1.E.

10.

SIGNIFICANCE OF GEOGRAPHICAL LOCATION.2Fo! $c pioblem of rhc .dli.st lisibility or the Ncw Cre$enr Moon $e

orientatiob ofthe paths of the Sun and the Moon Eladve lo mch othcr dM tlarile to lhehoizon h significdnl. They chatrgc scason lo seaon as \tll as year lo ,car a.d olsodepeod on thc latitudc of lhe place. Due 1o fte axial rctation of lhc lanh cvcrv object inlhe sly appsr 10 hvel along a cidular palh (The Diuftal Patb) ext nding frcm ealcmhorimn ro thc $'eslm borircn. Thn parh lics in a plae pellcl lo th Plane of thcequalo. Tlc obj*ts in ou sty sbos. delimtion is coNLnl (sld and other exd_$larsysreh objccq slw.ys remd. in a 6xed smallcircle in our 3ky with Norlh Ccleslial Polcbeing ren Pole i.e. rheit diurnal palhs cE no( only 6xed bul lic in plan.s parallel to eacholhr. Thc Plan. of the orbn of lhc ednh eund lhe Sun h inclined to lhe pla.e of lheEqualor .l s .ngl of eosd 21tr 26' 26" lhcefor tne deliMiion ot lhe Sun in ou tkvvdi.s frch 230 26' 26" $nth ro 230 26' 26 nonh oYr a y6. Thc declimtion ofrhcMoon vdi.s frem aoDd 280 S5 south to 280 35' ;onn duting. llMlion penod Thc


7/23

durtng ! dry rdlt.' dF pdh of dt Su B itt pdh of dp rroor @ bc 6Bid6.d rsnrll circl. tl[l h.rc th.i poLs tl rlE Nodt cdcdi.l Pol6 i.. rlE dil4l plrlr3 ofrhcSu .d th. MM do nd lie ii p||G F .lld (o tlE pLG of dE di!.nrt prtht ot d6Fo. ple3 on E!n[ wirh Lrtu L. 8ral6 (t.1 660 ]a' (Mln q ort) rlE! e d.yrdunls .sy rd, *ha Son tqn iB b.low th. hdid dl dry (d ltoe lh. horian dlda!). Sinilllr forplrc6wirh ld' h.gldrlbo6f 25'drhdorl! dE .I!d.ysdun.8 .tqy luu rEtl etE d! Moon n bdor rL hdizoi dl &y (or.borc tt&

Ap.n fion th. plr6 chc to dr Gqld bd! tt Su!d thc Mmmy16lv6y clole to tE ho.izon. Fo. pb. d$. lo tlE audor ilE Cclcs'rl Equnor p6ssdoe lo dE anh !d tlElfd! ti. pdnr of ft S{n.!d dE Moon |!mi. hiSh in $cdq. Bu in ph6 wnh higls hd L. ih. Cd.lrid Equdd i! clos lo th. Hdid oth.l th. p.tlB of tlE S!..!d tlE M6 my b. d@ d cq bclow th. horizon. Th.68|le @. 1.2.1 CFs lh. driw dirrdd of dE cliDlic.td cd.ri.l .quror ia6np.rid lo tlE hdid for. lLe rin hrgl |tin|d. d dr dm of hol o&r !r!d!d rsd .quimx (6liFic in ttl.) .n rorld aodd .qlircx (6liFi. in pinl) .

N\l.rsllldib 1l'

Figur.No.1.2I C.l.eid SdE fo. d obffi in hi8lj blnud.


8/23

Thi! figue clorly 3howr th.! for hid ldiod.! vhcr d'c dccli.dion oflhc Sun keulh (wirt r in th. mnltfr h.hirph6c bctBa Sumd $lnie rld th AuruhtulEquinox or ba96 &tunn l .quiu !d thc wint6 5lni@) tho p.th of rh. Su and@nt qu.dly rhd of ih. M@n |!Dir vrry okr!. ro th. hori4n. Ir rn F @nditi@3 th.w M@. .8q @nju.rion .nlE tlDitr h.low hqid or Ery cloF ro tlE ho.izonmatina it inposibl. to s da .nq lF d ilre d.y! nm djuncrio. Th. iu.rionb.con6 @e ifdldng lhb p.n ofdE yd th..L.li.dioo oftlE M@. is oti of0a&.linrlion of th. Su4 p|nia,brly clos to AduFnd Equircx (@nd S.prcmbd.ndtuob6} B.rsq vqol .quitux !d dF SUDE Solricc (t@d ,uc) th.dlipric isrehtircly high .trd th. p.O3 oflh. S{n .d th. Md e higls 10. Dtina it asid tos rel.dv.ly tqDgq cGsnB Th. enl.rid n aG!.d gedly for th $urn@

Hdnd \r'FiglE No. | 2.2: Cclcadd Sph@ &|r D obr.ffi in low lrlind.

