Visibility of Luna and Lunar Crescent

ON I'HE COMPARATIYE STADY OF MATHEMATICAL

MODELS FOR EARLIEST VISIBILITY OF THE

CRESCENT MOON AND THEIR MODIFICATION

Cdndidare:

M U I IA M MA D SITA H ID QU RESH I

lo

SupeNisor

(D..) N.sidddh Khan

Prcfesso.. Depadmenr ofMarlEma cs

Unive$ity of Karachi

Prcscntcd in prrtiatfulfitme ofrhe requircments for rle deerce

DOCTOR OF PHILOSOPHY

Ar Insliruk ofSpae & ptder&, Aslrophysics

unlversity of Kamchi, Kamchi

September, 2007

CIiII-TIFICA'TE

welcc.pllhe lhcsh as cortunrjns to the rcq0iEd $afttnrd

CERTIFICATE

c. incd thn th. ..ndidltc hr! @mplcr.d $c rh. T undr ny supervision

a-w"*

,/U*sql .',.hs la' t r- aanh,d$

,{' \4{t

Ur/

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iii

tnU'onaneofAltah

The md n 3}.'d/tul ard t to rno8t D3''cficent

Dedicated to my late mother

Anwer Sultana

whose patient struggle in life and passion for mathematics

motivated me to study

I am d.cply graleful to Prolcssor Dr. Nasituddin Kndn, my SrDervisol aor his patience.

inkluble sriNlla(on and counlloss sugg6tions dsht from the bcSinning rill rhG poirrt

*hen I .n ablc ro complerc lhis $ork.

I sould ole lile lo expess Dy CFlilude for tne valuble suSgestions polided by

Prcfessor Dr. Roshid Kadal Ansli of IcdeEl Urdu UniveBily. Karachi_ in rhe bcginnnrs

ol tlis prcjed My spccial $hks for Prcfcssor Dr. Mtrhamn)ad Ayub Khar you$rlzai ol

Urne6nj ofKonchi$no Emincd a nofiaror tor mc nbuglour rhis \ork I mus,tso

suhnrit nry hunblc tanks lo Prolcser Dr. Muhannad ltoys of rjnilcGny of Kuata

l-!mpur. Maliysia, Dr Abdul Haq Suhan ol Univeriry of Sanaa,yenren tor reir kind

cnconmAcment during $is sort. lhanls !E ats due aor Mr MulNflrnrad shotrl Odch

n nleins rhc wcbsllc NN\.icoon)ior! dnd Mr (hitid shorkor nriNgilg

\\\rr!r!]N4lt!!!!.9r1!rr. uorh lhc*

"ohsitc

rcnraincd ! eurcc ot infomllliotr. rcscd'lh

prpeu aDd.laln lbr rhn \orl lo hrve b..o posibt€. l_asr b nor rhc leasr; I subnjn m,

stccral rhdnlis llr Dr Shlbbir Ahs.n oi Lahorc Universny of M.meeNnr Scienccs

$hoe {aluablc nrge$iions helped nre in cdnins rhn dissedation.

I sish b cxtcnd my sirrccrc thanks to lhc University of Knrachi. i! pirticutar ns vicc

chrnc€ll . Prolcs$lpr Pecuda Qasinr Rm. lor rhc consislcnl DoRI suppon tor rho

My sife. cbildo. and in-laws enained under lols of prcsure durine lhis proloneed

periodofny\ort lor shich no gralirude nay bc enough.

MSQ

Abstract

The pmbl.m of delemining ftc day when Uc nev tunar crcs.nl can be 3en ti6t

ar an, site of observarion hs Efrained an opch pobtem since andquirr, The

phenon.non ImniB imFnel for beeioing ! luw mon$ in a puEly ob*rvariomt

lunar calcnd.i. In li. n6! chapler the asrrcnomicat pdamercn relare.i lolheprobtcnr are

!.!ieRtd alohe si$ a briefdcsriplion oasme rules oafitrnb arribut.d ro rhc ancicnr

and lhe nr!ie!.1 nodets lhc afecrs ot geognphical loc.rion on rhc problcr orc aho

dcscribed. A brief Evierv ofc.tendaB lnd.ssociared celestiat cycles h atso pescnled

$nh splri.l chphasis on |hc rulcs enurcjarot for Islamic ob*rarionat lu.arcalcnd,r a

.{rlr as lo'r cemu, AD. ln lhe end of the ch.prf Eviov or rhe conlrib[ion of thc

.strcnoncrs of 2orr' cenrurr is pesentcd $al b.8im rvirh thc etoFical toodct or

Solvins fte pnblm ol lhe ti6l visibilny oa lbe n.$. lumr crcsccnr inlolvcs

lenedy colcrlations and thc trsc ofatt ctcFenrary !slrononjic0t rechniqucs. A tovi.w ot

n e tcchniq,es and algorilhms is pEsc .d in rhc *cond chlpr. Sp$i.l emphNis is

Ailrn lo r|r dekminarion oa linr of oriunclion of Moon sirh rhe SuD. i c binh of nc*

Moon. rh. risinsand seu'hgol thr Smnnd rhe M@n_ posnionsofrhc sun,nd thc Moor

ar 2n! tnrc on rhc dll or rc day aftc. conjL .rion .t.tc ch.prr cnds rith a brict

dsscrption of the .ompurer prosranr Hitotot dsvebp.d r'.r compftrioN done ir rhis

\ort A n$v posam is deEloped in ordq lo prcdnce the dala Fquied aor rhis \ol[

$ar is Cencrally.or avlitabte troh orher sotlwarc lhat aheadycxrsr

ShniDg $ilh simtl€ Ulbytonian crncrion. fic lhnd chaprr eiptores rhe ancienr

dnd the Drdielal hlrbeoaricat hodets follo\€d b, rhe dcscriprion ol. narh.natical

trploEion of oedieval Mutim Aslomn6. Condricat considehions asiaGd

wilh the prcblen are.xplorcd in nrore dclaitin oder ro evalu0le the Lunlr Ripencss Las

o.0 sMcess, suggc$ed in lhc medieval e6. 'ls

Son nodincado.s to rh€ la$ aru ate

suggesled Thc shoncominss of the ripenc$ lav a,c then di*ussd and lieht is shcd oD

rhe Easns rhat lead ro lhe dcvelopnent of ARCV-DAZ ,.trrion b4d nodels b, the

cdly 2di cenlury astononeB. Thc significanr findin8 of the chaprer h $al borh lhc

simpk Babylonian c ldion and rhe Lunar RipeNss IEw a!€ morc successfutin tcrms of

thcn co.sistnct Nnh rn. posirile sghrings dords in comparison b fie hodeh

ddllopcd in th€ fiBr haliof2(]ucenrury Ascroa46Sobsonationsisuscdforleslinslll

modcls in thil {ork. nr.r incldc obseNations collccted since thc latcrhrlaof l9'i cemtry

ln conparison lo lh. empnical odcls nrd to prcdicr lhe visibilil) of lhe nc$

lu. cr€*enl iill rhe lid balfot $e l$ndelh c€nlur)- lhe 6od.ls dcleloped on rhe

b6is of phFical rheoriN of sl! brighhess and extincrion trc cxPloEd nr thc lol'nh

.[dfl*. i he* models incl"de l[o$ d.ralotrd by Bun aM sclDrftr scpamLl]. Btoin

bed his mddrl on lhe rveige briehhcss of lhc lull Mdon and lhe lriligh slv.

Schacfeis model calculales th. octual lin ins mdgnitud. oaihc skj and $c ntgnnud.

of rhc cftsdrl id tcsls a vkibilitt cldinr on rh. bnsis ol nugnirrd. coniGt and rs

difte{nr ir nrtuLc tio$ allother nodeh An.ra bdel deeriprion of Bruin s modcl ihe

scmi-cnrpiricrLnDdel olYillot is discu$ed in d{(.i1. Nhich isconsiderud to be rlt nDn

Nnrflchcnsi\c !trd aldrcnlic nrodcl. Y{llop dcducdd his busic d.h lrotrr lr ns

\isibilnJ curvcs. On nre brsis of Schadfcas tcchliqocs qc hd\c rlconsLruorcd Inui s

model rnd pruduccd lcw visibililt crNes Md. ncs scmi-enrpnicll nodel lbr rhe

\isibilnJ oanc$ llnar crcscenl fte delclopNcnl ofllris n.w nrodel is one oith( Naior

{chicve.nEoflhis$ort.AllIhemodelsarclestddnthssamcdaldselasisusedinrhe

!rc\ious chaplcr lhc nc$ nrodcldc\cloPLd in lhis qorl is found lo be rhe bcst aDonssl

rlB nlodcflr dr) dels in rcnrs olils cdnsnlcnct Nilh lh. nunrb.r olposirilc sishrinSs

in lhc d asduscd A compali$n ofsuccess oaqch nodcl h aho disNsed in tlns

Al $e end of $c founn chapler a emtce} is iEmed b *di dle tulhcnticirt of a

claim of siehlins or ne\ cte$e on lhe bsis of a rmi_enpnical modcl and th'

Nasnitudc conlras oodel -fhe sisnific.nce of such a stat€s) hN b€cn highlignred 4s

rherc ae found d nuob.r of authenric ne\r cG$c risibihv claims thlt are nol

coosisienr wnh a *ni-cmpilical nodel tn lhcse ces a semi'copnical nodel des not

allow visibilil, ofthe cltsm wilnoul oprical aid bu rhe magnitude conkasr is in f.vour

ofvisibility. This happens s aemi€npirical modcl dcs nor bl. i.to considcnrion lhc

elsvation of lhc sne of obseoation abole sed levsl md dr wcarher conditions t-he

maSn'lud. contBl model consided all lhese facto6.

Dcyond $eorctical considerarions a na$ematical model should possess poucr oa

appliabilily. The prine applidion of rhc mathemaical nodels cxplo&{. anatysed an,l

develoFd in this $ork is lo dermin€ rhc €adiesr visibihy ota ncN lunn cft*cniar 8J

location oflhc {orld and lo lerify a chim ofcre$on lhibiliry. Ap.n froh $is prirlc

applietion in the fiih chafler rhc sedi+mpi.ical modch arc applicd b devetop a

hchnique for cilculdring the leneti oan$v obsc(€d lun cascent. The phenom..on of

shonentu! otcrcscent lengrh is lino"n for centuries ond drrinS 20ri cenl!try a nrnrbc! of

4!$ns hale b(tn suSgcscd lor $e eme. Ilo\€rer. our sugg.srd rNhnique is rhc tld

ofilsn rc ihr pdvid. a simplc com brionili@lforcrlculatirglensrholob*ncd

.Nsccnl. Mooover. lhc *-nlichpincdl nlodch are nlso rfplicd r. lcril} thc act(dl

p@ti$'l obsen.t,onal lun.r cdendar i,r Paki$an for lhe lasr sc\cn )c s. rhc nDdcls

and thc tnctis.d

calcndar aN found to bc in asEemcnr in 95% oi$e ncs mooff duling

thc pciod of srudy. Morivaled by rhis hish rare of consh(cncl rve hrvc pNscntc(t a

''PEdicted Obscn arioml Lunar C.leod.i forPakisan.

A sumntr' ofrhis Nhole.flon h preseired in lhe lasr ch.prcr. v.nols ihpoarrr

issues aE hiehliSlt d $nh a di$ussion on rh. aurur. scope of Esearch nr the arcd

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CONTENTS

Urdu R.ndenng of Abslracl

2.

Introduction

Ll P@nd.6 for visibiliry olNry Lumr Cr$chr

L2 sienillcmce o| ceogdphical t calion

t.l Tbe Rut.s oaThunb: The Ani.m & fi. Medievrl

1.4 Crl.ndas and CelesdalCycles

L5 condbution oflh€ zod cotury Aslro.on6

Astronomicsl Algorithms & Techniqucs

2

l0

tl

20

24

2.1

2.2

2,t

2,5

2.6

2.1

2.8

2,9

Dyne cs ofMmn ed |he Eanh

Coordindtes of rhe M@n

Nes Lu@ Caenl visibihy Pard.ic'!

Thc Softve Hiblol.qpp

25

l0

4i

5l

55

58

63

3.

Ancicnt, Mcdicv{l & Etrly 20$ Ccntury Modcls

l. | 'ftc ltabylonian Crilcri!

3.2 Some Sph€ncd Trigonomebc Consi.lcations

l.l Thc Lud Ripehess I,3w md ils Modiicdlion

1.4 Ehpidcal Models of Edly 2d CenNry

1.5 ConposonahdDiscussion

15

76

82

t03

122

Phyaic.l Modala & their Evolution

4.1 Bruid s Physical Model

4.2 Y.llopl sinslc Pamel.r Mo.lel

4,1 Srch.f.ls Umitiog Magritudc Mo&l

4 4 A Nry Crirrion For vilibility of Nry Luc cEsdt

125

t27

l3?

ta6

tt9

t67

5.

Applicrtion

5.1 L€n$h of Cd!c.

5-2 Ob6.ri{tional L|lM C.lctrdrn of Pakkur

t'|2

l7l

l9l

6.

Dircusion

t9E

Th. sonwm Hihlol.cpp 205

fte Anci.nl Med'.valrnd E dy 2of 'enluo

Models 2ll

Th. PhysiolModcls 24!

Lwcd.nd.r2000'20{t 255

Futurccalendar2ooE-2ol0 264

21t

Chapter No. I

INTRODUCTION

Slnce rhe anclent ttmes rh. 6preaEn@ ot n€w tunar cr6cent marked lhe

beginningofa new monlh. With the d€votopment ot ctv zarions org.ntrtng lme tor

e(ended pe ods lnto weeks, months snd teaE, the tlnar phase cyctes tead to tho

evolutio. ot calendaB. lh6refore the pobtem ot dotermining the day of tiFt

sEhtingol n€w crescent moon arlbcted hlman betngr tt Invotves con stderatio n or

a nuhber of astronomlcat es wett as other facto6. On the orher hand th6 probtem

ol obserylng the new tunar cres.6nt at an eaniesr posstbte momenl ts chalenging

for bolh amateuE and professtonat .5tonome6. Th6 .st,onomtcat paEmetets on

whlch thesolution olthlsprobtem ts bas6d are b eflydtscussed in the ftsl .iicte ot

The circudstarces and parameteE assciated with th€ problem of sighting

new runar cresce.t Sreatty vrry wirh th€ v.rytng post o. or obseder on rhe globe.

The atfecls ot the geographicat tocation on the probtem ls briefiy discussed in rhe

next 6rtlcle. Atempts to determlne crle a tor the delermintngthe flrst ltslbitity ot

new runar c@scenr ai..y ptac€ rpp.ared as €any as the Babytonian ers (Faroohier

al, 1999, Bruin, 197?, tty.s, 19946). Signncanr adrances were made by Mustims

and AEbs during medlevgt perlod. A brt€l nccount ot thes6 ancient aid medieval

effons b discu$ed ln thtd.dtcte ofthts chapter.

Srnce anliqolty changing phas6 o, the Moon and a comptete cyct€ or rhese

vrriallorc has been ured a3 meansot k€€ptnEaccount ofcatendaG. |yas hasgiven

a delailed accounr of the history of ihe sci€nce of lunar crescent vlstblity and the

lslahic Calenda. (rryas, 1994.), Dogger h6 di$ussed lhe history and rhe

deveropn. of calenda6 (oo€get,1992), Fetngold &o€rshowitz hav. presenred a

(omparEllvesludy or morethan rw€nry catendaG or vsrtous (ypes both sncien, a;d

moden (Reingold & Dschowttz, 2OO1). A numbor otorhor rutnors navo contrtbur.d

on rcrai€d tssle. (atr.3hk, 1993, y.s, 1997, Odeh, 2oo4 etc.). A. ,ghr from rhe

beginnlng the c6tenda6 rE 6s$ctated wirh th. cyct€s ot rh€ heavens, rhe eme

.re Evi*ed atongvilh tno a$ociated catendaB in the nen anbb oi thts chapter

wlth speclal emphasts on the tst.mic Lunar catendai

Durlng the mod€rn rims tt wa3 onty in rhe tasl quaner of the 19rh century

lh.l weslem askorom66 stangd erpto,ing the pbblem of ea l6sr sigh trg o, new

lunar the tast antcte is a bnot luruoy 'oscent.

of lrorature that delcribed and

ttweropod vaious c,ire.t. o, models for lhe $tulto. ot probt€m ot d.t.rmining th€

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

Ll

PARAMETERS FOR VISIlIILITY OF NEW LUNAR

CRf,SCDNT

The Moon. the ody natunt sateuirc oflhe Eanh. is eoiog ound lhe [anh h an

orbit thot is a hiShly iftgrlar ellipe Thc i resutariries in lhe pd$ of rhe M@n a€ duc lo

llrc facl Et its norion is govemed by not onty tbe gEln ional pu oflhe Eanh. bur is

also aaf.clcd by toan) of rhe ncishholiing cetestial objecN (Danby, 1992). \\,nh r

Hadh s varying dhr&ce fron lhc Sun, thc e,Iecioa $e Sun,s gnvnodonat pull v,rics

srbstmti ly snh rheElarile posnionsofthe Eanh and the Moon. Moreov.!,lhe afcors

orthe confisur.lion oflheshole Sotrsrslem (losirions ofa| lhe major ptMcrs a.d $c

major ar.rcid9 on rhc morion oflhe Moon is oor nesligibte, Thus, accoding ro rhc

''Eplremirid€s Lun,irs Prisiemcs,, popularly tnoh d Chapont,s tuM tho,y ELp,

2000/82 (Chapronl-Touze & Chapronr. t 983, Cluprcnr-Touzc & Chaprcnt, t99l), tor lhe

b.sl possible plecision iD te lonSirude, hdnde md lhe disrance (bclsen lhc Mon and

the Eonh), dere aE equired as nany as 15i227 p€riodic r€rms. On lbe othf hand, for

dcteminhg 6e position ofrhe Sun b r simile d.8E of acc@cy one rqdB 2,425

periodic rems in vicw olrhe ,Variarions S.culair€s des Orbnes plandaiet, rhe Fiench

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

explorins |he circunsr.ee of rlE vGibihy of ihc !a lud .esnt, Handling sdh a

laqe hnmber of pdiodi. terms mak6 this fiEt slep very ciucial. The lest of the sludy

dep.nds on th. Elativc pcitio$ of lhe Ss and thc M@n ed rhe hod&i at md aner

thc aunsel for a.y place on the &nh.

These theori€s detemio. lhe c.l$i.l ecliptic coodiiates of the sold sy$en

obj*|' fid dc b6ed on sphencal polar c@rdinats. The fig Ll.l sho\s the spheiical

pol.r coordiMle sys&n, The origin of lh. syslen O is €nher lhe ccntrt of lh€ E nh ot

lhat ofthe Sun. The xy'plane k the plane olsliptic, the plane i. shich the E horbi6

iound the Su, The x-dis poin6 in the diEtion ol the vcmal Equimr l which is the

poinr of intersection of the Ecliptic (palh of rhe Eanh arcund rhe Sun) and lhd celesrial

equaror (rvhos plde coimides wnh rhe pl.ne oa rhc lereslrial equalor). P is $e objsl

llhc Moon in gcocemric sysem or thc Earth in llelioccn(ric sy(em) $hosc stheical

porq @dimrs tp. 0. 0) e p = lo4. 0 = 4oP 6d e-zzo?.\i\ee P ist\.

prcjccrion ollhe position P ollhe objccl on ro the xy-phne. In the celestial ecliptic

cmdinaB the cetesial loicitudc I = o dd rhe cel.$ial l6(ludeP= 9f - el= zPoP ).

whcn fie hcli@enilic eclipric coordinates ofthe E nh aa elaluared usins VSoP ft.y

aE $en tdslomed inlo lh€ gc@nric eliplic coodimles of lhe Sun The corjuncdon

or $e Bidh ofNe\ Men occu6 wben }{1 = Is.

Fig. No I I I

SuDpoe ('-.r",/.)!d (.,,r",t) E thc p6isc dist nc6, etipri. tonSnld.

..d Oc hnud. of lhc Moon .nd rt Sltr Bperiv.ty, Ff.Gd ro fi. nsr cquircx ol

rh. d.y of @njucrion. For .ny t@tor on th. Esnh wnh Gcri.l eotdimrcs. I rh.

rcGrdal lonsiud. ot rh. d@ .d t is rh. Lcid.l tlrirud., thc ti6t scD i. de

d.rcmimtion of dc visibihy ot ft. nd t!@ cE$.ni is ro d.r.diF rh. aclut

dyMicrl tin (TD oi rD T., of rlE @njwtion. N.xt, om Eqdc coBidcrine rh.

la.!l tima ofslrdns ofrt. Su {d rlE M@n. IfT, lrd Tm (CoordiDr.d Urivcdt Tin.

TUC) bc lhe rinB of rh. lcat sunsa ud rhc hooEt, ihcn d. i* G$cnr h.y b.

visiblc only ilT, < To, This La4 to lh. p.md.r LAG - Tr - Tr.

Using th. cclipric c@rdiNl6 ot |he sun 6d thc Mon drt.ut.l.d for rh. 1.. o,

dy orho noD.ni of tiD., Oc cqq|oi.t @rdin c! of rh. |m bodiB (a.,4)!nd

(a..t,) cd bc oh.ircd. Vhcdd. rhc nght aeBion k $c djspt@mcnr of$. objer

fbm wdal .qdnox .long lh. cctcsli.t cqu|or (in $. q. qudtut s I ). md d js $c

delin.rion or rhe displ.ccftnl of$. obj.cl fbm dc pt.E or ue.qullor t c.l lour

Aigl. tf is rh.i oblaircd fon Oc dilT.Ene of thc leEt Sid.c.t Tin. (r_SD and rh€

ishr e.6iotr. This liMuy gies ihc leat horianr.t c@diiat s, dimuft (,r). rhc

displ&.md ol rhe objer fbi rhc diEcdon of lh. Nonh rowrds E 5r ,tong $.

hodenlal in l@l sty .nd rlE ald|ud. (r), rh. hciehr or rh. objer lbov€ hori&o. Aftcr

.djustus for $. cf@rioi ed $. h.ishr of rh. obencr's to..rion .bovc s.. trycl rhc

ropo..nric c@rdiiaks (,4., rr ).nd (,{, .}. ) of lhe M@n and de sun, 6!..riv€ty aE

lo almosr all rhc nod.ls fo..&lid hoo!-nddng, rh. mj.nt s rcll .s lh.

mod.m, th. difieE@ ofeinurlrs (D,,tz = l!r. - /r.l. can.d rlodvc szimurh) .id thrt ot

rltiluda (ARcv - ,- -/'., c.llcd e of vision) $ rhoM in rhc fig 1t.2, at th. rin. of

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

show rhc *pa{ion bctwn ln. Sun lnd rh. M@ that is tsM !s 0E .rc of tilnl

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

lor €nnien visibility of luE d.gnr r€auircr ro aat it|ro @mid.6iion a @dba of

other pm$€rd. On. e.h p@n€tq t ihe As. ofthe Moon (AcE) = Tt To d.ffned

s th€ tine elaps.d sine th€ la.t onjuncli@ rill rh. doc of(rlE sn!.r or tlE any orhd

rclevut tioe) obsvltion Arbrhe inponDt lir.ror is dE Widft of Cres.r (w) thti

dep€Ids on rhc digre of$e Moor. As dE din@ of 6. Moon frm rhe €2trt veies

6on &@.d 0.14 nillion km to a@nd 0.4 nillid tD ln. Fni{i.rr€.d of rhe luw

disc vri6 liom I t e ninutF io | 65 e minur6.

Fig I L2

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

would b€ wid.g lnd th6 bndttrd Winh of tlE qtsfl i! di.6tly prcFrlo@t lo rhc

Ph.F (P) of th. M@, thll is r tunqio of dE ARCL. r\ @mpla. ltul of all rh.$

T.

2

2.

3

T.

t-

T-. T.

5.

6,

1.

E.

10.

B€dTimeofvhibilily Tr

Age oflhe M@netTr AC€

Arc'ofvision

Relative AziDurh

ARCV

DAZ

Arcorlight(Elo!8afion) ARCL

Pbseofcqc.n P

Widlhofcrcs@nl w

A conputarioml stnlegy may be lo stad b, deternining $e tim. of Bidh of lhe

New Moon or conjlnctio. or th. Moon *ith the snn. For .ny plee on the slobe

deremirc rh. dm€ ot suner |har follos rhe timc of bidn of ncw M@n. For this monenr

conpute thc e€@entic .cliplic coorditulcs of both lte Sun aod $. Moon. Tlmfom

the* coodinat.s to lhe lGal horizonbl c@rdinates. Tbereby coFPute tnc LAC,lhe Age,

ARCL, ARCV. DAZ and the width W. The ddoih of all $6e computations shau bc

t.2

SIGNIFICANCE OF GEOGRAPHICAL LOCATION

Fo! $c pioblem of rhc .dli.st lisibility or the Ncw Cre$enr Moon $e

orientatiob ofthe paths of the Sun and the Moon Eladve lo mch othcr dM tlarile to lhe

hoizon h significdnl. They chatrgc scason lo seaon as \tll as year lo ,car a.d olso

depeod on thc latitudc of lhe place. Due 1o fte axial rctation of lhc lanh cvcrv object in

lhe sly appsr 10 hvel along a cidular palh (The Diuftal Patb) ext nding frcm ealcm

horimn ro thc $'eslm borircn. Thn parh lics in a plae pellcl lo th€ Plane of thc

equalo. Tlc obj*ts in ou sty sbos. delimtion is coNLnl (sld and other exd_$lar

sysreh objccq slw.ys remd. in a 6xed smallcircle in our 3ky with Norlh Ccleslial Polc

being ren Pole i.e. rheit diurnal palhs cE no( only 6xed bul lic in plan.s parallel to each

olh€r. Thc Plan. of the orbn of lhc ednh eund lhe Sun h inclined to lhe pla.e of lhe

Equalor .l s .ngl€ of eosd 21tr 26' 26" lhcefor tne deliMiion ot lhe Sun in ou tkv

vdi.s frch 230 26' 26" $nth ro 230 26' 26 nonh oY€r a y6. Thc declimtion ofrhc

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

snrll circl. tl[l h.rc th.i poLs tl rlE Nodt cdcdi.l Pol6 i.. rlE dil4l plrlr3 ofrhc

Su .d th. MM do nd lie ii p||G F .lld (o tlE pLG of dE di!.nrt prtht ot d6

Fo. ple3 on E!n[ wirh Lrtu L. 8ral6 (t.1 660 ]a' (Mln q ort) rlE! e d.yr

dunls .sy rd, *ha Son tqn iB b.low th. hdid dl dry (d ltoe lh. horian dl

da!). Sinilllr forplrc6wirh ld' h.gldrlbo6f 25'drhdorl! dE .I!d.ys

dun.8 .tqy luu rEtl etE d! Moon n bdor rL hdizoi dl &y (or.borc tt&

Ap.n fion th. plr6 chc to dr Gqld bd! tt Su!d thc Mmmy16€l

v6y clole to tE ho.izon. Fo. pb€. d$. lo tlE audor ilE Cclcs'rl Equnor p6s€s

doe lo dE anh !d tlElfd! ti. pdnr of ft S{n.!d dE Moon |!mi. hiSh in $c

dq. Bu in ph6 wnh higls hd L. ih. Cd.lrid Equdd i! clos lo th. Hdid o

th.l th. p.tlB of tlE S!..!d tlE M6 my b. d@ d cq bclow th. horizon. Th.

