different methods and formulae

8/22/2019 DIFFERENT METHODS AND FORMULAE

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Chapter No. 3

ANCIENT, MEDIryAL & EARLY2OTH CENTURY MODEU;For calendarlcar purposes .s we as an Interes0n8 and chalte.gingaslbnonlc.l p,obleh, jn rhe hbrory ofmanktnd there have ben constsrenr efiortsro delermano when and whererhe nek tunarcro@nt wll be lrst seen, t]e,esutrs

of these ealons have boen numeroG 6nd have been based on dtlle.ent technlques6nd tools.lhose results can be temed as 't odets/Crireria tor the oadtest vtltbi|lryor nw runar cfescenr,. Most ot the earty modets are de ved dtioc y trom theobsoations of the new tunar crescent and are enDnjc.tin natlre. Some are basedon thorellcalconslderations. MGI tmpon.nt of these eftons hctud6:

)

The Aabyroni.n rule of thumb .when the age ot the .ew Moon is 24hooB only then the crescent can be seen,. fte redlscovery of themore mrrhematrc.lly lnvolv6d crfte on ot Etong. on + LAG > 21.lhal w6senunciatedw beforechisrian or. (Fatooht et at, 1999).

Tho cdena deveroped In ihe medtevat periods by Mus ms/Adbsba*d on obsenarion, sph6ri6t vtgonomtry and lhe ephemerid oflhs Moon and the Sun (8rutn, 1977). This happen.d as earty .s rhe

The exploration or Foth ngham (1910), Maunda (1911) and orheFln the fi6l quarter ot the twontl6th @ntury. These effort! were moGot slallstil in natur based on th obseda ons mos y in Room by

llr)

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In rhb ch.pter we stan wtth a b,t.f dtscGston on retativety Ecent Gdi


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abbelialed s ARCL but shoM 6 aL in th. figue. SD is the difidce b.isFn thealrirudes of the su dd rhc Moon lnow q "dc of dcs@nr" dnoid 6 ao (ako call.d.e of vision dd abbrevi.tcd s ARCV). Not that the aidnrde of lh. Esenl abovchodan is,l $ lnal a, = r + ,l As rh. ooitr( A is al lh. se altnude 6lhc cFsenl ddB at lh. sme depBion d th. su, A8 is .quabnal $pdti@ b.teen the Sun 4d rhcMoon called "arc of *pdrion ' ad is .qnivd6t b rhe Moon*r - SuNr tig shoM inthe fisld d as The fig@ cd bc u*d to h.aua fi 4.s ofdesenl ed liehl for aSivcn lalitude I once an accurare enough .phemetdes shows the dc of sepdalion andwheE to loot for the dimly illminaled cBcent againsl lhe briehl evening twiliSht.

lig No.3.1.1 Angule Pamcters as@ialed wilh sighling ofnew lunar crescenll, modem tines the earliesi Ffee.cc of ey syslemalic study of the earliesi

sigbring of nev lud cEs@nr is thrt du. rolhc.inShm (Fothainghd. l9l0). He refe6ro l2s enlury Jryish phildopher Mcinoni


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wA diislaled by S. Oddz 6 Crlr of Moituntd.s, Boot Thr4, treatite Eigtu,SmdiJicdtion ol the Nee M@a Yelc Judaic. Seris, volh. XI, Yatc Univesity pE$,No Hav6 CT, 1956). Maimodidcs mrk w studi.d by von Lifl.ow inSiEunghberichk da Wiehu Altd.nie, Math-Natrre, Cl6e,lxvi, (1872), pp.459-480,ln his *ort Maimonides nak6 rh. snalLsl visible phe ofthc M@n depqdenr on trcvariables, as claimed by Forh.in8hd. Th.se veiables bave ben temd ss rr? rretonAorion otthe Moon a\d tle aplbre.t ahgle of tision - F6\.noghe slarcs rhal ifby hgk ofrision Mai1onides m.es the diffeEncc of the Sun md the Moon in rc.irhdirlaccs then th. nle of Mainoid6 .nd his om rule for hinimlm visibt. pb4 ofMoon ae ndly the enc Howevcr he f!fther says lhat Maimonides rute sivcs slidrlylower oininun alliludes that could be duc tc, b.(.r ob$fring condirio.s in Jeruslemlh& in Atbos. Unfodu.alely Fohednghah .dmik that hc could not sumcienrlyundedand Maimodis' dithmdiol nelhod dd has not giecn a dchited @ounl of rb.same (Foihciinsham, l9l0).

Lder, Bruin(Bruin. 1977)gives a moe debiled eounrof any medievalatenprol solving fie probln of findin8 sronomial onditiotu for thc fi6t visibility ofcrscnl. De$ribinglhe Isl.mic astonomical procedurcs Btuin says $a1Al-Khurizinigives mathenadcal rules and rablcs lor pedicling thc new cre$e.t where6 AlBamnipeeds a conplerc soludon. He aho nenlions lh lato accounl ofMo$s ber Madoonlnd clains rhst ibn Mamoon larsely follos Al Balloi. One oflhe nost inponanl dp.clofBruin's accounr ofrhc eflods oftbe bedicval era is rhar Anbs^rulins orthis ea hadolEady rcalizd rhe sienitlcece of cE$cnr widrh. This inpoiant aFcl enaincdhissins frem all th. najor cooiribudons of lw.ntietn century bcloE Bruin. Bed on thisdscription ed thc modein loowl.dge in ihis work lhe sme h6 been explocd dor..xlecively od is pEsnt d latu id the chaPler.

lilcatue that app.ared duing 2od c.nrury lhe gencral rule ofBabylods for first visibilily ofnew lunecrescnlis:

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at the titu of lual su6.t shottd be g.ater than 2't howsthe su6el ohd the no6et thDuld be g.ot* than 12 tim(l.l.l)

l2 rim. deBes is cquivslenr ro ;$ ofo hour or48 ninu$ (Brura I9??.llv4 le944

lto$ev4, il wd Fat@hi, Stephe$on and At'darghdlli who havc 'xPlorol lhcflon5 of lhc Blbylonians dlsiv.lv &d have epond rnal eording to lhe histotielEords of re* c.Mnl siSlftings ol PG Christie m lhe cderion attibured hBabylonis is d ovr simplificalioi (Fabohi.t al 1999) Ac@tding lo $en $udv oi209 recoids of Posi{ve new crsceol 3ig} ing extdctd fom Babvloni&s Astronon'cardi&ies. the Babylonitus had succceded in ioduladng a lrulv mathem'lical lunar lheory$hich ihey 6ed to pEdicl !&ious par@ele$ of lumr notion Unfortumtelv wrlhoulpresnline my iheorerical dlaih of Babvlonian eE lh'v appear 10 be "ilicdl aboulBluin\ slgscsion thar Babvlonia criterion was whar is mentioncd

above (l l l)'lnstEd.1hey clain that Babvlonian ctilerion wEs as Follo*s:

Tnc ies cEsce.r is een it'' Elongutio, (ARCL) + nooe|st6't laE tine (Uc) (3.12)

Even, rhis cdrerion is apon d on b6sis of the sueestios ol Ncusebaud (Neue'ba@r'1955) &d nol 6 a rcsult of ov d'$rip$on of a madehalcal lh'orv Ttev havenehtioned two svstns of sol& moliors on *bich th lt@ rhsrv ol Babvlonids lasbased, but hoq lh new rul they hav. attdbded to the Babvlonims wD drived al $ nolgiv6, They have gone on lo plesd .lifleant valucs fo! th' @nslul on lhe 'kh h8d;ideof(3.1.2) Thse lalEs Engc fton 5 DiniDum of2loro a maximM ol23" lnlhep!ent work we adofl fte followins as a bener critcrion alfiibutcd

lo de Babvlo'ians as'slss.sted by Fatoohi el .l:

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(r. I r)

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The Mon ar divi.s this qitcda 6 d.sc'ib.d by th*. aurhos is thar (D ihc m of lishtd6dibes how bnghr is th. cFac.nl &d (ii) ihe moon*i*utuel lag de*db. hov lons thectcse @uld coaii abovc horian.n r th $n sl. MoE is rhe vql@ ofqch ofdrqcpaluelea moE is the po$ibilily ofsishdng offie ce*cni. Howver, drcr. h6 to b.5conbined ni.imun ofthe two dd the sigh{ne Eords wcre the only heans to verify lhccrilerion or diving al the (itrion. Using $is condition Fatoohi a al hsv Fponed thcaslhs of lhcir @mpu|,alioB for 39 rw cEsnt 3ighings rhal in lud. bolh lhcBab)lonirn cEscenr qighr ing Rordc tud thes'ghtine Ecords Eponed in $e 2Odcenruryliredtule. Th. aulhos have rcponcd ihat oul l99 cdcs the Babyloni.n c lerion wassuccestul in 98.7% of fic posilive siehliog {cE$e.r claioed to h.v. b.en sn) c45but fsiled i. 45.7'lo c66 of the nesalive siShine GF$.rt no( *o) ces