Sihihrly $. fg@ m. 1.2.2 dbE th. ddiE dndllod ofdc ..liptic !d@lqlirl equ|tioi ii oonpcilor to thc hodzon for i pLc. with low hritude ai {@rdwrul e{ui@r (eliplic in ligln blw) .rd |@rd auMMl .qui@x (alipiic in liglnpirl). ftir fise d$ 3ho* thd clo* io @tunr.l .quiDx t|| PrlB ofrh. Sun ed rh.M@n r. rchrivcly cloF ro th. hodan.rd if !h. D.clintti@ ofthc M@. is slh ofrnll ofthc Sun tn. drdhid & nd !6y lpod fd liSltilg of ! tl.riv.ly y@ng d*.nl.


9/23

Tbh is lhe hain rcason bcbind @nsidedng Age of Moon s not a vcry good indicabr fordt visibilily of cF56t. In sprinS.nd Mmr vcry t$utrg.ge cltg cd h. sn lnddudng th. {ullmn and *inle6 very old clescent may .scape sighling in th holthernhemisphed. Thc situalion is rcveB.d s6odlly for de sud@ hmisphec,

whcn lhe M@n is wsl of fte sun (el6rial loncnudc or rh. M@n is noc rh&$ar ofdre Sun) i. our sky it is Old Moon carching up wilh rhe Sun. The Old Cre$enr canb. en in lhe mominss befoE lhc suriF. An r $e binh of Ncw Mmn (c.lesliallongilude of rhe Moon b*omes jusl eratd lhe rbat of lhe sun) rhe Mooi Eehes e6roltheSun6dcan no$ b. *njustaftssun*t. How.vn dcpending on ih. d{limrioBolrhe Sun and the Moon and lhe localion of$e observer $e new cBcent My et wellbcfoE thc sunst in which cG it is iDFssible lo * th ncr crsceft Th8 the fisl andrhc Do$ nnporLnr clilerion aor lhe visihility of the .ev.$ccm Moon is ftar the Moon*$ anr $c sunscl. Alknately $c cnbrio. for thc visihilily of thc l5l cd$ent is dDtrhc Moon riss before lhe suni*.

Whcn thc cresent is very close ro th sun il rcmains invisiblc dE to the facl tbar$c ohospherc close to lhe sun remains highlt illnminated evcn afi( $e sunser'Thqefoe fi.rc n6 to be a frininum lhrcshold epaEtion or Elongarion between fic Sln,nd rhe Moon b.loN vhich ihe crecent cannol be sen. Thh minimud o. lhrsholdeloneation varies hotuh to oond 6 fie djsrance bctween lhc Earlh and $e Moon keepsvalyine. Vhcnclossl (at apoee) $e Moon h tround 350000 kn lrom thc Eanh ind !llhe fdhat (perig) il k aound 400000 lo. Thus a Ne\ Mon ar apogec mat be *cnwhennscbnsation fod rhc Suh is snallrnd a New Moon at periSce may nol b s.cn arnuch ld8cr elongaton. Thi. is du. ro the fst that closer is thc Mmn the larsr ddbrigbler ir appead in oui sky ahd wheo il k farrher il appeaB saller and less biShl atrhc se elongalion. Thts .nother inpona crir.rion is lhar $e New Mmn cm be *enfor a combinorion ofcenain optinlm valu.s ofelong ion and lhe distanccof$e Moonfom fi. E.nh. thi! conbiMlion Esults into tlE op(nm valu of thc widlh orcFvnlA cFsenr wi$ small longation $al appea6 lare.t md briSller al apogee may be seen


10/23

and a.rc$cnr snh ldgc clone.lion $ar apDcar snalle. snd laintr at p(riscc nar nol

RULTS OF THUMB: THE ANCIENT & TtII Mf,Dtt:vAL:'rhe edlist rcferehc. for ey rireno. for lhc visibitir! ofncw luntr cresccnt is

annburcd ro rhe BabJ,lonians (Foueringhm, r9r0. Aruin. 1977. Schaete!. lgttsa. yas.I 994. Faloohi er a1. I s99 elc.). Most oflhe cxplom a&ibute ttE tollowinS rutc of $unb

'Thc ,.n lunar .rescent i s.ek \ en its aAQNons4 loas 18 hihnles h.hi4d th. tns.! nnrc thdn 21 ho^ un1 L

h na3 bem pointcd oul ihar th. ,crual Dabytonu (lcnonephislicarcd ar coopEEd ro lhk siopl. tule (F.r@hi l ar 1999)_ ft is eilhs ou, trck otknostcdse oa lhen era or the nissing hjsbncd rccords $ar h6 rcsridd ourcompchos'on of lheir effons. Autcnric Ecods of sighljne of new cEscent s young as5 hours durirs Babylo.id c qisls (I6loohi el ai 1999. Andcrtic & Fihis. 2006) inlit 6turc. Sinilady. in modcm time ftsc.nc lagsjnS o.tyjr nrnucs behind $e su.*lharc bccn *m wi(h aec tcaslhan 20 n6uB. Thc dala ser of46j obsetuations considcarduflnC lhis $ork (b b. di*lssd in ddait in chaFcd j dd 4) inctldcs jt cascs whcn rhccrcscd $nh l, C tcss rhan 48 Dinutes w.s cpond to h.vc bNn *en wilhour any