68|le @. 1.2.1 CFs lh. driw dirrdd of dE cliDlic.td cd.ri.l .quror ia

6np.rid lo tlE hdid for. lLe rin hrgl |tin|d. d dr dm of hol o&r !r

!d!d rsd .quimx (6liFic in ttl.) .n rorld aodd .qlircx (6liFi. in pinl) .

N

\l.rsllldib

1l'

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

Thi! figue clorly 3howr th.! for hid ldiod.! vhcr d'c dccli.dion oflhc Sun k

eulh (wirt r in th. mnltfr h.hirph6c bctBa Sumd $lnie rld th€ Auruhtul

Equinox or ba96 &tunn l .quiu !d thc wint6 5lni@) tho p.th of rh. Su and

@nt qu.dly rhd of ih. M@n |!Dir vrry okr!. ro th. hori4n. Ir rn F @nditi@3 th.

w M@. .8q @nju.rion .nlE tlDitr h.low hqid or Ery cloF ro tlE ho.izon

matina it inposibl. to s da .nq lF d ilre d.y! nm djuncrio. Th. iu.rion

b.con6 @e ifdldng lhb p.n ofdE yd th..L.li.dioo oftlE M@. is oti of0a

&.linrlion of th. Su4 p|nia,brly clos to AduFnd Equircx (@nd S.prcmbd.nd

tuob6} B.rsq vqol .quitux !d dF SUDE Solricc (t@d ,uc) th.dlipric is

rehtircly high .trd th. p.O3 oflh. S{n .d th. Md e higls 10. Dtina it asid to

s rel.dv.ly tqDgq cGsnB Th. enl.rid n aG!.d gedly for th $urn@

Hdnd

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

Sihihrly $. fg@ m. 1.2.2 dbE th. ddiE dndllod ofdc ..liptic !d

@lqlirl equ|tioi ii oonpcilor to thc hodzon for i pLc. with low hritude ai {@rd

wrul e{ui@r (eliplic in ligln blw) .rd |@rd auMMl .qui@x (alipiic in ligln

pirl). ftir fise d$ 3ho* thd clo* io @tunr.l .quiDx t|| PrlB ofrh. Sun ed rh.

M@n r. rchrivcly cloF ro th. hodan.rd if !h. D.clintti@ ofthc M@. is slh of

rnll ofthc Sun tn. drdhid & nd !6y lpod fd liSltilg of ! tl.riv.ly y@ng d*.nl.

\r'

Tbh is lhe hain rcason bcbind @nsidedng Age of Moon s not a vcry good indicabr for

dt visibilily of cF56t. In sprinS.nd Mm€r vcry t$utrg.ge cltg cd h. sn lnd

dudng th. {ullmn and *inle6 very old clescent may .scape sighling in th€ holthern

hemisphed. Thc situalion is rcveB.d s6odlly for de sud@ hmisphec,

whcn lhe M@n is wsl of fte sun (el6rial loncnudc or rh. M@n is noc rh&

$ar ofdre Sun) i. our sky it is Old Moon carching up wilh rhe Sun. The Old Cre$enr can

b. en in lhe mominss befoE lhc suriF. An r $e binh of Ncw Mmn (c.leslial

longilude of rhe Moon b*omes jusl eratd lhe rbat of lhe sun) rhe Mooi Eehes e6r

oltheSun6dcan no$ b. *njustaftssun*t. How.vn dcpending on ih. d{limrioB

olrhe Sun and the Moon and lhe localion of$e observer $e new cBcent My et well

bcfoE thc sunst in which cG it is iDFssible lo * th€ ncr crsceft Th8 the fisl and

rhc Do$ nnporLnr clilerion aor lhe visihility of the .ev.$ccm Moon is ftar the Moon

*$ an€r $c sunscl. Alknately $c cnbrio. for thc visihilily of thc l5l cd$ent is dDt

rhc Moon riss before lhe suni*.

Whcn thc cresent is very close ro th€ sun il rcmains invisiblc dE to the facl tbar

$c ohospherc close to lhe sun remains highlt illnminated evcn afi( $e sunser'

Thqefoe fi.rc n6 to be a frininum lhrcshold epaEtion or Elongarion between fic Sln

,nd rhe Moon b.loN vhich ihe crecent cannol be sen. Thh minimud o. lhrshold

eloneation varies hotuh to oond 6 fie djsrance bctween lhc Earlh and $e Moon keeps

valyine. Vhcnclossl (at apoe€e) $e Moon h tround 350000 kn lrom thc Eanh ind !l

lhe fdhat (perig€) il k aound 400000 lo. Thus a Ne\ Mon ar apogec mat be *cn

whennscbnsation fod rhc Suh is snallrnd a New Moon at periSce may nol b€ s.cn ar

nuch ld8cr elongaton. Thi. is du. ro the fst that closer is thc Mmn the lars€r dd

brigbler ir appead in oui sky ahd wheo il k farrher il appeaB saller and less biShl at

rhc se elongalion. Thts .nother inpona crir.rion is lhar $e New Mmn cm be *en

for a combinorion ofcenain optinlm valu.s ofelong ion and lhe distanccof$e Moon

fom fi. E.nh. thi! conbiMlion Esults into tlE op(nm valu€ of thc widlh orcFv€nl

A cFsenr wi$ small €longation $al appea6 lare.t md briSller al apogee may be seen

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

RULTS OF THUMB: THE ANCIENT & TtII Mf,Dtt:vAL:

'rhe edli€st rcferehc. for ey rireno. for lhc visibitir! ofncw luntr cresccnt is

annburcd ro rhe BabJ,lonians (Foueringhm, r9r0. Aruin. 1977. Schaete!. lgttsa. yas.

I 994. Faloohi er a1. I s99 elc.). Most oflhe cxplom a&ibute ttE tollowinS rutc of $unb

'Thc ,.n lunar .rescent i s.ek \ en its aAQ

Nons4 loas 18 hihnles h.hi4d th. tns.!

nnrc thdn 21 ho^ un1 L

h na3 bem pointcd oul ihar th. ,crual Dabytonu (lcnon

ephislicarcd ar coopEEd ro lhk siopl. tule (F.r@hi €l ar 1999)_ ft is eilhs ou, trck ot

knostcdse oa lhen era or the nissing hjsbncd rccords $ar h6 rcsrid€d our

compchos'on of lheir effons. Autcnric Ecods of sighljne of new cEscent s young as

5 hours durirs Babylo.id c€ qisls (I6loohi el ai 1999. Andcrtic & Fih€is. 2006) in

lit 6turc. Sinilady. in modcm time ftsc.nc lagsjnS o.tyjr nrnucs behind $e su.*l

harc bccn *m wi(h aec tcaslhan 20 n6uB. Thc dala ser of46j obsetuations considcar

duflnC lhis $ork (b b. di*lssd in ddait in chaFcd j dd 4) inctldcs jt cascs whcn rhc

crcscd $nh l, C tcss rhan 48 Dinutes w.s cpon€d to h.vc bNn *en wilhour any

oplicalaid. rvhcEas.26 ca*s arc tncE in shich lhe age ofMoon ws lcs than 20 ho!6.

Rece rc-evatuation o|r(ords ofthe sigbdngs o|mscsrs in Babyton and Nineleh aho

sholv rhar cFscenh much younger rhd 20 hou* dd lho* lassrng bchind sunser muclr

l.ss rhln 40minures were sen (Andolic & Fimch.2006),

these rccenr compurarionat ef|ons for the recorded obseBarions oa lhc

Babylonian e6 ctearly indicarc lhar lhe lule of thmb asociared wnh rhn eD is an ovcr

simplificadon. lr has bcen eccndy ctaihed rh.t Babytonians had rodularcd I hrt,

nalhehalicdt tun& lheory which rhey uscd for prcdicring vartuus paramercBotth tunrr

l0

notion s iounded in thc lue €ph€mdjs ih.y pEpaed (Fat@hi d al, 1999). Ac@rding

lo lheF $udies it is pointed out lhal the noonset-su$et lag atone.ould nor have been

used as lhc visibilily dit.rion by rhe Babylo.ids. BabytoniaN sysren had $c tollowi.S

rlohedrior rL, - moon*r tasrme rin deCrees,,S) - conninr

In vdious $ud,cs lhe vrtue of$e conqant rs dedlce ftum | 7 degFes lo anyrhere

&ound 2l d.8R<. Hokercr on lhe basis ot rhc confimed 209 positile siehlings fmnl

,car .567to yetu .77 rhc \atue of rhc .o6ranr is deduced ru be 2l degrees (Far@hi cr at,

r999).

Sinilarly hosr of$c modcm aulhou arribule rules ot th€

ttpe lo fie Muslinrsand A6bsofmedieval tinrcj

''Th. t.latie attitLt! UnCy) > Ao ahd koans.t,ulet td|tin >.t2 nin cr_

lr has been indicared lhar Mrslins. reatization thar rhe Eddh-Moon dhlancc laries during

one comptet€ cyctc ot tlhar ph.ses and lhe hininuh moons€r lag lime considercd b,

Êbs vaiied fiom 42 ninures for Moon ar perigee.nd 48 ninurcs for Moon at aposee

Thonsh lhis -nrle of thunt. is no4 sophisricated 6 compNd ro $e one

art'buled ro |hc Babylonims sli jr does nor povidc rhe complere prcrup of rhe cffons ol

Arabs ad Mulin' h $c hedirwt ines exl€trir ur ofAolemaic sy{en ed rhe

spherical risonodeq devetop.d b, Ambs lead to the L!n& Rrp€.es flnclion (bat

depends on locatbritud. ofrheptace ofobrNadon dd lbe c.testiat to,gilude ofthe srn

and lhe Mooh) 6 eartr as lO cenrury AD (Bruin t97?)

Babylonie crirerion h indeed lery sinplc for praclcar purPoses and is supposed

to have ben cmpirical in

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)

hints towards thc idponance of$c Vidrh oflhc luM crc$ent (Bruii, 1977) Thus an

elabootc stsreo of calcularioc involvcd in delemining the tinc of €licsr visibilitt

appeds to have developed only abund AD 500. Morcexplicirnentiohsofrhcsederaited

calcularions are loud ar vuios plaes in the dly Islmic litenle (Brui., 197?).

Ohe ol thc ealiesl Muslin Aircnomcr eho developed lnc lables fo. se lidnC

the luna!crescenl s lisibilitywas Yaqub lbn Tdiq (Ke.nedy. 1968).lt has bcen eponcd

in $e litcraluE (Atuin, 1977) that lbn Taliq had r.cogniad the imponance oflhc Width

(w) oflhe crescenr. Thh not only shows rhat ar rhd rine rhe varying dislance ot rhc

Moon had been rcaliud bul it aho mdc ir possible 10 inpov€ upon dc ncs cr*cnl

vnibilily cinerion. Bruin hd Eponed th Al-Ailuni had $e Falizario. oflhe tong ind

dillicult calcularions inlolved in lhe dercnnimlion tor the nev lun,r aesccm vnibitiry

lnd in hG Crloro.,os " reconrmcnded rhc work ofMuhanm.d lbn crbn At B,llani

l ltnlhook oJAtto"o,'t t?rstard to Lario by Nalino, l9|]3).

'lhe sihplc *nclioh for cadiesr vhibiliry oflunarcrescchr evotved sjnce lhe riftes

of B6btlonids qas passd onb rnc Muslims drcueh Hindus wirh !e,)-' tinle

inprclemenr. Molilated by the euEnic iijuhctions a.d rhe salnlgs of rn prollrr

Muhamnad (PBUH) rh. Fobtem ofearliest sighring of hnar cE*enl \s lhomnehl).

Invesriglredby rlrc earlJ Mrslih aslronomcGotsLhto l0'tccnluryAt).

OD the bash ofreatiarion of the idjponance ofwid$ various Arab asllonomcrs

conclndcd td lor celics( vjsibilil, ofcrc$.nl thc minimun equabrial $pararion offie

Sun lnd the Moon varics from t00 whcn lhe crescent is widesr up to l2i whcn llr

cNcc.l is n@$cst Such der,il.d calculariotr \.rc wor\cd oul.s earty as 9ri cenrur)

AD by Muslinr asrrononers as$ciated wirh rhc Ab&sid coun of Al,Mahun. Fronl

monesr the* {stbnoncf Al-BaIsli kncw lh,r ihc cnleria that.gc of n@, shoutd b€

moe tnan 24 hou6 (or arc sepdarion belveeo $e Sun sd lhe Mooh) isa good sldtine

poinr but i A only m appnrnarion. H. b€t,.vcd LlEr lhc dcienl6hnomc6 d,d nor

understohd the ph€nonenon conpteretr, According to Bruih, AtDatani,s compuradonal

sork is o very elabo6re sysr€m of mrhehari€t catculalions (Bruin, | 977).

t2

'Ihis work is nor intcndcd lo explor lhc hisbry oa Aslonon! rulatn lo thc

canie$ sigltingoflhe new lunarcresccnt. Tbe Babylonia. and the nlcdielalcilons shall

nol bc dplocd frem a hi$orical FBpecrivc, wc shall rcstricl our explodlion onl) on

rlR codprison of $cse clTons sith $os ofthe moden ons. llowcvq $c mcdiclal

narh.nalical ide6 shall b. erploEd in morc dcoil in chaprc. :1.

I.,1 CALf,NDARS AND THE CELESTIAL CYCLES

Dogeel defines calendar 6 '! syslcm ol orgaizine ,ir4 tbr fi. pupos. of

@lonins im ow cxtend.d p€riodt (Doge( 1992). tl is a schcmc lor keping an

.ccounr ol da!i.'wceks . monrh. . 'yc!6 - ceolunes and "nillennia . Thc bsic

notion bchind I calendar isor8anizingr,la.olroairiconrinuous now rhal isindcp(ndcnr

ol any ehcn. of irs orsaniztion. Ahongst lhe divisions of timc in dayi. w.chs dc

eme aR dtccrly 4sociared snh $c celestial c_aclcs. the D,"r,at, \h. Ann@l ^d

rhc

/,rur. A @nplele daily rcvolurion of lhe s\t. de DildEl norion. is Efertsd b 4 a day.

lechnicallt an apparcnr r,/df.t,' h the inlcpal bcrseen rwo successivc rransiN oflhc

Sun ar any pllcc. Du. lo thc morion of rhc Eanh mund $e Sun rhe sky appc.m ro

rclolvc round tbe Eanh very slowly (lss rban a dcste€ pn da!) and one contterc

Evolurion of sky i. rhis way is Ffmd lo 6 o ,.ar. I {hni€lr! a t opictt y.ar k trc

time inleNal bctweh lwo successivc pseges oflh. Su rluough vcdat eqlinox. one oa

r\e porn6 ol inFMcrion ol rhc celB(dtequarorand rhe actiptic

Each calendar is dilided i.to yea6. yeas inb nonths. non$s inlo vNks ed

dals Most oflh€ know calmdm rha( mcn dcviFd bd sevcn da$ in a rvak. tn

diflaent Esions md cras 4 to lodot weks halc aho bccn considercd. ltowclcr rhc

nuobcr of days in a dond hd EDained variabl. in diltcc.i calcndars md wi$in a

paniculd calcndar. Tlr scbeme. il$ce isany. ofdift€!€nl DUmb$ofdays in a monlh of

a calendar is ba$d or rnc typ. ol cal4dar, Most of lhe knom atcndars.E clarsificd

iflo dtree major typcs. Solar Calcndm. Sriclly L@r C.tcnda6 and Luni-Solar

Calenda6. A blief description is Evi.$€d in rhe foltowinel

1l

Solsr Cal.nd6 m b4d on the ulrl motion of lhe F-!nh @und the Su..

''ycai in such calendds h ihc "Trcpial Yee, defined abolc. Thon the lcng$ ota

'tropical y€&" (Dogger, 1992) b givcn by:

365.2421396698-0,0000061s359r-7,29xr0_r0Ir +2,64x10'07r (1,4,1)

7 s $e tinc in Julian cenruics sincc fic epoch J2000.0 gilcn by:

T-(JD-245rt4t0!t6525 (1.4.2)

vherc JD is rhe Julid Dat€ which is th. dm€ in .mber ofdavs ctaD*d sine heo n@n

at Genwich on Janua.y l. 4712 in Julid Cal6d{. Cwnlly thc loglh of t@pi€l ycar

n 165.2421898 days or 365 days.5 hos.48 ninurcs ed 45.2 sends. As it is nor a whote

number on yer of a eld calodar @rsists of cirher_ 365 days ot t66. tn old Jution

crlendr cvery fourih yc& @.lained 166 days (i.e_ a teap yed) a otbq yff conhincd

165 days In lnc cuftntly us.d Cregoiian calendr! the leap ycd rutc is nodificd. A

century ycd y 0ike 1700, 1800 elc) whict is a lop yer i. Juli& calmdtr bur is nol

complerely divisible by 400 is nor a tep ycar in OEsorian mtcndar. Thus in crer,400

yeas theE ae I 00 l@p y€6 in Jnlitu Cal€rdff wiced in cESori& calqda, rb@ are

only 97. The s€etu od nfuy o1her natMl pbsomcna toltov lnis sotar cyclc. sp.cialty

haBcsing rides, lhe length ofdays,lbc dmes of imir, suser, sunds. etc. I$e JLrtian

calendar B insrilured on Jdu.ry l. 45 Bc by Juli$ caM. wirh rhc h€lp of

Alciandlie stronon€r Sosigm6 &d w r nodificadon ofdc Roman Republidn and

the ancicnr Egyptian calendaB (Michels, 196?). WtEcd lhe crcso.ian catendar was

n(csibrcd du. to rhc facl ln.r dqinS oe od . lDlf nil.mia lhe Julim otcndar w6

displaed from the ssonal vsiatioos by s much s l0 days. Thus pope Crcsory X t

corclnukd a conmission in l6rt €qrury AD for lh. csl€ndar Efoms. The dain anrhor da

dE nN sysrem w6 slononer Aloysiu Litju of Naplcs (Coync el ,t_ 1981. Dull4

1988, Moyer. 1982 ed Micb.k 196?). When it ws implcn.nted ofljcialy. rhe ddre

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

kecp accoul of thh skipping of days in r]te solar calendar. Difercnl counrries, culluics

and Fligiou omhMiiics adaprd to rhis DodilicElior ar diff@.1 !inc. Ir is lheEtorc a

high hrk for hisrorifls ro tccp racl of r]E apprepriare dalcs.

,n every solr qlendr thft @ iwctle mo.lbs. For simple dithmcric Ea$ns

theE colld hdve been seven months of 30 dars and fi ve of I I days (in a nomal ye.r and

six ofl0 days plw 3ix or 3l days in a lcap re&). How;ver in pncrice sincc rh€ ime of

CEek it w6s knoM lhar d€ wi.r6 h.lfofa nodat ,€u h6 l8l days whee6 rhc

summ.rhalf@ ains I84 days. The it&n for ih. sue is lie faler molio. of thc Eznh

rhc Frihelion. thc Eanh clossr ro $; Su dar oeun on aound January 4,i. Srdnins

with January cvery ancinale nonlh is of3t days tiu Juty. rcbru.ry h of28 days (o.29

days for a le.p yctu) sd lhe Br G ofto days.&h. The.ltdar€ nonrLs ofjl days

dd l0 dals omintB $.d frcm AuSul ro DMnber

ln a Strictly Lunar cat€ndd, lhar is ba$d on $e luharion pcriod in one year lhcle

arc eilher 354 days or 155 days. Tbe clrcnr ave!.8. of$e lunation period h 29.510589

days or 29 days, 12 hou6. 44 ninui4 dd 2.9 $con.l$(A$nrcmical Atnaiac. 2007).

HoEEr ihis aEase is.hrging, Aaodin8 ro rh. tun5. lltery of Chaprcnl-Tou&, dd

chapont lh.e vdiadons !E eoukd for by the tolowing cxpEssion (Chdpon!

'lbur'ind Chapro , 1988):

20.5305888511 + 0.o0oo0o2t62l, a_ I.o4! IO.

o ! a. Da)5 ( r.4 r)

whce T n givcn b, ( L4.2). An, rrEdicutu phe cycl. my v.ry fmn te nes hv uD to

seven hom. Tnus rhis priod vtuj.s froh !romd 29.2 days ro mor lnan 29.3 days fom

nontn lo nonlh. Tnerefore in aU Iutu calend&s rhe numbe! ofdays in a nomh is eithcr

29 or 10. l. Arithmetic Lunar calcndq rheE $ lllenate hontns of29 and 30 davs. tn

obiwrional lue Rt.ndd $.8 6 bc s mey 4 rhE. rcns{ulirc mon$s o, 2,

days qch od d mdy s folr conKud@ momh5 of 30 days e.ch { oys, I 994). In an

l5

Arnhn.tic calcndar ther. are cilh.r 6 monlhs of 29 dals md 6 nonrhs of tO days. (a

notml ycar) or 5 momlB of 29 days ad 7 nonlhs of 30 da's (! lep yd).

Tnerc is no fixed nle for leap ycs in r obseNarional tunar calcndar. rhcrccdn

nol be anr. HoEver in lbe Aritineric L@d cat€tutd our ol lO yea6. cyclc t I yc6 aE

hap tcds (3Js dayt (ly.s 1994. R.igold & Dcrshowirz 20Ol_ Tsybutsty, t9Z9). l.hc

rule is rhal lhe tear nunbf /is a teap yer it

( l4 + ll.))modi0)< lt (t.4A)

Olhwisc the year is nor a teap year. In such an a.ihmelic lunar catcndor alt rhe odd

.umbeEd mon$s conlain 30 days and lhe.vd nhb.ed nonibs conhin 29 days cach.

h re lcip yedr a day is added ro $e Meltih month. in gmeFl. a srrictl, tune oalendar

ldvanc$ by I I days againj lhe solar cat.ndd. Th€refoF rhe $asons aod a orhcr

pncnom€na rhzr dcpcnd on Ihe eld cycle {to ml follow a slmcfiy lu,ar catddsr Ttc

monlb numbq 9 (R nad&) in lhe Isl6mic qtods my f.ll in wi.l{. slmmd. aurmn

Thc Luni-Solar @tfldaB d b.sielly lufu bul ro kep r&k of$e *a.ons in

prrce of adding sin8le days in a lqp ye& a whol€ monfi k addcd (inrqcshlion) lo

followthe $ld! cyclc, The Hebrew dd the Hindn qlendaG fa into thG ctass(Rcideol.t

& De6howir2 2001 . Bushwic[ 1989, ScEIt & Ditshir 1896. at_Bituni r 000. at_Btuni

l0l0).In ce ofHindu calcnds (th.y have bo$ $e $lar tud rhe luni-sotar calcndaB) a

lunarmonrh is imclcalatcdiwhcneve,irnrsinroEconpr€resotarmoilh.tncascotrhc

HcbFw l!ni,$lr cole.dar d addniomt honrh of l0 da)6 rs mrcEararcd betorc rhc

usurl 12" nonlh oflbe yce. The En ofihe dd.its of th6c calodd i, morc of$c

soc'al ahd reliSious nlrurc aod is nor in tine with tbeprcsnt*o,k.

Calend.ical cab'narioB for cacn calhdlr ha€ 6eir oM $phkhadons_ bul

bcins phcnoncnoloSicat rtE obsctualionat lunr calend., is mosr chaltensins. A tunnr

month bas lo beein wib the actuat sighling of lbe new luns cresce d.d tne conditions

I6

ofits si8hling greatly vary not only longiludinally on rhc globc bd de[Md o. $e tarnude

ofplaces. ThB an obedarioMl lunar c.lendar may vsry along the se logirude. Apart

iron calenddcal aspects $c pobl.m ofsighting v.ry yolne cGsc€nt is on€ of lhe mosr

excitins fld challenging obseFations for both $c adareur md rhe pma€$ioml

asionofre6 Besides, thc pcdicion of fic vkibilily of a paniculer .erv cEsenr ar a

panicular placc is a long and inter.sling cohpuhrional exerci$ Thc sane was clliad

as early as rhe dedieul times by the grelr ostonomcs Al-Khwarizni, Al Batani. At

Farghani etc (Bruin 1977) Mocovcr. t|t prediclion foL nak€d eye obseivarion ha lwo

noE cohplicated hsues under srutiny rh.laresr esearch. Onc oathese is biologicat.

lhc abililt ofhunan eye to conrfa$ $e dimly illunrinatcd ciesceht in rbe bnght rsilighl.

The olher aspcct h ol physicrl n4lue ofalfiosph.ric conditions ftar c.n b.dtr aoecr rhe

risibilit! and rc contdsr h this \o.k c l'hdk h morc on thc $lrcnoNical asp..rs of

lhe eadiesl siebdrg olthe new luna!.rr$enl dnd ahosphcric €ondnion !.e onb plnlr

Opinions aro diided d tq thc oriSin of$c.llton olcomlins yer6 in an! tofrr

in rhe ARbian l,cninsula. Accordihg to At ttazsi il $anln as soon as $c cni rcnof

Prcph€l Adan nuliplicrl and spRod aouid rhc rw d (Ro$nlhal l9i2. l_mqi t979) A

calehdar sE originated wncn rhe Himyarires adoptcd one sjtn m epo.h lhal marked rhe

b.ginni.B or the eigns ol Tubbn. Cenerally it is b€licrcd thal lhc pnctice of 12 lunar

nontns lo a year cxisbd in pre,lslahic Ar.b c.lend.rs since rhe rime ofconsrruclion of

Kaba b) lhe prophd Abnham a.d conrinued in Islom (llyos. 1994). The namcs ol$c

months and thcii sequenc€ Mre thc snc as tho* used in lhc cuiicm tslamic lunar

calcndar Ibllowed by morc rhah on.lifrh of$e roralFopulation ofrhc world.