In rccnt lins a number of orsdialios haYe smnged for coll.cdon of r.*entsighting or non{ighline @ords. Th*e inclu& Isldic Ctffint Oberyalion Prcjdl lhSo h African aslononical Obsaatory and rheii websiFs Sinild rccords are reponcdro aid colhdd at lhe wbsile @!sS.h!i!s.eol1 Morevet, lhc ldrgest data elyd availabl. in lhe pubhh.d paFG is lhal by Odh (OdGh 2004) we hav' *lecred 463of lhes dords ofdening cEsnts el.dcd for tE compdison ofrhe nodels studied in

Usins (l.l.l) applid lo the dar. 9l cletd for this wlk the '6!lts upEs ed in the lable no. Ll.l. ln this lable ollv lhose rccords are prcscnled when thccrescent wa reponed to hare ben seen wilh or snhoul ev optical aid ad ARCV +LAG is les lha 22 degr6 Th. rabL is end on ARCV + LAc It shows lhal thcrc ft? claims of oplillt unaid.d sighinss of new ce$enl tol salisfving $e Babvloniancileion gi!.n by (l,l,l). The complG dola sel is P!sentd in the Appe'di\-ll Thctable i. App.nditll is en d on visibilily colum N in d$ending older, so thal alloflically @id.d visibility ces of crc$cnt sighing in ordtr of ARCV + LAG (indcSt*t app.d al lhe top of the llbl The table in Aptndia'll sho* S No ihcobseffation No. (odeh, 2OO4), dare ofobsryatio!, latitude ed lonsitude of rhe locationof ob6dtr, visibihy N' for uaided visibilitv' 'B' for visibilitv thrcugl binmld, T

8l


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for visibilny thftud rel.$ope. IlE vis'bility @lul6 @.mPty if rhe cjt*nt M notFen and co.tains V' in an apprcpriatc colLlm if the c!.sc.nl is sen. The esl of lhecoluens oflhe nbl @n!in age ofMoon, LAC, ARCL, ARCV md DAZ. Th lasl file@lmns arc ior the five models coroid.rcd in this chapler, The colllm h.dded A is for

This tabl shows $al out of 196 daims ofopdcally unaided sighlihSs oftbe nescresccnl there @ only ? cbes nol saisfyjn! lhc Babylonid critmon. HoEler, oul of267 cdes vhen 6e cc*cnt wd nor e.n wilho any opftal aid ARCV + LAC isgleate! thdn or 6qul to 22 deg@s i. l07.ses. Thus for pdniw siehli.8s rh. critno.is foutrd lob successlul in 96.4% c.ses dd !frongsl lhe hegalive sigbdn$ ii su@ssfulin 59.92% css The sucess of model in desciibihg posilile sighlings alonc is tul acal success ofa nodel. Thc model is not suned seu if n is nol able to dc$tibe thenegalilc sishfng 6 wcll as ir d@s the p$ilive siShtings.

SOME SPHERICAL TRIGONOMfTRIC CONSIDERATIONSThh hd be.n m.ntioned e&lio rhal ior rhe ploblcm ofsiglting of new lund

crescenl $e orienlation ofthe ecliplic plays md inportant lole (article 1.2). Thecfoc, ilis inportant b rrisil lhe ne|hods thai lead lo lhe ddelhimion oflhe mgle lhal the*lipric nlkes wilh the horircn al the tinc of snel on the *$em hotian (il ispanicularly significer in view oflhe fact that the new c!e*.nt appeds close lo n). Thisangle lalies as agaiNi rhe lixed celesrial equalor lhe orientation ofthe ecliPtic slowlylaries lhroueh the year. lis fislE no 3 2,I on lhe nexl pag. shovs lhe wcsrcm pan of$e cclcnial spheE wilh impodanl points &d angles m d de$nM below:

w' the west cardinal point,t, rh vml cquinox, inleclion of.cliptic dd cqulor.P, the Nonh c.l.sti.l pol.

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Z qv - r rb ouiq'iry of6..dinic,z 8}'r1 - 900 + 9, 9 bdte lb.lditd. od6. d...'Z ZSr - 9 + 4 lb dSb ofrb.cliP(ic xdft 6. v..dc.l,rs-d-G5tlirst d.clhnioofrn tu,PZ - 90'- o,St - ?e 6. lonSili. ofd! $'!

FrsNo 32lFMn 1! 6su! 3.2.1 in lhc tPhdic.l ttrtgL SPZ 'pplvi'g 1.l{ of ctsiE of3phicrt ti8oiodt tt.rdA non &. !id. PZ oe gBs:

tlne=cort.o.PSz'ft. g@d |!l ioo b..e..o d..lidlor 5 .d lonSind' I i3:

.int - doP co.. +.o.P.i!'rin r

(Jr.r)

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(3.2.2)(3.2.1)

@s(PSz +@+ A)=rer.@r,1.usins (3.2.2) !d (3.2.3):

FDm ridcle sh, st nir8 !r Pr:

6d (32.t) tosethd *irh (1.2.1) leds lo:

(3.2.5) sd (l.2.6) thd sivc:

(3.2.4)

0.2.5)

(1.2.6)

Ju"'This show that e + A or the .ngl. thlr ihc elipti. nEt s wirh thc v.nicsl is 5.e!dcp.rdcnt 6 n d.peo& on the lorgitud. ofthc su only for a fix.d place {or latirude 9).ln th. Esr of rh di$usion in rtis snicl. rhc anglc 900 - q + A is dnord s V. In dEncxl .!ticl. etr@ cv.r ? + A is L*d it h olol&Ld on ibe t 6is of (3.2.7).

In @ the dslinaion of th. Moon is $u1h of th.t of lhc Su in NonhqDH.nisphft (md north of the Su in th. rcuth@ hemisph*) it is possibl. thsr d.n.ncr conjuction th. tr.w llM cKar $i! b.f@ lhe slNr in which w ir ir simply

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impossibtc io 3a thc cr.!ent, TLse circmshnc.s aF shom in the 68ure 3.2.2 on the

iDiii)

Cl, lh. cclsli.l .q@lor, 1 i! th. vdal eqlinor.TS is the diuFal path of th. Sm thar is just *riirg al s.DE b thc diumal palh of the Moon th.l st before thc sUNl 3t E.Ds = 6M - 6s, letpendi.ula. to the clesial equlo. is lh dif.rence of lhedecliMtion 6M of ihe M@n &d the deli.ation 6s of rhc sun. Declinalion ofthe Moon t south of lh sun md rhe upper limn of Ds is 50 9'. lhe inclihalionoflund orbit to lhc trliplic.

v) zcvs = 9Oo - 4 is the agle bctween $e c.l.srial equ.tor lnd $e horihnNS whee C teprenls lhe lotitude oflh Pl&. TheztSD = Cvi) DM = dM - qs, lhe dilreEnce of risht a$cnsio. dM ofMoon and rhc rishreension os oflhc Sun. For higher lairudcs DE nav b. htgc allowing laigcvllues ol ncepliE LAG aftr co.junctionvii) sF = pM - 9s : 0M is p,Fndicultr ro fi. E lifljc is $e difiercnce bctw*nlhe c.l6ial taliluds of the Men and lhc Ss As for DS Sf never xcadsrhelinilof5'9'.viii) FE (E' .ol showo in rhc fisure 6 it almost coincides wilh E) h ptrallel lo the

ix) lislhe Vemal Equinox ed Z)6t =horizon. Dep.ndniS on laliude C and

v h !h. angle of lhe Ecliptic with $elh se@n, 'Y naY lary ftum zeto

(vhctr ecliptic is alons the botizo') lo 90 deB6 (when cliptic ispc+endicula! to lhe holzon) l! cas' when v is small DE thal has b bepaEllel lo the 4liplic is nuch ltrsei tha' DE (ftol is paEllcl io lh cqualo!)and hdc. E sd E aE mrch s.panted Tlle detmi'arion of$is oSle mdits sienifidc. shall be di$sed in the n'xt drcLx) I'lt figurc shos rh SM is jN| about o sel in a pl&e oI sdall b ndiudlaftude d , $e M@n havins ddlinarion eulh of il' sun ha al@dv *l (ddwd set at poinl E) $ th.l lh LAG = TiG of Mdnct Tim of suNer is

E'


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fig. 1.2.2: G@tn r.y ofP@riv. Ag. .rtd N.gaiE L.gA@rding lo rlE luN elqdr b.!.d @ IIE binh of Nd M@ b.bc tlE ssa

tha nd lun|r mn$ b.gio .l dr tirc of o.!a 6 $ii vrt Ming- How6, for tll.lu@ c.lsds b.!d d rn viltbilitr of $. n4 s@4 w hN mfli &e.olb.gii on thb cming B n b 3inply ioF{bl. lo $ tlE w luu c|!g in thiscinhtee. Fd plE witt lrSc ldind.. d|r dbw for llre v.l6 of DE |,d h.@DM ln. tri&gl6 uidd 6!&l6.riB o rbt tr rtrdl .dd.!xl ..otl b. tBt.d sryhdicrt r.iugl6. tlc! d y @ll lo tr|.diun hdnd6 @ qEidqld e th|r th.tdugl. EDS i! r 3rdl dgh rlgld oilElc o *y wih dt.!gl. tt D t..Ia E .d Stu 9oo !r!d rlE a.sl. d S b.rsa D !d E i! ?. $. ldind. oftlE pL& Th@foE:

