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ON I'HE COMPARATIYE STADY OF MATHEMATICAL<br />
MODELS FOR EARLIEST VISIBILITY OF THE<br />
CRESCENT MOON AND THEIR MODIFICATION<br />
Cdndidare:<br />
M U I IA M MA D SITA H ID QU RESH I<br />
<strong>lo</strong><br />
SupeNisor<br />
(D..) N.sidddh Khan<br />
Prcfesso.. Depadmenr ofMarlEma cs<br />
Unive$ity of Karachi<br />
Prcscntcd in prrtiatfulfitme ofrhe requircments for rle deerce<br />
DOCTOR OF PHILOSOPHY<br />
Ar Insliruk ofSpae & ptder&, Aslrophysics<br />
unlversity of Kamchi, Kamchi<br />
September, 2007
CIiII-TIFICA'TE<br />
welcc.pllhe lhcsh as cortunrjns to the rcq0iEd $afttnrd
CERTIFICATE<br />
c. incd thn th. ..ndidltc hr! @mplcr.d $c rh. T undr ny supervision<br />
a-w"*<br />
,/U*sql .',.hs la' t r- aanh,d$<br />
,{' \4{t<br />
Ur/<br />
6{"<br />
iii
tnU'onaneofAltah<br />
The md n 3}.'d/tul ard t to rno8t D3''cficent
Dedicated to my late mother<br />
Anwer Sultana<br />
whose patient struggle in life and passion for mathematics<br />
motivated me to study
I am d.cply graleful to Prolcssor Dr. Nasituddin Kndn, my SrDervisol aor his patience.<br />
inkluble sriNlla(on and counl<strong>lo</strong>ss sugg6tions dsht from the bcSinning rill rhG poirrt<br />
*hen I .n ablc ro complerc lhis $ork.<br />
I sould ole lile <strong>lo</strong> expess Dy CFlilude for tne valuble suSgestions polided by<br />
Prcfessor Dr. Roshid Kadal Ansli of IcdeEl Urdu UniveBily. Karachi_ in rhe bcginnnrs<br />
ol tlis prcjed My spccial $hks for Prcfcssor Dr. Mtrhamn)ad Ayub Khar you$rlzai ol<br />
Urne6nj ofKonchi$no Emincd a nofiaror tor mc nbug<strong>lo</strong>ur rhis \ork I mus,tso<br />
suhnrit nry hunblc tanks <strong>lo</strong> Prolcser Dr. Muhannad ltoys of rjnilcGny of Kuata<br />
l-!mpur. Maliysia, Dr Abdul Haq Suhan ol Univeriry of Sanaa,yenren tor reir kind<br />
cnconmAcment during $is sort. lhanls !E ats due aor Mr MulNflrnrad shotrl Odch<br />
n nleins rhc wcbsllc NN\.icoon)ior! dnd Mr (hitid shorkor nriNgilg<br />
\\\rr!r!]N4lt!!!!.9r1!rr. uorh lhc*<br />
"ohsitc<br />
rcnraincd ! eurcc ot infomllliotr. rcscd'lh<br />
prpeu aDd.laln lbr rhn \orl <strong>lo</strong> hrve b..o posibt€. l_asr b nor rhc leasr; I subnjn m,<br />
stccral rhdnlis llr Dr Shlbbir Ahs.n oi Lahorc Universny of M.meeNnr Scienccs<br />
$hoe {aluablc nrge$iions helped nre in cdnins rhn dissedation.<br />
I sish b cxtcnd my sirrccrc thanks to lhc University of Knrachi. i! pirticutar ns vicc<br />
chrnc€ll . Prolcs$lpr Pecuda Qasinr Rm. <strong>lo</strong>r rhc consislcnl DoRI suppon tor rho<br />
My sife. cbildo. and in-laws enained under <strong>lo</strong>ls of prcsure durine lhis pro<strong>lo</strong>need<br />
periodofny\ort <strong>lo</strong>r shich no gralirude nay bc enough.<br />
MSQ
Abstract<br />
The pmbl.m of delemining ftc day when Uc nev tunar crcs.nl can be 3en ti6t<br />
ar an, site of observarion hs Efrained an opch pobtem since andquirr, The<br />
phenon.non ImniB imFnel for beeioing ! luw mon$ in a puEly ob*rvariomt<br />
lunar calcnd.i. In li. n6! chapler the asrrcnomicat pdamercn relare.i <strong>lo</strong>lheprobtcnr are<br />
!.!ieRtd a<strong>lo</strong>he si$ a briefdcsriplion oasme rules oafitrnb arribut.d ro rhc ancicnr<br />
and lhe nr!ie!.1 nodets lhc afecrs ot geognphical <strong>lo</strong>c.rion on rhc problcr orc aho<br />
dcscribed. A brief Evierv ofc.tendaB lnd.ssociared celestiat cycles h atso pescnled<br />
$nh splri.l chphasis on |hc rulcs enurcjarot for Islamic ob*rarionat lu.arcalcnd,r a<br />
.{rlr as <strong>lo</strong>'r cemu, AD. ln lhe end of the ch.prf Eviov or rhe conlrib[ion of thc<br />
.strcnoncrs of 2orr' cenrurr is pesentcd $al b.8im rvirh thc etoFical toodct or<br />
Solvins fte pnblm ol lhe ti6l visibilny oa lbe n.$. lumr crcsccnr in<strong>lo</strong>lvcs<br />
lenedy colcrlations and thc trsc ofatt ctcFenrary !slrononjic0t rechniqucs. A tovi.w ot<br />
n e tcchniq,es and algorilhms is pEsc .d in rhc *cond chlpr. Sp$i.l emphNis is<br />
Ailrn <strong>lo</strong> r|r dekminarion oa linr of oriunclion of Moon sirh rhe SuD. i c binh of nc*<br />
Moon. rh. risinsand seu'hgol thr Smnnd rhe M@n_ posnionsofrhc sun,nd thc Moor<br />
ar 2n! tnrc on rhc dll or rc day aftc. conjL .rion .t.tc ch.prr cnds rith a brict<br />
dsscrption of the .ompurer prosranr Hitotot dsvebp.d r'.r compftrioN done ir rhis<br />
\ort A n$v posam is deE<strong>lo</strong>ped in ordq <strong>lo</strong> prcdnce the dala Fquied aor rhis \ol[<br />
$ar is Cencrally.or avlitabte troh orher sotlwarc lhat aheadycxrsr<br />
ShniDg $ilh simtl€ Ulbytonian crncrion. fic lhnd chaprr eiptores rhe ancienr<br />
dnd the Drdielal hlrbeoaricat hodets fol<strong>lo</strong>\€d b, rhe dcscriprion ol. narh.natical<br />
trp<strong>lo</strong>Eion of oedieval Mutim As<strong>lo</strong>mn6. Condricat considehions asiaGd<br />
wilh the prcblen are.xp<strong>lo</strong>rcd in nrore dclaitin oder ro evalu0le the Lunlr Ripencss Las<br />
o.0 sMcess, suggc$ed in lhc medieval e6. 'ls<br />
Son nodincado.s to rh€ la$ aru ate<br />
suggesled Thc shoncominss of the ripenc$ lav a,c then di*ussd and lieht is shcd oD<br />
rhe Easns rhat lead ro lhe dcve<strong>lo</strong>pnent of ARCV-DAZ ,.trrion b4d nodels b, the
cdly 2di cenlury astononeB. Thc significanr findin8 of the chaprer h $al borh lhc<br />
simpk Baby<strong>lo</strong>nian c ldion and rhe Lunar RipeNss IEw a!€ morc successfutin tcrms of<br />
thcn co.sistnct Nnh rn. posirile sghrings dords in comparison b fie hodeh<br />
ddl<strong>lo</strong>pcd in th€ fiBr haliof2(]ucenrury Ascroa46Sobsonationsisuscdforleslinslll<br />
modcls in thil {ork. nr.r incldc obseNations collccted since thc latcrhrlaof l9'i cemtry<br />
ln conparison <strong>lo</strong> lh. empnical odcls nrd to prcdicr lhe visibilil) of lhe nc$<br />
lu. cr€*enl iill rhe lid balfot $e l$ndelh c€nlur)- lhe 6od.ls dcle<strong>lo</strong>ped on rhe<br />
b6is of phFical rheoriN of sl! brighhess and extincrion trc cxP<strong>lo</strong>Ed nr thc <strong>lo</strong>l'nh<br />
.[dfl*. i he* models incl"de l[o$ d.ra<strong>lo</strong>trd by Bun aM sclDrftr scpamLl]. Btoin<br />
bed his mddrl on lhe rveige briehhcss of lhc lull Mdon and lhe lriligh slv.<br />
Schacfeis model calculales th. octual lin ins mdgnitud. oaihc skj and $c ntgnnud.<br />
of rhc cftsdrl id tcsls a vkibilitt cldinr on rh. bnsis ol nugnirrd. coniGt and rs<br />
difte{nr ir nrtuLc tio$ al<strong>lo</strong>ther nodeh An.ra bdel deeriprion of Bruin s modcl ihe<br />
scmi-cnrpiricrLnDdel olYil<strong>lo</strong>t is discu$ed in d{(.i1. Nhich isconsiderud to be rlt nDn<br />
Nnrflchcnsi\c !trd aldrcnlic nrodcl. Y{l<strong>lo</strong>p dcducdd his busic d.h lrotrr lr ns<br />
\isibilnJ curvcs. On nre brsis of Schadfcas tcchliqocs qc hd\c rlconsLruorcd Inui s<br />
model rnd pruduccd lcw visibililt crNes Md. ncs scmi-enrpnicll nodel lbr rhe<br />
\isibilnJ oanc$ llnar crcscenl fte delc<strong>lo</strong>pNcnl ofllris n.w nrodel is one oith( Naior<br />
{chicve.nEoflhis$ort.AllIhemodelsarclestddnthssamcdaldselasisusedinrhe<br />
!rc\ious chaplcr lhc nc$ nrodcldc\c<strong>lo</strong>PLd in lhis qorl is found <strong>lo</strong> be rhe bcst aDonssl<br />
rlB n<strong>lo</strong>dcflr dr) dels in rcnrs olils cdnsnlcnct Nilh lh. nunrb.r olposirilc sishrinSs<br />
in lhc d asduscd A compali$n ofsuccess oaqch nodcl h aho disNsed in tlns<br />
Al $e end of $c founn chapler a emtce} is iEmed b *di dle tulhcnticirt of a<br />
claim of siehlins or ne\ cte$e on lhe bsis of a rmi_enpnical modcl and th'<br />
Nasnitudc conlras oodel -fhe sisnific.nce of such a stat€s) hN b€cn highlignred 4s<br />
rherc ae found d nuob.r of authenric ne\r cG$c risibihv claims thlt are nol<br />
coosisienr wnh a *ni-cmpilical nodel tn lhcse ces a semi'copnical nodel des not
al<strong>lo</strong>w visibilil, ofthe cltsm wilnoul oprical aid bu rhe magnitude conkasr is in f.vour<br />
ofvisibility. This happens s aemi€npirical modcl dcs nor bl. i.to considcnrion lhc<br />
elsvation of lhc sne of obseoation abole sed levsl md dr wcarher conditions t-he<br />
maSn'lud. contBl model consided all lhese facto6.<br />
Dcyond $eorctical considerarions a na$ematical model should possess poucr oa<br />
appliabilily. The prine applidion of rhc mathemaical nodels cxp<strong>lo</strong>&{. anatysed an,l<br />
deve<strong>lo</strong>Fd in this $ork is <strong>lo</strong> dermin€ rhc €adiesr visibihy ota ncN lunn cft*cniar 8J<br />
<strong>lo</strong>cation oflhc {orld and <strong>lo</strong> lerify a chim ofcre$on lhibiliry. Ap.n froh $is prirlc<br />
applietion in the fiih chafler rhc sedi+mpi.ical modch arc applicd b devetop a<br />
hchnique for cilculdring the leneti oan$v obsc(€d lun cascent. The phenom..on of<br />
shonentu! otcrcscent lengrh is lino"n for centuries ond drrinS 20ri cenl!try a nrnrbc! of<br />
4!$ns hale b(tn suSgcscd <strong>lo</strong>r $e eme. I<strong>lo</strong>\€rer. our sugg.srd rNhnique is rhc tld<br />
ofilsn rc ihr pdvid. a simplc com brionili@lforcrlculatirglensrho<strong>lo</strong>b*ncd<br />
.Nsccnl. Mooover. lhc *-nlichpincdl n<strong>lo</strong>dch are nlso rfplicd r. lcril} thc act(dl<br />
p@ti$'l obsen.t,onal lun.r cdendar i,r Paki$an for lhe lasr sc\cn )c s. rhc nDdcls<br />
and thc tnctis.d<br />
calcndar aN found to bc in asEemcnr in 95% oi$e ncs mooff duling<br />
thc pciod of srudy. Morivaled by rhis hish rare of consh(cncl rve hrvc pNscntc(t a<br />
''PEdicted Obscn arioml Lunar C.leod.i forPakisan.<br />
A sumntr' ofrhis Nhole.f<strong>lo</strong>n h preseired in lhe lasr ch.prcr. v.nols ihpoarrr<br />
issues aE hiehliSlt d $nh a di$ussion on rh. aurur. scope of Esearch nr the arcd
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-at
CONTENTS<br />
Urdu R.ndenng of Abslracl<br />
2.<br />
Introduction<br />
Ll P@nd.6 for visibiliry olNry Lumr Cr$chr<br />
L2 sienillcmce o| ceogdphical t calion<br />
t.l Tbe Rut.s oaThunb: The Ani.m & fi. Medievrl<br />
1.4 Crl.ndas and CelesdalCycles<br />
L5 condbution oflh€ zod cotury Aslro.on6<br />
Astronomicsl Algorithms & Techniqucs<br />
2<br />
l0<br />
tl<br />
20<br />
24<br />
2.1<br />
2.2<br />
2,t<br />
2,5<br />
2.6<br />
2.1<br />
2.8<br />
2,9<br />
Dyne cs ofMmn ed |he Eanh<br />
Coordindtes of rhe M@n<br />
Nes Lu@ Caenl visibihy Pard.ic'!<br />
Thc Softve Hib<strong>lo</strong>l.qpp<br />
25<br />
l0<br />
4i<br />
5l<br />
55<br />
58<br />
63<br />
3.<br />
Ancicnt, Mcdicv{l & Etrly 20$ Ccntury Modcls<br />
l. | 'ftc ltaby<strong>lo</strong>nian Crilcri!<br />
3.2 Some Sph€ncd Trigonomebc Consi.lcations<br />
l.l Thc Lud Ripehess I,3w md ils Modiicdlion<br />
1.4 Ehpidcal Models of Edly 2d CenNry<br />
1.5 ConposonahdDiscussion<br />
15<br />
76<br />
82<br />
t03<br />
122
Phyaic.l Modala & their Evolution<br />
4.1 Bruid s Physical Model<br />
4.2 Y.l<strong>lo</strong>pl sinslc Pamel.r Mo.lel<br />
4,1 Srch.f.ls Umitiog Magritudc Mo&l<br />
4 4 A Nry Crirrion For vilibility of Nry Luc cEsdt<br />
125<br />
t27<br />
l3<br />
ta6<br />
tt9<br />
t67<br />
5.<br />
Applicrtion<br />
5.1 L€n$h of Cd!c.<br />
5-2 Ob6.ri{tional L|lM C.lctrdrn of Pakkur<br />
t'|2<br />
l7l<br />
l9l<br />
6.<br />
Dircusion<br />
t9E<br />
Th. sonwm Hih<strong>lo</strong>l.cpp 205<br />
fte Anci.nl Med'.valrnd E dy 2of 'enluo<br />
Models 2ll<br />
Th. PhysiolModcls 24!<br />
Lwcd.nd.r2000'20{t 255<br />
Futurccalendar2ooE-2ol0 264<br />
21t
Chapter No. I<br />
INTRODUCTION<br />
Slnce rhe anclent ttmes rh. 6preaEn@ ot n€w tunar cr6cent marked lhe<br />
beginningofa new monlh. With the d€votopment ot ctv zarions org.ntrtng lme tor<br />
e(ended pe ods lnto weeks, months snd teaE, the tlnar phase cyctes tead to tho<br />
evolutio. ot calendaB. lh6refore the pobtem ot dotermining the day of tiFt<br />
sEhtingol n€w crescent moon arlbcted hlman betngr tt Invotves con stderatio n or<br />
a nuhber of astronomlcat es wett as other facto6. On the orher hand th6 probtem<br />
ol obserylng the new tunar cres.6nt at an eaniesr posstbte momenl ts chalenging<br />
for bolh amateuE and professtonat .5tonome6. Th6 .st,onomtcat paEmetets on<br />
whlch thesolution olthlsprobtem ts bas6d are b eflydtscussed in the ftsl .iicte ot<br />
The circudstarces and parameteE assciated with th€ problem of sighting<br />
new runar cresce.t Sreatty vrry wirh th€ v.rytng post o. or obseder on rhe g<strong>lo</strong>be.<br />
The atfecls ot the geographicat tocation on the probtem ls briefiy discussed in rhe<br />
next 6rtlcle. Atempts to determlne crle a tor the delermintngthe flrst ltslbitity ot<br />
new runar c@scenr ai..y ptac€ rpp.ared as €any as the Babytonian ers (Faroohier<br />
al, 1999, Bruin, 197, tty.s, 19946). Signncanr adrances were made by Mustims<br />
and AEbs during medlevgt per<strong>lo</strong>d. A brt€l nccount ot thes6 ancient aid medieval<br />
effons b discu$ed ln thtd.dtcte ofthts chapter.<br />
Srnce anliqolty changing phas6 o, the Moon and a comptete cyct€ or rhese<br />
vrrial<strong>lo</strong>rc has been ured a3 meansot k€€ptnEaccount ofcatendaG. |yas hasgiven<br />
a delailed accounr of the history of ihe sci€nce of lunar crescent vlstblity and the<br />
lslahic Calenda. (rryas, 1994.), Dogger h6 di$ussed lhe history and rhe<br />
deveropn. of calenda6 (oo€get,1992), Fetngold &o€rshowitz hav. presenred a<br />
(omparEllvesludy or morethan rw€nry catendaG or vsrtous (ypes both sncien, a;d
moden (Reingold & Dschowttz, 2OO1). A numbor otorhor rutnors navo contrtbur.d<br />
on rcrai€d tssle. (atr.3hk, 1993, y.s, 1997, Odeh, 2oo4 etc.). A. ,ghr from rhe<br />
beginnlng the c6tenda6 rE 6s$ctated wirh th. cyct€s ot rh€ heavens, rhe eme<br />
.re Evi*ed atongvilh tno a$ociated catendaB in the nen anbb oi thts chapter<br />
wlth speclal emphasts on the tst.mic Lunar catendai<br />
Durlng the mod€rn rims tt wa3 onty in rhe tasl quaner of the 19rh century<br />
lh.l weslem askorom66 stangd erpto,ing the pbblem of ea l6sr sigh trg o, new<br />
lunar the tast antcte is a bnot luruoy 'oscent.<br />
of lrorature that delcribed and<br />
ttweropod vaious c,ire.t. o, models for lhe $tulto. ot probt€m ot d.t.rmining th€<br />
day or lh€ fl6t vlslblltty ot tun.r crescont at any ptac€ on th6 E<strong>lo</strong>be du nE 2olh<br />
Ll<br />
PARAMETERS FOR VISIlIILITY OF NEW LUNAR<br />
CRf,SCDNT<br />
The Moon. the ody natunt sateuirc oflhe Eanh. is eoiog ound lhe [anh h an<br />
orbit thot is a hiShly iftgrlar ellipe Thc i resutariries in lhe pd$ of rhe M@n a€ duc <strong>lo</strong><br />
llrc facl Et its norion is govemed by not onty tbe gEln ional pu oflhe Eanh. bur is<br />
also aaf.clcd by toan) of rhe ncishholiing cetestial objecN (Danby, 1992). \\,nh r<br />
Hadh s varying dhr&ce fron lhc Sun, thc e,Iecioa $e Sun,s gnvnodonat pull v,rics<br />
srbstmti ly snh rheElarile posnionsofthe Eanh and the Moon. Moreov.!,lhe afcors<br />
orthe confisur.lion oflheshole Sotrsrslem (<strong>lo</strong>sirions ofa| lhe major ptMcrs a.d $c<br />
major ar.rcid9 on rhc morion oflhe Moon is oor nesligibte, Thus, accoding ro rhc<br />
''Eplremirid€s Lun,irs Prisiemcs,, popularly tnoh d Chapont,s tuM tho,y ELp,<br />
2000/82 (Chapronl-Touze & Chapronr. t 983, Cluprcnr-Touzc & Chaprcnt, t99l), tor lhe<br />
b.sl possible plecision iD te <strong>lo</strong>nSirude, hdnde md lhe disrance (bclsen lhc Mon and<br />
the Eonh), dere aE equired as nany as 15i227 p€riodic r€rms. On lbe othf hand, for<br />
dcteminhg 6e position ofrhe Sun b r simile d.8E of acc@cy one rqdB 2,425<br />
periodic rems in vicw olrhe ,Variarions S.culair€s des Orbnes plandaiet, rhe Fiench<br />
pr.netary th€ory lnoM as VSOP87 (B&raSnon & Fl!Bo!, t988, Meus' 1998). The
accurat€ d€terni.alio. of lhc posiiiom of the Su dd tl'e Moon is tbe fisl slep towards<br />
exp<strong>lo</strong>rins |he circunsr.ee of rlE vGibihy of ihc !a lud .esnt, Handling sdh a<br />
laqe hnmber of pdiodi. terms mak6 this fiEt slep very ciucial. The lest of the sludy<br />
dep.nds on th. Elativc pcitio$ of lhe Ss and thc M@n ed rhe hod&i at md aner<br />
thc aunsel for a.y place on the &nh.<br />
These theori€s detemio. lhe c.l$i.l ecliptic coodiiates of the sold sy$en<br />
obj*|' fid dc b6ed on sphencal polar c@rdinats. The fig Ll.l sho\s the spheiical<br />
pol.r coordiMle sys&n, The origin of lh. syslen O is €nher lhe ccntrt of lh€ E nh ot<br />
lhat ofthe Sun. The xy'plane k the plane olsliptic, the plane i. shich the E horbi6<br />
iound the Su, The x-dis poin6 in the diEtion ol the vcmal Equimr l which is the<br />
poinr of intersection of the Ecliptic (palh of rhe Eanh arcund rhe Sun) and lhd celesrial<br />
equaror (rvhos plde coimides wnh rhe pl.ne oa rhc lereslrial equa<strong>lo</strong>r). P is $e objsl<br />
llhc Moon in gcocemric sysem or thc Earth in llelioccn(ric sy(em) $hosc stheical<br />
porq @dimrs tp. 0. 0) e p = <strong>lo</strong>4. 0 = 4oP 6d e-zzo.\i\ee P ist\.<br />
prcjccrion ollhe position P ollhe objccl on ro the xy-phne. In the celestial ecliptic<br />
cmdinaB the cetesial <strong>lo</strong>icitudc I = o dd rhe cel.$ial l6(ludeP= 9f - el= zPoP ).<br />
whcn fie hcli@enilic eclipric coordinates ofthe E nh aa elaluared usins VSoP ft.y<br />
aE $en tds<strong>lo</strong>med in<strong>lo</strong> lh€ gc@nric eliplic coodimles of lhe Sun The corjuncdon<br />
or $e Bidh ofNe\ Men occu6 wben }{1 = Is.<br />
Fig. No I I I
SuDpoe ('-.r",/.)!d (.,,r",t) E thc p6isc dist nc6, etipri. tonSnld.<br />
..d Oc hnud. of lhc Moon .nd rt Sltr Bperiv.ty, Ff.Gd ro fi. nsr cquircx ol<br />
rh. d.y of @njucrion. For .ny t@tor on th. Esnh wnh Gcri.l eotdimrcs. I rh.<br />
rcGrdal <strong>lo</strong>nsiud. ot rh. d@ .d t is rh. Lcid.l tlrirud., thc ti6t scD i. de<br />
d.rcmimtion of dc visibihy ot ft. nd t!@ cE$.ni is ro d.r.diF rh. aclut<br />
dyMicrl tin (TD oi rD T., of rlE @njwtion. N.xt, om Eqdc coBidcrine rh.<br />
la.!l tima ofslrdns ofrt. Su {d rlE M@n. IfT, lrd Tm (CoordiDr.d Urivcdt Tin.<br />
TUC) bc lhe rinB of rh. lcat sunsa ud rhc hooEt, ihcn d. i* G$cnr h.y b.<br />
visiblc only ilT, < To, This La4 to lh. p.md.r LAG - Tr - Tr.<br />
Using th. cclipric c@rdiNl6 ot |he sun 6d thc Mon drt.ut.l.d for rh. 1.. o,<br />
dy orho noD.ni of tiD., Oc cqq|oi.t @rdin c! of rh. |m bodiB (a.,4)!nd<br />
(a..t,) cd bc oh.ircd. Vhcdd. rhc nght aeBion k $c djspt@mcnr of$. objer<br />
fbm wdal .qdnox .<strong>lo</strong>ng lh. cctcsli.t cqu|or (in $. q. qudtut s I ). md d js $c<br />
delin.rion or rhe displ.ccftnl of$. obj.cl fbm dc pt.E or ue.qul<strong>lo</strong>r t c.l <strong>lo</strong>ur<br />
Aigl. tf is rh.i oblaircd fon Oc dilT.Ene of thc leEt Sid.c.t Tin. (r_SD and rh€<br />
ishr e.6iotr. This liMuy gies ihc leat horianr.t c@diiat s, dimuft (,r). rhc<br />
displ&.md ol rhe objer fbi rhc diEcdon of lh. Nonh rowrds E 5r ,tong $.<br />
hodenlal in l@l sty .nd rlE ald|ud. (r), rh. hciehr or rh. objer lbov€ hori&o. Aftcr<br />
.djustus for $. cf@rioi ed $. h.ishr of rh. obencr's to..rion .bovc s.. trycl rhc<br />
ropo..nric c@rdiiaks (,4., rr ).nd (,{, .}. ) of lhe M@n and de sun, 6!..riv€ty aE<br />
<strong>lo</strong> almosr all rhc nod.ls fo..&lid hoo!-nddng, rh. mj.nt s rcll .s lh.<br />
mod.m, th. difieE@ ofeinurlrs (D,,tz = l!r. - /r.l. can.d r<strong>lo</strong>dvc szimurh) .id thrt ot<br />
rltiluda (ARcv - ,- -/'., c.llcd e of vision) $ rhoM in rhc fig 1t.2, at th. rin. of<br />
lc.l su@t T, Th. e of vkion is ale l.m.d s F of d.fft$ion. TIF tig, |.t.2 de<br />
show rhc *pa{ion bctwn ln. Sun lnd rh. M@ that is tsM !s 0E .rc of tilnl<br />
abbrtviacd s ARCL .rd is .ls tioM $ .tong$ion.
Ap&1 frh rh. Elni* &idrh, rlE e ofvision !d tn. e oflighq dE dirc.it<br />
<strong>lo</strong>r €nnien visibility of luE d.gnr r€auircr ro aat it|ro @mid.6iion a @dba of<br />
other pm$€rd. On. e.h p@n€tq t ihe As. ofthe Moon (AcE) = Tt To d.ffned<br />
s th€ tine elaps.d sine th€ la.t onjuncli@ rill rh. doc of(rlE sn!.r or tlE any orhd<br />
rclevut tioe) obsvltion Arbrhe inponDt lir.ror is dE Widft of Cres.r (w) thti<br />
dep€Ids on rhc digre of$e Moor. As dE din@ of 6. Moon frm rhe €2trt veies<br />
6on &@.d 0.14 nillion km to a@nd 0.4 nillid tD ln. Fni{i.rr€.d of rhe luw<br />
disc vri6 liom I t e ninutF io | 65 e minur6.<br />
Fig I L2<br />
Ihun if rhc M@n ir dGc$ to in anh.r th. rirc of obs{tion rne c|ts@i<br />
would b€ wid.g lnd th6 bndttrd Winh of tlE qtsfl i! di.6tly prcFr<strong>lo</strong>@t <strong>lo</strong> rhc<br />
Ph.F (P) of th. M@, thll is r tunqio of dE ARCL. r\ @mpla. ltul of all rh.$<br />
T.<br />
2<br />
2.<br />
3<br />
T.<br />
t-<br />
T-. T.
5.<br />
6,<br />
1.<br />
E.<br />
10.<br />
B€dTimeofvhibilily Tr<br />
Age oflhe M@netTr AC€<br />
Arc'ofvision<br />
Relative AziDurh<br />
ARCV<br />
DAZ<br />
Arcorlight(E<strong>lo</strong>!8afion) ARCL<br />
Pbseofcqc.n P<br />
Widlhofcrcs@nl w<br />
A conputarioml stnlegy may be <strong>lo</strong> stad b, deternining $e tim. of Bidh of lhe<br />
New Moon or conjlnctio. or th. Moon *ith the snn. For .ny plee on the s<strong>lo</strong>be<br />
deremirc rh. dm€ ot suner |har fol<strong>lo</strong>s rhe timc of bidn of ncw M@n. For this monenr<br />
conpute thc e€@entic .cliplic coorditulcs of both lte Sun aod $. Moon. Tlmfom<br />
the* coodinat.s to lhe lGal horizonbl c@rdinates. Tbereby coFPute tnc LAC,lhe Age,<br />
ARCL, ARCV. DAZ and the width W. The ddoih of all $6e computations shau bc<br />
t.2<br />
SIGNIFICANCE OF GEOGRAPHICAL LOCATION<br />
Fo! $c pioblem of rhc .dli.st lisibility or the Ncw Cre$enr Moon $e<br />
orientatiob ofthe paths of the Sun and the Moon Eladve <strong>lo</strong> mch othcr dM tlarile to lhe<br />
hoizon h significdnl. They chatrgc scason <strong>lo</strong> seaon as \tll as year <strong>lo</strong> ,car a.d olso<br />
depeod on thc latitudc of lhe place. Due 1o fte axial rctation of lhc lanh cvcrv object in<br />
lhe sly appsr 10 hvel a<strong>lo</strong>ng a cidular palh (The Diuftal Patb) ext nding frcm ealcm<br />
horimn ro thc $'eslm borircn. Thn parh lics in a plae pellcl <strong>lo</strong> th€ Plane of thc<br />
equa<strong>lo</strong>. Tlc obj*ts in ou sty sbos. delimtion is coNLnl (sld and other exd_$lar<br />
sysreh objccq slw.ys remd. in a 6xed smallcircle in our 3ky with Norlh Ccleslial Polc<br />
being ren Pole i.e. rheit diurnal palhs cE no( only 6xed bul lic in plan.s parallel to each<br />
olh€r. Thc Plan. of the orbn of lhc ednh eund lhe Sun h inclined to lhe pla.e of lhe<br />
Equa<strong>lo</strong>r .l s .ngl€ of eosd 21tr 26' 26" lhcefor tne deliMiion ot lhe Sun in ou tkv<br />
vdi.s frch 230 26' 26" $nth ro 230 26' 26 nonh oY€r a y6. Thc declimtion ofrhc<br />
Moon vdi.s frem aoDd 280 S5 south to 280 35' ;onn duting. llMlion penod Thc
durtng ! dry rdlt.' dF pdh of dt Su B itt pdh of dp rroor @ bc 6Bid6.d r<br />
snrll circl. tl[l h.rc th.i poLs tl rlE Nodt cdcdi.l Pol6 i.. rlE dil4l plrlr3 ofrhc<br />
Su .d th. MM do nd lie ii p||G F .lld (o tlE pLG of dE di!.nrt prtht ot d6<br />
Fo. ple3 on E!n[ wirh Lrtu L. 8ral6 (t.1 660 ]a' (Mln q ort) rlE! e d.yr<br />
dunls .sy rd, *ha Son tqn iB b.<strong>lo</strong>w th. hdid dl dry (d ltoe lh. horian dl<br />
da!). Sinilllr forplrc6wirh ld' h.gldrlbo6f 25'drhdorl! dE .I!d.ys<br />
dun.8 .tqy luu rEtl etE d! Moon n bdor rL hdizoi dl &y (or.borc tt&<br />
Ap.n fion th. plr6 chc to dr Gqld bd! tt Su!d thc Mmmy16€l<br />
v6y c<strong>lo</strong>le to tE ho.izon. Fo. pb€. d$. <strong>lo</strong> tlE audor ilE Cclcs'rl Equnor p6s€s<br />
doe <strong>lo</strong> dE anh !d tlElfd! ti. pdnr of ft S{n.!d dE Moon |!mi. hiSh in $c<br />
dq. Bu in ph6 wnh higls hd L. ih. Cd.lrid Equdd i! c<strong>lo</strong>s <strong>lo</strong> th. Hdid o<br />
th.l th. p.tlB of tlE S!..!d tlE M6 my b. d@ d cq bc<strong>lo</strong>w th. horizon. Th.<br />
68|le @. 1.2.1 CFs lh. driw dirrdd of dE cliDlic.td cd.ri.l .quror ia<br />
6np.rid <strong>lo</strong> tlE hdid for. lLe rin hrgl |tin|d. d dr dm of hol o&r !r<br />
!d!d rsd .quimx (6liFic in ttl.) .n rorld aodd .qlircx (6liFi. in pinl) .<br />
N<br />
\l.rsllldib<br />
1l'<br />
Figur.No.1.2I C.l.eid SdE fo. d obffi in hi8lj blnud.
Thi! figue c<strong>lo</strong>rly 3howr th.! for hid ldiod.! vhcr d'c dccli.dion oflhc Sun k<br />
eulh (wirt r in th. mnltfr h.hirph6c bctBa Sumd $lnie rld th€ Auruhtul<br />
Equinox or ba96 &tunn l .quiu !d thc wint6 5lni@) tho p.th of rh. Su and<br />
@nt qu.dly rhd of ih. M@n |!Dir vrry okr!. ro th. hori4n. Ir rn F @nditi@3 th.<br />
w M@. .8q @nju.rion .nlE tlDitr h.<strong>lo</strong>w hqid or Ery c<strong>lo</strong>F ro tlE ho.izon<br />
matina it inposibl. to s da .nq lF d ilre d.y! nm djuncrio. Th. iu.rion<br />
b.con6 @e ifdldng lhb p.n ofdE yd th..L.li.dioo oftlE M@. is oti of0a<br />
&.linrlion of th. Su4 p|nia,brly c<strong>lo</strong>s to AduFnd Equircx (@nd S.prcmbd.nd<br />
tuob6} B.rsq vqol .quitux !d dF SUDE Solricc (t@d ,uc) th.dlipric is<br />
rehtircly high .trd th. p.O3 oflh. S{n .d th. Md e higls 10. Dtina it asid to<br />
s rel.dv.ly tqDgq cGsnB Th. enl.rid n aG!.d gedly for th $urn@<br />
Hdnd<br />
FiglE No. | 2.2: Cclcadd Sph@ &|r D obr.ffi in <strong>lo</strong>w lrlind.<br />
Sihihrly $. fg@ m. 1.2.2 dbE th. ddiE dndl<strong>lo</strong>d ofdc ..liptic !d<br />
@lqlirl equ|tioi ii oonpci<strong>lo</strong>r to thc hodzon for i pLc. with <strong>lo</strong>w hritude ai {@rd<br />
wrul e{ui@r (eliplic in ligln blw) .rd |@rd auMMl .qui@x (alipiic in ligln<br />
pirl). ftir fise d$ 3ho* thd c<strong>lo</strong>* io @tunr.l .quiDx t|| PrlB ofrh. Sun ed rh.<br />
M@n r. rchrivcly c<strong>lo</strong>F ro th. hodan.rd if !h. D.clintti@ ofthc M@. is slh of<br />
rnll ofthc Sun tn. drdhid & nd !6y lpod fd liSltilg of ! tl.riv.ly y@ng d*.nl.<br />
\r'
Tbh is lhe hain rcason bcbind @nsidedng Age of Moon s not a vcry good indicabr for<br />
dt visibilily of cF56t. In sprinS.nd Mm€r vcry t$utrg.ge cltg cd h. sn lnd<br />
dudng th. {ullmn and *inle6 very old clescent may .scape sighling in th€ holthern<br />
hemisphed. Thc situalion is rcveB.d s6odlly for de sud@ hmisphec,<br />
whcn lhe M@n is wsl of fte sun (el6rial <strong>lo</strong>ncnudc or rh. M@n is noc rh&<br />
$ar ofdre Sun) i. our sky it is Old Moon carching up wilh rhe Sun. The Old Cre$enr can<br />
b. en in lhe mominss befoE lhc suriF. An r $e binh of Ncw Mmn (c.leslial<br />
<strong>lo</strong>ngilude of rhe Moon b*omes jusl eratd lhe rbat of lhe sun) rhe Mooi Eehes e6r<br />
oltheSun6dcan no$ b. *njustaftssun*t. How.vn dcpending on ih. d{limrioB<br />
olrhe Sun and the Moon and lhe <strong>lo</strong>calion of$e observer $e new cBcent My et well<br />
bcfoE thc sunst in which cG it is iDFssible <strong>lo</strong> * th€ ncr crsceft Th8 the fisl and<br />
rhc Do$ nnporLnr clilerion aor lhe visihility of the .ev.$ccm Moon is ftar the Moon<br />
*$ an€r $c sunscl. Alknately $c cnbrio. for thc visihilily of thc l5l cd$ent is dDt<br />
rhc Moon riss before lhe suni*.<br />
Whcn thc cresent is very c<strong>lo</strong>se ro th€ sun il rcmains invisiblc dE to the facl tbar<br />
$c ohospherc c<strong>lo</strong>se to lhe sun remains highlt illnminated evcn afi( $e sunser'<br />
Thqefoe fi.rc n6 to be a frininum lhrcshold epaEtion or E<strong>lo</strong>ngarion between fic Sln<br />
,nd rhe Moon b.<strong>lo</strong>N vhich ihe crecent cannol be sen. Thh minimud o. lhrshold<br />
e<strong>lo</strong>neation varies hotuh to oond 6 fie djsrance bctween lhc Earlh and $e Moon keeps<br />
valyine. Vhcnc<strong>lo</strong>ssl (at apoe€e) $e Moon h tround 350000 kn lrom thc Eanh ind !l<br />
lhe fdhat (perig€) il k aound 400000 <strong>lo</strong>. Thus a Ne\ Mon ar apogec mat be *cn<br />
whennscbnsation fod rhc Suh is snallrnd a New Moon at periSce may nol b€ s.cn ar<br />
nuch ld8cr e<strong>lo</strong>ngaton. Thi. is du. ro the fst that c<strong>lo</strong>ser is thc Mmn the lars€r dd<br />
brigbler ir appead in oui sky ahd wheo il k farrher il appeaB saller and less biShl at<br />
rhc se e<strong>lo</strong>ngalion. Thts .nother inpona crir.rion is lhar $e New Mmn cm be *en<br />
for a combinorion ofcenain optinlm valu.s ofe<strong>lo</strong>ng ion and lhe distanccof$e Moon<br />
fom fi. E.nh. thi! conbiMlion Esults into tlE op(nm valu€ of thc widlh orcFv€nl<br />
A cFsenr wi$ small €<strong>lo</strong>ngation $al appea6 lare.t md briSller al apogee may be seen
and a.rc$cnr snh ldgc c<strong>lo</strong>ne.lion $ar apDcar snalle. snd laintr at p(riscc nar nol<br />
RULTS OF THUMB: THE ANCIENT & TtII Mf,Dtt:vAL:<br />
'rhe edli€st rcferehc. for ey rireno. for lhc visibitir! ofncw luntr cresccnt is<br />
annburcd ro rhe BabJ,<strong>lo</strong>nians (Foueringhm, r9r0. Aruin. 1977. Schaete!. lgttsa. yas.<br />
I 994. Fa<strong>lo</strong>ohi er a1. I s99 elc.). Most oflhe cxp<strong>lo</strong>m a&ibute ttE tol<strong>lo</strong>winS rutc of $unb<br />
'Thc ,.n lunar .rescent i s.ek \ en its aAQ<br />
Nons4 <strong>lo</strong>as 18 hihnles h.hi4d th. tns.!<br />
nnrc thdn 21 ho^ un1 L<br />
h na3 bem pointcd oul ihar th. ,crual Dabytonu (lcnon<br />
ephislicarcd ar coopEEd ro lhk siopl. tule (F.r@hi €l ar 1999)_ ft is eilhs ou, trck ot<br />
knostcdse oa lhen era or the nissing hjsbncd rccords $ar h6 rcsrid€d our<br />
compchos'on of lheir effons. Autcnric Ecods of sighljne of new cEscent s young as<br />
5 hours durirs Baby<strong>lo</strong>.id c€ qisls (I6<strong>lo</strong>ohi el ai 1999. Andcrtic & Fih€is. 2006) in<br />
lit 6turc. Sinilady. in modcm time ftsc.nc lagsjnS o.tyjr nrnucs behind $e su.*l<br />
harc bccn *m wi(h aec tcaslhan 20 n6uB. Thc dala ser of46j obsetuations considcar<br />
duflnC lhis $ork (b b. di*lssd in ddait in chaFcd j dd 4) inctldcs jt cascs whcn rhc<br />
crcscd $nh l, C tcss rhan 48 Dinutes w.s cpon€d to h.vc bNn *en wilhour any<br />
oplicalaid. rvhcEas.26 ca*s arc tncE in shich lhe age ofMoon ws lcs than 20 ho!6.<br />
Rece rc-evatuation o|r(ords ofthe sigbdngs o|mscsrs in Babyton and Nineleh aho<br />
sholv rhar cFscenh much younger rhd 20 hou* dd lho* lassrng bchind sunser muclr<br />
l.ss rhln 40minures were sen (Andolic & Fimch.2006),<br />
these rccenr compurarionat ef|ons for the recorded obseBarions oa lhc<br />
Baby<strong>lo</strong>nian e6 ctearly indicarc lhar lhe lule of thmb asociared wnh rhn eD is an ovcr<br />
simplificadon. lr has bcen eccndy ctaihed rh.t Babytonians had rodularcd I hrt,<br />
nalhehalicdt tun& lheory which rhey uscd for prcdicring vartuus paramercBotth tunrr<br />
l0
notion s iounded in thc lue €ph€mdjs ih.y pEpaed (Fat@hi d al, 1999). Ac@rding<br />
<strong>lo</strong> lheF $udies it is pointed out lhal the noonset-su$et lag atone.ould nor have been<br />
used as lhc visibilily dit.rion by rhe Baby<strong>lo</strong>.ids. BabytoniaN sysren had $c tol<strong>lo</strong>wi.S<br />
r<strong>lo</strong>hedrior rL, - moon*r tasrme rin deCrees,,S) - conninr<br />
In vdious $ud,cs lhe vrtue of$e conqant rs dedlce ftum | 7 degFes <strong>lo</strong> anyrhere<br />
&ound 2l d.8R Ao ahd koans.t,ulet td|tin >.t2 nin cr_<br />
lr has been indicared lhar Mrslins. reatization thar rhe Eddh-Moon dhlancc laries during<br />
one comptet€ cyctc ot tlhar ph.ses and lhe hininuh moons€r lag lime considercd b,<br />
^Ebs vaiied fiom 42 ninures for Moon ar perigee.nd 48 ninurcs for Moon at aposee<br />
Thonsh lhis -nrle of thunt. is no4 sophisricated 6 compNd ro $e one<br />
art'buled ro |hc Baby<strong>lo</strong>nims sli jr does nor povidc rhe complere prcrup of rhe cffons ol<br />
Arabs ad Mulin' h $c hedirwt ines exl€trir ur ofAolemaic sy{en ed rhe<br />
spherical risonodeq devetop.d b, Ambs lead to the L!n& Rrp€.es flnclion (bat<br />
depends on <strong>lo</strong>catbritud. ofrheptace ofobrNadon dd lbe c.testiat to,gilude ofthe srn<br />
and lhe Mooh) 6 eartr as lO cenrury AD (Bruin t97)<br />
Baby<strong>lo</strong>nie crirerion h indeed lery sinplc for praclcar purPoses and is supposed<br />
to have ben cmpirical in<br />
no significdt chahgc in rhh rill
Flarively r€c.nr rimes, ho$v€r,lhc atiesr Hildu tdis litc pa,ch Sidn it ( D 5OO)<br />
hints towards thc idponance of$c Vidrh oflhc luM crc$ent (Bruii, 1977) Thus an<br />
elabootc stsreo of calcularioc involvcd in delemining the tinc of €licsr visibilitt<br />
appeds to have deve<strong>lo</strong>ped only abund AD 500. Morcexplicirnentiohsofrhcsederaited<br />
calcularions are <strong>lo</strong>ud ar vuios plaes in the dly Islmic litenle (Brui., 197).<br />
Ohe ol thc ealiesl Muslin Aircnomcr eho deve<strong>lo</strong>ped lnc lables fo. se lidnC<br />
the luna!crescenl s lisibilitywas Yaqub lbn Tdiq (Ke.nedy. 1968).lt has bcen eponcd<br />
in $e litcraluE (Atuin, 1977) that lbn Taliq had r.cogniad the imponance oflhc Width<br />
(w) oflhe crescenr. Thh not only shows rhat ar rhd rine rhe varying dislance ot rhc<br />
Moon had been rcaliud bul it aho mdc ir possible 10 inpov€ upon dc ncs cr*cnl<br />
vnibilily cinerion. Bruin hd Eponed th Al-Ailuni had $e Falizario. oflhe tong ind<br />
dillicult calcularions in<strong>lo</strong>lved in lhe dercnnimlion tor the nev lun,r aesccm vnibitiry<br />
lnd in hG Cr<strong>lo</strong>ro.,os " reconrmcnded rhc work ofMuhanm.d lbn crbn At B,llani<br />
l ltnlhook oJAtto"o,'t trstard to Lario by Nalino, l9|]3).<br />
'lhe sihplc *nclioh for cadiesr vhibiliry oflunarcrescchr evotved sjnce lhe riftes<br />
of B6bt<strong>lo</strong>nids qas passd onb rnc Muslims drcueh Hindus wirh !e,)-' tinle<br />
inprclemenr. Molilated by the euEnic iijuhctions a.d rhe salnlgs of rn prollrr<br />
Muhamnad (PBUH) rh. Fobtem ofearliest sighring of hnar cE*enl \s lhomnehl).<br />
Invesriglredby rlrc earlJ Mrslih aslronomcGotsLhto l0'tccnluryAt).<br />
OD the bash ofreatiarion of the idjponance ofwid$ various Arab asl<strong>lo</strong>nomcrs<br />
conclndcd td <strong>lo</strong>r celics( vjsibilil, ofcrc$.nl thc minimun equabrial $pararion offie<br />
Sun lnd the Moon varics from t00 whcn lhe crescent is widesr up to l2i whcn llr<br />
cNcc.l is n@$cst Such der,il.d calculariotr \.rc wor\cd oul.s earty as 9ri cenrur)<br />
AD by Muslinr asrrononers as$ciated wirh rhc Ab&sid coun of Al,Mahun. Fronl<br />
monesr the* {stbnoncf Al-BaIsli kncw lh,r ihc cnleria that.gc of n@, shoutd b€<br />
moe tnan 24 hou6 (or arc sepdarion belveeo $e Sun sd lhe Mooh) isa good sldtine<br />
poinr but i A only m appnrnarion. H. b€t,.vcd LlEr lhc dcienl6hnomc6 d,d nor<br />
understohd the ph€nonenon conpteretr, According to Bruih, AtDatani,s compuradonal<br />
sork is o very elabo6re sysr€m of mrhehari€t catculalions (Bruin, | 977).<br />
t2
'Ihis work is nor intcndcd <strong>lo</strong> exp<strong>lo</strong>r lhc hisbry oa As<strong>lo</strong>non! rulatn <strong>lo</strong> thc<br />
canie$ sigltingoflhe new lunarcresccnt. Tbe Baby<strong>lo</strong>nia. and the nlcdielalci<strong>lo</strong>ns shall<br />
nol bc dp<strong>lo</strong>cd frem a hi$orical FBpecrivc, wc shall rcstricl our exp<strong>lo</strong>dlion onl) on<br />
rlR codprison of $cse clTons sith $os ofthe moden ons. l<strong>lo</strong>wcvq $c mcdiclal<br />
narh.nalical ide6 shall b. erp<strong>lo</strong>Ed in morc dcoil in chaprc. :1.<br />
I.,1 CALf,NDARS AND THE CELESTIAL CYCLES<br />
Dogeel defines calendar 6 '! syslcm ol orgaizine ,ir4 tbr fi. pupos. of<br />
@<strong>lo</strong>nins im ow cxtend.d p€riodt (Doge( 1992). tl is a schcmc <strong>lo</strong>r keping an<br />
.ccounr ol da!i.'wceks . monrh. . 'yc!6 - ceolunes and "nillennia . Thc bsic<br />
notion bchind I calendar isor8anizingr,la.olroairiconrinuous now rhal isindcp(ndcnr<br />
ol any ehcn. of irs orsaniztion. Ahongst lhe divisions of timc in dayi. w.chs dc<br />
eme aR dtccrly 4sociared snh $c celestial c_aclcs. the D,"r,at, \h. Ann@l ^d<br />
rhc<br />
/,rur. A @nplele daily rcvolurion of lhe s\t. de DildEl norion. is Efertsd b 4 a day.<br />
lechnicallt an apparcnr r,/df.t,' h the inlcpal bcrseen rwo successivc rransiN oflhc<br />
Sun ar any pllcc. Du. <strong>lo</strong> thc morion of rhc Eanh mund $e Sun rhe sky appc.m ro<br />
rc<strong>lo</strong>lvc round tbe Eanh very s<strong>lo</strong>wly (lss rban a dcste€ pn da!) and one contterc<br />
Evolurion of sky i. rhis way is Ffmd <strong>lo</strong> 6 o ,.ar. I {hni€lr! a t opictt y.ar k trc<br />
time inleNal bctweh lwo successivc pseges oflh. Su rluough vcdat eqlinox. one oa<br />
r\e porn6 ol inFMcrion ol rhc celB(dtequarorand rhe actiptic<br />
Each calendar is dilided i.to yea6. yeas inb nonths. non$s in<strong>lo</strong> vNks ed<br />
dals Most oflh€ know calmdm rha( mcn dcviFd bd sevcn da$ in a rvak. tn<br />
diflaent Esions md cras 4 to <strong>lo</strong>dot weks halc aho bccn considercd. ltowclcr rhc<br />
nuobcr of days in a dond hd EDained variabl. in diltcc.i calcndars md wi$in a<br />
paniculd calcndar. Tlr scbeme. il$ce isany. ofdift€!€nl DUmb$ofdays in a monlh of<br />
a calendar is ba$d or rnc typ. ol cal4dar, Most of lhe knom atcndars.E clarsificd<br />
if<strong>lo</strong> dtree major typcs. Solar Calcndm. Sriclly L@r C.tcnda6 and Luni-Solar<br />
Calenda6. A blief description is Evi.$€d in rhe foltowinel<br />
1l
Solsr Cal.nd6 m b4d on the ulrl motion of lhe F-!nh @und the Su..<br />
''ycai in such calendds h ihc "Trcpial Yee, defined abolc. Thon the lcng$ ota<br />
'tropical y€&" (Dogger, 1992) b givcn by:<br />
365.2421396698-0,0000061s359r-7,29xr0_r0Ir +2,64x10'07r (1,4,1)<br />
7 s $e tinc in Julian cenruics sincc fic epoch J2000.0 gilcn by:<br />
T-(JD-245rt4t0!t6525 (1.4.2)<br />
vherc JD is rhe Julid Dat€ which is th. dm€ in .mber ofdavs ctaD*d sine heo n@n<br />
at Genwich on Janua.y l. 4712 in Julid Cal6d{. Cwnlly thc <strong>lo</strong>glh of t@pi€l ycar<br />
n 165.2421898 days or 365 days.5 hos.48 ninurcs ed 45.2 sends. As it is nor a whote<br />
number on yer of a eld ca<strong>lo</strong>dar @rsists of cirher_ 365 days ot t66. tn old Jution<br />
crlendr cvery fourih yc& @.lained 166 days (i.e_ a teap yed) a otbq yff conhincd<br />
165 days In lnc cuftntly us.d Cregoiian calendr! the leap ycd rutc is nodificd. A<br />
century ycd y 0ike 1700, 1800 elc) whict is a <strong>lo</strong>p yer i. Juli& calmdtr bur is nol<br />
complerely divisible by 400 is nor a tep ycar in OEsorian mtcndar. Thus in crer,400<br />
yeas theE ae I 00 l@p y€6 in Jnlitu Cal€rdff wiced in cESori& calqda, rb@ are<br />
only 97. The s€etu od nfuy o1her natMl pbsomcna toltov lnis sotar cyclc. sp.cialty<br />
haBcsing rides, lhe length ofdays,lbc dmes of imir, suser, sunds. etc. I$e JLrtian<br />
calendar B insrilured on Jdu.ry l. 45 Bc by Juli$ caM. wirh rhc h€lp of<br />
Alciandlie stronon€r Sosigm6 &d w r nodificadon ofdc Roman Republidn and<br />
the ancicnr Egyptian calendaB (Michels, 196). WtEcd lhe crcso.ian catendar was<br />
n(csibrcd du. to rhc facl ln.r dqinS oe od . lDlf nil.mia lhe Julim otcndar w6<br />
displaed from the ssonal vsiatioos by s much s l0 days. Thus pope Crcsory X t<br />
corclnukd a conmission in l6rt €qrury AD for lh. csl€ndar Efoms. The dain anrhor da<br />
dE nN sysrem w6 s<strong>lo</strong>noner A<strong>lo</strong>ysiu Litju of Naplcs (Coync el ,t_ 1981. Dull4<br />
1988, Moyer. 1982 ed Micb.k 196). When it ws implcn.nted ofljcialy. rhe ddre<br />
October 4, I t82 (Thuday) in tne Jllie cal.ndar 6 fouosEd by rhc dale Ocrcb$ I 5.