oplicalaid. rvhcEas.26 ca*s arc tncE in shich lhe age ofMoon ws lcs than 20 ho!6.Rece rc-evatuation o|r(ords ofthe sigbdngs o|mscsrs in Babyton and Nineleh ahosholv rhar cFscenh much younger rhd 20 hou* dd lho* lassrng bchind sunser muclrl.ss rhln 40minures were sen (Andolic & Fimch.2006),these rccenr compurarionat ef|ons for the recorded obseBarions oa lhcBabylonian e6 ctearly indicarc lhar lhe lule of thmb asociared wnh rhn eD is an ovcrsimplificadon. lr has bcen eccndy ctaihed rh.t Babytonians had rodularcd I hrt,nalhehalicdt tun& lheory which rhey uscd for prcdicring vartuus paramercBotth tunrr

l0


11/23

notion s iounded in thc lue phmdjs ih.y pEpaed (Fat@hi d al, 1999). Ac@rdinglo lheF $udies it is pointed out lhal the noonset-su$et lag atone.ould nor have beenused as lhc visibilily dit.rion by rhe Babylo.ids. BabytoniaN sysren had $c tollowi.S

rlohedrior rL, - moon*r tasrme rin deCrees,,S) - conninrIn vdious $ud,cs lhe vrtue of$e conqant rs dedlce

&ound 2l d.8R Ao ahd koans.t,ulet td|tin >.t2 nin cr_

lr has been indicared lhar Mrslins. reatization thar rhe Eddh-Moon dhlancc laries duringone comptet cyctc ot tlhar ph.ses and lhe hininuh moonsr lag lime considercd b,^Ebs vaiied fiom 42 ninures for Moon ar perigee.nd 48 ninurcs for Moon at aposee

Thonsh lhis -nrle of thunt. is no4 sophisricated 6 compNd ro $e oneart'buled ro |hc Babylonims sli jr does nor povidc rhe complere prcrup of rhe cffons olArabs ad Mulin' h $c hedirwt ines exltrir ur ofAolemaic sy{en ed rhespherical risonodeq devetop.d b, Ambs lead to the L!n& Rrp.es flnclion (batdepends on locatbritud. ofrheptace ofobrNadon dd lbe c.testiat to,gilude ofthe srnand lhe Mooh) 6 eartr as lO cenrury AD (Bruin t97?)Babylonie crirerion

to have ben cmpirical inh indeed lery sinplc for praclcar purPoses and is supposed

no significdt chahgc in rhh rill


12/23

Flarively rc.nr rimes, ho$vr,lhc atiesr Hildu tdis litc pa,ch Sidn it ( D 5OO)hints towards thc idponance of$c Vidrh oflhc luM crc$ent (Bruii, 1977) Thus anelabootc stsreo of calcularioc involvcd in delemining the tinc of licsr visibilittappeds to have developed only abund AD 500. Morcexplicirnentiohsofrhcsederaitedcalcularions are loud ar vuios plaes in the dly Islmic litenle (Brui., 197?).

Ohe ol thc ealiesl Muslin Aircnomcr eho developed lnc lables fo. se lidnCthe luna!crescenl s lisibilitywas Yaqub lbn Tdiq (Ke.nedy. 1968).lt has bcen eponcdin $e litcraluE (Atuin, 1977) that lbn Taliq had r.cogniad the imponance oflhc Width(w) oflhe crescenr. Thh not only shows rhat ar rhd rine rhe varying dislance ot rhcMoon had been rcaliud bul it aho mdc ir possible 10 inpov upon dc ncs cr*cnlvnibilily cinerion. Bruin hd Eponed th Al-Ailuni had $e Falizario. oflhe tong inddillicult calcularions inlolved in lhe dercnnimlion tor the nev lun,r aesccm vnibitirylnd in hG Crloro.,os " reconrmcnded rhc work ofMuhanm.d lbn crbn At B,llanil ltnlhook oJAtto"o,'t t?rstard to Lario by Nalino, l9|]3).

'lhe sihplc *nclioh for cadiesr vhibiliry oflunarcrescchr evotved sjnce lhe riftesof B6btlonids qas passd onb rnc Muslims drcueh Hindus wirh !e,)-' tinleinprclemenr. Molilated by the euEnic iijuhctions a.d rhe salnlgs of rn prollrrMuhamnad (PBUH) rh. Fobtem ofearliest sighring of hnar cE*enl \s lhomnehl).Invesriglredby rlrc earlJ Mrslih aslronomcGotsLhto l0'tccnluryAt).