SDAf (328)

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sioiLdyrt rrtdSL AEFS b dlo.ndl r!d. t codrt.ld|plEdilati|lgbqi6

tilr,.= (3.2.e)Pcpl$iry SD by 5M - E!, SF by PM - !! lid .llrdrudi8 ES illn (3.2.8) ,nd (3.2.9) w

tN -ts-$N -6d,HAg.in in ISDE tm t!rc:

02.r0)

.DE-r=S, (3.2.U)A-@, -6,161

(l.2.ll)-

ln b frrr! !23 $,ldd It dlr d. Dlc ntDglc Ft sDE of6. Flvior6 68rtarEql'itld.ortun.8 dvcLAOi.i.i.EDM


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usirg (3.2.l0) il tlr4 lat s into:

(aM -as)


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Laliild. ofMoon in d8rc6,Angle of E liptic wilh Horizon in d8rees,LAG, Moonkt {u6el in oinut s.

rable No 3.2.1 : posirive Ase Nesaiifr&B-;s-i66l2oliT;R;;iifr;;3.3 LUNAR RIPENESS LAW & ITS MODIFTCATION

Befor |h. rin$ of ptolefry lh.rc ws no knoyn sy$enalrc descriprion ofthedynamics of thc planeb and rhe Moon, Theretoe pblemy,s explor.tion Che epicyclebased description ofthe pltret4y noiion) sens to be rhe tirst sciious. scientific 6dststemlic aucmpl to d$nbe 1he dymics oteld systed obl*is. Muslim 6ed lhistyiem b cxploc lhe condidoE for 1hc fid visibility of ns lutrd crsce . plolemicthory hal d.*rib.d th plderary nodon Eins rhe epicyctcs tcory rhal could Dddictthe pl4ehry, lunar .nd sol& ephenns to a g@d soush d.er. of accuacy. TneMuslins had put ro use lhis iheory weu ond had developed sotr dnd lu.ar tables. Thepos ion of$e Sun ws predictable !o a g@d dce@ ofprccisio. and @nsequendy rheaele !r lhat ih. *lidic hal6 with ihc hoizon ar the line ofsue( ould b calculardBing rne sphe.ical uigoroheFy that Muslim naomddn hav dv.loDed thn*1v6.As decribd by Bruin ( 1974 sing lhc &8lc of ectiplic ,y wid! the hoiizon al lhe tine ofsu*! the lalitude of th place of obseFarion, egle of sepsauoo brseen Moon dd

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Sm p.r.[d lo lqldo. (difrcqE of ddr dgii L!..dioB), tlE !sl. of d@!n(dife|i@ of a|lnuda o. ARCU !d ri tid[ ofsc.'n the M4lin aslorcEcrs$@ aU. to develop ! hisltly rdi.ble qilcion for tlE visibilry or invisibihv ofrh6 @

Figua (l.1. I ) bclow enicl ir r r.otuttu rin offg@ 4t ofBtui. ( le77) shovstl|st dF sD IEs jur !.r sin of r/t{. Thc liie \nX D85rg tb.o{8! $ts drllal poiltW, is th e{uror irclined d o dsL g (tlE ldihd. of lhe ple) fim tlE @rnd to rhehoriar AS k th. dimd pdn ofth. Sun !h!t is p66llel lo thc.qualor is shom asbroko 1i... TIt diurnrl BIh ofd. Md MZ is tls sboM wnh 3 3imilt trot.lincDdralll Io ih. .qudor. TIE lin. HS ir tL E lipaic lh.t drd e .ngl. A wfi th diurolpath of 6e Su d $..qulor ThBtL FiiiFb D!l6.td..8L q + d witl the Mnrl.MS t th. s.r.r.dd b.'ra rlF Mdn !d ttr Su (e of liSln).

Fig No l.l. | : Trigmrdic ddiFin of @rdni@ of w luE d6ern

\:\.


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MT i3 p.rDodoio to i[. ..lirdc rII8 ar.h l}a TS - 1 - 16. b 6. 'Etltinsb MIII, rn - fu d ft. erb a H b..r!.r Eclbrb (IIT9 jd &. Ho.te|.lolM) - 9d - (r + AI Tb - rilrb tm{ b lisb.raLd d T yE tq!.d3L i M!cvl.{HddT-9+A.Itrb.ioplyk &b6crdtio:

ra-Er +rS-t,6lp+D+L 4. (3.3.1)|hir Fldih n .e. .r .$|d@ O) of B.uh O9ZD. Irtia8b HgB b rlcd s . ph.

codr+ a){d ti.lgL ABS lad! lor

(3J:)

(3.33)(33.4)

(3.3.t'Il& 'slr (3.3,5) i! dimalot Aom eb &uii O9,D t& tih in hi! .qurid(a) |b h cbiD. nr b..o obi!.d by ld4 lhe tigoffiy. Ittt smc 6ru. i,t'.d.d ||3toglpt ric.l [email protected], turiotbADs 8it&.1

f|(!o .phqi6l trt|tEL tlTMr

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sin rr'r _ rh&sn(e+a) @s(e+a)erhlr (3.1.t) rs = sili(sin r!.r..(e + 6r+,L -,rr 0.l_6)I. sphoical iridglc ABS:

stnaD . stn 4r @ePand in spherial lridgle HCS:

(1.1.7)

sinfls=-l]1gll l].l.8lco(@ + a)

(3.3.9)

!.):9:.q+a)

!;t::!

!lj".((:

[.

:l

-t

laegles Hs dd ss @ snall cqua$ons (3.3.6) lnd (1.3.9) yhld alnost sme lesutB sthos by (1.3.1) ad (1.1.4). If the valuc of 9 b lege. then Hs dd as ae nol shallandth spherical trisonoderric rsults (3.3.5) and (3.3.9) should be ned for doF Muale

11'e sienificdce of (3.3-6) is thlt if th. cphcmedes of both rh Su and theMoor m tnoM @uErdy (aj th.y c rcw) ud rhe &gl. 900 (p + A) = v of etipri.{irh ihe honan lculalcd fom (1.2.7) rh. !.lE of HS @ bG aal@ted for rbc tire ofsUM for dt day of lhe yw !trd psrti.uldly for rh. dlt or rt y .Rc UE bnh ot ncq

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Moon. wlee6, oncc rhc hinimm &stc of sepaDrion !s (.quivd.nt ro LAc) fo, $evisibility of @w l@d ct.st is tnoM for thc day the @rcspondine agle HS sinS(1.1.9) cs als b. .vatua&d. Horev.r, rhc ue of (j,j.6) is ind.p.nd.nt ofany visibilitycondition ed is fixcd for rhe day ir d.pcnds only on rhe posirions oflhe Sun and lheM@n for fie rim. ofobsenato. &d the locaiio. ofrhe obF ei. On the o$( hahd rheu$ or (3.1.9) dep.nds on trE vjsibifty @ndiiioG, @cly lhe mininu del. ofs@ml'on tu, thar.& b. kmh only on lb. bdis ofa t gc nunbd of oboMrions torAt rhe rim of Muslih Gtrononcts rhe .phenerdes of $ Moo. oay nol haleben so aeudely tnown s ir is today, but rhe Eligiou tecnn.ss oflhe seing $e newlund crs@t nusr hlve tcad ro moc rcchre valu6 or,s. Bruin (t9?7) hs i.dicaredthat Muslim arbnoheG rveE welt awe of rbe f&! lhar d8Lnce oalhe Eanh ild rhMoon and hence rhe width ofcrescent for sde uc oflighr varies. Though exploiationsofdr ancienr considcred lhis variarion ro behdve tibearly we now know thal it inlolveslne lflgononefic func on.In lnis worL $is prcbleb is h&dlcd by considering rne actual eni_d.inler ofth. Mooh dd lhc fer $at MnslinvABbs ob*tued thal ar shone r .lisrances the crcscenrwasseenwhehtheLACo!arcofseparationaswaslOdelccsor40minulesoftimcand

al lae! dishcca thc cErcent ks seen Nhen rhe m ot seplFrion dJ B t2 d8res or48 ninules of timc. -r1li5 teads b a siople Eladon blwn /s md lhe aduat semi-^-- D22

Using rhis etalioo thc @ of *pdlion necded for ce$enr*p@tion felor is .alcutated in ihe sonwar. Hilalol.

ol r0)