1582 (rdday) ii the Crcgorian alendd. Ther.by aU dare conversioo algoilbms have to<br />
kecp accoul of thh skipping of days in r]te solar calendar. Difercnl counrries, culluics<br />
and Fligiou omhMiiics adaprd to rhis DodilicElior ar diff@.1 !inc. Ir is lheEtorc a<br />
high hrk for hisrorifls ro tccp racl of r]E apprepriare dalcs.<br />
,n every solr qlendr thft @ iwctle mo.lbs. For simple dithmcric Ea$ns<br />
theE colld hdve been seven months of 30 dars and fi ve of I I days (in a nomal ye.r and<br />
six ofl0 days plw 3ix or 3l days in a lcap re&). How;ver in pncrice sincc rh€ ime of<br />
CEek it w6s knoM lhar d€ wi.r6 h.lfofa nodat ,€u h6 l8l days whee6 rhc<br />
summ.rhalf@ ains I84 days. The it&n for ih. sue is lie faler molio. of thc Eznh<br />
rhc Frihelion. thc Eanh c<strong>lo</strong>ssr ro $; Su dar oeun on aound January 4,i. Srdnins<br />
with January cvery ancinale nonlh is of3t days tiu Juty. rcbru.ry h of28 days (o.29<br />
days for a le.p yctu) sd lhe Br G ofto days.&h. The.ltdar€ nonrLs ofjl days<br />
dd l0 dals omintB $.d frcm AuSul ro DMnber<br />
ln a Strictly Lunar cat€ndd, lhar is ba$d on $e luharion pcriod in one year lhcle<br />
arc eilher 354 days or 155 days. Tbe clrcnr ave!.8. of$e lunation period h 29.510589<br />
days or 29 days, 12 hou6. 44 ninui4 dd 2.9 $con.l$(A$nrcmical Atnaiac. 2007).<br />
HoEEr ihis aEase is.hrging, Aaodin8 ro rh. tun5. lltery of Chaprcnl-Tou&, dd<br />
chapont lh.e vdiadons !E eoukd for by the to<strong>lo</strong>wing cxpEssion (Chdpon!<br />
'lbur'ind Chapro , 1988):<br />
20.5305888511 + 0.o0oo0o2t62l, a_ I.o4! IO.<br />
o ! a. Da)5 ( r.4 r)<br />
whce T n givcn b, ( L4.2). An, rrEdicutu phe cycl. my v.ry fmn te nes hv uD to<br />
seven hom. Tnus rhis priod vtuj.s froh !romd 29.2 days ro mor lnan 29.3 days fom<br />
nontn <strong>lo</strong> nonlh. Tnerefore in aU Iutu calend&s rhe numbe! ofdays in a nomh is eithcr<br />
29 or 10. l. Arithmetic Lunar calcndq rheE $ lllenate hontns of29 and 30 davs. tn<br />
obiwrional lue Rt.ndd $.8 6 bc s mey 4 rhE. rcns{ulirc mon$s o, 2,<br />
days qch od d mdy s folr conKud@ momh5 of 30 days e.ch { oys, I 994). In an<br />
l5
Arnhn.tic calcndar ther. are cilh.r 6 monlhs of 29 dals md 6 nonrhs of tO days. (a<br />
notml ycar) or 5 momlB of 29 days ad 7 nonlhs of 30 da's (! lep yd).<br />
Tnerc is no fixed nle for leap ycs in r obseNarional tunar calcndar. rhcrccdn<br />
nol be anr. HoEver in lbe Aritineric L@d cat€tutd our ol lO yea6. cyclc t I yc6 aE<br />
hap tcds (3Js dayt (ly.s 1994. R.igold & Dcrshowirz 20Ol_ Tsybutsty, t9Z9). l.hc<br />
rule is rhal lhe tear nunbf /is a teap yer it<br />
( l4 + ll.))modi0)< lt (t.4A)<br />
Olhwisc the year is nor a teap year. In such an a.ihmelic lunar catcndor alt rhe odd<br />
.umbeEd mon$s conlain 30 days and lhe.vd nhb.ed nonibs conhin 29 days cach.<br />
h re lcip yedr a day is added ro $e Meltih month. in gmeFl. a srrictl, tune oalendar<br />
ldvanc$ by I I days againj lhe solar cat.ndd. Th€refoF rhe $asons aod a orhcr<br />
pncnom€na rhzr dcpcnd on Ihe eld cycle {to ml fol<strong>lo</strong>w a slmcfiy lu,ar catddsr Ttc<br />
monlb numbq 9 (R nad&) in lhe Isl6mic qtods my f.ll in wi.l{. slmmd. aurmn<br />
Thc Luni-Solar @tfldaB d b.sielly lufu bul ro kep r&k of$e *a.ons in<br />
prrce of adding sin8le days in a lqp ye& a whol€ monfi k addcd (inrqcshlion) <strong>lo</strong><br />
fol<strong>lo</strong>wthe $ld! cyclc, The Hebrew dd the Hindn qlendaG fa into thG ctass(Rcideol.t<br />
& De6howir2 2001 . Bushwic[ 1989, ScEIt & Ditshir 1896. at_Bituni r 000. at_Btuni<br />
l0l0).In ce ofHindu calcnds (th.y have bo$ $e $lar tud rhe luni-sotar calcndaB) a<br />
lunarmonrh is imclcalatcdiwhcneve,irnrsinroEconpr€resotarmoilh.tncascotrhc<br />
HcbFw l!ni,$lr cole.dar d addniomt honrh of l0 da)6 rs mrcEararcd betorc rhc<br />
usurl 12" nonlh oflbe yce. The En ofihe dd.its of th6c ca<strong>lo</strong>dd i, morc of$c<br />
soc'al ahd reliSious nlrurc aod is nor in tine with tbeprcsnt*o,k.<br />
Calend.ical cab'narioB for cacn calhdlr ha€ 6eir oM $phkhadons_ bul<br />
bcins phcnoncno<strong>lo</strong>Sicat rtE obsctualionat lunr calend., is mosr chaltensins. A tunnr<br />
month bas <strong>lo</strong> beein wib the actuat sighling of lbe new luns cresce d.d tne conditions<br />
I6
ofits si8hling greatly vary not only <strong>lo</strong>ngiludinally on rhc g<strong>lo</strong>bc bd de[Md o. $e tarnude<br />
ofplaces. ThB an obedarioMl lunar c.lendar may vsry a<strong>lo</strong>ng the se <strong>lo</strong>girude. Apart<br />
iron calenddcal aspects $c pobl.m ofsighting v.ry yolne cGsc€nt is on€ of lhe mosr<br />
excitins fld challenging obseFations for both $c adareur md rhe pma€$ioml<br />
asionofre6 Besides, thc pcdicion of fic vkibilily of a paniculer .erv cEsenr ar a<br />
panicular placc is a <strong>lo</strong>ng and inter.sling cohpuhrional exerci$ Thc sane was clliad<br />
as early as rhe dedieul times by the grelr ostonomcs Al-Khwarizni, Al Batani. At<br />
Farghani etc (Bruin 1977) Mocovcr. t|t prediclion foL nak€d eye obseivarion ha lwo<br />
noE cohplicated hsues under srutiny rh.laresr esearch. Onc oathese is bio<strong>lo</strong>gicat.<br />
lhc abililt ofhunan eye to conrfa$ $e dimly illunrinatcd ciesceht in rbe bnght rsilighl.<br />
The olher aspcct h ol physicrl n4lue ofalfiosph.ric conditions ftar c.n b.dtr aoecr rhe<br />
risibilit! and rc contdsr h this \o.k c l'hdk h morc on thc $lrcnoNical asp..rs of<br />
lhe eadiesl siebdrg olthe new luna!.rr$enl dnd ahosphcric €ondnion !.e onb plnlr<br />
Opinions aro diided d tq thc oriSin of$c.llton olcomlins yer6 in an! tofrr<br />
in rhe ARbian l,cninsula. Accordihg to At ttazsi il $anln as soon as $c cni rcnof<br />
Prcph€l Adan nuliplicrl and spRod aouid rhc rw d (Ro$nlhal l9i2. l_mqi t979) A<br />
calehdar sE originated wncn rhe Himyarires adoptcd one sjtn m epo.h lhal marked rhe<br />
b.ginni.B or the eigns ol Tubbn. Cenerally it is b€licrcd thal lhc pnctice of 12 lunar<br />
nontns <strong>lo</strong> a year cxisbd in pre,lslahic Ar.b c.lend.rs since rhe rime ofconsrruclion of<br />
Kaba b) lhe prophd Abnham a.d conrinued in Is<strong>lo</strong>m (llyos. 1994). The namcs ol$c<br />
months and thcii sequenc€ Mre thc snc as tho* used in lhc cuiicm tslamic lunar<br />
calcndar Ibl<strong>lo</strong>wed by morc rhah on.lifrh of$e roralFopulation ofrhc world.<br />
Fom historic pcspecrire lhc inpon.nce ofrhe lunarcalcnda in lbe pE-lslamic<br />
AEbia $€s the pilsrimage ro Kadba (Haj) rhar talls in $e monrh orzul-hajjah, rhc l2'"<br />
nrnlh oflhe hlmic and prelslamk lunai calendar, Althoush this event was a puelt<br />
religious evenl. il was also inportlDt ior kadc and busincss wnb b$ ol goods<br />
exchanging bands. ll w6 this econonic rcrivil, $ar w6s badly anected as rhe lunarrear<br />
ad!dced llxoush *&ns. PEcufnent ofcop and rhe ovailabilir) or.ac f.,ul anim...<br />
l1
8@rly vdied s6r <strong>lo</strong> s@n, 'fte t€en tha1 $e irbEl.tion s6 inl,ldud in de<br />
Arabim Peninsula w$ this .cooonic drivity rarh.r than dy sronomical t.son.<br />
''QalM" a nativ€ of Mecca is Fpuled to be the fiIsl petson assisned <strong>lo</strong> derrmine thc<br />
daca for thc coming yem pilgrinage and wheth.i $e inmalarion w6 drc or nol<br />
(Hahim, 198, Ahmed, l99l).<br />
Sincc lhe early days of lh. inception of lund cakndd in the Arabim P€ninsula<br />
four nonths including lhe Zil Hljiah wE o6i&r.d scEd md \@ {!e pDhibited<br />
dudnglhesc sacrcd non$s. The custon cari.d over to the post lslamic.ra in lhe lsladic<br />
cultu€. As with the Ronan ql.ndd thc i eEalalion ws abu*d i. ABbia io ordq <strong>lo</strong><br />
change rh. saqed nonfis inro noh{acred moorbs dd vicc-vc& Al thc eme rihe rhe<br />
lunarcalendt!us€d inMadi@rendined in itsoriginal lzmonthsay€arford.<br />
Muslihs fol<strong>lo</strong>{ed rhe calendar ot Mecca in the bceiNling. Rut afier the l,<strong>lo</strong>phcr<br />
Muhonmad migdred to Madin. a<strong>lo</strong>.8 wnh his compdions. Mustims adoFcd lh.<br />
cal.ndar usd in Madina. Aner $. conqu$tofMeccaby propbel Muhlmmad, Muslims<br />
conlinued 1o ue rhe calcnde of Madim bul rhe catcndar ofMecca @ in pmllcl. Wi$<br />
the lar pilgrimag€ ro Meca of Propher Mubmnad in rhe <strong>lo</strong>,r ye& aner miSr.rion ro<br />
Modina (AD 612) fequenlly abused pEctice of inicrcalation \m abolished thbush a<br />
QuEnic injuiction. Th. pEcrie ofnadin8 a lsd nonrh s.ith $e finl siebdne of nerv<br />
clesccnt MooD was iurhenticated by Qurtuic injuncrion and tne gayine of l,rcpher<br />
Muhmad, sith paniculd empbdis on b€Simine dd rhe cnding of lh€ nonth of<br />
fasting md the monrh of lhe pilgdmase.<br />
With adoption of puel! lunar cat.nd& wirh a luar nonrh bcginninS $ith $e<br />
fist siehdns ofrhe ncw IuMr crcscehr lhrcush euMic injuncrion dd lhe eyinSs or<br />
Pophet Muh.mmad |he .rolution of Isl.mic lunar catedar b.gd As for my calendar<br />
one r€qutr.s a staning point of rine Gpoch) or bceinnins of m en, tor couring yea6,<br />
Oe p€ople of MadiM ac h€li.v.d ro hoK u*d d.poch s ene dn6 ! mooin or tm<br />
aftcr Prcphet Muhdnad mignicd to Madina in AD 6t2 (Iys, 1994). However ! hore<br />
wid€ly acepGd lime ofomcial adoptjon of Ht6 s lhe bceinning ofrhc Ist@ic cR n<br />
t8
AD 617 during $c caliph.t of Umd bin Khafiab. Whltcvcr h. thc 1in. of adoplion, lhe<br />
lslMh cal.ndrr. or dc Hiji c.l.nd& s@ wi$ Fnday l6ri ruly 622 AD on Julitu<br />
calends which Moding b didft{c luMr or krdrhicslddd is l" diy ofMuha]m<br />
(1" nonth ofe klmic y.ar) ofth. tsleic I (R.in8old & ftrshowii2, 2001). The<br />
o,ficial dar. or adoprion oflhi!.h &d cal.rdd '@ b l'' Muhtu Il AH (and HijB)<br />
Simpl. 34heh.s or 4rolh odd appoxidlions b$cd on <strong>lo</strong>ng Lm 3vs.gs h.d<br />
b€.n dcvi*d to pcparc <strong>lo</strong>ng r.d cel.nd& in vid ol incrconv.uion ol lslmic drGs<br />
&d tne d!t6 in oth.r lei-$<strong>lo</strong> ad el& olcndr d.r.s.<br />
blanic legd syst h cgllcd Shdia , is lh. sou@ of lsldic tn+kcping ststn.<br />
whil. le8.l pc.pls .rc efeglnd.d by seking sisbne fom $. *i.nlilic kno*l.dge.<br />
Till th. rin. of sking of B.ddad by Hrl.su Xhu in AD 1258, Isluic law had<br />
cvolv.d .ldr guidclires ao. elcndriol oNidcFlio.s (llya 1994). The olcndric.l<br />
sddelircs ftlv.d undcr thc Islmic lN qn bc outlin d a fol<strong>lo</strong>ws:<br />
i)<br />
it<br />
iit<br />
l-.nsth of a lunar month is cnhe. 29 days or l0 days.<br />
L.nA$ ol a lumr tlar is .ith.r 154 dlts or 155 dayr.<br />
Tn rc can be . mdimuh of 4 cor*cutiE monrh. of l0 datr @h or 3<br />
conscutive nomhs of29 &ys each.<br />
E4h new nonth b.sins with fiBt sightns ot ncw lu@ cGccir ovd the<br />
k$em hori@n .n.r ft. l@l ss.i,<br />
v) Ailcmptr rhould bc nad. on 29u of ea.h honrh for sighring of ncw crcscnr.<br />
lf il is nol scn on the 294 duc <strong>lo</strong> any l€en (stmnohic.l condirions or<br />
w.thd connEints) thc noi$ should bc complckd a of l0 days.<br />
l9
visul sighring epo.t must b. suppon€d lhrcugh a wnness rcpon.<br />
The p.rson involvql in rcponing musl be eliable. adull, lruthful, sme<br />
virh good cyesielt !f it i! preved dat the Fen pioviding wihess<br />
puDo*ly nisled the pe6on ousl b. punished.<br />
!iii) Thelisual sighling Epon should nor coniict vilh basic scientific knowledgc.<br />
'lesling ofcvidcne ot sighting on scienlific srcunds<br />
includc 0hcckine oflhu<br />
shapc of crcscen! ils inclinalion. Poshion in sky. altitudc. imc ofoberyalion<br />
dd sky condnions.<br />
Sighting should be caried our in sh o.8ditd way for €dch moflh.<br />
Accunula(ion of emB @orriculally in view ofconsideling ! toonth ro bc ol<br />
io days due <strong>lo</strong> invisibilrty of ih. adcent of i,te 29'" ofoiscutire nonlht<br />
bd ro be avoidcn *hodd thc ns c6c.nl is sighted oo 2Etr ofa nonth. li<br />
such ascs con crions ec nadc b bcgituing of thar mo lr<br />
As the klamic lav and lhe Qudih injunctions depend hcovily on tbe fiNl<br />
sishling of new lunar cr€scent the eE ly hlanic slale plrced speial enphsis on $e<br />
Bsdch i. the lield of Asmhony. CoNcque ly €nomous onlribution! $n mde in<br />
$c deve<strong>lo</strong>pmcnt of $i€n€e of fie @lid visibility of new crcsnt dd pEdicrion<br />
CONTRIBUTION OF 2OIH CENTURY ASTRONOMERS<br />
Tne nod.h dcvc<strong>lo</strong>pmenl of lhe sciencc of erliesl noon_siehinS bc8iN vnh fic<br />
obFMtio.al work oa Scbnid vho Bordcd a ldge nmber of.cw and old cescmB<br />
fon Arhens io th.lal{ h.lfof lhe l9'i c.ntury. On rne basis of rhis obsdarioml dalt<br />
Fo$qinglm ([olhcinskm, l9l0) and Maunder (Mabd.r, l9ll) dwe<strong>lo</strong>ped the<br />
obscralion.l oirdia of ediest vhibility of ns lmd crescent. A similtr work is<br />
20
eponed in rhe Explanation Io the ln.l<strong>lo</strong>h Attranonical EPle,cfir $al is basd oi<br />
Schoch (Sch@b, l9l0). In allrhe* worts for dei€miriry lhe day ofrhe fi*r liribilir) ot<br />
new cE$o! ARCV (arc ofvirion) b shoM to be a sMnd d.8Fe pohomial runcrion<br />
of DAz (claive uinuth). The* edly €f<strong>lo</strong>rc !rc only empnical in natuF N rhe critcria<br />
dc!.<strong>lo</strong>ped de bsed on lluing rhe dau so rh8t mo$ ofthe obs.oaiions arc conshlcnl<br />
TlE* nelhods do not hke into cosideElion lhe vidth ofcE$enl Btuii (Bruii,<br />
1977) co.sidd rh€ ihpodahce oacE*cnt widtn fld his Dod.l desqibes ARCV (in<br />
rems ofannud. of cr.scent above horizon plus tn€ solar depRssion be<strong>lo</strong>v lhe horizln)<br />
os a funcion ofthe crescenr widtn. The Nodel ofBruio. ho\rever, kkes $e {idlh !s a<br />
funcion ofih€ RCV ond DAz md a fircd Ddnh-Mmn dislaftc (in lemsolfixcd * i'<br />
dirnrl'ler ol fte lu.ar disc). Hc sas ale thc fi6t \no consid.rcd phJsical .stkcs<br />
dsociakd *nh $c problen like blightnds ofskt and thar oflhe M@n. Thus tu nrodcl<br />
due <strong>lo</strong> Bruin is the lirsl drolelical node<strong>lo</strong>fr|t nodern timcs Using the nrodels tur thc<br />
briShhess oalhe lull M@n as a fuhcrion of altilude (Befrporad, 190,r) lnd $c s[t<br />
brighhe$ duri.g lNililhi (Kooncn ct al., 1952, Si€denbpl 1940) Bruin delc<strong>lo</strong>p lhc<br />
visibililt cu^6 r€lnrin! the aldudc olcrc*€d sirh the slar dcpGsion. A<strong>lo</strong>nS lhc$<br />
ctrrvcs the bdehb€ss ofcrcscem (modellcd on $e basis of lirll Moo.) is al leasl ds much<br />
os tlhl ofrhe twiliShl sky. He aho delc<strong>lo</strong>pcd cuNes sholvnu rel{lDn berwccn ARCV<br />
and sl.t depesion rhll ldd prorcd to bc crucial for lunhcr nodellihg ol canics'<br />
Yal<strong>lo</strong>p has edne lbr funho improvemenr ond consides cece *idth as a<br />
funclion ofARCV. DAZ and lhe acllal scmi-dianeter ol lhe Moon (rhar depends on Lhc<br />
Eanh-M@n disrucc) al $e time ofob6endion (Yal<strong>lo</strong>p. 1998). \vilh de!€<strong>lo</strong>pinS a model<br />
<strong>lo</strong>r lhe besl time of yisibiht YaUop s nod.l al$ onsideF ARCV 6 a ihi.d d.gtce<br />
polynomial tunction of $e actul cresc.nt oidrh ar the ben ime of visibihJ. Such<br />
polynofrial is oblain.d by applying leasr square appbxination on a basic data sel This<br />
blsic data s€l *qs d.duced bt Yal<strong>lo</strong>p fom lhe limiling lbibilil, .uvs of bnrin b!<br />
slctins ARCV for a Sircn sidtn from tlr ninin@ oh lhe componding limilin8<br />
2\
visibility curve. So fd th. mod€l due to Y.l<strong>lo</strong>p b lhe morl oulhfltic and dep€ndable<br />
slronohi€l cri|€na for cxper.d ali.sl visibilily ol new l$m ccenl. The modct due<br />
10 Yal<strong>lo</strong>p isa smi-empirical model based on $e lheor€dcal considedions of Bruin and<br />
.ftiteriononthebasisofabusicd.tadeduccdfonBrui. svisibilitycurvcs.<br />
Tnc mosr extctuive r.atm6t of rhe physicat dp
. Th. d.pend.nc. ofth€ dlendsB on th. cyclic moiion ofth. Moon sld $e Suh.<br />
. txB c.l.ndar of $. Mulih! .spei.lly a lh.i. b.li.rts lticrty dphais. u<br />
tltliarsidtingorrhetw<strong>lo</strong>Ncr$.a fois.niosaldcldilglun&ndrha<br />
. A! rccou ofrh€ ei<strong>lo</strong>ns ofrhe rsrorcmd oflhe oodc6 rim.s |o rd.lr.ss the<br />
prcbhm ofderemining$e fiisl day ofsishdnsofnew Iund c'.eenr<br />
Itr lh. b&ksound of tlE$ cfon! w tFrc qp<strong>lo</strong>r€d all lh. old ald thc modm<br />
mrhodi Mthcm.dol dDdel or crirdion dEr r.quiG ro be $tisfi.d for 0E lirsr vGibility<br />
ot ihc !w lu{ cEsdr, Duing lhi! qp<strong>lo</strong>Gtion rn E$ltr of lh.$ nod.ls N<br />
comFcd lnd nodifietios baw b..n 3uSgcrt d $,lEE fler posibl.. Funhd ft. mo$<br />
authentic of the hodeh hav€ b€en uscd to .xo<strong>lo</strong>d fie ob€crvationil lunar cal.hdar<br />
fouowEd in Pakhhn. A.sed o. the rcsulF ofthese exD<strong>lo</strong>Hions a luturc obervrtioml<br />
lus calcndd for Patislm is anDut d,<br />
23
Chapter No, 2<br />
ASTRONOMICAL ALGORITHMS &<br />
TECHNIQUES<br />
For lhe determlnat<strong>lo</strong>n ol th€ prsiso <strong>lo</strong>cal<strong>lo</strong>n ol the obJsts in th6 Solar<br />
SFtem, panbulany the Sun .nd rh6 Moon, ih6 Fr€nch planetary theory VSOPaT,<br />
(sretagnon & Francou, 1994) and th. lunrr rh€ory ELP-2000 (Chapront-Tou# .nd<br />
Chapron!1943, €91) rr. well sulted. A number of sotlwaE hav6 ben deve<strong>lo</strong>p€d<br />
tor lho shulat<strong>lo</strong>n of cel€stlal phenomona based on the$ th€o.16 and similar otner<br />
works.In the cunanl studylhe same lh€ori.s havs b6.n u*d <strong>lo</strong> <strong>lo</strong>l<strong>lo</strong>wthe pGit<strong>lo</strong>ns<br />
otthesun and rhe Moon.<br />
Moreoler, to conven the theorles Into computat<strong>lo</strong>nal iechnlques,<br />
malhematlcal lechnlques, tools and dlgo thms nave been dt<strong>lo</strong>ted. Moch ot the<br />
computai<strong>lo</strong>n.l work is ba*d on the algorithms devo<strong>lo</strong>ped by Me€us (Me€us,199a)<br />
but a subslantlal ahount or work on computat<strong>lo</strong>nal aEonthms has been done<br />
Ind€pendsnlly. For a ihorolgh und.Franding ol lhE compul.t<strong>lo</strong>n.l toob the<br />
problem of llme ls exp<strong>lo</strong>r6d rnd discused in d6tatl. ln this 6Ebrd the @nblbution<br />
of a numbd of authors has beon sludled in as much detail as ls required (Aokl et.<br />
al., 19aa. Sorkoskl, 1944, C<strong>lo</strong>men@ 194a, 1957, de ,ag6€, and ,lappel (Eds.),<br />
197l- Dl.k, 2ooo, E$€. and Parry, 1995, E3*n et, al., 195a, Gurnot dnd<br />
S6ldelmann, 1944, Markowiu er. Er. 1954, Mul<strong>lo</strong>r 6nd rappel, 1977, t9unk and<br />
lrlacoonald, 1975, Nel$n et, al., 2001, Newcomb, 1495, Sadler (Ed.), 1960,<br />
seldehann rnd Fukqshlmai 1992, Spencer, 1954, Stephenson and Moi$n, 1944,<br />
1995, Sleph6n$n, 1997, W€lls, 1963, otc.).
th. output ot ihesa €ftort! b a .omplter progr.m for .natysts ot ftst<br />
vblblllly ol luh.r cr€scent named Httatol wrttten In C.t.ngs.ge dt$u$ed 6t the end<br />
2.I INTRODUCTION<br />
For rhc derernination of lhc visibilily mnditions otNew lund Cre$€hl (or the<br />
old.st lutur cresccn0 over a tocal hori2on, lhe 6Bt rask h ro deremine rhe Unive6al<br />
Tine (uT) and dai. ofrbe scocenhic cobjucrion of thc Moo. &d rhe sb or thc Binh<br />
of New Mmn. In ils morion tuDnd rhe Eanh rhe Mmn tralcls doud 12 deg@s frcn<br />
rv€sr ro c6t cvery day sd iakes ovr rhe Sun in doud every 29.5 dars on thc avcm8c.<br />
When de Moon is vcry c<strong>lo</strong>* <strong>lo</strong> ve$ ot (he Sun it aprears betore the sunrie ed vhen il<br />
N elst oilhe Sui it appcmsjusr anq rhc sunset. The lunar crcscenr is vcly rarel) lkibtc<br />
on rbedayofthe conjLncton. Dqjon Linn(Danjon 1932. t9l6) has b.cn i.$rpreted as<br />
a limn on Ihe hininuo c<strong>lo</strong>ngalion ofrhe visible lu@ cE*cnr Accordine <strong>lo</strong> rhis lihir<br />
the ruMr ccsc€nr is nor visibb if lhc e<strong>lo</strong>nSadon is te$ lhd deSEes (Do88er &<br />
Sch.efer 194. Schaefe! 1991, yal<strong>lo</strong>p 1998). Thc hdimM e<strong>lo</strong>nSadon ot $e Moon ar<br />
lhe dtoe of$e Eeocentdc conjuncrion is same as the inctinarion otrhe lunar orbjt fbn<br />
rhe p<strong>lo</strong>. of ectipric (50 9). When rhe Moon lakes orr rh€ Sun ar ir ndimun<br />
e<strong>lo</strong>ngation $c ninimun time il lal€s b molc foo bein8 / fmm rhe $n (on the .6rem<br />
side) ro be o again {on &e w6om side) is eoud th,ee quans ofa day. Th6 it is<br />
theorcdcally possible lhrl rhe qesent is t6l sen on $e dly of conjunction or is tsr<br />
seen on the day ofconjunction. TheoEricalty n is aho possibte rbar if$e crescenr n hsl<br />
seno. the day ofconjuncrionad rhen rhe new cEscenr 6 seen on ln. da, iller, orthc<br />
crekenr rs rasr srn on rbe day b€to,e rhe conjurcrion .nd !h. rh. new c@dl n *en<br />
on the day oflhe conjunction. In th6e cM rhe cre$e r.mDs hv$ible for.oneahalf'<br />
day. Howevcr non€ of the$ lheoletical possibihia are rcatird in practice too<br />
freque.dy. Mosrty rhe cEscent Enains inlisible for .two-sd.a-holt, days at least,<br />
These condnions dcpehd on tc obse(ets t@a on.<br />
25
O@ thc rim. of thc g.cdric biil of th. Ncw MFn is d.lmin d, dE rcxl<br />
r.3l( ir b d.cdic dE <strong>lo</strong>cd circuhlt .c6 oflhc Su ud dE M@n tt dE tirc ofsu5.l<br />
on lhc day of rh. conjwrid or r &, .ncr thc djulction (or d lh. lirc of swi$ on<br />
lh. dly ol conjncdon or thc dly b.foE). Fd this i'lk ooc mEt d.r.miE th. l@l<br />
tim6 of thc ru$t a.d lhc noon3ct In tr. mmins in otdcr to b. vkiblc th. Su rhould<br />
tag b.hind thc Moon in odd rh|lth. @s i3 visiblc .d in thc dcnine! thc M@n<br />
sholld bc hgAitrg b.hind rh. Su. Clsiotly thc LAC of $c M@n h!5 Em.in d on<br />
ih!{ndr cosidcElion for lhc adi.sr vbibility of$. w lua ct6..nl. SiM thc<br />
rim6 of th. Baby<strong>lo</strong>ni.ns thtuu8h ni.tdl6 !96 .rd ill tlE 20d e uy it hs b.qr<br />
coNid.Ed I deisiv. aerd. Blby<strong>lo</strong>ni@ @Nidc!!d nininm LAG rcquiGd for tlF<br />
visibihy or new lund ca*nt to b. 48 miNt6 wh.@ thc MBlidAFbs co.sid.td il<br />
<strong>lo</strong> b. 42 to 48 minut s dcp.ndinS on th. E ih-M@n disl'ne. In Dod.m liis tlDogh<br />
thc lisibihy condnioB hlvc b.cn eFncd nainly d@ <strong>lo</strong> .ll kinds ot lnifi.ill<br />
pollurios dc ncw luM cGs6t hls bc.n rcpon d <strong>lo</strong> b. tishr.d vlEn ia LAC B<br />
mwh ls ths 42 hinut€s.<br />
TIE ddcmin.tion ot $c l@l riGs or rh. suEr ..d $. n@nsr, though statd<br />
!o b. s@nd t.st in *qu.na, i! dcp.nd.nt on rhc dct mimtion ol dt PEci*<br />
toldenkic @diMl* of th. Su ed thc M@n- h is rh.tfoE imFnliv. lhat b.foE rh.<br />
&t frinatioi of rhc LAG orc nusr find th. E.@ ric @rdiMl6 ed lh.n lhe<br />
bpo@ntdc c@rdinarcs foi $c l@don on lh. g<strong>lo</strong>bc fom sh.E ob6cNalion it to bc<br />
m&. of th.* bodic!. Th.* e dcrivcd fren th. two $.oti.s, dF VSOPET .rd th.<br />
ELP-2000 (di*.us.d lakr in $. ch!pr.o. As both rh* thod.s &*db. lh.luM.nd<br />
$. iol& c@dinatcs 6 .xplicit tin sid, mlkiig olt thc prilc "lim. usm.nt' is<br />
*nlid tor lh. appUcadon of tI6c fodul8. 'It. "ritu" co*idcrcd in thc* 0E@.s<br />
ed rh. orh.r thcod6, is . rim ird.Dcnd.nr oflh. dillioB of thc E nl ed ir g.tudlly<br />
Lfr.d s "Drnmial Timc . How.r dc limC' speificd bt ou clclc b ba*d on thc<br />
!sn8. norion of rhe Esnli .nd t. Sui 6d is lm.d .s thc "Md Sol& Tim." Th.<br />
titu cosi&cd in ihc applidion of dE rh.oncs b ihc B.lyc.ftic DyMic.l Tin.<br />
(tBD) or dE TcGrrirl Tirc (II) which ir 6sin d wirh th. C6.dl Th@ry of<br />
RGl ivily (Ch.po -To@a & Ch!!ro4 l9l, Ncl$n.t..1,2001, Guiml &<br />
26
Seid.lm@, 1988). The TT is d.fined in FlatioD witb the "lnt matioDl A<strong>lo</strong>nic Tine"<br />
T"I=TAI+]2F.IE4<br />
c.l.t)<br />
Btce TAI it dgul.t€d &cording ro atomic rimc, In TAI $c b6ic uoir of rime is thc SI<br />
sond {defined by Bueu Inremarional des Poids.l M.su6, BIPM, in t96, a<br />
dudlion of9,192,611,770 periods of rodiarions cooesponding ro lh. tmsition b€rween<br />
uo htFrfinc levcls ofd.gbund slat oflneCesinm lll abm G.t tenerat.,2OOt)).<br />
A day on rhk scale is 86400 SI seonds <strong>lo</strong>ng {Astonomicat Atndrc, 200). On rbe<br />
other hand rhe clet dm." is thc Univ.Ml Tin€ (Ul denned wirb rcf.r.nc€ ro nem<br />
sun and Nocialed vi1h thc Crccnwicn Md Sid@t Tim. (GMST). UT is detircd as<br />
$e hour angle ofrhe Med Sun ar cEenwicb ptus 12i6. Due ro rhe in€euta ries in the<br />
rctations of th€ Eanh ihcr N di$repancies b.reen the lwo rimes. rhe TT and rhc UT.<br />
This dillcrcrce is ruferred ro as $c delht (At):<br />
AI=TT-UT<br />
(2.t.2)<br />
Th.rcaore wh.nevcr w wanl b dd@irc th€ position of $. SM and lhe M@.<br />
lbr a panicular lime on ou! c<strong>lo</strong>cks w€ have to fomutate rhe rine argudenr using rhese<br />
cons'dcEiions othcnvi* fie c<strong>lo</strong>ck dn. of the phcnomena shaU nor be appropriat€.<br />
Finally. rhc lime arSunenl in drc rhcorics requi,es lhe delemimtioh otrh€ Jutim Dare of<br />
the UT in question. TIE Julim Dale is th. sys!€h ofconrinuols limc sle lhar begins o.<br />
Noon or crcen*i.h Jduary t, y6 -4t2 (cdlcd lhc cpoch oftne..r!lian dde,) ln rhis<br />
r,m€ scare tlre moment described by a date (CEeorie or Julie) ond rine (UT) is<br />
considercd as rhe "nmb.r of dayJ,, d.nor.
ine argunent md the explicii tin€ eries fomulas of the ELP ed lbe VSOP th.<br />
c@rdicrcs of lhe M@n ad rhe Sun m calcrl.t d for fte s.me i6i'nl of rhe day or th.<br />
-<br />
d.y aAq @njudion- In s. @ spheric.l pol& @ordimtes a th€ gtucsrdc disr.nce, X"<br />
th. ecliptic <strong>lo</strong>ngitud. and 9. the dcliplic ladtude, refeftd b as th. Ediprh Coordimles.<br />
Tn* c@rdinals i. lh ft sqt dsirely ro obrain rh. im€s of th. Su$.i &d dE<br />
Mooisr (or rhos of $e sun.ie ad noosi*) for 1hc day in qu6rion.<br />
TIE lune ca* (or rhc cE*enrs of M€rcury &d venls) k fomcd by $e<br />
region of $e lbn suface tow,rds fie Sun tnar fa.lls belwn rhe two plancs rhrcu8} the<br />
e E of rh. Moon, one perp.nd'.ulr ro rhe tine ol vi.w of fie obFn.. dd thc o$.r<br />
FrrEtrdicule to rhe di@rjotr of rhe Sun. Thc nrio of rh. ea of lhis crcw md lhc<br />
total dea of th€ Lund disc is called lhe.phs.. ofihc Moon, The phse ot th€ Moon is<br />
di@tty r€laLd ro rh. seperion bcrwen rne Sun and rhe M@n or .<strong>lo</strong>rgation.<br />
The Astronohical Alm$ac pubtished amually shtes lhar $e dew luna! cr€scenr<br />
is sencElly nor visibtc who irs phe is te$ lhan t% (Askomniqt Almde, 2007).<br />
- This ha prcv.d ro b. nisl@dine in vicw ofihe fac| rhar the brightn€$ oflhe csenr<br />
can sr.auy vary rbr the ehe vatue ofthe phas owins <strong>lo</strong> the varyin8 dislanc. ofrhe<br />
Mmr froh $. Eanh. The Elrri-Mmn disunft Eics foo t5O $o@d kitomcrrs <strong>lo</strong><br />
400 lholsdd ti<strong>lo</strong>mer$. Tnus when dosl <strong>lo</strong> ihe Earlh the luntr 6cshl hay be<br />
vGibl. wilh ns pnsc much l€s lhan r % and in cae of fanhd ir nsy not b€ vkible evdn<br />
wilh ph& g@ler thd l%. D@ ro rhis varying disr.rce rh. siz of rhe tuu dis i. fer<br />
cha.ges. C<strong>lo</strong>ser the Moon ih€ disc dppees larger. Th. Muslims had nodccd rhh vdhdon<br />
ii th€ sizc of rhe luie disc hund I OOO ye6 aeo. tn rbe Modm rimes il was nol before<br />
Bruin lhar lhc imporrace of 1he actul visibt. widrh of lh. lutu cllsqr M @liz4<br />
Uhinately n w6 yaltop who ued the widlh of luo{ cr*cent in his ohe_psmei€r<br />
nodel of tu4 cre$.nr vjsibitity etadng n b the ahnnd. of rh€ c@.nr on rh. <strong>lo</strong>cal<br />
orce rh€ sc@orric c@rdinar6 of thc sd md thc M@n e catcutar.d lh.<br />
.fieIs oi Refracrion, Abcration hd fic pddta @ @lculat.d for llc.@odinaies of<br />
2a
oth rhe Sun and thc Moon. Thse cor€ct d eclipiic coordi.ates of fie Sun and the M@n<br />
@ lhen arNtomcd into Equtodal @dinal6 a, the RiSht A*nsion 4d 6, lhe<br />
Delination. In order <strong>lo</strong> g.r th. tical Hodental coo.dimt€si Ahitude and Azimuth, of<br />
ihe Su ed the Moon, rhe ob*frers rcft$rial cooidinale dd the t cal SidftalTime<br />
(dcfin€d as lhe L@al Hou AnSle oflh. Equinox) aE rcquiEd. A simple alSorilhn lcrds<br />
ro fie Greensich Mean Sidereal Time (GMST).r lhe Oi'Unilesal Tinre for any given<br />
dsle. AddinS lhe <strong>lo</strong>cal <strong>lo</strong>ngilude 1o rbis GMST erv.s<br />
1he r,cal Md Sidereal Tihe for<br />
0i' Uhiv.sal Time for any eiv.n dale is oblained. finally to gct $€ Local SidcealTine<br />
for my nonent of the day day be obtain€d keepids in mind the f6ler pace of $e<br />
Oncc lhe tical SideEolTihe ofany moncnl is <strong>lo</strong>oM the Hour Anele oflhc any<br />
obj*t is oblained. The <strong>lo</strong>cal Euatorial coordiiarcs s, th. Hou Anele ed 6. the<br />
declinalion,lead to the Altnudc (heiehr obove lh€ holizon) dd Azidu$. n@ poiilsol<br />
tine @ imporrant <strong>lo</strong>r lhis slldy. when m objecl is al lhe <strong>lo</strong>cal meidian (i.e. TEnsi().<br />
when rhc objecl rbes ind when an object srs.<br />
Thc time oa tEnsil may be calcularcd using rhe Hour Angle to be eo, or rhe<br />
co.siderinS rhe tical Sidereallime 1o b.lhe Righr Ascension offte objccr. The *timate<br />
olthispointoflimc can b€ inpoled uing an ireraive pccs (hal inmlves Eadjustihg<br />
the lime arsumena'<strong>lo</strong> be lhis approximatc tim€ and rccalcuhting the coordinates offie<br />
objd at thisdme argumehr.<br />
ConsiderinS Hour Angle t io be 9Oo or 6i" approxidare rihe of rhe rkins<br />
(n€earive t, o. the se(ins (posnive 14 ae obraincd. R*rlculatins $e rimc dsunenrs<br />
tor approxidale limes of thcsc eve s $c coodi.Eres ofrhe object de deremined again<br />
ad the b.lter approximtion of$e inct oflh€s denrs ac ob!.ined. This giv.s the<br />
risins a.d the setins orthe cenres ol$e objers (rhe sun and the Moon) thar can b€<br />
adjnsred |o s€t $e actual isins (the first app€arance of the weslem limb of the objec,<br />
.nd acrual s.tti.s (homen! of dcappe&ance ofrh. cded linb of the objsrl.<br />
29
Onc. lhe lincs of fting edor selting ofbodr fie Sun 6d rh€ Moon are oblained<br />
on€ d work our att th. p.mete6 of fic import,nce for lhc atrsl (ot la0 visibilitv ot<br />
rhe n€v (or old) lutr crc$.nr.<br />
2.2 DYNAMICS OF THE MOON AND THE EARTH<br />
T!. dcv€<strong>lo</strong>pn€nt of nodem dy.di4l lhoics for $e sole sv$em beg& wnn<br />
$e discov€ry of Laws of Pleel,rf Motion by Jobannes Kepler in the 16('centurv At<br />
aboul rhe sm.lin€ haac N€Mo. cme !p wilh his tt$ of Motion dd the Univeisal<br />
Law of Cdvi|llion. Whar fouowd is a <strong>lo</strong>ng histo.y ol deve<strong>lo</strong>pool ol nathemtiql<br />
techniques lead i ng 10 the fo m! latioi o f Cd 6tial Dynam ics The e l<strong>lo</strong>is w.rc d ir4t€d to<br />
describcrh€dorionofplanctsandlheirsal.lliles,dteroidsand@nebinordcrtopledicl<br />
then positions in tuturc with accey of SEater 4d g@l€r d€sG Conuibntio.s of<br />
Eul€r, Laplace, Poison. Gauss, Olber, Cowll. Encke, Claittul. Hesn. D.launav. Hill<br />
and Brown. besides daoy olher nahemalici.ns and astdnoneb. bare ben oi grat<br />
si8nificancc. A nunbei ofcldsical md nodcn books de now available thal d.scribe thc<br />
d€rails ofd.F conuibutions (Sna41953. Deby, 1992, Plunmer, 1966. Pollard 1966.<br />
Woolard and Cl€nene 1966. Bouvct lnd Cl€mence, 196l elc.). Contibutions ae oho<br />
availabl€ rhat give delails ofthe lund dynamics (chap<strong>lo</strong>nt_Touze dd Chapront. l98l'<br />
1991. 1988. Chaproni er. al. I998, Srandish I981. 1998 etc) Einsrein's theolv of<br />
rldrivu suc(aded in dc$nbrns the molion olthe penhelion<br />
The satllite to planet nss !.tio in case of lhe Moon_E nll svst€n s larg€sl<br />
( 1.23 x l0-r ) in compdi$n ro ey o1h€r saEllite-ple.t pait (lh. nexr ldecar beins lhar of<br />
Triton-N.Drune l6s ntio = 2 i lO I ). Thccforc rhe Eanh do.s not prclid. lhe dominmt<br />
efecrive fore aclins oh lhc Mooh.It is not only lhe Sun bul all msjo!pltnels and la€€r<br />
of lh€ dtdoids lhat @nkibute 10 th. cffcctiv€ force ,cting on the Moon Thus d<br />
cph€ncredF prcpded *ilhout tdtinS into eout all lh.e contibudons b boud to be<br />
emn@u' Mey of the epheheedes of $. Moon of.dlv dsvs, both b.<strong>lo</strong>r' md aliet $e<br />
<strong>lo</strong>mJlauon of $e Neqonie ndhtuics. weE bded on obieBed sd <strong>lo</strong>mpukd<br />
averages of various Lnds r.laled 10 lhe dyndics of lhc Moon<br />
30
Today it is tnoM with a g@l degK of @!6cy lhoi lhe avetagc avrodic p.riod<br />
(intepal ber\,s t$o coMurive ne* M@m) is 29.510589 davs (29 dars, 12 hom 44<br />
ninul6 dd 2.9 sondt. The av€Bec uomalisd. morth (intcn.l ber@n lwo<br />
su(esive pa$ages of rh€ Moon throush it! perige is 27 5s4550 days (27 davs 13 hous<br />
18 oinutes ed 32.1 $cond, ed 4 aw'age sidercal honth (inLdal b.tw€en re<br />
succesiw p6eg€s of rhe M@n tnroueh ! fixed st!r) is 2 321662 days (27 davs houf<br />
4l minures and I1.6 secondt. Thus on thc b6is ofa sidcEal no h thc Moon rravel3<br />
abund 12.176158 degr.et p.r.lay on the av.rage Relaiv. to lhe Su. lhc M@n traveh<br />
12.19049 dtgc6 per day on lv€mee. Howvo, rhe mininu Bt 6 b. 12 08 deges<br />
Fr<br />
day and $e ndimum t2.41 deeees per day. Tnis oll happeis beau$ the orbil oi<br />
rhe Moon around tbe Ennh is a hiehly "i!rcCular' .lliPe whercs lho deviadons tre<br />
cau$d by p.nuibatios du. <strong>lo</strong> the S!n, lhe Planeb<br />
and olhs $lf svsten objsls<br />
Thc smi-major arh ollhc orbn ol M@n n on alcnge 384400 kn bul has a<br />
snaU oscilldlion aound thb valn€ whose period is hala lhe svnodic nonth Thc<br />
.@enrricily of the orbn is 0,t49 bul vdics d much sj 0 l lT The inclinalion of the<br />
luhr orbir from the ecliplic is 500 9' bul vdi6 up tol9 Elen the nod6 (poihts of<br />
inFdecrion ollhe lunar orbil and the ecliptic) of lhe orbn !re noi fixed 'nd<br />
go round ihc<br />
rcliptic in 18.6 yeaB with an o$illalion.bour dE $cular notion lhal moun|s to as<br />
nuch as I 6 dcsl6 Thc line ol apsd! tle eo rcund thc echpiic complelrne on'<br />
rouod in 8.85 yem and dciltations {iih udnude of 12'413 destccs Ths all lhc<br />
''elenentt' oi the orbn exhibn bo$ scular a ell as peiodic ldiarions lhis mkes rh'<br />
deimiMdon of lund ephem€Ed€s a daunling task<br />
The undeslandi!8 of lhe dynanics ol the Moon dd thal of th' planets in $e<br />
nod€m *lup b€sd vith lh€ doluliodv e4<strong>lo</strong>its oi Johees K'Pler $d lsc<br />
N.s,ror Kcpt.r cnpnically dcdu€ed his Lls of Pld.lary Molion on thc b6is ot o<br />
extensile study of the obsralional .lcta @llected ovcr ce'nies of lhc posilio' of<br />
pldets. Thcsc cfl be sbtcd 6 fol<strong>lo</strong>ws:<br />
3l
L<br />
2.<br />
L<br />
Pl&erary otbiB d. clliptic with lhe Sun a1 one ol lh€ foci.<br />
R.di6 vdtor of ! pldel (V@<strong>lo</strong>r dnM liom the sun <strong>lo</strong> rhe Pld€D<br />
sw*ps equal &.a5 in equl lihe jorensls.<br />
Th. sqlle of rhe Friod of rcvolulion of a Plder eund rhe Su is<br />
proponionalto lhe cube ofils dea. dblance ftom die Sun.<br />
N€fion nol only prcent€d his Law ofUnive6al Gralilaion but verified KePleas<br />
of Pleetary Motion usi.8 lh. tnw ol G6vilation. According to N€Mon s Larv rhe<br />
of atmcion beM.en two bodi6 wilh nas*s 2' ed ,, pl@d 6t a dista.ce /<br />
F =c!+i<br />
(2.2.r')<br />
The force is aflnclive and G is the PmPonio.ahy constant dllcd Univ.dl Cavilalioml<br />
Consranr (6.62x 10 'r-)*s rrr). i is enhs dit€ctcd ircn ,L to -r or rron ',rou|<br />
Frem rhe ddc of publicalion ot tte'Priripi.l bv NeMon in 1687 a nudbet o'<br />
ast<strong>lo</strong>noneB, physicisrs and na$cnalicians conribulcd sisnilicontlv in $e dcvc<strong>lo</strong>phenr<br />
of th. lndesl..dins of Lud sd Plm€hry dy.amra.<br />
Howele. at lh€ time of NeMon (and p€ftap3 lill <strong>lo</strong>dav) the dvnanics of lhe Moon<br />
pa*nred sear dilficuhy rhar forced Ne\\'<strong>lo</strong>n <strong>lo</strong> darc thal ",',e Ltmr theo'v nale his<br />
he.l ache dtut kept hih aeake so olen thut he vorld think o! it no norc " lDanbv 1992)<br />
He had tlitficulry in describing lh. modon of Fnge (the poill in <strong>lo</strong>d orbil c<strong>lo</strong>esr ro<br />
the &nh) and could explain it ro only withi. m accurev ot 8 percenl Clanaut (1749)<br />
appli.d ml)lical merhods ad succeded in explainilg d€ molion of p€ngee bv 6'ng<br />
seohd o.der approximation. He published his lraorP de Ia /,,e md a se! ofnumencal<br />
tablcs ih 1752 ior computation of lhe posnion of fie Moon. the Eost sisnificmr<br />
contribudon frcm Euler appered in l 72 shcn he tublish€d his srcond lL@ lh'ory<br />
32
kplM\ ficory of luM hotior, Publish.d in 1802. cnp<strong>lo</strong>y.d rrmfo@ing th.<br />
eqution ofmolions $ tlt.t the lruc <strong>lo</strong>ngltudc wa an independ.nt ldiable Hh wotk also<br />
povid.d e dpl.n tion of lh. sdo a@1.61io. of the M@! ripl@'s mdhods w.E<br />
cdied io a high degrce of accu!@y by sevetal malhedaicies One of then was<br />
Dmoi*e. who publthcd his th@ry dd r.bl6 in 182 that Gmaircd in *id' Ne until<br />
Ha.Fn! sork dppeaed. P. A. Hasen's wolk ext nded for over fonv veds fon 1829<br />
dd his r.bl4 w.re publkM in lE5. ThGe labl6 rcmain.d in s for w.ll ov'r finv<br />
yds. Delaunay publishcd his work ih 1860lhal M basd on disturbiry tunctions $ai<br />
in€ludcd 120 t€ms. By sn.ltti.al m.4 h. Gmov€d 0E teru of disorbins turulion on'<br />
by one ahd gEdually builds up th. $lution. Autho6 cl.im thar Delaunav s vorl( n thc<br />
mo$ Frf{t elulion of th.l@r problm v.t found {D&bv 1992)<br />
Thc posilion of $e M@n dodd th€ ElnI is desdbed bv spndical pold<br />
coordinatcs (r,,190 p) wnn r bci.s th. heli@.ntric disr:nce or rhe tltncr' $e<br />
eliflic <strong>lo</strong>nsitudc and I lhe ecliFic laliode The most commonlt Gd edv <strong>lo</strong> handl€<br />
lunai $bles dlrinB the nosr Pan of 20't c.n$ry \rcG dDc <strong>lo</strong> BrcM (Bmm 1 960) This<br />
ll]M $.ory w3 inpo!.d bv Eckefi &d ws tnown 6 ILE, sho^ tot Inprored Lrnr<br />
Epr,rdr,r. Tnc lh.ory coGtruct d bv ChcPont and ChaPon(_Toul is knosn 6 ELP<br />
(Chapo er. al,. 1983. 1988) shon tot Ephtina des I air's Pukienhes ln ELP<br />
simplifi.d tables have ben cxlEclqi from lhe lheorv <strong>lo</strong> rcplcsnt lh' luDa nolion in $e<br />
fom of explicit line sries fomule. Th6e tabl's @ b€ lsed to direcllv compute $e<br />
luntr coord'nabs. ELP h Dol onlt morc Peci* md complele in conpdien ro ILE it<br />
ale povides noe nodern valEs oi l(M pdmelels md olh€r Phvsical coislsts' For<br />
6000 y.4 on eacb side of J2ooo o ELP povid€s lund c@rdiMt€s that l4lv have eno$<br />
.xc.eding few arc secoo.ls Toeelher with th' d€v€<strong>lo</strong>pment of VSOP (Vdialrons<br />