OD the bash ofreatiarion of the idjponance ofwid$ various Arab asllonomcrsconclndcd td lor celics( vjsibilil, ofcrc$.nl thc minimun equabrial $pararion offieSun lnd the Moon varics from t00 whcn lhe crescent is widesr up to l2i whcn llrcNcc.l is n@$cst Such der,il.d calculariotr \.rc wor\cd oul.s earty as 9ri cenrur)AD by Muslinr asrrononers as$ciated wirh rhc Ab&sid coun of Al,Mahun. Fronlmonesr the* {stbnoncf Al-BaIsli kncw lh,r ihc cnleria that.gc of n@, shoutd bmoe tnan 24 hou6 (or arc sepdarion belveeo $e Sun sd lhe Mooh) isa good sldtinepoinr but i A only m appnrnarion. H. bt,.vcd LlEr lhc dcienl6hnomc6 d,d norunderstohd the phnonenon conpteretr, According to Bruih, AtDatani,s compuradonalsork is o very elabo6re sysrm of mrheharit catculalions (Bruin, | 977).

t2


13/23

'Ihis work is nor intcndcd lo explor lhc hisbry oa Aslonon! rulatn lo thccanie$ sigltingoflhe new lunarcresccnt. Tbe Babylonia. and the nlcdielalcilons shallnol bc dplocd frem a hi$orical FBpecrivc, wc shall rcstricl our explodlion onl) onrlR codprison of $cse clTons sith $os ofthe moden ons. llowcvq $c mcdiclalnarh.nalical ide6 shall b. erploEd in morc dcoil in chaprc. :1.

I.,1 CALf,NDARS AND THE CELESTIAL CYCLESDogeel defines calendar 6 '! syslcm ol orgaizine ,ir4 tbr fi. pupos. of

@lonins im ow cxtend.d priodt (Doge( 1992). tl is a schcmc lor keping an.ccounr ol da!i.'wceks . monrh. . 'yc!6 - ceolunes and "nillennia . Thc bsicnotion bchind I calendar isor8anizingr,la.olroairiconrinuous now rhal isindcp(ndcnrol any ehcn. of irs orsaniztion. Ahongst lhe divisions of timc in dayi. w.chs dceme aR dtccrly 4sociared snh $c celestial c_aclcs. the D,"r,at, \h. Ann@l ^d rhc/,rur. A @nplele daily rcvolurion of lhe s\t. de DildEl norion. is Efertsd b 4 a day.lechnicallt an apparcnr r,/df.t,' h the inlcpal bcrseen rwo successivc rransiN oflhcSun ar any pllcc. Du. lo thc morion of rhc Eanh mund $e Sun rhe sky appc.m rorclolvc round tbe Eanh very slowly (lss rban a dcste pn da!) and one conttercEvolurion of sky i. rhis way is Ffmd lo 6 o ,.ar. I {hnilr! a t opictt y.ar k trctime inleNal bctweh lwo successivc pseges oflh. Su rluough vcdat eqlinox. one oar\e porn6 ol inFMcrion ol rhc celB(dtequarorand rhe actiptic

Each calendar is dilided i.to yea6. yeas inb nonths. non$s inlo vNks eddals Most oflh know calmdm rha( mcn dcviFd bd sevcn da$ in a rvak. tndiflaent Esions md cras 4 to lodot weks halc aho bccn considercd. ltowclcr rhcnuobcr of days in a dond hd EDained variabl. in diltcc.i calcndars md wi$in apaniculd calcndar. Tlr scbeme. il$ce isany. ofdift!nl DUmb$ofdays in a monlh ofa calendar is ba$d or rnc typ. ol cal4dar, Most of lhe knom atcndars.E clarsificdiflo dtree major typcs. Solar Calcndm. Sriclly L@r C.tcnda6 and Luni-SolarCalenda6. A blief description is Evi.$d in rhe foltowinel

1l


14/23

Solsr Cal.nd6 m b4d on the ulrl motion of lhe F-!nh @und the Su..''ycai in such calendds h ihc "Trcpial Yee, defined abolc. Thon the lcng$ ota'tropical y&" (Dogger, 1992) b givcn by:

365.2421396698-0,0000061s359r-7,29xr0_r0Ir +2,64x10'07r (1,4,1)7 s $e tinc in Julian cenruics sincc fic epoch J2000.0 gilcn by:

T-(JD-245rt4t0!t6525 (1.4.2)vherc JD is rhe Julid Dat which is th. dm in .mber ofdavs ctaD*d sine heo n@nat Genwich on Janua.y l. 4712 in Julid Cal6d{. Cwnlly thc loglh of t@pil ycarn 165.2421898 days or 365 days.5 hos.48 ninurcs ed 45.2 sends. As it is nor a whotenumber on yer of a eld calodar @rsists of cirher_ 365 days ot t66. tn old Jutioncrlendr cvery fourih yc& @.lained 166 days (i.e_ a teap yed) a otbq yff conhincd165 days In lnc cuftntly us.d Cregoiian calendr! the leap ycd rutc is nodificd. Acentury ycd y 0ike 1700, 1800 elc) whict is a lop yer i. Juli& calmdtr bur is nolcomplerely divisible by 400 is nor a tep ycar in OEsorian mtcndar. Thus in crer,400yeas theE ae I 00 l@p y6 in Jnlitu Calrdff wiced in cESori& calqda, rb@ areonly 97. The setu od nfuy o1her natMl pbsomcna toltov lnis sotar cyclc. sp.cialtyhaBcsing rides, lhe length ofdays,lbc dmes of imir, suser, sunds. etc. I$e JLrtiancalendar B insrilured on Jdu.ry l. 45 Bc by Juli$ caM. wirh rhc hlp ofAlciandlie strononr Sosigm6 &d w r nodificadon ofdc Roman Republidn andthe ancicnr Egyptian calendaB (Michels, 196?). WtEcd lhe crcso.ian catendar wasn(csibrcd du. to rhc facl ln.r dqinS oe od . lDlf nil.mia lhe Julim otcndar w6displaed from the ssonal vsiatioos by s much s l0 days. Thus pope Crcsory X tcorclnukd a conmission in l6rt qrury AD for lh. cslndar Efoms. The dain anrhor dadE nN sysrem w6 slononer Aloysiu Litju of Naplcs (Coync el ,t_ 1981. Dull41988, Moyer. 1982 ed Micb.k 196?). When it ws implcn.nted ofljcialy. rhe ddreOctober 4, I t82 (Thuday) in tne Jllie cal.ndar 6 fouosEd by rhc dale Ocrcb$ I 5.