9l


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As mc ioned abov. dE Mulin sao&Ders have alaldy dduad fromobrpatios lhar for very thin but visibl. cE$ent mirimm ss @ 12 lime degds (48minureO dd for d {ider but bmly vbible cresent mininm as w6 only l0linedegl6 (40 minulcr. This nininm as ie acturlly dep.ndcnr on the width of $e crc*entrhar @ be dedvcd 6ins (2.E.E) dd (2.8.9).nd rhe Elarive atitud. (ARc9 thar cd beobtained using (2.8.5) for th. sun ed the Moon. Th. issue shall b. dieussed in noed.tail larer. Usins 0.3.10) rh.t shows 6s beins dependent o. the lisull dimers (in arcminulet of lh. M@n in our sky one ce compute lhc Lun Ripi.s Fudio. A(,i9)as henrioned by Btuin (Brui., 197?) or tha! is siven by (3.3.6). Aaodihslo Bruin rherul. ddrd by M6lims w6 thal if thc lalue of HS calculatd using (3.1.9) equals orexccds lhar @lcolared by (1.!.6) rhe c..s would be visibl. orhcNi* nor. In thnwork this rule has ben named M6lin Lunar tupness Law or sifrply Lumr Ripenss

According ro Bruin (19?7) $e valu6 ol HS de.ored a R()., 9) obkin.d fron(3 3.6) seE pGsenled in rhc $+ lled Lunat Ripenss Tables sDd fisr visibility ofne!Lunr cmcent ws walulcd by Mulin BltonomcB usiog rne sboE rul. In lhis workHS a3 deived tom (3.3.6) h dnokds,l4 and:

Fo, a fixed pl&e (e constal0 it is noted lhat forsDallvalues ofl,, sin ly is also smatl&d lko n ahosl sme as rv ,j. Bui when the *liplic is wll inclihel io$&ds lhehoien (q + A is large bu lcas rhan 9oo) 1r'4, nay becode noE rhan 1! ri(iflv > 0snd ecliptic is roweds nonh) or renain le$ rh& 1, - .ls (ir pr, < 0 and etipric isiowdds nonh). Ho@er,{da is endely independenl of Oe se@. aod depnds only onrhc rcladve coordinates of rhc sun md rhc Mood. As rhe major componen in (3.1.t l) islhc diferenc. ol longnudes of rhe su and lhc Mooh ,taa h closely linked to ihc ACE of

^d., - sin r[sin/, r.nre+^t+lM ts (l.r.r l)


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Mftovd. ir 9M . o, x&" * l,{ - k. Bur whcn boil 0M 6d I + A hav. the s4esign 0.3.1l) shows th &4 *ould b. mor. thd l, - & and if pM dd e + ^ havediffercnl signs &e would bc l.$ the ,L - .ts. Bul thcF v&iatioos u nor sMnal

The vahes for HS ohained fom (3,3,9) dc denolcd s X,b and:

^,,=.'""( (l I 12)

For a fi&d place (9 consrd.t) R,, dcPcnds on ae spa6tio. a, (rhe equabrial t'Acber@n the sun ed the Moon).nd thc s.en 6I + A is *son dependeot fiat shall bshoM larer. Using (he looh sod le hniqu.s di*ussd in.hapter 2 ie, fo! $e dav or lh.day alier conjunction aor s placc of ob*ryarion ii .valuaEd. Ille ninihud valueofas is@lculaled 6ing rne lrchnique d.srib.d al lh. .nd ol Ptviou .nicl $ lhat ,tB isdedu.ed. If lt- is calculat d using h csdml.d value ol a, rh.n w call il X." (mErimared valk ot Lunar Ripencit funclion), Il ir is .al.ulated uin8 an avedge vrlue10.5 degres of a, thcn *. c.ll il I- (d .vcaec wluc ol Lud tuFns nrtudo.)Ac@drnely A&" = i- - X,a,6d Alt-, = i^ - 1&) Thc sinplst fom ofrhe Ll|E

i3i

:) $i" a;t+,ry -rsj'o'"'( (3.1.r)Tlus Lunar Ripeness Law that piolidc a solution of the probleh oldelemining

rhe first day of visibility of nw lunar cre*enl h based on the Angle lhat the Eclipticnat6 wilh the bodzonlal or $e angle 0 + a ihal it dak.s wilh the venical. on the dav orthe day after conjunciion once lh. coordinatcs of the Sun dd the Moon at the tim ofsuns for &y leation.rc calcuhted. (3,3.1l) allo*s onc lo calculale,e,4 ed sine 6applopnab value oi4 d sivenby(3.3.10), in (3 3 12) the valu. ofx"i ce be tound ltr

95


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viw of $udying ihe behaliour of lhe Rip.nes iunction oler a yea! for any pl&ccalculding Ie, for cvcry day of the yd is not useful. This is bc@w of thc &ct lhatyar ro ye dd da, (o day vdiarions in lalitude oflhe Moo. dd hence nd, ats notdepndent on the rime of ye.!. However, *ith posiblc lalues of4, one cd calculateel.v val@ of l* for @h day ofa yd &d study ns veialioG scenally as sell d$irh chdging lalilude of pl.cc. As possiblc values of,, vary from l0 degcca lo 12degc.s an appopdate value for the day oi day after conjunctioh is obtained only on thebdis of the ule disraice o. sminimd.r of the Moon. Still tor a compdie. values oflS6. foi dill@lt ladrudes, se calculaled lor both $e cxrene valucs ,, = loo dd a, -120 and the curves,t,r aeainsr the longitude oflhe Suo (fiom March 2l) aft ploned inFiSuGs (3 3 2) lo (3.1.5) These cws snow inleFsting featres lbat cs b sunmdized

i) from fi8.1.1.2 il is clar rhar forg = O0 i... for a placc on $eequabr&i! issinusoid.l wirh mdina.t March 21, Septcmber 2l dd hinima at Ju.e 21.D4ember 21. Snalle! v.lues of iG nems shaller lalues ]?d4 or !larivlyrounger cEsent m.y b rn. L&gr values of x i" dcans largq valus iiaorrelaliv.ly oldcr crcscenl may b. seen. Thw close to equalor older cEscehtnay be visible near equi.oxes and Elatilely younger crescnl .ea! ehrlcs.If $c Moon is nc{ jls apogc lhen il is noving faner and apFaB rhicker insky Blatively younger cescents bcone ripe for visibilny. Funher rhepaencc of two maxim and two binima ihdicates four stong innecdonpoints. There arc rwo regions ot lpMd concaviry (douod solsdcet dd rsof downwdd concaviiy {&ound cquinoxs)As rhc larirude of rhe place in rc66 (one move ro rhe nonh ofequabt $emuinum ol1." at $e venal equjnox loseu nDline ir eard lo k d ayoungcr cescenr bul the ndimum of lhc veml equinox ris.s e $al itbecones norc difiiculr to ec a yodge! crc$ed (69. 3.3.3). On thc o$erhmd fte ninimM of lhe s|!md ehric mo6 rowuds sdng (v.malequinox) ad that of rhe wintd rchric. movs rowds autumn (aurunnaleqlinox) md in ei$er c& d.crcae funner making ir esier ro *e a younger

96


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15

1312

10

I

cwc ahaFs clos. ro ll alhlnn lm6d thc solsti.6 flltt ns.

F g. No. Ll.2 X* forg=00

Kg. ffi. . .r. rl, .,.- ; I210o 180Fi8 No.l l l rt'L forg= l0o

BcyoDd tropic of .alF two of the infl@don Poids re simPlv gone &d $ehsimu al lhc aotmnal quinor ds tunner malir,g n FoE difficuh lo siiD

91


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iv)

. yomg.r ct!.c. (f8; 3J.1). Tb nioidsn oftL qttc Ed,. tt !t ttl

.quinox mrli!8 it .did io r.c yomg.r @3@l! n tr ro ml .qu$oxrG frltd iF!.ic i. dr ldtr& &. D!!io@ bcoc tt}.' rd !i8lelrd dt. ril!i|!@ t cF &cr!.'nry tuvi!8 &n ftr biSba ld'ndd il i5g6.r.lty .si6 ro @ younScr cr!.c.nt3 cl@ to vlrnll equimx .ltd diffrct lrclqe b aull|nn l cquilor.