Sacnl.ifs des Olbires Ple€titt bv Bctagl<strong>lo</strong>n ed Fdncou (t9E8) lhe t!bl6 due ro<br />
Chapront er. al destib€ the motion ofall major bodies (€xcept Pluio) ih tbe solar<br />
sysrem. Both rh€ $€ori6, ELP &d rhc vsoP w dcv'<strong>lo</strong>pcd d rhe Buean d's<br />
ll
BdicaUy a nhcory of pldelary ed lund motion involvd inies ionolasysrcm<br />
of di0eGnial eqladoi dat constin&s th. major pad of th. sfudy of elcstial n46mic'<br />
Therc & i. gddt t$o.ppoachcs for slving such dyndical systctu, ealytol ud<br />
nusical, Anallri€l nerhods m b6ed on solutio. by Iouder sen€s ed thc Poiso.\<br />
S€lies. Thc ELP-2000-85 (Chdprcnl€r,al, 1988) is semidalttic sd has be€n obla'ncd<br />
frod a fil o f ELP-2000-82 (Cn!po!i cr. al. I 983) to th. nlncrical inrcer.tion of lhe .lei<br />
Propulsion Labomrory DE200/LE200 (Slstrdish l98l). P@$ion in this o@ry ha b€en<br />
$keh iion Lasrar Gsklr, 1986).<br />
For te an lylic par! ELP-2000 Fpet€s th€ nain lEblm fbm th.<br />
penurbarions. Th. 6ain problem tak s into accout thc &tio. of thc E nn's ce E of<br />
m6s aid lhe acion ofrhe Sun's orbir aound lhe Eorlh-Moon baycentr. such lhat Ihe<br />
Sun\ odit is asuned to be Kepl€rid cllipse. This rcrulls in<strong>lo</strong> Fouri.r series w'lh<br />
numdical c@fiicie s snd ar$mdb th.t m suns ol mulripls of fou fundade.tal<br />
paBmeted D (diffeence of ihe ned <strong>lo</strong>ngitudcs of the Sun ed $e M@n). / (mean<br />
monaly of rh€ Moon), l(hem anohaly of $e Sun) lnd a (M@n s argumen( ol<br />
h ude). Tlis main problem 6uhs in<strong>lo</strong> dne *.ies fomnld for Mmn\ <strong>lo</strong>ngitudc.<br />
ladtude and gocenrric disrscc @daining 2645 rcos in all. Apan f<strong>lo</strong>fr lh* *rics<br />
acrons of all rh€ other significant objects in solar sysren dt consideed N<br />
'Fnurbadons <strong>lo</strong> ihc mdin pbblen lhol include:<br />
L tndiKr pl&c(ary perturbarions rbar re induced by lne diff.tsoc.s b€lwecn<br />
the lrue orbit of the Sun aound lhe Eanh-Moo. bdycenue and tssuned<br />
Kepbnm Elliplhal orbil oflhc Sun *uned in the min Pmblem.<br />
2. Dir€ct planerary penub.tioN due <strong>lo</strong> etions of othr plan$ on ibe M@n<br />
<strong>lo</strong>r borh the direct ad rhc indt€ct planetlry penubalions lhe ELP-2000<br />
co6id6 th€ orbib ofthe pldct giv€n by BGlagnon\ VSOP82 rh@.y.<br />
L P.rurbations du. ln€ figues of $e Edh md thc Mon (Moons, 1982).<br />
34
4. R.larivislic p.nub.rions (Lsrnd. & Chlpronl-TouJ 1982).<br />
5- Penurb.tuG du. to tidd afccis (villim3.t. a! l97E).<br />
6. Molion of the efccnce frde conside&d wilh spect to m in€rrigl fime of<br />
consideradon ol all lhese pelturbarioB esuhs in<strong>lo</strong> rime series fomlla for geocentric<br />
<strong>lo</strong>ngitude, IatiMe .rd disra@ of 0E M@n at nds $e numh.r of lcm !o 15:37.<br />
An al€natc ro this ther€icrl appro&h is !o rcpr€*nl1hc coordinai.s explicilly<br />
6 tinc ledes fomule. This epEs.rolion of tlie tinc sed€s is dcr<strong>lo</strong>ped by Chap6n!<br />
Touzi ind Chapront (Chapront-Touu & Chaprcnq l9tl, pp. l0) und ha bcc. !*d in<br />
fiis work. The mjor fomula used in rhh vork due !o $es authob de lkrcd be<strong>lo</strong>Nl<br />
Tbe ge@entric <strong>lo</strong>ngitude<br />
'/h.xpesed<br />
as:<br />
'/=218.31665416+48126.8813424'r-0,00011268'r:<br />
+0.000001856tr'<br />
0.0000000r534.r1+,5. +(s; +/ 1si +r' xsi/1oo0o)/l0oo<br />
(2.2.t)<br />
wh€rc<br />
I = h in iulian centuies sinceJ2000.0<br />
s, =tv,sr,(dj') +d|),,+ao *r' xro I +ao)'/r xr0' +d|,.rr xro r)<br />
I = )':st(o;'' -a:"' '4<br />
(2.2.21<br />
(2 2.r\<br />
si = tv;,s,,(d;,o + dnD.r) (2.2 4)
$.t%'sr'(sfl + dI'.,)<br />
(r25)<br />
Tb vdB 6f tb.ootM v- r; ct, d!o8 eitb |b ot a'. G Sivto i! Chr.od-<br />
Tout.d Ct q.od (Clg!d.To@a.nd Ci{.Gi, l9l, rp a!-56}<br />
Ib gcoc.nttc ldhde UL riE b,:<br />
t/-s,, +(s:/ +r'$ +i's;/10000)/100 Q26)<br />
s,, .5,"si,'(rj". p:".,.p:!.rr xron +/i",r xr0r +r'r 'r'<br />
x ro ')<br />
s:, -'u.,&4fri.t + p{t. tl<br />
Q2.',)<br />
e2.8)<br />
.$ -tr:$4i:o +r4D.,) t229'<br />
q.t!;&,(r;o +r'o.,1<br />
Q2.tO)<br />
Th! y.lG of th c@trt ! t<br />
- !; .iq dotE eilh lho.c of 0!.t 8lt'.! in Clq<strong>lo</strong>t'<br />
Toudsrd Ctar@r (ct Fld-ToEald ftlnfi, 1991, I 5{a).<br />
rindly tl|c geo.6tic dllt n6 it dven by:<br />
x-315000.57rsr +sl +r.8i +y'.s;/10000<br />
Q2.rD
s- =t'"cdldj"'+djl xron)<br />
s;=:4cd(d;'+r;,''/l<br />
(2.2.t2)<br />
(2.2.t))<br />
s; = t,;c,rt6"4' + r,4, .4 (2.2.t4)<br />
si= +d;o).r) e.z.ts)<br />
i"-",,(u-''<br />
-fhc !!hEs of the consr.nls rz, /; etc, d<strong>lo</strong>ng with lhose oa6s ae given in Ch.prcnl-<br />
'fouzi and Chapronr {Chapo.r-TouzC and Chatronr. 1991. pp 65-73),<br />
Atlat0)s. /0)s and D{ors, ,', s,,,sl..U,su,si./.'r.r;.,". md r; @ in desrcB,<br />
arr\, y'r's, d1'\, si,si,.'; and ,; ac in d.s@Jcenrury, ar'\, y''\. irlrs, si.si.v;<br />
lnd r; e in desEs/6tury2, ao's, y'r\, ana d's m in aegeevcenruD r and a"rs, /'rs<br />
aid d'h de in deecs/centud. R, s, and Sh dc in ki<strong>lo</strong>melres, 5i md 4 .t in<br />
ki<strong>lo</strong>derrs/cenru.y. siand 4 sr. in ki<strong>lo</strong>metftVcenruly:.<br />
For Tbe detemimtion of plan.lary coodinstes lhe complele n-body problcn is<br />
requied to be slv€d. A. ml,,lic solution of planerary dotion wa3 pre*nted by<br />
8rct.S<strong>lo</strong>n (BElag<strong>lo</strong>n, | 982) of Burau d.s LonSilud€s of Fnce lhat described only thc<br />
.llipric @rdinates of the pldels. Th. elution is populdly knosn d VSOP82<br />
(Variations S.culai6 d6 Orbit€s Pldauirct. Latcr, BEtagnon and F6Nou of th. se<br />
Bur6u hodificd VSOP82 inro VSOP87 (Bcl8non & FMcoq 1988) in 3uch a uy lJllt<br />
th.i! eluio! povid6 both thc Catcsim (or tst&suld) @rdimtes 6 wll a th.<br />
sphcdcal pola coodiiates of $e pldels in a helimenlric syslei. Thcir slllion<br />
V SOP8t descibes ln€ el enehls of thc osc ulat ing or inst aneous orbit in lems of:<br />
t7
a - sdimjor uis of th. dbil<br />
l, = m@ ld8itlrL ordF ddt<br />
l= ,@$<br />
p= sincr,i Dsino<br />
c= si(Xi)coso<br />
whec. is thc.@eotriciry ofthe orbn, r is ln. bngnud. ofp.dhelion, i ir th. incliodon<br />
oflh. orbit fmm rh. plD. ofeliptic dd O is the <strong>lo</strong>gilud. oflh€ e.lding nod. oa th.<br />
olbit. €.cholth. rcctarsul$c@rdioat (X L a or &c aph.ic.l polar coordimt€s (2. I<br />
t) is u.xplicir tunctionofiime ed is inlhe forn of p.dodic eds md Poison $ies.<br />
Every lcd ot Ues $.i6 is in fic tod of:<br />
I"(ssii9+ I:cosp) or 1"..i€os(, +c)<br />
(2.2.t6J<br />
$,lEE a - 0, l, 2, l, 4, 5. I is tn. tire in lhoFnds of.,ulie )€s fod J2000.0, i.€.<br />
f=<br />
165250<br />
e=ia,1,, i<br />
= I to 8, ! reprsent rh. m.m <strong>lo</strong>ngnud.s of the plancc Mercury <strong>lo</strong><br />
= 9, l0 6d I Lc rcpllsfl th. Dclawy sgmen& of lh. Men D, F md<br />
The hsl of I is the na <strong>lo</strong>ngiMe of U. Mon Bivd $nh 6p€t to lh.<br />
tlay. In thc ah@te €xpcsion,<br />
B= Za,^," + P<br />
c=>d,N, 12.2.11,<br />
S - -,asinp,<br />
(2.2.r8)<br />
l8
,l md M dc Sivci<br />
in the lable 2 of (BdaBnon & Flmou l96E)<br />
These data series @ .vailabl€ on CD's dd laFs For r.cta.gular c@rdiml6 of<br />
rhe phf,€ls |he dlia llles VSOPSA. VSOPSTC dd VSOP8TE dc @d sd for the<br />
spherical polarcoordinaEs VSOPSTBand VSOPSTDa!. used. A shonerleBionofrhese<br />
data series isSilenby Meeus (M.eus, 1998)and lheff. tr u$d in this@r\ l. the*<br />
hbles firsr €olumn gives.4s, fie eco.d ,s dd the lhird sives Cs. The dats file hs 6<br />
sedes fo, €dh or rhe coodi.ates L (h€ helioccnlric Lonsnud.) and R (lne helioccntk<br />
dislanc. ofthe Eanh) and 5 for fte c@rdi.ale B (lhe hclioc..lric Lafiudc)<br />
Iiiclr s$ics lir L. ll .nd ll ne uscd as fol<strong>lo</strong>\s ro obhin thc heliccDkic polar<br />
t't = LA; co\B; . I jT t.<br />
r2.2 rol<br />
i 0. l . 2. :1. 4. and 5 <strong>lo</strong>r l- md I( and 0, l,2,3,4<strong>lo</strong>r B Ihc suFrrscnpt tsund for I<br />
(L). 2(lJ) and 3{R). x rus tolsh 0 <strong>lo</strong> dilfeonr incger <strong>lo</strong>r di(feEfl coordinaGs and<br />
their lssiatcd seri6- Fo. I - 0- /{i coftsponds ro <strong>lo</strong>nsirudc scri.s. I = l<br />
l/{iconsponds ro lari$de sri€s and I = 3, /r,coiiesponds to dist .ce serica. ll.ch<br />
coordin eislhcneuluatdas:<br />
r=ltcuil'<br />
) "=lz'',')<br />
=l:, ','' I<br />
(2 2 20)<br />
/. md , d. ir ndian nesuEs and X is in aslrohonical unils. These m as menrioned<br />
crlier rhe c@.dinares of $c Eanh in heliocsnkic c@rdinare sysrcm wnce6 for {he<br />
problem ofdckmrining posirion oflh€ Sun inour sky w. €ctually requiE rhe Cmcentric<br />
coodinales of lhe Sun instead. In case of rhe Eanh this uNformation is sinple:<br />
l9
,is = I +l8O! Fs= -e 12.2.2t)<br />
ond the hcli@entric dislanft ofthe E.nI is smc as th. s€@edtic<br />
distanc' oflh€ S!n'<br />
2.3 BIRTH OF NEW MOON<br />
A3 menioned earlier the Moon in ilsjounev dound the Eanh llavels arcund 12<br />
dcgrees cvery doy in our sky and $l.s ovcr tic Su in lround €vcry 29 5 davs whc' rhe<br />
Ccoc€ntnc Lonsnude of the Sun e.l lhe M@n N se rhe nonent is tnoM s tnc<br />
Tine ol Bidh of Ncs Moon Tbe dudion berve€n two succe$ive Binhs of Nev M@ns<br />
is called lhc Lun.lion Period Ho$ever dE Llnal'on petiod is n<br />
from 29.2 days <strong>lo</strong> 29 8 davs This is rh. cMn b€hind<br />
'ons<br />
irt tu'd mon$s of 29<br />
d.ys €ach or lhe con*culive luor monlh of lO davs cacn For fi€ tine ofBinh ofNew<br />
Moon one requiEs to find lhe noment when rhe seocentric <strong>lo</strong>nsilud's olilt Moon and<br />
rhc sun coiNide Thus one needs to lracl lhc <strong>lo</strong>'Siludcs of eeh of thcn rinc<br />
Coisidcrins $e najor rcms oa lhe line scrics fomulac fo' lhe <strong>lo</strong>nsiodes of the Sun md<br />
$c M@tr in th€ planeurv rhcorv VSOP'2000-8 and lhc l(nw th@rv ELP-2000'8 $e<br />
noment when thc rwo <strong>lo</strong>nsiludes aG sMc can be evaluatd An alsori$m due b Meeus<br />
(Mceus, l99E) for $e dclemiiation or $e rin' or Binh of Ns Mmn is as rol<strong>lo</strong>s:<br />
On a!c€8. lhe tnpic.l Y€d (dudiion betw'en t*o conseculive<br />
passages ol lhe<br />
Sun lhroush equ ox) iscudcnrlv 16524219 davs (from (l I l)) Thc avedge svnodrc<br />
Monrh (trre intedal bclwccn r$o cons{u'r\c t\e* Moonsr toten o\er a rcn'ur} ii<br />
29.530589dar(fon (1.3.3)). 'Ixus in oft topical vear rh're aG otr a\€Esc 12 1682664<br />
Synodic mo h5 Thqe<strong>lo</strong>e since lhe $ad of lhe vcd 2O0O i e J2O0O O the nunbei of<br />
synodic monlhs elapsd oregrveh bvl<br />
t = (f,' - 2000) x l2 3682664<br />
(2.1 l)<br />
and lhc time in lropical ceoruries el.Psed since J2000 0 is Siven bti
t236.42664<br />
(2 t.2)<br />
An approximarc valu€ ofth.Iulid Dale ofthe Ncw M@n ir $cn giv€n by:<br />
JDE - 2451550.09166 +29.5JO58886t.t+0.00015437..r -0.OOO00Ol5<br />
j.r<br />
+ 0.0000000003' r!<br />
(2.3.3)<br />
$herc t i. d inteSer Thus @ording ro this fomula rhe for t = O |h€ Jutiil Dar. of$c<br />
lid ces.nr ofycar 2OOO is 2451550.09766 lhar is J&ury 6.2000 ar I8n r4m, md<br />
4l *!. I of dynlmical lime. For a horc accmle valu€ oa ihe Julia D.te of thc New Moon<br />
lhe p€dlrbation tchs due <strong>lo</strong> lhe Sun md rhc planer ar€ added, Thc pertuibafions tenrs<br />
due to thc sun d€ given byl<br />
x= -0 t07 2.SlN(M )+ 0 I72J t<br />
T<br />
EISIN(M) +O.0l60A.SINOr M )+0 0lAj91stNe,F)<br />
+0.007i9.E"'lN(M -M)-O 0O5 1.t.ErSINtM + V +a 0a208'E^2istN Q,M)<br />
0 00| t t 'SlN(M -2.F)-0.000t7.StN(M + 2*F)+O OOO|6.E.SI^-(2+M, + M)<br />
-0 000J2.slN(3.M )+ A OOO.| 2+E.StN(M+ 2tF)+0,aoB8*E.SIN(M_2.F)<br />
-a 00021'E.stN(2. M'-!r' o 0u | 7,stN4.] 0.00007.stN(M'+ 2.io<br />
+0 0a0u.slN(2.M -2. F) +O.00001'S!N(1.M)+0 \oaol.stNi,,t-+ M-2+F)<br />
+0 0040j.SIN(2'M + 2.FrO 00O03.S|N(M +M.2.F)<br />
+0. 0000J.StN (M -M + 2.D -0. @OO2.S|N(M. -M). D -O.O0OO2.StN(r. M, + M)<br />
+0.00002.s1N(1'M)<br />
Q).4)<br />
wherc ,r,1 = rb. mce &omaly ofth. Sd al ihc JDE<br />
= 2. I SiJ +29 I A5 3567.k-0 ,OOO0 t I t -0 oaoooo I I I<br />
(2.3.5)<br />
M = thc mee anomly of de Moon d lhe JDE<br />
= 2A 1.5613+ J85 8 1693528'k+O.Ot 07j82N I +0 OAOA I :38. I<br />
-0 0A0A0A0,8. 11<br />
4l
F " M@'!.4ruqtof hind.<br />
- ! 6'0. ! aE+390.670t0264.1-0.00t 61 r 8. | - 0.0U00227. <br />
+ 0.0@w0 .f<br />
Q.3.1)<br />
g - lncinde of lsnding md. of {!. luar orbn<br />
- !24.7716-1.5637t588.t+0.0020672'1 + 0.OOOOO2 t 5r I<br />
(2.3.8)<br />
E = Eedtricity of $. oftit of Elni<br />
= 1-0.002t 16.T-0.0000074. <br />
(2.1.e)<br />
Th. pedurbltion |.tus de io pleas e€:<br />
v = A00032t.StN(A I )+A000165+sNLtr+a000 t6.trSIN(/r+0.0a01 26.stN(A1)<br />
+ 0. 0 00 t t.sr N (,4, + a 0,f06 2 +s t N (/tq + a 00006. s t N ( t7 ) + 0. 00u t 6.3 ! N ( / E)<br />
+ 0. 00001 7.s t N (,t 9) + 0. 0000 4 2.s N (/ t q + a 0000 1. s t N (t t t )<br />
+0.0000t7rsIN(AI2)+o0o00J5.StN(Ar i)+o W023.SlN(,| !4) (2 3 to)<br />
A I -299- 2 7 + 0. t O74o8.k-O,OO9 I 7 t. <br />
t2-25t.E8+A0t6J2t.N<br />
,13-25 LE|+266J IEA6.t<br />
a4-t19,42+J64t2478.*<br />
/5-U.66+ ! 8 206239.k<br />
A6-t4171+53.303771.t<br />
/7=207-14+2.153732.t<br />
19-34_t2+27.26t2391t<br />
A ! 0- 207. I I +0. | 2 1 824'k<br />
/1 t I = 29 t. 34+ l. 64a379.t<br />
(2.3.1r)<br />
12.t.t2)<br />
(2.l.lr)<br />
Q.3.t4)<br />
(2.1.15)<br />
(2.3.16)<br />
(2.1.17)<br />
(2.3.18)<br />
(2.3.19)<br />
Q3.20)<br />
Q.3.2tt<br />
42
A I 2= 1 6t. 7 2+24. I 98 I 54.k<br />
A I 3= 2 39. 5 6 + 2 t. 5 t t099.t<br />
at 1:33 t. t5+ 2.7925 I E.*<br />
(2.3.22)<br />
(2.3.23)<br />
(2.3.24)<br />
Tlus $c ,ulid Dlle of lh. Na M@n ie giv.n by<br />
JD=JDE+X+Y<br />
(2 r.2t<br />
Thc dm€ deeribcd by this dat is lhe DlaMnical Timc and lhc coretions for Al mcr be<br />
mad€ <strong>lo</strong> get $e Universal Time (discu*d in 0Encxtarticle). For anr l@al coopuralion,<br />
th. Ld.l Zonc lide and d!i. n6r b. calculated ftom |he UniveMl Tim. and dale<br />
ohaincd abovc on rhe bdis oatbe <strong>lo</strong>nsirudc ofany place on rhe Eanh. Thc darc h $en<br />
de day of conjuncrion fo! the place and rho rime of conjuncdon (he bnrh orNcw M@n)<br />
can b€ dy rinc fom 0- 10 23r" 59'" t9h on thd dav.<br />
2.4 THE TIME ARGUMEI{T<br />
ft w6 mentiohed earlicr thal rhedynahicsofatl soltr systcnr objec$ isdescibed<br />
by <strong>lo</strong>mula b6cd on 6adcs of Cl6sic.l Mcchdics and lhe Relarivisric Dyneics in<br />
t.m3 or rid€ serics. In order b ersriv.ly ue rhcsc fomulas an apprcpriate rrs<br />
argMerr cotrcsponding to rhe honeht of obseiving the lunar crescenr at my place on<br />
lhe surface of the E rth h6 io be €valualcd. Such a lime a.sudent h6 ro b€ co.inuou<br />
and mui halc a clearly dcUncd point of i's beeimine (|he aro rine). called .p€h.<br />
various theorics and problems uF difercnr .pochs depending on rhs co.rext for a<br />
geneEl consideFtion in pls.lary ad luE dynmics thft ,re lwo ihporlanl epochs.<br />
Th€ nr$ of lhese epochs is a oonefl in edote pdsl coftsponding to rhe Noon al<br />
G@nwich on Janlary l. 4 l2 B.C.E. on th. Julid c{l.ndar (or Nolember 24, 4 I I on<br />
Cdgode cal.ndtu) (R€ingold & D.Rhowila 001), Fren $is pojnt of timc rh. rine<br />
elapsed tiu my laler point oflime in number of days md a possiblc fierion ofa day h<br />
4l
elled th. Julid Dare abb@ialed B JD. So the ,D @Gponding <strong>lo</strong> thc 5i" 30.i. on<br />
Ocrober 5, 2004 ar Grcenwich is 245t284.2708t11t.. Thls rhe Jutim D!i. i3. mesE<br />
of tih. clapsd sine thn €pocl 6id is cxpEss.d in nmber of n@ eld d.y5_<br />
The orhcr epoch of inpori,lcc <strong>lo</strong> rh€ €lftnl work k fie noncnt of dme €ell.d<br />
J2000 0 a.d il Eprcseds lhe l2ln TDTon J.nuary l,2O0O i.€. (Abononhal Almahac,<br />
2007). Tbe JD conesponding ro rhis hom.nt dl C@nwicb is 2451545 days. This is the<br />
.poch or 2eo me for borh rhe Luq Th@iy ELP,2000 (Chopo.lTouz{ & Chaponr,<br />
l9l) dd th€ pl&elary th@ry VSOP,87 {Brc|,gnon & Fmcou, 1988). In bo$ rhes<br />
0leodcs lih. m@!red frcn J2000.0 bo foMrd od b6ct*ards. In ELP thir rimc is in<br />
is coBidcrcd in rulian CentJies (i.c.16525 me.n sle days) md i! VSOP il is inJulie<br />
Millemia (365250 mce solrdays).<br />
BoththeseepocbsmbasedonrheinreRa<strong>lo</strong>irimealted..toedn$hrdat, qhich<br />
is defincd a fie int rval berween lw slccessile rhsits (r6sdge rh<strong>lo</strong>ugh rhe <strong>lo</strong>cll<br />
meddim) ofrhc fic ious body knoM as ft. mee Se. This ficlitious body novcs sirh<br />
uifom sF.d a<strong>lo</strong>ry lhc cel6ial cquabr .hd is consider.d in place of lh. acrul Sun $ar<br />
oovcs vith non-unifom sped (dw ro rhe .llipli€ orbn of tne Efin) abnS lne Ectipd€.<br />
The tmsit of $c aclual Sm over a <strong>lo</strong>cal heridi& vdes up ro I I minurs over a penod<br />
of on€ lrcpical ye (Astmnooical AlFanac, 2O()7), Thereby atl civil rim. reckoning<br />
have b€eh a$ocialed with the hedn Sln lhot consisrl of 24 mean solar hous, The<br />
beeinningofa civilday, i.e. zro hou* on civil c<strong>lo</strong>cks occu^ at nidniehl when the nour<br />
dgle of rh. Mean Sun is rw€Irc hou6 lccodin8 to the <strong>lo</strong>cal or sisddd hcidia..<br />
Th. tiDe describ€d by lh. cl@k showing d. mm $lf rinc is not wilhoul its<br />
di*Ep.n i6. In fd it is lhe E€rtn, $e gtob€, ilsctf rhal is ou cl@t lnd $c n.m solat<br />
tine B suppos.d to b. basd on ths a!€ragc rate al which lhe Eadh n spinning a@und irs<br />
dn. Howeverlhn considearion is only with rcspecl !o lh€ Med Sun. Du.<strong>lo</strong> the orbilal<br />
noionoflh. Eanh beins in rhe sme direction as its dis ofm<strong>lo</strong>lion (ton qesl <strong>lo</strong> EEst)<br />
on€ ubl totaiion coDpldes in l6s thm ihis nee sold day. So the actuat nle of dial<br />
rchtion is b.t r ralized by rhe No sw;sivc resirs of a sior. Dis in|cdal is lemed
a r Si&F.l Dry ud rh. timc masuld @rdi.s to thb $d. is rlrc Sid.Flt Timc_<br />
Aglin du.<strong>lo</strong> lhc clliptird orbit ofr[. E lrh ttris D.riod i!.le nor uifom e w h!rc ro<br />
@tuid* "M.rn Sidccal Day" rnd !@diiSly M6 Sidaql Tirc. Oc md el& dly<br />
cquls |.002390915 !|16 .i&Gd dlys (or 24B 03i" 56.t553* on m@ sid.Mt<br />
tim.). Altcturcly oE md 3idcEd dly cquals 0.926956633 ns st.r d.ys (or 2lb<br />
56'r" 04.09053* on md sol$ rimc).<br />
In gcrcr.l rh. o@ eld {n ir rhc rinc |.t n inro &@6r in bolh lb. civil tu.<br />
ekoning a sll a tlE $tromnic.l. Wh.Ed lh. dylmiclt rh6d* denbc |h.<br />
norions of th. objers ii solr srsr.m on th. bair of rhc coninuoBly nowins tinc<br />
dc$dbcd a Dyimiol Tim.. T|| UniwBl Timc .d rlE DyiMiel Tim. E nor<br />
@Biri. .nd rh. dilTelwc bdw6 dF tw is <strong>lo</strong>t ! trom fumrioi of ritu dd @utd<br />
bc <strong>lo</strong>und only bt high pcirion ob&dados of th. skics. Th. diffm. of thc lrc AT is<br />
rlbull|.d i. Asnonomicrl Aln.t@ for th. t.tqopic G6 (AD 1620 ri <strong>lo</strong>dar.). For rh.<br />
cd prior <strong>lo</strong> lic rcl.spic .n rhc v!l@i ol dT G cdcll.r.d on thc blsis ot rnc<br />
qlculatioB of .cliter @ultrrion lnd 0E rin6 of th* d.nt. @rdert jn lh. tmwn<br />
history. Th. val@s of AT e giEn fq only |h. nan of.eh cd.nd& y.r ,nd $os for<br />
olh.r dn6 ofFd ce b. incrlDlar.
How.r s noE @nht<strong>lo</strong>n l !'pNch ir ro @!qt <strong>lo</strong>€l =24) ...... 1rcft6. dar<br />
( UDD-LDD+I<br />
tf (UDD>dorqLMM)) //darr is nMber of days in a donrh @y<br />
I<br />
t)]<br />
upHH
AYY-LW-l;<br />
J<br />
UDD-&tr(UtlM)<br />
olr@ l[. UEircc.l d.rc dd tirc is lppllFi.Lly dju{.d @ pcad! <strong>lo</strong><br />
c.lcula& rhe Julih Dar. for 0E d.lc {d rhc ritu. For rlis calcularion inirially if<br />
ihc nondr is Jmqry o! fcbnoy, it i! co id.Ed morh .mbq 13 or 14.<br />
Esp@tiv€ly, ofthe ptwiou trd ad rh. t@ i! sle d.crcNed by l.<br />
UW-WY-l<br />
Thc nmbd ofeftry yd (likc ,E | 100, 1700 .rc) d[ ln. yd UYY<br />
ir Fquicd <strong>lo</strong> t@6t for !mh6 of mhd rap y@.<br />
Stcp4: 1=INr(Wt00)<br />
ln lml of 6e $tommic.l dlculariotu ! daL d and .na Fliday.<br />
Octoba 15, 1582 is coaid.tld io b. I dlc of Grgodd caldd.r {d a darc toni<br />
Thsdly, Octob* 4, l5E2 ..d prior to rhb datc i. coGiderd s a dale i! Jutie<br />
cal€ndd. Ilru ifa dsl. i! fton Crc8ode cal.dde ndh.. EquiEs m accout<br />
ofnon-tlap yed tom.mong3i 0F nomal lcap yds due to th€ modified rule<br />
ofliad y.rinthe Grc8onlr qal.nd$(y.$divisible by 100 but rot diviribte<br />
by 400 e @r L@! yes).<br />
d"@ledq it casdtb'<br />
I<br />
a=2-/t+rMf(1/1)<br />
.l*1I "@Ldo i.t IdlM"<br />
( B_0 J<br />
47
Now orc n6ds ro c@t 6. nub.r of diys .laps.d si@ the Julid DaL<br />
epoch (Jeulry l, 4712) iill Ihc .nd of th. PEviou yc& .rd lhe trmb€r of dltr<br />
€laled f<strong>lo</strong>m ln. n61 d.y of rh. PEviou yd till the cnd of lhe c|lml monlh<br />
Meu @dider this sMbl ianing fom ys -4716 dD( adds addilional davs<br />
thal @ balmed by subt@lion of lh. co6t ni 1524.5.<br />
JD=\NTQ65.2t(UW+ 17 16)+ tl]/r(30.600t (uMM+ 1))+ UDD+ B-l 524 5<br />
+ (u HH + (UMIN + USEq60)/60)/21<br />
(2.4.r)<br />
ln both th. lh.orics VSOP8 tnd ELP2000 lhe €Poch is the J2000 0<br />
@ftsponding ro rhe Julie Dsr.2451545 theEfot one tin lly c.t5 the lin€<br />
JD 245t545 Ar<br />
16525 1155760000<br />
(242)<br />
Notc rtEt in rhc l6t st@ lh. lid l.B on 0E lighl hfld side is the nmbd of<br />
lulie entude elap*d sine 12000.0 ad th. s@sd l@ is for At which is @allv<br />
siven in mobd of.eonds od h.rc w. nccd to @nved n i o nmb.r of c€ntuies (lio<br />
ELP &d Millemia for vsoP) due to which il hN <strong>lo</strong> be divided by the nunbq of s6onds<br />
in a Juli$ c€nrury ifor ELP md Julim Mill.inia fot VSOP).<br />
lf and when ttie dynmical tim. of d cv.nt is *rom we need io @hv.n rl b<br />
U.ivesal dd Lhen into Local Zon. tim.. Thc dylmical tihe is oblain d s the nmbet<br />
of Julie cenluries sin@ J2000.0 e that rhc Julisn dare of tbe evmt cm b€ calculated 6:<br />
48
1p - 15575s.(, -<br />
=-4-), usrsts<br />
t 3155@@00J<br />
ea.t,<br />
TIlc int Sr.l Ft ofdF Juliu D.!. *ill b. onEtud io Cdddr Dd. .nd<br />
t|| hclion l Frt ro th Unirwl Tim:<br />
St F2:<br />
Z-INT(JD)<br />
St d-3: F-JDZ<br />
Fd &rB Fior ro ocrobcr 15, | 582 (JD = 229161) rhc inl.Sq p.rr of.ID<br />
is !.oircd ! il ii olhdtuc ldjulr.d for th. @diti@ of lhc GGgorid cdqd{<br />
St.pa: ifz
LMU-UMM<br />
stcFr2: alMM>2 { UW-C-1716 }<br />
Etr. t UW-C-17tt )<br />
LW-UW<br />
steFf3: hout(=drylM(dqr))r21<br />
Stepr4: UHR-tM(tbu) LHR-UHR-ZON<br />
If (LHR-2,t ... ... tt@t et)<br />
{ LHR-LHR-21: LDD-UDD+,,<br />
4t6-DD>day1vut4<br />
ste!'rs: htn .=Aov-L| 9.60<br />
'|<br />
LDD-|: LMM-UMM+I:<br />
y(LMM>t2)<br />
{ LW-UMM+I; L tN-l;<br />
)))<br />
sreFl6: UMN-IM(nlnut.) LW-UL{N<br />
steplT: tu.ond-(ntr.-LMIN),60<br />
50
SGpf8: USEC=lNr(taond) LSEC-USEC<br />
Sr.pr9l Outpd UW. UMM IJDD, UHR, tjMIN, USEC<br />
A"d LW, LMM AD LHR, LMIN, LSEC<br />
In thc Nd-M@n Algorilhn thc our pur is thc dynmi@t lin. &d this dgon$D<br />
is p.niculaly usfll in @nvcning Uis rinc <strong>lo</strong> U v.Mt od &y l@.t @G rin .<br />
2.5 COORI}INATES OF THE MOON<br />
For rh. dckmination of lh. c@rdinrcs ot 1h. M@n .r ey Sivcn lc.l tim. .rd<br />
dlt liB| scp is ro fomular th. rinc a4um.nt a dieNcd in diclc 2.4. So lh. pcc$<br />
bcsins by s.l*ftg pl.c. of ob6.rq (Ljnsirude &d La ud.), te.l darc ard rim. rhrl<br />
lcds to lhc rim. &Bum. t eco.dinS ro th. atgorjlhn dc$ib.d ,bovc, s:<br />
Ju<strong>lo</strong>n Do'. - {51545 N<br />
16525 115560000<br />
(r 5.t)<br />
Using Uis rim. arsumcnr th. @Bhcrio! ot rhc dh. *n6 d.*ribing rh. twr<br />
c@'dinar.s is do.c (chapo -Touza ed Chaponr t99t) s dieled in aniclc 2.2<br />
StF||<br />
Fo..dipdc <strong>lo</strong>ngirudcofth. Moon us.2.2.2 ro2.2.5 ed suhdrut lh.ir<br />
csulrsi.2.2,l.<br />
Sr.F2r For cclipdc t.hu& of U. M@! E 2.2. 10 2.2. t0 &d sub$nuE th.i<br />
Bul{.in2.2.6,<br />
St.Fl:<br />
For gec.nr.ic dilllre of rhc M@n uc 2.2. l2 ro 2.2. | 5 ed subsritut<br />
lh.nrcsulr.in2.2.1L<br />
5t
DE to lh. nodotr of thc ob6*d. th. diual .nd lhe mual motior of thc Eatth.<br />
poiilior of €E.y objer in th. sky is afrctcd by 0!. phcmmn of Abcmdon- Th.<br />
fol<strong>lo</strong>winS @cid.ntion is only for rhe Eaih'Moon pls.t.ry Ab.dlion sd do.s <strong>lo</strong>l<br />
includ. the dimal nolion of thc ob6*er (W@lad & Cl.mcnce, I %6).<br />
S1.t-4r CORREC'TION FOR ABERMTION<br />
v-y4.000 t 9521-0.0&nl059.skQ 25 1177 198 9\) 12 5.2)<br />
U- U-0.40001 754t31h(183 J +48J202't)<br />
(2.s.3)<br />
R- R + 0.0708.C6Q 2 5 + 177 I 96. 91r)<br />
(2_5.4)<br />
Fimlly a 1he lrue €quinox of ft. d.y and dE mcm €quinox of lhe dly de<br />
difiedt dE ro thc phenom.non ofNuiation the F€.is cerdinab on nol be ob|dircd<br />
without th€ nul'lion in <strong>lo</strong>nS itudc AV and the nubtioh inobliquityAs.<br />
St pS:<br />
CORRECTION FOR NUTATPN<br />
a'r =l0r 't(y, + y:..)'si,{/,I" +/d' '. +/d' '.r '<strong>lo</strong> r)<br />
(2.s.t<br />
Y=V+ L,t 12-5.6)<br />
. -22.63928-0 0ll\+0.555'<strong>lo</strong>' t.r -0,0141'l0j<br />
r tl<br />
(2.5.7)<br />
^' =<strong>lo</strong> I 'I(., +,' '.). cdko) + /,r' . ' +rdr ..).<strong>lo</strong> 1)<br />
(2.5.8)<br />
(2.5.9)
Tn€ v.lu. of F s, !,'3 md G's Bd in lh. qp63iont ahovc tt! gt!€n in<br />
thc T$1. 9 in Chlprcnr-Tooze 6d Chlpotrt (Ch.pre -<strong>lo</strong>uz; &d Ch.pon!<br />
l99l,pp.l9):<br />
Oft. rh. @retion dE to .ubrion is dre onc my so ro 6trd da Eq@tori.l<br />
c@dihal.s, the Rjght Asnsion d dd thc dcclin tion 6:<br />
St.p6:<br />
EQUATOI.ULCOORDIANIES<br />
"=r-'(<br />
c6(tpriio")Si4v)C6(U) -.t4 Eps i t d) Si dlu<br />
'l<br />
(2.5.r0)<br />
6 = sn't l3ih(Epsitm)Si"(v)cos(UJ + cB(Eptiron)sinei (2.5. I I )<br />
Thc* re rh. nle cqubri.l c@'dinar6 of lh. Men wnh EfcEn.. b th. toe<br />
L<br />
cqu<strong>lo</strong>r of $e dare ahd the trua dyunical equioox of the dar€_ Th€$ m stiu lhe<br />
s.oqtic c@diMl* md lhc afr.cr of lhc D6nion of lh€ ob*rcr on rh. g<strong>lo</strong>b. is ycr ro<br />
b. lalfl ifto @ounr so tb.t rhc ..iopoccnt.ic" c@rdid., (coordin t€s relariw <strong>lo</strong> thc<br />
posnion ofth. ob*fr.r) nay be obl.iied_ In odd rcd5 rh€ !ffftrj oftE ..paraltd" arc<br />
to be t t.n in<strong>lo</strong> &count. Th€ Pdttd r i! givcn by:<br />
12.s.tz)<br />
wh@ 4 ir rhe g€catdc dist4. of dE Mer HowEr rhis q@nry paml* d.Fnds<br />
on rhe Hoq Ancle (ihe since ibe object cosad thc l@al n.ridim) tor whicn w. ncd<br />
thc lad sidc@l tin. LST.<br />
St FTl<br />
Fqhtlate tiw dgM't t<strong>lo</strong>r t UTlq !h. dat. ,tdet cwidqatio,.<br />
5l
I. =6r4t-50'.54841+E tot84,.s12s66I,+0'.093104.r, -0,.0000062.rr<br />
(2.5.1t<br />
gits th. Grcawich Mcan Sid€E5l Ti6. .t oir UT of rhe drte. Th.n lh€<br />
Ctqwich Si&@l Tio. for &e rirc ug@nt for ih. rimc of ob6qvation is:<br />
T - To + (UHR + (UMLN +USEC/60y60)a0.997269j66JJ (2.5.t4)<br />
'Iijen fic Hou anglc at ihis moncnt ofthe Moon is:<br />
H=T a<br />
(2.5 rt<br />
srep.sl tllec^ofPatala,<br />
If p is th. goc€rntic 6dius ofrh. Esdh, p,b ihe S.ocenkic hrirudc<br />
of irE ohsftr ih rh. dsh1 .s6io! a' ed rhc dEtiEtjon d, aft.r<br />
co@lion for prhllar &€ obr.ined $:<br />
@sd - p@sp'sin rcos l/<br />
(2 5.16)<br />
.Aliitu& r=sinpUnrlcdp'cosr'costl (2.5.20)<br />
This complcGs ihe der@inarion of dlipric, €qurqid .d Oe hondnIll<br />
@rdid6 of rh. Mor<br />
2.6 COORDINATES OF TI{E SUN<br />
Foi rhe d.t.mimtion of 0E c@rdiDre of rh. Sun on. p@..& in €runy rhe<br />
se say a for lhc @rdi!!re of rhc Moon.<br />
slcFl:<br />
Scldt ple oroh6dvd (Longiru& ed tilirud.). lcal d ..dd rine rhat<br />
lqds to rh. thc egment t &coiding ro thc algolirhm dceiib.d lbove, s:<br />
Julian Dde )451545 Ll<br />
1652t0 I | 55600000<br />
12.6.1)<br />
Using lhis lire argment lh€ connrudion of d. (m. eriB &enbins rh.<br />
c@rdinates of the Elnh s givd by Br.raSnon & Fdcou (1988) thc h.lie.nrric<br />
coodinales of lhc E nb de obtained lhar ft lder trusfoftcd into fte gre.tdc<br />
c@rdimles ofth. Sun s fol<strong>lo</strong>sl<br />
SreFl:<br />
For heliohllic dlipric <strong>lo</strong>nsitud. of lh. Flnll' in lirc wit[ (2.2.19) &d<br />
2.2.20 w h!rc:<br />
2^,<br />
"e"(,,<br />
.r^, )<br />
(2.6.2)<br />
2""<br />
"."6"<br />
t r,,l (2.6.1)<br />
55
t,-'i^,-"(t"*", \<br />
2t<br />
L,-ZA,s\8,+c,. I<br />
c2.6.4)<br />
(2.6.5)<br />
L. =ZA,@l\tr+crt )<br />
(r5.6)<br />
!,.:r,cdt r +cir ,<br />
c2.5.)<br />
Ttr. l!l!.3 or A'', B'r .d C s for {i! d$n r 6ior c dB b, Mc.!i<br />
(Meu!, 1998, A!p.ndir-[], pp. 4lt.a2t). Finlly t!. lt lio..onic <strong>lo</strong>Bitd. of |h E lt<br />
i, -.<br />
t0'<br />
(2.5.8)<br />
St p2:<br />
Fd Fliptic ldintdc ofrlF E nh;<br />
lut<br />
Bo-Z/,.d\8,+c,. l<br />
Q.5.9)<br />
,r =:/,dF" +c"4<br />
(2.6.10)<br />
,, . X,{" "o.(r"<br />
+c,r J<br />
(2.6.l|)<br />
Bt-244a.+c"r I<br />
(2.6.r2)<br />
8. = t,l,6.F" +c"' J<br />
(2.6.|])
Tb v.lu.t of A's, B't .rd C'! 6. l} {oi.t nds tt giE bv Mo.u!<br />
(M6n, l9S, A!0.odtr-Itr, !P. 4lS-421). Fidlv lh. t lbc.otic ldlidt of th' E ib it<br />
l0'<br />
(26.r1)<br />
$.Fl:<br />
For lElioc.dric .lisl&e of th. E|nb<br />
&-X,i"6!p"+c,r J<br />
zt2 t<br />
xr - I,r,6lp" +c"'J<br />
(2.6.15)<br />
(2.6.r6)<br />
(2.6.17)<br />
nt<br />
13 = t,r"co't +c"' " J<br />
(2.6.rt)<br />
t0,<br />
h=>an(fr]\B,+c{<br />
(2.6.19)<br />
)<br />
<strong>lo</strong>/<br />
xr = t,."co4p" +c"' I<br />
(2.6rI))<br />
19 l<br />
x, = t,,r" o.[8i + c,' ]<br />
TIE vds of 4.. B's !d C" b. lb eorlr Ei6 & tih b, M6!<br />
(Mc.q, 199E, Aprcndix-lu, pp. 4l E-421). Fi!.lly tlE helio@tric di!l&c. of th. Elnh<br />
ZL,r<br />
" = ''id,<br />
Q.62t)<br />
A!" Bs lnd C! N all in hdiM for <strong>lo</strong>oSin|d! , a|(l ldltt|d. t B6 od Ct c in ttdieE<br />
.d A! i! .*!doiql uitr tu Llio.ldtc diaec X.<br />
51
The coftcctions for Abe@don md Nuradon de done in dc sanc way as St p4<br />
and Slep-5 b€fore. Fi@lly te conveBion to rhc equaronat @ordinates dd then b<br />
horizonlal coordiDr.s is als done lhc sme way 6 ws done ror fte Moon.<br />
2. RISING AND SETTINC<br />
For the d.refti@tion ofprecise rihings of rhc setrnE or nsing of& obFcl one<br />
requircs pEcis c.l€stial coordinales of $eF objech 6i lh€ insht of1he occunine ofrhe<br />
phehomem However! these insranh dre the points ofime that we rcqlic ro find our so<br />
rhar a process oasuccessive appoxinadon is needcd <strong>lo</strong> ahv€ al lhee dhes. Such an<br />
iterarive p<strong>lo</strong>cess is nec€ssry b4au$ rhe objccis under con$dsaion (lbe Sun Md the<br />
Moon) significantly changc thei position relalive to the ixed cel€sriat sDhere durihg m<br />
i.terval around ha a ddy. The whols process shns sm m esehare ror the ritoc of<br />
'ransn ofthe objed (ove.lhe <strong>lo</strong>catneridian) wbich lben reads ro,nlial esrinars for de<br />
holrmghs ar th. appoxihare line ofrhe rising orselling or thc objer Th€e aB in facr<br />
the csimares for rhe tocat sidereal rinres ofdc phenonena. rrco these esfnales of rtE<br />
sdceal rimcsoflhe eveors lhe unilecat hean slarrimeardlheh lhe toql lihescan be<br />
cakuhted. ths. fist approximarions fo! lhe ransii rbe nsihg ed rbe sedne are<br />
obtajned usinglhc cetestirtcoordjnars ofthe obj(r evaluard ar Oh. Ur. ar any poinr or<br />
rhe g<strong>lo</strong>b€ and ao! any otrhe evenrs under considecdon lhis nomeir ((]i, Ul oay be m<br />
cail.r or a tater nom€nr. This is the rson lhar the mrfial catcularions are onlv<br />
rppo\imaro cdlcutduon fo.hcle appo\,dar. ,t.., "t<br />
,h. .,.",. ,h. ."i..;;:.<br />
coordnats of thc objecr hNc to bc catcutaled asain and wholc.atculalions henLion€d<br />
above !!€ repealed for bekr esrimates, The details of rh€e catculalions arc describ.d in<br />
(he fol<strong>lo</strong>winS p@sraphs.<br />
, . Forrhe sh. rhccompnaaon. &e simptes conp@ rornore tor lhe Moon. the<br />
,(o nme or hNir ot rhe Sun can be hirra ) coturdered ar t2-,tocsl rzonc, rime<br />
wnecas for the Moon n dEily vdi*. Th6 Unileisat Tihe or <strong>lo</strong>ql tuir is simply 12<br />
Zl wheE ZT (ene ne) is posirive for the east <strong>lo</strong>ns,udq ed nesarivc for Resr<br />
58
<strong>lo</strong>nSitudcs. At uivcrssl rime 12 - zT wilt b. a dn€ of 0E se darc in O@wich s Ih.<br />
<strong>lo</strong>ql dae. Th. rime dgu. for $is rine ed date is rhen fomulatcd dd th€<br />
coordiMtes offie sunar.obaincd. wlm m objdr h in rmsil irs hour dgtc (HA) is<br />
zo ed its dsh ecdion (RA) is se s ln. Loc.l Sidqdt Tin. (LSn siNc:<br />
p. l)<br />
As3Ming dar lhe obj*t hs r{A - a !r da rine of t@t Fa6n e.7,I ) shos thal:<br />
d= LSTt<br />
If JD is rhe Julie ddc for $e dly ar d' UT 0En wnn , = eD _ 2451545y36525<br />
mcaur.d in Julian centuri6 lh. c€enwich Mqn Sidc,@l Tide I, {CMST) b siven by:<br />
4 =6141-50,.s4841+8640184'.812866.r+0'.093t0,t.r: _0,.000006rr,r<br />
(2.7.t)<br />
Thch for the observer ar rhe pla.e vilh eeographic <strong>lo</strong>nerrude , (negative fo. west<br />
<strong>lo</strong>nenud.s a
FimUy the coordides ofthc Ss @ Edcdar.d for UTE qd 0r fol<strong>lo</strong>wins cdculadon<br />
I,sT = a,<br />
csnr-LST-L,<br />
n_GSTtr-To<br />
(2.1.7)<br />
{2.7,8)<br />
(2.7.9J<br />
uTr=T,'L00273790935<br />
(2.7.10)<br />
So ihat theE is a dilfer€ne of l€ss drs a *cond b.tween one v.lue ofuTr ed ib.ext<br />
[or lhe <strong>lo</strong>cat sunrise and sunset $c base vatue $ the UTtr ahd inirial<br />
apptoxmalDns for the sunnse k UTr - 6h = UTn and lhat ot tnc suMt is UTr + 6r -<br />
UTst. Dcpendi.8 on rhe tocat <strong>lo</strong>ngirude L, UTs ed Ursl nay ti.I on pr.viou or rexr<br />
day csp.clively so lbar a necese,y dare adjutmeir mul b. dorc. runher ddpe.ding on<br />
the <strong>lo</strong>cal lairude il is funher possible thar these pbenomcna sihply donl occur.<br />
The pbc.s bcgins<br />
Urs0. The holr angl€ HA<br />
de c@rdi@res of rh. Sun for UT6 (or<br />
stt'ne or nsing ovcr a l@al horjzon k<br />
(2.7 r)<br />
stEE I is rh€ eeghphic laritude of rh. obsedcr lnd lhe au nd€ of d. poinr of sky is<br />
a$med to b. ero, How.ver owirS to th€ phdohcnon ofEri&tion a ,r.r, rhe Su dd s<br />
pld.l ae wll b.<strong>lo</strong>w th. hode. std rhcy @ a€ru.lly seo sdrjng or risin8. The<br />
t\@g€ rFe.t of EFadioD is rhar . srat E@iN visibte de ir ft hs gone j4 e Dinu!€s<br />
be<strong>lo</strong>w hori^n. 'Ilis allitudc is kmh s srand.rd drilud. dcnoEo s 4, ed is @nsid@.I<br />
6t)
<strong>lo</strong> bc -50 N sords on aveas. f6r rlc Su, 0Ei itulud.s ihc affet of dE Ef4clion<br />
and thc sni dian.ter bo$. For a mor€ accwtc valuc the &rul s.mi dimet r SD,<br />
rhould be calculal.d fmo dE didea of1h. Su ed suh.actcd tioh d..veagc afi*l<br />
of Glhction, fte chag. in l.hp€dl@ in ihe niddle laritDd.s my lary thjs by @ud<br />
20 seconds of tinc ed tbe baromctric pres6res nay caue a veiaion of dother 12<br />
*con& of time. Ho{€v.r 6 rhcse varia{otu cd nol LE d.temired a priori the avcEgc<br />
afects m co.nd.ed in crlculalions. Unog fte $..drd slrjM. 01h. hou anqlc oflh.<br />
object is then e<strong>lo</strong>luared as:<br />
(21.t 4<br />
Thus rh€ liBr dppro\,maion for lhe rhe ofrisng of$e obled h:<br />
T, =UTO - E.<br />
12.1tJ)<br />
And $at ofrhe s. in8 isl<br />
T, =Wtr+ Ho<br />
(2.7.t4\<br />
Thes. e only !h. n6t app,oxinarion3 for lh€ tines of swise ahd the sunsel<br />
Esp.ctirely. Fomularing ihe lin a8unors for €.ch of fien Fpaan.l, rh. c@rdinar.s<br />
of lbe Su ha\e b be catculared aglin. The<br />
Grecrwich sid.El rinc orrcspondjns ro<br />
borh t &d I h6 ro be obained as:<br />
r-Ta+J6O.98564TTy'15<br />
12.7.\5)<br />
,1 or I e rlEr l@at lFu ogte<br />
4d ihc &imurh of th. obj..t is<br />
12.1.t6)
Si"At = Sinpstnq +C6e,Cot6,CotH,<br />
(21.11)<br />
Th. cordiG for rhe tuirg ! siring d:<br />
AT, =<br />
(2.7 l8)<br />
Adding these A_, ifto rh. appropriarc I, givcs ibpDvcd valu.s<br />
tunhq inpov€d r:lucs catcutal. $e @.dinaas of rhc Ss fd<br />
&d rcpeat (2.6. r 2) ed (2.6.1 8) ro obrain lh. UT for d€ wau.<br />
<strong>lo</strong>r Ue Moon rhc i$uc is mt! complided s af€.r of panlta h signiticor.<br />
Th€ !ffrr ofrcfddjon h ro dcc'le the anirh dist ne !o rhar the objed is visibl. dei<br />
if n is th@Edcat gonc dom $. horizon bln rhe aflccr of pa6 d b to inc@sc rhe zcnid<br />
d6t fte e lhe obj.cr is wlt .hovc lhc horj@n dd n a!,p.4 ro Mvc et (or rcl n*n<br />
still), Thus for fie Moon the si&dard attjtude is Civcn by:<br />
ao =E - 0-2n5. t -10134''<br />
(2.1.te)<br />
qb.G r is dc pa&Id of rb. Moon eivd byi<br />
(2.7.20)<br />
p s $. g€@eniric dist ncc oflhe<br />
obsrvr &d X, rhe g@ce ric dislece otrhe Moon, p<br />
\2.7.21'<br />
62
is th€ equrorirl ndiu5 of the Eard,<br />
dd ., is th. pold mdids of the<br />
E nL The resl of thc calculatiohs for rhe irlnsil, rising ed lhe stling of lhe Moon arc<br />
th. se d 1hat for thc Sun.<br />
2.8 I{EW LUNAR CRESCENT VISIBILITY PARAMETERS<br />
A nunber ofpdrameleB have been corsidered imponanl for deteminin8 *belber<br />
the new cr€$ent would bc lisible al a lcaiion oo the Eanh oi ool. Tbese weE briely<br />
di*u$ed ad lblcd in thc b€eimins of chaper l. Once tr tiD. of th. Binn or a<br />
paniculd Ncw M@i or th€ conjucrion hs ben delmined a numb.r of p,@eGB @<br />
rquned to h€ d€t mincd. Th€* includ. (i) Tinc of Suroet T". {ii) Tinc of M6Mr T..<br />
(iii) LAcTn - Ti (i!) Bcs| Time orvisibiliry Tr,(!) Ase ofrh€ Moon ar Tb. AcE. (vi)<br />
Arc of visior ARCV, (vii) Reladle Azimulh DAZ, (viii) Arc of Lishr (E<strong>lo</strong>nsatioi)<br />
ARCL. (ix) Ph@ ofcr€s.nt P, md (x) width ofcrcsent w. In vhw of the di$u$ion<br />
ofth. s&onomic.l alSorirhns ed rehniqud in tbis chaptd the* circmst nc6 de c-<br />
co$iderEd ro .xp<strong>lo</strong>E compurdiom of vdbN psm€re6 $al e inponmr ao. $e<br />
lnolysis of l@al visibility oflh€ ncs lunr cr.$nt on $e day ofconjuhdion or d. doy<br />
The fiBl of th.s paludd ii rh. coiju.tiotr of lhe M@. wnn ihe Sm or tn.<br />
rime ofBinh ofNcv hootr. The alsdirlm for lh. conpuraion of (his tim. w6 pE*nl.d<br />
in anicle 2.3. U3ing $is algoddm thc riic of g@centic binh of lhc Ns m@n is<br />
obtain€d. The algorilho tates yee s input and giles lhe Julian datc oflhe tine of the<br />
binn ofnew boon. Il is imporrmt ro nole $ar the i.pur iiy€ai is ior a whole iMbs, it<br />
is o Eal nMb.r €alculat€d on the bdb of lhe exp€cled dale of th. New M@h. For<br />
insran@ the Nry M@n in $e nodn ofApri!,2007 i. dp.ctd msd l7s d.y of thc<br />
,e0 = 2047 +atJ0+t7)/J65<br />
63
An |pDDxiE.r. valu. of "'*" wirh e G@ of f.e d.,s m*! wll. If UE "yd" it r<br />
vholc nmba $c algoriihn si!4 lhc ,!lie D.rG to! $. Nd Men th.t @6 cl@n<br />
ro thc b.gimine ofthc !d".<br />