Fom historic pcspecrire lhc inpon.nce ofrhe lunarcalcnda in lbe pE-lslamic

AEbia $€s the pilsrimage ro Kadba (Haj) rhar talls in $e monrh orzul-hajjah, rhc l2'"

nrnlh oflhe hlmic and prelslamk lunai calendar, Althoush this event was a puelt

religious evenl. il was also inportlDt ior kadc and busincss wnb b$ ol goods

exchanging bands. ll w6 this econonic rcrivil, $ar w6s badly anected as rhe lunarrear

ad!dced llxoush *&ns. PEcufnent ofcop and rhe ovailabilir) or.ac f.,ul anim...

l1

8@rly vdied s6r lo s@n, 'fte t€en tha1 $e irbEl.tion s6 inl,ldud in de

Arabim Peninsula w$ this .cooonic drivity rarh.r than dy sronomical t.son.

''QalM" a nativ€ of Mecca is Fpuled to be the fiIsl petson assisned lo derrmine thc

daca for thc coming yem pilgrinage and wheth.i $e inmalarion w6 drc or nol

(Hahim, 198?, Ahmed, l99l).

Sincc lhe early days of lh. inception of lund cakndd in the Arabim P€ninsula

four nonths including lhe Zil Hljiah wE o6i&r.d scEd md \@ {!e pDhibited

dudnglhesc sacrcd non$s. The custon cari.d over to the post lslamic.ra in lhe lsladic

cultu€. As with the Ronan ql.ndd thc i eEalalion ws abu*d i. ABbia io ordq lo

change rh. saqed nonfis inro noh{acred moorbs dd vicc-vc& Al thc eme rihe rhe

lunarcalendt!us€d inMadi@rendined in itsoriginal lzmonthsay€arford.

Muslihs follo{ed rhe calendar ot Mecca in the bceiNling. Rut afier the l,lophcr

Muhonmad migdred to Madin. alo.8 wnh his compdions. Mustims adoFcd lh.

cal.ndar usd in Madina. Aner $. conqu$tofMeccaby propbel Muhlmmad, Muslims

conlinued 1o ue rhe calcnde of Madim bul rhe catcndar ofMecca @ in pmllcl. Wi$

the lar pilgrimag€ ro Meca of Propher Mubmnad in rhe lo,r ye& aner miSr.rion ro

Modina (AD 612) fequenlly abused pEctice of inicrcalation \m abolished thbush a

QuEnic injuiction. Th. pEcrie ofnadin8 a lsd nonrh s.ith $e finl siebdne of nerv

clesccnt MooD was iurhenticated by Qurtuic injuncrion and tne gayine of l,rcpher

Muhmad, sith paniculd empbdis on b€Simine dd rhe cnding of lh€ nonth of

fasting md the monrh of lhe pilgdmase.

With adoption of puel! lunar cat.nd& wirh a luar nonrh bcginninS $ith $e

fist siehdns ofrhe ncw IuMr crcscehr lhrcush euMic injuncrion dd lhe eyinSs or

Pophet Muh.mmad |he .rolution of Isl.mic lunar catedar b.gd As for my calendar

one r€qutr.s a staning point of rine Gpoch) or bceinnins of m en, tor couring yea6,

Oe p€ople of MadiM ac h€li.v.d ro hoK u*d d.poch s ene dn6 ! mooin or tm

aftcr Prcphet Muhdnad mignicd to Madina in AD 6t2 (Iys, 1994). However ! hore

wid€ly acepGd lime ofomcial adoptjon of Ht6 s lhe bceinning ofrhc Ist@ic cR n

t8

AD 617 during $c caliph.t of Umd bin Khafiab. Whltcvcr h. thc 1in. of adoplion, lhe

lslMh cal.ndrr. or dc Hiji c.l.nd& s@ wi$ Fnday l6ri ruly 622 AD on Julitu

calends which Moding b didft{c luMr or krdrhicslddd is l" diy ofMuha]m

(1" nonth ofe klmic y.ar) ofth. tsleic I (R.in8old & ftrshowii2, 2001). The

o,ficial dar. or adoprion oflhi!.h &d cal.rdd '@ b l'' Muhtu Il AH (and HijB)

Simpl. 34heh.s or 4rolh odd appoxidlions b$cd on long Lm 3vs.gs h.d

b€.n dcvi*d to pcparc long r.d cel.nd& in vid ol incrconv.uion ol lslmic drGs

&d tne d!t6 in oth.r lei-$lo ad el& olcndr d.r.s.

blanic legd syst h cgllcd Shdia , is lh. sou@ of lsldic tn+kcping ststn.

whil. le8.l pc.pls .rc efeglnd.d by seking sisbne fom $. *i.nlilic kno*l.dge.

Till th. rin. of sking of B.ddad by Hrl.su Xhu in AD 1258, Isluic law had

cvolv.d .ldr guidclires ao. elcndriol oNidcFlio.s (llya 1994). The olcndric.l

sddelircs ftlv.d undcr thc Islmic lN qn bc outlin d a follows:

i)

it

iit

l-.nsth of a lunar month is cnhe. 29 days or l0 days.

L.nA$ ol a lumr tlar is .ith.r 154 dlts or 155 dayr.

Tn rc can be . mdimuh of 4 cor*cutiE monrh. of l0 datr @h or 3

conscutive nomhs of29 &ys each.

E4h new nonth b.sins with fiBt sightns ot ncw lu@ cGccir ovd the

k$em hori@n .n.r ft. l@l ss.i,

v) Ailcmptr rhould bc nad. on 29u of ea.h honrh for sighring of ncw crcscnr.

lf il is nol scn on the 294 duc lo any l€en (stmnohic.l condirions or

w.thd connEints) thc noi$ should bc complckd a of l0 days.

l9

visul sighring epo.t must b. suppon€d lhrcugh a wnness rcpon.

The p.rson involvql in rcponing musl be eliable. adull, lruthful, sme

virh good cyesielt !f it i! preved dat the Fen pioviding wihess

puDo*ly nisled the pe6on ousl b. punished.

!iii) Thelisual sighling Epon should nor coniict vilh basic scientific knowledgc.

'lesling ofcvidcne ot sighting on scienlific srcunds

includc 0hcckine oflhu

shapc of crcscen! ils inclinalion. Poshion in sky. altitudc. imc ofoberyalion

dd sky condnions.

Sighting should be caried our in sh o.8ditd way for €dch moflh.

Accunula(ion of emB @orriculally in view ofconsideling ! toonth ro bc ol

io days due lo invisibilrty of ih. adcent of i,te 29'" ofoiscutire nonlht

bd ro be avoidcn *hodd thc ns c6c.nl is sighted oo 2Etr ofa nonth. li

such ascs con crions ec nadc b bcgituing of thar mo lr

As the klamic lav and lhe Qudih injunctions depend hcovily on tbe fiNl

sishling of new lunar cr€scent the eE ly hlanic slale plrced speial enphsis on $e

Bsdch i. the lield of Asmhony. CoNcque ly €nomous onlribution! $n mde in

$c developmcnt of $i€n€e of fie @lid visibility of new crcsnt dd pEdicrion

CONTRIBUTION OF 2OIH CENTURY ASTRONOMERS

Tne nod.h dcvclopmenl of lhe sciencc of erliesl noon_siehinS bc8iN vnh fic

obFMtio.al work oa Scbnid vho Bordcd a ldge nmber of.cw and old cescmB

fon Arhens io th.lal{ h.lfof lhe l9'i c.ntury. On rne basis of rhis obsdarioml dalt

Fo$qinglm ([olhcinskm, l9l0) and Maunder (Mabd.r, l9ll) dweloped the

obscralion.l oirdia of ediest vhibility of ns lmd crescent. A similtr work is

20

reponed in rhe Explanation Io the ln.lloh Attranonical EPle,cfir $al is basd oi

Schoch (Sch@b, l9l0). In allrhe* worts for dei€miriry lhe day ofrhe fi*r liribilir) ot

new cE$o! ARCV (arc ofvirion) b shoM to be a sMnd d.8Fe pohomial runcrion

of DAz (claive uinuth). The* edly €florc !rc only empnical in natuF N rhe critcria

dc!.loped de bsed on lluing rhe dau so rh8t mo$ ofthe obs.oaiions arc conshlcnl

TlE* nelhods do not hke into cosideElion lhe vidth ofcE$enl Btuii (Bruii,

1977) co.sidd rh€ ihpodahce oacE*cnt widtn fld his Dod.l desqibes ARCV (in

rems ofannud. of cr.scent above horizon plus tn€ solar depRssion belov lhe horizln)

os a funcion ofthe crescenr widtn. The Nodel ofBruio. ho\rever, kkes $e {idlh !s a

funcion ofih€ RCV ond DAz md a fircd Ddnh-Mmn dislaftc (in lemsolfixcd * i'

dirnrl'ler ol fte lu.ar disc). Hc sas ale thc fi6t \no consid.rcd phJsical .stkcs

dsociakd *nh $c problen like blightnds ofskt and thar oflhe M@n. Thus tu nrodcl

due lo Bruin is the lirsl drolelical nodelofr|t nodern timcs Using the nrodels tur thc

briShhess oalhe lull M@n as a fuhcrion of altilude (Befrporad, 190,r) lnd $c s[t

brighhe$ duri.g lNililhi (Kooncn ct al., 1952, Si€denbpl 1940) Bruin delclop lhc

visibililt cu^6 r€lnrin! the aldudc olcrc*€d sirh the slar dcpGsion. AlonS lhc$

ctrrvcs the bdehb€ss ofcrcscem (modellcd on $e basis of lirll Moo.) is al leasl ds much

os tlhl ofrhe twiliShl sky. He aho delclopcd cuNes sholvnu rel{lDn berwccn ARCV

and sl.t depesion rhll ldd prorcd to bc crucial for lunhcr nodellihg ol canics'

Yallop has edne lbr funho improvemenr ond consides cece *idth as a

funclion ofARCV. DAZ and lhe acllal scmi-dianeter ol lhe Moon (rhar depends on Lhc

Eanh-M@n disrucc) al $e time ofob6endion (Yallop. 1998). \vilh de!€lopinS a model

lor lhe besl time of yisibiht YaUop s nod.l al$ onsideF ARCV 6 a ihi.d d.gtce

polynomial tunction of $e actul cresc.nt oidrh ar the ben ime of visibihJ. Such

polynofrial is oblain.d by applying leasr square appbxination on a basic data sel This

blsic data s€l *qs d.duced bt Yallop fom lhe limiling lbibilil, .uvs of bnrin b!

slctins ARCV for a Sircn sidtn from tlr ninin@ oh lhe componding limilin8

2\

visibility curve. So fd th. mod€l due to Y.llop b lhe morl oulhfltic and dep€ndable

slronohi€l cri|€na for cxper.d ali.sl visibilily ol new l$m ccenl. The modct due

10 Yallop isa smi-empirical model based on $e lheor€dcal considedions of Bruin and

.ftiteriononthebasisofabusicd.tadeduccdfonBrui. svisibilitycurvcs.

Tnc mosr extctuive r.atm6t of rhe physicat dp<l3 eciared wit th. pDblm

is due lo SchNld (schacftr, 1988(a). I988(b)) who has considercd tlt Poblem of

bidtn€ss contEl isoroudt. He ale rcali2ca jnpoitance ofthe physiolosicol aspecrs

like $e abilirt oahunan.yc ro eoe th. limili4 co.kast. Soh&reas nodel is ! pdrel!

lhoerical n\odcl. Recntlt Odeb (2004) hls claim€d to harc dev€loPed a nc* ctnenon

for the.arli.sr vnibihr oineq luffi cle$ent bur his cinqion isjust ano$o lom ol

Olh.r authoE have contributed on cloted nsues (Ashbrcot, lt7!a. I97lb.

Cald$ell a|]d t-anct.2000. I0loohi c. al.. l99lt. lltas. l98l!. l98lb. l9li'll, l93lb

1985. 1988. 199,1r. 1994b, M.N5l1r_. 1983, Schaefler, 1988r. 1988b, 1996- Sch!{ffcr er'

!l 1991, Suhan.2005, Queshi& Khan.2005.2007 elc) ln fih \ork. €och ancior Md

nodciD halhc'hadcrl mod.l aor detmnrnB $e day ofltu fi6l siehing ofne* lunar

cEsce h cNploied A comparative study oi lhes nodch is canicd out and each

crncrion is rrlsfornrd in(o a one pdamcter visibilily crilcriotr like th.t duc lo Yall.p

Th€ resulrinS cllendaricdl imp licariohs .rc aho cxplorcd

lr hos been poinlcd our in lhe bLSinnine of $e chlpld thar the prublem ol

determinihg $e dat when the nes luar (esc€nl $onld bc \isible has its calcndarical

inplicarion. The earli$t siShtine of new luMr calcnde is ale a cballmging l.st lor bolh

maleuB and Dlolessional ashhoncB. In \ic$ olrhis rhc chafler hd niehli8hled:

Paranere6 on which visibiliry of new lunar cescent depends

Thc cfons ofrhe 6tro&meB iD lhc lncie.l and lhe n di€ul tlnes.

22

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

. txB c.l.ndar of $. Mulih! .spei.lly a lh.i. b.li.rts lticrty dphais. u

tltliarsidtingorrhetwloNcr$.a fois.niosaldcldilglun&ndrha

. A! rccou ofrh€ eilons ofrhe rsrorcmd oflhe oodc6 rim.s |o rd.lr.ss the

prcbhm ofderemining$e fiisl day ofsishdnsofnew Iund c'.eenr

Itr lh. b&ksound of tlE$ cfon! w tFrc qplor€d all lh. old ald thc modm

mrhodi Mthcm.dol dDdel or crirdion dEr r.quiG ro be $tisfi.d for 0E lirsr vGibility

ot ihc !w lu{ cEsdr, Duing lhi! qploGtion rn E$ltr of lh.$ nod.ls N

comFcd lnd nodifietios baw b..n 3uSgcrt d $,lEE fler posibl.. Funhd ft. mo$

authentic of the hodeh hav€ b€en uscd to .xolod fie ob€crvationil lunar cal.hdar

fouowEd in Pakhhn. A.sed o. the rcsulF ofthese exDloHions a luturc obervrtioml

lus calcndd for Patislm is anDut d,

23

Chapter No, 2

ASTRONOMICAL ALGORITHMS &

TECHNIQUES

For lhe determlnatlon ol th€ prsiso locallon ol the obJsts in th6 Solar

SFtem, panbulany the Sun .nd rh6 Moon, ih6 Fr€nch planetary theory VSOPaT,

(sretagnon & Francou, 1994) and th. lunrr rh€ory ELP-2000 (Chapront-Tou# .nd

Chapron!1943, €91) rr. well sulted. A number of sotlwaE hav6 ben develop€d

tor lho shulatlon of cel€stlal phenomona based on the$ th€o.16 and similar otner

works.In the cunanl studylhe same lh€ori.s havs b6.n u*d lo lollowthe pGitlons

otthesun and rhe Moon.

Moreoler, to conven the theorles Into computatlonal iechnlques,

malhematlcal lechnlques, tools and dlgo thms nave been dtloted. Moch ot the

computailon.l work is ba*d on the algorithms devoloped by Me€us (Me€us,199a)

but a subslantlal ahount or work on computatlonal aEonthms has been done

Ind€pendsnlly. For a ihorolgh und.Franding ol lhE compul.tlon.l toob the

problem of llme ls explor6d rnd discused in d6tatl. ln this 6Ebrd the @nblbution

of a numbd of authors has beon sludled in as much detail as ls required (Aokl et.

al., 19aa. Sorkoskl, 1944, Clomen@ 194a, 1957, de ,ag6€, and ,lappel (Eds.),

197l- Dl.k, 2ooo, E$€. and Parry, 1995, E3*n et, al., 195a, Gurnot dnd

S6ldelmann, 1944, Markowiu er. Er. 1954, Mullor 6nd rappel, 1977, t9unk and

lrlacoonald, 1975, Nel$n et, al., 2001, Newcomb, 1495, Sadler (Ed.), 1960,

seldehann rnd Fukqshlmai 1992, Spencer, 1954, Stephenson and Moi$n, 1944,

1995, Sleph6n$n, 1997, W€lls, 1963, otc.).

th. output ot ihesa €ftort! b a .omplter progr.m for .natysts ot ftst

vblblllly ol luh.r cr€scent named Httatol wrttten In C.t.ngs.ge dt$u$ed 6t the end

2.I INTRODUCTION

For rhc derernination of lhc visibilily mnditions otNew lund Cre$€hl (or the

old.st lutur cresccn0 over a tocal hori2on, lhe 6Bt rask h ro deremine rhe Unive6al

Tine (uT) and dai. ofrbe scocenhic cobjucrion of thc Moo. &d rhe sb or thc Binh

of New Mmn. In ils morion tuDnd rhe Eanh rhe Mmn tralcls doud 12 deg@s frcn

rv€sr ro c6t cvery day sd iakes ovr rhe Sun in doud every 29.5 dars on thc avcm8c.

When de Moon is vcry clo* lo ve$ ot (he Sun it aprears betore the sunrie ed vhen il

N elst oilhe Sui it appcmsjusr anq rhc sunset. The lunar crcscenr is vcly rarel) lkibtc

on rbedayofthe conjLncton. Dqjon Linn(Danjon 1932. t9l6) has b.cn i.$rpreted as

a limn on Ihe hininuo clongalion ofrhe visible lu@ cE*cnr Accordine lo rhis lihir

the ruMr ccsc€nr is nor visibb if lhc elonSadon is te$ lhd ? deSEes (Do88er &

Sch.efer 194. Schaefe! 1991, yallop 1998). Thc hdimM elonSadon ot $e Moon ar

lhe dtoe of$e Eeocentdc conjuncrion is same as the inctinarion otrhe lunar orbjt fbn

rhe plo. of ectipric (50 9). When rhe Moon lakes orr rh€ Sun ar ir ndimun

elongation $c ninimun time il lal€s b molc foo bein8 / fmm rhe $n (on the .6rem

side) ro be ?o again {on &e w6om side) is eoud th,ee quans ofa day. Th6 it is

theorcdcally possible lhrl rhe qesent is t6l sen on $e dly of conjunction or is tsr

seen on the day ofconjunction. TheoEricalty n is aho possibte rbar if$e crescenr n hsl

seno. the day ofconjuncrionad rhen rhe new cEscenr 6 seen on ln. da, iller, orthc

crekenr rs rasr srn on rbe day b€to,e rhe conjurcrion .nd !h?. rh. new c@dl n *en

on the day oflhe conjunction. In th6e cM rhe cre$e r.mDs hv$ible for.oneahalf'

day. Howevcr non€ of the$ lheoletical possibihia are rcatird in practice too

freque.dy. Mosrty rhe cEscent Enains inlisible for .two-sd.a-holt, days at least,

These condnions dcpehd on tc obse(ets t@a on.

25

O@ thc rim. of thc g.cdric biil of th. Ncw MFn is d.lmin d, dE rcxl

r.3l( ir b d.cdic dE locd circuhlt .c6 oflhc Su ud dE M@n tt dE tirc ofsu5.l

on lhc day of rh. conjwrid or r &, .ncr thc djulction (or d lh. lirc of swi$ on

lh. dly ol conjncdon or thc dly b.foE). Fd this i'lk ooc mEt d.r.miE th. l@l

tim6 of thc ru$t a.d lhc noon3ct In tr. mmins in otdcr to b. vkiblc th. Su rhould

tag b.hind thc Moon in odd rh|lth. @s i3 visiblc .d in thc dcnine! thc M@n

sholld bc hgAitrg b.hind rh. Su. Clsiotly thc LAC of $c M@n h!5 Em.in d on

ih!{ndr cosidcElion for lhc adi.sr vbibility of$. w lua ct6..nl. SiM thc

rim6 of th. Babyloni.ns thtuu8h ni.tdl6 !96 .rd ill tlE 20d e uy it hs b.qr

coNid.Ed I deisiv. aerd. Blbyloni@ @Nidc!!d nininm LAG rcquiGd for tlF

visibihy or new lund ca*nt to b. 48 miNt6 wh.@ thc MBlidAFbs co.sid.td il

lo b. 42 to 48 minut s dcp.ndinS on th. E ih-M@n disl'ne. In Dod.m liis tlDogh

thc lisibihy condnioB hlvc b.cn eFncd nainly d@ lo .ll kinds ot lnifi.ill

pollurios dc ncw luM cGs6t hls bc.n rcpon d lo b. tishr.d vlEn ia LAC B

mwh ls ths 42 hinut€s.

TIE ddcmin.tion ot $c l@l riGs or rh. suEr ..d $. n@nsr, though statd

!o b. s@nd t.st in *qu.na, i! dcp.nd.nt on rhc dct mimtion ol dt PEci*

toldenkic @diMl* of th. Su ed thc M@n- h is rh.tfoE imFnliv. lhat b.foE rh.

&t frinatioi of rhc LAG orc nusr find th. E.@ ric @rdiMl6 ed lh.n lhe

bpo@ntdc c@rdinarcs foi $c l@don on lh. globc fom sh.E ob6cNalion it to bc

m&. of th.* bodic!. Th.* e dcrivcd fren th. two $.oti.s, dF VSOPET .rd th.

ELP-2000 (di*.us.d lakr in $. ch!pr.o. As both rh* thod.s &*db. lh.luM.nd

$. iol& c@dinatcs 6 .xplicit tin sid, mlkiig olt thc prilc "lim. usm.nt' is

*nlid tor lh. appUcadon of tI6c fodul8. 'It. "ritu" co*idcrcd in thc* 0E@.s

ed rh. orh.r thcod6, is . rim ird.Dcnd.nr oflh. dillioB of thc E nl ed ir g.tudlly

Lfr.d s "Drnmial Timc . How.r dc limC' speificd bt ou clclc b ba*d on thc

!sn8. norion of rhe Esnli .nd t. Sui 6d is lm.d .s thc "Md Sol& Tim." Th.

titu cosi&cd in ihc applidion of dE rh.oncs b ihc B.lyc.ftic DyMic.l Tin.

(tBD) or dE TcGrrirl Tirc (II) which ir 6sin d wirh th. C6.dl Th@ry of

RGl ivily (Ch.po -To@a & Ch!!ro4 l9l, Ncl$n.t..1,2001, Guiml &

26

Seid.lm@, 1988). The TT is d.fined in FlatioD witb the "lnt matioDl Alonic Tine"

T"I=TAI+]2F.IE4

c.l.t)

Btce TAI it dgul.t€d &cording ro atomic rimc, In TAI $c b6ic uoir of rime is thc SI

sond {defined by Bueu Inremarional des Poids.l M.su6, BIPM, in t96?, a

dudlion of9,192,611,770 periods of rodiarions cooesponding ro lh. tmsition b€rween

uo htFrfinc levcls ofd.gbund slat oflneCesinm lll abm G.t tenerat.,2OOt)).

A day on rhk scale is 86400 SI seonds long {Astonomicat Atndrc, 200?). On rbe

other hand rhe clet dm." is thc Univ.Ml Tin€ (Ul denned wirb rcf.r.nc€ ro nem

sun and Nocialed vi1h thc Crccnwicn Md Sid@t Tim. (GMST). UT is detircd as

$e hour angle ofrhe Med Sun ar cEenwicb ptus 12i6. Due ro rhe in€euta ries in the

rctations of th€ Eanh ihcr N di$repancies b.reen the lwo rimes. rhe TT and rhc UT.

This dillcrcrce is ruferred ro as $c delht (At):

AI=TT-UT

(2.t.2)

Th.rcaore wh.nevcr w wanl b dd@irc th€ position of $. SM and lhe M@.

lbr a panicular lime on ou! clocks w€ have to fomutate rhe rine argudenr using rhese

cons'dcEiions othcnvi* fie clock dn. of the phcnomena shaU nor be appropriat€.

Finally. rhc lime arSunenl in drc rhcorics requi,es lhe delemimtioh otrh€ Jutim Dare of

the UT in question. TIE Julim Dale is th. sys!€h ofconrinuols limc sle lhar begins o.

Noon or crcen*i.h Jduary t, y6 -4?t2 (cdlcd lhc cpoch oftne..r!lian dde,) ln rhis

r,m€ scare tlre moment described by a date (CEeorie or Julie) ond rine (UT) is

considercd as rhe "nmb.r of dayJ,, d.nor.<t 6 JD, (consisting of a whole number

indicali.g rhc nuober of days .tapsed sj.ce tbe ep@h of Juli& Dar€ dd a filclion

d.sribine the flacrio.s ofa day after lhe whole number of dayt since the ep@h of

Julie Daie. Using rhjs JD for &y Domol lh. 1 is tten .ddcd lo .c@unr fo, th.

icguldnies in the oradon of thc Edrrb. The theories use tinc arghehl I lhat is on lhe

$al. of".umber ofJulio centudcj ehps€d rinc rh. epoch r2OOO.O. Thus usine tbn

21

ine argunent md the explicii tin€ eries fomulas of the ELP ed lbe VSOP th.

c@rdicrcs of lhe M@n ad rhe Sun m calcrl.t d for fte s.me i6i'nl of rhe day or th.

-

d.y aAq @njudion- In s. @ spheric.l pol& @ordimtes a th€ gtucsrdc disr.nce, X"

th. ecliptic longitud. and 9. the dcliplic ladtude, refeftd b as th. Ediprh Coordimles.

Tn* c@rdinals i. lh ft sqt dsirely ro obrain rh. im€s of th. Su$.i &d dE

Mooisr (or rhos of $e sun.ie ad noosi*) for 1hc day in qu6rion.

TIE lune ca* (or rhc cE*enrs of M€rcury &d venls) k fomcd by $e

region of $e lbn suface tow,rds fie Sun tnar fa.lls belwn rhe two plancs rhrcu8} the

e E of rh. Moon, one perp.nd'.ulr ro rhe tine ol vi.w of fie obFn.. dd thc o$.r

FrrEtrdicule to rhe di@rjotr of rhe Sun. Thc nrio of rh. ea of lhis crcw md lhc

total dea of th€ Lund disc is called lhe.phs.. ofihc Moon, The phse ot th€ Moon is

di@tty r€laLd ro rh. seperion bcrwen rne Sun and rhe M@n or .lorgation.

The Astronohical Alm$ac pubtished amually shtes lhar $e dew luna! cr€scenr

is sencElly nor visibtc who irs phe is te$ lhan t% (Askomniqt Almde, 2007).