'17

t613

1o

toFis. No. l.l.4r X"i for g - 25'

1}. siuDtidr r!\/{r!6 .dirlly i! fivou ofam|'llll cquiM ft. lb -ttld!For the Arctic Circle .nd to it3 lorth (latitu


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a0

50

20

Fi&No. 1.1.5. i^fore=61 50

Howevr lhe narcr i! lrot $ sinplc beca6' tuc olc of latitude of rhe cKdt'i-., whctht thc cc$eni is !o!ih or north of ihc Sun plavs e impona role lhaidercmin6 rhe v.luc ofrtdo lhal is lronglv btnude dcFndst Now lh' stDns

nuimumat lhe v.oit .qrinox for hi8h.r lttitud6 indicat's thtt e old'r M@n rov rc1b' vbibl'b s the AGE offt. Mo@ iNrrsd $ do6 iG elo'sdi@ ad brighlress ThtebE nis posibL ftat invisibilitv $g86l.d bv $' Rrpcnas tun.lid valucs m'v b' nisledingThh inly l.!d lo ltlollN for sdalld LAG vd6 drd sm lqrlEof x'L h rlnt$!d( th. RiD.G$ tulcrion vtl@s for dE ob!'dltioml dab avlilablc in lilettrurt e.dculacd sd pr6.ntcd in ApFndirlt Out oflhis dx' s't th' c!56 whd thc cscenls Eporr.dly sd bui .rc nor h lg.mdt *nn dE L|M RipGs ltw rE also s||osin nblc rc. 3J.1. TtG obae liontl dah k $lccLd AoF lh4 tPo :d bv Od'h (Odeh';0{a) fron *hich c.*s or cvdi!8 '6cnr obs'naiioN re cosidercd onlv Ttblc3.1.1 shows o. No. (th. oberysti@ nunb'r (Od'h' 20Oa))' dtt' of obseNttion ldnud,ftt longin/. of ih. t@dron of obaa$s' M@n\ !8e and LAG' snele v of ecliPlic sitt'rrorLo, t"lio,a" or Uo.,, toogirudes oI the Moon ed th' 3d' Rd giv'n bv (13 12)*ill !6 siv@ bv (3.310), Re sivn bv (3 3 12) *ith 's = 10 5 d'8r*s' Ri' sircn bvO.3.I l) and a&. = Rdq - &"

I


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'Ihe table in Appn.tix'U shos S. No , the obseNation No (Odh 2004)' date orobseNalio., latiiude dd longilude of the locaiion ol obserur, visibilitv 'N' for lbaidedlisibility,'B' fot visibiliiy lhouel binGuld''T for visibihv thbush lelescope Thevisibility coluds d empty if the.res.ent wd nol s*n sd conlaitu iV' in an.ppropriltc colmn ifth qsftnt is sen The rcsl ollhe coluons oflhe t8ble contarnage ofMoon , LAG. ARCL, ARCV dd DAZ The ld file columns de fot the livemodels consideted in this chrpter The colu. headed A is for the Babvlonid model'and A for the Lunar Ripeness model The colnmn headed B cod'ins AR"

The lobl i! ,Appendixll shows lhat oul of 196 cdcs in shicb $e crese hasben reponed to hove been een wilhoul oplicdl aid thee tle onlv 14 cases hat do notobey thc LuMr Ripencss La* stalcd above The debils or rhcs obscrydlions de tintherexploEd below *ltn a Modi6ed L@r Ripeness Law is sugeestd The nodificalion isneedcd in ordet 10 sepdarc tbc caes of dked eve visibilitv and rhe caes when optical isused for clescent visibilitv

A tot.l number of l2 posiliv obsedations ih tbis dala hav b'en wnh the helP ofbin@ules ed klcscopes when $e cr.$enl ws nor visible vilhout oplical aid dnd theLuai Ripe.ess Law is .ot elisfied 'nth is logicallv valid as the Lua' Ripencss Lawwas deduced onlt for mlcd eve ob*tvatiotr Tlis bas he'n the dolivadon durine thN$ ork lo ood i fy th Lunft Rip.ne ss Law ro e ocompass the otricallv aided obse d'tions

The columN ofihe table contain Ob$flario' Ssial Nunber as led bv Odeh(2004), Date of ob*tvation titilude oi lhe Place Longitud ol the Place VisibililvcolunN N (lor unaided visibilirv), B (wirh bimculd) and T (wirh iele*ope)' The*visibilily colums d. mptv io! invisible cresent md co ain v for lisible Rst of lhecolws contalns Ase of ctesce.t in hou6 fo' the best ime (Y6llop' 1998)' Lae innin(es, S.padtion btwcn the Su ed tbe M@n At of LiShl (ARCL)' the angle v =900 - (9 + d), that thc clipiic mai


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the lonsirnde orthe sd, Arc{f-*pdtion fa.tor, Sitd by (3.3.10), Esdnat d RipencssIuclion, ,{.r olcularcd eqution (3.4.2), Ar6of-epdalion ractor, Aversge Ripen.ssFwrion X., calcllared Ning avcds. tu = 10.5 d.gEs dd lh quarion (1.4.2), AclulRiDene$ Fucrio. ,Rdy calculated sins equtioh (3.4.I ), AP-" the difiencc of AveEgeRiFn s Fwrion & th. Acrul one. Th. ribL l.l. I des nor show thc visibility @luonss tbis table @mprises of css wher ihe cesftnt Ms rPodedly seen withont oflical

The ncxl I27 c6es dc grcuprd 6 coup-B. Therc de la (llolo) mked cyeviibility cdes, 25 (19.?%)binoculd visibihy cesdd l6 ( 12.6%) rlesopic visibilirycses for values of AFw lying betwen -1.58 dd 0.0. Oul oflhe 14 natred cye lisibiliryca* in thn range, threc lery low AR{yfvalue cdcs have comon chancterisdcs. Th.*e No. 286. 2 dd 2?2 witb AR.!i valu. -3-5, -3.4? md -2.88 All lh* c4s m nerautumnal.equiiox (Sepr.20. Oct.2l &d Ocl. t epecdvety), r$enl hs oldc. age(39.1 l, 39.24 dd 4l .91 hous Elpectiv.ly) md hav consqueitly larger phe.

The lsbl is soned on the values ofd&y. On the basis of a clo* dalrsn of theresults of conparing ARNi ihe dilleences of Av.lage Ripeness Funclion valus End lheActual Ri!..ess Fuclion v.lucs we ob*de dEt:

The Tablc shows ft.t lh.E are only 14 pGitive sidrrines out ofa rolal nmber of196 positive siShlings that are nor according lo lhe Lunar Ripe.ess t { (ARo, = Rn",

The a!. no posilire siSndne wirh A&q < -1.58 wirh or wirhour oprical aid theEns in wnich l8 aft.npb have ben nention.d in the lne6ud. We considr it asCoup-A for thc Modificd Muslin Lund Ripencas Las lhal we slal. ih ihis erlq 6:

''tt th. tlill..e,ce ofAv.rase Np.ts Fanctior R-, omt the Actuot Rip.h6sFu"cion Rro F /Re) is l.ss nnh -J.6 it is ihDossibl. to see nE n.N luat .rcsc.nltor att ldtitula uith or Nilhott optical aid'.

l0t


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All lhe* conditions falolr rhe lisibility and it wds ncntDncd above in view of(1.41) that aor oldcr a8e cresccnb (/.M ls tae) sh! er arc or scparation may beauowed. Thisexplainsrhe very sm.ll LAG (29.s, 33.62 and 32 07 respecrivly) in rhesecas. Modovr, in all $.s cases lhe arc ofvbion is snau (ARcv 6.85, 6.8 dd 7.34deg!es rcspectively) buftelorile uihulhs rc taree (DAZ = 18.4,2O.j and 18.l dgreesr$perilely) aioh the sun and tosr widlhs (51,65 ed 5r dc seconds respccrilely). Atllbese lacloB suppolt thc ctaims of lisibitjry md weE anlrcrpatcd above whcn il sassuggesred rhat smalter L,{C valnes oa!, be alosed in such cases. All dF clains defon latirudejusr moE rhm 30 des*s. The shath, values ot Rj"y in conpdison to R.,,h lnes cscs is due ro larsc lalues ofa + A (noe th& 54 deee$),lhe &gle oalheeclipric with venical ed large nesaivc vatu$ of tarnude of Mooo LM (tes than _4.5degrcs) lhar rcduces Rla to hate n nuch shalter lnu /w _ 1.. Alt rhe* rhrobsen.rion de rncgR.nedqrrh rhe Babytonru cr.rerion

Therc is no fudhq c6e ofcesenr visibitity tjl.dong$ otherposilile cases wilh -1.6 < AR."r < O, two de varue of dRq < -1,6. Fbhvery young ccenr' Theet02


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m obddation no.274 (ARm = -1.19 ) drd 416(A&tr- -l_06) i{i$ age 14.8 sd t5.9houE rcsredvely. Despile being lery yorS having FLlively l&aer LAGS (39.j md17.7 ninulcs sperively) in both c&s rhe Moon ws very close ro the Eanh rcsuhinein ldse visua.l dimcler bur lh cltsnl widrhs wcrc sb.ll (10_? &d ll.9 e. s@ndronly). with $all the rclalivc eimurhs (4.6 a l.8 dese, the crcsent eas atnostve'liMlly abov. the su rhat bring it in ihe ided cordnion foi visibiliry bul lh snauelative altitudes (8.5 ed 9.1 deele) nake rhe$ claims highly oplini$ic. Borh th6eobse^ario6ec in d$dgreementwirh Lh BdbyloniM qirenon.