otuc rhc ,ulid D.rc of$c binl ofrlE Nq M@n cl@n b lh. .xpdrcd d.y ha<br />
b..i found tb. m. b inni.lly conwn.d <strong>lo</strong> lhc UniEEal rift dd darc ed<br />
coneqE ly rhc le.l lih. and ddc. For ln.s convffirioE ftom Julis Dat. to rh.<br />
le.l rim. ed drlc th. tehniqu6 of rhc .rricl. 2.4 6 u$d, Bcfoc thb im. lh. Ew<br />
'lun.rion" h6 nor b.!un s no qNnid ofrlr visibiliry ofrhc Bw l|!w cEghl on rh.<br />
A6ins bcfoE this rin . B.foE this tin. only th.'bld cG*dl" ce b. lat wn b.foa<br />
rh. sutuie on rhc dry of onjudion or a dly o! tvo b.foa-<br />
Ar dr rimc ot@dudion dc M@n n.y b. &rwhcrc wi$in . strip ot widd <strong>lo</strong>o<br />
18 eud dG elipric, i-.. wirhin 50 9 of rhc Sun. ap.n fom |h. ecume of . slar<br />
dlips $G "@.nl'cxisl5 bul dE <strong>lo</strong> irs.xlrEh. c<strong>lo</strong>*ncs ro lh. slai!8 su ir co nor<br />
b. s..n ed ha ndcr ben sn,<br />
Thc ndr inpondr p.md4 is $c l@.1 lim. of suel ed th. M@n $t Th.*<br />
can be conput d for ey day of lhc yd Bins th€ r.chniq!.! of $c anicl. 2.7. Howvr<br />
$ce rehniqEs 'lquiE<br />
th. dctmin ion of th. c@diEtcs of both lh. Sln &d th.<br />
M@n for D dFckd rihe of lrsi! dsing dd s.ltinS of dh of thd. Thc c@diDL3<br />
of th. su !rc obl.incd usi.s dE VSOPE fi6ry of Brdagnon ad fEncou {or a<br />
sihplificd v6ion sivcn by MeE) &d bricfly prc*ntcd in aniclc 2.2 Dd 2.6. Sihildly<br />
rh. D@iF @rdidtcs of thc M@n c oh.in d aing th. ELP2000 of ChrFo -Tot<br />
ed Ch.pont pr*nrcd in dticL 2.2 and 2.5. Th. .oddinal6 ofrh. Sun &d lh. M@n<br />
lhis wr e th. SMtric sph.dcd pold c@dinrl.s (dis1@ P. alatial <strong>lo</strong>nsilud. r<br />
dd d€ c.l.nial l! rud. n. And @n cdon for .bdnlion thc sc e lMtfomd <strong>lo</strong><br />
th. g.on.rric .q@torial @tdidlcs, dght s.cioi o &d &cliDlion 6. Firlly th.<br />
roFcc'tric dSlu e.sion ed d.cliMdon !r! oh.incd itld.s irto cosi&tstion thc
d[dl! of pdallq. Using lhe <strong>lo</strong>cal siddnat timc tb. @rdina€s e then trdsfomcd in<strong>lo</strong><br />
Ioc.i hori@ntd @ordinalcs dr. ahitud. ud &imuth,<br />
Once $e tine of leal sunsct (T) and th.t of Moon sr (I.) d obt in.d lhe<br />
o@ctcr LAG = Tn - Tr is elculat€d. Unlc$ lh. LAC is posnive for the New Moon<br />
dd ncSrdv. for old Mmn th€ cFsent cd ncv.. b€ sen.<br />
Suppoe (..,,r.,r.)ed t,.j,,r.) d ftc p@ie dislanes, aliflic <strong>lo</strong>!8it!d.<br />
sd th. lalirudc of rhe M@n and 6c Su, Esp.ctivcly. af@d ro the ft4 .qtinox of<br />
rhc dD. or sunst at any lMtio. on $c Elnh with tftstiat coordinaies(1.r) on rh.<br />
day, or day aftcr. lhe binh ofNew Msn (al$ oll.d the Csenlric Coijliction oi th.<br />
Moon).Thefi'slsteDinthedelemimtionofih.visibililyoflh€nevlunarcresce,on<br />
thc day of conjuncrion or the day affer, is io dcternihe lh€ actual dy.anical ihe (TD o!<br />
TT) T", oflhc coojuction. Ncxl one requires considcrins the <strong>lo</strong>cal rines ofsening ofth.<br />
Sun &d tb. Moon. Le! Tt dd Tn (Coordimi.d Univdsal Tine TUC) b€ th. tihes of lh.<br />
l@.1 sunsel dd the nooi.eq vi|h Tj < T..<br />
Using rnc .cliptic @rdimLs oa lh. $n ad rh. Men E alculared for th. T6 thc<br />
egutodal @rdimrs of rhe two bodics (d.,4) dd (d,,t,) e c.<strong>lo</strong>lat d usins<br />
(Me$,1998):<br />
s t h\^ )c os\r ) - Tdnl P)sth(E)<br />
cas6)<br />
(2.8 )<br />
st(d) = si,,(l)co(.J + codB)st(€)st(r)<br />
o.8.3)<br />
wl@d, fi. nd emid is in 0r s. qudtul6 t, ed ',<br />
the obliquily orrh.<br />
dlipri. is d$ adjug€d for lh. d, . ed thc b.9 rine Th. tsal Hou Atgl. H is th.n<br />
oblaincd tion rhc difreren@ ofth. l@l Sidftal Tiru (Zsl) 4d th. rishl 6c.nsion<br />
Tnis finally gjves lhe l@d nonzontal c@dinal6 amulh (,1) $d the allilud. (ir) bv:<br />
65
anlAt =<br />
c6(Ir)s,,{t) rz,(d)co{l)<br />
(2.8.4)<br />
si,(r) = sr,(c)si'(6) + c4(d)ca(t)c,r(fl)<br />
(2.8.t<br />
And adjusi.g for lhc rciacdon 6d $e heigh of $e obFNd's l@aiio. above *. Id'l<br />
rbc topoentic @oilinares (,r,,r")ed (,i,,7t,) of the Moon dd fte Sun. esFclivtrv<br />
In alnost all lhc nodels for cdlicst m@mid ins $c ancidl d w'll ts ibe nod'm'<br />
ln. dir@. of eim hs (DAZ = lA,-A-1.<br />
@lled El.tive @'ihuth) ard thn of<br />
aldrudB (ARCV - ," - t.. cdl..l m of lision) d shoM in rhe lisr I, 'r<br />
the iimc or<br />
<strong>lo</strong>c.l 3uEl T, sd/or al lhe b$l Tb Play a vilil ole:<br />
As ihe dgulu sePar.tions invol!€d b.iwen $. Sutr md lh. lunar cesenl at<br />
$6. dnes @ sdaU, wi6ou1 dEh mt th. e of lighr (ARCL) h giEn bv:<br />
(2.8.6)<br />
Wh.@ for larSct ugl6 or moF ffi.rc Gults th. d of lieltr should b. eldl.Ld<br />
ARCL = cdr(co(r-)co{rr,)cd(tra - s,'(r.)si(4)) (2 E )<br />
66
Fi&28!<br />
A@d tun lhc Ft.tiv€ .rlmuh tbc !D of vil<strong>lo</strong>n and th. e of lighr Uc cdlqir<br />
for dli6r ytuibrliry of t|ld qqc@t rlqoits io irL ie c@li
"='"[<br />
(289)<br />
Sinc. ihc acienl dd thc nedieval dnes {E ACE, thc time elapscd siie fie<br />
birlh oI New Moon lill thc sunsi of lh. d.y in quesdon, w4 eo@lly coNidecd a<br />
signindt p&Met€r. Howev€!, in ihe tinc of MuslinvABb $lronom€rs jl had already<br />
b.en c.lied tal 0E ACE is nor a nD&hc al padndcr, stll thc pardet* is<br />
inporlrnl 1o calculat€. This is bccaue oflhe fact that depending on rhe aordinats of the<br />
Mooo dd lhc seaens at lime a very "young' cEsanl ce b. ee.. Amongsl bo$ rbe<br />
anat€u s vcll a rhe prcfcssioBl Grronom.u $@ is aluyr a conFlilion for havinS<br />
$e €ord <strong>lo</strong> s the "youneesr" crcsr eiihd wilh oFical aid or wilhout i1.<br />
Amongst the early nod€ls ofnew cc*.!t vnibility asltunomeB sed $c !.larion<br />
betren rh. rcldrive altitudc (ARCV = alriMc of rhc Mmn - ahilude of lhl Sun) dd rhe<br />
rclrriv. uimurh rDA-/ - e'murh of Lh. Su - uimdh of de M@n' Th. rso<br />
parmeleG arc slill consideFd inporl&l as if DAZ = 0. the crcscenl is venically abole<br />
$e point of 3uEel dd und.r such a cncM$dce the youg.st a @ll as lhc thimesl<br />
crc$qt cd bc ecn. Wirh lugd DAZ wles only oldtr md thc thictd ccs @ be<br />
How lhick or lhin is lh€ cEsenl al any lime cs b€ ddemined once lhe<br />
s.pmriotr (c<strong>lo</strong>nsaiion) b€twn rhc su ed $c Mmn hd bccn dchi'cd usi.g (2 8 )<br />
sith €nh€r horizontal coordin.les or rh€ €quatorial aordintle! This e<strong>lo</strong>ngation or Atc or<br />
LiSht (ARCL) hads to lh. Phde (P fiaction of the illuninatcd lund disc facing fie<br />
obsdr) of lb. Mftn using (2.8-E). Ho*ev.a the lhic<strong>lo</strong>N or $e widlh (w) or rhe<br />
c..rral pon ol the creot des not only dcFnd on thc phe of the M@!,<br />
't<br />
also<br />
depclds on rhc Elnh'Moon disrece, rh. sMc cm be ohaitcd usins (2 8 9) Dte besr<br />
inc of lbc crcsdt visibihy sqgg€ned by Yauop is criical fot siShting lhc crcsn1<br />
u'dd mdgjml conditions &d ce be @mpur€d sine (2 E l) sisndne i5 crirical when<br />
68
THE SOFTWARf, HILAI-OI.CPP<br />
In this worl( a softwde k deve<strong>lo</strong>p.d for the odysh of rhe fisr visibiliry of new<br />
lue crc*dt dEr is simil& io mturc 6 rh. M@'calc by M@ur (Msru, 2001) dd<br />
Accurare Times by odeh (odcb, 2006) but rhar can b. lscd ro compae aI rhe<br />
compur.rional and thc ecie.l dd Doden visibiliry nod€ls. T1| lisdng of the poelm<br />
Hila<strong>lo</strong>|.cpp is giyco i! ApFndix 4. Tn progm f€nrca d bri€fly de*dbed b.to*:<br />
Th. prog@ LinAnal ucs ih&e dala filcs, rwo for input ed one for oulplt. The<br />
two inpul dala il6 e (he files For.ad.rv thal conrains rhe padrcr€E 4, ,, and., (s<br />
described inanicle2,6 lbove ). The pdmetcs oflhis dala file de adtu8€d in rabtes2.l<br />
(A) <strong>lo</strong> 2.1 (F) in appcndix 2. 'Ih. ol]],er itpnt tie is tlc .tp2MA& rhat conlaiE the<br />
paluercs as de$rib.d in rhc anicle 2.5 above. The pdmeles of rhis dara file d€<br />
dds.d inbbles Ll (A) 10 1.3 (D) in appendix L<br />
Thc third fiI. u*d by lhe prcgrd is *..t lhal is optional dd is n*d only vhen<br />
fie Esults of fi. cobput2rions in Oc preetm m EquiEd to be sbEd. Tl!. vcGion of<br />
lh€ progr.n Hila<strong>lo</strong>l giv€n in appendix 4 hs the lile nanc scrrrgaA, rhar sbres lhe<br />
onpulalional Esulrs of a sinelc .x@urior of $e prcgrm. fte i.fomations stoBt in<br />
o. No., rhc obsenarion nhber thar n scncally digned by Odch (odeh,<br />
2004).<br />
Date. datc of obwatioo,<br />
Long., th. <strong>lo</strong>ngitudc of 0E ple ftom wh€E rh. clesnr is ob6eN.d<br />
Latil., th. ladMe oflhc ple<br />
Eld., the .lcvarion of rh. place above s! levcl,<br />
T.mp., cnimred rcmFdturc of$e riDe of obswarion,<br />
Hmid.. .stibaied relative humidny oflh. plac. at rhe rine of obs.Nadon<br />
S@.i, lhe l6al tinc of sNet<br />
69
9. JD of Conjunct, rh€ Julie Dat of the binh of.q M@n or the lsr<br />
10. As., lhc a8c ofrhc Moon.r rh€ besr tim. e@ding 10 Yal<strong>lo</strong>p (Yal<strong>lo</strong>p, 1998),<br />
ll.<br />
L{C, the diff€rcncc berNcn th€ Moo. *r dd rh€ sunsel. it minucr,<br />
I 2 ARCL, dc ot light or e<strong>lo</strong>nsetion of lhe M@n fion th. su at the b.st tme, in<br />
ll, ARCV, the eladve altitudc of fie crc*ol .t the b.st 1ioe, it d.crees<br />
1,4. D,AZ the rclaiile @ibulh ai the best time, in degrc.s,<br />
15. Widtb, the cenlral widlh of lhc Fsr€nt al bcs1 rime, in dc minues.<br />
16. q-val. rh. visibihy ptue1q dcfined by Yal<strong>lo</strong>P (Yal<strong>lo</strong>p, 1998) <strong>lo</strong> h.<br />
discssd in d€rail in chapl.r 4.<br />
17, Phi+della, fie dglc thal lhc.cliPlic dalcs with lh€ venical on th. wslem<br />
hoii2on, in d.sE.s.<br />
I E. Mlalit. I,<strong>lo</strong>o. s 4liptic latiiude in d.8e'<br />
19. M<strong>lo</strong>ngi! Moon's.cliptrc <strong>lo</strong>n8itudc. in d.gtus-<br />
20. S<strong>lo</strong>ngil., Sun\ *liptic <strong>lo</strong>ngitud., in dccres<br />
21. M-SD, dgule s.minid.tcr of th. M@|L in @ oinut s.<br />
22. As-Fac! e of sep@lioh fsctor d€fined in l.ler chap1e6, in dcgrc6<br />
21. R-En, cslimat€d Rip.n $ Fucrion vatue dclin€d h chapler I<br />
4, R-A!er, averas€ Ripeness funcrion valu€ dellned in chapl{ 3<br />
25. R-actual, acbal Rip.ns Fsction vtlue dcnled in chapls 3<br />
26 DR+s! $e diflecnc€ of R-rclualand R-Enimaled<br />
27. DRnc! thc difrccne of R-a.tu4l ed RnErage<br />
28. MoonS M.g, Moon\ mogniiude al the besl lne of Yal<strong>lo</strong>p,<br />
29. Lio-Mae(lin€), lh. LiDiting Mt8nitud. of the stv ned |ne c€s6t s<br />
defioed by Scha€ff.r (Schaefd, 1988b) a<strong>lo</strong>Ds wnh ih€ Yal<strong>lo</strong>p's besl dne in<br />
udveEal dne dis.ts.d in cha .r 4.<br />
30. SI'(DME8), tbe univ.ssl tine wh.n the conrrdt ofthe stv bdelhes dd<br />
rh. Mmn\ bljehBt just tum in favou of thc Moon dd 0E ditr€Gnc' of dE<br />
70
m.gnitude of the Moon dd $c linning m.gnitutle of rhe sky ai thal monen!<br />
I_<br />
B6(Dmag), fic divdsd rimc *h.o U. @ntrasl of rhe sry bnSh,ES ed<br />
th. Moon\ bdghret is b6r in favou of rhe Moon ald thc diflcn@ of $e<br />
ha9irude of rhe M@D ed thc lim'ring m.gn'nrd. ofih. sly al rhar monen<br />
32.<br />
Lsr(Dmae), ihe univdal tme wh€n rh€ conrmst orrhe sky bnshhess ed<br />
tbc Moon\ biehd is lal io &vou of th. Moon ad rbc difcrcn@ of $e<br />
nagoirude ofthe Moon ad rhc liniring magnitude of$e sly ar th.r momenr.<br />
A3 thc .x€cution oflhe proghh b.gins ir prompts for ob*Nadon nMb.r, dale,<br />
nonlh, y.&, <strong>lo</strong>ngirude, Iaritude od cl.vation abole sea l€vel of th€ phcc and $e<br />
esrinated ttmperaturc ahd estimrd r.larilc humidiiy of lhe place. This p<strong>lo</strong>mpr is<br />
iniliared by rh. lunction tzrdd,r.rtnd callcd by<br />
the tufuri,on nt i n n u t i<br />
"..<br />
Ticn rh. prcg@ c,lls for $e tundi@ math .h4tg.. This tunc on n6r<br />
ddmin€s lh. Julia Date of dE dDc of n.esr @njwdon or rhe bini of ncw M@n<br />
crlling thc fucrion /t ]@_D@r. Th. tu dion r'r_r@_r@r is bdcd on .tgorithn due<br />
ioMeeus(1998)discqsedinadicl€2.3abovc(fomu162.3.1.ro2.3.25).'Iletuncrion<br />
rorr, cna,a. $en calh the tu.ctior renir&rr lhal dekmines the tihcs of lhc sunsel<br />
dd the Moon set tttroud fiD.rioB tun-q.t 6d mon_se.Ih€ tunciions rzu_r.r sd<br />
tutu_sa t. b6ed on lh€ alsorithn dhcus$d in aaicle 2.7 abov.. Dr functio.<br />
rdttirgt.le calculars rhe b6r rimc for casent visibitiiy aeording io @ndnion du€<br />
to Yal<strong>lo</strong>p (1998) di*ussd h a larq cha .r Tnis is fol<strong>lo</strong>wd by @hpul!fion of the<br />
Julie Dat 6d dB Exlcnd€d Juli@ dat corcsponding ro rhe best rim. 6ing rh€<br />
tunctiolirri.ndrl. rhat t bs*d on rl| algorirhn pEdred above in anicl. 2.4. This<br />
Iesds ro fomulation orlhe rime {sM€n1GIs dipused in adiclc 2.1) fo' rhc Elp2ooi)<br />
md VSOPS lheories for the c.lculation of coordimres of lhe Moon and ihc Su<br />
respectively. Th.e coordinates e calcul.r.d according b the algorithDs in anictes 2.5<br />
dd 2.6 iopl.m.n&d in fucdoB a@r_.ou.t ed t\. su"_.erd Bpeilely. B.fore<br />
7l
h*.@rdin l€s @ c.bdat.d th. etr€c1oftuhio. dd the sidd.d tihc coftspondins<br />
10 zdo hou uiversal tide, for the dat€ consideed. are calculted using the tuncdois<br />
,ttttid ad srl_tiw, E#iv.ly, The infomatioi/d.tA thu fe Scn nted is displayed<br />
od *'€.n usins rhc t$aiqs ouin<strong>lo</strong>, rtbpt t scootd (.@fitn^.s of 1he su.) ed<br />
dbrldr_n rord (coodinatcs of tbe Moon).<br />
Finally, fhc tunction nainrcutir. crLtl^r.s md dbplaF on $en aU rhe<br />
visibiliiy parmetcrs dircused in pFiou.rricle (2.8) &d lisred ss l0 to 16 dd 2l<br />
abov. in lnc clm anicle. O$d pffi.r.c (l7<strong>lo</strong> 20 dd 22 1o 12) re @mpui.d in<br />
orher functions (r to 20 in tunction r@r_cMtt a\d 22 to 32 tt liM'si\ brl<br />
display€d a<strong>lo</strong>.g with these pdmet.rs. All thc Ddmd.s lisbd as I <strong>lo</strong> 32 d€ wiren <strong>lo</strong><br />
the outp dda file J.r"8ear in nm<strong>lo</strong>ulrro4, (pam€rs I to 29) ed tim._nnse<br />
(p@ncle6 30 to 32), Tle tundion /ltr,r,, is ih€ reproducrion ot Scba.fer's pog,aD<br />
(S€hs.fer, 1998, Bogd, 2004) <strong>lo</strong>r delediiirg th€ liniting hagnitude of dy point of<br />
The Esl of the functions used in lhc pregrm aE lisLd dd briety dcscibed<br />
/.ar.r..l chects lhe1hd the y.e u$d is lcap yed (ac@rdi.8 ro crcsorid<br />
rule) or rcr lfrhe yd is lap lhc tuncdon rerums I orh€fli$ O.<br />
.orwt_r'ru convens an eelc inro dcgres, arc minutes do arc *cond3.<br />
.,,rzt_rtu convens lime in hours into hoE, ninuIs &d s*onds.<br />
hodlw Btms rh. @jnd.r .ff4 dividing th€ inpur b, 160. 6is fucdon<br />
pamt<br />
h 6ed ohain 6e mst. in the tuB€ 0 deercs b 360 degE6.<br />
calculat s rh. afr*tl ot psrlts in i8hl acEion ad decu.adon<br />
to cct th. ropcdFic ddr as1fuion 6d d.ctinarion and is bas.d<br />
on step 8 of alsori$n in anicte 2.5.<br />
itr._r.. rhrcud ,rr.-do, ,nd d"c_sc thrcugh d.._r,,: Ttcs. tundions altows<br />
chdging tine (al ld.b ofhourr ninure ad ssnds) and dare<br />
(at lev.ls ofdale dd honrh).<br />
72
In fer sihin lhc tuldion di.i@{tt. lhc prcgr&n Hil.<strong>lo</strong>! ha<br />
^<br />
do-|9kL l@p<br />
rhar remiDrd *tEtr rhe iddrifid za,r rcccivB V' (pl*in8 dld le,) fion thc Br.<br />
If my oths key is prcsed lh. pregm rmdN in wait ndq PEsing Panicdd k ls a<br />
is obvioN in the er.l{ar. cohbilsriotr vdio{s&lions e iniii.l€d likc inc@ing or<br />
deoeasiry tim€ dd datc, chegilg tmpdal@ or hlmidity o. wiling to th€ output nb<br />
r.rngafir. bitiad.g any ofthFc.ctioro Gp..G th. *ftution ofall lhe conpulalions<br />
wilb new timc, dale,lcmp.dture or huDidily. In cd. ofpre$ingP all lh€ data is winen<br />
ro fie out file s.r'"a4.t in one lin€ a.d prc$ins 4 Mit s only rhe $lected !alu$ oI<br />
tine and conesponding value of thc diffelcnce of doo.\ magnilude and $e liniling sky<br />
masnitude. Tlese f€atues of thc progm Hih<strong>lo</strong>l (varyin8 dne, dale and !@lhcr<br />
conditio6) nEle tbe progrm norc dynoiq .s compdd <strong>lo</strong> Mooncalc of M&ru<br />
iMdd, 2001 ) ad A@uai. TiF. of Odeh (Od.h, 2006).<br />
In view of lhc prcbls of d.rminiig th. dlr of the fi6l sier ing of new llld<br />
c@.nt in this cnaFd w cxp<strong>lo</strong>Ed.ll thc Mjor 6pet of conp arional cForls. Th€*<br />
IE sm!.ndylicd dyn.Dic.l lhddcs VSOP87 and ELP2000 d'at dryib. th.<br />
motion ofplseB <strong>lo</strong>ud thc Su sd oflh. M@n reund the Eanh. The* e m.<br />
mosl rent dd nost accunt. avlilabl. <strong>lo</strong>oh for the delmi.ation ot<br />
ephendb oflhc Sun dd th. Mootr.<br />
Ite ateonlhm lhat lelds <strong>lo</strong> lbe dct min.lion oftb. dynmical lime ofde lud<br />
@njucdor or lh. binh ot ncw Moon.<br />
Tn€ pmblms ssociarcd wi|b th. dynamical &d uiv€Eal lim€ ed related<br />
isues. Wiihout having a @mpl.l. kno* bow of lhse isues apprclrjalc tbe<br />
argmcnt for the d.lminadon of lud ed dE $l& cmrdimres co not be<br />
1)
TL dl irpde dgsitor tu d. ddroni4 m. tr..r,r.l od <strong>lo</strong>cd roc<br />
dn . o{ th. @ rd 6. md* lllth! s* d6itr of ti.<br />
dD.r lb pcr@i !rd&d tflt t Fb&o of di- dShdrS o{ rtr<br />
le or*dt ci rdt!. da-!.d.<br />
lrib I tdrwh ||I&tbt r of d| 6..c tt{itr d .|6mi.l G.hiS..<br />
erl .ltdl6or | .orio!.. F!!im ir redd b &r .U t ,.w lull<br />
Tto 1b ct# &ir_.r .ll tt cqdrilrl ,trrit r@id.d eith it.<br />
p<strong>lo</strong>bLa ofib. d.ol<strong>lo</strong>ilirS tl. dry of6rr jihli!8 ofew lurt.!..c.d<br />
71
Chapter No. 3<br />
ANCIENT, MEDIryAL & EARLY<br />
2OTH CENTURY MODEU;<br />
For calendarlcar purposes .s we as an Interes0n8 and chalte.ging<br />
aslbnonlc.l p,obleh, jn rhe hbrory ofmanktnd there have ben constsrenr efiorts<br />
ro delermano when and whererhe nek tunarcro@nt wll be lrst seen, t]e,esutrs<br />
of these ea<strong>lo</strong>ns have boen numeroG 6nd have been based on dtlle.ent technlques<br />
6nd tools.lhose results can be temed as 't odets/Crireria tor the oadtest vtltbi|lry<br />
or n€w runar cfescenr,. Most ot the earty modets are de ved dtioc y trom the<br />
obs€oations of the new tunar crescent and are enDnjc.tin natlre. Some are based<br />
on th€orellcalconslderations. MGI tmpon.nt of these eftons hctud6:<br />
The Aabyroni.n rule of thumb .when the age ot the .ew Moon is 24<br />
hooB only then the crescent can be seen,. fte redlscovery of the<br />
more mrrhematrc.lly lnvolv6d crfte on ot Etong. on + LAG > 21.<br />
lhal w6senunciatedw€ beforechisrian or. (Fatooht et at, 1999).<br />
)<br />
Tho cdena deveroped In ihe medtevat periods by Mus ms/Adbs<br />
ba*d on obsenarion, sph6ri6t vtgonom€try and lhe ephemerid of<br />
lhs Moon and the Sun (8rutn, 1977). This happen.d as earty .s rhe<br />
llr)<br />
The exp<strong>lo</strong>ration or Foth€ ngham (1910), Maunda (1911) and orheF<br />
ln the fi6l quarter ot the twontl6th @ntury. These effort! were moG<br />
ot slallsti€l in natur€ based on th€ obseda ons mos y in Room by<br />
75
In rhb ch.pter we stan wtth a b,t.f dtscGston on retativety Ecent G<br />
di
abbelialed s ARCL but shoM 6 aL in th. figue. SD is the difidce b.isFn the<br />
alrirudes of the su dd rhc Moon lnow q "dc of dcs@nr" d€noi€d 6 ao (ako call.d<br />
.e of vision dd abbrevi.tcd s ARCV). Not that the aidnrde of lh. €Esenl abovc<br />
hodan is,l $ lnal a, = r + ,l As rh. ooitr( A is al lh. se altnude 6lhc cFsenl dd<br />
B at lh. sme depBion d th. su, A8 is .quabnal $pdti@ b.teen the Sun 4d rhc<br />
Moon called "arc of *pdrion ' ad is .qnivd6t b rhe Moon*r - SuNr tig shoM in<br />
the fisld d as The fig@ cd bc u*d to h.aua fi€ 4.s ofdesenl ed liehl for a<br />
Sivcn lalitude I once an accurare enough .phemetdes shows the dc of sepdalion and<br />
wheE to <strong>lo</strong>ot for the dimly illminaled cBcent againsl lhe briehl evening twiliSht.<br />
lig No.3.1.1 Angule Pamcters as@ialed wilh sighling ofnew lunar crescenl<br />
l, modem tines the earliesi Ffee.cc of ey syslemalic study of the earliesi<br />
sigbring of nev lud cEs@nr is thrt du. rolhc.inShm (Fothainghd. l9l0). He refe6<br />
ro l2s enlury Jryish phildopher Mcinoni
wA diislaled by S. Oddz 6 Crlr of Moituntd.s, Boot Thr4, treatite Eigtu,<br />
SmdiJicdtion ol the Nee M@a Yelc Judaic. Seris, volh. XI, Yatc Univesity pE$,<br />
No Hav6 CT, 1956). Maimodidcs mrk w studi.d by von Lifl.ow in<br />
SiEunghberichk da Wiehu Altd.nie, Math-Natrre, Cl6e,lxvi, (1872), pp.459-480,<br />
ln his *ort Maimonides nak6 rh. snalLsl visible phe ofthc M@n depqdenr on trc<br />
variables, as claimed by Forh€.in8hd. Th.se veiables bave ben tem€d ss rr rr€<br />
etonAorion otthe Moon a\d tle aplbre.t ahgle of tision - F6\.noghe slarcs rhal if<br />
by hgk ofrision Mai1onides m.es the diffeEncc of the Sun md the Moon in rc.irh<br />
dirlaccs then th. nle of Mainoid6 .nd his om rule for hinimlm visibt. pb4 of<br />
Moon ae n€dly the enc Howevcr he f!fther says lhat Maimonides rute sivcs slidrly<br />
<strong>lo</strong>wer oininun alliludes that could be duc tc, b.(.r ob$fring condirio.s in Jeruslem<br />
lh& in Atbos. Unfodu.alely Fohednghah .dmik that hc could not sumcienrly<br />
undedand Maimodis' dithmdiol nelhod dd has not giecn a dchited @ounl of rb.<br />
same (Foihciinsham, l9l0).<br />
Lder, Bruin(Bruin. 1977)gives a moe debiled eounrof any medievalatenpr<br />
ol solving fie probl€n of findin8 sronomi€al €onditiotu for thc fi6t visibility of<br />
crsc€nl. De$ribinglhe Isl.mic astonomical procedurcs Btuin says $a1Al-Khurizini<br />
gives mathenadcal rules and rablcs <strong>lo</strong>r pedicling thc new cre$e.t where6 AlBamni<br />
peeds a conplerc soludon. He aho nenlions lh€ lato accounl ofMo$s ber Madoon<br />
lnd clains rhst ibn Mamoon larsely fol<strong>lo</strong>s Al Bal<strong>lo</strong>i. One oflhe nost inponanl dp.cl<br />
ofBruin's accounr ofrhc ef<strong>lo</strong>ds oftbe bedicval era is rhar Anbs^rulins orthis ea had<br />
olEady rcaliz€d rhe sienitlcece of cE$cnr widrh. This inpoiant aFcl enaincd<br />
hissins frem all th. najor cooiribudons of lw.ntietn century bc<strong>lo</strong>E Bruin. Bed on this<br />
d€scription ed thc modein <strong>lo</strong>owl.dge in ihis work lhe sme h6 been exp<strong>lo</strong>cd dor.<br />
.xlecively od is pEsnt d latu id the chaPler.<br />
lilcatue that app.ared duing 2od c.nrury lhe gencral rule of<br />
Baby<strong>lo</strong>ds for first visibilily ofnew lunecresc€nlis:<br />
78
at the titu of lual su6.t shottd be g.ater than 2't hows<br />
the su6el ohd the no6et thDuld be g.ot* than 12 tim<br />
(l.l.l)<br />
l2 rim. deBes is cquivslenr ro<br />
;$ ofo hour or48 ninu$ (Brura I9.llv4 le944<br />
lto$ev4, il wd Fat@hi, Stephe$on and At'darghdlli who havc<br />
'xP<strong>lo</strong>rol<br />
lh€<br />
cf<strong>lo</strong>n5 of lhc Blby<strong>lo</strong>nians dlsiv.lv &d have epon€d rnal eording to lhe histotiel<br />
Eords of re* c.Mnl siSlftings ol PG Christie m lhe cderion attibured h<br />
Baby<strong>lo</strong>nis is d ov€r simplificalioi (Fabohi.t al 1999) Ac@tding <strong>lo</strong> $en $udv oi<br />
209 recoids of Posi{ve new cr€sceol 3ig} ing extdctd fom Babv<strong>lo</strong>ni&s Astronon'car<br />
di&ies. the Baby<strong>lo</strong>nitus had succceded in ioduladng a lrulv mathem'lical lunar lheory<br />
$hich ihey 6ed to pEdicl !&ious par@ele$ of lumr notion Unfortumtelv wrlhoul<br />
presnline my iheorerical d€laih of Babv<strong>lo</strong>nian eE lh'v appear 10 be<br />
"ilicdl<br />
aboul<br />
Bluin\ slgscsion thar Babv<strong>lo</strong>nia criterion was whar is mentioncd above (l l l)'<br />
lnst€Ed.1hey clain that Babv<strong>lo</strong>nian ctilerion wEs as Fol<strong>lo</strong>*s:<br />
Tnc ies cEsce.r is een it<br />
'' E<strong>lo</strong>ngutio, (ARCL) + nooe|st6't laE tine (Uc)<br />
(3.12)<br />
Even, rhis cdrerion is apon d on b6sis of the suee€stios ol Ncusebaud (Neue'ba@r'<br />
1955) &d nol 6 a rcsult of ov d'$rip$on of a madehalcal lh'orv Ttev have<br />
nehtioned two svst€ns of sol& moliors on *bich th€ lt@ rhsrv ol Babv<strong>lo</strong>nids las<br />
based, but hoq lh€ new rul€ they hav. attdbded to the Babv<strong>lo</strong>nims wD drived al $ nol<br />
giv6, They have gone on <strong>lo</strong> plesd .lifleant valucs fo! th' @nslul on lhe<br />
'kh<br />
h8d<br />
;ideof(3.1.2) Th€se lalEs Engc fton 5 DiniDum of2<strong>lo</strong>ro a maximM ol23" lnlhe<br />
p!€ent work we adofl fte fol<strong>lo</strong>wins as a bener critcrion alfiibutcd <strong>lo</strong> de Babv<strong>lo</strong>'ians as '<br />
slss.sted by Fatoohi el .l:<br />
79
80<br />
(r. I r)
The Mon ar divi.s this qitcda 6 d.sc'ib.d by th*. aurhos is thar (D ihc m of lisht<br />
d6dibes how bnghr is th. cFac.nl &d (ii) ihe moon*i*utuel lag de*db. hov <strong>lo</strong>ns the<br />
ctcse @uld coaii abovc horian.n r th€ $n sl. MoE is rhe vql@ ofqch ofdrqc<br />
paluelea moE is the po$ibilily ofsishdng offie ce*cni. Howver, drcr. h6 to b.5<br />
conbined ni.imun ofthe two dd the sigh{ne Eords wcre the only heans to verify lhc<br />
crilerion or diving al the (it€rion. Using $is condition Fatoohi a al hsv€ Fponed thc<br />
aslhs of lhcir @mpu|,alioB for 39 rw cEsnt 3ighings rhal in lud. bolh lhc<br />
Bab)<strong>lo</strong>nirn cEscenr qighr ing Rordc tud thes'ghtine Ecords Eponed in $e 2Odcenrury<br />
liredtule. Th. aulhos have rcponcd ihat oul l99 cdcs the Baby<strong>lo</strong>ni.n c lerion was<br />
succestul in 98.7% of fic posilive siehliog {cE$e.r claioed to h.v. b.en sn) c45<br />
but fsiled i. 45.7'<strong>lo</strong> c66 of the nesalive siShine GF$.rt no( *o) ces<br />
In r€ccnt lin€s a number of orsdialios haYe smnged for coll.cdon of €r.*ent<br />
sighting or non{ighline @ords. Th*e inclu& Isldic Ctffint Oberyalion Prcjdl lh<br />
So h African as<strong>lo</strong>nonical Obsaatory and rheii websiFs Sinild rccords are reponcd<br />
ro aid colhd€d at lhe wbsile @!sS.h!i!s.eol1 Morevet, lhc ldrgest data el<br />
yd availabl. in lhe pubhh.d paFG is lhal by Od€h (OdGh 2004) we hav' *lecred 463<br />
of lhes dords ofdening cEsnts el.dcd for tE compdison ofrhe nodels studied in<br />
Usins (l.l.l) appli€d <strong>lo</strong> the dar. 9l clet€d for this wlk the '6!lts<br />
u€<br />
pEs ed in the lable no. Ll.l. ln this lable ollv lhose rccords are prcscnled when thc<br />
crescent wa reponed to hare ben seen wilh or snhoul ev optical aid ad ARCV +<br />
LAG is les lha 22 degr€6 Th. rabL is en€d on ARCV + LAc It shows lhal thcrc ft<br />
claims of opli€llt unaid.d sighinss of new ce$enl tol salisfving $e Babv<strong>lo</strong>nian<br />
cileion gi!.n by (l,l,l). The compl€G dola sel is P!€sent€d in the Appe'di\-ll Thc<br />
table i. App.nditll is en d on visibilily colum N in d€$ending older, so thal all<br />
oflically @id.d visibility ces of crc$cnt sighing in ordtr of ARCV + LAG (in<br />
dcSt*t app.d al lhe top of the llbl€ The table in Apt€ndia'll sho* S No ihc<br />
obseffation No. (odeh, 2OO4), dare ofobsryatio!, latitude ed <strong>lo</strong>nsitude of rhe <strong>lo</strong>cation<br />
of ob6dtr, visibihy N' for uaided visibilitv' 'B' for visibilitv thrcugl binmld, T<br />
8l
for visibilny thftud rel.$ope. IlE vis'bility @lul6 @.mPty if rhe cjt*nt M not<br />
Fen and co.tains V' in an apprcpriatc colLlm if the c!.sc.nl is sen. The esl of lhe<br />
coluens oflhe nbl€ @n!in age ofMoon, LAC, ARCL, ARCV md DAZ. Th€ lasl file<br />
@lmns arc ior the five models coroid.rcd in this chapler, The colllm h.dded A is for<br />
This tabl€ shows $al out of 196 daims ofopdcally unaided sighlihSs oftbe nes<br />
cresccnl there @ only cbes nol saisfyjn! lhc Baby<strong>lo</strong>nid critmon. HoEler, oul of<br />
267 cdes vhen 6e cc*cnt wd nor e.n wilho any opftal aid ARCV + LAC is<br />
gleate! thdn or 6qul to 22 deg@s i. l07.ses. Thus for pdniw siehli.8s rh. crit€no.<br />
is foutrd <strong>lo</strong>b€ successlul in 96.4% c.ses dd !frongsl lhe hegalive sigbdn$ ii su@ssful<br />
in 59.92% css The sucess of€ model in desciibihg posilile sighlings a<strong>lo</strong>nc is tul a<br />
cal success ofa nodel. Thc model is not suned seu if n is nol able to dc$tibe the<br />
negalilc sishfng 6 wcll as ir d@s the p$ilive siShtings.<br />
SOME SPHERICAL TRIGONOMfTRIC CONSIDERATIONS<br />
Thh hd be.n m.ntioned e&lio rhal ior rhe p<strong>lo</strong>blcm ofsiglting of new lund<br />
crescenl $e orienlation ofthe ecliplic plays md inportant <strong>lo</strong>le (article 1.2). Thecfoc, il<br />
is inportant b rrisil lhe ne|hods thai lead <strong>lo</strong> lhe ddelhimion oflhe mgle lhal the<br />
*lipric nlkes wilh the horircn al the tinc of snel on the *$em hotian (il is<br />
panicularly significer in view oflhe fact that the new c!e*.nt appeds c<strong>lo</strong>se <strong>lo</strong> n). This<br />
angle lalies as agaiNi rhe lixed celesrial equa<strong>lo</strong>r lhe orientation ofthe ecliPtic s<strong>lo</strong>wly<br />
laries lhroueh the year. lis fislE no 3 2,I on lhe nexl pag. shovs lhe wcsrcm pan of<br />
$e cclcnial spheE wilh impodanl points &d angles m d de$nM be<strong>lo</strong>w:<br />
w' the west cardinal point,<br />
t, rh€ v€ml cquinox, inl€eclion of.cliptic dd cqu<strong>lo</strong>r.<br />
P, the Nonh c.l.sti.l pol€.<br />
82
Z qv - r rb ouiq'iry of6..dinic,<br />
z 8}'r1 - 900 + 9, 9 bdte lb.lditd. od6. d...'<br />
Z ZSr - 9 + 4 lb dSb ofrb.cliP(ic xdft 6. v..dc.l,<br />
rs-d-G5tlirst d.clhnioofrn tu,<br />
PZ - 90'- o,<br />
St - e 6. <strong>lo</strong>nSili. ofd! $'!<br />
FrsNo 32l<br />
FMn 1! 6su! 3.2.1 in lhc tPhdic.l ttrtgL SPZ<br />
'pplvi'g<br />
1.l{ of ctsiE of<br />
3phicrt ti8oiodt tt.rdA non &. !id. PZ oe gBs:<br />
tlne=cort.o.PSz<br />
(Jr.r)<br />
'ft. g@d |!l ioo b..e..o d..lid<strong>lo</strong>r 5 .d <strong>lo</strong>nSind' I i3:<br />
.int - doP co.. +.o.P.i!'rin r<br />
83
(3.2.2)<br />
(3.2.1)<br />
FDm ridcle sh, st nir8 !r Pr:<br />
@s(PSz +@+ A)=rer.@r,1.<br />
(3.2.4)<br />
usins (3.2.2) !d (3.2.3):<br />
0.2.5)<br />
6d (32.t) tosethd *irh (1.2.1) leds <strong>lo</strong>:<br />
(1.2.6)<br />
(3.2.5) sd (l.2.6) thd sivc:<br />
Ju"'<br />
This show that e + A or the .ngl. thlr ihc elipti. nEt s wirh thc v.nicsl is 5.e!<br />
dcp.rdcnt 6 n d.peo& on the <strong>lo</strong>rgitud. ofthc su only for a fix.d place {or latirude 9).<br />
ln th. Esr of rh€ di$usion in rtis snicl. rhc anglc 900 - q + A is d€nor€d s V. In dE<br />
ncxl .!ticl. etr@ cv.r + A is L*d it h o<strong>lo</strong>l&Ld on ibe t 6is of (3.2.7).<br />
In @ the dslinaion of th. Moon is $u1h of th.t of lhc Su in NonhqD<br />
H.nisphft (md north of the Su in th. rcuth@ hemisph*) it is possibl. thsr d.n<br />
.ncr conjuction th. tr.w llM cKar $i! b.f@ lhe slNr in which w ir ir simply<br />
84
impossibtc io 3a thc cr.!ent, TLse circmshnc.s aF shom in the 68ure 3.2.2 on the<br />
iD<br />
iii)<br />
Cl, lh. cclsli.l .q@<strong>lo</strong>r, 1 i! th. vdal eqlinor.<br />
TS is the diuFal path of th. Sm thar is just *riirg al s.<br />
DE b thc diumal palh of the Moon th.l s€t before thc sUNl 3t E.<br />
Ds = 6M - 6s, letpendi.ula.<br />
to the c€lesial equ<strong>lo</strong>. is lh€ dif.rence of lhe<br />
decliMtion 6M of ihe M@n &d the deli.ation 6s of rhc sun. Declinalion of<br />
the Moon t south of lh€ sun md rhe upper limn of Ds is 50 9'. lhe inclihalion<br />
oflund orbit to lhc trliplic.<br />
v) zcvs = 9Oo - 4 is the agle bctween $e c.l.srial equ.tor lnd $e horihn<br />
NS whee C teprenls lhe <strong>lo</strong>titude oflh€ Pl&. TheztSD = C<br />
vi) DM = dM - qs, lhe dilreEnce of risht a$cnsio. dM ofMoon and rhc rishr<br />
eension os oflhc Sun. For higher lairudcs DE nav b. htgc al<strong>lo</strong>wing laigc<br />
vllues ol ncepliE LAG aftr co.junction<br />
vii) sF = pM - 9s : 0M is p€,Fndicultr ro fi. E lifljc is $e difiercnce bctw*n<br />
lhe c.l6ial taliluds of the Men and lhc Ss As for DS Sf never €xcads<br />
rhelini<strong>lo</strong>f5'9'.<br />
viii) FE (E' .ol showo in rhc fisure 6 it almost coincides wilh E) h ptrallel <strong>lo</strong> the<br />
ix) lislhe<br />
Vemal Equinox ed Z)6t =<br />
horizon. Dep.ndniS on laliude C and<br />
v h !h. angle of lhe Ecliptic with $e<br />
lh€ se@n, 'Y<br />
naY lary ftum zeto<br />
(vhctr ecliptic is a<strong>lo</strong>ns the botizo') <strong>lo</strong> 90 deB€6 (when €cliptic is<br />
pc+endicula! to lhe holzon) l! cas' when v is small DE thal has b be<br />
paEllel <strong>lo</strong> the 4liplic is nuch ltrsei tha' DE (ftol is paEllcl io lh€ cqua<strong>lo</strong>!)<br />
and hdc. E sd E aE mrch s.panted Tlle det€mi'arion of$is oSle md<br />
its sienifidc. shall be di$sed in the n'xt drcL<br />
x) I'lt figurc shos rh€ SM is jN| about o sel in a pl&e oI sdall b n€diud<br />
laftude d , $e M@n havins ddlinarion eulh of il' sun ha al@dv *l (dd<br />
wd set at poinl E) $ th.l lh€ LAG = TiG of Mdnct Tim of suNer is<br />
E'
fig. 1.2.2: G@tn r.y ofP@riv. Ag. .rtd N.gaiE L.g<br />
A@rding <strong>lo</strong> rlE luN elqdr b.!.d @ IIE binh of Nd M@ b.bc tlE ssa<br />
tha nd lun|r mn$ b.gio .l dr tirc of o.!a 6 $ii vrt Ming- How6, for tll.<br />
lu@ c.lsds b.!d d rn viltbilitr of $. n4 s@4 w hN mfli &e.ol<br />
b.gii on thb cming B n b 3inply ioF{bl. <strong>lo</strong> $ tlE w luu c|!g in this<br />
cinhtee. Fd plE witt lrSc ldind.. d|r dbw for llre v.l6 of DE |,d h.@<br />
DM ln. tri&gl6 uidd 6!&l6.riB o rbt tr rtrdl .dd.!xl ..otl b. tBt.d s<br />
ryhdicrt r.iugl6. tlc! d y @ll <strong>lo</strong> tr|.diun hdnd6 @ qEidqld e th|r th.<br />
tdugl. EDS i! r 3rdl dgh rlgld oilElc o *y wih dt.!gl. tt D t..Ia E .d S<br />
tu 9oo !r!d rlE a.sl. d S b.rsa D !d E i! . $. ldind. oftlE pL& Th@foE:<br />
SD<br />
Af<br />
(328)<br />
86
sioiLdyrt rrtdSL AEFS b d<strong>lo</strong>.ndl r!d. t codrt.ld|plEdilati|lgbqi6<br />
tilr,.=<br />
(3.2.e)<br />
Pcpl$iry SD by 5M - E!, SF by PM - !! lid .llrdrudi8 ES illn (3.2.8) ,nd (3.2.9) w<br />
tN -ts-$N -6d,H<br />
02.r0)<br />
Ag.in in ISDE tm t!rc:<br />
.DE<br />
-r=S,<br />
DA-@, -6,161<br />
(3.2.U)<br />
ln b frrr! !23 $,ldd It dlr d. Dlc ntDglc Ft sDE of6. Flvior6 68rta<br />
rEql'itld.ortun.8 dvcLAOi.i.i.EDM
usirg (3.2.l0) il tlr4 lat s into:<br />
(aM -as)
Laliild. ofMoon in d€8rc6,<br />
Angle of E liptic wilh Horizon in d€8rees,<br />
LAG, Moonkt {u6el in oinut s.<br />
rable No 3.2.1 : posirive Ase Nesaiifr&B-;s-i66l2oliT;R;;iifr;;<br />
3.3 LUNAR RIPENESS LAW & ITS MODIFTCATION<br />
Befor€ |h. rin$ of ptolefry lh.rc ws no knoyn sy$enalrc descriprion ofthe<br />
dynamics of thc planeb and rhe Moon, Theretoe pblemy,s exp<strong>lo</strong>r.tion Che epicycle<br />
based description ofthe pltret4y noiion) sens to be rhe tirst sciious. scientific 6d<br />
ststemlic aucmpl to d€$nbe 1he dymics oteld systed obl*is. Muslim 6ed lhis<br />
tyiem b cxp<strong>lo</strong>c lhe condidoE for 1hc fid visibility of ns lutrd cr€sce . p<strong>lo</strong>lemic<br />
th€ory hal d.*rib.d th€ plderary nodon Eins rhe epicyctcs tcory rhal could Dddict<br />
the pl4ehry, lunar .nd sol& ephen€ns to a g@d soush d.er€. of accuacy. Tne<br />
Muslins had put ro use lhis iheory weu ond had deve<strong>lo</strong>ped sotr dnd lu.ar tables. The<br />
pos ion of$e Sun ws predictable !o a g@d dce@ ofprccisio. and @nsequendy rhe<br />
aele !r lhat ih. *lidic hal6 with ihc hoizon ar the line ofsue( ould b€ calculard<br />
Bing rne sphe.ical uigoroheFy that Muslim naomddn hav€ d€v.<strong>lo</strong>Ded th€n*1v6.<br />
As decrib€d by Bruin ( 1974 sing lhc &8lc of ectiplic ,y wid! the hoiizon al lhe tine of<br />
su*! the lalitude of th€ place of obseFarion, egle of sepsauoo b€rseen Moon dd<br />
a9
Sm p.r.[d <strong>lo</strong> lqldo. (difrcqE of ddr dgii L!..dioB), tlE !sl. of d@!n<br />
(dife|i@ of a|lnuda o. ARCU !d ri€ tid[ ofs€c.'n the M4lin as<strong>lo</strong>rcEcrs<br />
$@ aU. to deve<strong>lo</strong>p ! hisltly rdi.ble qilcion for tlE visibilry or invisibihv ofrh6 @<br />
Figua (l.1. I ) bc<strong>lo</strong>w enicl ir r r.otuttu rin offg@ 4t ofBtui. ( le77) shovs<br />
tl|st dF sD IEs jur !.r sin of r/t{. Thc liie \nX D85rg tb.o{8! $ts drllal poilt<br />
W, is th€ e{uror irclined d o dsL g (tlE ldihd. of lhe ple) fim tlE @rnd to rhe<br />
horiar AS k th. dimd pdn ofth. Sun !h!t is p66llel <strong>lo</strong> thc.qua<strong>lo</strong>r is shom as<br />
broko 1i... TIt diurnrl BIh ofd. Md MZ is tls sboM wnh 3 3imilt trot€.linc<br />
Ddrall€l Io ih. .qudor. TIE lin. HS ir tL E lipaic lh.t drd e .ngl. A wfi th€ diurol<br />
path of 6e Su d $..qu<strong>lo</strong>r ThBtL FiiiFb D!l6.td..8L q + d witl the Mnrl.<br />
MS t th. s.r.r.dd b.'ra rlF Mdn !d ttr Su (e of liSln).<br />
\:\.<br />
Fig No l.l. | : Trigmrdic ddiFin of @rdni@ of w luE d6ern
MT i3 p.rDodoio to i[. ..lirdc rII8 ar.h l}a TS - 1 - 16. b 6. 'Etl<br />
tinsb MIII, rn - fu d ft. erb a H b..r!.r Eclbrb (IIT9 jd &. Ho.te|.l<br />
olM) - 9d - (r + AI Tb - rilrb tm{ b lisb.raLd d T yE tq!.d3L i M<br />
!cvl.{HddT-9+A.Itrb.ioplyk &b6crdtio:<br />
ra-Er +rS-t,6lp+D+L 4. (3.3.1)<br />
|hir Fldih n .e. .r .$|d@ O) of B.uh O9ZD. Irtia8b HgB b rlcd s . ph.<br />
codr+ a)<br />
(3J:)<br />
{d ti.lgL ABS lad! <strong>lo</strong>r<br />
(3.33)<br />
(33.4)<br />
(3.3.t<br />
'Il&<br />
'slr<br />
(3.3,5) i! dima<strong>lo</strong>t Aom eb &uii O9,D t& tih in hi! .qurid<br />
(a) |b h cbiD. nr b..o obi!.d by ld4 lhe tigoffiy. Ittt smc 6ru. i,<br />
t'.d.d ||3toglpt ric.l !i!@.tty, turiotbADs 8it&.1<br />
f|(!o .phqi6l trt|tEL tlTMr<br />
9l
sin rr'r _ rh&<br />
sn(e+a) @s(e+a)<br />
erhlr (3.1.t) rs = sili(sin r!.r..(e + 6r+,L -,rr<br />
0.l_6)<br />
I. sphoical iridglc ABS:<br />
stnaD . stn 4r @eP<br />
(1.1.7)<br />
and in spherial lridgle HCS:<br />
sinfls=-l]1gll<br />
co(@ + a)<br />
l].l.8l<br />
:l<br />
glj!<br />
".(<br />
(:<br />
t::!<br />
!<br />
;<br />
!.<br />
)<br />
:9:.q<br />
+a)<br />
-t<br />
[.<br />
(3.3.9)<br />
laegles Hs dd ss @ snall cqua$ons (3.3.6) lnd (1.3.9) yhld alnost sme lesutB s<br />
thos by (1.3.1) ad (1.1.4). If the valuc of 9 b lege. then Hs dd as ae nol shalland<br />
th€ spherical trisonoderric rsults (3.3.5) and (3.3.9) should be ned for doF Muale<br />
11'e sienificdce of (3.3-6) is thlt if th. cphcme€des of both rh€ Su and the<br />
Moor m tnoM @uErdy (aj th.y c rcw) ud rhe &gl. 900 (p + A) = v of etipri.<br />
{irh ihe honan €lculalcd fom (1.2.7) rh. !.lE of HS @ bG aal@ted for rbc tire of<br />
sUM for dt day of lhe yw !trd psrti.uldly for rh. dlt or rt y .Rc UE bnh ot ncq<br />
92
Moon. wlee6, oncc rhc hinimm &stc of sepaDrion !s (.quivd.nt ro LAc) fo, $e<br />
visibility of @w l@d ct€.st is tnoM for thc day the @rcspondine agle HS sinS<br />
(1.1.9) cs als b. .vatua&d. Horev.r, rhc ue of (j,j.6) is ind.p.nd.nt ofany visibility<br />
condition ed is fixcd for rhe day ir d.pcnds only on rhe posirions oflhe Sun and lhe<br />