15/23

1582 (rdday) ii the Crcgorian alendd. Ther.by aU dare conversioo algoilbms have tokecp accoul of thh skipping of days in r]te solar calendar. Difercnl counrries, culluicsand Fligiou omhMiiics adaprd to rhis DodilicElior ar [email protected] !inc. Ir is lheEtorc ahigh hrk for hisrorifls ro tccp racl of r]E apprepriare dalcs.

,n every solr qlendr thft @ iwctle mo.lbs. For simple dithmcric Ea$nstheE colld hdve been seven months of 30 dars and fi ve of I I days (in a nomal ye.r andsix ofl0 days plw 3ix or 3l days in a lcap re&). How;ver in pncrice sincc rh ime ofCEek it w6s knoM lhar d wi.r6 h.lfofa nodat ,u h6 l8l days whee6 rhcsumm.rhalf@ ains I84 days. The it&n for ih. sue is lie faler molio. of thc Eznhrhc Frihelion. thc Eanh clossr ro $; Su dar oeun on aound January 4,i. Srdninswith January cvery ancinale nonlh is of3t days tiu Juty. rcbru.ry h of28 days (o.29days for a le.p yctu) sd lhe Br G ofto days.&h. The.ltdar nonrLs ofjl daysdd l0 dals omintB $.d frcm AuSul ro DMnber

ln a Strictly Lunar catndd, lhar is ba$d on $e luharion pcriod in one year lhclearc eilher 354 days or 155 days. Tbe clrcnr ave!.8. of$e lunation period h 29.510589days 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&, ddchapont lh.e vdiadons !E eoukd for by the tolowing cxpEssion (Chdpon!'lbur'ind Chapro , 1988):

whce T n givcn b, ( L4.2). An, rrEdicutu phe cycl. my v.ry fmn te nes hv uD toseven hom. Tnus rhis priod vtuj.s froh !romd 29.2 days ro mor lnan 29.3 days fomnontn lo nonlh. Tnerefore in aU Iutu calend&s rhe numbe! ofdays in a nomh is eithcr29 or 10. l. Arithmetic Lunar calcndq rheE $ lllenate hontns of29 and 30 davs. tnobiwrional 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

20.5305888511 + 0.o0oo0o2t62l, a_ I.o4! IO. o ! a. Da)5 ( r.4 r)

l5


16/23

Tnerc is no fixed nle for leap ycs in r obseNarional tunar calcndar. rhcrccdnnol be anr. HoEver in lbe Aritineric L@d cattutd our ol lO yea6. cyclc t I yc6 aEhap tcds (3Js dayt (ly.s 1994. R.igold & Dcrshowirz 20Ol_ Tsybutsty, t9Z9). l.hcrule is rhal lhe tear nunbf /is a teap yer it

Arnhn.tic calcndar ther. are cilh.r 6 monlhs of 29 dals md 6 nonrhs of tO days. (anotml ycar) or 5 momlB of 29 days ad 7 nonlhs of 30 da's (! lep yd).

( l4 + ll.))modi0)< lt (t.4A)Olhwisc the year is nor a teap year. In such an a.ihmelic lunar catcndor alt rhe odd.umbeEd mon$s conlain 30 days and lhe.vd nhb.ed nonibs conhin 29 days cach.h re lcip yedr a day is added ro $e Meltih month. in gmeFl. a srrictl, tune oalendarldvanc$ by I I days againj lhe solar cat.ndd. ThrefoF rhe $asons aod a orhcrpncnomna rhzr dcpcnd on Ihe eld cycle {to ml follow a slmcfiy lu,ar catddsr Ttcmonlb numbq 9 (R nad&) in lhe Isl6mic qtods my f.ll in wi.l{. slmmd. aurmn

Thc Luni-Solar @tfldaB d b.sielly lufu bul ro kep r&k of$e *a.ons inprrce of adding sin8le days in a lqp ye& a whol monfi k addcd (inrqcshlion) lofollowthe $ld! cyclc, The Hebrew dd the Hindn qlendaG fa into thG ctass(Rcideol.t& De6howir2 2001 . Bushwic[ 1989, ScEIt & Ditshir 1896. at_Bituni r 000. at_Btunil0l0).In ce ofHindu calcnds (th.y have bo$ $e $lar tud rhe luni-sotar calcndaB) alunarmonrh is imclcalatcdiwhcneve,irnrsinroEconprresotarmoilh.tncascotrhcHcbFw l!ni,$lr cole.dar d addniomt honrh of l0 da)6 rs mrcEararcd betorc rhcusurl 12" nonlh oflbe yce. The En ofihe dd.its of th6c calodd i, morc of$csoc'al ahd reliSious nlrurc aod is nor in tine with tbeprcsnt*o,k.