Out of th 14 positive obseru.tioi wnh A&r < O, ihte werc very f.ini cEscenrsi.e. obsrarion n@be6 389 (d&. = {.94), 341 (AR^, = -0.87) ed a55 c0 62) The*crscent werc low in ottitude (?.2, ?.8 and 8.j deg,es respdively) ed hrd snaerongarion (10.9. 13.3 dd 9 deg@s Esp.cively), This hakes these ct6i6s lo be higblyopr'n'strc os well. Two otthcsc {389 and 455) are sko in djeereocot viri Babyloniancnlenon {heec 341 is a mdSiMl6e in Babylonian qirerion (with ARCV + LAC =

Howvr, for all $ee I I 9 c6es when $e crescnt wd daoed ro hdve been senwirh naked eye,lhe conmo. ieatule w6 relarivty hi8h l&uoes {generalty g@&r thqS0 degrres on ilher side of rhe quator) except fo! obFNatbn no. 416. Th cbim 4l6 ntioh latitud 6.5 degres nonh Apan fmm rhh lonely c6e it appe&s lhat fo, A&r < O nis ihpossible to see lh cE*entatpt&es witlllatiruds less fian 30 desEes (borhNodh

Inlhe l19caksofsroup B,lhe frequency for optica y aiddyisibiliry (botb wnnbinocula and retescopcs) ihceses ed one ce cdily gnerau lhal whcn Atqr, tiesbetween -3,58 and O.O theE is s hjgh rDssibility of crescenr visibilhy wilh soe opricalsid for bod lh high latifude as wel s low l.rilude obefres. I hereby jn this wort thes4ondpd of rhe Modilied t\4ujlih Lus RiFne$ LaB $$Rds

t0l


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''if h. nns tizt b.nv.r -3,t otd 0.0 th. posibili9 oJ rhibilit! ollittt crcnetl*ith and *itio optical did ln re6s vnh ihcE6iig tolu6 o!.4R-. fot hlEh.rlatitqda, gq*o ! gedt.r thon 30 d.erc$ notth a4d souk ard ZR*. b.l"g h th.ronq. -3.5 b A0 h. postibility ol nak.tt erz visinlw ako irceqe llowvq, in t'.tong. oI th.G wluzs oJ r'RnIor tMllo lotitad6 th. "oh.d .tz vitibilitr h dlMt

Nxt is the Oroup-C $at mnrains 76 cases wilh AFrv in rhe dnB.0.0 ro L6, InlhB goup rh.r. e 12 naked cye visibihy.as.s (15.8ol.),26 binocuta (14.7%) and 15&l6copic visibilily cs4 (19.?%) e lhar vGibility with both naled cye and with opdcalrid b.com$ ooe probablc. UnfonuMtty rhe dala is h.avily inclined lowads the highhtnude clesanda clrdenarcation for u.aided visibilit, for snalter laritudc ob*Fe6cd nor be made. Srill, the thid pan oftbe [,todilied Muslim Lund Ripeness Law is

''fth. wtu6 oJ/R@ tk 6ztwa 0.0 an.t r.6 ttt. pEribitiiu of,i,n i, oJJintcftsceat teith and tuithout oplicdt k ,trong fo, higlrq latitud.d,.The Croup-D containine a ro|at nlmber of

unaided vhibilitr oflunar cscs forr'i!' > 1.6,Il is^il r'B@ >t.6 ttt. posnbitu, oJ ,bibititr

siorg lot boh towt aMt hiEha totitd.r".

221 c6es has 170 cases ofoprically

of li61 c6c.nt fitttott opti.dt b

2,

Finally, the 3umnarized Moditied Muslim r_umr Ripencs l-awr:ZR-< -3.5 ihpNibL to tee th. n.v luaar c.6c."tlo, a tdrru!., ,ith o,-3.5 < /R@ < A0 ihposibt. to ee the neN crcsce"t with ot *,tthout oplicnt oit!Jo, tnt .r latitutl$. For hiehe, t,titnds therc b n high pnsibiti,r ol gibitittotli6t cE .nt dth oyial

104


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t 0.0 < .1R,. < 1.6 p8tbiti,tr o! ,hltw o! Ji6t M@nt btrh ond withouloptt ol h staq !o, tigt.r tdlituds. Lo.a!|rA N6t tuith opticd! a ! a".!th.a try nelnE lt Nlth haL.d e!. h6 a eood .ranc. o! opticdlr u"tidedne,> 1.6 th. pNtlbnrr oJ' tbi!i!, oifust cnse wtuhout optcat t st ona

Jor bo& Iow.r and highu tatttud.t,,Another way oI lootjng inio ih delaih of Lue Ripene$ hodel is lo look inro

the plots ofaverage ripehess fmcrion md rhe acod ,ipenes tuncion for borh vhibh &dinvisiblc crcsei$ for single talirude, U.forluarcly, rhe dru avaitable dd considercd inlhis work h Esrficlcd in rhc F6e lhat scieltifically re@rded ob*dalions fo, a sinetchnud a nor found lery frqucnrly xcept tor Athens (ladrude 38 dc8,as .orrh) edCape Tou (laritude 13.9 degr4s so!rh). !n panicula, for sn,Ir tatitudes. plac6 cto*IoequarorobseruatiouarelolvcryfFquenuyav.ilable.Anunbrofplaccsecsetecredherc od tbeir d.h pto(cd for Alerage Ripenes Fundion &,, and the Acllal Ripen*Fuiclion R6y for borh rhe Eponedty inlisibte dd the vilibtc oe*mr! Thesc jnctudcplees with laritudes L8N, 33 95, 6.5N dd 38N (FiguB 3.3.6 1o 1.1.9).

Figure 3.j.6 for tadtude 3 L8N snows tbe bsl slcsieh,incs ou, of 6 (83.37") in asrcen wnh $" ,_J:'iff:ln:"11,'#:J:obscdat'ons (6 our of6) e in agrccmnl wilh the ta* toi this larilud. Nexr is Cap.'rown-(hnnde ll.9 S) wirh succcss percenhgc 65% (ll posrlire siehings in sgenenrod, ol 20, the needrvc obseryauons fo, Cap Toq are In agrehenr sirh la\ tor9lJ% c66. This 6 ao owed by ,40r.6 (ladrlde 39N) w,rh 2 oul of3 {66.7910) posirivceghlingsad II o of l8 (61.l%) neSariv sighdnes a8rce snh fic lav. Ior laritude 6.5deg!.es, I our of2 (50%) posilive sightings od 4 oul of6 (66 ?y.) asE. wirh rhe tan.

I05


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I

415't0

0 loo zlo^ actual RF br in\isibte . e.lrrr nr Or ri"tOre

400b

Fia No. 3 3.6: Rjp.@r Fundion for r-ditute 3 L8 dg.c Nonrr

---r;-+:;b.,RF6.'"".b,. .acrld R

Fig. No I 1.7: RiplB Fudid for C.p. Tow4 Sqth Anic!, Lgnbde r3.9 itlr*t06


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f

201al6

10

-111+.-Ig'1.rl'1 t r.tl'::.gE::. fFig. No l.l.8: Ripdess Fudi@ for Lrtibde 6.j d|:gG Non\

20nJ.E""'g:f*,","..,v"ltr"

300 /tooiU,ar nr b. rnti"itrc ]

Fig. No 3.1.9: Ripdcas luDrid for Arhdq lrriMe t8 d@ Non[Loogitudc 21 7 d.grca Easrt01


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All thes. figu6 show ! lend for lhe Aveog. RiFns Funcrion lhal is indicaL


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rd t mwh thidd fd !.d FlriFM@ dir.E. So n n DosibL thlr $e d6acrr eirhm.lld & md @.r.qudtly strdla .o mr b. vi3ibt . Ilis i! p@j$ly thc ce for rheobgMtaom m. 286,2.rd 272, lna @.ctn s ct.itried ro b. sl eirh nrld eycbut the Lury RiFnB te i.di.d6 rh. otsuriB P@ imf.sibt. (My lo, A&,vlluet wh@ the Blbylonia. dilsio. atlos fi.n. So lb. quGrion lriss, for pt.aswith hign6 bdlud.q etipric mch inclin.d rorsds hdia Dd dE qegn binA o.hori@n !id. oI ilE eliptic ah@ld w giv. do.! .Uolle for !c? Shoutd rhe p.I! oltlE Rip.B Fundion {q be low th.. $c, e &cotding io 6glB 3.1.2 to 1.3 5 clos.to tlE rltmr.l .quimr? TrF luF ii ddd, lcr.rt jn lh. dilosio. of thc@nsloinrs ofrhc Lu@ &F|3s tuturion wtEn ir i5 poi.t d od rhr for rcry low AR",v.ru.3 rh. d*qn of bw LAc hn okt agc.td hjc DAz i, E ofl.d ro ben sd. thjsnay b. th. po$ibl l6$n for rt. nrort n sirob@! tik. MDidd !.d Fothdiigh|ntu qploriig .el.ri@ b.tqq ARCV .d DAz for the ft51 vhibitity of tu@ cresn

o,^. --.ig

Fig No 3 4.t: Spncriql Trigonomfiic ih..iFion of 6ndiiid, of nq lu@ de*nt

t09


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In cae ofMoon nonh ofediflh:

DAZ ! ES = EJ J JS lassine I es -elr@teand 'n c4c ofM@n soulhof(tipric:

o,lz - o5 - p1g , 115 = o, "in, *6s - 6ncos p

- -[(DAZ)

{r.4.r)

(1.4.2)

(J4.3)

DAZ siven by (l.a.l) js much sfraller $& rhar eivcn by (1.4.2) for th. r.son tar in(3.4.1) rhe difirEicc of dqliiatioD of rh Sun qd rhe M@n (\ _ 6M) is sna er |hanrr{ in (3.4.2). wnh k4er aRcL (SM') for M, (3.4.2) as compared |o M (1.4.r) &ry j,rar8.r bul &[ or Rd. is sdc. Ahhongh A&,i (Ra, - &ryJ r sme tdr the two ca$s burcrc$cd al M. is older, rhick.. and brighrei mucb soaler rhan for M. Esutts intodill.Enr ofarcs oftishr (ARCL) or older cE$eit with large! ph6e. Thus in case of incase ofM it hst be difijcult to ke the crescqr s conp&.o ro rne c6eofM..Whenler as is v.niet (perpodicuttr to rhc horiz!) DAZ vanishs tudoptihue- aidnioo of I Oo (when Moon is ctosesr !o rhe t dnh) b l2v (,hen lhe M@i isf.nhd frcn rh Ea.rh) ccu_ As and whei tu i; nor vniet lhe ol. DaZ cohs inroplay lnd Oe oplimum condiroN for as ca b retded. For hucn ls8er valus of DAZ4d older dd wjder cescnr may b. visibte wirh s@ er vahca of ARCV or aD. So rheoodels inlolvine ARCV-DAZ rclarions comc into play. h these hodel ARCV is aruncrron ofDAZ so as should aho be a fuction ofDAZj

4,. = 4.. cosa ad ARcy = ftDAz)=aL

Il0


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This n@s rhar for coBtaor LAC (e a) ARCV decE|scom|anr 9, t.AG ed ARcv e di@tty lotDnionat ro echktnude L lrgc DAZ dd taige ARCL m@s mallr ARCVLAC.Duing Elarivety ucnI lioes rhe dploorions of rhe @tist visibiliry oa ndrutu crsc.nt or rh lat visibjtity of rhe old cFscnr b.gd wnh lhe obstuations oadin Alhens and iG viciniry by Schnidt md othes. fteoretical exploradon was iniriaredrurc du ro @leldtui@t E6otu ll)e ey paniculd sbononiqt qusdon(Foth*inshm 1903).h lhc beginning of th. trcnlietb ceirury n was eatized rhat ncrhods of verifyingdates, panicltarly tum, dotes, wcrc nor avrjtable md pspje were @ncehed 6bout the6r6noorc.t ondniom thar govn dE nsted rc visibilny of $e rs lun.r .,esccnr(FodE nsllah, t90l). To evatuare anonohicrl condirions for rhc ktiesl risibitny of:::::: *'*!' "-" " *.ber ofsrudies ,ppeaEd. .rhe slrk ofr. K. Forhe,iishd(1910) hinsell4d that ofE. W. Mau.dq (l9ll) is of vjtal importance. aou rhcsconu'bur'ons *eE bed on fie nald er. obcenarions ot new luna cr*enr nade byAugusl Momeseo, JutiN schhidt sd Friedricb schn@nriburo^ ee .,p,.r*, i" "'* ";; ;;;;:;Til:,#:lTJ:,i:;;

^, . lonn n.ghao claim ro havc sueg.sEd (in his d,crc aprHed in rh roumat ofhilqophx 1 90) lhat in ord.r to catcularc thc rrue date ol phasis on ough ro have alable of te requisne depre$ion of the Sun betow honzon at fic 6oonkt, or of lheaurtude ol the M@n at ttE $tuer for difre o8uld dishnces of the M@r fom rhesun. He h6 .l$ adnitted abou his hl'en* .Monhsn in ,his rcead. Motre.-;;;rJ,'* T,,'1:::;:;llTfmoo. hade h the laGr half of thc nine&flti century by Jutic Schnjd| IriedrjchSchmidr and Morn*n himslr n hblcs menlioned e;liq euld be @nslrEred on thebdh of such obFrystios dut were povided in Monm*n, s chromtogie lt8a3), W_lll


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69{0. Fo(brilgb.(l (t910) hs &!.rd,!od lt !. oh.{trdo.! giviq cilit drB ofot66htio! od ir! r!$lnc Fsvit d Dy tioi.iaod ll. !E!Xind.,nd thaidltrhof t! M@ Ebiis ro lh. Stl! r drc lim of Ms.r (or !ui!.) crrqld.d byFo$dDgh.n tin!.lt Tt4 ob6.'vdo$.r. ,!lrrr!8ed i! rh. T$tc No. 3.a.2 { hsb. rddidool .rlc-rr@.. Tb culs &! .ts Fs&d oo o ARCV-DAZ chti inFi8@ No 3.4.2. Fo{t.iishn (t9t0) .t$ Gidna I ddrrEy r.u. lhar isr.prottuc.d a Tibt.3.4.1 b.lo% rhd sivB rt nisinM lltiidc for lh. !.nibt. @*a|lfot rr|ridl! valu.s ofrcldirc .?hn'h.

H. .!o it"EtoFd . mlrhoni.d Gt rion lo dc'qitc rtc k,E:Mioioue Atrtr* - 12..0 _ 0r.0OrZ, (3.a.a)

]T1,:.* y* *** ni5 cod. d.fn* a r.sioD or sry lroud rh6 poiot of$Ds a $ow! u n6 tgu 3.a.2 b.!oe. If, .l 6. d4 of su!.! rL crBc..l i! .boEd! cu.rc ir ltould tc viliuc orh.eirc iot T[. cruE h crinlit to. Fo6qingh.n,s

T.bL No. 3.t. t: Fortsiryhrn,.""-r,rffi

Fit. No. 3.al Fortdbgh!$! clrl!lt2


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T.ble No. 1.4.2 Forh.dng[d\ Rutc

d66- A@-on ii- '3' 1,1 lo&!

&ia

I|l


40/50

w. d.firc a pa@et . !r , .,visibihy pamerd,, accordiDg ro Forheringhao slrt = (aRcv - t2 + 0.00s21) | tO (1.4.s)

rn lhc T!bl. No. 3.4.2 rhe tsl colhn @nrajd ihe v!l@ of ,. fo, mhoD.eryaxon Epond by Forherineh@ ed rhq rh.lablc is $n d in rh. inccasiag ordcrofyl . ln !i.w of{1.4.4) iflnc lund attitlde is lcss rba. 12,-0-0..OO8Zr lhe qcscentshould nol bc visible_ Alrmatct, in vjw of(1.4.5) if rhe value or ,ts is rcSalive theccsnl shoutd nor be visible, Thc vdB of Elarivc altiiude of the Moon dd irs Gtalivainuths aE rho* calcul.&d by Fothqinghah. Ir is e.s y nobd thal out of 20obsetuations ao. which lhe varueof,r is negatilc two obseryario$ ar posilivc: One. onOci 27, 1859 (obseru.don no.2) od tbe 0lh0 on tlE nohmg ot scpr. 14, t87troDsnar'oo no.4t). Hosrver he hin*lfdnilslhat his marheh.dcl elarion (3.4.4) jsnor $ Etiabtca tE sumary labte.bla. Thc eo is exhibile{r i.lhe [email protected] ir'c esl ofdE posnjvc ob$dadons e aborc FotheringtEn 3 vnibitiry cw..

hsread ofEtying on lh. ..bueb,. rylDxihale natbeharicalnay consider the su6My tabte dala and ltr a quadratc cuneapproxrhation. Following rhis we oblliEd lhe tolosins @.a@nARCV dd DAZ jor rhe shmarJ rabt. r.4.r d,h:/RCv = -0 0ot2g Da z' +o.o7442aDAZ +|.86429 (3.4.6)

ed dennc e dlkrure visrbitiry pamercr ya ba*d orapProx'&afior d folto$:I L\ = URcl/ +a.00929D,42, '.O.O14429DAZ _|t.s642g)I |) (j.4_7)

Seond d88 L@r squd

lt4


41/50

Tabl.No.1.4.3

5


42/50

fhis nbdi6c.tio! i! rn oL by Fo$dnBltn it lpplid to dE obseutio4repsrd by Forh.ingh.n .rd e$lrs e pc5I.d in Tlble m. I 4 I TIE dat! of thelnbL n d$ pr*ftd in FiA No I4.1 !t ir 4ily sa th3t rh l@t sq@e ntins ro as@nd dgE polynohisl h.5 id inprovcd $ylhin8 6ttF tN nore positiveob*N.don have lilln into th mg ofn g{ive vd!$ of vd Th* ee Nmbred 6?(ofo.t 27, 1878) a$d l7 (F.b 20, l87l).