M@n for fie rim. ofobsenato. &d the <strong>lo</strong>caiio. ofrhe obF ei. On the o$( hahd rhe<br />
u$ or (3.1.9) dep.nds on trE vjsibifty @ndiiioG, @cly lhe mininu del. of<br />
s@ml'on tu, thar.& b. kmh only on lb. bdis ofa t gc nunbd of obo€Mrions tor<br />
At rhe rim€ of Muslih Gtrononcts rhe .phener€des of $€ Moo. oay nol hale<br />
ben so aeudely tnown s ir is today, but rhe Eligiou tecnn.ss oflhe seing $e new<br />
lund cr€s@t nusr hlve tcad ro moc rcchre valu6 or,s. Bruin (t97) hs i.dicared<br />
that Muslim arbnoheG rveE welt awe of rbe f&! lhar d8Lnce oalhe Eanh ild rh€<br />
Moon and hence rhe width ofcrescent for sde uc oflighr varies. Though exp<strong>lo</strong>iations<br />
ofdr ancienr considcred lhis variarion ro behdve tibearly we now know thal it in<strong>lo</strong>lves<br />
lne lflgononefic func on.<br />
In lnis worL $is prcbleb is h&dlcd by considering rne actual eni_d.in€ler of<br />
th. Mooh dd lhc fer $at MnslinvABbs ob*tued thal ar shone r .lisrances the crcscenr<br />
wasseenwhehtheLACo!arcofseparationaswaslOd€elccsor40minulesoftimcand<br />
al la€e! dishcca thc cErcent ks seen Nhen rhe m ot seplFrion dJ B t2 d€8res or<br />
48 ninules of timc. -r1li5 teads b a siople Eladon b€lwn /s md lhe aduat semi-<br />
^-- D 22<br />
ol r0)<br />
Using rhis etalioo thc @ of *pdlion necded for ce$enr<br />
*p@tion fe<strong>lo</strong>r is .alcutated in ihe sonwar. Hila<strong>lo</strong>l.<br />
9l
As mc ioned abov. dE Mulin sao&Ders have alaldy d€duad from<br />
obrpatios lhar for very thin but visibl. cE$ent mirimm ss @ 12 lime degds (48<br />
minureO dd for d {ider but bmly vbible cresent mininm as w6 only l0line<br />
degl6 (40 minulcr. This nininm as ie acturlly dep.ndcnr on the width of $e crc*ent<br />
rhar @ be dedvcd 6ins (2.E.E) dd (2.8.9).nd rhe Elarive atitud. (ARc9 thar cd be<br />
obtained using (2.8.5) for th. sun ed the Moon. Th. issue shall b. dieussed in noe<br />
d.tail larer. Usins 0.3.10) rh.t shows 6s beins dependent o. the lisull dimers (in arc<br />
minulet of lh. M@n in our sky one ce compute lhc Lun Rip€i.s Fudio. A(,i9)<br />
as henrioned by Btuin (Brui., 197) or tha! is siven by (3.3.6). Aaodihs<strong>lo</strong> Bruin rhe<br />
rul. d€dr€d by M6lims w6 thal if thc lalue of HS calculat€d using (3.1.9) equals or<br />
exccds lhar @lcolared by (1.!.6) rhe c..s would be visibl. orhcNi* nor. In thn<br />
work this rule has ben named M6lin Lunar tup€ness Law or sifrply Lumr Ripen€ss<br />
According ro Bruin (197) $e valu6 ol HS de.ored a R()., 9) obkin.d fron<br />
(3 3.6) seE pGsenled in rhc $+ lled Lunat Ripenss Tables sDd fisr visibility ofne!<br />
Lunr cmcent ws walulcd by Mulin BltonomcB usiog rne sboE rul€. In lhis work<br />
HS a3 deived tom (3.3.6) h d€nokds,l4 and:<br />
^d.,<br />
- sin r[sin/, r.nre+^t+lM ts (l.r.r l)<br />
Fo, a fixed pl&e (e constal0 it is noted lhat forsDallvalues ofl,, sin ly<br />
is also smatl<br />
&d lko n ahosl sme as rv ,j. Bui when the *liplic is wll inclihel io$&ds lhe<br />
hoien (q + A is large bu lcas rhan 9oo) 1r'4, nay becode noE rhan 1! ri(iflv > 0<br />
snd ecliptic is roweds nonh) or renain le$ rh& 1, - .ls (ir pr, < 0 and etipric is<br />
iowdds nonh). Ho@er,{da is endely independenl of Oe se@. aod dep€nds only on<br />
rhc rcladve coordinates of rhc sun md rhc Mood. As rhe major componen in (3.1.t l) is<br />
lhc diferenc. ol <strong>lo</strong>ngnudes of rhe su and lhc Mooh ,taa h c<strong>lo</strong>sely linked to ihc ACE of
Mftovd. ir 9M . o, x&" * l,{ - k. Bur whcn boil 0M 6d I + A hav. the s4e<br />
sign 0.3.1l) shows th &4 *ould b. mor. thd l, - & and if pM dd e +<br />
^<br />
have<br />
differcnl signs &e would bc l.$ the ,L - .ts. Bul thcF v&iatioos u€ nor sMnal<br />
The vahes for HS ohained fom (3,3,9) dc denolcd s X,b and:<br />
^,,=.'""(<br />
(l I 12)<br />
For a fi&d place (9 consrd.t) R,, dcPcnds on ae spa6tio. a, (rhe equabrial t'Ac<br />
ber@n the sun ed the Moon).nd thc s.en 6I + A is *son dependeot fiat shall b€<br />
shoM larer. Using (he <strong>lo</strong>oh sod le hniqu.s di*ussd in.hapter 2 ie, fo! $e dav or lh.<br />
day alier conjunction aor s placc of ob*ryarion ii .valuaEd. Ille ninihud valueofas is<br />
@lculaled 6ing rne lrchnique d.srib.d al lh. .nd ol Ptviou .nicl€ $ lhat ,tB is<br />
dedu.ed. If lt- is calculat d using h csdml.d value ol a, rh.n w call il X." (m<br />
Erimared valk ot Lunar Ripencit funclion), Il ir is .al.ulated uin8 an avedge vrlue<br />
10.5 degres of a, thcn *. c.ll il I- (d .vcaec wluc ol Lud tuFns nrtudo.)<br />
Ac@drnely A&" = i- - X,a,6d Alt-, = i^ - 1&) Thc sinpl€st fom ofrhe Ll|E<br />
'"'(<br />
i3i<br />
:) $i" a;t+,ry -rsj'o<br />
(3.1.r)<br />
Tlus Lunar Ripeness Law that piolidc a solution of the probleh oldelemining<br />
rhe first day of visibility of n€w lunar cre*enl h based on the Angle lhat the Ecliptic<br />
nat6 wilh the bodzonlal or $e angle 0 + a ihal it dak.s wilh the venical. on the dav or<br />
the day after conjunciion once lh. coordinatcs of the Sun dd the Moon at the tim€ of<br />
suns for &y leation.rc calcuhted. (3,3.1l) al<strong>lo</strong>*s onc <strong>lo</strong> calculale,e,4 ed sine 6<br />
app<strong>lo</strong>pnab value oi4 d sivenby(3.3.10), in (3 3 12) the valu. ofx"i ce be tound ltr<br />
95
vi€w of $udying ihe behaliour of lhe Rip.nes iunction oler a yea! for any pl&c<br />
calculding Ie, for cvcry day of the yd is not useful. This is bc@w of thc &ct lhat<br />
y€ar ro y€e dd da, (o day vdiarions in lalitude oflhe Moo. dd hence nd, ats not<br />
dep€ndent on the rime of ye.!. However, *ith posiblc lalues of4, one cd calculate<br />
el.v val@ of l* for @h day ofa yd &d study ns veialioG scenally as sell d<br />
$irh chdging lalilude of pl.cc. As possiblc values of,, vary from l0 degcca <strong>lo</strong> 12<br />
degc.s an appopdate value for the day oi day after conjunctioh is obtained only on the<br />
bdis of the ule disraice o. sminimd.r of the Moon. Still tor a compdie. values of<br />
lS6. foi dill@lt ladrudes, se calculaled <strong>lo</strong>r both $e cxrene valucs ,, = <strong>lo</strong>o dd a, -<br />
120 and the curves,t,r aeainsr the <strong>lo</strong>ngitude oflhe Suo (fiom March 2l) aft p<strong>lo</strong>ned in<br />
FiSuGs (3 3 2) <strong>lo</strong> (3.1.5) These cw€s snow inleFsting featres lbat cs b€ sunmdized<br />
i) from fi8.1.1.2 il is cl€ar rhar forg = O0 i... for a placc on $eequabr&i! is<br />
sinusoid.l wirh mdina.t March 21, Septcmber 2l dd hinima at Ju.e 21.<br />
D4ember 21. Snalle! v.lues of iG nems shaller lalues ]d4 or !€lariv€ly<br />
rounger cEsent m.y b€ rn. L&g€r values of x i" dcans largq valus iia<br />
orrelaliv.ly oldcr crcscenl may b. seen. Thw c<strong>lo</strong>se to equa<strong>lo</strong>r older cEsceht<br />
nay be visible near equi.oxes and Elatilely younger cresc€nl .ea! ehrlc€s.<br />
If $c Moon is nc{ jls apog€c lhen il is noving faner and apFaB rhicker in<br />
sky Blatively younger cescents b€cone ripe for visibilny. Funher rhe<br />
paencc of two maxim and two binima ihdicates four stong innecdon<br />
points. There arc rwo regions ot lpMd concaviry (douod solsdcet dd rs<br />
of downwdd concaviiy {&ound cquinoxs)<br />
As rhc larirude of rhe place in rc66 (one move ro rhe nonh ofequabt $e<br />
muinum ol1." at $e venal equjnox <strong>lo</strong>seu nDline ir eard <strong>lo</strong> k d a<br />
youngcr cescenr bul the ndimum of lhc veml equinox ris.s e $al it<br />
becones norc difiiculr to ec a yodge! crc$ed (69. 3.3.3). On thc o$er<br />
hmd fte ninimM of lhe s|!md ehric€ mo6 rowuds sdng (v.mal<br />
equinox) ad that of rhe wintd rchric. mov€s rowds autumn (aurunnal<br />
eqlinox) md in ei$er c& d.crcae funner making ir esier ro *e a younger<br />
96
cwc ahaFs c<strong>lo</strong>s. ro ll€ alhlnn l<br />
m6d thc solsti.6 flltt ns.<br />
F g. No. Ll.2 X* forg=00<br />
15<br />
13<br />
Kg. ffi<br />
12<br />
10<br />
I<br />
. . .r. rl,<br />
9o 180<br />
.,.-<br />
; I<br />
210<br />
Fi8 No.l l l rt'L forg= l0o<br />
iiD<br />
BcyoDd tropic of .alF two of the infl@don Poids re simPlv gone &d $e<br />
hsimu al lhc aotmnal €quinor ds tunner malir,g n FoE difficuh <strong>lo</strong> s<br />
91
. yomg.r ct!.c. (f8; 3J.1). Tb nioidsn oftL qttc Ed,. tt !t ttl<br />
.quinox mrli!8 it .did io r.c yomg.r @3@l! n tr ro €ml .qu$ox<br />
iv)<br />
rG frltd iF!.ic i. dr ldtr& &. D!!io@ bcoc tt}.' rd !i8le<br />
lrd dt. ril!i|!@ t cF &cr!.'nry tuvi!8 &n ftr biSba ld'ndd il i5<br />
g6.r.lty .si6 ro @ younScr cr!.c.nt3 cl@ to vlrnll equimx .ltd diffrct lr<br />
clqe b aull|nn l cqui<strong>lo</strong>r.<br />
'17<br />
t6<br />
13<br />
1o<br />
to<br />
Fis. No. l.l.4r X"i for g - 25'<br />
1}. siuDtidr r!\/{r!6 .dirlly i! fivou ofam|'llll cquiM ft. lb -ttld!<br />
For the Arctic Circle .nd to it3 <strong>lo</strong>rth (latitu
a0<br />
50<br />
20<br />
Fi&No. 1.1.5. i^fore=61 50<br />
Howev€r lhe narcr i! lrot $ sinplc beca6' tuc olc of latitude of rhe cKdt'<br />
i-., whcth€t thc cc$eni is !o!ih or north of ihc Sun plavs e impona role lhai<br />
dercmin6 rhe v.luc ofrtdo lhal is lronglv btnude dcFndst Now lh' stDns nuimum<br />
at lhe v.oit .qrinox for hi8h.r lttitud6 indicat's thtt e old'r M@n rov rc1b' vbibl'<br />
b s the AGE offt. Mo@ iNrrsd $ do6 iG e<strong>lo</strong>'sdi@ ad brighlress ThtebE n<br />
is posibL ftat invisibilitv $g86l.d bv $' Rrpcnas tun.lid valucs m'v b' nisleding<br />
Thh inly l.!d <strong>lo</strong> lt<strong>lo</strong>llN for sdalld LAG vd6 drd sm lqr€lEof x'L h rlnt<br />
$!d( th. RiD.G$ tulcrion vtl@s for dE ob!'dltioml dab avlilablc in lilettrurt e<br />
.dculacd sd pr6.ntcd in ApFndirlt Out oflhis dx' s't th' c!56 whd thc cscenl<br />
s Eporr.dly sd bui .rc nor h lg.mdt *nn dE L|M RipGs ltw rE also s||os<br />
in nblc rc. 3J.1. TtG obae liontl dah k $lccLd AoF lh4 tPo :d bv Od'h (Odeh'<br />
;0{a) fron *hich c.*s or cvdi!8 '6cnr<br />
obs'naiioN re cosidercd onlv Ttblc<br />
3.1.1 shows o. No. (th. oberysti@ nunb'r (Od'h' 20Oa))' dtt' of obseNttion ldnud€<br />
,ftt <strong>lo</strong>ngin/. of ih. t@dron of obaa$s' M@n\ !8e and LAG' snele v of ecliPlic sitt'<br />
rrorLo, t"lio,a" or Uo.,, toogirudes oI the Moon ed th' 3d' Rd giv'n bv (13 12)<br />
*ill !6 siv@ bv (3.310), Re siv€n bv (3 3 12) *ith 's<br />
= 10 5 d'8r*s' Ri' sircn bv<br />
O.3.I l) and a&. = Rdq - &"<br />
I
'Ihe table in App€n.tix'U shos S. No , the obseNation No (Od€h 2004)' date or<br />
obseNalio., latiiude dd <strong>lo</strong>ngilude of the <strong>lo</strong>caiion ol obseru€r, visibilitv 'N' for lbaided<br />
lisibility,'B' fot visibiliiy lhouel binGuld''T for visibihv thbush lelescope The<br />
visibility coluds d€ empty if the.res.ent wd nol s*n sd conlaitu iV' in an<br />
.ppropriltc colmn ifth€ q€sftnt is sen The rcsl ollhe coluons oflhe t8ble contarn<br />
age ofMoon , LAG. ARCL, ARCV dd DAZ The ld file columns de fot the live<br />
models consideted in this chrpter The colu. headed A is for the Babv<strong>lo</strong>nid model'<br />
and A for the Lunar Ripeness model The colnmn headed B cod'ins AR"<br />
The <strong>lo</strong>bl€ i! ,Appendixll shows lhat oul of 196 cdcs in shicb $e crese has<br />
b€en reponed to hove been een wilhoul oplicdl aid thee tle onlv 14 cases hat do not<br />
obey thc LuMr Ripencss La* stalcd above The debils or rhcs obscrydlions de tinther<br />
exp<strong>lo</strong>Ed be<strong>lo</strong>w *ltn a Modi6ed L@r Ripeness Law is sugeest€d The nodificalion is<br />
needcd in ordet 10 sepdarc tbc caes of dked eve visibilitv and rhe caes when optical is<br />
used for clescent visibilitv<br />
A tot.l number of l2 posiliv€ obsedations ih tbis dala hav€ b'en wnh the helP of<br />
bin@ules ed klcscopes when $e cr.$enl ws nor visible vilhout oplical aid dnd the<br />
Luai Ripe.ess Law is .ot elisfied 'nth is <strong>lo</strong>gicallv valid as the Lua' Ripencss Law<br />
was deduced onlt for mlcd eve ob*tvatiotr Tlis bas he'n the dolivadon durine thN<br />
$ ork <strong>lo</strong> ood i fy th€ Lunft Rip.ne ss Law ro e ocompass the otricallv aided obse d'tions<br />
The columN ofihe table contain Ob$flario' Ssial Nunber as led bv Odeh<br />
(2004), Date of ob*tvation titilude oi lhe Place Longitud€ ol the Place Visibililv<br />
colunN N (<strong>lo</strong>r unaided visibilirv), B (wirh bimculd) and T (wirh iele*ope)' The*<br />
visibilily colums d. €mptv io! invisible cresent md co ain v for lisible R€st of lhe<br />
colws contalns Ase of ctesce.t in hou6 fo' the best ime (Y6l<strong>lo</strong>p' 1998)' Lae in<br />
nin(es, S.padtion b€twc€n the Su ed tbe M@n At of LiShl (ARCL)' the angle v =<br />
900 - (9 + d), that thc €clipiic mai
the <strong>lo</strong>nsirnde orthe sd, Arc{f-*pdtion fa.tor, Sitd by (3.3.10), Esdnat d Ripencss<br />
Iuclion, ,{.r olcularcd eqution (3.4.2), Ar6of-epdalion ractor, Aversge Ripen.ss<br />
Fwrion X., calcllared Ning avcds. tu = 10.5 d.gEs dd lh€ €quarion (1.4.2), Aclul<br />
RiDene$ Fucrio. ,Rdy calculated sins equtioh (3.4.I ), AP-" the difie€ncc of AveEge<br />
RiFn s Fwrion & th. Acrul one. Th. ribL l.l. I des nor show thc visibility @luons<br />
s tbis table @mprises of css wher ihe cesftnt Ms r€Podedly seen withont oflical<br />
The lsbl€ is soned on the values ofd&y. On the basis of a c<strong>lo</strong>* dalrsn of the<br />
results of conparing ARNi ihe dilleences of Av.lage Ripeness Funclion valu€s End lhe<br />
Actual Ri!..ess Fuclion v.lucs we ob*de dEt:<br />
The Tablc shows ft.t lh.E are only 14 pGitive sidrrines out ofa rolal nmber of<br />
196 positive siShlings that are nor according <strong>lo</strong> lhe Lunar Ripe.ess t { (ARo, = Rn",<br />
The€ a!. no posilire siSndne wirh A&q < -1.58 wirh or wirhour oprical aid the<br />
Ens€ in wnich l8 aft.npb have ben nention.d in the lne6ud. We consid€r it as<br />
Coup-A for thc Modificd Muslin Lund Ripencas Las lhal we slal. ih ihis erlq 6:<br />
''tt th. tlill..e,ce ofAv.rase Np.ts Fanctior R-, omt the Actuot Rip.h6s<br />
Fu"cion Rro F /Re) is l.ss nnh -J.6 it is ihDossibl. to see nE n.N luat .rcsc.nl<br />
tor att ldtitula uith or Nilhott optical aid'.<br />
The ncxl I27 c6es dc grcuprd 6 coup-B. Therc de la (l<strong>lo</strong><strong>lo</strong>) mked cye<br />
viibility cdes, 25 (19.%)binoculd visibihy cesdd l6 ( 12.6%) r€lesopic visibiliry<br />
cses for values of AFw lying betwen -1.58 dd 0.0. Oul oflhe 14 natred cye lisibiliry<br />
ca* in thn range, threc lery <strong>lo</strong>w AR{yfvalue cdcs have comon chancterisdcs. Th.*<br />
e No. 286. 2 dd 22 witb AR.!i valu. -3-5, -3.4 md -2.88 All lh* c4s m ner<br />
autumnal.equiiox (Sepr.20. Oct.2l &d Ocl. t epecdvety), €r€$enl hs oldc. age<br />
(39.1 l, 39.24 dd 4l .91 hous Elpectiv.ly) md hav€ consqueitly larger phe.<br />
l0t
All lhe* conditions fa<strong>lo</strong>lr rhe lisibility and it wds ncntDncd above in view of<br />
(1.41) that aor oldcr a8e cresccnb (/.M ls ta€e) sh! er arc or scparation may be<br />
auowed. Thisexplainsrhe very sm.ll LAG (29.s, 33.62 and 32 07 respecriv€ly) in rhese<br />
cas. Modov€r, in all $.s cases lhe arc ofvbion is snau (ARcv 6.85, 6.8 dd 7.34<br />
deg!€es rcspectively) bufte<strong>lo</strong>rile uihulhs rc taree (DAZ = 18.4,2O.j and 18.l d€grees<br />
r$perilely) aioh the sun and tos€r widlhs (51,65 ed 5r dc seconds respccrilely). Atl<br />
lbese lac<strong>lo</strong>B suppolt thc ctaims of lisibitjry md weE anlrcrpatcd above whcn il sas<br />
suggesred rhat smalter L,{C valnes oa!, be a<strong>lo</strong>sed in such cases. All d€F clains de<br />
fon latirudejusr moE rhm 30 des*s. The shath, values ot Rj"y in conpdison to R.,,<br />
h lnes cscs is due ro larsc lalues ofa + A (noe th& 54 deee$),lhe &gle oalhe<br />
eclipric with venical ed large nesaivc vatu$ of tarnude of Mooo LM (tes than _<br />
4.5degrcs) lhar rcduces Rla to hate n nuch shalter lnu /w _ 1.. Alt rhe* rhr€<br />
obsen.rion de rncgR.nedqrrh rhe Babytonru cr.rerion<br />
Therc is no fudhq c6e ofcesenr visibitity tjl. varue of dRq < -1,6. Fbh<br />
dong$ otherposilile cases wilh -1.6 < AR."r < O, two de<br />
very young ccenr' Thee<br />
t02
m obddation no.274 (ARm = -1.19 ) drd 416(A&tr- -l_06) i{i$ age 14.8 sd t5.9<br />
houE rcsredvely. Despile being lery yorS having FLlively l&aer LAGS (39.j md<br />
17.7 ninulcs sperively) in both c&s rhe Moon ws very c<strong>lo</strong>se ro the Eanh rcsuhine<br />
in ldse visua.l dimcler bur lh€ cltsnl widrhs wcrc sb.ll (10_ &d ll.9 e. s@ndr<br />
only). with $all the rclalivc eimurhs (4.6 a l.8 dese, the crcsent eas atnost<br />
ve'liMlly abov. the su rhat bring it in ihe ided cordnion foi visibiliry bul lh€ snau<br />
elative altitudes (8.5 ed 9.1 deele) nake rhe$ claims highly oplini$ic. Borh th6e<br />
obse^ario6ec in d$dgreementwirh Lh€ Bdby<strong>lo</strong>niM qirenon.<br />
Out of th€ 14 positive obseru.tioi wnh A&r < O, ihte werc very f.ini cEscenrs<br />
i.e. obsrarion n@be6 389 (d&. = {.94), 341 (AR^, = -0.87) ed a55 c0 62) The*<br />
crscent werc <strong>lo</strong>w in ottitude (.2, .8 and 8.j deg,es resp€dively) ed hrd sna<br />
erongarion (10.9. 13.3 dd 9 deg@s Esp.cively), This hakes these ct6i6s <strong>lo</strong> be higbly<br />
opr'n'strc os well. Two otthcsc {389 and 455) are sko in djeereocot viri Baby<strong>lo</strong>nian<br />
cnlenon {heec 341 is a mdSiMl6e in Baby<strong>lo</strong>nian qirerion (with ARCV + LAC =<br />
How€v€r, for all $ee I I 9 c6es when $e crescnt wd daoed ro hdve been sen<br />
wirh naked eye,lhe conmo. ieatule w6 relariv€ty hi8h l&uoes {generalty g@&r thq<br />
S0 degrres on €ilher side of rhe €quator) except fo! obFNatbn no. 416. Th€ cbim 4l6 n<br />
tioh latitud€ 6.5 degres nonh Apan fmm rhh <strong>lo</strong>nely c6e it appe&s lhat fo, A&r < O n<br />
is ihpossible to see lh€ cE*entatpt&es witlllatirud€s less fian 30 desEes (borhNodh<br />
Inlhe l19caksofsroup B,lhe frequency for optica y aid€dyisibiliry (botb wnn<br />
binocula and retescopcs) ihceses ed one ce cdily g€nerau€ lhal whcn Atqr, ties<br />
between -3,58 and O.O theE is s hjgh rDssibility of crescenr visibilhy wilh soe oprical<br />
sid for bod lh€ high latifude as wel s <strong>lo</strong>w l.rilude obefres. I hereby jn this wort the<br />
s4ondpd of rhe Modilied t\4ujlih Lus RiFne$ LaB $$Rds<br />
t0l
''if h. nns tizt b.nv.r -3,t otd 0.0 th. posibili9 oJ rhibilit! ollittt crcnetl<br />
*ith and *itio optical did ln re6s vnh ihcE6iig tolu6 o!.4R-. fot hlEh.r<br />
latitqda, gq*o ! gedt.r thon 30 d.erc$ notth a4d souk ard ZR*. b.l"g h th.<br />
ronq. -3.5 b A0 h. postibility ol nak.tt erz visinlw ako irceqe l<strong>lo</strong>wvq, in t'.<br />
tong. oI th.G wluzs oJ r'RnIor tMl<strong>lo</strong> <strong>lo</strong>titad6 th. "oh.d<br />
.tz vitibilitr h dlMt<br />
N€xt is the Oroup-C $at mnrains 76 cases wilh AFrv in rhe dnB.0.0 ro L6, In<br />
lhB goup rh.r. e 12 naked cye visibihy.as.s (15.8ol.),26 binocuta (14.7%) and 15<br />
&l6copic visibilily cs4 (19.%) e lhar vGibility with both naled cye and with opdcal<br />
rid b.com$ ooe probablc. UnfonuMt€ty rhe dala is h.avily inclined <strong>lo</strong>wads the high<br />
htnude clesanda cl€rdenarcation for u.aided visibilit, for snalter laritudc ob*Fe6<br />
cd nor be made. Srill, the thid pan oftbe [,todilied Muslim Lund Ripeness Law is<br />
''fth. wtu6 oJ/R@ tk 6ztwa 0.0 an.t r.6 ttt. pEribitiiu of,i,n i, oJJint<br />
cftsceat teith and tuithout oplicdt k ,trong fo, higlrq latitud.d,.<br />
The Croup-D containine a ro|at nlmber of 221 c6es has 170 cases ofoprically<br />
unaided vhibilitr oflunar cscs forr'i!' > 1.6,Il is<br />
^il r'B@ >t.6 ttt. posnbitu, oJ ,bibititr<br />
siorg <strong>lo</strong>t boh towt aMt hiEha totitd.r".<br />
of li61 c6c.nt fitttott opti.dt b<br />
Finally, the 3umnarized Moditied Muslim r_umr Ripencs l-awr:<br />
ZR-< -3.5 ihpNibL to tee th. n.v luaar c.6c."t<strong>lo</strong>, a tdrru!., ,ith o,<br />
2,<br />
-3.5 < /R@ < A0 ihposibt. to ee the neN crcsce"t with ot *,tthout oplicnt oit!<br />
Jo, tnt .r latitutl$. For hiehe, t,titnds therc b n high pnsibiti,r ol gibititt<br />
otli6t cE .nt dth oyial<br />
104
t<br />
0.0 < .1R,. < 1.6 p8tbiti,tr o! ,hltw o! Ji6t M@nt btrh ond withoul<br />
optt ol h staq !o, tigt.r tdlituds. Lo.a!|rA N6t tuith opticd! a ! a".!<br />
th.a try nelnE lt Nlth haL.d e!. h6 a eood .ranc. o! opticdlr u"tided<br />
ne,> 1.6 th. pNtlbnrr oJ' tbi!i!, oifust cnse wtuhout optcat t st ona<br />
Jor bo& Iow.r and highu tatttud.t,,<br />
Another way oI <strong>lo</strong>otjng inio ih€ delaih of Lue Ripene$ hodel is <strong>lo</strong> <strong>lo</strong>ok inro<br />
the p<strong>lo</strong>ts ofaverage ripehess fmcrion md rhe acod ,ipenes tuncion for borh vhibh &d<br />
invisiblc crcsei$ for single talirude, U.forluarcly, rhe dru avaitable dd considercd in<br />
lhis work h Esrficlcd in rhc F6e lhat scieltifically re@rded ob*dalions fo, a sinetc<br />
hnud€ a€ nor found lery fr€qucnrly €xcept tor Athens (ladrude 38 dc8,as .orrh) ed<br />
Cape Tou (laritude 13.9 degr4s so!rh). !n panicula, for sn,I€r tatitudes. plac6 cto*<br />
Ioequarorobseruatiouare<strong>lo</strong>lvcryfFquenuyav.ilable.Anunb€rofplaccsecsetecred<br />
herc od tbeir d.h pto(cd for Alerage Ripenes Fundion &,, and the Acllal Ripen*<br />
Fuiclion R6y for borh rhe Eponedty inlisibte dd the vilibtc oe*mr! Thesc jnctudc<br />
plees with laritudes L8N, 33 95, 6.5N dd 38N (FiguB 3.3.6 1o 1.1.9).<br />
Figure 3.j.6 for tadtude 3 L8N snows tbe b€sl slc<br />
sieh,incs ou, of 6 (83.37") in asrcen€ wnh $" ,_J:'iff:ln:"11,'#:J:<br />
obscdat'ons (6 our of6) e in agrccm€nl wilh the ta* toi this larilud€. Nexr is Cap.<br />
'rown-(hnnde<br />
ll.9 S) wirh succcss percenhgc 65% (ll posrlire siehings in sgenenr<br />
od, ol 20, the needrvc obseryauons fo, Cap€ Toq are In agrehenr sirh la\ tor<br />
9lJ% c66. This 6 ao owed by ,40r.6 (ladrlde 39N) w,rh 2 oul of3 {66.7910) posirivc<br />
eghlingsad II o of l8 (61.l%) neSariv€ sighdnes a8rce snh fic lav. Ior laritude 6.5<br />
deg!.es, I our of2 (50%) posilive sightings od 4 oul of6 (66 y.) asE. wirh rhe tan.<br />
I05
I<br />
4<br />
15<br />
't0<br />
0 <strong>lo</strong>o z<strong>lo</strong><br />
3b<br />
400<br />
^ actual RF br in\isibte . e.lrrr nr Or ri"tOre<br />
Fia No. 3 3.6: Rjp.@r Fundion for r-ditute 3 L8 d€g.c Nonrr<br />
---r;-+:<br />
;b.,RF6.'"".b,.<br />
.acrld R<br />
Fig. No I 1.7: Rip€lB Fudid for C.p. Tow4 Sqth Anic!, Lgnbde r3.9 it€lr*<br />
t06
f<br />
-111+.-Ig'1.rl'1 t r.tl'::.gE::. f<br />
Fig. No l.l.8: Ripdess Fudi@ for Lrtibde 6.j d|:gG Non\<br />
20<br />
1a<br />
l6<br />
10<br />
20n<br />
J.E""'g:f*,","..,v"ltr"<br />
300 /too<br />
iU,ar nr b. rnti"itrc ]<br />
Fig. No 3.1.9: Ripdcas luDrid for Arhdq lrriMe t8 d€@ Non[<br />
Loogitudc 21 7 d.grca Easr<br />
t01
All thes. figu6 show ! lend for lhe Aveog. RiFn€s Funcrion lhal is indicaL
d t mwh thidd fd !.d FlriFM@ dir.E. So n n DosibL thlr $e d6acrr eirh<br />
m.lld & md @.r.qudtly strdla .o mr b. vi3ibt . Ilis i! p@j$ly thc ce for rhe<br />
obgMtaom m. 286,2.rd 272, lna @.ctn s ct.itried ro b. sl eirh nrl€d eyc<br />
but the Lury RiFnB te i.di.d6 rh. otsuriB P@ imf.sibt. (My <strong>lo</strong>, A&,<br />
vlluet wh€@ the Blby<strong>lo</strong>nia. dilsio. at<strong>lo</strong>s fi.n. So lb. quGrion lriss, for pt.as<br />
with hign6 bdlud.q etipric mch inclin.d rorsds hdia Dd dE qegn b€inA o.<br />
hori@n !id. oI ilE eliptic ah@ld w giv. do.! .Uolle for !c Shoutd rhe p€.I! ol<br />
tlE Rip.B Fundion {q be <strong>lo</strong>w th.. $c, e &cotding io 6glB 3.1.2 to 1.3 5 c<strong>lo</strong>s.<br />
to tlE rltmr.l .quimr TrF luF ii ddd, lcr.rt jn lh. di<strong>lo</strong>sio. of thc<br />
@ns<strong>lo</strong>inrs ofrhc Lu@ &F|€3s tuturion wtEn ir i5 poi.t d od rhr for rcry <strong>lo</strong>w AR",<br />
v.ru.3 rh. d*qn of bw LAc hn okt agc.td hjc DAz i, E ofl.d ro ben sd. thjs<br />
nay b. th. po$ibl€ l6$n for rt. nrort n sirob@! tik. MDidd !.d Fothdiigh|n<br />
tu qp<strong>lo</strong>riig .el.ri@ b.tqq ARCV .d DAz for the ft51 vhibitity of tu@ cresn<br />
o,^. --.<br />
ig<br />
Fig No 3 4.t: Spncriql Trigonomfiic ih..iFion of 6ndiiid, of nq lu@ de*nt<br />
t09
In cae ofMoon nonh ofediflh:<br />
DAZ ! ES = EJ J JS lassine<br />
es I<br />
-elr<br />
@te<br />
{r.4.r)<br />
and c4c ofM@n soulhof(tipric:<br />
'n<br />
o,lz - o5 - p1g , 115<br />
= o, *6s - 6n<br />
(1.4.2)<br />
"in, cos p<br />
DAZ siven by (l.a.l) js much sfraller $& rhar eivcn by (1.4.2) for th. r.son tar in<br />
(3.4.1) rhe difirEicc of dqliiatioD of rh€ Sun qd rhe M@n (\ _ 6M) is sna er<br />
rr{ |han<br />
in (3.4.2). wnh k4er aRcL (SM') for M, (3.4.2) as compared |o M (1.4.r) &ry j,<br />
rar8.r bul &[ or Rd. is sdc. Ahhongh A&,i (Ra, - &ryJ r sme tdr the two ca$s bur<br />
crc$cd al M. is older, rhick.. and brighrei mucb soaler rhan for M. Esutts into<br />
dill.Enr ofarcs oftishr (ARCL) or older cE$eit with large! ph6e. Thus in case of in<br />
case ofM it hst be difijcult to ke the crescqr s conp&.o ro rne c6eofM..<br />
When€ler as is v.niet (perpodicuttr to rhc horiz!) DAZ vanish€s<br />
optihuetud<br />
aidnioo of I Oo (when Moon is ctosesr !o rhe t dnh) b l2v (,hen f.nhd lhe M@i<br />
frcn is<br />
rh€ Ea.rh) ccu_ As and whei tu i; nor v€niet lhe ol. DaZ coh€s<br />
play<br />
inro<br />
lnd Oe oplimum condiroN for as ca b€ retded. For hucn ls8er valu€s of<br />
4d DAZ<br />
older dd wjder cesc€nr may b. visibte wirh s@ er vahca of ARCV or aD. So rhe<br />
oodels in<strong>lo</strong>lvine ARCV-DAZ rclarions comc into play. h these hodel ARCV is a<br />
runcrron ofDAZ so as should aho be a fuction ofDAZj<br />
4,. = 4.. cosa ad ARcy = ftDAz)=aL<br />
- -[(DAZ)<br />
(J4.3)<br />
Il0
dh of such obFrystios dut were povided in Monm*n, s chromtogie lt8a3), W_<br />
This n@s rhar for coBtaor LAC (e a) ARCV decE|s<br />
com|anr 9, t.AG ed ARcv e di@tty <strong>lo</strong>tDnionat ro ech<br />
ktnude L lrgc DAZ dd taige ARCL m@s mall€r ARCV<br />
LAC.<br />
Duing Elarivety ucnI lioes rhe dp<strong>lo</strong>orions of rhe @tist visibiliry oa nd<br />
rutu cr€sc.nt or rh€ lat visibjtity of rhe old cFsc€nr b.gd wnh lhe obstuations oad€<br />
in Alhens and iG viciniry by Schnidt md othes. fteoretical exp<strong>lo</strong>radon was iniriared<br />
rurc du€ ro @leldtui@t E6otu ll)e ey paniculd sbononiqt qusdon<br />
(Foth*inshm 1903).<br />
h lhc beginning of th. trcnlietb ceirury n was eatized rhat ncrhods of verifying<br />
dates, panicltarly tum, dotes, wcrc nor avrjtable md pspje were @ncehed 6bout the<br />
6r6noorc.t ondniom thar gov€n dE nsted €rc visibilny of $e rs lun.r .,esccnr<br />
(FodE nsllah, t90l). To evatuare anonohicrl condirions for rhc ktiesl risibitny of<br />
:::::: *'*!' "-" " *.ber ofsrudies ,ppeaEd. .rhe slrk ofr. K. Forhe,iishd<br />
(1910) hinsell4d that ofE. W. Mau.dq (l9ll) is of vjtal importance. aou rhcsc<br />
onu'bur'ons *eE bed on fie nal€d er. obcenarions ot new luna cr€*enr nade by<br />
Augusl Momeseo, JutiN schhidt sd Friedricb schn<br />
@nriburo^ ee .,p,.r*, i"<br />
"'* ";; ;;;;:;Til:,#:lTJ:,i:;;<br />
^, . <strong>lo</strong>nn n.ghao claim ro havc sueg.sEd (in his d,crc aprHed<br />
Philqophx in rh€ roumat<br />
1 90)<br />
of<br />
lhat in ord.r to catcularc thc rrue date ol phasis on ough ro have a<br />
lable of te requisne depre$ion of the Sun betow honzon at fic 6oonkt, or of lhe<br />
aurtude ol the M@n at ttE $tuer for dif€re o8uld dishnces of the M@r fom rhe<br />
sun. He h6 .l$ adnitted abou his hl'en* .<br />
Monhsn in ,his rcead. Motre.-;;;rJ,'* T,,'1:::;:;llTf<br />
moo. hade h the laGr half of thc nine&flti century by Jutic Schnjd| Iriedrjch<br />
Schmidr and Morn*n himslr n€ hblcs menlioned e;liq euld be @nslrEred on the<br />
lll
69{0. Fo(brilgb.(l (t910) hs &!.rd,!od lt !. oh.{trdo.! giviq cilit drB of<br />
ot66htio! od ir! r!$lnc Fsvit d Dy tioi.iaod ll. !E!Xind.,nd thaidltrh<br />
of t! M@ Ebiis ro lh. Stl! r drc lim of Ms.r (or !ui!.) crrqld.d by<br />
Fo$dDgh.n tin!.lt Tt4 ob6.'vdo$.r. ,!lrrr!8ed i! rh. T$tc No. 3.a.2 { h<br />
sb. rddidool .rlc-rr@.. Tb culs &! .ts Fs&d oo o ARCV-DAZ chti in<br />
Fi8@ No 3.4.2. Fo{t.iishn (t9t0) .t$ Gidna I ddrrEy r.u. lhar is<br />
r.prottuc.d a Tibt.3.4.1 b.<strong>lo</strong>% rhd sivB rt nisinM lltiidc for lh. !.nibt. @*a|l<br />
fot rr|ridl! valu.s ofrcldirc .hn'h.<br />
T.bL No. 3.t. t: Fortsiryhrn,.""-r,rffi<br />
H. .!o it"EtoFd . mlrhoni.d Gt rion <strong>lo</strong> dc'qitc rtc k,E:<br />
Mioioue Atrtr* - 12..0 _ 0r.0OrZ,<br />
(3.a.a)<br />
]T1,:.* y* *** ni5 cod. d.fn* a r.sioD or sry lroud rh6 poiot of<br />
$Ds a $ow! u n6 tgu€ 3.a.2 b.!oe. If, .l 6. d4 of su!.! rL crBc..l i! .boE<br />
d! cu.rc ir ltould tc viliuc orh€.eirc iot T[. cruE h crinlit to. Fo6qingh.n,s<br />
Fit. No. 3.al Fortdbgh!$! clrl!<br />
lt2
T.ble No. 1.4.2 Forh.dng[d\ Rutc<br />
- A@-on ii-<br />
'<br />
d66<br />
3' 1,1 <strong>lo</strong>&!<br />
&<br />
ia<br />
I|l
w. d.firc a pa@et . !r ,<br />
.,visibihy pamerd,, accordiDg ro Forheringhao sl<br />
rt = (aRcv - t2 + 0.00s21) | tO<br />
(1.4.s)<br />
rn lhc T!bl. No. 3.4.2 rhe tsl colhn @nrajd ihe v!l@ of ,. fo, €mh<br />
oD.eryaxon Epon€d by Forherineh@ ed rhq rh.lablc is $n d in rh. inccasiag ordcr<br />
ofyl . ln !i.w of{1.4.4) iflnc lund attitlde is lcss rba. 12,-0-0..OO8Zr lhe qcscent<br />
should nol bc visible_ Alr€matct, in vj€w of(1.4.5) if rhe value or ,ts is rcSalive the<br />
ccsnl shoutd nor be visible, Thc vdB of Elarivc altiiude of the Moon dd irs Gtaliv€<br />
ainuths aE rho* calcul.&d by Fothqinghah. Ir is e.s y nobd thal out of 20<br />
obsetuations ao. which lhe varueof,r is negatilc two obseryario$ ar€ posilivc: One. on<br />
Oci 27, 1859 (obseru.don no.2) od tbe 0lh0 on tlE nohmg ot scpr. 14, t87t<br />
roDsnar'oo no.4t). Hosrver he hin*lfdnilslhat his marheh.dc€l elarion (3.4.4) js<br />
nor $ Etiabtca tE sumary labte.bla. Thc eo€ is exhibile{r i.lhe FiC@No.1.4.j.<br />
All ir'c esl ofdE posnjvc ob$dadons e aborc FotheringtEn 3 vnibitiry cw..<br />
hsread ofEtying on lh. ..bueb,. rylDxihale natbeharical<br />
nay consider the su6My tabte dala and ltr a quadratc cune<br />
approxrhation. Fol<strong>lo</strong>wing rhis we oblliEd lhe to<strong>lo</strong>sins @.a@n<br />
ARCV dd DAZ jor rhe shmarJ rabt. r.4.r d,h:<br />
/RCv = -0 0ot2g Da z' +o.o7442aDAZ +|.86429 (3.4.6)<br />
ed dennc e dlkrure visrbitiry pamercr ya ba*d or<br />
apProx'&afior d folto$:<br />
Seond d€88 L@r squd<br />
I L\ = URcl/ +a.00929D,42, '.O.O14429DAZ _|t.s642g)I |) (j.4_7)<br />
lt4
Tabl.No.1.4.3<br />
5
fhis nbdi6c.tio! i! rn oL by Fo$dnBltn it lppli€d to dE obseutio4<br />
repsr€d by Forh€.ingh.n .rd e$lrs e pc5I.d in Tlble m. I 4 I TIE dat! of the<br />
lnbL n d$ pr*ft€d in FiA No I4.1 !t ir 4ily sa th3t rh€ l@t sq@e ntins ro a<br />
s@nd d€gE polynohisl h.5 id inprovcd $ylhin8 6ttF tN nore positive<br />
ob*N.don have lill€n into th€ mg€ ofn g{ive vd!$ of vd Th* ee Nmb€red 6<br />
(ofo.t 27, 1878) a$d l7 (F.b 20, l87l).<br />
Dunng th€ sme.6 Mdundg (l9ll) @Nid€rcd &olh€r b6io data s.i given in<br />
the hbl. .o 3 4 4 b nr $e obwnioml drl..<br />
20 30<br />
hvl3bL cro.c.nt3 . villbL cr.8c€nb . Forh6rinoham tute<br />
Fig. No 343 Fothringhan s Rule<br />
l16
25<br />
F-d;:<br />
Fi8 No 144<br />
A last sque qu.d6dc poltDnnl 6ncd b $jt d.L iGld! [E fi)l<strong>lo</strong>wing cl.rion:<br />
at-v DAz'1 PAzl + tt (r.4 s,<br />
lm 20<br />
Usiig lbls polynoni.t ed $e @nditid $d the (,B@tt would be visiblc if:<br />
ARC1'> !4: V1+r<br />
(34e)<br />
a.d lppli.d i to dd. us€d by rdtsirghd $e @h3 obrlid 4c prc&tn d in t$lc m<br />
3 4.4 !dfig.no 3 44 In lh. tabb t t lh. 'visibility p.m€rcl dcfin.d asl<br />
,,fl!0o20)), =f no", -l DAzl -P4.,\\<br />
(3 4 r0)<br />
II
35<br />
30<br />
25<br />
20<br />
15<br />
10<br />
5<br />
0<br />
-10<br />
. hisible Cre€cents . Visibt€ Cr€sconts<br />
fig. No 1.4.5<br />
Th. rrbl. 1.45 3how U[r !E @ft! br.d on Medd,s oaM &. such<br />
inprevcd Md rhs. b dly * obqvdid j.. m 4j d lel t4, tS7l, thd dqin6<br />
eon rhc @rdidoa (3.4 9) IE irbt. j 4.4 and dr fg!rc ooln sr|N rh|r ood.t du€ ro<br />
M.udq is nu.h inpotld s ohprld <strong>lo</strong> rh. Dodet (!l. b ho Ein*hrn.<br />
rin.lly in rhis vort bolh the @ddq .trl. Forhdi|ghe .tr rh.r due |o M&i
E- 6i<br />
6i<br />
intTii:i<br />
'E<br />
s<br />
na<br />
n3<br />
E<br />
ll9
:i::<br />
alF{s_{";+E:'-.-l {-:<br />
'l<br />
Fia. No l4 7<br />
On lh€ orher han4 figure 3 4 ,nd rh. tdlte i. 6ppe.dix tl<br />
od ol 196 pGniv! nShd.Ss o,ly lO o6@tirotu d*i.1e |ion<br />
Ihur M.unders etton is mrch b.rid lho dd of Forb.gnant<br />
t20
lh. €f<strong>lo</strong>ft e nol s su@esstul s lh. Baby<strong>lo</strong>nia cdlcnon ed Lud RiFnc$ Law<br />
coBider.d in the prelioN two articl€s 6 fd s lb. nMbq ofdeviorion fon the law in<br />
the ob*tued @seni d€ @ren.d.<br />
Anolher eforr ofgEar significance rnar h found in tir.6tuE is based on rh€ wft<br />
of Schoch ( I93O) and h known d lndie Melhod give. in the Exp tnlnion ro The trdian<br />
Astronohi.at Ephenetis , t^ ttlh n€lhod dr ba5ic dala u!€d rs givs hee il lablc no<br />
3_4.6,<br />
A leasr square quadralic polynoniat fincd b rhis dala yields lhc fo owing Etaionj<br />
A nC t = l0 t74l - u Ot 3l)DAZI _ O.@aj DAZI (1.4 r r)<br />
UsinS lhBpotynomiaI od rhc condirion dur rhecre*enr $outd be \6rble ii<br />
/ RC v >1e it4i - o.ot lTDAzl_ o.@st DAZ. (t.4.12)<br />
@d applied il to dara *l<br />
Pc*nled in fis. no.1.j.8. For 6is figuE ,// Bed G<br />
Ihrs hrl( $e Esutrr ohained aE<br />
''visibilily peamelef defined sl<br />
\ =(ARCV _ lt}.3:,43 _ O.o137DAzl _ o.!n97 D,12..)) (3.4.1:l)<br />
,,,,,.i." "" -O 19 our 196 porir.\e sishrinss Dar devidrc rbn l,ll. condi on<br />
TIs rill rhe hrer hatfofrhc 2Od c.otury rh. Indiao n.rh@ ws consid.red to be rhe besr.<br />
Horck in vi€w of rhe analysis of Baby<strong>lo</strong>nian cnlcrion and the Lund Ripen s Law<br />
exp<strong>lo</strong>Fd 6d prsenred in this voik. dot w.d rhe Indim mehod rs s sle6sful in |.ms<br />
l2l
of Fsnivo 3igidier dwidirB froo rbd.l. Dring rlE nodsa tis6 rh. Bely<strong>lo</strong>nim<br />
cniqioo dd Ripenss tuaction h.! no( ben exp<strong>lo</strong>Ed s rho@8fly 4 is done in rhe<br />
wott This cxp<strong>lo</strong>orion hs lqd ro a sianili@1flnding rhai rh. eient md rh€ n€nievat<br />
nE
Tne ddc oa aliflic wirh lnc horizon plats d impondt olc for th€ condirioE of<br />
edli.st visibility ofn.w tun( cese.i. For norlhd h.misrhcre if. conjunctio.<br />
falls nes aururual equinox this ssle is snall for borh hiddle md high.r<br />
latitudes md fi€rcfoE cBeol is eidlcr c<strong>lo</strong>e to rhe horian or .vcn b€<strong>lo</strong>w lhe<br />
hori&o .r rhe lin ofsu*!. Th.EfoG old.r @scenG may *ape si8hli.g.<br />
Th€ acicnt Bobl<strong>lo</strong>nie crildion for the €elid visibitit, or new tunar ccscenr<br />
h6 the hienc$ suee$ pcencge (96.4rr'o) monesr alt rhc nodets consid.Ed in<br />
lhis cbapr€r for posiljve sighdigs coGidered in lhjs tr!,k. Ho$!ver, $c su@e$<br />
perccnlage fof neSalive lidri.gs is nol good enouSh (59.9%). Th$ the o!e6[<br />
succ.ss pe rcen rzge ot lhe Baby<strong>lo</strong>nie c rir.rion is 75.4%.<br />
The ideas retated b lh. Luar Ripeness tuncrion thal deletoped dlrin8 $e<br />
nedi.val ca ae lhobughly invcsligared. Wirh ftoucm rahnques of<br />
ohpulltioB rhis hs Esultcd i o a useful ne6od for det mrnrng lhe day ofrhe<br />
fi6t sighdlg of @w lumrc&$enr. The onl/ probteh thar sdaced againsr lhis<br />
nethod i3 lhc sightings that deviared from rhe nod€l i. hisher tatlludes. Thee are<br />
older c,€*ents a.d brighler crcscentr h,v€ <strong>lo</strong>wq Rip.ness<br />
11"' ly:. rh€ s@ess per.nlas. of Lunar RiFnB rarv for posnive<br />
siel incs G 9.8% (bcrb rhan alt hod.ls coNidcEd excepl Baby<strong>lo</strong>nitu cril.rion)<br />
bul rhal fo regadve sightings n only 57.% (wose lhan all ofter crneria<br />
consideEd in this .hapte.). The o!.hlt succe$ p.,cenbg€ is 72.l./o.<br />
Theadw.rage ofnelhods lhar sre bakd on Etarion beNeen ac ofvision &d<br />
!€hdve .zinulhs ed thal dc moE lhomu8hly invesric"ted durjng Eod.rn.6 is<br />
a6o exptored. Il is fomd lh, rhe Indie Dcrhod b.sed on dr b.sic dala of Scho.h<br />
is rhe besl aDones! lhe ARCV_DAZ ba*d Dedods<br />
amon3st the €npidel Dodels ofte e&ly 20,r cenrur', rh€ rndid De$od be.l<br />
uccss peEenlaec tor posnrve srghrings {eo.l"b)<br />
rollw.d b) rhe Mlmdets herhod r84%r and rh. me$od due ro Maundcr<br />
t23
(54.lYo). Howv.r, in t m of th. su.6e pc@nla8e {o. eSatve sighdnes<br />
Forhdinghm\ crilfion is rlE bcn (@ons$ slt m.thods consid@d in th.<br />
chaptet wnh 94.7% fotlMd by Momdeas (82%) ed th€n rhe Indid ednod<br />
/67.8r/'.<br />
The oveEll success pcrced.ge of rhe In
Chapter No. 4<br />
PHYSICAL MODEIS & THEIR EVOLUTION<br />
AI thblclh rhe twen €rh .entury and Inro $e ie€nry ftrst century a tot o,<br />
wotk h.s been done on vanous aspects ot the probtefr of vtstli ty ot new tqnaf<br />
ce$€.t. th€se other t$u65 In.tud6 (t) ton€th o, tunar cr6sc6hi {Danjon 1932.<br />
1936, yas 19a3b, 1garta, McNa y, schaeisr, 199ft, McN.xy, 19a3, sotran, 2005,<br />
Qu6hl & khan, 2006, .rc,) (ia) the mintmun o, timtthg €tonga(on of n6w vistbr.<br />
lunar cfescenr (Danjon, 1932, ras 19a3b 6tc,), (thr $asohat varaa ons In the<br />
€anbt vjstblliry oi .ew luna,.r.scenr (llyas, 19A5, Crrdwel & !an€y, 2OOO erc.).<br />
On€ ot rhe mosr stgntfi.ant or these and olher sffons B rhe hrroduction of rhe<br />
-|rrohattomt<br />
Lmr D6te Ltne" or |LDL (contras0na rh. |nr.ma onat date ltno<br />
(solar)) by lltas (ltyas! 19A6b). Thoogh the td6a h.s n€ve, D.en !s6d h pbctic€ of<br />
lunar cat€.daE but rhe !€me has b€en erten.ivety usoc |n <strong>lo</strong>nrare (ror insrance<br />
luooioat by t{an4r lnd A@urat. llmo by Odeh) as a gu|d€ for the rcgtons of<br />
vlslblllty or i.vtltbllty oi ths n6w l!..f cr€s6nt, How€ver, In thi. wort the main<br />
emphasts ls on th6 modets rhat deat wlrh rh€ probt.m o, eaniest vlsrbi ry of new<br />
lun.r cesco.t so that other tssues 6r€ nor constdered.<br />
The ri6t asr@phlstc.t modet fo, ervtng thls polreh was rhat o, aruin<br />
{8tu1n, 1977),Ihis kas ba$d on tho av.rage bdgttness hodetrorru tltoon, rh6<br />
averaSo b.tghtnoss ot the twlighr sty and lhe theory or extlnction ((ooman, 1952,<br />
B6mporad, 1904, Sted€ntopf, 1940). Sruh was ,bo rhe rid jn modern rimes ro<br />
exprott lho v.rtaf<strong>lo</strong>ns of tunrr s.miii.meter wlth rhe Eaft]vtoon dlsrance,<br />
Afiotu€rds, app€ared lh6 .xtenstve u!6 o,lho phFlca .nd scionce ot vtslblity by<br />