- This ha prcv.d ro b. nisl@dine in vicw ofihe fac| rhar the brightn€$ oflhe csenr

can sr.auy vary rbr the ehe vatue ofthe phas owins lo the varyin8 dislanc. ofrhe

Mmr froh $. Eanh. The Elrri-Mmn disunft Eics foo t5O $o@d kitomcrrs lo

400 lholsdd tilomer$. Tnus when dosl lo ihe Earlh the luntr 6cshl hay be

vGibl. wilh ns pnsc much l€s lhan r % and in cae of fanhd ir nsy not b€ vkible evdn

wilh ph& g@ler thd l%. D@ ro rhis varying disr.rce rh. siz of rhe tuu dis i. fer

cha.ges. Closer the Moon ih€ disc dppees larger. Th. Muslims had nodccd rhh vdhdon

ii th€ sizc of rhe luie disc hund I OOO ye6 aeo. tn rbe Modm rimes il was nol before

Bruin lhar lhc imporrace of 1he actul visibt. widrh of lh. lutu cllsqr M @liz4

Uhinately n w6 yaltop who ued the widlh of luo{ cr*cent in his ohe_psmei€r

nodel of tu4 cre$.nr vjsibitity etadng n b the ahnnd. of rh€ c@.nr on rh. local

orce rh€ sc@orric c@rdinar6 of thc sd md thc M@n e catcutar.d lh.

.fieIs oi Refracrion, Abcration hd fic pddta @ @lculat.d for llc.@odinaies of

2a

both rhe Sun and thc Moon. Thse cor€ct d eclipiic coordi.ates of fie Sun and the M@n

@ lhen arNtomcd into Equtodal @dinal6 a, the RiSht A*nsion 4d 6, lhe

Delination. In order lo g.r th. tical Hodental coo.dimt€si Ahitude and Azimuth, of

ihe Su ed the Moon, rhe ob*frers rcft$rial cooidinale dd the t cal SidftalTime

(dcfin€d as lhe L@al Hou AnSle oflh. Equinox) aE rcquiEd. A simple alSorilhn lcrds

ro fie Greensich Mean Sidereal Time (GMST).r lhe Oi'Unilesal Tinre for any given

dsle. AddinS lhe local longilude 1o rbis GMST erv.s

1he r,cal Md Sidereal Tihe for

0i' Uhiv.sal Time for any eiv.n dale is oblained. finally to gct $€ Local SidcealTine

for my nonent of the day day be obtain€d keepids in mind the f6ler pace of $e

Oncc lhe tical SideEolTihe ofany moncnl is looM the Hour Anele oflhc any

obj*t is oblained. The local Euatorial coordiiarcs s, th. Hou Anele ed 6. the

declinalion,lead to the Altnudc (heiehr obove lh€ holizon) dd Azidu$. n@ poiilsol

tine @ imporrant lor lhis slldy. when m objecl is al lhe local meidian (i.e. TEnsi().

when rhc objecl rbes ind when an object srs.

Thc time oa tEnsil may be calcularcd using rhe Hour Angle to be eo, or rhe

co.siderinS rhe tical Sidereallime 1o b.lhe Righr Ascension offte objccr. The *timate

olthispointoflimc can b€ inpoled uing an ireraive pccs (hal inmlves Eadjustihg

the lime arsumena'lo be lhis approximatc tim€ and rccalcuhting the coordinates offie

objd at thisdme argumehr.

ConsiderinS Hour Angle t io be 9Oo or 6i" approxidare rihe of rhe rkins

(n€earive t, o. the se(ins (posnive 14 ae obraincd. R*rlculatins $e rimc dsunenrs

tor approxidale limes of thcsc eve s $c coodi.Eres ofrhe object de deremined again

ad the b.lter approximtion of$e inct oflh€s denrs ac ob!.ined. This giv.s the

risins a.d the setins orthe cenres ol$e objers (rhe sun and the Moon) thar can b€

adjnsred |o s€t $e actual isins (the first app€arance of the weslem limb of the objec,

.nd acrual s.tti.s (homen! of dcappe&ance ofrh. cded linb of the objsrl.

29

Onc. lhe lincs of fting edor selting ofbodr fie Sun 6d rh€ Moon are oblained

on€ d work our att th. p.mete6 of fic import,nce for lhc atrsl (ot la0 visibilitv ot

rhe n€v (or old) lutr crc$.nr.

2.2 DYNAMICS OF THE MOON AND THE EARTH

T!. dcv€lopn€nt of nodem dy.di4l lhoics for $e sole sv$em beg& wnn

$e discov€ry of Laws of Pleel,rf Motion by Jobannes Kepler in the 16('centurv At

aboul rhe sm.lin€ haac N€Mo. cme !p wilh his tt$ of Motion dd the Univeisal

Law of Cdvi|llion. Whar fouowd is a long histo.y ol developool ol nathemtiql

techniques lead i ng 10 the fo m! latioi o f Cd 6tial Dynam ics The e llois w.rc d ir4t€d to

describcrh€dorionofplanctsandlheirsal.lliles,dteroidsand@nebinordcrtopledicl

then positions in tuturc with accey of SEater 4d g@l€r d€sG Conuibntio.s of

Eul€r, Laplace, Poison. Gauss, Olber, Cowll. Encke, Claittul. Hesn. D.launav. Hill

and Brown. besides daoy olher nahemalici.ns and astdnoneb. bare ben oi grat

si8nificancc. A nunbei ofcldsical md nodcn books de now available thal d.scribe thc

d€rails ofd.F conuibutions (Sna41953. Deby, 1992, Plunmer, 1966. Pollard 1966.

Woolard and Cl€nene 1966. Bouvct lnd Cl€mence, 196l elc.). Contibutions ae oho

availabl€ rhat give delails ofthe lund dynamics (chaplont_Touze dd Chapront. l98l'

1991. 1988. Chaproni er. al. I998, Srandish I981. 1998 etc) Einsrein's theolv of

rldrivu suc(aded in dc$nbrns the molion olthe penhelion

The satllite to planet nss !.tio in case of lhe Moon_E nll svst€n s larg€sl

( 1.23 x l0-r ) in compdi$n ro ey o1h€r saEllite-ple.t pait (lh. nexr ldecar beins lhar of

Triton-N.Drune l6s ntio = 2 i lO I ). Thccforc rhe Eanh do.s not prclid. lhe dominmt

efecrive fore aclins oh lhc Mooh.It is not only lhe Sun bul all msjo!pltnels and la€€r

of lh€ dtdoids lhat @nkibute 10 th. cffcctiv€ force ,cting on the Moon Thus d

cph€ncredF prcpded *ilhout tdtinS into eout all lh.e contibudons b boud to be

emn@u' Mey of the epheheedes of $. Moon of.dlv dsvs, both b.lor' md aliet $e

lomJlauon of $e Neqonie ndhtuics. weE bded on obieBed sd lompukd

averages of various Lnds r.laled 10 lhe dyndics of lhc Moon

30

Today it is tnoM with a g@l degK of @!6cy lhoi lhe avetagc avrodic p.riod

(intepal ber\,s t$o coMurive ne* M@m) is 29.510589 davs (29 dars, 12 hom 44

ninul6 dd 2.9 sondt. The av€Bec uomalisd. morth (intcn.l ber@n lwo

su(esive pa$ages of rh€ Moon throush it! perige is 27 5s4550 days (27 davs 13 hous

18 oinutes ed 32.1 $cond, ed 4 aw'age sidercal honth (inLdal b.tw€en re

succesiw p6eg€s of rhe M@n tnroueh ! fixed st!r) is 2? 321662 days (27 davs ? houf

4l minures and I1.6 secondt. Thus on thc b6is ofa sidcEal no h thc Moon rravel3

abund 12.176158 degr.et p.r.lay on the av.rage Relaiv. to lhe Su. lhc M@n traveh

12.190?49 dtgc6 per day on lv€mee. Howvo, rhe mininu Bt 6 b. 12 08 deges

Fr

day and $e ndimum t2.41 deeees per day. Tnis oll happeis beau$ the orbil oi

rhe Moon around tbe Ennh is a hiehly "i!rcCular' .lliPe whercs lho deviadons tre

cau$d by p.nuibatios du. lo the S!n, lhe Planeb

and olhs $lf svsten objsls

Thc smi-major arh ollhc orbn ol M@n n on alcnge 384400 kn bul has a

snaU oscilldlion aound thb valn€ whose period is hala lhe svnodic nonth Thc

.@enrricily of the orbn is 0,t49 bul vdics d much sj 0 l lT The inclinalion of the

luhr orbir from the ecliplic is 500 9' bul vdi6 up tol9 Elen the nod6 (poihts of

inFdecrion ollhe lunar orbil and the ecliptic) of lhe orbn !re noi fixed 'nd

go round ihc

rcliptic in 18.6 yeaB with an o$illalion.bour dE $cular notion lhal moun|s to as

nuch as I 6? dcsl6 Thc line ol apsd?! tle eo rcund thc echpiic complelrne on'

rouod in 8.85 yem and dciltations {iih udnude of 12'413 destccs Ths all lhc

''elenentt' oi the orbn exhibn bo$ scular a ell as peiodic ldiarions lhis mkes rh'

deimiMdon of lund ephem€Ed€s a daunling task

The undeslandi!8 of lhe dynanics ol the Moon dd thal of th' planets in $e

nod€m *lup b€sd vith lh€ doluliodv e4loits oi Johees K'Pler $d lsc

N.s,ror Kcpt.r cnpnically dcdu€ed his Lls of Pld.lary Molion on thc b6is ot o

extensile study of the obsralional .lcta @llected ovcr ce'nies of lhc posilio' of

pldets. Thcsc cfl be sbtcd 6 follows:

3l

L

2.

L

Pl&erary otbiB d. clliptic with lhe Sun a1 one ol lh€ foci.

R.di6 vdtor of ! pldel (V@lor dnM liom the sun lo rhe Pld€D

sw*ps equal &.a5 in equl lihe jorensls.

Th. sqlle of rhe Friod of rcvolulion of a Plder eund rhe Su is

proponionalto lhe cube ofils dea. dblance ftom die Sun.

N€fion nol only prcent€d his Law ofUnive6al Gralilaion but verified KePleas

of Pleetary Motion usi.8 lh. tnw ol G6vilation. According to N€Mon s Larv rhe

of atmcion beM.en two bodi6 wilh nas*s 2' ed ,, pl@d 6t a dista.ce /

F =c!+i

(2.2.r')

The force is aflnclive and G is the PmPonio.ahy constant dllcd Univ.dl Cavilalioml

Consranr (6.6?2x 10 'r-)*s rrr). i is enhs dit€ctcd ircn ,L to -r or rron ',rou|

Frem rhe ddc of publicalion ot tte'Priripi.l bv NeMon in 1687 a nudbet o'

astlononeB, physicisrs and na$cnalicians conribulcd sisnilicontlv in $e dcvclophenr

of th. lndesl..dins of Lud sd Plm€hry dy.amra.

Howele. at lh€ time of NeMon (and p€ftap3 lill lodav) the dvnanics of lhe Moon

pa*nred sear dilficuhy rhar forced Ne\\'lon lo darc thal ",',e Ltmr theo'v nale his

he.l ache dtut kept hih aeake so olen thut he vorld think o! it no norc " lDanbv 1992)

He had tlitficulry in describing lh. modon of Fnge (the poill in lod orbil cloesr ro

the &nh) and could explain it ro only withi. m accurev ot 8 percenl Clanaut (1749)

appli.d ml)lical merhods ad succeded in explainilg d€ molion of p€ngee bv 6'ng

seohd o.der approximation. He published his lraorP de Ia /,,e md a se! ofnumencal

tablcs ih 1752 ior computation of lhe posnion of fie Moon. the Eost sisnificmr

contribudon frcm Euler appered in l 7?2 shcn he tublish€d his srcond lL@ lh'ory

32

kplM\ ficory of luM hotior, Publish.d in 1802. cnploy.d rrmfo@ing th.

eqution ofmolions $ tlt.t the lruc longltudc wa an independ.nt ldiable Hh wotk also

povid.d e dpl.n tion of lh. sdo a@1.61io. of the M@! ripl@'s mdhods w.E

cdied io a high degrce of accu!@y by sevetal malhedaicies One of then was

Dmoi*e. who publthcd his th@ry dd r.bl6 in 182? that Gmaircd in *id' Ne until

Ha.Fn! sork dppeaed. P. A. Hasen's wolk ext nded for over fonv veds fon 1829

dd his r.bl4 w.re publkM in lE5?. ThGe labl6 rcmain.d in s for w.ll ov'r finv

yds. Delaunay publishcd his work ih 1860lhal M basd on disturbiry tunctions $ai

in€ludcd 120 t€ms. By sn.ltti.al m.4 h. Gmov€d 0E teru of disorbins turulion on'

by one ahd gEdually builds up th. $lution. Autho6 cl.im thar Delaunav s vorl( n thc

mo$ Frf{t elulion of th.l@r problm v.t found {D&bv 1992)

Thc posilion of $e M@n dodd th€ ElnI is desdbed bv spndical pold

coordinatcs (r,,190 p) wnn r bci.s th. heli@.ntric disr:nce or rhe tltncr' ? $e

eliflic lonsitudc and I lhe ecliFic laliode The most commonlt Gd edv lo handl€

lunai $bles dlrinB the nosr Pan of 20't c.n$ry \rcG dDc lo BrcM (Bmm 1 960) This

ll]M $.ory w3 inpo!.d bv Eckefi &d ws tnown 6 ILE, sho^ tot Inprored Lrnr

Epr,rdr,r. Tnc lh.ory coGtruct d bv ChcPont and ChaPon(_Toul is knosn 6 ELP

(Chapo er. al,. 1983. 1988) shon tot Ephtina des I air's Pukienhes ln ELP

simplifi.d tables have ben cxlEclqi from lhe lheorv lo rcplcsnt lh' luDa nolion in $e

fom of explicit line sries fomule. Th6e tabl's @ b€ lsed to direcllv compute $e

luntr coord'nabs. ELP h Dol onlt morc Peci* md complele in conpdien ro ILE it

ale povides noe nodern valEs oi l(M pdmelels md olh€r Phvsical coislsts' For

6000 y.4 on eacb side of J2ooo o ELP povid€s lund c@rdiMt€s that l4lv have eno$

.xc.eding few arc secoo.ls Toeelher with th' d€v€lopment of VSOP (Vdialrons

Sacnl.ifs des Olbires Ple€titt bv Bctagllon ed Fdncou (t9E8) lhe t!bl6 due ro

Chapront er. al destib€ the motion ofall major bodies (€xcept Pluio) ih tbe solar

sysrem. Both rh€ $€ori6, ELP &d rhc vsoP w dcv'lopcd d rhe Buean d's

ll

BdicaUy a nhcory of pldelary ed lund motion involvd inies ionolasysrcm

of di0eGnial eqladoi dat constin&s th. major pad of th. sfudy of elcstial n46mic'

Therc & i. gddt t$o.ppoachcs for slving such dyndical systctu, ealytol ud

nusical, Anallri€l nerhods m b6ed on solutio. by Iouder sen€s ed thc Poiso.\

S€lies. Thc ELP-2000-85 (Chdprcnl€r,al, 1988) is semidalttic sd has be€n obla'ncd

frod a fil o f ELP-2000-82 (Cn!po!i cr. al. I 983) to th. nlncrical inrcer.tion of lhe .lei

Propulsion Labomrory DE200/LE200 (Slstrdish l98l). P@$ion in this o@ry ha b€en

$keh iion Lasrar Gsklr, 1986).

For te an lylic par! ELP-2000 Fpet€s th€ nain lEblm fbm th.

penurbarions. Th. 6ain problem tak s into accout thc &tio. of thc E nn's ce E of

m6s aid lhe acion ofrhe Sun's orbir aound lhe Eorlh-Moon baycentr. such lhat Ihe

Sun\ odit is asuned to be Kepl€rid cllipse. This rcrulls inlo Fouri.r series w'lh

numdical c@fiicie s snd ar$mdb th.t m suns ol mulripls of fou fundade.tal

paBmeted D (diffeence of ihe ned longitudcs of the Sun ed $e M@n). / (mean

monaly of rh€ Moon), l(hem anohaly of $e Sun) lnd a (M@n s argumen( ol

h ude). Tlis main problem 6uhs inlo dne *.ies fomnld for Mmn\ longitudc.

ladtude and gocenrric disrscc @daining 2645 rcos in all. Apan flofr lh* *rics

acrons of all rh€ other significant objects in solar sysren dt consideed N

'Fnurbadons lo ihc mdin pbblen lhol include:

L tndiKr pl&c(ary perturbarions rbar re induced by lne diff.tsoc.s b€lwecn

the lrue orbit of the Sun aound lhe Eanh-Moo. bdycenue and tssuned

Kepbnm Elliplhal orbil oflhc Sun *uned in the min Pmblem.

2. Dir€ct planerary penub.tioN due lo etions of othr plan$ on ibe M@n

lor borh the direct ad rhc indt€ct planetlry penubalions lhe ELP-2000

co6id6 th€ orbib ofthe pldct giv€n by BGlagnon\ VSOP82 rh@.y.

L P.rurbations du. ln€ figues of $e Edh md thc Mon (Moons, 1982).

34

4. R.larivislic p.nub.rions (Lsrnd. & Chlpronl-Tou?J 1982).

5- Penurb.tuG du. to tidd afccis (villim3.t. a! l97E).

6. Molion of the efccnce frde conside&d wilh spect to m in€rrigl fime of

consideradon ol all lhese pelturbarioB esuhs inlo rime series fomlla for geocentric

longitude, IatiMe .rd disra@ of 0E M@n at nds $e numh.r of lcm !o 15:37.

An al€natc ro this ther€icrl appro&h is !o rcpr€*nl1hc coordinai.s explicilly

6 tinc ledes fomule. This epEs.rolion of tlie tinc sed€s is dcrloped by Chap6n!

Touzi ind Chapront (Chapront-Touu & Chaprcnq l9tl, pp. l0) und ha bcc. !*d in

fiis work. The mjor fomula used in rhh vork due !o $es authob de lkrcd beloNl

Tbe ge@entric longitude

'/h.xpesed

as:

'/=218.31665416+48126?.8813424'r-0,00011268'r:

+0.000001856tr'

0.0000000r534.r1+,5. +(s; +/ 1si +r' xsi/1oo0o)/l0oo

(2.2.t)

wh€rc

I = h in iulian centuies sinceJ2000.0

s, =tv,sr,(dj') +d|),,+ao *r' xro I +ao)'/r xr0' +d|,.rr xro r)

I = )':st(o;'' -a:"' '4

(2.2.21

(2 2.r\

si = tv;,s,,(d;,o + dnD.r) (2.2 4)

$.t%'sr'(sfl + dI'.,)

(r25)

Tb vdB 6f tb.ootM v- r; ct, d!o8 eitb |b ot a'. G Sivto i! Chr.od-

Tout.d Ct q.od (Clg!d.To@a.nd Ci{.Gi, l9l, rp a!-56}

Ib gcoc.nttc ldhde UL riE b,:

t/-s,, +(s:/ +r'$ +i's;/10000)/100 Q26)

s,, .5,"si,'(rj". p:".,.p:!.rr xron +/i",r xr0r +r'r 'r'

x ro ')

s:, -'u.,&4fri.t + p{t. tl

Q2.?',)

e2.8)

.$ -tr:$4i:o +r4D.,) t229'

q.t!;&,(r;o +r'o.,1

Q2.tO)

Th! y.lG of th c@trt ! t

- !; .iq dotE eilh lho.c of 0!.t 8lt'.! in Clqlot'

Toudsrd Ctar@r (ct Fld-ToEald ftlnfi, 1991, I? 5?{a).

rindly tl|c geo.6tic dllt n6 it dven by:

x-315000.57rsr +sl +r.8i +y'.s;/10000

Q2.rD

s- =t'"cdldj"'+djl xron)

s;=:4cd(d;'+r;,''/l

(2.2.t2)

(2.2.t))

s; = t,;c,rt6"4' + r,4, .4 (2.2.t4)

si= +d;o).r) e.z.ts)

i"-",,(u-''

-fhc !!hEs of the consr.nls rz, /; etc, dlong with lhose oa6s ae given in Ch.prcnl-

'fouzi and Chapronr {Chapo.r-TouzC and Chatronr. 1991. pp 65-73),

Atlat0)s. /0)s and D{ors, ,', s,,,sl..U,su,si./.'r.r;.,". md r; @ in desrcB,

arr\, y'r's, d1'\, si,si,.'; and ,; ac in d.s@Jcenrury, ar'\, y''?\. irlrs, si.si.v;

lnd r; e in desEs/6tury2, ao's, y'r\, ana d's m in aegeevcenruD r and a"rs, /'rs

aid d'h de in deecs/centud. R, s,? and Sh dc in kilomelres, 5i md 4 .t in

kiloderrs/cenru.y. siand 4 sr. in kilometftVcenruly:.

For Tbe detemimtion of plan.lary coodinstes lhe complele n-body problcn is

requied to be slv€d. A. ml,,lic solution of planerary dotion wa3 pre*nted by

8rct.Slon (BElaglon, | 982) of Burau d.s LonSilud€s of Fnce lhat described only thc

.llipric @rdinates of the pldels. Th. elution is populdly knosn d VSOP82

(Variations S.culai6 d6 Orbit€s Pldauirct. Latcr, BEtagnon and F6Nou of th. se

Bur6u hodificd VSOP82 inro VSOP87 (Bcl8non & FMcoq 1988) in 3uch a uy lJllt

th.i! eluio! povid6 both thc Catcsim (or tst&suld) @rdimtes 6 wll a th.

sphcdcal pola coodiiates of $e pldels in a helimenlric syslei. Thcir slllion

V SOP8t descibes ln€ el enehls of thc osc ulat ing or inst aneous orbit in lems of:

t7

a - sdimjor uis of th. dbil

l, = m@ ld8itlrL ordF ddt

l= ?,@$

p= sincr,i Dsino

c= si(Xi)coso

whec. is thc.@eotriciry ofthe orbn, r is ln. bngnud. ofp.dhelion, i ir th. incliodon

oflh. orbit fmm rh. plD. ofeliptic dd O is the logilud. oflh€ e.lding nod. oa th.

olbit. €.cholth. rcctarsul$c@rdioat (X L a or &c aph.ic.l polar coordimt€s (2. I

t) is u.xplicir tunctionofiime ed is inlhe forn of p.dodic eds md Poison $ies.

Every lcd ot Ues $.i6 is in fic tod of:

I"(ssii9+ I:cosp) or 1"..i€os(, +c?)

(2.2.t6J

$,lEE a - 0, l, 2, l, 4, 5. I is tn. tire in lhoFnds of.,ulie )€s fod J2000.0, i.€.

f=

165250

e=ia,1,, i

= I to 8, ! reprsent rh. m.m longnud.s of the plancc Mercury lo

= 9, l0 6d I Lc rcpllsfl th. Dclawy sgmen& of lh. Men D, F md

The hsl of I is the na longiMe of U. Mon Bivd $nh 6p€t to lh.

tlay. In thc ah@te €xpcsion,

B= Za,^," + P

c=>d,N, 12.2.11,

S - -,asinp,

(2.2.r8)

l8

,l md M dc Sivci

in the lable 2 of (BdaBnon & Flmou l96E)

These data series @ .vailabl€ on CD's dd laFs For r.cta.gular c@rdiml6 of

rhe phf,€ls |he dlia llles VSOPS?A. VSOPSTC dd VSOP8TE dc @d sd for the

spherical polarcoordinaEs VSOPSTBand VSOPSTDa!. used. A shonerleBionofrhese

data series isSilenby Meeus (M.eus, 1998)and lheff. tr u$d in this@r\ l. the*

hbles firsr €olumn gives.4s, fie eco.d ,s dd the lhird sives Cs. The dats file hs 6

sedes fo, €dh or rhe coodi.ates L (h€ helioccnlric Lonsnud.) and R (lne helioccntk

dislanc. ofthe Eanh) and 5 for fte c@rdi.ale B (lhe hclioc..lric Lafiudc)

Iiiclr s$ics lir L. ll .nd ll ne uscd as follo\s ro obhin thc heliccDkic polar

t't = LA; co\B; . I jT t.

r2.2 rol

i 0. l . 2. :1. 4. and 5 lor l- md I( and 0, l,2,3,4lor B Ihc suFrrscnpt tsund for I

(L). 2(lJ) and 3{R). x rus tolsh 0 lo dilfeonr incger lor di(feEfl coordinaGs and

their lssiatcd seri6- Fo. I - 0- /{i coftsponds ro lonsirudc scri.s. I = l

l/{iconsponds ro lari$de sri€s and I = 3, /r,coiiesponds to dist .ce serica. ll.ch

coordin eislhcneuluatdas:

r=ltcuil'

) "=lz'',')

=l:, ','' I

(2 2 20)

/. md , d. ir ndian nesuEs and X is in aslrohonical unils. These m as menrioned

crlier rhe c@.dinares of $c Eanh in heliocsnkic c@rdinare sysrcm wnce6 for {he

problem ofdckmrining posirion oflh€ Sun inour sky w. €ctually requiE rhe Cmcentric

coodinales of lhe Sun instead. In case of rhe Eanh this uNformation is sinple:

l9

,is = I +l8O! Fs= -e 12.2.2t)

ond the hcli@entric dislanft ofthe E.nI is smc as th. s€@edtic

distanc' oflh€ S!n'

2.3 BIRTH OF NEW MOON

A3 menioned earlier the Moon in ilsjounev dound the Eanh llavels arcund 12

dcgrees cvery doy in our sky and $l.s ovcr tic Su in lround €vcry 29 5 davs whc' rhe

Ccoc€ntnc Lonsnude of the Sun e.l lhe M@n N se rhe nonent is tnoM s tnc

Tine ol Bidh of Ncs Moon Tbe dudion berve€n two succe$ive Binhs of Nev M@ns

is called lhc Lun.lion Period Ho$ever dE Llnal'on petiod is n

from 29.2 days lo 29 8 davs This is rh. cMn b€hind

'ons

irt tu'd mon$s of 29

d.ys €ach or lhe con*culive luor monlh of lO davs cacn For fi€ tine ofBinh ofNew

Moon one requiEs to find lhe noment when rhe seocentric lonsilud's olilt Moon and

rhc sun coiNide Thus one needs to lracl lhc lo'Siludcs of eeh of thcn rinc

Coisidcrins $e najor rcms oa lhe line scrics fomulac fo' lhe lonsiodes of the Sun md

$c M@tr in th€ planeurv rhcorv VSOP'2000-8? and lhc l(nw th@rv ELP-2000'8? $e

noment when thc rwo lonsiludes aG sMc can be evaluatd An alsori$m due b Meeus

(Mceus, l99E) for $e dclemiiation or $e rin' or Binh of Ns Mmn is as rollos:

On a!c€8. lhe tnpic.l Y€d (dudiion betw'en t*o conseculive

passages ol lhe

Sun lhroush equ ox) iscudcnrlv 16524219 davs (from (l I l)) Thc avedge svnodrc

Monrh (trre intedal bclwccn r$o cons{u'r\c t\e* Moonsr toten o\er a rcn'ur} ii

29.530589dar(fon (1.3.3)). 'Ixus in oft topical vear rh're aG otr a\€Esc 12 1682664

Synodic mo h5 Thqeloe since lhe $ad of lhe vcd 2O0O i e J2O0O O the nunbei of

synodic monlhs elapsd oregrveh bvl

t = (f,' - 2000) x l2 3682664

(2.1 l)

and lhc time in lropical ceoruries el.Psed since J2000 0 is Siven bti

t236.42664

(2 t.2)

An approximarc valu€ ofth.Iulid Dale ofthe Ncw M@n ir $cn giv€n by:

JDE - 2451550.09166 +29.5JO58886t.t+0.00015437..r -0.OOO00Ol5

j.r

+ 0.000000000?3' r!