Calend.ical cab'narioB for cacn calhdlr ha 6eir oM $phkhadons_ bulbcins phcnoncnoloSicat rtE obsctualionat lunr calend., is mosr chaltensins. A tunnrmonth bas lo beein wib the actuat sighling of lbe new luns cresce d.d tne conditionsI6


17/23

ofits si8hling greatly vary not only longiludinally on rhc globc bd de[Md o. $e tarnudeofplaces. ThB an obedarioMl lunar c.lendar may vsry along the se logirude. Apartiron calenddcal aspects $c pobl.m ofsighting v.ry yolne cGscnt is on of lhe mosrexcitins fld challenging obseFations for both $c adareur md rhe pma$iomlasionofre6 Besides, thc pcdicion of fic vkibilily of a paniculer .erv cEsenr ar apanicular placc is a long and inter.sling cohpuhrional exerci$ Thc sane was clliadas early as rhe dedieul times by the grelr ostonomcs Al-Khwarizni, Al Batani. AtFarghani etc (Bruin 1977) Mocovcr. t|t prediclion foL nakd eye obseivarion ha lwonoE cohplicated hsues under srutiny rh.laresr esearch. Onc oathese is biologicat.lhc abililt ofhunan eye to conrfa$ $e dimly illunrinatcd ciesceht in rbe bnght rsilighl.The olher aspcct h ol physicrl n4lue ofalfiosph.ric conditions ftar c.n b.dtr aoecr rherisibilit! and rc contdsr h this \o.k c l'hdk h morc on thc $lrcnoNical asp..rs oflhe eadiesl siebdrg olthe new luna!.rr$enl dnd ahosphcric ondnion !.e onb plnlr

Opinions aro diided d tq thc oriSin of$c.llton olcomlins yer6 in an! tofrrin rhe ARbian l,cninsula. Accordihg to At ttazsi il $anln as soon as $c cni rcnofPrcphl Adan nuliplicrl and spRod aouid rhc rw d (Ro$nlhal l9i2. l_mqi t979) Acalehdar sE originated wncn rhe Himyarires adoptcd one sjtn m epo.h lhal marked rheb.ginni.B or the eigns ol Tubbn. Cenerally it is blicrcd thal lhc pnctice of 12 lunarnontns lo a year cxisbd in pre,lslahic Ar.b c.lend.rs since rhe rime ofconsrruclion ofKaba b) lhe prophd Abnham a.d conrinued in Islom (llyos. 1994). The namcs ol$cmonths and thcii sequenc Mre thc snc as tho* used in lhc cuiicm tslamic lunarcalcndar Ibllowed by morc rhah on.lifrh of$e roralFopulation ofrhc world.

Fom historic pcspecrire lhc inpon.nce ofrhe lunarcalcnda in lbe pE-lslamicAEbia $s the pilsrimage ro Kadba (Haj) rhar talls in $e monrh orzul-hajjah, rhc l2'"nrnlh oflhe hlmic and prelslamk lunai calendar, Althoush this event was a pueltreligious evenl. il was also inportlDt ior kadc and busincss wnb b$ ol goodsexchanging bands. ll w6 this econonic rcrivil, $ar w6s badly anected as rhe lunarrearad!dced llxoush *&ns. PEcufnent ofcop and rhe ovailabilir) or.ac f.,ul anim...

l1


18/23

8@rly vdied s6r lo s@n, 'fte ten tha1 $e irbEl.tion s6 inl,ldud in deArabim Peninsula w$ this .cooonic drivity rarh.r than dy sronomical t.son.''QalM" a nativ of Mecca is Fpuled to be the fiIsl petson assisned lo derrmine thcdaca for thc coming yem pilgrinage and wheth.i $e inmalarion w6 drc or nol(Hahim, 198?, Ahmed, l99l).

Sincc lhe early days of lh. inception of lund cakndd in the Arabim Pninsulafour nonths including lhe Zil Hljiah wE o6i&r.d scEd md \@ {!e pDhibiteddudnglhesc sacrcd non$s. The custon cari.d over to the post lslamic.ra in lhe lsladiccultu. As with the Ronan ql.ndd thc i eEalalion ws abu*d i. ABbia io ordq lochange rh. saqed nonfis inro noh{acred moorbs dd vicc-vc& Al thc eme rihe rhelunarcalendt!usd inMadi@rendined in itsoriginal lzmonthsayarford.