Schaetor donng rh6 t6st quarte, o, th€ twe. eth @ ury Gchaefrer, 19s6, gaaa,<br />
19aab, 1989, 1990, 1991., 1993) ba$d o. vrdous racto6 ike armosohoric
e{inctton .nd sky briShtn* ds€ to vanou. objocrs l6adtn€ to th€ ttmtttng<br />
ma€l tude or the sky. H6 al$ lhe.att2od rhe hportance ot (a) tack ot hrorm.lton<br />
about seather predrclton st iems a.d (b) necd ot tunher qt<strong>lo</strong>.6 on ot th6<br />
phys<strong>lo</strong><strong>lo</strong>s/ of huhan vi3<strong>lo</strong>n c.pabthbs. Thus,3tnc6$€ rheoFltcat modet teadi€<strong>lo</strong><br />
Lunar Rlp€nes law by modtqvat Mus ms th6 only rheoreltcat mod€ts a€ du€ ro<br />
Bruln and Schaeter In rhl. york Schaoter'. t$hntque! .16 .pptied ro th6 ,6cent<br />
obs.aalion.l data and a.e toond to b6 In good aet!€metrt wtrh the obsetoa<strong>lo</strong>nal<br />
The exptotts or yattop (ya[op, 199a) whlch was .gatn mor€ of enoticat tn<br />
.aruE b ba36d on the ob$da{onat dsta and p.n o, Brutn,. mod6t bur wtrh rhe<br />
srmprrcrt ot a 5tngl6 par.meref crito.ton tor the n.w oes.enr usibtr(v, Thls<br />
Yal<strong>lo</strong>p s modet c.. b€ termed as a s.mbmpt car moder. one o, rh€ m6t<br />
srgnlllc.nt connbutons ot y6<strong>lo</strong>p ts hts concept of b€st th€ of vistbfiiy, Tne<br />
sonware Hrtatol computes borh th. q_vatues (ya op, 199a) and rhe naCrltod€<br />
@ntrst (rhe rerm cotned In rhts work) rhal tsthe dnf.ron@ o, rhe r{agnjtud€ ot tho<br />
llroon and th. timtflnS hagnltud6 of rhe sky ctos6 to cr€*enr. Ihe conpartson ot<br />
lh€ tso i3 di$ussed ..d som. ot th6 extra ordtnary oDsendtbns are cdflc.flr<br />
analy$d. rlle mosl stgniftcant part ol thts ch6pter ts the d€vetopnenr ot a new<br />
srnge psrsmeter crttodon for th€ fl6r vistb lty o, new tuhar crsc6nt, we have<br />
con3ld€red the acruatbnghrne$ ot the cr6c6nt rhat b pnase dep.nd€nt(instead of<br />
aver.€. bd€ntnBs ot ihe tul Moon ctos. to horizon usod by sruh).nd rho.ctual<br />
b ghrne$ ot lh€ iw ght 3ky c<strong>lo</strong>se <strong>lo</strong> rhe potnt whee rn€ cesce rs prese( For<br />
lh€ bighhe$ of bolh (the 63conr and th€ sky) the <strong>lo</strong>ots devo<strong>lo</strong>ped Schr.fer.nd<br />
otheE hav6 been osed, thts has ,6!utted h<strong>lo</strong> new vlrtb tty and lmtring vbtb iry<br />
cunqr lhb te6ds to a n€w s€t of basic dat. whtch In run E.onvened hro a rcw<br />
slnge paramoter cdredo. ba*d on . retarion het e€n aRcv and width o, crescenl.<br />
lh6 osr mod.t ts anorh€r somfempnbat modot, Our cdterion is <strong>lo</strong>{.(t to havo<br />
bettor succe$ percentago than any orh.r crile on devetopod dq ng rh€ 2Oh<br />
t26
4.1 BRUIN'S PHYSICAL MODEL<br />
Atuin bsed his wolk (BNin, 1977) on 1he obseded avedge brigbh.ss of sky<br />
agaiBl rhe posirion of th. sun b.<strong>lo</strong>s ho,iD, ,ner ssel (0ut natched tnc resuls ol<br />
K@h{ et. al. 0 952)) and &e bdshrB of rh. M@n s a tu&tio. of ddud., b6ed or<br />
lhe 1h6ry ote inclion dw B€npoFd (Benpomd, l9O4). The fi8ures gi!e, by him, Fie.<br />
$d 3 (Bruin. 1977, pp.339) arc r.poduced here in Fi8 No.4.t,1. On rh. b6h ofthcse<br />
sludies Aruin deve<strong>lo</strong>ped the Lund visibilily cuFes (relarin8 atdtude, , of qeeenr<br />
p<strong>lo</strong>ttcd asainst s, lhe els depE$ion b€<strong>lo</strong>w hoizon) 6d fie Lioiring Visibitiry curyes<br />
{Elatine, + r,gai6r r) ed pEenled in fis. no. 9 (in€tude
cxnibircd in n8 9 ir is mr.d lhnl aI $le dip r = 0 d€gees lh. ninimM dliNd€s for<br />
dif,cFnt vidrh ces@t3 @ girq<br />
a2<br />
2 3<br />
Tbh hcms Aar rbe bndrh$ of lh. sky ar rhe* atrirudes (r) G we .s rhe brishrne$ of<br />
th€ cBent of@ftsponding widrh when lhe su hdjul $r. I. ord{ dul l,rtc ce$. is<br />
rt 16r a bright s the sty wilh dftEding ahiludc rh. c@e nusl b. wider ud sidc,.<br />
Tnes ar€ thc sanins poinls of rh. vnibilny c!ryes fial aG aU shaqty d(rtsing<br />
f!.crions ol the sotar dip, That heons lhar not only the brighh*s oi cc$e bul $e<br />
brighr&s of sky (har cqul a<strong>lo</strong>ns lbcs cwa) borh diminish sh!+t, wirh $e<br />
inc@ine st& deFEssion.<br />
Th€r.forc, for larger values ot r rh. ee$ent of sne brightne$ @ b. $en at<br />
<strong>lo</strong>wer od <strong>lo</strong>er alftudc ,. Tlw arc al$ the sianing points or rn€ cums ttat show lhc<br />
b.h.viou of, + s {sh ofglribd. of M@n ed lh. sot{ depb) agaiGr th. $tar dept r<br />
Ahhou8h the sm of th. atlitude /, of $e cresced dd the r me $lar dip Eoains alnost<br />
consbrt 4 $e ce*enr gos doM dese cw6 coftsponding to fix€d crcsnt width<br />
ed thcEby ro nx€d brighhess. As (he atrirudc of ce$.hl daca*s 0r. sky brighrness<br />
n4l d
Fig. No. 4.l - ! fic fi8u6 fron BNin s Pap.' ( t 977)<br />
rrr, , rc rddb-r rr oi *l<br />
t29
For thimer c6c.nr rh€ ,r + s aSaitur r p<strong>lo</strong>ts Sive thc b.st time of vhibility 6 the<br />
poinr whi.h is the midhum of rhe cwe, h rlso 3ueSens a tug. of tim. for *hich Oe<br />
crc*dt my Ehajn visibl€- Th* de excellent id6 rhar @uld help cre$ent-hunre6<br />
bul untdtunably Btuin has nol pesenl€d a clear cul ompuratond lrchriqu€. Brs.d on<br />
sihil& ided Schaefer ha worled out Enorher schehe of computario. <strong>lo</strong> b€ discu$.d<br />
latcr. The schcnc for computrtiohs is d€duccd frch the visibility cuNes olBruin sd is<br />
bsed on ll)e dinimum point of rbe r + It p<strong>lo</strong>r againn s. A relarion belken rhe cE$.nl<br />
vidth cor.sponding to d€ r + , cwe sd rhe valu. of , + r ar rhe minimum oi lh.<br />
cufl. nay be d.duced fron abularing rheF ratucs. For lhis purpo* rhc data deducrd<br />
fiom 6g 9 b, YaUop (Yal<strong>lo</strong>p,1998) is d fol<strong>lo</strong>ws:<br />
Table 4. L2<br />
2 3<br />
The valu€s of ARCV ore the valucs of , + r picked fbo rh. ninimln of rhe, + r cufl es<br />
asaisl r whi.h coqespon.ls to the b.sl -tine ofvisibihy ofcresent. Tlh dar! 6 edto<br />
a dird degd polynonial usinS lean sq@. apptuximrion lqdr <strong>lo</strong> the fol<strong>lo</strong>vine r.larion<br />
ARCV =12.4023 9.487arv +3.9512W' O.5632tr'<br />
(4rr)<br />
Tnis ( an be tEns<strong>lo</strong>med ro lh. \ Fibitrly oaEmero . r, lunc,ion as to'toq:<br />
v), =(ARCV -(12.4021 9.481W +3.95t2t/) _0.5632W)Dtrc<br />
(4.1.2)<br />
Tnis crn be le$6d on rhc obscnarional dara with the visibitiry condilio thal a cssce.l<br />
should be visible if,, > 0 olheNi* it should E@in invhible. Using condiion (a.1.2) on<br />
lhe data *t ed ro evatuar. n rhods in ch.prr 3 dlc p<strong>lo</strong>r w obhined is shoM in ligu.<br />
no. 4.1 ,2. ln this fig@ lhe cue nmed ..Boin\ Limit,, is the p<strong>lo</strong>l of $e equrion 4. t .2.<br />
|]0
Th. r€d of th. p<strong>lo</strong>r CD$ th. rc|t|iE.Xirtde (in d.8r€).gaiMr th. cltsdr wirhh (i.<br />
I snd9 €l@ld€d ar dE b€s tinc of visibility for @l obsd8rioi of ttE &tl sel<br />
coNid.r.d ii th€ peioos chl9iq. All !h. pd iE siglding cas .rd ihe @lrs of<br />
cdlculalimr b.$d d (4. L2) @ 31Em in Trbta 4 | 3 TtE @not.r. dda sr b shom i,<br />
R.--E-'_;"_<br />
2<br />
r-r*.rl.sc.G,wu.ca*<br />
Fig. No 4 1.2<br />
ApFidix.|u slDB rl| @hs of .rptyin8 Btuin.s [bn, yttrop,s ondid) €itdi@ (ro<br />
be disss€d in rexr rnicte) .nd th€ w dftdim d€vetoped in rhj3 woli (ro be repon€d<br />
in anicl. 4 4 betoe) undq rh. hQding -Mod.r tne firn @Enn otriaitu lh. vat!* of<br />
visibilily pmmetd d.fn€d by (4.12). TrE Ddl lh cotunr, e ror the orhq rh<br />
ltl
t32
lll
tJ4
tlq<br />
$I;;<br />
135
1lz<br />
Our ofth. 196 positiv€ sighring in the @pt. t4 cs6 d.qarc rron th. Linjr d@<br />
!o Btuin. TheEfoE rh. nodet du! b Bruin is thc best mongsl ql ihe orhe, 20o ccntury<br />
nod.k ad is s good $ lh. Lufu RiFn* Law. Ar nx es rhai daiar. f.oo<br />
Brby<strong>lo</strong>nid (io.rion atso dcviak froE Bruin,s nodet. A ctoer <strong>lo</strong>ok in b ttE derails of<br />
14 caes $at deviab fron Bruin,s mod€t sve.ls oEt rhe la cses devi.rilg frcn the<br />
Luar Rip.ness Law and lh.e 14 caes have 8 common cascs. rn. fihl I cs.s ofvisible<br />
cNolntable t.3.1 (ob*nation no.286,2ed 272) ofoncrec cros to auruDnal<br />
.qunox have !alu6 ofve srlt aboE Bruin.s timi! siyen by (4.r,2). A. n @ di*ussed<br />
'n snicle 1.3 the ce!ce.& in 6ese .6es eE older o<br />
vi'ibihypo$ibre. rh. Bruin,slt.i,i,b.,.d..rh"b,ich.::,:;f,J:1':;T::<br />
cases ae erisfying 1ne Btui.\ lioir despite having sEEll las.<br />
Our otlhe six cdes lhat d€viat. fror Btuilt linit bnr not fbn LUM Ripenes<br />
raw{obs.4o.413,434,I19,ll5,2S5 ed 290) fiEi fiG havc qalt valu6 (o.ol,<br />
0.061 and 0.07 rcspediv€ly) so dc hsginal for Led Ripen*s<br />
^Rm<br />
Law. Sihitdly si, of the<br />
Ijg that deviat fbh rhe Lu@ Rip€n $ t.w but rc! f@h Btuin,, ljnir (obwlrioa<br />
numb6 286,2,22,314,6t3 ed 716) firsl lbG have b*n dscussed befoE. The n.xr<br />
t36
two ll4 ad 613 have smaU v, v.lu6 $ dc m&gitul €ds in BNint Mod.l. Tn hsl<br />
one (obs. No.7l6) is asain u old ag., wide ond bdClt crceoi<br />
Ths sc ob*rve that 6c Bruio's Linit md the Lund Ripen.ss hw ee nol<br />
exaclly supplcm.ndng eh oth.r blr thcy @ rh.rically cquiql. Howevd a<br />
Brui. s linit is rakin8 brighhe$ into considqation theE<strong>lo</strong>G Bruin's limit h moE<br />
<strong>lo</strong>gicat ud noE su€6sful.<br />
4,2 YALLOP'S SINCLE PARAMETER MODEL<br />
To devc<strong>lo</strong>p his onc pdndd nod.l of firsr visibihy of luMr clt$nl Yal<strong>lo</strong>p<br />
used Bruin s wort in order to .xi6ct optimum crc*enls *idth for vdious rehtive<br />
ahnud$ o. @s of v'sion ARCV, nedion.d in ptvio6 anicle. Like Folhcdnghan<br />
deve<strong>lo</strong>ped a <strong>lo</strong>mula cllline ARCV wilh DAZ on lhe bask ofht sunnary ldble da|a,<br />
Yal<strong>lo</strong>p coroider.d fie bsic dau for d€ve<strong>lo</strong>ping a Elalion b€lren ARCV md lhe Nidth<br />
of crescent W, Hh data (froft pase 2 ol NAO T*hrical Note No. 69 (1998)) is<br />
Epioduc€d in the oble no. 4. | .2.<br />
Hoselcr. d slal.d hy Yal<strong>lo</strong>p, f6m 1996 Mdh HM Nautical Aldanac Ofllcc<br />
ddided ro abndon i!3 tcsl bared on the Bruin m€lhod (4.2.1) for lh. one baed on lhc<br />
''Indi.n" melhod d thc lndie m€thod produced mor€ scnsiblc rcsults fo! old .gc<br />
s'-shrings al high alirnd6 thai occuB at l.rsl once d y.r fo.lalnudd rtomd 55 d.sd'<br />
Bed on fie sorli of schoch (1930) th. b6ic dara ued in the Indis. melhod h siven in<br />
DAZ 0' I0. 15. 20.<br />
I00 30<br />
Tatrl.No42l<br />
As rhe width of rh€ qesent is sivcn by (2.E.9) dd thst @ abo 6e wifien as:<br />
tt7
' = ls
*lich l6ds ro tlE visibiliiy pabr..ld:<br />
,ip = (,{XCr. . 01.t733-9.0312W +2011J4W" 013601/ )/10<br />
(42t<br />
Fig No 421<br />
. lnda Umir . htsibl.c€.e<br />
ApplyinS Mendd's nodifi€d ondition to thc daia sr of Ch.pLr I it is fou.d<br />
thal oul of 196 positiw sightings @ iher.27 clg th,l dwiat€ fron the 6nd;lion<br />
d.fined by (4 2 5) rhe se e shoM in 6g@ 4 2 2 Thus Me 6s nodited<br />
@tdition is dill mr bdd th3r th. ltdiu lr|d rn Btuit's @rditios All thc 14 {,g<br />
rnat ddilr. iom Bruin s linit tte d6nd. lion rh. Indie linn Ttu rvo tddidonll<br />
wr io!! m. 3r4 .rd 33) ttd ddid. fron lndis $nn & Idtd to b. tugjn.l cet<br />
(vr = 0 ol8 and o oo2 repedively) in Y.l<strong>lo</strong>P! Indiln dEd.l Thu3 thc t@ hod.ls ar€ in<br />
c<strong>lo</strong>re sgr4rcot s Iney re both bas.d on sinilu lpprs.hes a.d onlv slishtlv dillq itr
i<br />
{<br />
Fig No 4'2 2<br />
lD addirion 10 @Ndting .ll visibiliiy otditions from DAZ-bas.d <strong>lo</strong> th. Widlhb!*d<br />
Y.l<strong>lo</strong>p achiev€d tqo orhd rdeltbl. t8ks. Oft oftheh is hh d.ducion of BBt<br />
ine of Visibihy The otht r.Mbble ontdbllion liom Yal<strong>lo</strong>p is io tkaw lincs<br />
140
.twe.'r€giotrs of v&ids vbibility condilions on lh. b.s of viribihv pdma.. ,/}<br />
used in lhe Indian mdlDd th|t be cals 6.8tu .<br />
Fis.No423<br />
Fdthc bd rinc ofq.sit lisibililv' hc nor6 ih.t h fg 9 of Btuin tlE siniM<br />
of viribilitv c!tues of , t r agaiin s <strong>lo</strong>r different ges.ent widths, fom a slrsisl]t line<br />
which whs Drcj.cl€d n€els tn orisin of the @rdiMt€ svst'm (" r) The 3{m h<br />
shom h6e in fi8u.. 4 2 3 s a G{t lir. PNing ttroSli niiinun of @L @rc Th'<br />
s<strong>lo</strong>p. olrhis saSl lin.i3(, trta/4d,rs 5/{ Itis dnimun @@sponds to lh'<br />
b€d vilibility @dnio|\ thc b.d cerlln bd}6 rlE N6'ee briahtns ortlt d(v !fl|<br />
the b.iShtness oa th. c..stl ..dding <strong>lo</strong> Bruin Yall@ inrdPr'tt it a 'poinr in tm'<br />
rhNt rlivi.!€s "the tim€ lin€" beiv@ the Sunst ts ed lhe M@n!€t l, in a nlio 5 4 Thus<br />
rhie "poi.t in tin€" i3 th. b.!i tine I, of c..s..ot vitibilitv Sivd bv:<br />
t,=<br />
5\ + 4\rs + LlG) ,<br />
9<br />
l4l<br />
T. *!1,4c<br />
(427)
In all cr<strong>lo</strong>ldjoE in rhis rcrk rhe coiDur.rioa have b.6 dotu for this besl tim.<br />
sivcn by (4.2.7). EsFcially the calculatiotu fo. rhe iable no. 4.2.2 iD $ltich rbe Indid<br />
condition (4.2.4) c used lhc @mputations ac done for this best tim..<br />
Finally, Y.l<strong>lo</strong>p d.duccs rhe visibihy @ndilios for dillcrcnt M8es oI q_vatu4<br />
aic. a detailed dltysk of rhe dlta *r of Nund 256 obsdado6 avaihble in his timc.<br />
Our Esulti in lh. sdond l$1 colmn of rlblc in app€ndix-lll, e lPpliqtion of lhc<br />
Yal<strong>lo</strong>o s condnion (bascd on bsic dala of schoch, l9l0 or lhe Indid ndhod) io! $e<br />
doia selcted inchapter 3 wbich is taLh mostly iion Od€h (Odeh,2004) Theconditions<br />
d.duced by Yal<strong>lo</strong>p d. Eprodu€ed heE in labl.4 2 4 Ihese conditio.s e iddica<strong>lo</strong>6 for<br />
lisibility eiih or wilhout oPlica! aid.<br />
Tlble 4.2.4r Thc t-test crileria.<br />
{A) q> 0 216 ENily visible (,4ICl 2 I2'r {EVl<br />
(B) 02le,l> {014 visible und( p.rfd condnioN (v U PC)<br />
(c)<br />
The linilins ralucs of t<br />
(Yal<strong>lo</strong>p,1998):<br />
May nedoptrtl lid'o find cerdIMNOA)<br />
(D) { 160:4 > -0 232 w-itt neo oprielaia o frna crescent tRoet<br />
(E) -0 2l>4'-0 29q froivsrbG viii[lcmpe,4nC. < 8 5'tlt (D<br />
(F)<br />
4-299>q<br />
weE chosn for lhc aix cnbria a to F for the fol<strong>lo</strong>wing reens<br />
-t<strong>lo</strong>r<br />
'sitk,<br />
ueto" Dujon tinir;Rcl < 8' I<br />
A <strong>lo</strong>@ limit is rcquir€d to spdl. ob*talio6 $at d uivial 6on rhos l'ul<br />
nave ed. .len.nt ofdifiqitv. Accotding 10 Yal<strong>lo</strong>p il wc foud that lhe id'al<br />
sirr^ion ARCL = l2o nd DAZ = O" prcducd a Fnsible cut-otl poin! for shich q<br />
= +0 216, Tnere aG Ill exmDles i! Tabl. in appendixlll G'cond lai column)<br />
wh€n q exc.eds this value, ad i. 8.Mal it should be v€rv ..sv <strong>lo</strong> se 1he new<br />
(v)<br />
142
c@dt in ln.e m, Providcd th.c is tro obscaing c<strong>lo</strong>ud in lhc sky The<br />
rponed posilive sightings i. dr6e cascs e I 14. Tns it is @mble to consider<br />
cscs wilh I > +0.216 10 b. lhose wbcn the cccnl is esily visible.<br />
(B)<br />
Fon obseryc^ rcporls it has been fou.d that, in gcneral, t = 0 is c<strong>lo</strong>se to lhe<br />
<strong>lo</strong>wr linit for fi6t visibility ud.r p€lfecr ahosphenc co.ditio.s 3l s.. l€vel<br />
withour Equirine op1ic.! aid. Yat<strong>lo</strong>P u$d his Tlble 4<strong>lo</strong> st lhis <strong>lo</strong>sd linn for<br />
vhibility mot p@helyand he says thal from inspstion of Tabl. 4 ihe<br />
siSnilidcc ofq - 0 @ b. str bul t = -{ 014 is &olhd Possible cul_of val@'<br />
Thce @ 68 €Aes in Tabl.4 sith q in ihis Ens. in the daia u*d bv Ytl<strong>lo</strong>t wirh<br />
48 positive siShtings. Th. data sed in this qotk (in app.ndixlll second lst<br />
@lu.) thec aE lll ca*s with q > 4Ol4 bul les lhan +0216 Oll of th€*<br />
l1l cse) rhe!. ae 6a posirive sighnngs withoul oPtical aL<br />
(c)<br />
Yal<strong>lo</strong>pusd hisia6L4<strong>lo</strong> fi.d|hecut ffpoinl *hs opti@laid isal@vs ne€ded<br />
<strong>lo</strong> Iind the crcse.t moon by rutching lhe q_test visibilirv codc wilh scha'Iecs<br />
.od.. The rcund€.I value of 4 = _O 160 @ cho*n for lhc cut_olT dirclion ln<br />
Tablc 4 (Yal<strong>lo</strong>P, 1998), thcc \,w 26 cNs thar s Gfv rhis diterion t0'16 < q <<br />
-0.014), only lhree ces out oflh.se wre positive unaided siSltings tbe tcst were<br />
se.n vith hiMul6 or r.lceoFs ln Tlble i. appendixlll' *cond lat cormn<br />
useii in this *ork therc 62 cses in rhis tuge ol q_valu€s oul of vhich I 0 s€E<br />
unaided si8ltinSs dd 29 times lh€ cresced *as seen either vith bin@uktr oi s'lh<br />
(D)<br />
ln lhis ca* (-0160 > t > -0232) YaUop s Table 4 ha' roo lew enli's fton<br />
which <strong>lo</strong> estinale a <strong>lo</strong>wt limit for 4<br />
The situalion is nade NGe bv thc fact lhat<br />
whe€ thse is an €nl!v. in non cas.s' the Moon ss not scen even wi$ oplical<br />
aid. ln f4t it h @€ fot lhc cc.dl to be obedcd lE<strong>lo</strong>* 6 a9pae't c<strong>lo</strong>'ga<strong>lo</strong>n<br />
ofabout7c5 (Fatoohi d al, 1998) Yd<strong>lo</strong>p's Tablc 4 h6 14 cs's in lhis dg€ oul<br />
wbicb 6 re positiv€ sightirgs thdough binoculss o! &le$ope ed one exl6<br />
oiilinary ce of uaid.d siShing ln th' r'bl' i' app'ndixlll Gecold 16l<br />
141
@l@) of lhb wolk lb@ e 33 .M our of which 18 @ potiliv. sidines<br />
with binoculd or 1el.$op6 dd th@ extra otdinary ces of unaid.d sidli.g<br />
(ob$flalion no. 189,455 add 24) lh.t is dillcral fren fi. @ @sidercd bv<br />
Yal<strong>lo</strong>p, Al lhe tine of Yal<strong>lo</strong>p {1998) lhis wG thc linil he<strong>lo</strong>w shich n ws<br />
NW.d that il is nol Posible 10 c thc lhin .t snl n@n .v.n vilh a lel4oF<br />
Al<strong>lo</strong>eing I' for hoiienlal panllax of lhc Moon, and ienoring th' eff'cl of<br />
rcfa.liol\ fo. & aptent .<strong>lo</strong>ngadon of q5, ,'{XC, = 8c5 lf ,'lZ = 0p rhis<br />
coresponds to . <strong>lo</strong>wr linit of I = -{ 2l2 Wilhout good fidding lgles@p* md<br />
posidoml infomation, obsne6 aa unlikelv [o w lhc crc$"t he<strong>lo</strong>w this limn<br />
(E)<br />
(r)<br />
TheE is a lheoEical cul{fi Point *hen th. appatcnl e<strong>lo</strong>nSdlion of lhe Moo'<br />
frem the sm is ", knom 6 lhe Ddjon linir (Ddjon l9l2 1936, llvas' l98lb<br />
nabohi el al, l99S) This limit is obtained bv extlapolatine obedalions nade al<br />
lager e<strong>lo</strong>ngtions. Al<strong>lo</strong>wing t" for hori@nul p@lld ofth' Moon a'd i8f,olrng<br />
rhe effect ofrefEcion, ah.ppor€nr e<strong>lo</strong>ngarion o' 70 is eqnivalent 1o<br />
''{'R'1 = 8'<br />
Wirh ,,llcl, = 8'ed D,42 = O! the corespondine <strong>lo</strong>w€r linil on I is -O 291'<br />
However, in Yal<strong>lo</strong>pt lable 4 th€E tte 2l enfiies witl onlv ] posilive sightine<br />
with a bin@uld fot I < 4 212 ed no sighting claim wilh s_aided cve ln lable<br />
in apFndix-lll (se@nd lst colw) th* ae l8 c*s ir the mse -0 212 2 4<br />
><br />
-0 299 ou<strong>lo</strong>fwhich there atc 7 sishdng snh binocuhr ot tlescopes and onlv 2<br />
claims of un_aided siShdngs (obsdation <strong>lo</strong>s 389 and 455) Both the*<br />
obfdrtio.s dcviak from all the crit.ria considered up b lns po'nl<br />
The table in .ppendixlll G4ond last cotumn) or this wod( shoss 66 cases wtrr<br />
-0 299 > q ed therc is one.xtra ordiMry claim of sishtins<br />
opti*l aid (obs .o. 189) Ap&t fron $is lheE is no posilive siehting silb o!<br />
wilhoul opdcal sid. In lable 4 of Yat<strong>lo</strong>p tnee is no clain of €Esce sisblins $<br />
A@oding |o the visibilitv dNifiorion shos<br />
divid.d into 5 tgions bv four consi&t_q val@s As lne<br />
abovc, rhc suf4 of E nb b<br />
aclual visibihY of lhe crescent
depends on ils width dd oi iis dltilude .bov. hori@n d th. rioc of su.sd accodi.g <strong>lo</strong><br />
Bruin (Bruin, l9) dd Yal<strong>lo</strong>p (Y.l<strong>lo</strong>p, 1998) a @nsht q'valu. describes a cufre on<br />
th. g<strong>lo</strong>be i.di@lin8 sinile visibiliiy @nditio.s a<strong>lo</strong>ng all poinc of$e curye' Such a<br />
clne is a psldcr@bolic cwe wi$ v€icx on the 6l-dost <strong>lo</strong>isitude Thc<br />
<strong>lo</strong>npludi.al p6nion of lbis ven x vei* nonih 1o nonlh 6nd th' hnude of |he vqar<br />
d.Fn& on lhe d.cliMtion of th. Moon ed lhe Sun on the ccle$itl sphere Duing<br />
sumheG in nonhem henisph€lc lhe sun\ declimtion h extren€ nodh dd if ihe<br />
d.climtion of ths Moon is norrh of $c su lhis ledex no!.s to exlEne nonb and lhe<br />
c6@ visibilily is edier i. rhc nonn hitudes Duing summer of the nodnem<br />
ladud.s if lhe Mmn is eufi of lhe s6 then ftis v.nex d@s not @ch irs<br />
'xcm€<br />
nonnern posilion dd still tne .e{ cE*mt vhibilitv is better in 1ne norlhen laitudcs<br />
Thc situaion h revese for lhe $liheal henisph€te Tne palabolu opens westMd above<br />
(nonhwardt and be<strong>lo</strong>w Goutbwrdt fion th€ vcnex The CuR' A is rhe collection of<br />
points on rhe 8tob. fot which rhe q_value is 0.216 All rc8io's $ithin lhc lwo b6nches ot<br />
$. p&abola wesl of rhc vertcx e the EgioB wh.e |h. q_vdhe is s@ler<br />
lhm 0-216 md<br />
the cle$ent h ea3ily visible to ln. Mlcd eve in lhis rcgion' Csc B is the colleclion or<br />
ed B lhe<br />
all points where the q-value is -0014. In all tbe Eeions be$€e! cuftes<br />
cicsc.nl is lisible to thc nai.d ete onlv undet pcrfed visibililv condnions<br />
The Fgions b.(wen cuN. B and C (q_value _0-16) de $e tgio's io wbich d<br />
obsen.r suld Gquire opdcal did 10 <strong>lo</strong>catc ihe cr*c'nl dd th€n it mav be visible <strong>lo</strong><br />
naked cye. For reSio.s wi$ q-vdlue I€$ t\ai _0 16 rhe cresenl wolld not be visible to<br />
the .akcd eye For a comnon. untaincd obsePer it h highl) unlikelv thal ihe cFscem<br />
*ould bc seen in rgions ed oflhe aNe A Thc scienificallt ecoded obsaliom (oh<br />
ehich all $c sludy of the rwnli€th cdturv i3 bNd) do oot Pohibil obedalion of<br />
.r.sc.nl wiln mkcd €ye in region belwn cwet A 6d C In sucn t'gioN' in facr' lhc<br />
probabilily of obsenation incFdes whh the nlmbef of keen tdined od expenenced<br />
145
4J<br />
SCHAEFFER'S LIMITING MAGNITUDE MODEI,<br />
Ilruin (197) ued only avfrgc brislhcrs ofsk, duinsrviliehl and rh' vanarion<br />
of brishtrcs oinE full Moon ro obEin appNximlc cohlrsl lb! (csccnls of$rtuus<br />
sidths ro deve<strong>lo</strong>p hh crcsccnl lisibility cuncs dhcussed a6o!c ftc t'srltins nodel<br />
fornalized in lms of , rclation berwen cRsenl width md thc Elalilc tltiludc ol<br />
€{*dr ar "tal rime dcduccd by Yat<strong>lo</strong>p (1998) pro!.d 1o b€ hiShlv succcssful'<br />
lihough, the basic data cxtacled by Yal<strong>lo</strong>p frcm Druin s visibililv cuBcs- was<br />
rcplaced bt rhc basic das due to Scboch (1930) (lhe Indiah nclhod) <strong>lo</strong> aFivc ar his q_<br />
value coodidons (di$nsed abovc) for rh€ nN ces€ vhibilil! n h6 bccn secn rhlr<br />
rcsulls iom Bruin s d,tt .re mrlgi.ally bctter dan thc lndian mcthod<br />
'or<br />
lhc ddo scl<br />
Schacllcr (1988a. 1988b) oh lhc o$er hand has uscd $c phvsics of \isihilirv'<br />
cxr.nsivclt ih rcsulrcd inro a <strong>lo</strong>ol (6at S€hacrdconvencd in<strong>lo</strong> a compubr Dmgmm) to<br />
ddmine $c briebhcss ol sky at anv poi ol tinc dd for di{Icrcnl aftosphenc<br />
rcmp€*tutes aDd rclarile humidil!. Ih rhis {ork Scbaeficls progrm h ep$duccd a'd<br />
frade pad of the lundr crc$eni visibililt software dcvc<strong>lo</strong>pcd. Ilila<strong>lo</strong>! ro clal@re<br />
lisibilily cohdnions. Tlis pan of the sftld is rsi b conpurc thc sk! briSbhcss (or<br />
linni.B magnirude) al points c<strong>lo</strong>s to $e cre$eht md the aPPatunt nagniiudc or lh'<br />
lunar crcscenr ro sludy $c laryins conmsr dunng twilighr for vanous tcnpcrat!rcs and<br />
olativc fiunidny. If rhe appaFnl brighmes of the crc$e.l h morc ddi tlE bighhcs of<br />
lhc xrilishi sty lhc cre$e should b. visible o$cNne nor Schacflcr (1988a) himsell'<br />
hasap!li.d a sinilar rcchnique <strong>lo</strong> @lyscthc visibililvor invhibilnv data avnilablc in his<br />
riFc. -l he @hnique is ,Pplied to $e d a*<strong>lo</strong>f rh. chap161dd the rcsulB obbiocd arc<br />
pnqntcd in rabt No 4l l Bcforc a discussion 6n rhcs' Esuhs Schrcli'is<br />
nldhodo<strong>lo</strong>gy t-or compulins sky briShhess undc! difeEnl ahosphcnc condithns 's<br />
Aner rhc conjunclion thc ne* lumrcrescen{cd bc seen in rh. scslcm skv ck'sto<br />
tne non&n md lhe Poi oa $@r Simildly the last crc$enr cm bc secn i. lbc
casrcrn st! bc forc sunsd. l hc conlEi bc(wce n lhe briebtness o f crcsccni Nd thai o r thc<br />
rwilislr sky dcp{nds on d number of f.c<strong>lo</strong>u. Thcsc incllde:<br />
P6nio! of the c@nl which n.elf is significsntlv {fTctcd bv Lhc<br />
Tbc $.nerine orthe tieii ftom cEee fld thc sunliShr dle to (a) thc<br />
roial amount oaair-nass the ligbi tdvels-lhrcugh. (b) thc rotal amounr ol<br />
aeosol pesnt in lhis air. and (c) lhc slratosphctic ozonc thoush shich<br />
rhc lighl bas <strong>lo</strong> ftvel. Th6c *!lt ring souEes cauF th. inlcnsnv of liehr<br />
sourccs of liehr rhal includc $c S!n. lhc M@h and o$r sourcG (like<br />
anilicial light lbat G nol considcfd in this work 8lh'v m noi s<br />
affc(i!c during lwilisht.<br />
The amosph{ic ldp@tuE and the relaive hunidilv<br />
Taling in|o considmrion all lh* alldls |he to|,al brighlncss of lh ikv is<br />
conpded at lh€ point whcE cmcol is pae lf rhe bnsl i$s of $c atv is noE than<br />
or equal to tbe briShine$ of the ccs. ihc cGcent En nor be s(n' Eten ii rhe<br />
brishtness of the crcsocnr is marginallv moe $an the bflghtncss ot ihc skv it is vcrv<br />
difilculi <strong>lo</strong> <strong>lo</strong>c.c lhc (erc6t withour tny oplictl lid ln $e fol<strong>lo</strong>winS thc q@'rilalivc<br />
bols aR di*ussed bricfly for .ll dc$ @nnuaiions:<br />
For alirudc w.ll lbo€ horian $e lpFcnt Fsnion<br />
anamon.tR. thc angl. ofcfmdion (Snan l953.G@n 1985). givcnby:<br />
ofrhc crc*e isEi*d bv<br />
x=5s.{#]*' (4.i r)<br />
whec P is the ahoq'hcic P6$rc. / is rh. |cnFatuc and z<br />
lhe cEscenr. for ahiildcs c<strong>lo</strong>sr ro rhs horizin rhe fol<strong>lo</strong>wing<br />
t47
i = l,.cor ,+ 1i - l<br />
L I+441<br />
(4 t.2)<br />
r rnr I<br />
x- t.ot.corlA,+ ."- | (4.ltj<br />
I n'+s.tt I<br />
whcre, 90'r : lnd , - 9ou : R The sollwde Hila<strong>lo</strong>l wo deve<strong>lo</strong>i,cd in $is work<br />
ror lh€ delcrminarion of lisual linitiie naehitude thc major nePs ol calcularions<br />
iadoplcd lion schacf4 s pogBE) rc lised b.<strong>lo</strong>w wih bncf de*riPtion Rel4nccs<br />
and dcrailed d.sc'iptions can bc folnd in Scherd (1993)<br />
Thc prcgEn funcion /drat r'l(,) 1at6 s inpu/pE-c.lculatcd valnes listcd bc<strong>lo</strong>w:<br />
. a4r. thcal(iludcofnoon abovehonzon. tdu. ftc azinnth oithc noon<br />
. par. the alrillde oflhc place abovc sea kvelih ndcB, Pn,z. cslimlted rclalivc<br />
hunidny of the place. p/a/. lairud€ of the plde. PratP. esrinatcd lcnD€dlrrc of<br />
. sd/pn4. lhe right Nensioh ofthe $n at the inc of ob.cnstion.<br />
. \eshlil (= 0.365.0.44,0.55, 0.. 0.9) $. {av.lcngths @@spondinS <strong>lo</strong> U- B. v.<br />
R md I bands<br />
. b,r.r[t (* Sxl(]iJ. x10 i. lx10fr. I'<strong>lo</strong>i3.3xlOI) p!3ncl$ !.lues in lho<br />
nish dhc brigihss sssilled snh .rch wvelenglh $lccrcd<br />
. a!r1t (= 0, 0.0.01!. 0.008.0) p@eler values in $e cxrinction cctlicicnr<br />
corespondins <strong>lo</strong> lhe ozrne factor ssdialed wirh ceh wavel.neth sclccled<br />
. rr.!.r/, (- 0.04- 0.045. 0.031. 0.02. 0.015) p.Bmcrer values in $c cxrinclion<br />
ceficicnr coftspondi.e to ihe w@ficr fac<strong>lo</strong>a (hmidnx lcmpedurc elc )<br />
associatcd wilh each wavelengh sel{tcd<br />
. @vh[j] l. -lA.%. -10,45. -11.05, -11.9. _12.7) lhc nagnitude of fuu moon<br />
coftsponding ro dififtnt slel.d warclen€lh b6nd5<br />
. mschtil (= -25.96. -26.09. -26.74, -27.26,- -27.55) fic $lar magnilLrdc<br />
coftspondinS to diff€rc sel*ted w6v.le41h bmds<br />
148
cnschtl ( t.36.0.s1. 0.00, -0.6. -1.17) @Nrion for lund masoirudcs<br />
!! csponding ro difiere.l sleded wavelcngih b6nds<br />
r"a.. the ler ofde obseMtion,<br />
c,,ap.lhc e<strong>lo</strong>nsalion oalhc n@n fon lhc sun al lhe line ofob*ryorion<br />
'lhc funcrion shs Nnh $lcclinS a poinl I wnh sky position sitnhr lult mdh<br />
I O.l.lM tum + 0.r-i-e. a poinl c<strong>lo</strong>* <strong>lo</strong> th. cstrc of $e tu.e disc Thc<br />
^.ilh<br />
dishnce ofrhis poid.:ed,sr is scd <strong>lo</strong> calcul.k lhc 8as. aeiosl ed lhe o7..c ndss<br />
xr = {cos(cdig )+0.0286 icxp( to.5'cos(zsrd,ri )r-l<br />
(4.3.4)<br />
x4 = bo(ud'i )+o.or I 1€xp( -2a.s'cos( rd,u/,Jl ))l-r (4l.t<br />
. L lsintf,",./,r I l' I<br />
(4.t.6)<br />
"'=l lr.:oro;rrl ]<br />
comspondine ro live diff@r \Ev€l€nglhs cleted in the @v Ydc,ll/:<br />
J, *orr'.^-,0*r!!\<br />
' ''o' t<br />
This is fol<strong>lo</strong>wd by rhe calculaio. of th€ cx nction @flicicnls componcnls<br />
& =o 1066 '*e[- s4J'("qrrtl'r) 4 14i7)<br />
r, =o r,(xru;'".-,(-#).(' -a##-*,) ",",,,<br />
Ka = o.'chlil'O +0 4'lpto! '<br />
co{satpha ) - cosl3' Ptat )))/ .<br />
(.r.3.9)<br />
*,, =,,,,,,r,,.. *.(,i#<br />
).".(tf ).".(- #il )<br />
(4.i.10)<br />
For cqch wav€<strong>lo</strong>slh bed rh.* eidnction ccmcicnts a'c ecumulalcd in<br />
krchli)= K, + Kd + Ko + K" (4l l l)<br />
And rheir linearconbinalio. wiih mas componcnls tE gathcrcd inro n v dnvhJl:
,.tdchltl = K, r ^ s + K,' x, + Ko' x o + X-' xs (4I 12)<br />
lbe.qurioft (4.3.) to (4-3.12) aE Pleed in a <strong>lo</strong>op thal rum nv€ ddes oncc for €ach i<br />
O. 1.2,I and 4. Aner$eexccution oflhb <strong>lo</strong>op lhc rir m4ss al poin r, thc tosnion of<br />
rhe M@nandfiarof theSunacc.lculal.dusinS:<br />
rrr@/ = [cos{'drd'Jt )+o.o2s icxp(-II'co(z"dirt.)]<br />
(4.l.ti)<br />
il (nofu < 0) nnposl = 40<br />
ar nnpoeJ -lcos(go - nul )+0025 r.xp(-ll<br />
- . (4.1.14)<br />
il (\ah
lf thc hviligh briebhes rv/, doninats ovc. tne day lieh briehhcs dat, rhc. thc sud<br />
.i r,grrb and .' ,1, are s<strong>lo</strong>red in ,r.r/t otncffi$ rhe sun of ,'s&r, md etb rR noE'd<br />
in L.r/t Morcover. ifrhe Moon is.bov. horizln thm de,, C lho addcd to bs.r/t/.<br />
l-iially, fic brighbes ,.\rrlt is @neened in<strong>lo</strong> .m-tanb.6. fon cquaon (4l.16)<br />
till rhis point all conpulalion is done id a <strong>lo</strong>oP that €recutes Ilve timcs again, once for<br />
sch wavel€rylh bdd *l4td fnc €aicula$on ol 6e limitins naghitudc /€, is donc .s<br />
la (tel < l5o0) ic,,e=<strong>lo</strong>at..eo <strong>lo</strong>-re)<br />
Els { co,a - I o ! rr . .n o " I 0 r ' }<br />
kh = tum'\t + Jctu" tb.t r<br />
(4.1.24)<br />
(4.3.25)<br />
(4.1.26)<br />
t.. = t6 i1 -2 t,IVD) d.".htzl<br />
\ In(I0) l<br />
(4.) 21)<br />
Schaelcr (Scha.fer. 1988) in his drtshold co.tdsl nodel calculates /i as rhe <strong>lo</strong>g<br />
of $c ratio of rhe acrual tohl brigltoBs of th. M@n ad $e tolal biSlttnss oi $c Moon<br />
neded for visibiht fo. rhc givcn obsping condiions ln this rc|k we considr rhc<br />
mlg.itude oflhe Moon and $e lisual limiliog naennude slculatcd frcm thc alsoinhm<br />
givcn abovc- The diftcmc. of Moo.s naenitude z"d!A and lhc visutl limitins<br />
nalnitude le- is consideted .s nugnitude .onxas de"or.d 4 zDzq A p<strong>lo</strong>l (Iig 4l I<br />
bc<strong>lo</strong>w) sbows dirercncc ofscbacfcls thEshold cohtrasl I and $c nasnnudo contasl<br />
/r43 Scries I sho*s ,1aas <strong>lo</strong>r crcsmls $ wca not $cn and erics 2 shows ,'lu zg ror<br />
crusoent ihrl \vcre sccn Sdies I and series 4 sbow ,4 cot.spondins <strong>lo</strong> dcsccnc $al<br />
wce not sen d lhal scte sen Esp.clively.lheda1a <strong>lo</strong>r this fig is talcn lrom Sch.cfc!<br />
(1988). P$nive vales ol cont6$ fo. crcscns $al wrc nol *o ad lhc nesrtilc<br />
vrlues <strong>lo</strong>r conlrdt lhal *erc sccn snoN lhc inconsistcncics ol thc modcLs qirh lhc<br />
ob*ndion. Th6e inconsisteoci.s may E$ll fDn .stiDatcd lalues of t€npcaorc a.d<br />
Elative hlmidill adopt€d for lhe calcllalions.<br />
l5r
Thdhold conExl vr| f.e.ltld. conbtt<br />
i-<br />
Fig No 4l l<br />
Ta!t.4.1.1 is d.v.<strong>lo</strong>pcd sina the sc prcssn Hildol i4 o!&r !o sDlv* $'<br />
crts@t ob*frarion @rds t Ln tod ti|c E (S.h&Lr' 1988t Yd<strong>lo</strong>t' l99E Odeh<br />
2004). In @h mw of thc l'bl. obsfrrion Dmbq (s sign d bv Odch (Od'lL 2004).<br />
dde of ob*darion. ldrlud€, <strong>lo</strong>ngrld. &d.lcvstion t<strong>lo</strong>ve s l.v.l of lh' ple ftos<br />
wh.d th. c'€l6l is ob*dc4 fol<strong>lo</strong>wEd bv $. diMr.d ldp.tat@ 3nd 6ri6aldl<br />
Elarivc h@idny. Th€ .ext thF @luld @nr.in $c uit.6,l lim' &d th'<br />
ldf,rdlig Ms,in4le .dr6t wh.n th. ru&ttn d" co,rr/dt b4om$ j6t f!vo@bl'<br />
for siShrinS of c'lsdt (@lum vith hddinS .3l!n) *id n is b€sl for sidlinS<br />
(@lwn wi$ lsdins 'bd-) ad whd i is favo@ble io' sidtina ror th' lar rin'<br />
(6ltm wilh hddirg 'ta'r). This 8iv6 lh. tine Engc <strong>lo</strong>r $e posibl' Et'd cve<br />
visibihy ot $. c'ls6! TIE rus,,'& .otdr i3 @Nid.ied onlv fot @id'd visibilit,<br />
oa crcsdt. Th. lan $E 6ll,!m @Duin th. i.fonMdo. E$ditg *tEths lhc<br />
clt* M claincd b b. vi3ible widout dv opdcal ai{ wi$ a bi'@ulu ot eith a<br />
t57
Tt. ob*Mtio6 @sidered in th. llblc 4.3.1 e ody rno* *lq il wr claim<br />
rhar rh. cc@ 6 sn by dy n@s .s ll|e r€cordi enen rhe<br />
de not El.ver. Moeov.r, by vrryiog thc Grinared t€npdt@ dd Glatirc huidily il<br />
n €valual.d 10 oblain oplimm @ndilios for nr}€d eye visibihy ol $. asc.nl, If th€<br />
ntaEnituda co"na is obtairea ro b€ i. favour of visibihy onlt dcn lhe lim. ra.8. of<br />
naked cye crescenr lisibility Ee delemined md included in thc rzble.lt thc nagnitude<br />
co"rf"rr is nol found b be in favou. of u.aid.d visibility the l6t posiliv. value of lhe<br />
nagritutu conn$t is c5lculaled md th. se vatue is included in aU the thR<br />
coGpoodins columns. 'rhe s,m positivc vahe in all rh* @tM.s it rhc indicalion<br />
that under lhe walh.r ondnioN consid.Ed the cr€sccm ws Ev€r visible to lhc naked<br />
]-his table shows tbal $erc d€ ll cases ofposidve sighting claihs {ilhout opiical<br />
aid when nagnilude contmt was nclq in iavour otunaided visibility. All tsc positile<br />
css e aho nor in aet€emenl wilh tbr conditions due io Folhqi.ghm dnd nalndeu<br />
Lunar Ripeoes las d@s nol al<strong>lo</strong>w l0 of thcse dd the two of lhen e oily ntr8'hal<br />
cses a.cordins b n. Only one 6e mlcdly diff€D fiom Lllm RiF@ss law. Onl, I of<br />
$csc 13 ces ue not al<strong>lo</strong>wed by B.by<strong>lo</strong>.id cdknon. To or thesc ces m not<br />
al<strong>lo</strong>wcd by lndim n€lhod ed rhe !6t of th. thrce m n&giMl c66 i. Indid method.<br />
Boih Bruin's limit 6d Yal<strong>lo</strong>p s critcrio. do not al<strong>lo</strong>w.ine ol rh.s cases ln cae or<br />
orher tou dlaims Ellfoutcs* are marginally ll<strong>lo</strong>w€d by Bruih's critelion bul Yal<strong>lo</strong>p s<br />
dlsion difieB a <strong>lo</strong>t in one oflhem.<br />
Thc liBt pase of the lahh 4.1.1 shows t$o etaa odinarv claims of nakcd cte<br />
vi.ibihy of crc$enl, ob6 No. 189 &d 455 wilh q_values -029 dd _0 216 Thc Yal<strong>lo</strong>p s<br />
c.it.don d@s nor al<strong>lo</strong>w naked €y. visibitilt for lhc$ q Qlrcs 8d tlE naanittde<br />
.o"r'4r, b ale nor favoMble ev.n wirh hiSlly exassenled wealh.r condnions The<br />
Modifi€d Ripene$ Fuciion (chapb 3) !alu6 corespoldine to lh€s oberations e<br />
aho noi favoudbte G0.95 md -0.62 in tablc 1.5.1) for cEsc€ni vkibililv Therefole th.*<br />
obsedations, as lhey fdil to etisry evoy model, de highly melilble ahd ar. outlie*<br />
l5l
TheE is only onc noc clain ofnajcd eye vilibility of ffiant wilh t-Elue lcs<br />
rnd -0.16, obeMtion no. 24, with q-BlE -0.22 | (AR- = - l .l9)- I1tis obs.dalion it<br />
not al<strong>lo</strong>wed by both the Yal<strong>lo</strong>p s crilerion dd th. Lunar Ripenes Law but wnh highd<br />
.tevariotr (1524 n 16 ahove s lcvel) dd <strong>lo</strong>e hmidity (6linat d to bc <strong>lo</strong>'ld lhe<br />
tuEnitude cohftast is favourabl€ for uaided visibilily lnd $e ob*nolion is not<br />
uNlidble. In rll the esl of lhe cB@nl oben rions *ith I-valB ls lnd -0 16 lhe<br />
claih ofvisibilily of cressnt is with binocular or wit lelescope. Thee claiN de also<br />
nor ucliable 6 out of25 s@h claim 8 harc favounble uas,itt.L cotun <strong>lo</strong>r n*.d<br />
ete visibility with oplimm 4tinales of lehpcdtwcs ed rclative hunidiry'<br />
The hagnilude conrAr for sme other r.porledly Positivc cE$€nl sidrinss<br />
{ithoul oplical aid n nol favoudble. Th6e e obeFalion nunbe6 14l, 3 19, 416, I16,<br />
315. 2a6. 611, 314,272, 2. The q-values (and A\"J fo! $ese descents & _0 153 C<br />
0.8), -0.1r (0.07),.0.101 c1.06). -0.047 (-1.t. -0.02(0.167), 0.00 (-3-5),0.01 c0.7),<br />
0.012 (-1.01).0.018 (-2-88),0.109 G1.47). For 341.416.Il6 rhe thrc. crireda (Yal<strong>lo</strong>p\.<br />
Lumr Rrpcoess Law md ihe nagnihde €oi!r60 e. co6islenl- For 286,633,114,212<br />
dd 2 Y.l<strong>lo</strong>p's citerion al<strong>lo</strong>ws.alcd cye vGibilily ud€r "perf*l lkibility condnionJ<br />
but bolh lhc Lumr Rip€ness Ls* and lh€ naerrilude conltdt are nnfavoulable for nk.d<br />
eye visibilily. Thu ir apFd thll Lus RiFmss L3w is nole coNistcnl with the<br />
m.Snilude conlrdt rcsulis.<br />
Io rhis work limiling t€lesopic hagnitudes &e not coGidcrcd as lhe r€poned<br />
ccsem ob$fr.tions wnh bineuls {d lclcsP.s do.ol pevid. approPriale d.iails.<br />
ThcrefoE lhe appopiale limiling lelescop.s cm not be conpnled MoEover, out wo*<br />
is horc conccmed wilh visibility of n w casd witboul dy otircal aid<br />
154
t!.2131)<br />
-'.gr""il-<br />
,,;* I " ,.;;; I<br />
,.!('!,"]<br />
;;; t"<br />
r-tp!!<br />
,,;p,{ I<br />
155
t56
t58
4,4 A NEW CRITERION TOR NEW CRtrSCENT IISIBILITY<br />
while d€vc<strong>lo</strong>ping rhe "vbibilitv cudcs" (it againsl 5) $d the limi$n8 visibilitv<br />
c!ru€J (, + r against r) for.onstad brightn.ss Bruin @nsidcrcd the avedgc brightnes<br />
of w$m horian duiog rwiligh od fi. ldistion of lh. bdgltN of lh' full M@n<br />
with tlE alitude above hotian c me ion d @lier' Instcad of @cid'dtrg av'ase<br />
brightness oi sky we hav€ coNideEd dual blighrnds of skv and fte cr'scent calculakd<br />
sing the techniques deve<strong>lo</strong>pcd by S€hefet sd olh'6 (Schmier' 1988b, l99l) i' lhe<br />
$nw@ Hila<strong>lo</strong>l. W. $lslcd c@ent vi.ibilitv circUmtarcls of ldious rcw Mm6<br />