(2.3.3)

$herc t i. d inteSer Thus @ording ro this fomula rhe for t = O |h€ Jutiil Dar. of$c

lid ces.nr ofycar 2OOO is 2451550.09766 lhar is J&ury 6.2000 ar I8n r4m, md

4l *!. I of dynlmical lime. For a horc accmle valu€ oa ihe Julia D.te of thc New Moon

lhe p€dlrbation tchs due lo lhe Sun md rhc planer ar€ added, Thc pertuibafions tenrs

due to thc sun d€ given byl

x= -0 t07 2.SlN(M )+ 0 I72J t

T

EISIN(M) +O.0l60A.SINOr M )+0 0lAj91stNe,F)

+0.007i9.E"'lN(M -M)-O 0O5 1.t.ErSINtM + V +a 0a208'E^2istN Q,M)

0 00| t t 'SlN(M -2.F)-0.000t7.StN(M + 2*F)+O OOO|6.E.SI^-(2+M, + M)

-0 000J2.slN(3.M )+ A OOO.| 2+E.StN(M+ 2tF)+0,aoB8*E.SIN(M_2.F)

-a 00021'E.stN(2. M'-!r' o 0u | 7,stN4.] 0.00007.stN(M'+ 2.io

+0 0a0u.slN(2.M -2. F) +O.00001'S!N(1.M)+0 \oaol.stNi,,t-+ M-2+F)

+0 0040j.SIN(2'M + 2.FrO 00O03.S|N(M +M.2.F)

+0. 0000J.StN (M -M + 2.D -0. @OO2.S|N(M. -M). D -O.O0OO2.StN(r. M, + M)

+0.00002.s1N(1'M)

Q).4)

wherc ,r,1 = rb. mce &omaly ofth. Sd al ihc JDE

= 2. I SiJ +29 I A5 3567.k-0 ,OOO0 t I t -0 oaoooo I I I

(2.3.5)

M = thc mee anomly of de Moon d lhe JDE

= 2A 1.5613+ J85 8 1693528'k+O.Ot 07j82N I +0 OAOA I :38. I

-0 0A0A0A0,8. 11

4l

F " M@'!.4ruqtof hind.

- ! 6'0.? ! aE+390.670t0264.1-0.00t 61 r 8. | - 0.0U00227. ?

+ 0.0@w0 .f

Q.3.1)

g - lncinde of lsnding md. of {!. luar orbn

- !24.7716-1.5637t588.t+0.0020672'1 + 0.OOOOO2 t 5r I

(2.3.8)

E = Eedtricity of $. oftit of Elni

= 1-0.002t 16.T-0.0000074. ?

(2.1.e)

Th. pedurbltion |.tus de io pleas e€:

v = A00032t.StN(A I )+A000165+sNLtr+a000 t6.trSIN(/r+0.0a01 26.stN(A1)

+ 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)

+ 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 )

+0.0000t7rsIN(AI2)+o0o00J5.StN(Ar i)+o W023.SlN(,| !4) (2 3 to)

A I -299- 2 7 + 0. t O74o8.k-O,OO9 I 7 t. ?

t2-25t.E8+A0t6J2t.N

,13-25 LE|+266J IEA6.t

a4-t19,42+J64t2478.*

/5-U.66+ ! 8 206239.k

A6-t4171+53.303771.t

/7=207-14+2.153732.t

19-34_t2+27.26t2391t

A ! 0- 207. I I +0. | 2 1 824'k

/1 t I = 29 t. 34+ l. 64a379.t

(2.3.1r)

12.t.t2)

(2.l.lr)

Q.3.t4)

(2.1.15)

(2.3.16)

(2.1.17)

(2.3.18)

(2.3.19)

Q3.20)

Q.3.2tt

42

A I 2= 1 6t. 7 2+24. I 98 I 54.k

A I 3= 2 39. 5 6 + 2 t. 5 t t099.t

at 1:33 t. t5+ 2.7925 I E.*

(2.3.22)

(2.3.23)

(2.3.24)

Tlus $c ,ulid Dlle of lh. Na M@n ie giv.n by

JD=JDE+X+Y

(2 r.2t

Thc dm€ deeribcd by this dat is lhe DlaMnical Timc and lhc coretions for Al mcr be

mad€ lo get $e Universal Time (discu*d in 0Encxtarticle). For anr l@al coopuralion,

th. Ld.l Zonc lide and d!i. n6r b. calculated ftom |he UniveMl Tim. and dale

ohaincd abovc on rhe bdis oatbe lonsirudc ofany place on rhe Eanh. Thc darc h $en

de day of conjuncrion fo! the place and rho rime of conjuncdon (he bnrh orNcw M@n)

can b€ dy rinc fom 0- 10 23r" 59'" t9h on thd dav.

2.4 THE TIME ARGUMEI{T

ft w6 mentiohed earlicr thal rhedynahicsofatl soltr systcnr objec$ isdescibed

by lomula b6cd on 6adcs of Cl6sic.l Mcchdics and lhe Relarivisric Dyneics in

t.m3 or rid€ serics. In order b ersriv.ly ue rhcsc fomulas an apprcpriate ?rrs

argMerr cotrcsponding to rhe honeht of obseiving the lunar crescenr at my place on

lhe surface of the E rth h6 io be €valualcd. Such a lime a.sudent h6 ro b€ co.inuou

and mui halc a clearly dcUncd point of i's beeimine (|he aro rine). called .p€h.

various theorics and problems uF difercnr .pochs depending on rhs co.rext for a

geneEl consideFtion in pls.lary ad luE dynmics thft ,re lwo ihporlanl epochs.

Th€ nr$ of lhese epochs is a oonefl in edote pdsl coftsponding to rhe Noon al

G@nwich on Janlary l. 4? l2 B.C.E. on th. Julid c{l.ndar (or Nolember 24, 4? I I on

Cdgode cal.ndtu) (R€ingold & D.Rhowila ?001), Fren $is pojnt of timc rh. rine

elapsed tiu my laler point oflime in number of days md a possiblc fierion ofa day h

4l

elled th. Julid Dare abb@ialed B JD. So the ,D @Gponding lo thc 5i" 30.i. on

Ocrober 5, 2004 ar Grcenwich is 245t284.2708t11t.. Thls rhe Jutim D!i. i3. mesE

of tih. clapsd sine thn €pocl 6id is cxpEss.d in nmber of n@ eld d.y5_

The orhcr epoch of inpori,lcc lo rh€ €lftnl work k fie noncnt of dme €ell.d

J2000 0 a.d il Eprcseds lhe l2ln TDTon J.nuary l,2O0O i.€. (Abononhal Almahac,

2007). Tbe JD conesponding ro rhis hom.nt dl C@nwicb is 2451545 days. This is the

.poch or 2eo me for borh rhe Luq Th@iy ELP,2000 (Chopo.lTouz{ & Chaponr,

l9l) dd th€ pl&elary th@ry VSOP,87 {Brc|,gnon & Fmcou, 1988). In bo$ rhes

0leodcs lih. m@!red frcn J2000.0 bo foMrd od b6ct*ards. In ELP thir rimc is in

is coBidcrcd in rulian CentJies (i.c.16525 me.n sle days) md i! VSOP il is inJulie

Millemia (365250 mce solrdays).

BoththeseepocbsmbasedonrheinreRaloirimealted..toedn$hrdat, qhich

is defincd a fie int rval berween lw slccessile rhsits (r6sdge rhlough rhe locll

meddim) ofrhc fic ious body knoM as ft. mee Se. This ficlitious body novcs sirh

uifom sF.d alory lhc cel6ial cquabr .hd is consider.d in place of lh. acrul Sun $ar

oovcs vith non-unifom sped (dw ro rhe .llipli€ orbn of tne Efin) abnS lne Ectipd€.

The tmsit of $c aclual Sm over a local heridi& vdes up ro I I minurs over a penod

of on€ lrcpical ye (Astmnooical AlFanac, 2O()7), Thereby atl civil rim. reckoning

have b€eh a$ocialed with the hedn Sln lhot consisrl of 24 mean solar hous, The

beeinningofa civilday, i.e. zro hou* on civil clocks occu^ at nidniehl when the nour

dgle of rh. Mean Sun is rw€Irc hou6 lccodin8 to the local or sisddd hcidia..

Th. tiDe describ€d by lh. cl@k showing d. mm $lf rinc is not wilhoul its

di*Ep.n i6. In fd it is lhe E€rtn, $e gtob€, ilsctf rhal is ou cl@t lnd $c n.m solat

tine B suppos.d to b. basd on ths a!€ragc rate al which lhe Eadh n spinning a@und irs

dn. Howeverlhn considearion is only with rcspecl !o lh€ Med Sun. Du.lo the orbilal

noionoflh. Eanh beins in rhe sme direction as its dis ofmlolion (ton qesl lo EEst)

on€ ubl totaiion coDpldes in l6s thm ihis nee sold day. So the actuat nle of dial

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_

Aglin du.lo lhc clliptird orbit ofr[. E lrh ttris D.riod i!.le nor uifom e w h!rc ro

@tuid* "M.rn Sidccal Day" rnd !@diiSly M6 Sidaql Tirc. Oc md el& dly

cquls |.002?3?90915 !|16 .i&Gd dlys (or 24B 03i" 56.t553?* on m@ sid.Mt

tim.). Altcturcly oE md 3idcEd dly cquals 0.9?26956633 ns st.r d.ys (or 2lb

56'r" 04.09053* on md sol$ rimc).

In gcrcr.l rh. o@ eld {n ir rhc rinc |.t n inro &@6r in bolh lb. civil tu.

ekoning a sll a tlE $tromnic.l. Wh.Ed lh. dylmiclt rh6d* denbc |h.

norions of th. objers ii solr srsr.m on th. bair of rhc coninuoBly nowins tinc

dc$dbcd a Dyimiol Tim.. T|| UniwBl Timc .d rlE DyiMiel Tim. E nor

@Biri. .nd rh. dilTelwc bdw6 dF tw is lot ! trom fumrioi of ritu dd @utd

bc lound only bt high pcirion ob&dados of th. skics. Th. diffm. of thc lrc AT is

rlbull|.d i. Asnonomicrl Aln.t@ for th. t.tqopic G6 (AD 1620 ri lodar.). For rh.

cd prior lo lic rcl.spic .n rhc v!l@i ol dT G cdcll.r.d on thc blsis ot rnc

qlculatioB of .cliter @ultrrion lnd 0E rin6 of th* d.nt. @rdert jn lh. tmwn

history. Th. val@s of AT e giEn fq only |h. nan of.eh cd.nd& y.r ,nd $os for

olh.r dn6 ofFd ce b. incrlDlar.<l. VeioG .u0!oc havc giwn veio6 lehniqu.s fot

obt inins llF* slq (Di.k y 195, M.!!t, l99E, Motri$n & Scoh.Nr, 2004.

Mofti$n & W.d, 1975, Scph.@n & MoEien. 1984 ald 195. Sr.ph.nsn, 1997,

lsl.m .t d. 2001.

Wh.ncv.r a c.lcul.tion for e.wnl ir to bG p. iomcd s ed( oul.v.rging

b.sd on l@l civil dh. sd dac. ror iBlhe vhcs llE rim ed darc of rh. binn or

nd M@i is compucd rh. lqlr cone our ro b. in dynmictl rim. This dyMical

dmc should h. conv.ncd ro thc Univ@l tinc by adding rhc c!frnr wt* or & ud 0En

lo th. l@l civil lim. by addine lh. Equbit &n tirc (@BidcEd posiliv. fo..6t

bnsnudB and rcg.dv. for Br lon8itud.s) for lh. l@tion of ob*rci Simil&ly if oie

wrs ro olculnc to ric Dcition of rh. M@n fo! {y l@rion of ob*rer, rh. ld.l zoi.

tin. hts to b. @nvcn d io uiwrsd tiD. by subr&ri.a thc aE tin. &d rh.n rhc

crent vduc or 41 should b. $bt .cr.d iom n ro gd tnc Dylmiql Tim.

How.r s noE @nhtlon l !'pNch ir ro @!qt lo€l <bre.rd ar rim ro

uiv.Eal dalc ed time which is ihcd cotrrcncd b rh. ,ulid .!ate dd fmUy th. .fi6r of

dt is taln inio ecoul. Tlc s. lpprorch b u*d in lhe fonowilg atgodrhn foi

c.lcul.tioo of the dn dgme l

Srep-l: l8el Local Doaroh ercgorin Cal.nd@ LZ LMM, LDD

Sr.F2: INrtIBalTin. &7awTih. LHH, LMN, LSEC, ZON

S4 LryY=LIY, UMM-LMM UDD.LDD

UHH= LHH + ZO N, U M I N = LMIN, LIs EC = I.S EC

ln eedeEl rhe only dif.rc.c. bclwen local hne tinc a.d dale ed lhe

Univcsal dme md d.t is th. diff.cn . bel@n hou6 rhat cq@h zon time

(considercd in inresal ho6 hcr). Duc b rhis ditf.rene th. t@al d.rc hd lbe

Univc6d dare mar difLr by onc &d tr.!& .djcthsf.

I[(UHH>=24) ...... 1rcft6. dar

( UDD-LDD+I

tf (UDD>dorqLMM)) //darr is nMber of days in a donrh @y

I

t)]

upHH<o)

LDD=I:

'UMM-LMM+Ii

rf (uMM> t2)

I UMM-|:

UW-LYY+ 1:

... ... d..t @ do,

{ UDD=LDD.!:

{ If(uDD-o)

( UMM.LMM.I

If(UMM-1)

46

AYY-LW-l;

J

UDD-&tr(UtlM)

olr@ l[. UEircc.l d.rc dd tirc is lppllFi.Lly dju{.d @ pcad! lo

c.lcula& rhe Julih Dar. for 0E d.lc {d rhc ritu. For rlis calcularion inirially if

ihc nondr is Jmqry o! fcbnoy, it i! co id.Ed morh .mbq 13 or 14.

Esp@tiv€ly, ofthe ptwiou trd ad rh. t@ i! sle d.crcNed by l.

UW-WY-l

Thc nmbd ofeftry yd (likc ,E | 100, 1700 .rc) d[ ln. yd UYY

ir Fquicd lo t@6t for !mh6 of mhd rap y@.

Stcp4: 1=INr(Wt00)

ln lml of 6e $tommic.l dlculariotu ! daL d and .na Fliday.

Octoba 15, 1582 is coaid.tld io b. I dlc of Grgodd caldd.r {d a darc toni

Thsdly, Octob* 4, l5E2 ..d prior to rhb datc i. coGiderd s a dale i! Jutie

cal€ndd. Ilru ifa dsl. i! fton Crc8ode cal.dde ndh.. EquiEs m accout

ofnon-tlap yed tom.mong3i 0F nomal lcap yds due to th€ modified rule

ofliad y.rinthe Grc8onlr qal.nd$(y.$divisible by 100 but rot diviribte

by 400 e @r L@! yes).

d"@ledq it casdtb'

I

a=2-/t+rMf(1/1)

.l*1I "@Ldo i.t IdlM"

( B_0 J

47

Now orc n6ds ro c@t 6. nub.r of diys .laps.d si@ the Julid DaL

epoch (Jeulry l, 4712) iill Ihc .nd of th. PEviou yc& .rd lhe trmb€r of dltr

€laled flom ln. n61 d.y of rh. PEviou yd till the cnd of lhe c|lml monlh

Meu @dider this sMbl ianing fom ys -4716 dD( adds addilional davs

thal @ balmed by subt@lion of lh. co6t ni 1524.5.

JD=\NTQ65.2t(UW+ 17 16)+ tl]/r(30.600t (uMM+ 1))+ UDD+ B-l 524 5

+ (u HH + (UMIN + USEq60)/60)/21

(2.4.r)

ln both th. lh.orics VSOP8? tnd ELP2000 lhe €Poch is the J2000 0

@ftsponding ro rhe Julie Dsr.2451545 theEfot one tin lly c.t5 the lin€

JD 245t545 Ar

16525 1155760000

(242)

Notc rtEt in rhc l6t st@ lh. lid l.B on 0E lighl hfld side is the nmbd of

lulie entude elap*d sine 12000.0 ad th. s@sd l@ is for At which is @allv

siven in mobd of.eonds od h.rc w. nccd to @nved n i o nmb.r of c€ntuies (lio

ELP &d Millemia for vsoP) due to which il hN lo be divided by the nunbq of s6onds

in a Juli$ c€nrury ifor ELP md Julim Mill.inia fot VSOP).

lf and when ttie dynmical tim. of d cv.nt is *rom we need io @hv.n rl b

U.ivesal dd Lhen into Local Zon. tim.. Thc dylmical tihe is oblain d s the nmbet

of Julie cenluries sin@ J2000.0 e that rhc Julisn dare of tbe evmt cm b€ calculated 6:

48

1p - 15575s.(, -

=-4-), usrsts

t 3155?@@00J

ea.t,

TIlc int Sr.l Ft ofdF Juliu D.!. *ill b. onEtud io Cdddr Dd. .nd

t|| hclion l Frt ro th Unirwl Tim:

St F2:

Z-INT(JD)

St d-3: F-JDZ

Fd &rB Fior ro ocrobcr 15, | 582 (JD = 229161) rhc inl.Sq p.rr of.ID

is !.oircd ! il ii olhdtuc ldjulr.d for th. @diti@ of lhc GGgorid cdqd{

St.pa: ifz<229161 ( t-Z )

Ert t a-ln((ZlE672l6.2tyt652 t.2t)

A-Z+I +GINT(d/0 )

Stcp5: B-A+1521

sr.p6: C-INT((B-122. ! y36t.2t)

SGp?: D-INT(J65.25.C)

Stcps: E-IM((I-DyJ0.6001)

Stcp9: dar-EDLIItT(30.6001.8)+F

StcFf0: UDD-tNTldat) LDD-UDD

sr.Frr: ttE<l4 ( UMll-El )

Els. ( UMt'r-ElJ )

a9

LMU-UMM

stcFr2: alMM>2 { UW-C-1716 }

Etr. t UW-C-17tt )

LW-UW

steFf3: hout(=drylM(dqr))r21

Stepr4: UHR-tM(tbu) LHR-UHR-ZON

If (LHR<o) ... ... d'o@. dar

( UIR-UIR+2I: LDD-UDDJ:

r!(LDD-0)

{ LMM-UMM-|; LDD-dasOMg:

It&t4M4')

( Ltl:'l'|2;

. LW-UW-I;

}D

It(LHl>-2,t ... ... tt@t et)

{ LHR-LHR-21: LDD-UDD+,,

4t6-DD>day1vut4

ste!'rs: htn .=Aov-L| 9.60

'|

LDD-|: LMM-UMM+I:

y(LMM>t2)

{ LW-UMM+I; L tN-l;

)))

sreFl6: UMN-IM(nlnut.) LW-UL{N

steplT: tu.ond-(ntr.-LMIN),60

50

SGpf8: USEC=lNr(taond) LSEC-USEC

Sr.pr9l Outpd UW. UMM IJDD, UHR, tjMIN, USEC

A"d LW, LMM AD LHR, LMIN, LSEC

In thc Nd-M@n Algorilhn thc our pur is thc dynmi@t lin. &d this dgon$D

is p.niculaly usfll in @nvcning Uis rinc lo U v.Mt od &y l@.t @G rin .

2.5 COORI}INATES OF THE MOON

For rh. dckmination of lh. c@rdinrcs ot 1h. M@n .r ey Sivcn lc.l tim. .rd

dlt liB| scp is ro fomular th. rinc a4um.nt a dieNcd in diclc 2.4. So lh. pcc$

bcsins by s.l*ftg pl.c. of ob6.rq (Ljnsirude &d La ud.), te.l darc ard rim. rhrl

lcds to lhc rim. &Bum. t eco.dinS ro th. atgorjlhn dc$ib.d ,bovc, s:

Julon Do'. - ?{51545 N

16525 1155?60000

(r 5.t)

Using Uis rim. arsumcnr th. @Bhcrio! ot rhc dh. *n6 d.*ribing rh. twr

c@'dinar.s is do.c (chapo -Touza ed Chaponr t99t) s dieled in aniclc 2.2

StF||

Fo..dipdc longirudcofth. Moon us.2.2.2 ro2.2.5 ed suhdrut lh.ir

csulrsi.2.2,l.

Sr.F2r For cclipdc t.hu& of U. M@! E 2.2.? 10 2.2. t0 &d sub$nuE th.i

Bul{.in2.2.6,

St.Fl:

For gec.nr.ic dilllre of rhc M@n uc 2.2. l2 ro 2.2. | 5 ed subsritut

lh.nrcsulr.in2.2.1L

5t

DE to lh. nodotr of thc ob6*d. th. diual .nd lhe mual motior of thc Eatth.

poiilior of €E.y objer in th. sky is afrctcd by 0!. phcmmn of Abcmdon- Th.

followinS @cid.ntion is only for rhe Eaih'Moon pls.t.ry Ab.dlion sd do.s lol

includ. the dimal nolion of thc ob6*er (W@lad & Cl.mcnce, I %6).

S1.t-4r CORREC'TION FOR ABERMTION

v-y4.000 t 9521-0.0&nl059.skQ 25 1177 198 9\) 12 5.2)

U- U-0.40001 754t31h(183 J +48J202't)

(2.s.3)

R- R + 0.0708.C6Q 2 5 + 177 I 96. 91r)

(2_5.4)

Fimlly a 1he lrue €quinox of ft. d.y and dE mcm €quinox of lhe dly de

difiedt dE ro thc phenom.non ofNuiation the F€.is cerdinab on nol be ob|dircd

without th€ nul'lion in lonS itudc AV and the nubtioh inobliquityAs.

St pS:

CORRECTION FOR NUTATPN

a'r =l0r 't(y, + y:..)'si,{/,I" +/d' '. +/d' '.r 'lo r)

(2.s.t

Y=V+ L,t 12-5.6)

. -22.63928-0 0ll\+0.555'lo' t.r -0,0141'l0j

r tl

(2.5.7)

^' =lo I 'I(., +,' '.). cdko) + /,r' . ' +rdr ..).lo 1)

(2.5.8)

(2.5.9)

Tn€ v.lu. of F s, !,'3 md G's Bd in lh. qp63iont ahovc tt! gt!€n in

thc T$1. 9 in Chlprcnr-Tooze 6d Chlpotrt (Ch.pre -louz; &d Ch.pon!

l99l,pp.l9):

Oft. rh. @retion dE to .ubrion is dre onc my so ro 6trd da Eq@tori.l

c@dihal.s, the Rjght Asnsion d dd thc dcclin tion 6:

St.p6:

EQUATOI.ULCOORDIANIES

"=r-'(

c6(tpriio")Si4v)C6(U) -.t4 Eps i t d) Si dlu

'l

(2.5.r0)

6 = sn't l3ih(Epsitm)Si"(v)cos(UJ + cB(Eptiron)sinei (2.5. I I )

Thc* re rh. nle cqubri.l c@'dinar6 of lh. Men wnh EfcEn.. b th. toe

L

cqulor of $e dare ahd the trua dyunical equioox of the dar€_ Th€$ m stiu lhe

s.oqtic c@diMl* md lhc afr.cr of lhc D6nion of lh€ ob*rcr on rh. glob. is ycr ro

b. lalfl ifto @ounr so tb.t rhc ..iopoccnt.ic" c@rdid., (coordin t€s relariw lo thc

posnion ofth. ob*fr.r) nay be obl.iied_ In odd rcd5 rh€ !ffftrj oftE ..paraltd" arc

to be t t.n inlo &count. Th€ Pdttd r i! givcn by:

12.s.tz)

wh@ 4 ir rhe g€catdc dist4. of dE Mer HowEr rhis q@nry paml* d.Fnds

on rhe Hoq Ancle (ihe since ibe object cosad thc l@al n.ridim) tor whicn w. ncd

thc lad sidc@l tin. LST.

St FTl

Fqhtlate tiw dgM't tlor t UTlq !h. dat. ,tdet cwidqatio,.

5l

I. =6r4t-50'.54841+E tot84,.s12s66I,+0'.093104.r, -0,.0000062.rr

(2.5.1t

gits th. Grcawich Mcan Sid€E5l Ti6. .t oir UT of rhe drte. Th.n lh€

Ctqwich Si&@l Tio. for &e rirc ug@nt for ih. rimc of ob6qvation is:

T - To + (UHR + (UMLN +USEC/60y60)a0.997269j66JJ (2.5.t4)

'Iijen fic Hou anglc at ihis moncnt ofthe Moon is:

H=T a

(2.5 rt

srep.sl tllecôfPatala,

If p is th. goc€rntic 6dius ofrh. Esdh, p,b ihe S.ocenkic hrirudc

of irE ohsftr ih rh. dsh1 .s6io! a' ed rhc dEtiEtjon d, aft.r

co@lion for prhllar &€ obr.ined $:

@sd - p@sp'sin rcos l/

(2 5.16)

<2.5 t1)

d'="".I

Gind -psin9 sintr)cosaa

ccd-pcGg'sinr@3tt

(2.5.tE)

Fin lly ro otltain rh. Lad Hdianrrl C@diDtcs A2inuth.nd rt AlriMe E

haY€ lhc followi.g lissfomalio$:

SrcFg: Azimu$

(2.5.19)

54

.Aliitu& r=sinpUnrlcdp'cosr'costl (2.5.20)

This complcGs ihe der@inarion of dlipric, €qurqid .d Oe hondnIll

@rdid6 of rh. Mor

2.6 COORDINATES OF TI{E SUN

Foi rhe d.t.mimtion of 0E c@rdiDre of rh. Sun on. p@..& in €runy rhe

se say a for lhc @rdi!!re of rhc Moon.

slcFl:

Scldt ple oroh6dvd (Longiru& ed tilirud.). lcal d ..dd rine rhat

lqds to rh. thc egment t &coiding ro thc algolirhm dceiib.d lbove, s:

Julian Dde )451545 Ll

1652t0 I | 55?600000

12.6.1)

Using lhis lire argment lh€ connrudion of d. (m. eriB &enbins rh.

c@rdinates of the Elnh s givd by Br.raSnon & Fdcou (1988) thc h.lie.nrric

coodinales of lhc E nb de obtained lhar ft lder trusfoftcd into fte gre.tdc

c@rdimles ofth. Sun s follosl

SreFl:

For heliohllic dlipric lonsitud. of lh. Flnll' in lirc wit[ (2.2.19) &d

2.2.20 w h!rc:

2^,

"e"(,,

.r^, )

(2.6.2)

2""

"."6"

t r,,l (2.6.1)

55

t,-'i^,-"(t"*", \

2t

L,-ZA,s\8,+c,. I

c2.6.4)

(2.6.5)

L. =ZA,@l\tr+crt )

(r5.6)

!,.:r,cdt r +cir ,

c2.5.?)