Dunng th sme.6 Mdundg (l9ll) @Nidrcd &olhr b6io data s.i given inthe hbl. .o 3 4 4 b nr $e obwnioml drl..

20 30hvl3bL cro.c.nt3 . villbL cr.8cnb . Forh6rinoham tute

Fig. No 343 Fothringhan s Rule

l16


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25

F-d;:

Fi8 No 144A last sque qu.d6dc poltDnnl 6ncd b $jt d.L iGld! [E fi)llowing cl.rion:

at-v DAz'1 PAzl + tt (r.4 s,lm 20Usiig lbls polynoni.t ed $e @nditid $d the (,B@tt would be visiblc if:

ARC1'> !4: V1+r (34e)a.d lppli.d i to dd. usd by rdtsirghd $e @h3 obrlid 4c prc&tn d in t$lc m3 4.4 !dfig.no 3 44 In lh. tabb t t lh. 'visibility p.mrcl dcfin.d asl

, , =f no", -l DAzl -P4.,\\fl!0o20)) (3 4 r0)II?


44/50

35302520151050

-10. hisible Crecents . Visibt Crsconts

fig. No 1.4.5Th. rrbl. 1.45 3how U[r !E @ft! br.d on Medd,s oaM &. suchinprevcd Md rhs. b dly * obqvdid j.. m 4j d le?l t4, tS7l, thd dqin6eon rhc @rdidoa (3.4 9) IE irbt. j 4.4 and dr fg!rc ooln sr|N rh|r ood.t du roM.udq is nu.h inpotld s ohprld lo rh. Dodet (!l. b ho Ein*hrn.rin.lly in rhis vort bolh the @ddq .trl. Forhdi|ghe .tr rh.r due |o M&i


45/50

E- 6i

6i intTii:i 'Ena

s n3E

ll9


46/50

:i::alF{s_?{";+E:'-.-l {-:

'l

Fia. No l4 7On lh orher han4 figure 3 4 ? ,nd rh. tdlte i. 6ppe.dix tlod ol 196 pGniv! nShd.Ss o,ly lO o6@tirotu d*i.1e |ion

Ihur M.unders etton is mrch b.rid lho dd of Forb.gnantt20


47/50

lh. floft e nol s su@esstul s lh. Babylonia cdlcnon ed Lud RiFnc$ LawcoBider.d in the prelioN two articls 6 fd s lb. nMbq ofdeviorion fon the law inthe ob*tued @seni d @ren.d.

Anolher eforr ofgEar significance rnar h found in tir.6tuE is based on rh wftof Schoch ( I93O) and h known d lndie Melhod give. in the Exp tnlnion ro The trdianAstronohi.at Ephenetis , t^ ttlh nlhod dr ba5ic dala u!d rs givs hee il lablc no3_4.6,

A leasr square quadralic polynoniat fincd b rhis dala yields lhc fo owing EtaionjA nC t = l0 t74l - u Ot 3l)DAZI _ O.@aj DAZI (1.4 r r)

UsinS lhBpotynomiaI od rhc condirion dur rhecre*enr $outd be \6rble ii/ RC v >1e it4i - o.ot lTDAzl_ o.@st DAZ. (t.4.12)

@d applied il to dara *lPc*nled in fis. no.1.j.8. For 6is figuE ,// Bed G

Ihrs hrl( $e Esutrr ohained aE''visibilily peamelef defined sl\ =(ARCV _ lt}.3:,43 _ O.o137DAzl _ o.!n97D,12..)) (3.4.1:l)

,,,,,.i." "" -O 19 our 196 porir.\e sishrinss Dar devidrc rbn l,ll. condi onTIs rill rhe hrer hatfofrhc 2Od c.otury rh. Indiao n.rh@ ws consid.red to be rhe besr.Horck in viw of rhe analysis of Babylonian cnlcrion and the Lund Ripen s LawexploFd 6d prsenred in this voik. dot w.d rhe Indim mehod rs s sle6sful in |.ms

l2l


48/50

of Fsnivo 3igidier dwidirB froo rbd.l. Dring rlE nodsa tis6 rh. Belylonimcniqioo dd Ripenss tuaction h.! no( ben exploEd s rho@8fly 4 is done in rhewott This cxploorion hs lqd ro a sianili@1flnding rhai rh. eient md rh nnievatnE


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Tne ddc oa aliflic wirh lnc horizon plats d impondt olc for th condirioE ofedli.st visibility ofn.w tun( cese.i. For norlhd h.misrhcre if. conjunctio.falls nes aururual equinox this ssle is snall for borh hiddle md high.rlatitudes md fircfoE cBeol is eidlcr cloe to rhe horian or .vcn blow lhehori&o .r rhe lin ofsu*!. Th.EfoG old.r @scenG may *ape si8hli.g.Th acicnt Bobllonie crildion for the elid visibitit, or new tunar ccscenrh6 the hienc$ suee$ pcencge (96.4rr'o) monesr alt rhc nodets consid.Ed inlhis cbaprr for posiljve sighdigs coGidered in lhjs tr!,k. Ho$!ver, $c su@e$perccnlage fof neSalive lidri.gs is nol good enouSh (59.9%). Th$ the o!e6[succ.ss pe rcen rzge ot lhe Babylonie c rir.rion is 75.4%.The ideas retated b lh. Luar Ripeness tuncrion thal deletoped dlrin8 $enedi.val ca ae lhobughly invcsligared. Wirh ftoucm rahnques ofohpulltioB rhis hs Esultcd i o a useful ne6od for det mrnrng lhe day ofrhefi6t sighdlg of @w lumrc&$enr. The onl/ probteh thar sdaced againsr lhisnethod i3 lhc sightings that deviared from rhe nodl i. hisher tatlludes. Thee areolder c,*ents a.d brighler crcscentr h,v lowq Rip.ness11"' ly:. rh s@ess per.nlas. of Lunar RiFnB rarv for posnivesiel incs G 9.8% (bcrb rhan alt hod.ls coNidcEd excepl Babylonitu cril.rion)bul rhal fo regadve sightings n only 57.?% (wose lhan all ofter crneriaconsideEd in this .hapte.). The o!.hlt succe$ p.,cenbg is 72.l./o.

Theadw.rage ofnelhods lhar sre bakd on Etarion beNeen ac ofvision &d!hdve .zinulhs ed thal dc moE lhomu8hly invesric"ted durjng Eod.rn.6 isa6o exptored. Il is fomd lh, rhe Indie Dcrhod b.sed on dr b.sic dala of Scho.his rhe besl aDones! lhe ARCV_DAZ ba*d Dedodsamon3st the npidel Dodels ofte e&ly 20,r cenrur', rh rndid De$od be.luccss peEenlaec tor posnrve srghrings {eo.l"b)rollw.d b) rhe Mlmdets herhod r84?%r and rh. me$od due ro Maundcr

t23


50/50

(54.lYo). Howv.r, in t m ofForhdinghm\ crilfion is rlEchaptet wnh 94.7% fotlMd by/67.8r/'.

th. su.6e pc@nla8e {o. eSatve sighdnesbcn (@ons$ slt m.thods consid@d in th.Momdeas (82%) ed thn rhe Indid ednod

The .uth@ ciry or succ.s of .ach is m.6uEd in rems of nhb.r of cGscnrsiShtirgs wirhour oprical aid (posi,jve sigtrings) rhal ae In aEreehent qnh $ecnl.non.Some aurhorc have sre$ed on lesdng cdt.ria on rhc basis oI nunbcr of 6es whcn rh.cnlqon prcdicts sighrins ed dlc qe*ent i, nor s@n (nee.rv. sghrncs) (Far@hi e1 sl,1999) 6 wctl. Howvq odlcB hav indicated rhar cbaninoe.s wirh insca* in moe .,"","J;; ;;::,;:T:::"::i1",#;1988a). -IheEfoE our eftpbdis is on .xploring @nditions uder ehich cE$enr cm bescn od not on whcrh.r il is &rE y stu or noi e rhat lDNo.s may be 3tlchcd forJudeing ft-eliabiliry ofthe cl.jG of siShtin8e A3 menrioned cadier thee c& be aro ngadve sightings. rfa c terion pedicb sighilg sndmc @rccnr E not 6crldly *en d@s nol sl dll ncm lhat 1h. $irerion is not rcliabic. TIE:::lq condi,roro 8@,ry str6, vis,brny.!cn ,f 'nc sry,{ no, ovft$r as sh,rr br *.nrn rhc nxt chapG. B.tole authcndcn, of a cdledon is ,:jTt ":t, ::_ * *; ; ;;;;".;" *:lj:: :"ffj;j:Trwcn the dimtt illmimied Ihin cecot ad lhe bignrnes ot IM&h sky,

The oveEll success pcrced.ge of rhe In

different methods and formulae

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