css wh€n lhe crcscenl Ms rponed b hav. beo sn For cFs415 ofa pan'culd<br />
width we found lhe altiludes I oi skv points with btighhess equivalent <strong>lo</strong> that ol tho<br />
particlle crc$. at dill€tnt $l& depEsio.s r' Th. ovcdgd of the<br />
'ltiludes<br />
of skv<br />
Doinrs for difl'rnl $ln dctr.$ions for plirhuld sidlh d Lbllat€d in T'blc no 4 4 l<br />
The left most column of lhe lablc contains ihc solar dept€ssions r and ihe toP rov grv€s<br />
rh€ {idrhs r oflhe crcsenrs sclaled ed thc n xl one giv.s its na8lnude Thc enties of<br />
$e 6r of rhe €blc ue lhc aldtu&s , shcE lhe stv hs rh' sm€ brighhAt 4 lhc<br />
briehtness of tnc cresknl ofthc width al th. top of lh€ @luon.<br />
It should b. nol€d thal duins ihe twilighl 1ne widrh of the crc*dt vd'es up b a<br />
dc seconds for very wide d.sc.nls. Se6on to eason ed for dif|ere'r lalitldes the skv<br />
bddrlnes fot th. sdde altitu.l€ cl6e <strong>lo</strong> th. poid ldee ihe sun sts also vdi's <strong>lo</strong>r qch<br />
column of th. lable 4.4.1 a numbq of c!*s of alnosl eme cE$enl widtb wE<br />
@nsidered md each ahitudc is aveEg€ of lh.se cdes Thc dtla of lhe tabl' 4 4 1 is $e'<br />
p<strong>lo</strong>tt€d on a sraph shown i. fsw 4 4 1.lb lhis figure n as a fuhclion ofr (i' =/6,) @d r<br />
+ ,' = s 6 a fuclion ofr (rl = sat) aG bofi p<strong>lo</strong>ned /ft represnl the "visibilitv<br />
cwea &d the s(, rc9dnc the liniting visibilitv cudcs" sinild <strong>lo</strong> wh'l Btuin<br />
(Bruin. 1977) deve<strong>lo</strong>ped.<br />
159
T$LNa43.l<br />
tf t li rl!.<br />
a.l a, -r.8<br />
1l<br />
1 tt.a<br />
t3 4.5<br />
95<br />
5 2a<br />
5_a<br />
5 32 2.1 1<br />
,<br />
2,5 1.f 1,1 0,8<br />
3.05 0a ot5<br />
2f<br />
255<br />
r,e 0.t 0.55<br />
r.t<br />
0l o5 o3<br />
o,7 o25<br />
0,2<br />
r.t o2<br />
2 1 oi5 o.t<br />
Fia. r!b.4.,1.1<br />
Altdav.r 6ot Oap,Ebo<br />
!I<br />
5<br />
3<br />
2<br />
12<br />
t@
Alttudo v.6 Sohr Oottesa<strong>lo</strong>n<br />
19<br />
13<br />
€l:<br />
!9<br />
Th. @rdin rs oarn ninid ofafi, Ghom i. figurc 4.a.t) e lhd b.sic dals for lhe<br />
@del pe nrv€ dse<strong>lo</strong>Ded Sou tu dE i.ble 4 4 2l<br />
Using tubi. l€.e sque appronn rion * oh.ir€d lh. fol<strong>lo</strong>wing El.ro. ben*cn<br />
rchtiE .ltilud. ot@rf ARCV !d tu widd r<br />
ARCv = 435t 37tvt +2222O75O57W2 5.42264J1tt! + t0.4341159<br />
(4.4 D<br />
I6l
On lhe bais ofthh dlatioo wc tlcnre th. visibil,ty p.m.t€r v, 6 fol<strong>lo</strong>ws:<br />
v<br />
e = QRCV - ('-O.35lg$7W3 + 2.22207 5057W2 - 5.422643U t/ + 10.4341759) / l0<br />
(4 4.2)<br />
Our model for .adi.sl visibility of n w ls& *s6! ir that if rp > 0 (w @ll the<br />
visibility paEmctcr v, in (4.4.2) s th. r-vtl@) lhc ceml moy be lisiblc withonl<br />
opdcal aid olheei* noi Applting this €o.dition on thc dat set ued in chaflet I &d in<br />
lhb chaptcr qe present thc rsulls obtained for whole data set of463 cd* in d.ltn<br />
colunn of table in appe.dixlll. Fid fd .ss when rhe visibilily is claiocd wjlhoul<br />
optical aid in odcr of inccGing r-ralrcs @ sho$ in 1abl. 4.4-1. Oui ol0tcs cses<br />
only I I 66 d4iatc ton ou nodel. our of rhese I I c6es 8 de coosisteni sill the<br />
nagnitud€ conlrast 9 arc consisle.t wirh Yal<strong>lo</strong>p s crirerion and 8 ats consislcnt Mth the<br />
Lund Rip.o6s Law, Tbe oberyaiion Nnbes 3E9,455,274, 341 dd 316lhal d.viale<br />
from lhe Lmr Ripen.$ I-aw, ihc Yal<strong>lo</strong>pt crilerion ed 1be Dagnnude conlrdt d also<br />
negrdve i. our modcl. How€vcr the obseorion nwber 4161hat is negalile in orher<br />
models is al<strong>lo</strong>wcd by ou!nodel. ille Een is th.t in this ce ARCV b redonably hish<br />
(9.41 .l€g@9. Txe widdi h small (eund la @ sondt bur |hc M@. is rcry c<strong>lo</strong>se ro<br />
perige s c<strong>lo</strong>sr ro $e Efih.<br />
In fisuE 4.4.2 the minima of each visibilily cw. is joincd resulrins inro o<br />
nFi8ht linc stich $nen .x<strong>lo</strong>dcd inr.e.l tn€ oigin of rhe (r. t) c@rdjnde stseo. The<br />
srope of rhis linc is found ro b€ (lh + sys -) 9-3t5 ot t'/s - 4.)15. This lqds ro a Dodi6.d<br />
''b*ttine" of cesenlvisibility a:<br />
,3 = -=-i:<br />
s1:s+4.3(Ts+LAC)<br />
rBin conp&ison ro l(2.25)rh panoflAc in<br />
(4.4 )<br />
9.3<br />
l(2.163)rhpanofLAG<br />
besl tih. of crcscenr visibiliry is give.<br />
t62
l6l
In gendl l@kingd Lhe @npt.rcdsl! $ rn ubtr jn atFndrxtlt rl (m b€ nor.d<br />
'Ihr€ t no clain of visibiliy by ey n€{s who slalue < -O.t9l. In f{r as<br />
obrdolion nunbf 389 is not coDsin n( wjth .ny nod.l w dject n ed<br />
ihercrore we claim $at th@ js no aulhotic obstuation (wi1h or wi$our optical<br />
.id) ofneq lunar cEsenr for s-vatu. < 4,162. Thus cFsccn! can not be sen<br />
wn€n vcr r-value < -0. l6 evs wirh a rel*cop€.<br />
For -0.16 < r-value < -0,061h.rc e 26 (45olt clains ofcrcs vrs,bihy wirh<br />
opncd aid our of58 rcporled sd coNidercd obse(arions.<br />
^s<br />
unaided visibilny<br />
.laims 45J, 274 and l4l d <strong>lo</strong>l coNnlenl wirh ey nodet s w @nclude lbal<br />
fo. lhis tu8e ofr-utues the q€scenr cs be secn wirh opdcal ad only.<br />
Ih 49 c6es wi$ -0.06 <<br />
sighting with opricll aid<br />
&cmne crescenr vi0r a<br />
r{!lu. < 0.05 theG e<br />
(41%) we conctudc rhar<br />
or.eutd or a Glespe<br />
u..ided sishring, Unaided sightine b nol impossibt€<br />
ll ueided (25_5./0) ed 2l<br />
thqe ac stonS chdces of<br />
oii very slih chmc6 for<br />
For 0 05 < r-vatue < o. I 5, rhce dc j5 sishtincs wirh dtuar ai d (1[r/r) and 14<br />
$'thour oprical ad t2ra,. I hus rh. tuflr hay b€ easit) *.n qrh opucdt aid<br />
nr h! r4se ot s-htr &d !m be seer wirhour opticaj dn under ver) Com<br />
condnion (sarh{ coldiriotrs and heichr above *. lerct).<br />
For r-ulue > O. t 5, oul of nexl 2 t3 obeR.rions th.<br />
oplrcal aid I 65 timcs (77.5yo). Th@foE wc conclude<br />
cEeent can bc esity seen.<br />
c@kenr was s.en wirhour<br />
th.t for r-valu > 0. 15 rh.<br />
Thus ou oodcl $ar we @ll .eu€hi & (lh critcnon..cd be sleheiz€d 4<br />
164
l<br />
Calculate r-vdu. (or vp @odins ro 44.2) fo. th. c|lsdl a1 ilc besr tine<br />
2.<br />
Th. vbibilily condnior a girn by our 6odcl @ ihq giv€n in Tabte<br />
vhiutc uaer pertco onairiii'lili[<br />
May rcquirc opricar aia o iiiiFitiiMFoli<br />
Requi( ophol ad (ROA)<br />
Nor vis,ure wrrr opriciisia-li j<br />
.0.16
Wlen a ctirerion al<strong>lo</strong>ws oprielty uaid.d visibitiry of rhc ncw crcsnr dd ihe<br />
cFsnt is nol sn thcn it is a ncgativc.@t. Th.re c& be a.mbd of ree.s tor<br />
negativ€ em6. D.Airc rhe f&r rhal s obsd€i may b€ .xFrioed ed r.ined<br />
d<strong>lo</strong>noner &d lmws the <strong>lo</strong>erion offie cEsent rhe armosphdic €ondilions od the<br />
Physio<strong>lo</strong>ey of dc obw.at cy6 @t stil lead <strong>lo</strong> non-vbibilny of ihe crent Thes<br />
fac<strong>lo</strong>B ae sdlt nol wll exp<strong>lo</strong>Ed rhus the high frequency ofrhc..gaive crc6 shos<br />
Oil the probLm is srill nol eh.d conpl.Iety. A posniv. emr @cG when a nodel do*<br />
al<strong>lo</strong>w visibihy oa cesc. sd thc visibitiry is nor claimed Snaltq rhc n!frb€r of<br />
posirive ercB dd b.ttq is a visibility c.irerion.<br />
The hbl€ 4.4.5 slmdises the posnile ed negative obseNarioB in 6greoe.t<br />
or d'esrem. with diafcEnr oilcrion fmn ft€ dara sr w h.ve choen for $is M,k,<br />
Tte labl€ is aEanScd wirh dercsing succe$ pcrce age in tems of vjsibiliry claihs<br />
aruirenl wirh lhe $ircrion. Surprisingly lhe Baby<strong>lo</strong>nim cnrenon ha rhe besr succcs<br />
percenh8e ti)ttow€d by our r-vatu. cril€rioo, Th. l.btc funher shows lh Btuin,s<br />
cnrenon (4,1_2) and th. L@r Ripens law e.qudly suae$tuI, Thse @ fol<strong>lo</strong>wed<br />
by $. q-valu. critcnon of ya<strong>lo</strong>p dd rnc ARCV,DAZ_bdcd Indid ncrhod (1,6.t3)<br />
Tlle crn*ion duc ro Ma@dd (3.6.10) od <strong>lo</strong>$erinsnd,s crjterion aE dD lc6t<br />
sucessful ofde mdhods shom in rhc |2bte, Ir sbould i, nor.d lhar tess succdstut a<br />
modd is in d€$nbine pGilive sighdnS gncEr it is. Moreoler, strcrer a hodel jt should<br />
be nore coosislenr wirh .cgaljve obs€ryatioDs (wl|en fte cGsc.hl ,s nor sn).<br />
,<br />
The number of iegarive obsrnadons in ag@mc urh lhc diterioo is at$<br />
omrdkd 6 a resl of a cnr.non by sodc alrhos (Fatooni et ot, 1999). Howe!e!! ii<br />
should be notcd $ar rhe exp<strong>lo</strong>ration of Scha€fcr shows fior $e bneh|ness or nagnnude<br />
contra$ is highty dcpendenr on rhe w.alher .ondir io.s. All rhe single pameter cn&ria<br />
consdsed in the rable 4.4.5 do nor conlider wea$er ondiriotu of indilidu.l<br />
obsw.'r'on.If ay ofd* qitcdon a<strong>lo</strong>m viribility in soh€ cae il ,. sl,U possibte lhal<br />
lne setbq @ddirioN &c rcr f.vour.bk for vkibiliry &q oe crc*en is not acrualty<br />
166
Tl|rcforc, if Fothcdnghd's criGdon is nosl s@4stul i. b.i4 cotui$dr fot<br />
negalive obedadon rhG dcs not at dl htu thrl it ii h.tcr or noa d.p.dabl. thd<br />
Yd<strong>lo</strong>p'3 mod.l. Bolh Mah&E $d FolhdinSldt hod.k @<br />
oa resative oh6d6tioB (whd ctlsdr is mt *n) but e lc.st sussful for pciliv.<br />
ob*d.lion. Ihh is tre sinply b..!4 the criteri! a. sficlcr s @nFGd <strong>lo</strong> o$d<br />
crireria. On th. oih6 h.nd Baby<strong>lo</strong>nib @ndition, i-v.lue criedon ed lhe Luar<br />
Riperess law e highlt cotuistent with rhe positive obervalions but l4t coNislent wilh<br />
the ncgativ. ob*Nations. The* dd other crit.ria (Bruin's limn dnd Yal<strong>lo</strong>p\ 4.valuc<br />
onedon) arc noF conc.hed with condnions ud€r rh. visibihy o{ n w lutu oesenl i3<br />
tossible .nd not *nh fte .onditions un
visibihy cws inii.lly corceived by BNin, Thb clss includes Bruin's lidit, Yal<strong>lo</strong>p\<br />
a-EIu€ cnt€rion md th. s-value cilerion th.l is d.ve<strong>lo</strong>p€d in dis work<br />
w. hdc ucd Schaf€ls algo.nhn ro cxp<strong>lo</strong>E rhe Epoaed ob$natoN out do<br />
not cone uder ftc "Edily visible ' @ndition duc io Yol<strong>lo</strong>P md when $. cEse m<br />
r€ponedly sen, AnohSsl th€s ob$palio$ ces lhal have ufavouiablc ha8litude<br />
cont6l are critically exoined md some sft Gjccted (esp4iallv obs.nation nlnbcts<br />
389 od 455) s $ey de <strong>lo</strong>l consislent wnh.ny of lhe visibililv ctit rion consideEd in<br />
rbis work. Bo$ lhce rcponed cses m nol .oDsi.lercd Fliable as .ven undq niehlv<br />
exrgeerated .hosphdic t nFEtw md Glativ. humidny rhc magiudc contrdt is<br />
foud ro b. unfavoubL for cE$ent visibilily ve foud al lcdl onc ob*oation wh€n<br />
ile crc*enr sd Gpon€dly *n wi$olt.ny oflhal oid (obs. no. 274) ed thal is nol<br />
co.sistent wilh any cilcrion bul havc 6 la<strong>lo</strong>urable nagnifude co 6t fo! tlalivc<br />
humidiry less than 50% and atnospheric lefrpcrature aound l0 deglee centiSBde<br />
Apan fron obervarion nDmb.6 389 sd 455 1nee e ll olh.r caes shen the<br />
cEse w4 r.poncdly s@ without oPlical .id but ihe ma8lilud. .ont6l ws not<br />
favowbt.. Howvcr,olee m ofFsitiv. ob$nado$ re consined wilh ar lcsl on€<br />
oder crilerion consitleed in 6is sork A3 the brighhess model tE still nol pedecl<br />
rneEfore thcs. I I obsenalio.s ce nol bc dl.d out .s umliable.<br />
The models $at @ dedued Iioh rh€ Bruin's isibilitv cud6 ud liniling<br />
visibility cud6, th. Bruin's linil md Yal<strong>lo</strong>P s single p&meter (it .ion drc found to be<br />
consisr.d qith.{h other. Bolh hav. stmosl.quivaldt succGs Frc€ntagc br Bruin s<br />
We ha!. deve<strong>lo</strong>ped new visibihy curves and limiting visibililv curlcs using the<br />
brid nss mod.h due 10 schaeler md o$es O. the basis ofthese n'* cuBes a new<br />
dara *t ad a ncw sinsle pmetei cril.rion b d.duced The .s liniting visibililv<br />
cufl* havc lcad <strong>lo</strong> a slishdy nodifi.d "h.st lime of cE$nt visibilnv Our nw<br />
visibiliry cril..ion, thc r-value ;ilaion, is found <strong>lo</strong> be norc succ6sful fot posnive<br />
168
obsMriom bul lcs sue.senn in @mpdien 10 thc Bruin's I'mir dd Yal<strong>lo</strong>p'3 ctil.tion<br />
for n.g.tiv. obsdaliors.<br />
In vieq ofthe facl rh.t all th€ vkibilily oiroia de ai6ed al exp<strong>lo</strong>rin8 conditiohs<br />
undd which rhe new luM rcsert may b. seen the suc@$ oI a modcl <strong>lo</strong>r positive<br />
obsrv{ion is hor€ inporlrnl d i$uc as cohparcd to ils succe$ fo! ncradle<br />
obs.F iions. None of lh€ sodel h limcd ai deducin8 condnioa und.r which lhc<br />
vieibihy of ftw lbr cJ€sMt is Dposibl.. ThcEfo&, a nodcls see$ for bcing<br />
consblcnl wift $c positirc obsmlion b lh€ sses ofthe nodel.<br />
In vi.s of the nagnitude @nl@r model bed on brienb6s of c@.nt dd of<br />
rwiligbr sky il h6 ben *en lllar rhc lbibiliiy of .ew lun& cr$enl is Sreatly aff.cted by<br />
(i) $e eleelion above sca l€v.l oflhe obs.Nation sne, (ii) rhe almospheric lenpcdurc<br />
and (iii) lhe relalive humidily. Higher is the.levation morc islhe maghiiudc contrasl rn<br />
favou ofvhibilitt. Lowd is lhc lenpcEiuF or hunidily the nagnilude contdt is mote<br />
favouBblc for cEse sighdne. Evcn if *niampidal diteri likc Yal<strong>lo</strong>p's g-vtlu.<br />
cril.rion or ou s-value critdiotr
nay be considced for an adv&k pGdicliotr. I. cdc of vdifyins a visibilig claim $.<br />
dual ahosbh.ric conditions mv b. 6oded.<br />
Thercforc, a possiblc slral€gy for vedliatio. day be to usc eslituted alevalio.,<br />
t, esriMtcd tcmp.rarurc, 7, dd $dnakd Glaive huidity, l/0 md get thc rsults for<br />
g-value or r.value. lf n al<strong>lo</strong>ws crcsccnr visibilny for lh. eve.ing in question dd fte claim<br />
h nade, tbe clai6 is ac€pted lf1h. s.mi-.mpiricsl cribnon d@s nor.l<strong>lo</strong>ws ccsc.nr<br />
visibility dd cresccnl sishtins is claim.d, .voluare mleniludc contBsl M,. Ift<strong>lo</strong> > 01he<br />
progran Hila<strong>lo</strong>l al<strong>lo</strong>qs for vElidtio. i. el€valion, rempc€tur€ dd Elrtive hunidny e<br />
onc con adjust for thse quanfii* and qiry whether a favoulable ma8nirude contmsr (M<br />
< 0) is obtaincd or nor If rhe new mgnilude contmsr is favoldblc dcepl lhe claim<br />
On bsis of the d!r. generaled by pDgEn Hila<strong>lo</strong>l GhoM io ngu6 on the n xt<br />
page) Elations berween magnitude coht6$ and rhe quantiries on wbich ir dep.nds {t I<br />
ahd I, ar€ obtained 6 aol<strong>lo</strong>ws:<br />
M = i 8177 -1.autE + O.0Ao000ze<br />
M= I t626e40n22)<br />
M = -A 4 368 + 0 Ai26T _ A 0008f +A OAOA6t'<br />
ThG for meter inc.edc in elevalion M dece66 by O.OO0OO0I4' _ O,0O4l, for €ach<br />
pcF€nlagc ircre4 in relaive hmidity rl' vdies by incrss by O.O3l 56e(0 o,o:d) ed for<br />
€ach degce cenrigrade inc@sc in t€npqarue / incrcdes by 0.0126 - O.OO08T +<br />
0.000006T. This n6y ledd to the alproprjat€ elcvalio,! rehriv€ hhidiry dd<br />
athospheric lenperarure tequjEd for fa<strong>lo</strong>ldbl€ nagnituo€ conrsr An app<strong>lo</strong>pnate<br />
erclation my b€ tbe el€vation of . hill top o. buildirg Mf fbm wher€ m obseFalion<br />
nay bc @de or oay hale be. @dq The apprcpiare lchlsar@ ed rclalive hMidity<br />
t70
may 6ose that nay b. lhe ay€rlse for lhc eson or lhc valB $at wE acNally<br />
re@rded ar rh€ $ne of obsc(ation,<br />
ELEVATION VRS NAGXIIUOE CONTRAST<br />
y= E.ri.odtr+ 5 af<br />
3.<br />
E3<br />
$,<br />
3.<br />
33<br />
HUMIDIIY VRS IIAONTUOE CONTRAS1<br />
t<br />
TEIIIPERATURE VRS MAGNITUOE CONTRAST<br />
8<br />
B<br />
2<br />
t7l
Chapter No.5<br />
APPLICATIONS<br />
Durin8 rhls work lt ls obs€oed thar since th6 Babytontan eE flt recenty a<br />
nomber ol pr.diction crltsrta, mathehatic.t .s we as obsewational. were<br />
deve<strong>lo</strong>ped ro detemln€ whon the new tunardesc€nl woutd be flrsr seen tor a gtven<br />
rocataon. as tho fist appea.ance ot new tunar crescent marks tho beginning ot a<br />
new month in obseryarionat tunar catendaE thele crireria and modets are<br />
sagnirrca.t fo, calendaicat purposes_ wh€ther an actuat obsrya ohat tuna,<br />
carendar, like the tstamlc Lunar catendar, ui izes lh.se crireria tor arangtng its<br />
carendar of not rhe36 c.trerta provtd6s a €uidance for both teslhg an evtdenco ot<br />
crescenl si8lting by common peopte and tracing down ihe dares ot a catendar jr<br />
hErory where app.opriate dates are not we recoded. Ttus the main uti ty oi the<br />
prediction c terta tor the eartiest visibiny of new crescent ts to regutate tho<br />
obsefr at<strong>lo</strong>n.l lunar catendaf .<br />
Although ftrst order approximations, rike Anrhmetc LunarCatenda, thar are<br />
oased on lh6concept of LeapyeaFand rh6 averag6 mooon or rhe Moon fave been<br />
in use, Nluslims hEve been to owing dctuat si€htng o, crescenls ar teast tor the<br />
monrhs of fas n8 (Ramazan) and pitgnmage (z hajjah), the acruat moion oi the<br />
Moon varies grea y due to vartous tactoB whtch causo rne obseryarionat catendar<br />
<strong>lo</strong> be ditfered froh the arithmelc c6t6trdar.rh6 Catendars if based on a prediction<br />
c ler<strong>lo</strong>n llko thar otya op or ihe one devo<strong>lo</strong>ped In this work are rhe cto56t to rhe<br />
ob*pat<strong>lo</strong>nat catendai In this chapter, wecohp6rethese carendaBwith th€ a.tual<br />
obseryatronat catend6r in pracrice In paktstan tor th6 years 2ooo ro 2OOz, ft ts<br />
found rhst on avorage 93.7% obseryadons ,16 ac@rdtng to rho ya<strong>lo</strong>p,s o_v6tue<br />
t72
criterion or our s-value (or Q&K) cnterion. The disagr.ededt ls lhe resull ot eilher<br />
the bad weather du.<strong>lo</strong> whlch the.ew <strong>lo</strong>narcr6cent could not be slghted 6nd th.<br />
lunar nonlh b68an one day lare, or too opilmjsrlc clalms ol obse at<strong>lo</strong>h and tho<br />
Lunarmonthbeganonedayearllerthanpredlcted.<br />
Funher, In ihls work another aoollcar<strong>lo</strong>n ol these models is consldered. Tnls<br />
is lhe use ot lhese modelsto dwe<strong>lo</strong>p a compui.tional roolro detemlnethe lenglh<br />
of cesce.t from cusp to cus!. flie n€w lun.. c.scent as well as crescent on next<br />
few evenan$ b obsened io bs shorter tha. its rheoreli.ar length Le. 180 degree<br />
IromonecusD<strong>lo</strong>theother.Anumberoialrho6haved€scribedthereasonsforthe<br />
shortenln€ of the obsenBd crescent (Dan<strong>lo</strong>n, 1932, Schaefer, 1991, McNally,<br />
1943). However, Jew have atlempted devislng a mathemallcal technlque <strong>lo</strong><br />
delenino the exrenr of rhis shorrening ol rength ot crescent. On the basis ol one<br />
pafamelef dlleia ee have used cresent ot minimum visible widlh as limil on lhe<br />
le.$h or .re$enr and devrs6d a simple rechnlque to calculate lr. The chapter<br />
beginswilh a desc ption ofthesme.<br />
5.1 LINCTH OF LUNAR CRESCENT<br />
Ihe hct,lld rltc nc\v luhlrclesor appcds stroncr llEn 180. terA r.islnorn<br />
ldt cenlrics. Il wns Denj.h wlD fi^r savc rn exptturarnr for the pbenoneion (Dor.ru_<br />
lt32&1936)andalributcdi(o$etunarteniinc<strong>lo</strong>seto$ecuspsMcN! yanribuluda<br />
dillerenl reNon snh tis pbenom€ion disadine Danjon\ hlpo$esis (McNa y, 198:t).<br />
McNall) proved rhar lh€ tcngrh ofrhc shadoss ctose to $€ hrnar cNp ond fic d.panolc<br />
oflunar suiface IoD beins pertecr spherc coutdnl,jimiiish rtre brignllrss oflhe regions<br />
ofcresent c<strong>lo</strong>se ro cusps <strong>lo</strong> bc rhe caNc ol rhe phenomenon. IIe arribured ttrc tength<br />
shorlcdng ofcNscent to lbe,.seing afeca,due <strong>lo</strong> $e turbrlcDe of lne ,rhosplrere.<br />
McNally has also deve<strong>lo</strong>ped a limula for calcutalnlg te knBh of rhc desceDl, Larcly<br />
Sull (Sulta., 2005) has anribured 0E shortening of lengtn of crcscenr duc to the<br />
Blackwell Conr.st Threshold (Btackwclt, 1946) and has d€le<strong>lo</strong>ped I forhula <strong>lo</strong><br />
c.lcurale ciesenfs lcngrh. W. have aho devctopcd a sinpte &chniqne tor catcul.ring<br />
173
l.nglh of new lum ccscenl (Qucshi and Khrh, 2007). In the fouowins we reprcduce<br />
this cf<strong>lo</strong>rr wilh slighr nodiilcation md a codection,<br />
Schrrer cjecred McNall! s expldaion on $e bais or his Yiw thar rhc<br />
shonening oflhe cr{ent length is situply b.ca6c oflh€ sharP d*line offie brigntn€ss<br />
of rhe cr€scenr c<strong>lo</strong>* ro rhe cusps ( SchaelTer, 199 ! ). Usins rh. dcudte dodel of Hatl@<br />
(llapk , 1984) for calcularing rhc surface brighhcs ollhe cesnt Scbaetrer claims that<br />
Daljon\ collected obse aions anJ his own n€* dara fic tbe nodel Howewr, nenhet<br />
Danjon nor Schaefer have sneecsred a ddhod <strong>lo</strong>r calculaling cacehl ldg$ Hapk€t<br />
rDdcl nat be accurat for $eoreical setling rclatcd <strong>lo</strong> thc c<strong>lo</strong>ngalion ofrhe M@n bul as<br />
frr as the obscryed cresc€nls are conconcd dEre oughl <strong>lo</strong> be a depanur€ frod snooth<br />
relarion berween e<strong>lo</strong>ngation and lhe crescenl lcngth. Thc tason we co.sider n baFd on<br />
obsenarions ofsone norning and {enihs cesccDh.<br />
ltosl of the edly desfiption of the phononen, concenuated on rclatine it to the<br />
plEs (or dongation) of the Moon ihal is gcnet lly th€ !e6on b.hind the Phenonenon<br />
As rhc c<strong>lo</strong>ngalion increass rne lengrh of rhe cesccnl fronr cusp to cusP k€eps ioc€Ning<br />
Nhich a connon obseryalion. The nathemalicd dc*riprion ol lhe phenomenon lems 'n<br />
ufdeficicncy arc. br Daijon Nd !ho\t to bc incoftd bv McNallv Hosev€r. e or€r<br />
esrimrrcd limil oo rh€ ninimu \idlh of visibl. cEse (2 to 6 arc seconds) bv<br />
VcNJlly le ro ren .mdll "dlu*<br />
lbr DJnjotr Liilil lhe d*np'ion due <strong>lo</strong> McNallv i:<br />
<strong>lo</strong>gicall, souDd so is $at of Suh dd bolh rcsulkd in$ tcchniques for calculatng<br />
hDglh of se$oa. HoNerer bolh llcNally dnd Sullan hrve nol reponed lhe aptlicalion<br />
of rlEn desription o. lh€ Ecorded his<strong>lo</strong>ricll dria found in lii€iatuE (Yal<strong>lo</strong>p 1998.<br />
Schaeffer l99l elc ). According<strong>lo</strong> Danjon, d desctibed by F.l@hiet al<br />
si,(a) = st(.) cot(@) (5'l l)<br />
Wh.c $e @ PQ = o is fie deli.icncy at in fiS l (Fatoobi e|' Al" 198) a is rhe<br />
e<strong>lo</strong>ngltion dd o i5 half the cccenl l€nElh tr apP€6 rhal Dojon tr*d lhe Sine<br />
t14
fomulE. by Nunins sphdicEl 6gL at Q to b. idr egle McNallv rejeded Dmjoo s<br />
dSmcnt ed 6ing fou-Pafl fomuh adiv6 .l:<br />
(5.1)<br />
Nlserically, for small ddgtes d sd a thee is onlv . muejn't dillemce b'lsFn lhe lso<br />
Esults. Generally thc .<strong>lo</strong>nsation (4) cd be cahulaled and the cre$ent l'nrth (2o) is<br />
obede.!. e the rw fomuls 6 b. LPd <strong>lo</strong> find fi' dclici'ncv arc' Hosder' ro<br />
calcul.rc the c6..1 lenelh nonc of these @ be ued- McNally daetoPs a fomula fot<br />
angula! rpdlion 9 ftoD a cusp in tems ofclescent width R d:<br />
I<br />
SinE<br />
-T *t'<br />
t Rl<br />
(5.r r)<br />
nininum lisibl. *id$ n.sured ih ddid dn€cdon 'wtv<br />
lron dE<br />
disc. Thercforc the <strong>lo</strong>gh of th€ ce*ent h€ oblaits is l8O0 _ 29<br />
Usiry (he Blackqell $rcshold con(6t Sutan (sultan 2oo5) adres at tlE<br />
ninimun vhibG *id$ (in |ems of didetcr of rh' smalld eqlival€nl Blackselr<br />
(Bhckstll, 1946) di*) of c@enl ar Frigee dd apoge al his l@rion of ob$Parion'<br />
Tbe fomula rhal hc dele<strong>lo</strong>ped for calculali'8 thc crc$61 lenglh rsi<br />
,=f!1.'to"<br />
l2t )<br />
o.l4)<br />
whde r h th€ sdldidetd of tbe Moo[ md<br />
o5
l2t +w \<br />
{5.1.5)<br />
wirh w is the dim€ter of e snallsr visible .quivalent Bl&kqcll dnc and l/ h cental<br />
widtholrhecre$e.t. (5.1 5) is$ecorsted fomof (1.5)inQureshi a Knan(Qutshi &<br />
Kian, 2007). Sultd.onside6 minimum diameter of Blacliwell dGc <strong>lo</strong> bc 0.14 alc<br />
minucs Nho the Moon b near peris€e md 0.16 m ninul€s wh€n the Mmn is n.or<br />
The presenl $ork is based on rhe obseNalion or the last (old) rcsceit oD<br />
February 26,2006. Duling this obsenation rhat sGned from lhc bcginhing ofnorning<br />
oviliglr rill \ell past sutrie, il was noriced rhar for rhis 48 holrs Mon. noE tlrn 2<br />
dcFcs aMy ton lhe sun ftc ce$ent l€n8fi srancd ro derede \rith lhe rising suD<br />
simihr obsnalion on Mdch 28'' shen lhe aee of Moon wds dound 3l tous !.d<br />
arolnd 18.5 degrees asay lron rhe sun. rhe cdscenr leneth d*leased nrorc npidly qi(h<br />
d(E6ing co.ra$, The l.n rine thb ciescent wJs $en lvilhou optical aid well at$<br />
sunriF ss less lhd 90 dege* in l€ngh. Tso dtys later n€w cc$enl with agc 27 6<br />
hols ar 15 degees a\ey fom lhe sun Nas olsned till setling C<strong>lo</strong>se <strong>lo</strong> lhe lDtiab<br />
though rhick humid a(mosphde the crcs€nt lenglh Ms aganr obscrved to bed4rcsnrg<br />
Thee obsFations cleaily demonslrare thai lheF isnuch notc 1o b€ exp<strong>lo</strong>ad abolr lhe<br />
phcnohenon ol shonening of cc$ent lcngth apan fom tlapke s .c€urate model thrl<br />
shows detendoce ofcB*cnt lcnsth on c<strong>lo</strong>ngalion t<strong>lo</strong>ne.<br />
TherefoE, inrhis qorl,<strong>lo</strong> besin si$ qecoisidera simPl. seodeuical<br />
model ior<br />
rhe ce$e.r. fu model dFsibes the phenoncnon of shortnine of lcn8lh that depends<br />
on th. aclual seniSiaocLr of lh. Moon d *ll a the rclaliv. alliiude (ARCV) of th€<br />
csc.nl o\fr tftal sky. This is denved fro,n thc sinste pmnct.t (q-vatue or !!al!c)<br />
crilerion of edlie$ visibiliry or cre$ent (chapler 4) fld is bascd on th€ facl thai<br />
whe.cve! rhe widlh (o' btigb$e$) of cF*c c<strong>lo</strong>s to cusp is be<strong>lo</strong>w lhe mininuto<br />
vhiblc c€nrnl widrh (or b ghtn6, or rh. dcscent the vhol€ l.hsth or ccscent wirh<br />
t76
s.lld width eould nor b. visibl.. Applying ou mod.l on lhe rc@ded vbibihv rnd<br />
iNisibiliiy dala lvaitablc fic ldgth ofcEsenl in 4ch ce is calculaiql Thc calcnlaled<br />
lenSrhs of c$en1 @ al$ omParcd wih thc obseded leheths dd wi$ thc lengds<br />
catculsred uing fodlls (t.1.3) de b McNally md (s.1.4 & 5 1 5) due b Slltan. ln<br />
cs ofusing McNally's fomuh AA is @nsidded liom lhe cilerioD used in lhis work<br />
&d for Sul|!n's fomula the ninimM width or Bl&kwll disc (,) it 66ideEd b be in<br />
the d.ge 0.14 alc minul€ io 0.16 arc Finut€ and dep€nds on the dislane b.tqeen Eanh<br />
&d th. Mmn obtained using simpl. lined inieQolalion<br />
Irvi€wofQueshi & Khan (QuEshi & Khan,2007) tlre bLishrn€s ofcfsenr ar<br />
displacement v frcn a cusp is gvei bY<br />
8.. = tF...Cdv tr -cos t)+<br />
(5.r.6)<br />
sh* t G th€ radius of Moon, F. is fie mdimun nux of sunlight<br />
olVoon.l islhc 'ePar-rion<br />
berqecn lhe Sunand 'te<br />
Moon in our<br />
the lutu suface dd X B th€ disldc. of the M@n ftom the Eanh'<br />
lhe cescenl at angular sepaElion V fom lhe cusP is eivcn bv<br />
lV,r, =rcosWll CosE) = tl ccatv<br />
(5.!.7)<br />
whc t/. is lh. width of fie cdnr i! lne diddle ln bolh the* €quttioc !' vari6 lmh<br />
oo ro 9Oo a<strong>lo</strong>.g fie lenerh of rhe cFscent frcn centE ro a cusp respeclverv<br />
For rhc deve<strong>lo</strong>p'nen( ofanv nodel thai dcsibes $e ni'inun Po$ibl' sidlh rhtt<br />
cd bc visiblc lttrough naked eve one requi.es 10 s'ek guidance fron $e aclual<br />
obedalioN. ln $e history of *idtjficatlv rcponcd obeNation of thc srv voug<br />
cr.sent Moon, lh€ ecord is that due <strong>lo</strong> Pi.rce od February 25' 1990 (eporled bv<br />
s.h!.fd ad Yat<strong>lo</strong>p) Thc cl€*nt he claift |o have sn wirh narcd ev' B jusl 14 E<br />
houB and ils widrh wa3 O 18 dc minut€ Amonesl<br />
'll rfie '*oded ob*flalios lh'<br />
t11
siglling of such a youg ed rhin cesc.nr wb n€vd rcpon.d- In lhe norlel $ar is<br />
deve<strong>lo</strong>ped in this sork thc <strong>lo</strong>sct limil ofihe *idth ofvisibl. ctsent is considered ro be<br />
o.l8 aE minut€. However this ninimum is not th€ abelule ninimm for all c6@nls for<br />
all posible el ile altitudes (ARcv). I. fiis work *e consider rhh ninimun of 0. 1 8<br />
. arc-ninutcs of crcscenl width shen fic Elaliw dinulh. DAZ of $c M@n is ao md<br />
rlr r.larir. alliiudc ARcv gives lhe q-lalueof_0.22. Y.l<strong>lo</strong>p s ditedon (4 2 3) for $n q-<br />
,1R(y =9 6311-6J226W +0 7319 t'i2 ol0ls,t/3 6'18)<br />
Thc values of ARCV according <strong>lo</strong> lhis *iteion giving $e q_rdlue equal to '0 22<br />
Nould ield ! difcre <strong>lo</strong>tr€rlihiion lhc visiblc width olthe crescenr' Tbis is causcd bv<br />
dilTriem relarivc dzinulhs DAZ FoT lh. lasl posibl' ARCV (4o) th€ *id(h ol lbe<br />
iNisibk ae$erl Nould bearcund 108 alc{cconds $at occur dl a large valucofDAZ ln<br />
\i.\r ol diis $ilcrion ihi nininud {idth of lisible cr€$eol for @v ARcv is Gm'd trand<br />
s te *esent isjusi invisible for$is{idth:<br />
(tIa)<br />
Whqrc l'l. is tlE tlmrclical enlEl sidlh of he ces'enr and 4costl<br />
is $e r€duced<br />
rvidth al aogte V Irom lhe center of $e de$ent<br />
'/-<br />
n dre sidth r€duced bv the chnve<br />
ahnude ARCV li ordcr thal soEe pan ol dsenl is visibl€ 4(!'rvl<br />
> l'l' ar sonc<br />
\alue V = Vn 'llrus thc eftstive visibte widlh of$e ce$cnl tor v nnsing fon 0" ro<br />
(5.1.r0)<br />
Th. brishlness of crescenl falls sha4lv s 0 approaches 900 - t th' aclul visible reidrh<br />
of cE$.nt at any val!. of V m6r tE less tbd d€ e'onetic valu€ of vidh eiven bv<br />
equatioh (5.1.8) Therefore rhc etr6tiv' v6ible width siven bv (5 1 10) is justiced<br />
178
IE it ir ma dly lh koglh of.ilsc.d 6d nodo! htr lh. vilibL with of {tt*.al<br />
b! <strong>lo</strong> diminjrl rlso. Wh@\tr lhc cr.rcdt it ilviliblc 'n<br />
vi.w of (5.19) 4r'bs to<br />
c*v-fr=r<br />
(5.1.Il)<br />
I!.ll oOE qscr. ia vi.r6e lb .tleI it vit'bL ia sftlth tl to'E 'rgh<br />
!'n Dlrt<br />
dilh!6l!.cc..ttti3ncv.r!d.conDldc 180!irlcrgh Tbu!'tV=V.<br />
(5.r.r2)<br />
Tladc i. (51.12) ||,! i! ! ddr.. of Idf l!! l6€tt of ih' cr'!c'dt $ itll lt' iol'l<br />
lcogth oflh. ctelcai k Sivcn bY:<br />
t;rr*"CY'*)<br />
(t.l.lr)<br />
D<br />
mt
'I}lus $e ae$c len$h cd be dalualed whelever $e fieoelical sidlh lr'<br />
€xcecds the mi.ioM width ,/, visible acording b Yal<strong>lo</strong>Pt q'lalue ctn'rion oi our s_<br />
value criterion, for $. palticular lalues of ARCV In $c Fis 5. l l the *8nent [D or<br />
AC is the ninimm width lil- at dv ARCV inlisibl€ according to Yal<strong>lo</strong>p's cliterion<br />
The scgnenl AB is rhe thorcdcal widlh ltl" at fie cenrE of fie cresccr! Al sngular<br />
spaBrion vr from tbc centre olc€sent ED equh4(nrl/" 'lheefore the ponnt on<br />
rhe ouler linb oi rlc cresceni tbal las dguhr separaion fon cerrc geal$ ihan<br />
s. = ZDOA should noi be visibl€ TIE lisiblc cls€ then crtnds fom D o D' @d<br />
has letrBlh 2!rn. One should nole lhal Nheneve! ttl" (nininum vhible wid$ accordnr8 b<br />
Yal<strong>lo</strong>p s nitrion) is Srearer tan ttlq ($coetical widtb) {5 1 13) can nor bc sed ind the<br />
cresceDi isnotlisible, ie iihas m lengdr'<br />
The tr1odel deve<strong>lo</strong>ped nl this Nork <strong>lo</strong> cotrlpute tlB lenglhoflhe crcsccnl hrs been<br />
applied <strong>lo</strong> a nuobc. ol ob*rvalions lepon'd in IneralE (Schaeffd 1984 Yll<strong>lo</strong>p'<br />
1998) Tbe resuls foi ihe crescenl length against lhe e<strong>lo</strong>ngatio' also kno*n as irc of<br />
lighl orARCL,ae ptscnled in lable 51 1and in Fig t'l2 Tlre <strong>lo</strong>Blhs mcalculaFd<br />
by scleciing lhc nnrinum vhiblc Nidrh Vn of dcscenl foi tlrc valLE ofARcv 'c'ordrg<br />
b {t.1.8) in rh€ elerv ca* and ihen usnrg (5 1 13)<br />
The colunns or rhe hble 5 l ' I sl<strong>lo</strong>w rhc darc ot ob*ivirion thc @dinares of<br />
rhe lcatioD of obseryer (Ldlilude and <strong>lo</strong>ngitud€ Fspeclilclv) tbe rclatilc umull)<br />
PAZ). rlre dladve .hnude (ARCV) lhe c<strong>lo</strong>n8alion (ARCL) of $e Moon at lhc be'(<br />
'r:d<br />
rinc ofobstvalion. lbl<strong>lo</strong>wcd bv setr1i_didetr and lhc cental widh ol cr'scnl in arc<br />
ninules. rlt 4_valu., lhe niDimum cF$enl sidlh viiihle at lbe rclatile alritude<br />
c.lculated <strong>lo</strong>m (5.1 8) and finallt the cE$e'r lensth cdlculared usiie (5 1 11) Thetablc<br />
is aFmged in chmnotogi@t or
200<br />
F 180<br />
fi reo<br />
3ro<br />
Era<br />
i roo<br />
o80<br />
i;@<br />
JM<br />
=ao<br />
0<br />
20 30<br />
E mno<br />
tig- 5. | ,2: Crlgn Lcogll! tr' vdi.Ii@ dE ARCV<br />
Tlr! Fi8 5 1 2 slD{! lhai lh. nbdioel ctdid bdwen ct!*4t lcntlh<br />
'd lhc<br />
ctdgltion h not snolh s Eporr.d bv sd|Itrcr on {E b$i! of H@k''t mdcl Th'<br />
tui! r.rlds fd tl 5 N (i) dE cr.rc.nt hngth ha to bc tff"t'd bv dE ElihMoon<br />
dist ncc .! cl'in d abos (ii) 0F dttdgbdic n!td4' c<strong>lo</strong>e ro lqizd trN|l tfrecr li'<br />
cr$ent <strong>lo</strong>srh., cl.imcd bv McN.[v ald Silrd MoEov{ v'rv wi'lc cle<br />
'ot! ftal<br />
c invilibte dn <strong>lo</strong> $.it d4 vicitritv qftb $t hori4o mull mt v|tilh $ddolv for s<br />
invisiblc crcsc.nr the e<strong>lo</strong>ngaiion mv ba ttiSc $d nult not h'r'' as lcngh (i3 msl'<br />
dt b. i isiblc) !.dding to cLi$ of scn*&t Ttt invisibilitv of t eidt '!l*al<br />
!t<br />
de to ii! c<strong>lo</strong>..n6 io lFdzon (sd[ ARCV) Ho*E' sftn dft$L diiod' (ARCV)<br />
$e rnc G..nt 3lEI bc d!ibl.. Ot
182
T.bl. 5.1.1 {Continued<br />
r8l
Tabl€ 5.'1.1 (Conllnued)<br />
Durins rhis {ork $e O obw.tions of Dfljon (ne ionc{ bv Scha'lTs<br />
and Fabohi €1. al.) @uld nol he acc6sd, howvc. the pictotjal data of cdccnt ltngth ir<br />
beins eeneraied at fte Asao<strong>lo</strong>nical Obsdl<strong>lo</strong>ry al Univdilv of Kdachi. Th! obseNed<br />
cBcell lengh fon pho<strong>lo</strong>gEphic Ecoids i5 given in Tabl€ 5 1 2. This inclldes sone<br />
obsryadons made by olhe^ durirg past fe{ ycas and tbeir pictutes ate avai<strong>lo</strong>ble liom<br />
M.icproi.on maiDtaind by oteh,<br />
184
Tne nodel dcvc<strong>lo</strong>ped in lhis e for borh Ih€ brigh$6s ad th. lenglt of<br />
cmce.i is nainly geomerric, suppl€n .led by rhc single pmercr crilcrjon for $e<br />
earlien visibility of ncq lunar cFsc€n1, Thc model prcvid6 a simpte method of<br />
cdculaling lenglh of lunar ce$eot and tak€s in<strong>lo</strong> accounl lhe atnospheric afects<br />
indiicclly through singlc pdaneler crit ria for th. vnibility of new llnar cressenr only.<br />
whenerer t/. < n/- fi€ €r€$ent l6grh is nor cakulaled dd rh€<br />
according <strong>lo</strong> eordcd obsedation (lable 5.1,1), TL nodel hd been Gn.d in nw $ays<br />
I:i6r. ior so6e of $e rcccnt obeflarions whose pho<strong>lo</strong>glaphic Ecoids arc available $e<br />
cilculated dd obscrvcd cescenr le.erhs tre compared and shown in jablc 5.1 2 and nr<br />
lig 5.1 7. The cFscent l.nerhs calculared ushg ou! <strong>lo</strong>rnula (5.1.13) arc SEaler thm lhc<br />
obseoed values dd rhose due ro Sultan srechnique arc genedllt c<strong>lo</strong>ser to the obefled<br />
!alu6. In calcul{ions using th. fomnla of McNrlly A-t = ,r- dd X = seni'diafreter ol<br />
The colu$ns nr rable 5.12 show rhe dalc ofobseruation lhc coor\iiMlcs ol rhc<br />
<strong>lo</strong>catior ofthe obseNer (lalirode md <strong>lo</strong>ngitudes ii degr€e resp*li!ely),lhe e<strong>lo</strong>ngation o1'<br />
rhe Moon fod rhe sun in d€8rees, the seni-diamcrer oflhe Moon. thc c.nlral rvidt[ ol<br />
lh cEsce.l. ninioum visible width of cre*.nl (all in dc minul6). <strong>lo</strong>l<strong>lo</strong>sed bv lhe<br />
crescenl length crlculated by our nodel ,id $e obsded crc*chl l€nglh. lhc .a'nc ordrc<br />
obseNer Th€ l.( No colunns contain cGsce|n lcngtlrs as calculaled usirr8 rlE nrodek<br />
rlu" ro sutron onj Uctlutty .,p.ctvely. Thc du(a in rhh ubl€ has bccnlnlnucd iD ordcr<br />
oiincre.sine e<strong>lo</strong>ngalion ot ARCL.<br />
To d€Gmine rh. l€ngh ol! crcsenl fom picroial record the dieital phoroemph<br />
is opened in any sraphic $nw l@l- SeleclinS coordinales of lhe oa the poin6 (*o<br />
ofr&n do*roeachoftu cusp md on€ c<strong>lo</strong>sc ro supposed cenftof$ccrc*enoon the<br />
vhible oute! liob ot lhe oe$eol an equation of circ<strong>lo</strong> h deve<strong>lo</strong>ped $at leads to dre<br />
coordinales of lhe centre of the luft disc, toining lhe cenlre of the lund disc with lh€<br />
two yisible cusps of thc crsce.l tbe dsle made al fie centre bv thc tso cnspc is<br />
mc6u€d. Tn€ picue of one of tlE cre$ent m.$u.d in this Mv is shoM in Fi8 5 l <br />
185
Fig. No. 5. L7: Mesuemcnr of Length for Cterent of Mav 28. 2006 as pho<strong>lo</strong>eraphed rr<br />
Xffiltr UtoveElty obsefrd@ry.<br />
'the dlta aor the obsiedcEscehl lenglhshoM in |able 5 12 and the cbart i. fig<br />
5. L shows inlqesrine paficn. Clain d by Schacfer (Sch@fer, l99l ) fie crc$ent lcngh<br />
is a sn@$ fiDction of e<strong>lo</strong>nSatio.. but lhis drE er sho*s a tMd that cl*lv exhibns<br />
delialion tion oy such sn@$ rel.tion. I]lc dala sdple is snall bnl lheE de lwo<br />
subsets eech having neally sE@th relations, separatelv. b€twcc. lhc crescent length tnd<br />
the €<strong>lo</strong>ngation. Hovever, rhen consideed s a conbiNd dala sel lh€ obsflatios uith<br />
e<strong>lo</strong>ngarion ll93 de8lees (by AnFa} l44l de8@s (bv Oner) 1667 degEes (bt<br />
Ralini) md 20.12 degrees (by Qwshi) deviarc Mk€dlv lbm $e apPaEnl snoofi<br />
rclarion exhibiled by the rcsl oflhe dala set Cecchl l€nsths in rhcse four cNs are huch<br />
snallcr thm lhe uend shown by the rcsl ofob$wationsas wellss thcresultsofe&h rhc<br />
mod.ls co.sid€red abovc fot calculaline lhs le.gtb of $e ciesccnl<br />
186
AII the* fout deviati.S cdes ae photogdphically Ecoded dd have the ledi<br />
posibiliry o| obseNational ercs. Out of |he t€s1 of th. €ight ca*s anothcr six @<br />
plD<strong>lo</strong>snphic. Thc rccords due <strong>lo</strong> Sch&fq de lhc only cag 0tar "ob$nation.l lcngrh<br />
of .ew lunr cE$ent is @nMed ils Eladon wift e<strong>lo</strong>.Salion mul nol bc sneln 6<br />
shoq in flsuE 5. L2 as w€u 6 table 5. l2 (or nsurc 5. L7).<br />
It is tunher nored $ar rhe rool mear square erot calculated foi lhe lhree<br />
conputational frdhods (Ouls, Sultmt and McNally t it n found $at Suhan s melhod<br />
h6 rhe led( cror (4.76 degr€et fol<strong>lo</strong>sd by McNally's thal ha m .mr of 6 16 decrs<br />
dd our model hd an eror of 22 degR' The najo. difer€nce h€lwen our model dd<br />
rhN Sulbn's dd McNaUy s is that olr nodel 8iv* coosislmtlv hisher valucs ol the<br />
lenefi oi crc$ent, wliets $e orner lNo nodcls havc bolh posidve lnd n.8flive emn<br />