Ttr. l!l!.3 or A'', B'r .d C s for {i! d$n r 6ior c dB b, Mc.!i

(Meu!, 1998, A!p.ndir-[], pp. 4lt.a2t). Finlly t!. lt lio..onic loBitd. of |h E lt

i, -.

t0'

(2.5.8)

St p2:

Fd Fliptic ldintdc ofrlF E nh;

lut

Bo-Z/,.d\8,+c,. l

Q.5.9)

,r =:/,dF" +c"4

(2.6.10)

,, . X,{" "o.(r"

+c,r J

(2.6.l|)

Bt-244a.+c"r I

(2.6.r2)

8. = t,l,6.F" +c"' J

(2.6.|])

Tb v.lu.t of A's, B't .rd C'! 6. l} {oi.t nds tt giE bv Mo.u!

(M6n, l9S, A!0.odtr-Itr, !P. 4lS-421). Fidlv lh. t lbc.otic ldlidt of th' E ib it

l0'

(26.r1)

$.Fl:

For lElioc.dric .lisl&e of th. E|nb

&-X,i"6!p"+c,r J

zt2 t

xr - I,r,6lp" +c"'J

(2.6.15)

(2.6.r6)

(2.6.17)

nt

13 = t,r"co't +c"' " J

(2.6.rt)

t0,

h=>an(fr]\B,+c{

(2.6.19)

)

lo/

xr = t,."co4p" +c"' I

(2.6rI))

19 l

x, = t,,r" o.[8i + c,' ]

TIE vds of 4.. B's !d C" b. lb eorlr Ei6 & tih b, M6!

(Mc.q, 199E, Aprcndix-lu, pp. 4l E-421). Fi!.lly tlE helio@tric di!l&c. of th. Elnh

ZL,r

" = ''id,

Q.62t)

A!" Bs lnd C! N all in hdiM for looSin|d! , a|(l ldltt|d. t B6 od Ct c in ttdieE

.d A! i! .*!doiql uitr tu Llio.ldtc diaec X.

51

The coftcctions for Abe@don md Nuradon de done in dc sanc way as St p4

and Slep-5 b€fore. Fi@lly te conveBion to rhc equaronat @ordinates dd then b

horizonlal coordiDr.s is als done lhc sme way 6 ws done ror fte Moon.

2.? RISING AND SETTINC

For the d.refti@tion ofprecise rihings of rhc setrnE or nsing of& obFcl one

requircs pEcis c.l€stial coordinales of $eF objech 6i lh€ insht of1he occunine ofrhe

phehomem However! these insranh dre the points ofime that we rcqlic ro find our so

rhar a process oasuccessive appoxinadon is needcd lo ahv€ al lhee dhes. Such an

iterarive plocess is nec€ssry b4au$ rhe objccis under con$dsaion (lbe Sun Md the

Moon) significantly changc thei position relalive to the ixed cel€sriat sDhere durihg m

i.terval around ha a ddy. The whols process shns sm m esehare ror the ritoc of

'ransn ofthe objed (ove.lhe locatneridian) wbich lben reads ro,nlial esrinars for de

holrmghs ar th. appoxihare line ofrhe rising orselling or thc objer Th€e aB in facr

the csimares for rhe tocat sidereal rinres ofdc phenonena. rrco these esfnales of rtE

sdceal rimcsoflhe eveors lhe unilecat hean slarrimeardlheh lhe toql lihescan be

cakuhted. ths. fist approximarions fo! lhe ransii rbe nsihg ed rbe sedne are

obtajned usinglhc cetestirtcoordjnars ofthe obj(r evaluard ar Oh. Ur. ar any poinr or

rhe glob€ and ao! any otrhe evenrs under considecdon lhis nomeir ((]i, Ul oay be m

cail.r or a tater nom€nr. This is the rson lhar the mrfial catcularions are onlv

rppo\imaro cdlcutduon fo.hcle appo\,dar. ,t.., "t

,h. .,.",. ,h. ."i..;;:.

coordnats of thc objecr hNc to bc catcutaled asain and wholc.atculalions henLion€d

above !!€ repealed for bekr esrimates, The details of rh€e catculalions arc describ.d in

(he followinS p@sraphs.

, . Forrhe sh. rhccompnaaon. &e simptes conp@ rornore tor lhe Moon. the

,(o nme or hNir ot rhe Sun can be hirra ) coturdered ar t2-,tocsl rzonc, rime

wnecas for the Moon n dEily vdi*. Th6 Unileisat Tihe or loql tuir is simply 12

Zl wheE ZT (ene ne) is posirive for the east lons,udq ed nesarivc for Resr

58

lonSitudcs. At uivcrssl rime 12 - zT wilt b. a dn€ of 0E se darc in O@wich s Ih.

loql dae. Th. rime dgu. for $is rine ed date is rhen fomulatcd dd th€

coordiMtes offie sunar.obaincd. wlm m objdr h in rmsil irs hour dgtc (HA) is

zo ed its dsh ecdion (RA) is se s ln. Loc.l Sidqdt Tin. (LSn siNc:

p.? l)

As3Ming dar lhe obj*t hs r{A - a !r da rine of t@t Fa6n e.7,I ) shos thal:

d= LSTt

If JD is rhe Julie ddc for $e dly ar d' UT 0En wnn , = eD _ 2451545y36525

mcaur.d in Julian centuri6 lh. c€enwich Mqn Sidc,@l Tide I, {CMST) b siven by:

4 =6141-50,.s4841+8640184'.812866.r+0'.093t0,t.r: _0,.000006rr,r

(2.7.t)

Thch for the observer ar rhe pla.e vilh eeographic lonerrude , (negative fo. west

lonenud.s a<l posirive for rhe con) rhe ccnwich sid.rea! lme ot r,asit is:

csTrr=LsTtr_L

(2.7.4)

And lhe SideEat fme elaped siNe rhe OL UT of fie dar. is,

T,=CSTtr_To

(2.7.5)

This rim. is rh.n converrd b lh. UT by:

UTU =Tlt.0Or?l?909175

(2.7 6)

59

FimUy the coordides ofthc Ss @ Edcdar.d for UTE qd 0r followins cdculadon

I,sT = a,

csnr-LST-L,

n_GSTtr-To

(2.1.7)

{2.7,8)

(2.7.9J

uTr=T,'L0027379093?5

(2.7.10)

So ihat theE is a dilfer€ne of l€ss drs a *cond b.tween one v.lue ofuTr ed ib.ext

[or lhe locat sunrise and sunset $c base vatue $ the UTtr ahd inirial

apptoxmalDns for the sunnse k UTr - 6h = UTn and lhat ot tnc suMt is UTr + 6r -

UTst. Dcpendi.8 on rhe tocat longirude L, UTs ed Ursl nay ti.I on pr.viou or rexr

day csp.clively so lbar a necese,y dare adjutmeir mul b. dorc. runher ddpe.ding on

the local lairude il is funher possible thar these pbenomcna sihply donl occur.

The pbc.s bcgins

Urs0. The holr angl€ HA

de c@rdi@res of rh. Sun for UT6 (or

stt'ne or nsing ovcr a l@al horjzon k

(2.7 r)

stEE I is rh€ eeghphic laritude of rh. obsedcr lnd lhe au nd€ of d. poinr of sky is

a$med to b. ero, How.ver owirS to th€ phdohcnon ofEri&tion a ,r.r, rhe Su dd s

pld.l ae wll b.low th. hode. std rhcy @ a€ru.lly seo sdrjng or risin8. The

t\@g€ rFe.t of EFadioD is rhar . srat E@iN visibte de ir ft hs gone j4 e Dinu!€s

below hori^n. 'Ilis allitudc is kmh s srand.rd drilud. dcnoEo s 4, ed is @nsid@.I

6t)

lo bc -50 N sords on aveas. f6r rlc Su, 0Ei itulud.s ihc affet of dE Ef4clion

and thc sni dian.ter bo$. For a mor€ accwtc valuc the &rul s.mi dimet r SD,

rhould be calculal.d fmo dE didea of1h. Su ed suh.actcd tioh d..veagc afi*l

of Glhction, fte chag. in l.hp€dl@ in ihe niddle laritDd.s my lary thjs by @ud

20 seconds of tinc ed tbe baromctric pres6res nay caue a veiaion of dother 12

*con& of time. Ho{€v.r 6 rhcse varia{otu cd nol LE d.temired a priori the avcEgc

afects m co.nd.ed in crlculalions. Unog fte $..drd slrjM. ?01h. hou anqlc oflh.

object is then eloluared as:

(21.t 4

Thus rh€ liBr dppro\,maion for lhe rhe ofrisng of$e obled h:

T, =UTO - E.

12.1tJ)

And $at ofrhe s. in8 isl

T, =Wtr+ Ho

(2.7.t4\

Thes. e only !h. n6t app,oxinarion3 for lh€ tines of swise ahd the sunsel

Esp.ctirely. Fomularing ihe lin a8unors for €.ch of fien Fpaan.l, rh. c@rdinar.s

of lbe Su ha\e b be catculared aglin. The

Grecrwich sid.El rinc orrcspondjns ro

borh t &d I h6 ro be obained as:

r-Ta+J6O.98564TTy'15

12.7.\5)

,1 or I e rlEr l@at lFu ogte

4d ihc &imurh of th. obj..t is

12.1.t6)

Si"At = Sinpstnq +C6e,Cot6,CotH,

(21.11)

Th. cordiG for rhe tuirg ! siring d:

AT, =

(2.7 l8)

Adding these A?_, ifto rh. appropriarc I, givcs ibpDvcd valu.s

tunhq inpov€d r:lucs catcutal. $e @.dinaas of rhc Ss fd

&d rcpeat (2.6. r 2) ed (2.6.1 8) ro obrain lh. UT for d€ wau.

lor Ue Moon rhc i$uc is mt! complided s af€.r of panlta h signiticor.

Th€ !ffrr ofrcfddjon h ro dcc'le the anirh dist ne !o rhar the objed is visibl. dei

if n is th@Edcat gonc dom $. horizon bln rhe aflccr of pa6 d b to inc@sc rhe zcnid

d6t fte e lhe obj.cr is wlt .hovc lhc horj@n dd n a!,p.4 ro Mvc et (or rcl n*n

still), Thus for fie Moon the si&dard attjtude is Civcn by:

ao =E - 0-2n5. t -10134''

(2.1.te)

qb.G r is dc pa&Id of rb. Moon eivd byi

(2.7.20)

p s $. g€@eniric dist ncc oflhe

obsrvr &d X, rhe g@ce ric dislece otrhe Moon, p

\2.7.21'

62

r is th€ equrorirl ndiu5 of the Eard,

dd ., is th. pold mdids of the

E nL The resl of thc calculatiohs for rhe irlnsil, rising ed lhe stling of lhe Moon arc

th. se d 1hat for thc Sun.

2.8 I{EW LUNAR CRESCENT VISIBILITY PARAMETERS

A nunber ofpdrameleB have been corsidered imponanl for deteminin8 *belber

the new cr€$ent would bc lisible al a lcaiion oo the Eanh oi ool. Tbese weE briely

di*u$ed ad lblcd in thc b€eimins of chaper l. Once tr tiD. of th. Binn or a

paniculd Ncw M@i or th€ conjucrion hs ben delmined a numb.r of p,@eGB @

rquned to h€ d€t mincd. Th€* includ. (i) Tinc of Suroet T". {ii) Tinc of M6Mr T..

(iii) LAcTn - Ti (i!) Bcs| Time orvisibiliry Tr,(!) Ase ofrh€ Moon ar Tb. AcE. (vi)

Arc of visior ARCV, (vii) Reladle Azimulh DAZ, (viii) Arc of Lishr (Elonsatioi)

ARCL. (ix) Ph@ ofcr€s.nt P, md (x) width ofcrcsent w. In vhw of the di$u$ion

ofth. s&onomic.l alSorirhns ed rehniqud in tbis chaptd the* circmst nc6 de c-

co$iderEd ro .xploE compurdiom of vdbN psm€re6 $al e inponmr ao. $e

lnolysis of l@al visibility oflh€ ncs lunr cr.$nt on $e day ofconjuhdion or d. doy

The fiBl of th.s paludd ii rh. coiju.tiotr of lhe M@. wnn ihe Sm or tn.

rime ofBinh ofNcv hootr. The alsdirlm for lh. conpuraion of (his tim. w6 pE*nl.d

in anicle 2.3. U3ing $is algoddm thc riic of g@centic binh of lhc Ns m@n is

obtain€d. The algorilho tates yee s input and giles lhe Julian datc oflhe tine of the

binn ofnew boon. Il is imporrmt ro nole $ar the i.pur iiy€ai is ior a whole iMbs, it

is o Eal nMb.r €alculat€d on the bdb of lhe exp€cled dale of th. New M@h. For

insran@ the Nry M@n in $e nodn ofApri!,2007 i. dp.ctd msd l7s d.y of thc

,e0 = 2047 +atJ0+t7)/J65

63

An |pDDxiE.r. valu. of "'*" wirh e G@ of f.e d.,s m*! wll. If UE "yd" it r

vholc nmba $c algoriihn si!4 lhc ,!lie D.rG to! $. Nd Men th.t @6 cl@n

ro thc b.gimine ofthc !d".

otuc rhc ,ulid D.rc of$c binl ofrlE Nq M@n cl@n b lh. .xpdrcd d.y ha

b..i found tb. m. b inni.lly conwn.d lo lhc UniEEal rift dd darc ed

coneqE ly rhc le.l lih. and ddc. For ln.s convffirioE ftom Julis Dat. to rh.

le.l rim. ed drlc th. tehniqu6 of rhc .rricl. 2.4 6 u$d, Bcfoc thb im. lh. Ew

'lun.rion" h6 nor b.!un s no qNnid ofrlr visibiliry ofrhc Bw l|!w cEghl on rh.

A6ins bcfoE this rin . B.foE this tin. only th.'bld cG*dl" ce b. lat wn b.foa

rh. sutuie on rhc dry of onjudion or a dly o! tvo b.foa-

Ar dr rimc ot@dudion dc M@n n.y b. &rwhcrc wi$in . strip ot widd loo

18 eud dG elipric, i-.. wirhin 50 9 of rhc Sun. ap.n fom |h. ecume of . slar

dlips $G "@.nl'cxisl5 bul dE lo irs.xlrEh. clo*ncs ro lh. slai!8 su ir co nor

b. s..n ed ha ndcr ben sn,

Thc ndr inpondr p.md4 is $c l@.1 lim. of suel ed th. M@n $t Th.*

can be conput d for ey day of lhc yd Bins th€ r.chniq!.! of $c anicl. 2.7. Howvr

$ce rehniqEs 'lquiE

th. dctmin ion of th. c@diEtcs of both lh. Sln &d th.

M@n for D dFckd rihe of lrsi! dsing dd s.ltinS of dh of thd. Thc c@diDL3

of th. su !rc obl.incd usi.s dE VSOPE? fi6ry of Brdagnon ad fEncou {or a

sihplificd v6ion sivcn by MeE) &d bricfly prc*ntcd in aniclc 2.2 Dd 2.6. Sihildly

rh. D@iF @rdidtcs of thc M@n c oh.in d aing th. ELP2000 of ChrFo -Tot

ed Ch.pont pr*nrcd in dticL 2.2 and 2.5. Th. .oddinal6 ofrh. Sun &d lh. M@n

lhis wr e th. SMtric sph.dcd pold c@dinrl.s (dis1@ P. alatial lonsilud. r

dd d€ c.l.nial l! rud. n. And @n cdon for .bdnlion thc sc e lMtfomd lo

th. g.on.rric .q@torial @tdidlcs, dght s.cioi o &d &cliDlion 6. Firlly th.

roFcc'tric dSlu e.sion ed d.cliMdon !r! oh.incd itld.s irto cosi&tstion thc

d[dl! of pdallq. Using lhe local siddnat timc tb. @rdina€s e then trdsfomcd inlo

Ioc.i hori@ntd @ordinalcs dr. ahitud. ud &imuth,

Once $e tine of leal sunsct (T) and th.t of Moon sr (I.) d obt in.d lhe

o@ctcr LAG = Tn - Tr is elculat€d. Unlc$ lh. LAC is posnive for the New Moon

dd ncSrdv. for old Mmn th€ cFsent cd ncv.. b€ sen.

Suppoe (..,,r.,r.)ed t,.j,,r.) d ftc p@ie dislanes, aliflic lo!8it!d.

sd th. lalirudc of rhe M@n and 6c Su, Esp.ctivcly. af@d ro the ft4 .qtinox of

rhc dD. or sunst at any lMtio. on $c Elnh with tftstiat coordinaies(1.r) on rh.

day, or day aftcr. lhe binh ofNew Msn (al$ oll.d the Csenlric Coijliction oi th.

Moon).Thefi'slsteDinthedelemimtionofih.visibililyoflh€nevlunarcresce,on

thc day of conjuncrion or the day affer, is io dcternihe lh€ actual dy.anical ihe (TD o!

TT) T", oflhc coojuction. Ncxl one requires considcrins the local rines ofsening ofth.

Sun &d tb. Moon. Le! Tt dd Tn (Coordimi.d Univdsal Tine TUC) b€ th. tihes of lh.

l@.1 sunsel dd the nooi.eq vi|h Tj < T..

Using rnc .cliptic @rdimLs oa lh. $n ad rh. Men E alculared for th. T6 thc

egutodal @rdimrs of rhe two bodics (d.,4) dd (d,,t,) e c.lolat d usins

(Me$,1998):

s t h\^ )c os\r ) - Tdnl P)sth(E)

cas6)

(2.8 )

st(d) = si,,(l)co(.J + codB)st(€)st(r)

o.8.3)

wl@d, fi. nd emid is in 0r s. qudtul6 t, ed ',

the obliquily orrh.

dlipri. is d$ adjug€d for lh. d, . ed thc b.9 rine Th. tsal Hou Atgl. H is th.n

oblaincd tion rhc difreren@ ofth. l@l Sidftal Tiru (Zsl) 4d th. rishl 6c.nsion

Tnis finally gjves lhe l@d nonzontal c@dinal6 amulh (,1) $d the allilud. (ir) bv:

65

ranlAt =

c6(Ir)s,,{t) rz,(d)co{l)

(2.8.4)

si,(r) = sr,(c)si'(6) + c4(d)ca(t)c,r(fl)

(2.8.t

And adjusi.g for lhc rciacdon 6d $e heigh of $e obFNd's l@aiio. above *. Id'l

rbc topoentic @oilinares (,r,,r")ed (,i,,7t,) of the Moon dd fte Sun. esFclivtrv

In alnost all lhc nodels for cdlicst m@mid ins $c ancidl d w'll ts ibe nod'm'

ln. dir@. of eim hs (DAZ = lA,-A-1.

@lled El.tive @'ihuth) ard thn of

aldrudB (ARCV - ," - t.. cdl..l m of lision) d shoM in rhe lisr I, 'r

the iimc or

loc.l 3uEl T, sd/or al lhe b$l Tb Play a vilil ole:

As ihe dgulu sePar.tions invol!€d b.iwen $. Sutr md lh. lunar cesenl at

$6. dnes @ sdaU, wi6ou1 dEh mt th. e of lighr (ARCL) h giEn bv:

(2.8.6)

Wh.@ for larSct ugl6 or moF ffi.rc Gults th. d of lieltr should b. eldl.Ld

ARCL = cdr(co(r-)co{rr,)cd(tra - s,'(r.)si(4)) (2 E ?)

66

Fi&28!

A@d tun lhc Ft.tiv€ .rlmuh tbc !D of villon and th. e of lighr Uc cdlqir

for dli6r ytuibrliry of t|ld qqc@t rlqoits io irL ie c@li<lddio t.qbcr of

othd p!@der. One swh p€mcr{ is thc A8. of th. M@! (ACE) d!6n.d as thc tiEe

Gl4a.d .iG th. h4 cor{l|!cdo. til lt. lim of ob...Idtm Anothq idDoiird fr.io. i.

thc Wi&h of CF$.dt (VD fin d.!.nd! on rhc dida@ of fie Moor 'IrE gtel

aoom@ Al-gltllti of Brsldd hod rdi4d lt iopott c of @'@l {idth I

thou$nd r.s.go (BdiD lt7) At l[. dili.c oftlc MM nom Ot c|tih wi6 nm

eud O34 nillid 16 !o Nutd 0.4 mlllid b! th. leni{i!rcEr of th€ lunlt die

{rh. Am 15 I! oinu..lo 16.5 e ninn .. Tbut if{F Moot! is clo€.d lo F.flh.l

ih. tim. of ohdarior th! cn .!nl *ould b€ wid6t &d thu brightn wid0r of lh.

s!.c.d i3 din dy FDpo.tidtl lo lt Pte (P) ofih.ltloon thd i! t tmlid of rtc

ARCL:

P-

lr-cg?aRcLn

2

(2.E.8)

Ad tholhc c!!!6r width is givm bt:

"='"[

(289)

Sinc. ihc acienl dd thc nedieval dnes {E ACE, thc time elapscd siie fie

birlh oI New Moon lill thc sunsi of lh. d.y in quesdon, w4 eo@lly coNidecd a

signindt p&Met€r. Howev€!, in ihe tinc of MuslinvABb $lronom€rs jl had already

b.en c.lied tal 0E ACE is nor a nD&hc al padndcr, stll thc pardet* is

inporlrnl 1o calculat€. This is bccaue oflhe fact that depending on rhe aordinats of the

Mooo dd lhc seaens at lime a very "young' cEsanl ce b. ee.. Amongsl bo$ rbe

anat€u s vcll a rhe prcfcssioBl Grronom.u $@ is aluyr a conFlilion for havinS

$e €ord lo s the "youneesr" crcsr eiihd wilh oFical aid or wilhout i1.

Amongst the early nod€ls ofnew cc*.!t vnibility asltunomeB sed $c !.larion

betren rh. rcldrive altitudc (ARCV = alriMc of rhc Mmn - ahilude of lhl Sun) dd rhe

rclrriv. uimurh rDA-/ - e'murh of Lh. Su - uimdh of de M@n' Th. rso

parmeleG arc slill consideFd inporl&l as if DAZ = 0. the crcscenl is venically abole

$e point of 3uEel dd und.r such a cncM$dce the youg.st a @ll as lhc thimesl

crc$qt cd bc ecn. Wirh lugd DAZ wles only oldtr md thc thictd ccs @ be

How lhick or lhin is lh€ cEsenl al any lime cs b€ ddemined once lhe

s.pmriotr (clonsaiion) b€twn rhc su ed $c Mmn hd bccn dchi'cd usi.g (2 8 ?)

sith €nh€r horizontal coordin.les or rh€ €quatorial aordintle! This elongation or Atc or

LiSht (ARCL) hads to lh. Phde (P fiaction of the illuninatcd lund disc facing fie

obsdr) of lb. Mftn using (2.8-E). Ho*ev.a the lhicloN or $e widlh (w) or rhe

c..rral pon ol the creot des not only dcFnd on thc phe of the M@!,

't

also

depclds on rhc Elnh'Moon disrece, rh. sMc cm be ohaitcd usins (2 8 9) Dte besr

inc of lbc crcsdt visibihy sqgg€ned by Yauop is criical fot siShting lhc crcsn1

u'dd mdgjml conditions &d ce be @mpur€d sine (2 E l) sisndne i5 crirical when

68

THE SOFTWARf, HILAI-OI.CPP

In this worl( a softwde k develop.d for the odysh of rhe fisr visibiliry of new

lue crc*dt dEr is simil& io mturc 6 rh. M@'calc by M@ur (Msru, 2001) dd

Accurare Times by odeh (odcb, 2006) but rhar can b. lscd ro compae aI rhe

compur.rional and thc ecie.l dd Doden visibiliry nod€ls. T1| lisdng of the poelm

Hilalo|.cpp is giyco i! ApFndix 4. Tn progm f€nrca d bri€fly de*dbed b.to*:

Th. prog@ LinAnal ucs ih&e dala filcs, rwo for input ed one for oulplt. The

two inpul dala il6 e (he files For.ad.rv thal conrains rhe padrcr€E 4, ,, and., (s

described inanicle2,6 lbove ). The pdmetcs oflhis dala file de adtu8€d in rabtes2.l

(A) lo 2.1 (F) in appcndix 2. 'Ih. ol]],er itpnt tie is tlc .tp2MA& rhat conlaiE the

paluercs as de$rib.d in rhc anicle 2.5 above. The pdmeles of rhis dara file d€

dds.d inbbles Ll (A) 10 1.3 (D) in appendix L

Thc third fiI. u*d by lhe prcgrd is *..t lhal is optional dd is n*d only vhen

fie Esults of fi. cobput2rions in Oc preetm m EquiEd to be sbEd. Tl!. vcGion of

lh€ progr.n Hilalol giv€n in appendix 4 hs the lile nanc scrrrgaA, rhar sbres lhe

onpulalional Esulrs of a sinelc .x@urior of $e prcgrm. fte i.fomations stoBt in

o. No., rhc obsenarion nhber thar n scncally digned by Odch (odeh,

2004).

Date. datc of obwatioo,

Long., th. longitudc of 0E ple ftom wh€E rh. clesnr is ob6eN.d

Latil., th. ladMe oflhc ple

Eld., the .lcvarion of rh. place above s! levcl,

T.mp., cnimred rcmFdturc of$e riDe of obswarion,

Hmid.. .stibaied relative humidny oflh. plac. at rhe rine of obs.Nadon

S@.i, lhe l6al tinc of sNet

69

9. JD of Conjunct, rh€ Julie Dat of the binh of.q M@n or the lsr

10. As., lhc a8c ofrhc Moon.r rh€ besr tim. e@ding 10 Yallop (Yallop, 1998),

ll.