Muslihs follo{ed rhe calendar ot Mecca in the bceiNling. Rut afier the l,lophcrMuhonmad migdred to Madin. alo.8 wnh his compdions. Mustims adoFcd lh.cal.ndar usd in Madina. Aner $. conqu$tofMeccaby propbel Muhlmmad, Muslimsconlinued 1o ue rhe calcnde of Madim bul rhe catcndar ofMecca @ in pmllcl. Wi$the lar pilgrimag ro Meca of Propher Mubmnad in rhe lo,r ye& aner miSr.rion roModina (AD 612) fequenlly abused pEctice of inicrcalation \m abolished thbush aQuEnic injuiction. Th. pEcrie ofnadin8 a lsd nonrh s.ith $e finl siebdne of nervclesccnt MooD was iurhenticated by Qurtuic injuncrion and tne gayine of l,rcpherMuhmad, sith paniculd empbdis on bSimine dd rhe cnding of lh nonth offasting md the monrh of lhe pilgdmase.

With adoption of puel! lunar cat.nd& wirh a luar nonrh bcginninS $ith $efist siehdns ofrhe ncw IuMr crcscehr lhrcush euMic injuncrion dd lhe eyinSs orPophet Muh.mmad |he .rolution of Isl.mic lunar catedar b.gd As for my calendarone rqutr.s a staning point of rine Gpoch) or bceinnins of m en, tor couring yea6,Oe pople of MadiM ac hli.v.d ro hoK u*d d.poch s ene dn6 ! mooin or tmaftcr Prcphet Muhdnad mignicd to Madina in AD 6t2 (Iys, 1994). However ! horewidly acepGd lime ofomcial adoptjon of Ht6 s lhe bceinning ofrhc Ist@ic cR n

t8


19/23

AD 617 during $c caliph.t of Umd bin Khafiab. Whltcvcr h. thc 1in. of adoplion, lhelslMh cal.ndrr. or dc Hiji c.l.nd& s@ wi$ Fnday l6ri ruly 622 AD on Julitucalends which Moding b didft{c luMr or krdrhicslddd is l" diy ofMuha]m(1" nonth ofe klmic y.ar) ofth. tsleic '@ I (R.in8old & ftrshowii2, 2001). Theo,ficial dar. or adoprion oflhi!.h &d cal.rdd b l'' Muhtu Il AH (and HijB)

Simpl. 34heh.s or 4rolh odd appoxidlions b$cd on long Lm 3vs.gs h.db.n dcvi*d to pcparc long r.d cel.nd& in vid ol incrconv.uion ol lslmic drGs&d tne d!t6 in oth.r lei-$lo ad el& olcndr d.r.s.

blanic legd syst h cgllcd Shdia , is lh. sou@ of lsldic tn+kcping ststn.whil. le8.l pc.pls .rc efeglnd.d by seking sisbne fom $. *i.nlilic kno*l.dge.Till th. rin. of sking of B.ddad by Hrl.su Xhu in AD 1258, Isluic law hadcvolv.d .ldr guidclires ao. elcndriol oNidcFlio.s (llya 1994). The olcndric.lsddelircs ftlv.d undcr thc Islmic lN qn bc outlin d a follows:

i)itiit

l-.nsth of a lunar month is cnhe. 29 days or l0 days.L.nA$ ol a lumr tlar is .ith.r 154 dlts or 155 dayr.Tn rc can be . mdimuh of 4 cor*cutiE monrh. of l0 datr @h or 3conscutive nomhs of29 &ys each.E4h new nonth b.sins with fiBt sightns ot ncw lu@ cGccir ovd thek$em hori@n .n.r ft. l@l ss.i,

v) Ailcmptr rhould bc nad. on 29u of ea.h honrh for sighring of ncw crcscnr.lf il is nol scn on the 294 duc lo any len (stmnohic.l condirions orw.thd connEints) thc noi$ should bc complckd a of l0 days.

l9


20/23

visul sighring epo.t must b. suppond lhrcugh a wnness rcpon.The p.rson involvql in rcponing musl be eliable. adull, lruthful, smevirh good cyesielt !f it i! preved dat the Fen pioviding wihesspuDo*ly nisled the pe6on ousl b. punished.

!iii) Thelisual sighling Epon should nor coniict vilh basic scientific knowledgc.'lesling ofcvidcne ot sighting on scienlific srcunds includc 0hcckine oflhushapc of crcscen! ils inclinalion. Poshion in sky. altitudc. imc ofoberyaliondd sky condnions.Sighting should be caried our in sh o.8ditd way for dch moflh.Accunula(ion of emB @orriculally in view ofconsideling ! toonth ro bc olio days due lo invisibilrty of ih. adcent of i,te 29'" ofoiscutire nonlhtbd ro be avoidcn *hodd thc ns c6c.nl is sighted oo 2Etr ofa nonth. lisuch ascs con crions ec nadc b bcgituing of thar mo lr

As the klamic lav and lhe Qudih injunctions depend hcovily on tbe fiNlsishling of new lunar crscent the eE ly hlanic slale plrced speial enphsis on $eBsdch i. the lield of Asmhony. CoNcque ly nomous onlribution! $n mde in$c developmcnt of $ine of fie @lid visibility of new crcsnt dd pEdicrion