Th€functionalrel.tionbet*eene<strong>lo</strong>ngationardlhecrcscenll€nglhissinli<strong>lo</strong>rinolrtuodcl<br />
aDd $ar due to Srll.n s. bul lhe oneerhibi(ed by McNally's model is m.rkedly difitrcnl<br />
McNally s mod.l is Sivins betre esuhs for larser e<strong>lo</strong>.gaion but a3 $. c<strong>lo</strong>.Sation<br />
becond snaller the emr given by McNally s nod€l beconcs leser and ltrger<br />
Tdhl. No t.1-2tth:.^r"la r'!h,lak.t Lngths o[CrctehB<br />
t87
180<br />
160<br />
'110<br />
1m<br />
100<br />
80<br />
60<br />
z<strong>lo</strong><br />
15<br />
ELOIGAN()|{<br />
+OBSERVED FOURESHTS r SULTAMS o<br />
MCMI.LYS<br />
Fia. No 51.7<br />
200<br />
i60<br />
160<br />
E.ro<br />
:100<br />
380<br />
360<br />
20<br />
0<br />
15 20 25 3)<br />
ELq{GAIOI{<br />
Fig 5 | 8r C'lsat tdgo5 $rinc E<strong>lo</strong>qiion withdn wi.rioa dc ro aRCv<br />
188
Tlre modcl for rhe calculation of cc*ot lenglh may be u$d d rh. .dli6t<br />
visibiliry ctuerion 6uch in the smc say 6 Yal<strong>lo</strong>p\ cilcdon cm b. usd, Hos.ver, our<br />
€nphasis *6 nor b dcv€<strong>lo</strong>p d allcm e cdr€.ion for $c s€. This *ork ws i .nded<br />
for a b€rter und.htdding ofthe grcme$y of the lunar cEscent md <strong>lo</strong> dele<strong>lo</strong>p a ne$od<br />
for cllcuhting iis l€nsrh.<br />
The scond md indi@t tesi of the 6odel is its conparison qitb the rcsuhs of<br />
Ddjon ncnliored in Fal@hi el- al. Ifthe niiimlm visiblc eidlh ,/, for vdious ARCV<br />
according <strong>lo</strong> Yal<strong>lo</strong>p's critcrioh is r.placed by rhe nininum ever visible cenralwid$ of<br />
0.18 aa ninuks tlEt is equiva<strong>lo</strong>r b igoring $e atmospheric allst for <strong>lo</strong>wer ARCV<br />
|hen the Eladon b.tsen cEscdr lcngrh and .<strong>lo</strong>n9rioo bccon€s snoolh as pE*nrcd by<br />
Schaefer (Schaeftr. l99l). The smc is shoM ii Fis. 5.1.8. The chart in figwc 5.1.8 aho<br />
sho\s lhat limiling value ofe<strong>lo</strong>.gation <strong>lo</strong>r po$ible lisibilily oflh€ cEscsr is areund 8<br />
d.grees. As our compurlrions do considerthe an'cctofparallaxrhis limil is equivatenr <strong>lo</strong><br />
te degrce linil very popultrly r<strong>lo</strong>wn $ DanjoD\ linit.<br />
On thc b6is of $e calcuhred lenghs of {escenr and ns e<strong>lo</strong>nearior using ou<br />
hodel the Ddjon deficicncy arcs aE calculrted igno.ing the af&ch ol ARCV by<br />
aomula giv€n by Danjon and thal given by McNally (shown in Fig. 5.1.9 and s.t.<strong>lo</strong><br />
Bpedi!€ly) These resuhs ae in c<strong>lo</strong>s agreencnr wilh fte Fig I in Flroohi el al (lbar is a<br />
^rrcduoion of Danjon\ lis 2)- 189
*.<br />
Fi8. 5.l.9i D.fici@y Arc rg.ind E<strong>lo</strong>ngtiod f.dding to D6jon<br />
;r<br />
3.<br />
Pig.5.1.l0: D.tr idct A!!.gd!d Eld$tLa...qdiosb LlcNdly<br />
190
6.2 LUNAR CALENDAR FOR PAKISTAN<br />
Mosl of the hlami. counri* fol<strong>lo</strong>w d ob*wadonal lu.d cal€nde, ar leasl fot<br />
ftn rcligious purposes, Although subslantial work is done to cvolle a pediction<br />
ciiie on ro dcvc<strong>lo</strong>p a univerel calenda! by llys (1984b, 1988, 1991, 1994a, 199,1b,<br />
1997) ed otheB, r truly lnilersd calendd could nor be d€ve<strong>lo</strong>p.d. As according Io<br />
coDmon Islmic b€liefactual sighline of$c n6! L6d crceenl is n*essaty to b€gin a<br />
Lund monlh, such a univesl calend& $cns <strong>lo</strong> be inpo$ible. This b |fue €speci.lly<br />
b€cau* just aner conjuncion $e new luna! crcscenr is nol visibl€ on rhe sme day<br />
lhroughoullh€ worl,l, eveD ift*o places ofobseNatio.areoD sm€ <strong>lo</strong>Dgirude. According<br />
1o recent dele<strong>lo</strong>pnenb (Scabefer. 1991. Yal<strong>lo</strong>p, I998, Qurcshi &Khao.2007)<br />
ob*ryational l@r calendaB for each Isl.mic counlry m be d.ve<strong>lo</strong>ped bur dc ooi<br />
s.ncally acceptcd b)'' rarious Isl.nic conmunities.<br />
ln Pakisian an Offrcial Commiirle (Ruer-e-Hilal Comnft(ec) decidcs about when<br />
1o beg$ a Lunar nFdrh on tbc bdis ofpublic cvidence and obseNatioos. This connilre<br />
sa$e6 infodaioi 3bour thc claim ofsishriig rhe csce Thc clrims arejudged oi<br />
rhc basis oflhe diFdncsof$e Islmic Laws. Som€ FpEsenlalivesofvdousscienrific<br />
organiatiom de cle consnhed. Oncc $c clain/s is jNritled relisiously andor<br />
soientilicauy, $e Ennoucemenl is nDde rboul begiming ofthe nexr Lun,r m.nrh In x<br />
way,lhe beginning ofE\v lunemond is bascd on public obscBations ' verined ii licw<br />
ofde pnncipals laid dosn br lslamic shariaat laws (lisled inanicle 1.3).<br />
'lle advmrae. is rhal a larse numb$ of ob*Ne6" trle pan in $e exeai$ and<br />
wilh it the pmbability of sighlins .ew cBscenl is incesed. Morcover, hon oa thcsc<br />
observeis de iionr rural arcas sherc thde is leasl ihduriallraffic 6nd liSht pollution,<br />
aid n k higlJy prob.blc thal lhe obseNing condilions d€ n€ar perfect. TheEfore in this<br />
sluljy lhe* public obs.rvations e considcrcd |o bave a hki degE. of autnendciry. ln<br />
lhis \ork all lhcsc public ob$dation-b.*d d.res ofsiart of€ach Lunu nonth frcn ye.r<br />
2000 b 2007 ae reproduced and 4 srudi.d in conparison wirh thc Yal<strong>lo</strong>p\ q value and<br />
l9l
ours-value crirerion-baed crireria. A similar wolk for the period2000-2005 has akeady<br />
bcr reFned (Qu€shi snd Kh0, 2005)<br />
TIE resuhs of E 95 lnDation during the Jmualy 2000 <strong>lo</strong> Scprenbd 2OO7<br />
(Shawl, 1420 AH <strong>lo</strong> RaM, 1427 AH) .tong *nh obsRalional data 4 presenred<br />
in yeady lables in appcndixlv All rhe conpuklio.s fo! the* tables de basd for the<br />
city of KaEchi (Larirud. 24o 56, Longitude 60 l.). The fisl cotom indicales 16r<br />
conjDct'on wjrh nonrh, day, hour, ninde and seco.d for rhe l@al rim€ of conjuictioi<br />
in the sub-columhs. Thc calcularions are done for rhe day oftast codlncioi, fo! $e nexr<br />
day and vherc Equird two days tater, indicaled undc. colunn hcaded .Ldt Coij<br />
fol<strong>lo</strong>wed by rhe Dale in Oregorian catendar. Nexl $o cotlmns givc lhe r€larive ahnude<br />
(ARCV) and reladve azimuth (DAZ) in degrecs foltowed by nooDset-sunscr tag ir<br />
m,nules. ftc age ofM@n (in houc), rhe arc or tignt or €tonsalion {ARCL in degrt<br />
and fie cBscenhvidrl (irr arc minlret apper $e iexr rhre. cotunns, iolhsed by<br />
colhns indicaling the r-value, visibilily condition otr q-value. lhc .elahc and the<br />
visibihy conditio.s on s,value. Thc visibitiry condirions are lho* de$nbed in hblc<br />
4.2.4 (for Yal<strong>lo</strong>p\ oiterion) dd labt€ 4.4.4 (tor euEshi & Khan qirerioD). Under rhe<br />
hcrding of 'Momh $c colmn givs rhe nrhc of Isl.nic mondr $at begi$ anq a<br />
sighring reponed on $€ previous evening foltowed by $c CreBorion d.te ot slan ofrbh<br />
lDnlh.In lhis 16l coluDn und$ Cregori& dare ofsiafling lunnnDnrh, numbsofdar-s<br />
in dE nonrh, iE also 8nd Ttu la{ colunnr conr.ins a comnenl. If rc dccision oa<br />
sl.nrng new lunar noDlh is in agrccment vi$ the prediotion critsioD Gratur q.ilqioi)<br />
thc column conrains I,ROPER. tle ..ob*ryalional decision .nd tbc Nodet arc<br />
consideed <strong>lo</strong> b. i..gEeme.t only $ne. thc s-value clirerion show Ev, i.e. crcscenr is<br />
clsily visiblc. Tbe conrhenr in ttE tasl cotuDn cobrains ,LATE, if thc dcchion of<br />
startng new lund nonrh is one day lare 6 indicaled by rhe dat of csy visibitiry on r_<br />
lalue and EARLY iI the lunar nonlh has been naned ooe day edlier rhan lhar<br />
indisted by the prediclion crn€ on. A5 staled eelier all de* calculariohs @ doDe for<br />
KaEchi and for rhe b.sr <strong>lo</strong>cal lime for €E$.nr lisibitily. Hower, ir should b. nored<br />
lhal observarion claios are couected from al<strong>lo</strong>verlhe counry.<br />
192
Out of rhe 95 public obsesalioDs reponed jn rhis \o*, rher wre o.ly six<br />
occ6ions (Jbe,2000, Juq200l, Juty 2004, ApnL 2006, Jury,2006 and Juty,2o0)<br />
vh.n th€ sighling war Epon€d one day latc ii compdison b rhc pcdictio, crn€ria<br />
according ro which the New CEscent was lisible on rhe pevious day bd *s not<br />
reponed. Tbere was oDty one oeasion {hor rhe earty sighrnrg is rcponcd Nov. D,<br />
2004), hus,1herc weE only 6.3% enors in rhe decisio.s oflh€ moon sightingcomine<br />
of Pakistd duing ld sven yas. llese resul$ e ba$d on verifi.d chims Tbe data<br />
B8arding nnmberofcE*cnl siendnB d.ins $ar sre notrc ficd is nol available. The<br />
only case of LARLY'sighring acepred by $c audonties was for rhe nonrb of<br />
The main<br />
'e6oi<br />
tor lare sishlinss or rtr five indicated occdsions is th€ overBr<br />
sky all o\€r lhe country in.lllbese caes.In case of$e onl)-' earty sigh ns for rh.Ian<br />
scven yea6 alt€npts for cRscenr siehring \ere n,ad€ rhe Obscdatory ofu.ireBity oa<br />
Karuchi using 6 CodC reliacror rclescopc.'the prediction crnerion based on r-value<br />
dl<strong>lo</strong>wcd cescent visibility with oprical aid and rhat based on 4-vatue did hot .l<strong>lo</strong>wed<br />
visibilily eleo silh lelcsope. We failed in sighinS of the crsccnt bul rhe omcials<br />
acccpted clains of mlied eye sighli.ss fron aR. ctoe to Kamhi. Akhough,(he d\o<br />
$n$il cosidercd did nol al<strong>lo</strong>w naked ete visibitny on rhisoccasion lhc misconc+tion<br />
lhat if rhe ph4se ol rhc cresenr is noE rh.n l% n shoutd be lisible (Asrmnonicul<br />
AhnrnJ..:l]O7r n ayha\c leJd he,te.i. or D.Jtcr rorcer rt,e.tdtrn;.<br />
The <strong>lo</strong>w percenbgc ofee6 in rhis obscryalion.l elfo,1 is much noE pbnisi.g d<br />
compaGd b lhe 6ul$ of m@n wlch, progEms cohdEted in Unncd Slales in thc tar.<br />
1960s and rcpon€d by Dogger & Schsller ( 1994) Accordi.erorhh rcporl l5%positite<br />
€rcs (qons clains of observing $€ cresceno and 2yo negadve eriou (cescenr was<br />
visible bur rhe obsrveF nissd then) {qc iound monssr $c cxperienced ftooi<br />
s"lche6. Thqe exF i.nccd h@n watcheu \v.rc geneElly consideFd b know wne,e <strong>lo</strong><br />
lind $c crcsce.r dd whar dc oiolatioo of rhe ..homs" wa.<br />
l9l
Freque.dy. claims of lery dy sighiiog e made in sonc rcgioG of paknb!<br />
(pardculely for ianins rhe dontbs of Rd.a ad Shawat 4s ienrioned beforc) $at<br />
ad nor accefled by rhe au|hodd€s. Oie offic Dajor @!s of rh4 dly clains of<br />
sighting ofncw se$cnt in Pakisran is lh. kenness otlh€ clainr dd lh€ nisco.epdon<br />
.mongst trasses ftar when lhe 'tighting" hos been eponed in the Kingdom of Saodi<br />
Arabia lhe cr$€nt nrusl be sighred the ncxt day in palistan, However. tbe oilcill<br />
d
lhe lmd ft*e.t. C€ncrally thc crnical valuc of thc Phsc b consid€Ed <strong>lo</strong> b. t%<br />
(Astrononicrl AlMac, 200). Duing rhc p€riod f<strong>lo</strong>m Jduary,2000 b Sepr., 200 $ar<br />
were consid€.ed in lhis 3tudy rhe'e qs nor a sinsl. incidencc when rhe sidd.s ot<br />
cEscenl with phN l.$ lhan l% ws eponed and am€/.d. There weE two @c6io.s<br />
when lhe ph6e m grcarer rhan l% bur ihrr. ws no claim of sightings a.d sighting ws<br />
not possible accoding ro bo$ lh. oitclia considqed (Oct. 23,2006 and Dec. 2l,2006).<br />
In lhc cae ofNov, 13,2004 th. phase ws I l9% lhal may have been lhc Eason rhat rtre<br />
siehting.lain qas accepred (as ne.don.d earlie4.<br />
TheyounB€slcrescentof agc t9.l hou6(!r bstdheoivisibilny)eendurinClhe<br />
period of study was that of May l'7, 2OO. Apan from b.in8 $e youngesl cre$eht seen<br />
duing th€ Fnod of study lhc ob$rvltion has anottrcr inleresing teatur€. Thi5 Ms $e<br />
or y case when thc cr.eent was sccn on ft. enc cEgodm date a it N6 born<br />
ac@rding ro Patiskn sblded line. Apari ton $is record obstualion th@ are fou.<br />
orhd clains ofyoung Moons sighling,Nov. 11.2004(22.1hours),tan.t0.2006(21.45<br />
ho6), June 26, 2006 (22.76 houre) dd Fcb. 17. 2007 (2 t .6 hous). ln aI $eE vee 22<br />
(21% of Ihe studied) ascs of sishdnS daiN when $c .se of M@n @ les rhfl l0<br />
The obsemtion of Feb. | 7, 2007 @ nadc ar fic slronomical ob*Fabry ar rh€<br />
Instrule of Spsce & PlanetJry Astophts'cs. Unrvcrsily ol Kr',(hr *. ,pou.a U.<br />
cre$ent nsing lhe 6' Codd Refractor T€lescoF. Both rhe prcdidion crileria alto$ed<br />
crc*enr sighri.g wirboul oplical aid but wc could se rh. cr€scenl wi$oul tetescope. No<br />
other claim of qesenr siShring wa ac.iv.d by $€ aulho ies on rhis day, Tte<br />
aulho lies acceptcd ou €laim. This was $c young*r ccseft sen ar ou obseoa<strong>lo</strong>ry<br />
d$ng lhe period ofsludy, It is a Ko.d for ou! obs.ryarory and rhc vorld recod for rhn<br />
part'culaf clescenfs eeli.st obervarion (sw,ftoonsiahlina,con dd<br />
sv.icoprciecr.op). Our phoroSraphic r.cord h also posred on retevanr websnes.<br />
!95
Tlrc'€ seE 6 c66 when thc ag. of M@n @ bciwen 24 hou's dnd 30 hous<br />
ed rhe crescenl w4 not seen. Thus, ihis srudy also suppons th. idea that age is not a<br />
depcndablc l&to. for any pEdiction crildion.<br />
Anodier inrporbnt pdMeler concened is rhe noonFt-sun*l LAG. There are 22<br />
occasions *hen the crescenr of lag 60 minures or less was Fen. Oul of rhese only two<br />
*ft wnh lag less rhan 50 hinules, Nov. l], 2004 with lag ]5 mi.ur€s a.d Aprit t0,<br />
2005 Nith lsg .l,l minut€s. In ihe *co.d of thcse c6es rhe prediclion c (ia a o$€d<br />
naked ele lisibility. There wer€ 15 cases when the tag sas b€lveen 40 and 50 minules<br />
and bolh dic pEdic on crileria did nor altov crcscent<br />
visibility *ilhoui optic.l aid, Tbls lh. n@Mr-sN4lag a<strong>lo</strong>ne is also not a dependabte<br />
parametef for any prediclioh criterion. Out of 95 new Moons rhe predic on crileria difer<br />
from $e Baby<strong>lo</strong>nian clnerion (Faroonhi et al, 1999) only g tines when Daby<strong>lo</strong>nian<br />
crnerion (ARCL + l"Ac(in deg.ect < 22) alto$ nalcd eye visibility but ,{luc dd<br />
lhe s-laluc cnteria do tur Howld, on rhese nine ccas<br />
I'r view of tn6c fads we conclude rhat bod q-vatue ed r-value cilcda arc N€I<br />
suilcd predicrior cireria with r !.tue clireion delc<strong>lo</strong>ped in $is work has mdginalty<br />
be cr success percenrage ibr posnive ob$ralions (chapter 4). porricutarly in viewofrh€<br />
facl dar $cs fire.ia do not march acrual cr.senr visibitiry onty in one ou 95 cas<br />
(1.057q).nd thal @casion Ns conkoversial in liew ofrhe above discussioD. Thc nve<br />
cases Nhcn the cEscent $as seen hler tan prcdicrd aE not conside!€d as an ero! 0s thc<br />
probl€n occun d d!. <strong>lo</strong> ore(sr skies. TtrGfoF, tor lhe tomulaion oa turue<br />
obenarional IuMrqlendar $ese Fediction dreia de nidly dependable_<br />
TIE € has ben only o.e occasion whd lherc mrc three co.*curivc nonrhs of<br />
29 dars elcb (during June, July and Aususl 2ooo). This is rhe ndinum fo. Fpelnion of<br />
29 days lund nondrs anticipoied s e&ly a <strong>lo</strong>ricentury AD by Muslims ( yas, 1994a).<br />
The li6t (Rabi ul Awal, I42l) of$is lriphr ofnonthsbeema day talerrhan p!.dicled<br />
196
oftr\i'i$ lbir tNinun rcldirio! @!H ml lLv. tc.n th.Ia llxr! vt&r D ac&noo of<br />
frt! (diD|d) coorc.rlir.! tdt! of 30 d!y! 6d h 6b Fiod of<br />
'lldy.<br />
b. sighio8' ca c.r Morlly o.cN!nn8 Ep.ririon of fou 30 d.F' @d! io<br />
f'v! o! sil Afthow\ rh. kludc shd. Lsw (cb{i.! t) d<strong>lo</strong>s for con .don a! lnd<br />
et o oh..wrrion idic:!. m m., bur ftr.dvmcc ptslirg nd dcv.toFEr 6tcdf<br />
m ohsE|ioo l|E .j-n/- h.!G.t d Fcdidioo diEir b{' grarty bclp. Tt<br />
AFadix-V rhosa suh r ltnr c.ldr. in qdich .oryurrioB s! bed i! Knstti d<br />
lhe +v.lu. qilai@<br />
tvl
Chapaer No. 6<br />
Dtsct ssroN<br />
'ftis rvork $is inrsd.d to qp<strong>lo</strong>rc fld.ompae fic mathcmaical nodels for the<br />
crneds under which rhe new lund cresenl could be lnible at silen <strong>lo</strong>cdion on th€<br />
Eanh, Mormk!, it was intendcd that acomparhoD offiese n<strong>lo</strong>dcl is conducled and the<br />
models are nodified if posible, The lask of compdien od nodiEcadon of the nodel<br />
hs b€en succcsstuuy achielcd dd a neN model the r.vahe criterion has heen<br />
dcve<strong>lo</strong>ped. A sumdary ofthis wo is presnted be<strong>lo</strong>w sith r dkcusion on the najor<br />
achicvedenG ofllrc Nhole eihll.<br />
Fi6t ofalla bener undcFt.nding ol$c issues. conpubrional, aslrononrical dd<br />
obrerarional, associated wilh rhe prcblem;f l|t firet sishri,lg of$e new lLmtr ssenr<br />
is devc<strong>lo</strong>ptd. Wc have €xp<strong>lo</strong>dd the comPulalional tchniqucs and tlrc asrcnomical<br />
alSornhm dd rhcir applicadon (o $e extcnt that is .ecsary aor rhe calculations<br />
irrvolved in solving lie problem desribed in nr€ pEviors paraeraph lrritially lhc<br />
rcchtriques wcre nnptmenled on Miciosoft Drcd work slcels bui duc <strong>lo</strong> leDglh,<br />
calculalions and lhe use of<strong>lo</strong>ng fomuld Ne \cre for.ed to dcrc<strong>lo</strong>p a cortputfr prcgnD<br />
tlila<strong>lo</strong>l. T]le prognm has been used <strong>lo</strong> do allcrlculations lbt detemining c@ldinalcs of<br />
$c sun sd $c Moon.s well as the padmelcs in<strong>lo</strong>lved in oor pobhm lhe rc$'lts ol'<br />
nr. applicalion of $ne of tne modcls @ obunEd $ithin ln. pro8nn and lltosc for other<br />
modcls arc done on fic basn of $e dala generaled by pngEm thal is salcd as ao ouqul<br />
fllc. This file is $e. kans<strong>lo</strong>m.d into an MS-Dxcel rork shet md lhe resuhs of orhcr<br />
modeh re oblained therc. Tbc hbl€s comp sing thse ts!l$ tifiin dre m.in t€xl and<br />
thar apF{ in |hc apFndiG rc all d€ve<strong>lo</strong>p€d fon rhes *otk sh€ets<br />
The modcl due to Baby<strong>lo</strong>nies as d*dibed by Fatoohi et<br />
$at wa modilied by the MuslintAFbs is bricfly desribcd ir<br />
198
1..2) dd is ns applicarion is srudicd jn companson with other nod€h in cbapls ] (anicle<br />
Ll) and chaptq .t This tule is based on lhe sun of$c e<strong>lo</strong>nca(ion ahd the dc oflision<br />
-the Lunar Ripcness rule dednced by Muslins of rh. m€dieval eh in exp<strong>lo</strong>Ed more<br />
errensirely. h is <strong>lo</strong>un'i rhal though, $e Lutur Rip.!6s function is fo Dorc sophisl,caled<br />
a compaEd ro the Brby<strong>lo</strong>nian rulc, the luo mclhods prcducc almost equivalent Esuhs<br />
when applied to rhe re{nl obseRational ccords.<br />
'lhe $ct|rds h3sed oD relalions bclwe$ arc ofvision (ARCV) and $e tclJrivc<br />
azimurhs (DAZ) rhar \ere extcnsivcly deve<strong>lo</strong>p.d durinerh€€arlypanoflhe 20- ccDrutv<br />
ae <strong>lo</strong>ud noE sucesful dlrins lhis study in compdison <strong>lo</strong> U€ ancienr and Dcdielal<br />
Derhods The suoce$ in m€asuEd in ho* nany obseNalions are in agrcened wnh lhe<br />
models. I <strong>lo</strong>rv mdny rines lh. cFsccnt is seen wlEn fi. mod€ls suggcsls ils visibihy.nd<br />
hoN nan! tiDcs lbe c:.sceol is nol seen shen lhe nodcl aho suCgesls inrisibilny.<br />
lhc rcnen rhl fie oodels blrd I'n ARCV-DAZ tlalions are norc $cccsslul is<br />
irar wilh ihcrcasins rrlative azinu$ thc biCllNs corrdsl oflhc crcsccnt aalnrn rho<br />
skybrilhhess impro\.s and the crcscentsof<strong>lo</strong>seirclariveallitude ure visible. Sm.lleris<br />
rlc DAZ lhis b e]trr$ com6l dclerionlcs lnd lhc crcscn( is only \isible al hiel'cr<br />
,^RCv. h comprnotr to lhcse models lhc Lunlt Ripcncss tutrclior is slronglv buscd on<br />
rc of scpdlion. lh. toblcm $nh arc ol sparaiion is th $nh l ge DAz n tan tinruch<br />
snaller rhdn su-lgesrcd by lnc Lunft Rip$es law (10 to t2 d$recs) <strong>lo</strong>i a lisibld<br />
cr.sceit. ln rl$e cas.s \vilh large DAZ tlrc Ripene$ luncliorr ,4,r can bc llrgc so lhrl<br />
rhe larger valLEs of R,,, arc eq Ed. Conseqlendy. aor lrse. lrtirudcs ldgc .rc of<br />
scparalion is EqutcJ But lescr DAZ for larger laliodes auoNs snaller ARCV rtrd<br />
con*qucnrlt smrll( rc of scpration. fhercfore, cspeci.lly for <strong>lo</strong>calioDs Nit largcr<br />
hdudes Lunar Ripcn.ss law beomcs more inconsnrc s comparcd b the obscNtrions<br />
and thc ARCV-DAZ nodels.<br />
ll is sftn du ng $c di*usion ar rhe .nd of lh. 4ri cbapl€r lhal lhd ARCV_DAZ<br />
ba*d nodeb, the ForherihShmr's modcl. th€ Maundeas mojel and lhe lndian nrodel aE<br />
successiv€ly belr€r. These imprct.menls ae due to deducrion of b.ner and b.trer basic<br />
199
dau ofnininum aE ofvision for difcrc el.tiv. uinurhs- t<strong>lo</strong>wrcr. all th.* modcls<br />
isDrc the ridrh of fi. n.w lba cas4.d ftar vdie SMrly with thc Eanh-M@n<br />
diskrcc for rhc saec e<strong>lo</strong>ngllion. Thk c.!s!s vdiatioB in thc elully bdshh.s or th€<br />
.Eeent of emc donsdion. Cons.qEntly, ft. s. pln of ARCV ad DAZ $c<br />
t'i8hh.s of casnt v&i6 for dif.rcnt Eanh-M@n disranc.s. Th€ lsk of this<br />
considdrion is rh. Fain caus. of l.ss.r succ.ss pec a8. of $.* n.thods N<br />
conDlEd to the later ftod.ls foi posnilc ob*nations.<br />
Th. Glliaion of vdyiiS bid nas of crc$crl si$ fic widrh or cE*.nt ro!<br />
se €<strong>lo</strong>nSa on (and $m. pih ofARCV-DAZ €luct thc phriical d.$nPlion otth.<br />
poblcm by Bruin lead Yal<strong>lo</strong>p ro dcduce bdic dala €lating widtu ol cErent to $c arc<br />
ol visioi. Yal<strong>lo</strong>p d.dwed this dll! fod $c minima of lhe linning visibihy .ur$ of<br />
Inuin. Cons.qu.ndy. Yd<strong>lo</strong>p @s su.cc$ful in dcducins hh3inel. p.6nttc' r.sr. rhc 4_<br />
ulue ciledon. This crir.ion Produed bdler Gsuhs i. compari$n to atl th. ptcvious<br />
mdhod!. Th. dcdudion ofvadous visibiliry condirionson lh. bsis otdircBnt dngcs o!<br />
.r.valu.s $d rh2t ol rhc besr rimc of visibility' of .een frem rh. limitins risibilirv<br />
cunes of ENin !'e thc motr markable of Yal<strong>lo</strong>p\ contibution l-hc \isibili'!<br />
condirions pro\ idc ruidclinca lbour uds *h.t condnioos rlt co*cnr \ould hL casilv<br />
risibl€. $hcn n $ill b. visiblc undo P.rfcd st.rhct condnions whc. Ue opricrl \o(ld<br />
b< Eqlircd .d when fte cB$ent $ould bc sinPlv nor visible $ith ot Nnhout atrv<br />
opti.al aid. rfu* condilio* har. poKd ro b. noN and morc Eli.blc $i$ incBtsnB<br />
Thc najor co bution ol or e<br />
is rhc comp.ri$tr ol .ll fi. htljcls<br />
@rclically, frathcm.ricElly, phytictlly.nd in vi.w of$eir success p.rcentagcs<br />
s.r of ob*wations colled.d tom d. l.t. l9ri c.ntorv till dcntlv. Anorhcr 3isnillcst<br />
contribudon ol lhh wott is <strong>lo</strong> convcn aU th. mo
", =(^* - l- # -e4,:!.,'l) f "<br />
v, =ltncv .<strong>lo</strong>.no - o.orn1o.tz1-<br />
o.ooot uaz2\lto<br />
tt =(^RCV - lrt.8311.- 6.3226tv + O]llgr2 -O.tOl8V3rttO<br />
\t, -t5 + py rxtr+,pfl<br />
v | = tARcv - \t2.4023 - 9.4818W + 3.9512W | - 05612Wt D t rc<br />
vr=ARCL r !rc olscpaBlion - 21"<br />
ln rhis {ork schactels modcl of dldritc bdshrn€$ is ale erP<strong>lo</strong>nd in t.ms of<br />
nasnitud€ contrrst tnd thc dsllE aF in lsRD.nr $i$ his wo* (Schelet 1988.) -n'h<br />
has b€en ehieved by iniplcmenrine lh. rshliqus d.r.<strong>lo</strong>Ped bv Schel.r and o$ets |o<br />
ev.luarc sky brighhes (in tems of linninB mgnitud.) and r brishtocss ol lhe<br />
cE$€nL Biglrness ofsky and that otcE$cnt dcp.nd heavilv on various ihosphenc<br />
arsrs Amongsr ths lenpeBluE and c<strong>lo</strong>livc humidnv arc lcpl \2riabl. in the<br />
posram Hila<strong>lo</strong>l in ord.r to exP<strong>lo</strong>r posiblc condilions undd which claims of nlted ctc<br />
v'sibihy m.y b. Br.d for dill€Ent condilions fhis lc.ds <strong>lo</strong> wh.t Ne h!v. r€mcd 4<br />
Magdlude Conl6r (Amag _ maenilud. of Moon - skr's linritnl8 a!!iNdr) lf<br />
mognnrd€ .onl6{ is n.galit n is la<strong>lo</strong>ur of cN*tnr visibililv o'tunvie Dol [or<br />
cxftnEly crilical n.kcd cyc obserylrion ckn$ of Nw lunlr ce*cN lhe magnitude<br />
co 6l lE ber dallaled nrinut lv h i5 tolnd $d $he of ftcs cdcs appear doublful<br />
6 fie oasniude conrEst @ n.v.r in Lvout or cc*ent tisibililv cv'tr with hidlv<br />
ex.8g.ra&d wath€r coDdnioB (v.ry tow clttivc humidnv and lcmPtdur) <strong>lo</strong>r the<br />
vhole dudtion of th. ooonsd-suet hg |inc. Using th. pogim Hilatol the tin$ have<br />
ben evaluat d of(i) \$en $c nasnitud€ contGi is ndinn (ii) whcn rh' nasnilude<br />
conlldr jnrl bccodc ia<strong>lo</strong>uEble for crcsa risibililv and (iii) {hen rhc masnirudc<br />
conm was last in fa<strong>lo</strong>lr or ccsenr vkibilirv Th.s. esulN .r. sinild $ whar is<br />
de*ribed by. veniccl line ovr. liditing lisibili(y cur. ofBruin<br />
201
'fte najor achjevem€nt of this m* is th. romulation of t neq sincle pstuerer<br />
cireion on lhe b6is of rhe Fchniqus dele<strong>lo</strong>ped bv Bruin and Yal<strong>lo</strong>p using lhe<br />
bngnrrc$ modcls deve<strong>lo</strong>Fd bv Sch&fer dd othe6 rhc visibilirv cuFes and thc<br />
liniting visibilily cwes aE deve<strong>lo</strong>ped for crceenls ol diffeEnl widrhs ll'!l wcrc<br />
actually obened tnd hav. bcen rcpodcd in lile6tue Bruin dcve<strong>lo</strong>pcd lhe$ cllvcs on<br />
|he bdis or{i) lhe ale6se bdshlnss ofthc tult M@n md thc wv i dalt,s *nh rhe<br />
decr€sing alftude abole hoi@h and (ii) tbe avedge brighiness ot the skv dudrg<br />
t*ilid &d fic way n depends on lhc dolld d'prc$ion b'<strong>lo</strong>v horizo' On lhc olher<br />
hand tc bave usd rhe adual brightncss for Ihe obscb-ed cEsc€nB ola fixed wid r llEn<br />
<strong>lo</strong>r tn sne ofobselvation we hare calculared th' al ude of skv poins bavins rtrc sms<br />
bridft$ d $ar ol lh€ dcsenr for difrercht solar depcsions such compuratios m<br />
Ep€ai.d for a nunber ofobseryed cE$e 5 of the sahe wjd$ al dilIelenl <strong>lo</strong>calions and<br />
dmes. The aliudes or thc skv poitrts lhu obt'ined re ten 'vc6Bcd<br />
our Thc wnole<br />
prccess h rhe. rcpeaFd for cnsce.t ofdilTereni widlhs<br />
These conpulations tesulled in<strong>lo</strong> our lisibililv curycs ed $c linnins visibihv<br />
curyes. The linitihB lisibilitv cses we have obEined ae slishtlt diferenl<br />
'iod<br />
$ose<br />
of Bruii with nini@ slightlv displ&cd The ntlighl linc joining thc ninina oi dEs<br />
cudes has a s<strong>lo</strong>pe of9 3/5 as compaftd to 9/5 for Bruin s cutves Tnis lcads <strong>lo</strong> slighd,<br />
differe b.sl dme of visibililv" which is 4 l/9 1 rine th€ noo'elsuosl lag aller thc<br />
suet (6 cohPdcd <strong>lo</strong> Yal<strong>lo</strong>p s be$ tine which 4/g rincs $e noonst-sunscl lae afiet<br />
rhe suset). Th€ bdic dala ob$inod fion the ninim o<strong>lo</strong>r liniringlisibilnv cu8e l-rttd<br />
b a fiird desee Polvnodial tsuls in<strong>lo</strong> the fouowine visibihv pdamslcr:<br />
s =(.4RCV - (4351 31 /3 + 2222075057t4! - 5 422641ritf + l04l4ltt)/l0<br />
we have als dedeed visibilitt condi$ons in a tum€t simild io lbar of Yal<strong>lo</strong>p'<br />
Our vhibihy condilions tre sliCh v difiere lron those of Yal<strong>lo</strong>p Howevet' ltplj_'n8<br />
our model on the obseNatioml dal! the succcss pcrcenrage is iound <strong>lo</strong> b' thc besl<br />
dongn all the dot nodels tll erist and lhar dc €ncd dd conpaed in lhis \orr'<br />
Thc e6on is tbat our model is baed on $e acrual skv brighhre$ and lhe btislrhss oi<br />
201
(escenr $idr varying wea$cr condilions whercas Bruin s nodel h based on aleragd<br />
brishlncssofsky dd rhe FullMooo.<br />
As clain€d by Sctaefd th€ brightn s modcls @v b€ in mt by a mucb a 2e<strong>lo</strong><br />
still our Drodcl yielded be .t resul$ s comPsed 1o tho* due to Bdin ad Yal<strong>lo</strong>P as lar<br />
as rhe posirive observatiohs arc conemed lt is lroped rhat wilh beller models of skv<br />
brighhcss slill beiterclitrion ior fiGt !isibility ofnew lund descent can be delc<strong>lo</strong>pcd<br />
sonE of$e lpplicarions of {he cnbda of.atliesr visibililv of ne\f l<strong>lo</strong>d 'rc$ctrt<br />
havc becn considered in $is $ork The fid ond $c mosl impoddnt applicalion lre ro<br />
d€(eaniDc Nhcn tbe new lunar cresceht would b€ visible ar soDe ldalion on lhe e<strong>lo</strong>bc on<br />
rhe elennrg aicr rh€ oonjundion Anolher ar.a ol appl'calion is <strong>lo</strong> deduce d<br />
''obscNnrioMl lunar c.lcndii- tor a 'eeion<br />
Yct anofier aBa ol apdic'tion \e hn\c<br />
e\p<strong>lo</strong>reJ '.roJdrnrine<br />
lhe lcnslh ofne\ ldDrr crcs(nr<br />
t( !1ay be recalled (hal lhe liret appcardnce of new llnn crcscctrt nrarlis r<br />
b€sinning ola new monrh in obsenalional lunor calendarq rhere<br />
'ril'ria<br />
and models drc<br />
siBnillcor br calendatical Purposs whefier 'n aduul o6€n:ri'nol lunar calendai likc<br />
rlc hliNic Lunar calendar' ulilizrs tb6e cril.da for a!6nging its cltndar ot Dol lhcsc<br />
cLirdia ptuvidcs a guidancc Ibr bolb tesing an tvidcnce oi oescenl sightirrg bv connnon<br />
peotlcdnd tacingdowndtdtrlesofac.l€ndar inhhrorv{herc approprirle dates arc rol<br />
\€ll Ncordcd. rhus $c frtin ililt of the pcdicrio! crileria $e carliesr visibiln)' ol<br />
' 'or<br />
n€rv crc$.m is <strong>lo</strong> egntalc tnc obsefl.lional lunar calendtr dd tcsrifv lh€ clainrs ol<br />
risibility of ne\' lunar crcsccm<br />
Although fiIsl ordc! iPporirutions like Arnh'neLic Lunar Calendtr tlrat di'<br />
based on lle conc€pr of Le.p Yea$ md rne a!.rage oorion o' rnc Moon havt be'n nr lse<br />
Muslims h.vc been fol<strong>lo</strong>sin8 acoal sighing of cresc€DLs l@l <strong>lo</strong>r lhe monlhs ol<br />
'r<br />
i6rii8 (R.maan) $d pilsinage (zil hajjah) Thc calendars if bled o' a PEdicrion<br />
oir*ion like tbat of Yal<strong>lo</strong>p or dre ohe dcve<strong>lo</strong>ped in this work arc $e cLosest <strong>lo</strong> rnc<br />
obervarional calendd Conpan$n of rhese criteiia wiih lhc actual obFNsdonll<br />
201
old! b !|dc! in ruirD frr lh. Ft! 2000 io 2005 blc Da dc (q!!thi sd<br />
Klte, 2005). h lbir q'!rt 6. mrdy it cid.!d.d <strong>lo</strong> tb y{ 20(I|. It i! biDd lb. d<br />
lvdsc 9J% obtlnltiol! lrc m.oding to th! Y.l<strong>lo</strong>p'! r-uh. €dt rion d our GEluc<br />
(d Q&K) dilrio D. diltgt@t is lb. !!3rlt of lithd th. t d wtt|t r dM !o *hi'h<br />
&e rw |E|r c{&. codd nol t tigln d dd tb Lud E olh b.t!! oE &v l.|c qr<br />
bo AdrDirtic cLiu6 of otsElion tld {t L|e dot:t t g.! oE &t dlicr U|n<br />
Tbit tt|ldttbL !|gs for th. vi.ibnft, (ritsi. for ol@t'.ridl pltrPo" hB<br />
oorivcd |' to d.dE e "bt 6v.liodl lE[ c.lct b." for P'lii..r 0|al ts tctr<br />
204
APPEDNIX-I<br />
COMPUTER PROGRAM<br />
HILALOl<br />
205
{ cl6c{): maime.u0; cltsqo; Daiuourineo;<br />
fc<strong>lo</strong>se(fptr4)t<br />
{ gotory(]0,4hout
gotox(52,12):p.i {"Ase = '),sororr(6t,I 2ripnr'4 ./o7.llf h .0de<br />
mjd).24.0):<br />
Sobry(5z,11);pdn("Qvalue = ):go<strong>lo</strong>xy(65,13)prinl("%.llf',qval);<br />
soioxy(52,1a)ipn (visib. =)rso<strong>lo</strong>xy(65,14);@ut
vhile(nex|sl=V');<br />
Eo<strong>lo</strong>ry(50,29):@ur>ptari<br />
eotoxy(I0.|6);coul>plenp;<br />
gotoxy( 10,20);cour
i(ldltohonthdayslin(lnonlhl.0)l)<br />
{ ldaiFl.0:<br />
lmonth=ldonth+ L0i<br />
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lyeFlyea$l ,0;<br />
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gobxy(2,21);cout
)<br />
{<br />
gmr4c6-0r{41.0+50.5484V60.0y60,0+(6640184-812866'si+0.093104'st'st-<br />
0.0000062rst'srlsr)6600.0r<br />
east4@=eEst Erc{€lltsi'cos(epsi<strong>lo</strong>n'PUl 80.0yt 5.0i<br />
smstz€rc-lGmrzro/24.0ldoubl{<strong>lo</strong>ng(8tui4b24-0)))'24.0r<br />
gdu€re1Gastu o/24.0!doubL(<strong>lo</strong>os(ssrz'o24.0)))'24 0;<br />
if(smEm
{ fs@f(fpt'%lf yolf y.lf,&sx,&bx,&c);<br />
rmr=l€mF4.cd(bx+c.1);<br />
)<br />
s<strong>lo</strong>ng+{tmp.po*{!i):<br />
)<br />
s<strong>lo</strong>nrs<strong>lo</strong>ng/l 00000000.0;<br />
s<strong>lo</strong>ng=jon8. | 80.0/PIi<br />
s<strong>lo</strong>ng=G<strong>lo</strong>n g/360.Odouble(<strong>lo</strong>ngG<strong>lo</strong>ns/360.0)));<br />
s<strong>lo</strong>ngl=360.0 s<strong>lo</strong>ne+-180.0<br />
i(s<strong>lo</strong>ne350.0)<br />
s<strong>lo</strong>ng=s<strong>lo</strong>ng-360.0:<br />
// L.firurL OF EARTH<br />
fo(i=0:i
iisha
+alp3'pow(!4)/100000000.0)rPI/180.0);<br />
)<br />
fo{i=oji44;i+f)<br />
{ fed(fpa2,"o4lf%lf%lf',&v,&rip0,&alpl);<br />
smlT=s@vpf v'sitr((alpot.lp l'l)'Pvl 80.0);<br />
)<br />
fo(i=oii
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fo(i=o;i
mcela=sin(!lar*PI/180.0)'sin(mdcliaprPvl80.0)1si.(mall'PV180.0);<br />
nft la=merdcos(ndellarPvl80.0)'@s(mlt'PUl80 0));<br />
i(fah6(nela)
l<br />
i((dnon8100.0))<br />
ddone-n<strong>lo</strong>n8+360.0_s<strong>lo</strong>n8;<br />
dc<strong>lo</strong>ng=m<strong>lo</strong>ng'5<strong>lo</strong>n8;<br />
Bky lal'tan(phitehda'PVl 80 o)+de<strong>lo</strong>nsi<br />
doubl€ modtunc(double xY)<br />
I<br />
xFxy/l6o:<br />
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forg nodf6c(160.<strong>lo</strong>8+390.67050284rtoy'0.00161I81rcc're'<br />
0.00000227'powtle .3)-0.00000001 l'por(1e. 4)):<br />
omes=modtun(124.46_<br />
I 56175588'kav'0002062'l.e'tc. 0 000002I5'po$(tee3)l:<br />
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I(24299161.0) e=z; el$<br />
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denn
)<br />
nmFsinc(so 0/60 ofV l 80 0)-<br />
sin(;lar'Pvl80 o)rsin(sd€l$prPVl 80 0);<br />
dem=o(iL'Pvl80 0)t@s(sdel6P'Pvl80 0)i<br />
iflfab(nud/d.M)
mooi coord(i),<br />
llii'i--.Ji. i:s'pv r ro.or-'i"olat'Pvr 80 o<br />
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dem
d.m=@s(ptar'PvlE00)'6(md't!prPvl80 0);<br />
,nfabsrnutr/d.M)
gobxy(l 6,1 9);conven-dns(fabs(sdelap));<br />
iotoxy(16,20);co.v.d-hds(sha)i<br />
coroxv{ 16,21)i@nren dmstsla)i<br />
;oblt(I6.22 ):@ven d6l rabslsalt Di<br />
i(elr
)<br />
{<br />
ip nt(iprr4,""/od:%d{'/o8.3<strong>lo</strong>\t",in(thou),in((lbou'in(lhou))'60),nma8-len)i<br />
'<br />
{<br />
falt=malt+.1 ; fam=hem+.1:<br />
kndisl=9o.o-fali:<br />
G<strong>lo</strong>monlaco(sin(matl'PUl 80 0)rsin(fal1'Ptl80-0)+co(malt'Pv1 80 0)'co(i<br />
ah.P/180.0.)r@s((n'a-faa)rPvl80.)))r l80.o/lli<br />
re<strong>lo</strong>nsui=(acos(si nGan' P V 1 80.0)* sin(lalt'P V l E 0.0)+cos(salr' PI/ 1 80 0)'cos(ral(<br />
iPl/180.0)'co(G@-ran)'Pvl80.)))1 180.o/PI;<br />
sacon=1.0/(cos(zendist'P/180.0)10.0286rq(-10-51@s(andist1Pvl80.0)));<br />
aacom-1.o/(cos(rndisl.Ptl80.0)+0.0l2lrexpc24.5xcos(4ndistiPI/180.0)));<br />
oacon=pow(1.0-posGin(Endisl'PVl 80.0)/(1.0+20.0/6378.0),2.0),-0.5)i<br />
fo(i=ori
fem-Doq(IO0,(o I5'te<strong>lo</strong>morv400)|'62'pov]l100 O'/po$rle<strong>lo</strong>mor20l:<br />
I80 0) 2 oi)i<br />
ii'-i..-po*tro.o.s xr, r<br />
'oo+Po$kos(fe<strong>lo</strong>mon'Pv<br />
m@nFrou4<strong>lo</strong> 0,(_0 4'(Fmal-noscnlrl4r z/ r"i ..<br />
doonb=;oonb 1l O pova 10 0'( '0 4+[shtr I'stpos'r I ))i<br />
';j,f"nn",li.ilk##ir#niti#rr'niffi ;".r"''<br />
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lcs+=rov'( I O 0.(6.Itf.<strong>lo</strong>du 400)rl<br />
ili=,J"-'oi" rid 6.i:e"<br />
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da\bno$( |0 0,( 0 4'(mshtrl_o6(nlrl*) zr,n<br />
davb=
APPENDI'(.U<br />
ANCIENT, MEDIEVAI, AND EARLY 2OM<br />
CENTTTRYMODELS<br />
23r
2!2
23J
234
215
236
2]E
2t9
240
24r<br />
I
APPENDIX.III<br />
PHYSICAL MODELS<br />
211
244<br />
!.I9
245
241
244
249
250
9!<br />
!!!<br />
!.39<br />
!4<br />
4!C<br />
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q4!!<br />
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,-9!l<br />
9lJ1<br />
9-41<br />
25t
2r2
253
NB<br />
91'!<br />
254
APPENDIX-IV<br />
OBSERVATIONAL LIJNAR CALENDAR OT'<br />
. PAKISTAN<br />
200G2007<br />
AND ITS COMPARISON WTIE<br />
TSE VISIDILNY CBITERIA<br />
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APPENDIX-V<br />
F'UTURECALENDAR<br />
DATA FOR TFE OESERVATIONAL LUNAR CAIEIIDAR FOR<br />
PAXTSTIN<br />
BASED FOR COORDINATES OT KARA'CEI, PAKISTAN<br />
LATITUDE2"5I, I,()NGITUDE6/3'<br />
PREDICTEI' OBSERVATTONAI, LUNAR CAI.ENI,AR<br />
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RETtrRENCtrS<br />
L<br />
Alned, M., 1991. Oigin of the Hijd Calende, ,4l-,Vardar, Decenb€r<br />
l99l.pp.51-54.<br />
2. al-Bcruni, Abu Raihan Mubmoad, 1000. "Al-Atnd afBaqiya! m alQu.an<br />
al Khaliyal" Tmlaled.nd dnotated by Sachau, C E. d 'fte Chrono<strong>lo</strong>gy<br />
of Ancient Nations', willian H. Alle! & Co. London, 1879; Eprinted by<br />
Ilijd Inrenqrioml Publishcrs, Lahoe, Paki$ao. I 981.<br />
L<br />
alBenni, Ab! RaihaMlhannad. t010.'l.dia: An acclrale Desciiflion of<br />
all Cabgoncs of l{indu Thoughl. s Well dD* Which arc Adnhsible s<br />
rosc Which Musl be Reje.ted ' l ranslatcd lnd annolaGd by Sachau, C. E. !s<br />
'Albcruni s lndia: An Accoud ol rlE Religion. I'hi<strong>lo</strong>etlry, Lndtuc-<br />
Ceogbpht. Chrono<strong>lo</strong>8y, AnronoDy, Cus<strong>lo</strong>nrs. La$s hd Anro<strong>lo</strong>sy of Ind i! .<br />
Williah H. Allen & Co., London, Itl0: rcpFi edunde!lheAuthorityofthc<br />
Govemm€nl of w.st Pokish. Lahorc, 1962, and by S. Chnd & Co.. N.1v<br />
AndLrlia. U..dd Firneis. Nl. C.. 2006. Fisl l.unar Ccsccols tbr BabJ<strong>lo</strong>n<br />
t2000 -599) .<br />
hup:/^\1\ \v.univi..Jc J'/F l,l I Ceschichtr l iFr_l urur_Cl*cnrs4r40ir.hrm.<br />
5.<br />
Aoki, S., Cuinol, B, Kaplan G. H., Kinoshila, H.. Mccarlhy, D. D.. &<br />
Seideld,nn. P. K.. 1982, Th€ Ne* Dcfionion of Universal Tine , ,.lrr'o,r<br />
AT rophts., rns, i59-36 1<br />
Ashbrook, J, l92. Sohc v€ry<br />
n.<br />
Thin Lnnar Cr.senu". Sb, d L/cs.,p., tl2,<br />
271
Asbbrcok, J, 1972, "Moc rbour visibihy<br />
or LtM C@.na St/ 4<br />
3_<br />
Aslrononi@l Almeac, 2007.<br />
Obs.tutory, WtllinCkJ\.<br />
Na orul Almde Omce, United Slars Nalat<br />
Ka n i st I uh l - H e idet b e ry, 1, 69.<br />
_O. the Thory of Eirinclioi.', n_/i,r S/r@.<br />
t0,<br />
Blackwell, H, R.,I946, Conrdr Th4sbotds of Hunan Eyc_,I cpriialSoc<br />
A"Et., 36lrr),624.<br />
lL<br />
t2.<br />
Compaalive sludy ofth.<br />
Tltousd y€as", Mazd.<br />
lianid, Mnstin Lua! and<br />
rl<br />
Bor<strong>lo</strong>wsti. K, M., 1t88. ..ELp 2OOO-s5 ahd<br />
TinE Rcl.rion..,,1n .ar 1 .,h..2Oa.L8.<br />
rhu D)@ic6t Trnc Untre.sl<br />
Bowcr, F A.ndwali,R D t982. -sthbsphdic<br />
15.<br />
BcIasrcn, p. & Fmnco!, C.<br />
sp[ericol vdiabta. VSOPBT<br />
1988, -ptdebry Theries jn<br />
tol\tions.', A tuo n A, tu phrr., 202, W.lOg-<br />
16. Bala&on, P., 1982, ."Ihaon. du hovehent de<br />
S olut i@ V SOp 82,,, As ro h.,4 y r oph, I 14, 27 a -2A8.
17.<br />
Bouwcr,D&Ctcmence,c.<br />
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