L{C, the diff€rcncc berNcn th€ Moo. *r dd rh€ sunsel. it minucr,

I 2 ARCL, dc ot light or elonsetion of lhe M@n fion th. su at the b.st tme, in

ll, ARCV, the eladve altitudc of fie crc*ol .t the b.st 1ioe, it d.crees

1,4. D,AZ the rclaiile @ibulh ai the best time, in degrc.s,

15. Widtb, the cenlral widlh of lhc Fsr€nt al bcs1 rime, in dc minues.

16. q-val. rh. visibihy ptue1q dcfined by YalloP (Yallop, 1998) lo h.

discssd in d€rail in chapl.r 4.

17, Phi+della, fie dglc thal lhc.cliPlic dalcs with lh€ venical on th. wslem

hoii2on, in d.sE.s.

I E. Mlalit. I,loo. s 4liptic latiiude in d.8e'

19. Mlongi! Moon's.cliptrc lon8itudc. in d.gtus-

20. Slongil., Sun\ *liptic longitud., in dccres

21. M-SD, dgule s.minid.tcr of th. M@|L in @ oinut s.

22. As-Fac! e of sep@lioh fsctor d€fined in l.ler chap1e6, in dcgrc6

21. R-En, cslimat€d Rip.n $ Fucrion vatue dclin€d h chapler I

?4, R-A!er, averas€ Ripeness funcrion valu€ dellned in chapl{ 3

25. R-actual, acbal Rip.ns Fsction vtlue dcnled in chapls 3

26 DR+s! $e diflecnc€ of R-rclualand R-Enimaled

27. DRnc! thc difrccne of R-a.tu4l ed RnErage

28. MoonS M.g, Moon\ mogniiude al the besl lne of Yallop,

29. Lio-Mae(lin€), lh. LiDiting Mt8nitud. of the stv ned |ne c€s6t s

defioed by Scha€ff.r (Schaefd, 1988b) aloDs wnh ih€ Yallop's besl dne in

udveEal dne dis.ts.d in cha .r 4.

30. SI'(DME8), tbe univ.ssl tine wh.n the conrrdt ofthe stv bdelhes dd

rh. Mmn\ bljehBt just tum in favou of thc Moon dd 0E ditr€Gnc' of dE

70

m.gnitude of the Moon dd $c linning m.gnitutle of rhe sky ai thal monen!

I_

B6(Dmag), fic divdsd rimc *h.o U. @ntrasl of rhe sry bnSh,ES ed

th. Moon\ bdghret is b6r in favou of rhe Moon ald thc diflcn@ of $e

ha9irude of rhe M@D ed thc lim'ring m.gn'nrd. ofih. sly al rhar monen

32.

Lsr(Dmae), ihe univdal tme wh€n rh€ conrmst orrhe sky bnshhess ed

tbc Moon\ biehd is lal io &vou of th. Moon ad rbc difcrcn@ of $e

nagoirude ofthe Moon ad rhc liniring magnitude of$e sly ar th.r momenr.

A3 thc .x€cution oflhe proghh b.gins ir prompts for ob*Nadon nMb.r, dale,

nonlh, y.&, longirude, Iaritude od cl.vation abole sea l€vel of th€ phcc and $e

esrinated ttmperaturc ahd estimrd r.larilc humidiiy of lhe place. This plompr is

iniliared by rh. lunction tzrdd,r.rtnd callcd by

the tufuri,on nt i n n u t i

"..

Ticn rh. prcg@ c,lls for $e tundi@ math .h4tg.. This tunc on n6r

ddmin€s lh. Julia Date of dE dDc of n.esr @njwdon or rhe bini of ncw M@n

crlling thc fucrion /t ]@_D@r. Th. tu dion r'r_r@_r@r is bdcd on .tgorithn due

ioMeeus(1998)discqsedinadicl€2.3abovc(fomu162.3.1.ro2.3.25).'Iletuncrion

rorr, cna,a. $en calh the tu.ctior renir&rr lhal dekmines the tihcs of lhc sunsel

dd the Moon set tttroud fiD.rioB tun-q.t 6d mon_se.Ih€ tunciions rzu_r.r sd

tutu_sa t. b6ed on lh€ alsorithn dhcus$d in aaicle 2.7 abov.. Dr functio.

rdttirgt.le calculars rhe b6r rimc for casent visibitiiy aeording io @ndnion du€

to Yallop (1998) di*ussd h a larq cha .r Tnis is followd by @hpul!fion of the

Julie Dat 6d dB Exlcnd€d Juli@ dat corcsponding ro rhe best rim. 6ing rh€

tunctiolirri.ndrl. rhat t bs*d on rl| algorirhn pEdred above in anicl. 2.4. This

Iesds ro fomulation orlhe rime {sM€n1GIs dipused in adiclc 2.1) fo' rhc Elp2ooi)

md VSOPS? lheories for the c.lculation of coordimres of lhe Moon and ihc Su

respectively. Th.e coordinates e calcul.r.d according b the algorithDs in anictes 2.5

dd 2.6 iopl.m.n&d in fucdoB a@r_.ou.t ed t\. su"_.erd Bpeilely. B.fore

7l

rh*.@rdin l€s @ c.bdat.d th. etr€c1oftuhio. dd the sidd.d tihc coftspondins

10 zdo hou uiversal tide, for the dat€ consideed. are calculted using the tuncdois

,ttttid ad srl_tiw, E#iv.ly, The infomatioi/d.tA thu fe Scn nted is displayed

od *'€.n usins rhc t$aiqs ouinlo, rtbpt t scootd (.@fitn^.s of 1he su.) ed

dbrldr_n rord (coodinatcs of tbe Moon).

Finally, fhc tunction nainrcutir. crLtl^r.s md dbplaF on $en aU rhe

visibiliiy parmetcrs dircused in pFiou.rricle (2.8) &d lisred ss l0 to 16 dd 2l

abov. in lnc clm anicle. O$d pffi.r.c (l7lo 20 dd 22 1o 12) re @mpui.d in

orher functions (r? to 20 in tunction r@r_cMtt a\d 22 to 32 tt liM'si\ brl

display€d alo.g with these pdmet.rs. All thc Ddmd.s lisbd as I lo 32 d€ wiren lo

the outp dda file J.r"8ear in nmloulrro4, (pam€rs I to 29) ed tim._nnse

(p@ncle6 30 to 32), Tle tundion /ltr,r,,? is ih€ reproducrion ot Scba.fer's pog,aD

(S€hs.fer, 1998, Bogd, 2004) lor delediiirg th€ liniting hagnitude of dy point of

The Esl of the functions used in lhc pregrm aE lisLd dd briety dcscibed

/.ar.r..l chects lhe1hd the y.e u$d is lcap yed (ac@rdi.8 ro crcsorid

rule) or rcr? lfrhe yd is lap lhc tuncdon rerums I orh€fli$ O.

.orwt_r'ru convens an eelc inro dcgres, arc minutes do arc *cond3.

.,,rzt_rtu convens lime in hours into hoE, ninuIs &d s*onds.

hodlw Btms rh. @jnd.r .ff4 dividing th€ inpur b, 160. 6is fucdon

pamt

h 6ed ohain 6e mst. in the tuB€ 0 deercs b 360 degE6.

calculat s rh. afr*tl ot psrlts in i8hl acEion ad decu.adon

to cct th. ropcdFic ddr as1fuion 6d d.ctinarion and is bas.d

on step 8 of alsori$n in anicte 2.5.

itr._r.. rhrcud ,rr.-do, ,nd d"c_sc thrcugh d.._r,,: Ttcs. tundions altows

chdging tine (al ld.b ofhourr ninure ad ssnds) and dare

(at lev.ls ofdale dd honrh).

72

In fer sihin lhc tuldion di.i@{tt. lhc prcgr&n Hil.lo! ha

^

do-|9kL l@p

rhar remiDrd *tEtr rhe iddrifid za,r rcccivB V' (pl*in8 dld le,) fion thc Br.

If my oths key is prcsed lh. pregm rmdN in wait ndq PEsing Panicdd k ls a

is obvioN in the er.l{ar. cohbilsriotr vdio{s&lions e iniii.l€d likc inc@ing or

deoeasiry tim€ dd datc, chegilg tmpdal@ or hlmidity o. wiling to th€ output nb

r.rngafir. bitiad.g any ofthFc.ctioro Gp..G th. *ftution ofall lhe conpulalions

wilb new timc, dale,lcmp.dture or huDidily. In cd. ofpre$ingP all lh€ data is winen

ro fie out file s.r'"a4.t in one lin€ a.d prc$ins 4 Mit s only rhe $lected !alu$ oI

tine and conesponding value of thc diffelcnce of doo.\ magnilude and $e liniling sky

masnitude. Tlese f€atues of thc progm Hihlol (varyin8 dne, dale and !@lhcr

conditio6) nEle tbe progrm norc dynoiq .s compdd lo Mooncalc of M&ru

iMdd, 2001 ) ad A@uai. TiF. of Odeh (Od.h, 2006).

In view of lhc prcbls of d.rminiig th. dlr of the fi6l sier ing of new llld

c@.nt in this cnaFd w cxploEd.ll thc Mjor 6pet of conp arional cForls. Th€*

IE sm!.ndylicd dyn.Dic.l lhddcs VSOP87 and ELP2000 d'at dryib. th.

motion ofplseB loud thc Su sd oflh. M@n reund the Eanh. The* e m.

mosl rent dd nost accunt. avlilabl. looh for the delmi.ation ot

ephendb oflhc Sun dd th. Mootr.

Ite ateonlhm lhat lelds lo lbe dct min.lion oftb. dynmical lime ofde lud

@njucdor or lh. binh ot ncw Moon.

Tn€ pmblms ssociarcd wi|b th. dynamical &d uiv€Eal lim€ ed related

isues. Wiihout having a @mpl.l. kno* bow of lhse isues apprclrjalc tbe

argmcnt for the d.lminadon of lud ed dE $l& cmrdimres co not be

1)

TL dl irpde dgsitor tu d. ddroni4 m. tr..r,r.l od locd roc

dn . o{ th. @ rd 6. md* lllth! s* d6itr of ti.

dD.r lb pcr@i !rd&d tflt t Fb&o of di- dShdrS o{ rtr

le or*dt ci rdt!. da-!.d.

lrib I tdrwh ||I&tbt r of d| 6..c tt{itr d .|6mi.l G.hiS..

erl .ltdl6or | .orio!.. F!!im ir redd b &r .U t ,.w lull

Tto 1b ct# &ir_.r .ll tt cqdrilrl ,trrit r@id.d eith it.

plobLa ofib. d.olloilirS tl. dry of6rr jihli!8 ofew lurt.!..c.d

71

Chapter No. 3

ANCIENT, MEDIryAL & EARLY

2OTH CENTURY MODEU;

For calendarlcar purposes .s we as an Interes0n8 and chalte.ging

aslbnonlc.l p,obleh, jn rhe hbrory ofmanktnd there have ben constsrenr efiorts

ro delermano when and whererhe nek tunarcro@nt wll be lrst seen, t]e,esutrs

of these ealons have boen numeroG 6nd have been based on dtlle.ent technlques

6nd tools.lhose results can be temed as 't odets/Crireria tor the oadtest vtltbi|lry

or n€w runar cfescenr,. Most ot the earty modets are de ved dtioc y trom the

obs€oations of the new tunar crescent and are enDnjc.tin natlre. Some are based

on th€orellcalconslderations. MGI tmpon.nt of these eftons hctud6:

The Aabyroni.n rule of thumb .when the age ot the .ew Moon is 24

hooB only then the crescent can be seen,. fte redlscovery of the

more mrrhematrc.lly lnvolv6d crfte on ot Etong. on + LAG > 21.

lhal w6senunciatedw€ beforechisrian or. (Fatooht et at, 1999).

)

Tho cdena deveroped In ihe medtevat periods by Mus ms/Adbs

ba*d on obsenarion, sph6ri6t vtgonom€try and lhe ephemerid of

lhs Moon and the Sun (8rutn, 1977). This happen.d as earty .s rhe

llr)

The exploration or Foth€ ngham (1910), Maunda (1911) and orheF

ln the fi6l quarter ot the twontl6th @ntury. These effort! were moG

ot slallsti€l in natur€ based on th€ obseda ons mos y in Room by

75

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

di<ov.ryofthe Babllorlan cdrorla for tho 66t 6t vbtbltty ol new tunar crcscenl. A

sel or obsenatlonal d.ta solectld from rhe r*6nt tlteratore ts lsed to examtnerhe

ments orlhe Sabylonlan cnbrbn.Thts ts folowed by 3ome sphencat trigo.omeuc

conslderations ihat are llgnlllcant tor th€ condilloN on whtch the vlsibitiry or the

invisiblllty of th6 n€w lunar'o6cont haydepend.

On lhe basls of lhe t,igonom.tdc cofttd€htion the Lunar Ripene$ funclion

altrlhuled ro rheArabs rs.xplored and a tvodlfled Lunar Ripeness raw is suggested

and examrned in vlew ot th€ ,€.ordod oEerva{ons h rhe dara sel men oned

above. Thi3 Is dono uslng iho curient k.owtedgs, loots 6nd rechniques and the

.6uts of rhls erprorarlon ar6 present6d. a[ codputational wolk ts done Gtng the

In the end ot the ch6pter the gfrpktc.t modeL due io moden asldnomeG

olrhe early 20s century ar. retstt€d and compartson ot ihe same is done wilh the

Babyronlan and the medievat mod6ts.

BABYLONIAN CRITERIA

No lrisonobery or spheriqt rrigonomelly existed till rhe halheharical

developnenb in,\rabia in lhe 9u dd rhc lorh cenlu.y AD; the probteh w6 o.ly

descnbed i. rems ofcertain obseNable paranercs shown in rhc Fieure No Ll I in lhe

dcienr and lbe early Chrhlian e6.lfdny visibitiry $ireion was deduced musl have been

basd on the observations and nedsulemenrs ot rhes observable qumtities. The figure

snows lhe sesiern bolizon a dnc when the sun S h6 gon€ r deerees bebw norizon

(the sole deprc$ion) and rhe crcsenr at M isjusr visible. AB is rhe equaror lhat nakes

an agle q, $e geogBphical latitude oflhc ptace, wirh rhe nomat ro the horizon. SM n

rhe sepeadon b€teen lhe Sun tud $e tvl@n comhonly knoM s .@ of lishl. and

16

abbelialed s ARCL but shoM 6 aL in th. figue. SD is the difidce b.isFn the

alrirudes of the su dd rhc Moon lnow q "dc of dcs@nr" d€noi€d 6 ao (ako call.d

.e of vision dd abbrevi.tcd s ARCV). Not that the aidnrde of lh. €Esenl abovc

hodan is,l $ lnal a, = r + ,l As rh. ooitr( A is al lh. se altnude 6lhc cFsenl dd

B at lh. sme depBion d th. su, A8 is .quabnal $pdti@ b.teen the Sun 4d rhc

Moon called "arc of *pdrion ' ad is .qnivd6t b rhe Moon*r - SuNr tig shoM in

the fisld d as The fig@ cd bc u*d to h.aua fi€ 4.s ofdesenl ed liehl for a

Sivcn lalitude I once an accurare enough .phemetdes shows the dc of sepdalion and

wheE to loot for the dimly illminaled cBcent againsl lhe briehl evening twiliSht.

lig No.3.1.1 Angule Pamcters as@ialed wilh sighling ofnew lunar crescenl

l, modem tines the earliesi Ffee.cc of ey syslemalic study of the earliesi

sigbring of nev lud cEs@nr is thrt du. rolhc.inShm (Fothainghd. l9l0). He refe6

ro l2s enlury Jryish phildopher Mcinoni<16. Asordine to lorhcringhm. the work

.ppearcd in the TEti$ by Mlimonid.s (Moshc bcn Md@n) o\ \\. SahcliJi.otion oI

th. Ntu Mo@ 0",islre''.'rotalt: Scfq z.hnim - Hilhot Kiddush H'Hodcsh, I l?8 fial

77

wA diislaled by S. Oddz 6 Crlr of Moituntd.s, Boot Thr4, treatite Eigtu,

SmdiJicdtion ol the Nee M@a Yelc Judaic. Seris, volh. XI, Yatc Univesity pE$,

No Hav6 CT, 1956). Maimodidcs mrk w studi.d by von Lifl.ow in

SiEunghberichk da Wiehu Altd.nie, Math-Natrre, Cl6e,lxvi, (1872), pp.459-480,

ln his *ort Maimonides nak6 rh. snalLsl visible phe ofthc M@n depqdenr on trc

variables, as claimed by Forh€.in8hd. Th.se veiables bave ben tem€d ss rr? rr€

etonAorion otthe Moon a\d tle aplbre.t ahgle of tision - F6\.noghe slarcs rhal if

by hgk ofrision Mai1onides m.es the diffeEncc of the Sun md the Moon in rc.irh

dirlaccs then th. nle of Mainoid6 .nd his om rule for hinimlm visibt. pb4 of

Moon ae n€dly the enc Howevcr he f!fther says lhat Maimonides rute sivcs slidrly

lower oininun alliludes that could be duc tc, b.(.r ob$fring condirio.s in Jeruslem

lh& in Atbos. Unfodu.alely Fohednghah .dmik that hc could not sumcienrly

undedand Maimodis' dithmdiol nelhod dd has not giecn a dchited @ounl of rb.

same (Foihciinsham, l9l0).

Lder, Bruin(Bruin. 1977)gives a moe debiled eounrof any medievalatenpr

ol solving fie probl€n of findin8 sronomi€al €onditiotu for thc fi6t visibility of

crsc€nl. De$ribinglhe Isl.mic astonomical procedurcs Btuin says $a1Al-Khurizini

gives mathenadcal rules and rablcs lor pedicling thc new cre$e.t where6 AlBamni

peeds a conplerc soludon. He aho nenlions lh€ lato accounl ofMo$s ber Madoon

lnd clains rhst ibn Mamoon larsely follos Al Balloi. One oflhe nost inponanl dp.cl

ofBruin's accounr ofrhc eflods oftbe bedicval era is rhar Anbs^rulins orthis ea had

olEady rcaliz€d rhe sienitlcece of cE$cnr widrh. This inpoiant aFcl enaincd

hissins frem all th. najor cooiribudons of lw.ntietn century bcloE Bruin. Bed on this

d€scription ed thc modein loowl.dge in ihis work lhe sme h6 been explocd dor.

.xlecively od is pEsnt d latu id the chaPler.

lilcatue that app.ared duing 2od c.nrury lhe gencral rule of

Babylods for first visibilily ofnew lunecresc€nlis:

78

at the titu of lual su6.t shottd be g.ater than 2't hows

the su6el ohd the no6et thDuld be g.ot* than 12 tim

(l.l.l)

l2 rim. deBes is cquivslenr ro

;$ ofo hour or48 ninu$ (Brura I9??.llv4 le944

lto$ev4, il wd Fat@hi, Stephe$on and At'darghdlli who havc

'xPlorol

lh€

cflon5 of lhc Blbylonians dlsiv.lv &d have epon€d rnal eording to lhe histotiel

Eords of re* c.Mnl siSlftings ol PG Christie m lhe cderion attibured h

Babylonis is d ov€r simplificalioi (Fabohi.t al 1999) Ac@tding lo $en $udv oi

209 recoids of Posi{ve new cr€sceol 3ig} ing extdctd fom Babvloni&s Astronon'car

di&ies. the Babylonitus had succceded in ioduladng a lrulv mathem'lical lunar lheory

$hich ihey 6ed to pEdicl !&ious par@ele$ of lumr notion Unfortumtelv wrlhoul

presnline my iheorerical d€laih of Babvlonian eE lh'v appear 10 be

"ilicdl

aboul

Bluin\ slgscsion thar Babvlonia criterion was whar is mentioncd above (l l l)'

lnst€Ed.1hey clain that Babvlonian ctilerion wEs as Follo*s:

Tnc ies cEsce.r is een it

'' Elongutio, (ARCL) + nooe|st6't laE tine (Uc)

(3.12)

Even, rhis cdrerion is apon d on b6sis of the suee€stios ol Ncusebaud (Neue'ba@r'

1955) &d nol 6 a rcsult of ov d'$rip$on of a madehalcal lh'orv Ttev have

nehtioned two svst€ns of sol& moliors on *bich th€ lt@ rhsrv ol Babvlonids las

based, but hoq lh€ new rul€ they hav. attdbded to the Babvlonims wD drived al $ nol

giv6, They have gone on lo plesd .lifleant valucs fo! th' @nslul on lhe

'kh

h8d

;ideof(3.1.2) Th€se lalEs Engc fton 5 DiniDum of2loro a maximM ol23" lnlhe

p!€ent work we adofl fte followins as a bener critcrion alfiibutcd lo de Babvlo'ians as '

slss.sted by Fatoohi el .l:

79

80

(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

d6dibes how bnghr is th. cFac.nl &d (ii) ihe moon*i*utuel lag de*db. hov lons the

ctcse @uld coaii abovc horian.n r th€ $n sl. MoE is rhe vql@ ofqch ofdrqc

paluelea moE is the po$ibilily ofsishdng offie ce*cni. Howver, drcr. h6 to b.5

conbined ni.imun ofthe two dd the sigh{ne Eords wcre the only heans to verify lhc

crilerion or diving al the (it€rion. Using $is condition Fatoohi a al hsv€ Fponed thc

aslhs of lhcir @mpu|,alioB for 39 rw cEsnt 3ighings rhal in lud. bolh lhc

Bab)lonirn cEscenr qighr ing Rordc tud thes'ghtine Ecords Eponed in $e 2Odcenrury

liredtule. Th. aulhos have rcponcd ihat oul l99 cdcs the Babyloni.n c lerion was

succestul in 98.7% of fic posilive siehliog {cE$e.r claioed to h.v. b.en sn) c45

but fsiled i. 45.7'lo c66 of the nesalive siShine GF$.rt no( *o) ces

In r€ccnt lin€s a number of orsdialios haYe smnged for coll.cdon of €r.*ent

sighting or non{ighline @ords. Th*e inclu& Isldic Ctffint Oberyalion Prcjdl lh

So h African aslononical Obsaatory and rheii websiFs Sinild rccords are reponcd

ro aid colhd€d at lhe wbsile @!sS.h!i!s.eol1 Morevet, lhc ldrgest data el

yd availabl. in lhe pubhh.d paFG is lhal by Od€h (OdGh 2004) we hav' *lecred 463

of lhes dords ofdening cEsnts el.dcd for tE compdison ofrhe nodels studied in

Usins (l.l.l) appli€d lo the dar. 9l clet€d for this wlk the '6!lts

u€

pEs ed in the lable no. Ll.l. ln this lable ollv lhose rccords are prcscnled when thc

crescent wa reponed to hare ben seen wilh or snhoul ev optical aid ad ARCV +

LAG is les lha 22 degr€6 Th. rabL is en€d on ARCV + LAc It shows lhal thcrc ft

? claims of opli€llt unaid.d sighinss of new ce$enl tol salisfving $e Babvlonian

cileion gi!.n by (l,l,l). The compl€G dola sel is P!€sent€d in the Appe'di\-ll Thc

table i. App.nditll is en d on visibilily colum N in d€$ending older, so thal all

oflically @id.d visibility ces of crc$cnt sighing in ordtr of ARCV + LAG (in

dcSt*t app.d al lhe top of the llbl€ The table in Apt€ndia'll sho* S No ihc

obseffation No. (odeh, 2OO4), dare ofobsryatio!, latitude ed lonsitude of rhe location

of ob6dtr, visibihy N' for uaided visibilitv' 'B' for visibilitv thrcugl binmld, T

8l

for visibilny thftud rel.$ope. IlE vis'bility @lul6 @.mPty if rhe cjt*nt M not

Fen and co.tains V' in an apprcpriatc colLlm if the c!.sc.nl is sen. The esl of lhe

coluens oflhe nbl€ @n!in age ofMoon, LAC, ARCL, ARCV md DAZ. Th€ lasl file

@lmns arc ior the five models coroid.rcd in this chapler, The colllm h.dded A is for

This tabl€ shows $al out of 196 daims ofopdcally unaided sighlihSs oftbe nes

cresccnl there @ only ? cbes nol saisfyjn! lhc Babylonid critmon. HoEler, oul of

267 cdes vhen 6e cc*cnt wd nor e.n wilho any opftal aid ARCV + LAC is

gleate! thdn or 6qul to 22 deg@s i. l07.ses. Thus for pdniw siehli.8s rh. crit€no.

is foutrd lob€ successlul in 96.4% c.ses dd !frongsl lhe hegalive sigbdn$ ii su@ssful

in 59.92% css The sucess of€ model in desciibihg posilile sighlings alonc is tul a

cal success ofa nodel. Thc model is not suned seu if n is nol able to dc$tibe the

negalilc sishfng 6 wcll as ir d@s the p$ilive siShtings.

SOME SPHERICAL TRIGONOMfTRIC CONSIDERATIONS

Thh hd be.n m.ntioned e&lio rhal ior rhe ploblcm ofsiglting of new lund

crescenl $e orienlation ofthe ecliplic plays md inportant lole (article 1.2). Thecfoc, il

is inportant b rrisil lhe ne|hods thai lead lo lhe ddelhimion oflhe mgle lhal the

*lipric nlkes wilh the horircn al the tinc of snel on the *$em hotian (il is

panicularly significer in view oflhe fact that the new c!e*.nt appeds close lo n). This

angle lalies as agaiNi rhe lixed celesrial equalor lhe orientation ofthe ecliPtic slowly

laries lhroueh the year. lis fislE no 3 2,I on lhe nexl pag. shovs lhe wcsrcm pan of

$e cclcnial spheE wilh impodanl points &d angles m d de$nM below:

w' the west cardinal point,

t, rh€ v€ml cquinox, inl€eclion of.cliptic dd cqulor.

P, the Nonh c.l.sti.l pol€.

82

Z qv - r rb ouiq'iry of6..dinic,

z 8}'r1 - 900 + 9, 9 bdte lb.lditd. od6. d...'

Z ZSr - 9 + 4 lb dSb ofrb.cliP(ic xdft 6. v..dc.l,

rs-d-G5tlirst d.clhnioofrn tu,

PZ - 90'- o,

St - ?e 6. lonSili. ofd! $'!

FrsNo 32l

FMn 1! 6su! 3.2.1 in lhc tPhdic.l ttrtgL SPZ

'pplvi'g

1.l{ of ctsiE of

3phicrt ti8oiodt tt.rdA non &. !id. PZ oe gBs:

tlne=cort.o.PSz

(Jr.r)

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

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

83

(3.2.2)

(3.2.1)

FDm ridcle sh, st nir8 !r Pr:

@s(PSz +@+ A)=rer.@r,1.