CONTRIBUTION OF 2OIH CENTURY ASTRONOMERSTne nod.h dcvclopmenl of lhe sciencc of erliesl noon_siehinS bc8iN vnh fic

obFMtio.al work oa Scbnid vho Bordcd a ldge nmber of.cw and old cescmBfon Arhens io th.lal{ h.lfof lhe l9'i c.ntury. On rne basis of rhis obsdarioml daltFo$qinglm ([olhcinskm, l9l0) and Maunder (Mabd.r, l9ll) dweloped theobscralion.l oirdia of ediest vhibility of ns lmd crescent. A similtr work is

20


21/23

reponed in rhe Explanation Io the ln.lloh Attranonical EPle,cfir $al is basd oiSchoch (Sch@b, l9l0). In allrhe* worts for deimiriry lhe day ofrhe fi*r liribilir) otnew cE$o! ARCV (arc ofvirion) b shoM to be a sMnd d.8Fe pohomial runcrionof DAz (claive uinuth). The* edly florc !rc only empnical in natuF N rhe critcriadc!.loped de bsed on lluing rhe dau so rh8t mo$ ofthe obs.oaiions arc conshlcnl

TlE* nelhods do not hke into cosideElion lhe vidth ofcE$enl Btuii (Bruii,1977) co.sidd rh ihpodahce oacE*cnt widtn fld his Dod.l desqibes ARCV (inrems ofannud. of cr.scent above horizon plus tn solar depRssion belov lhe horizln)os a funcion ofthe crescenr widtn. The Nodel ofBruio. ho\rever, kkes $e {idlh !s afuncion ofih RCV ond DAz md a fircd Ddnh-Mmn dislaftc (in lemsolfixcd * i'dirnrl'ler ol fte lu.ar disc). Hc sas ale thc fi6t \no consid.rcd phJsical .stkcsdsociakd *nh $c problen like blightnds ofskt and thar oflhe M@n. Thus tu nrodcldue lo Bruin is the lirsl drolelical nodelofr|t nodern timcs Using the nrodels tur thcbriShhess oalhe lull M@n as a fuhcrion of altilude (Befrporad, 190,r) lnd $c s[tbrighhe$ duri.g lNililhi (Kooncn ct al., 1952, Sidenbpl 1940) Bruin delclop lhcvisibililt cu^6 rlnrin! the aldudc olcrc*d sirh the slar dcpGsion. AlonS lhc$ctrrvcs the bdehbss ofcrcscem (modellcd on $e basis of lirll Moo.) is al leasl ds muchos tlhl ofrhe twiliShl sky. He aho delclopcd cuNes sholvnu rel{lDn berwccn ARCVand sl.t depesion rhll ldd prorcd to bc crucial for lunhcr nodellihg ol canics'

Yallop has edne lbr funho improvemenr ond consides cece *idth as afunclion ofARCV. DAZ and lhe acllal scmi-dianeter ol lhe Moon (rhar depends on LhcEanh-M@n disrucc) al $e time ofob6endion (Yallop. 1998). \vilh de!lopinS a modellor lhe besl time of yisibiht YaUop s nod.l al$ onsideF ARCV 6 a ihi.d d.gtcepolynomial tunction of $e actul cresc.nt oidrh ar the ben ime of visibihJ. Suchpolynofrial is oblain.d by applying leasr square appbxination on a basic data sel Thisblsic data sl *qs d.duced bt Yallop fom lhe limiling lbibilil, .uvs of bnrin b!slctins ARCV for a Sircn sidtn from tlr ninin@ oh lhe componding limilin8

2\


22/23

visibility curve. So fd th. modl due to Y.llop b lhe morl oulhfltic and depndableslronohil cri|na for cxper.d ali.sl visibilily ol new l$m ccenl. The modct due10 Yallop isa smi-empirical model based on $e lheordcal considedions of Bruin and.ftiteriononthebasisofabusicd.tadeduccdfonBrui. svisibilitycurvcs.

Tnc mosr extctuive r.atm6t of rhe physicat dp


23/23

. Th. d.pend.nc. ofth dlendsB on th. cyclic moiion ofth. Moon sld $e Suh.

. txB c.l.ndar of $. Mulih! .spei.lly a lh.i. b.li.rts lticrty dphais. utltliarsidtingorrhetwloNcr$.a fois.niosaldcldilglun&ndrha

. A! rccou ofrh eilons ofrhe rsrorcmd oflhe oodc6 rim.s |o rd.lr.ss theprcbhm ofderemining$e fiisl day ofsishdnsofnew Iund c'.eenrItr lh. b&ksound of tlE$ cfon! w tFrc qplord all lh. old ald thc modm

mrhodi Mthcm.dol dDdel or crirdion dEr r.quiG ro be $tisfi.d for 0E lirsr vGibilityot ihc !w lu{ cEsdr, Duing lhi! qploGtion rn E$ltr of lh.$ nod.ls NcomFcd lnd nodifietios baw b..n 3uSgcrt d $,lEE fler posibl.. Funhd ft. mo$authentic of the hodeh hav ben uscd to .xolod fie obcrvationil lunar cal.hdarfouowEd in Pakhhn. A.sed o. the rcsulF ofthese exDloHions a luturc obervrtiomllus calcndd for Patislm is anDut d,

23

visibility of crescent-1

Documents