(3.2.4)

usins (3.2.2) !d (3.2.3):

0.2.5)

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

(1.2.6)

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

Ju"'

This show that e + A or the .ngl. thlr ihc elipti. nEt s wirh thc v.nicsl is 5.e!

dcp.rdcnt 6 n d.peo& on the lorgitud. ofthc su only for a fix.d place {or latirude 9).

ln th. Esr of rh€ di$usion in rtis snicl. rhc anglc 900 - q + A is d€nor€d s V. In dE

ncxl .!ticl. etr@ cv.r ? + A is L*d it h olol&Ld on ibe t 6is of (3.2.7).

In @ the dslinaion of th. Moon is $u1h of th.t of lhc Su in NonhqD

H.nisphft (md north of the Su in th. rcuth@ hemisph*) it is possibl. thsr d.n

.ncr conjuction th. tr.w llM cKar $i! b.f@ lhe slNr in which w ir ir simply

84

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

iD

iii)

Cl, lh. cclsli.l .q@lor, 1 i! th. vdal eqlinor.

TS is the diuFal path of th. Sm thar is just *riirg al s.

DE b thc diumal palh of the Moon th.l s€t before thc sUNl 3t E.

Ds = 6M - 6s, letpendi.ula.

to the c€lesial equlo. is lh€ dif.rence of lhe

decliMtion 6M of ihe M@n &d the deli.ation 6s of rhc sun. Declinalion of

the Moon t south of lh€ sun md rhe upper limn of Ds is 50 9'. lhe inclihalion

oflund orbit to lhc trliplic.

v) zcvs = 9Oo - 4 is the agle bctween $e c.l.srial equ.tor lnd $e horihn

NS whee C teprenls lhe lotitude oflh€ Pl&. TheztSD = C

vi) DM = dM - qs, lhe dilreEnce of risht a$cnsio. dM ofMoon and rhc rishr

eension os oflhc Sun. For higher lairudcs DE nav b. htgc allowing laigc

vllues ol ncepliE LAG aftr co.junction

vii) sF = pM - 9s : 0M is p€,Fndicultr ro fi. E lifljc is $e difiercnce bctw*n

lhe c.l6ial taliluds of the Men and lhc Ss As for DS Sf never €xcads

rhelinilof5'9'.

viii) FE (E' .ol showo in rhc fisure 6 it almost coincides wilh E) h ptrallel lo the

ix) lislhe

Vemal Equinox ed Z)6t =

horizon. Dep.ndniS on laliude C and

v h !h. angle of lhe Ecliptic with $e

lh€ se@n, 'Y

naY lary ftum zeto

(vhctr ecliptic is alons the botizo') lo 90 deB€6 (when €cliptic is

pc+endicula! to lhe holzon) l! cas' when v is small DE thal has b be

paEllel lo the 4liplic is nuch ltrsei tha' DE (ftol is paEllcl io lh€ cqualo!)

and hdc. E sd E aE mrch s.panted Tlle det€mi'arion of$is oSle md

its sienifidc. shall be di$sed in the n'xt drcL

x) I'lt figurc shos rh€ SM is jN| about o sel in a pl&e oI sdall b n€diud

laftude d , $e M@n havins ddlinarion eulh of il' sun ha al@dv *l (dd

wd set at poinl E) $ th.l lh€ LAG = TiG of Mdnct Tim of suNer is

E'

fig. 1.2.2: G@tn r.y ofP@riv. Ag. .rtd N.gaiE L.g

A@rding lo rlE luN elqdr b.!.d @ IIE binh of Nd M@ b.bc tlE ssa

tha nd lun|r mn$ b.gio .l dr tirc of o.!a 6 $ii vrt Ming- How6, for tll.

lu@ c.lsds b.!d d rn viltbilitr of $. n4 s@4 w hN mfli &e.ol

b.gii on thb cming B n b 3inply ioF{bl. lo $ tlE w luu c|!g in this

cinhtee. Fd plE witt lrSc ldind.. d|r dbw for llre v.l6 of DE |,d h.@

DM ln. tri&gl6 uidd 6!&l6.riB o rbt tr rtrdl .dd.!xl ..otl b. tBt.d s

ryhdicrt r.iugl6. tlc! d y @ll lo tr|.diun hdnd6 @ qEidqld e th|r th.

tdugl. EDS i! r 3rdl dgh rlgld oilElc o *y wih dt.!gl. tt D t..Ia E .d S

tu 9oo !r!d rlE a.sl. d S b.rsa D !d E i! ?. $. ldind. oftlE pL& Th@foE:

SD

Af

(328)

86

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

tilr,.=

(3.2.e)

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

tN -ts-$N -6d,H

02.r0)

Ag.in in ISDE tm t!rc:

.DE

-r=S,

DA-@, -6,161

(3.2.U)

ln b frrr! !23 $,ldd It dlr d. Dlc ntDglc Ft sDE of6. Flvior6 68rta

rEql'itld.ortun.8 dvcLAOi.i.i.EDM<aEhr:

DM - (d{-!.)

(l.2.ll)-

DM -(au-asr<Qu -,s)r$a

4.2.t2,

Pig.3.2.3

87

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

(aM -as)<tPu - Ps).inttsinv

(r.2.8)

ThN whcmvd ih. Ncw M@n is bod jsl bcfoG le.l su!54 ln. LAc .hould bc

tu8lliv. if condidon (3.2. | 2) o. (3.2.1l) is s.lkfi.d if thc ce.d i3 $ulh of th. Sw.

ln rh. hod.m s.tup oE csjud olcul.€ rh. Elarivc ,ltirudc ARCV offic M@n

d th. tim of S!Mr. lf ARCV is nc8.tiv. rh. LAG ha ro bc rceaiivc. Still lhc

conditioN (3.2.12) d (1.2.13) show lh. d.Fn&nce of thc ph.mmenon on dE (i)

.q@to'ial Mdidle. of lh. Su $d th. Moon, (ii) thc hnudc e of lhc ple, (iii) thc

aliptic hlitud.s ot th. Su. srd ih. M@n .nd (iv) lh. egl. v or.cliplic with $.

hoden. Thc Tlble 1.2. I sho*s smc of lhc n gltis LAG ces during the ye6 2000 to

2010 AD tor Kech'. Paki$d (lariud. 24.85 d.g6. lonenudc 6?.05 d.e@t.

Ne8{iv. LAG ces on thc da, olconjun tiotr Frlly oeu tom smdl !o n di6

llritld.s bul @ sis.ili@l a. cld dividine linc b.l$q conju.dioMl luN qlcn&6

ed th. obacMiioMl lune cd. t!3. for hirl| htitud. pl*ca rcearirc LAG 6cn anc.

conjuictioi dly @cu moE f€qu.ntly. The coluntu of 0E r.blc in scqucncc fom bn

ro dsht e d.*dH b.low:

tsrl dllc of coijuiction.

tncd Timc of Conjmtion,

Zorc (PST) tihe of locd su!a,

As. of M@tr .t le.l suet in ho6,

Elongadon or Moon rroo tlF 56 in dcrFs,

Delimtion of M@n in dceE€$,

Delidlio! of Su itr &88,

Righr AsEion of M@n i. dcgG,

Rieh AsEid of Su in dcgr6,

EE

Laliild. ofMoon in d€8rc6,

Angle of E liptic wilh Horizon in d€8rees,

LAG, Moonkt {u6el in oinut s.

rable No 3.2.1 : posirive Ase Nesaiifr&B-;s-i66l2oliT;R;;iifr;;

3.3 LUNAR RIPENESS LAW & ITS MODIFTCATION

Befor€ |h. rin$ of ptolefry lh.rc ws no knoyn sy$enalrc descriprion ofthe

dynamics of thc planeb and rhe Moon, Theretoe pblemy,s explor.tion Che epicycle

based description ofthe pltret4y noiion) sens to be rhe tirst sciious. scientific 6d

ststemlic aucmpl to d€$nbe 1he dymics oteld systed obl*is. Muslim 6ed lhis

tyiem b cxploc lhe condidoE for 1hc fid visibility of ns lutrd cr€sce . plolemic

th€ory hal d.*rib.d th€ plderary nodon Eins rhe epicyctcs tcory rhal could Dddict

the pl4ehry, lunar .nd sol& ephen€ns to a g@d soush d.er€. of accuacy. Tne

Muslins had put ro use lhis iheory weu ond had developed sotr dnd lu.ar tables. The

pos ion of$e Sun ws predictable !o a g@d dce@ ofprccisio. and @nsequendy rhe

aele !r lhat ih. *lidic hal6 with ihc hoizon ar the line ofsue( ould b€ calculard

Bing rne sphe.ical uigoroheFy that Muslim naomddn hav€ d€v.loDed th€n*1v6.

As decrib€d by Bruin ( 1974 sing lhc &8lc of ectiplic ,y wid! the hoiizon al lhe tine of

su*! the lalitude of th€ place of obseFarion, egle of sepsauoo b€rseen Moon dd

a9

Sm p.r.[d lo lqldo. (difrcqE of ddr dgii L!..dioB), tlE !sl. of d@!n

(dife|i@ of a|lnuda o. ARCU !d ri€ tid[ ofs€c.'n the M4lin aslorcEcrs

$@ aU. to develop ! hisltly rdi.ble qilcion for tlE visibilry or invisibihv ofrh6 @

Figua (l.1. I ) bclow enicl ir r r.otuttu rin offg@ 4t ofBtui. ( le77) shovs

tl|st dF sD IEs jur !.r sin of r/t{. Thc liie \nX D85rg tb.o{8! $ts drllal poilt

W, is th€ e{uror irclined d o dsL g (tlE ldihd. of lhe ple) fim tlE @rnd to rhe

horiar AS k th. dimd pdn ofth. Sun !h!t is p66llel lo thc.qualor is shom as

broko 1i... TIt diurnrl BIh ofd. Md MZ is tls sboM wnh 3 3imilt trot€.linc

Ddrall€l Io ih. .qudor. TIE lin. HS ir tL E lipaic lh.t drd e .ngl. A wfi th€ diurol

path of 6e Su d $..qulor ThBtL FiiiFb D!l6.td..8L q + d witl the Mnrl.

MS t th. s.r.r.dd b.'ra rlF Mdn !d ttr Su (e of liSln).

\:\.

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

MT i3 p.rDodoio to i[. ..lirdc rII8 ar.h l}a TS - 1 - 16. b 6. 'Etl

tinsb MIII, rn - fu d ft. erb a H b..r!.r Eclbrb (IIT9 jd &. Ho.te|.l

olM) - 9d - (r + AI Tb - rilrb tm{ b lisb.raLd d T yE tq!.d3L i M

!cvl.{HddT-9+A.Itrb.ioplyk &b6crdtio:

ra-Er +rS-t,6lp+D+L 4. (3.3.1)

|hir Fldih n .e. .r .$|d@ O) of B.uh O9ZD. Irtia8b HgB b rlcd s . ph.

codr+ a)

(3J:)

{d ti.lgL ABS lad! lor

(3.33)

(33.4)

(3.3.t

'Il&

'slr

(3.3,5) i! dimalot Aom eb &uii O9,D t& tih in hi! .qurid

(a) |b h cbiD. nr b..o obi!.d by ld4 lhe tigoffiy. Ittt smc 6ru. i,

t'.d.d ||3toglpt ric.l !i!@.tty, turiotbADs 8it&.1

f|(!o .phqi6l trt|tEL tlTMr

9l

sin rr'r _ rh&

sn(e+a) @s(e+a)

erhlr (3.1.t) rs = sili(sin r!.r..(e + 6r+,L -,rr

0.l_6)

I. sphoical iridglc ABS:

stnaD . stn 4r @eP

(1.1.7)

and in spherial lridgle HCS:

sinfls=-l]1gll

co(@ + a)

l].l.8l

:l

glj!

".(

(:

t::!

!

;

!.

)

:9:.q

+a)

-t

[.

(3.3.9)

laegles Hs dd ss @ snall cqua$ons (3.3.6) lnd (1.3.9) yhld alnost sme lesutB s

thos by (1.3.1) ad (1.1.4). If the valuc of 9 b lege. then Hs dd as ae nol shalland

th€ spherical trisonoderric rsults (3.3.5) and (3.3.9) should be ned for doF Muale

11'e sienificdce of (3.3-6) is thlt if th. cphcme€des of both rh€ Su and the

Moor m tnoM @uErdy (aj th.y c rcw) ud rhe &gl. 900 (p + A) = v of etipri.

{irh ihe honan €lculalcd fom (1.2.7) rh. !.lE of HS @ bG aal@ted for rbc tire of

sUM for dt day of lhe yw !trd psrti.uldly for rh. dlt or rt y .Rc UE bnh ot ncq

92

Moon. wlee6, oncc rhc hinimm &stc of sepaDrion !s (.quivd.nt ro LAc) fo, $e

visibility of @w l@d ct€.st is tnoM for thc day the @rcspondine agle HS sinS

(1.1.9) cs als b. .vatua&d. Horev.r, rhc ue of (j,j.6) is ind.p.nd.nt ofany visibility

condition ed is fixcd for rhe day ir d.pcnds only on rhe posirions oflhe Sun and lhe

M@n for fie rim. ofobsenato. &d the locaiio. ofrhe obF ei. On the o$( hahd rhe

u$ or (3.1.9) dep.nds on trE vjsibifty @ndiiioG, @cly lhe mininu del. of

s@ml'on tu, thar.& b. kmh only on lb. bdis ofa t gc nunbd of obo€Mrions tor

At rhe rim€ of Muslih Gtrononcts rhe .phener€des of $€ Moo. oay nol hale

ben so aeudely tnown s ir is today, but rhe Eligiou tecnn.ss oflhe seing $e new

lund cr€s@t nusr hlve tcad ro moc rcchre valu6 or,s. Bruin (t9?7) hs i.dicared

that Muslim arbnoheG rveE welt awe of rbe f&! lhar d8Lnce oalhe Eanh ild rh€

Moon and hence rhe width ofcrescent for sde uc oflighr varies. Though exploiations

ofdr ancienr considcred lhis variarion ro behdve tibearly we now know thal it inlolves

lne lflgononefic func on.

In lnis worL $is prcbleb is h&dlcd by considering rne actual eni_d.in€ler of

th. Mooh dd lhc fer $at MnslinvABbs ob*tued thal ar shone r .lisrances the crcscenr

wasseenwhehtheLACo!arcofseparationaswaslOd€elccsor40minulesoftimcand

al la€e! dishcca thc cErcent ks seen Nhen rhe m ot seplFrion dJ B t2 d€8res or

48 ninules of timc. -r1li5 teads b a siople Eladon b€lwn /s md lhe aduat semi-

^-- D 22

ol r0)

Using rhis etalioo thc @ of *pdlion necded for ce$enr

*p@tion felor is .alcutated in ihe sonwar. Hilalol.

9l

As mc ioned abov. dE Mulin sao&Ders have alaldy d€duad from

obrpatios lhar for very thin but visibl. cE$ent mirimm ss @ 12 lime degds (48

minureO dd for d {ider but bmly vbible cresent mininm as w6 only l0line

degl6 (40 minulcr. This nininm as ie acturlly dep.ndcnr on the width of $e crc*ent

rhar @ be dedvcd 6ins (2.E.E) dd (2.8.9).nd rhe Elarive atitud. (ARc9 thar cd be

obtained using (2.8.5) for th. sun ed the Moon. Th. issue shall b. dieussed in noe

d.tail larer. Usins 0.3.10) rh.t shows 6s beins dependent o. the lisull dimers (in arc

minulet of lh. M@n in our sky one ce compute lhc Lun Rip€i.s Fudio. A(,i9)

as henrioned by Btuin (Brui., 197?) or tha! is siven by (3.3.6). Aaodihslo Bruin rhe

rul. d€dr€d by M6lims w6 thal if thc lalue of HS calculat€d using (3.1.9) equals or

exccds lhar @lcolared by (1.!.6) rhe c..s would be visibl. orhcNi* nor. In thn

work this rule has ben named M6lin Lunar tup€ness Law or sifrply Lumr Ripen€ss

According ro Bruin (19?7) $e valu6 ol HS de.ored a R()., 9) obkin.d fron

(3 3.6) seE pGsenled in rhc $+ lled Lunat Ripenss Tables sDd fisr visibility ofne!

Lunr cmcent ws walulcd by Mulin BltonomcB usiog rne sboE rul€. In lhis work

HS a3 deived tom (3.3.6) h d€nokds,l4 and:

^d.,

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

Fo, a fixed pl&e (e constal0 it is noted lhat forsDallvalues ofl,, sin ly

is also smatl

&d lko n ahosl sme as rv ,j. Bui when the *liplic is wll inclihel io$&ds lhe

hoien (q + A is large bu lcas rhan 9oo) 1r'4, nay becode noE rhan 1! ri(iflv > 0

snd ecliptic is roweds nonh) or renain le$ rh& 1, - .ls (ir pr, < 0 and etipric is

iowdds nonh). Ho@er,{da is endely independenl of Oe se@. aod dep€nds only on

rhc rcladve coordinates of rhc sun md rhc Mood. As rhe major componen in (3.1.t l) is

lhc diferenc. ol longnudes of rhe su and lhc Mooh ,taa h closely linked to ihc ACE of

Mftovd. ir 9M . o, x&" * l,{ - k. Bur whcn boil 0M 6d I + A hav. the s4e

sign 0.3.1l) shows th &4 *ould b. mor. thd l, - & and if pM dd e +

^

have

differcnl signs &e would bc l.$ the ,L - .ts. Bul thcF v&iatioos u€ nor sMnal

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

^,,=.'""(

(l I 12)

For a fi&d place (9 consrd.t) R,, dcPcnds on ae spa6tio. a, (rhe equabrial t'Ac

ber@n the sun ed the Moon).nd thc s.en 6I + A is *son dependeot fiat shall b€

shoM larer. Using (he looh sod le hniqu.s di*ussd in.hapter 2 ie, fo! $e dav or lh.

day alier conjunction aor s placc of ob*ryarion ii .valuaEd. Ille ninihud valueofas is

@lculaled 6ing rne lrchnique d.srib.d al lh. .nd ol Ptviou .nicl€ $ lhat ,tB is

dedu.ed. If lt- is calculat d using h csdml.d value ol a, rh.n w call il X." (m

Erimared valk ot Lunar Ripencit funclion), Il ir is .al.ulated uin8 an avedge vrlue

10.5 degres of a, thcn *. c.ll il I- (d .vcaec wluc ol Lud tuFns nrtudo.)

Ac@drnely A&" = i- - X,a,6d Alt-, = i^ - 1&) Thc sinpl€st fom ofrhe Ll|E

'"'(

i3i

:) $i" a;t+,ry -rsj'o

(3.1.r)

Tlus Lunar Ripeness Law that piolidc a solution of the probleh oldelemining

rhe first day of visibility of n€w lunar cre*enl h based on the Angle lhat the Ecliptic

nat6 wilh the bodzonlal or $e angle 0 + a ihal it dak.s wilh the venical. on the dav or

the day after conjunciion once lh. coordinatcs of the Sun dd the Moon at the tim€ of

suns for &y leation.rc calcuhted. (3,3.1l) allo*s onc lo calculale,e,4 ed sine 6

applopnab value oi4 d sivenby(3.3.10), in (3 3 12) the valu. ofx"i ce be tound ltr

95

vi€w of $udying ihe behaliour of lhe Rip.nes iunction oler a yea! for any pl&c

calculding Ie, for cvcry day of the yd is not useful. This is bc@w of thc &ct lhat

y€ar ro y€e dd da, (o day vdiarions in lalitude oflhe Moo. dd hence nd, ats not

dep€ndent on the rime of ye.!. However, *ith posiblc lalues of4, one cd calculate

el.v val@ of l* for @h day ofa yd &d study ns veialioG scenally as sell d

$irh chdging lalilude of pl.cc. As possiblc values of,, vary from l0 degcca lo 12

degc.s an appopdate value for the day oi day after conjunctioh is obtained only on the

bdis of the ule disraice o. sminimd.r of the Moon. Still tor a compdie. values of

lS6. foi dill@lt ladrudes, se calculaled lor both $e cxrene valucs ,, = loo dd a, -

120 and the curves,t,r aeainsr the longitude oflhe Suo (fiom March 2l) aft ploned in

FiSuGs (3 3 2) lo (3.1.5) These cw€s snow inleFsting featres lbat cs b€ sunmdized

i) from fi8.1.1.2 il is cl€ar rhar forg = O0 i... for a placc on $eequabr&i! is

sinusoid.l wirh mdina.t March 21, Septcmber 2l dd hinima at Ju.e 21.

D4ember 21. Snalle! v.lues of iG nems shaller lalues ]?d4 or !€lariv€ly

rounger cEsent m.y b€ rn. L&g€r values of x i" dcans largq valus iia

orrelaliv.ly oldcr crcscenl may b. seen. Thw close to equalor older cEsceht

nay be visible near equi.oxes and Elatilely younger cresc€nl .ea! ehrlc€s.

If $c Moon is nc{ jls apog€c lhen il is noving faner and apFaB rhicker in

sky Blatively younger cescents b€cone ripe for visibilny. Funher rhe

paencc of two maxim and two binima ihdicates four stong innecdon

points. There arc rwo regions ot lpMd concaviry (douod solsdcet dd rs

of downwdd concaviiy {&ound cquinoxs)

As rhc larirude of rhe place in rc66 (one move ro rhe nonh ofequabt $e

muinum ol1." at $e venal equjnox loseu nDline ir eard lo k d a

youngcr cescenr bul the ndimum of lhc veml equinox ris.s e $al it

becones norc difiiculr to ec a yodge! crc$ed (69. 3.3.3). On thc o$er

hmd fte ninimM of lhe s|!md ehric€ mo6 rowuds sdng (v.mal

equinox) ad that of rhe wintd rchric. mov€s rowds autumn (aurunnal

eqlinox) md in ei$er c& d.crcae funner making ir esier ro *e a younger

96

cwc ahaFs clos. ro ll€ alhlnn l

m6d thc solsti.6 flltt ns.

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

15

13

Kg. ffi

12

10

I

. . .r. rl,

9o 180

.,.-

; I

210

Fi8 No.l l l rt'L forg= l0o

iiD

BcyoDd tropic of .alF two of the infl@don Poids re simPlv gone &d $e

hsimu al lhc aotmnal €quinor ds tunner malir,g n FoE difficuh lo s

91

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

.quinox mrli!8 it .did io r.c yomg.r @3@l! n tr ro €ml .qu$ox

iv)

rG frltd iF!.ic i. dr ldtr& &. D!!io@ bcoc tt}.' rd !i8le

lrd dt. ril!i|!@ t cF &cr!.'nry tuvi!8 &n ftr biSba ld'ndd il i5

g6.r.lty .si6 ro @ younScr cr!.c.nt3 cl@ to vlrnll equimx .ltd diffrct lr

clqe b aull|nn l cquilor.

'17

t6

13

1o

to

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

1}. siuDtidr r!\/{r!6 .dirlly i! fivou ofam|'llll cquiM ft. lb -ttld!

For the Arctic Circle .nd to it3 lorth (latitu<tc| Sn cr ih& 63.5 &8t a!) il k

liDplt ml polribL to s rh. s9 loM q.!..oi cloa. ro .utmt .quinox 6

th. vrl'r of&i t @06 gt lddra 9d. Tb vdE G snll.! 50 ofx*

cl@ to tqDd .quid indielc riqlt &.t cr!.c.nts y@gq tle 6

c@!d!d !o thor. clde 10 lutunn l cquiaox N bc !.d

9E

a0

50

20

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

Howev€r lhe narcr i! lrot $ sinplc beca6' tuc olc of latitude of rhe cKdt'

i-., whcth€t thc cc$eni is !o!ih or north of ihc Sun plavs e impona role lhai

dercmin6 rhe v.luc ofrtdo lhal is lronglv btnude dcFndst Now lh' stDns nuimum

at lhe v.oit .qrinox for hi8h.r lttitud6 indicat's thtt e old'r M@n rov rc1b' vbibl'

b s the AGE offt. Mo@ iNrrsd $ do6 iG elo'sdi@ ad brighlress ThtebE n

is posibL ftat invisibilitv $g86l.d bv $' Rrpcnas tun.lid valucs m'v b' nisleding

Thh inly l.!d lo ltlollN for sdalld LAG vd6 drd sm lqr€lEof x'L h rlnt

$!d( th. RiD.G$ tulcrion vtl@s for dE ob!'dltioml dab avlilablc in lilettrurt e

.dculacd sd pr6.ntcd in ApFndirlt Out oflhis dx' s't th' c!56 whd thc cscenl

s Eporr.dly sd bui .rc nor h lg.mdt *nn dE L|M RipGs ltw rE also s||os

in nblc rc. 3J.1. TtG obae liontl dah k $lccLd AoF lh4 tPo :d bv Od'h (Odeh'

;0{a) fron *hich c.*s or cvdi!8 '6cnr

obs'naiioN re cosidercd onlv Ttblc

3.1.1 shows o. No. (th. oberysti@ nunb'r (Od'h' 20Oa))' dtt' of obseNttion ldnud€

,ftt longin/. of ih. t@dron of obaa$s' M@n\ !8e and LAG' snele v of ecliPlic sitt'

rrorLo, t"lio,a" or Uo.,, toogirudes oI the Moon ed th' 3d' Rd giv'n bv (13 12)

*ill !6 siv@ bv (3.310), Re siv€n bv (3 3 12) *ith 's

= 10 5 d'8r*s' Ri' sircn bv

O.3.I l) and a&. = Rdq - &"

I

'Ihe table in App€n.tix'U shos S. No , the obseNation No (Od€h 2004)' date or

obseNalio., latiiude dd longilude of the locaiion ol obseru€r, visibilitv 'N' for lbaided

lisibility,'B' fot visibiliiy lhouel binGuld''T for visibihv thbush lelescope The

visibility coluds d€ empty if the.res.ent wd nol s*n sd conlaitu iV' in an

.ppropriltc colmn ifth€ q€sftnt is sen The rcsl ollhe coluons oflhe t8ble contarn

age ofMoon , LAG. ARCL, ARCV dd DAZ The ld file columns de fot the live

models consideted in this chrpter The colu. headed A is for the Babvlonid model'

and A for the Lunar Ripeness model The colnmn headed B cod'ins AR"

The lobl€ i! ,Appendixll shows lhat oul of 196 cdcs in shicb $e crese has

b€en reponed to hove been een wilhoul oplicdl aid thee tle onlv 14 cases hat do not

obey thc LuMr Ripencss La* stalcd above The debils or rhcs obscrydlions de tinther

exploEd below *ltn a Modi6ed L@r Ripeness Law is sugeest€d The nodificalion is

needcd in ordet 10 sepdarc tbc caes of dked eve visibilitv and rhe caes when optical is

used for clescent visibilitv

A tot.l number of l2 posiliv€ obsedations ih tbis dala hav€ b'en wnh the helP of

bin@ules ed klcscopes when $e cr.$enl ws nor visible vilhout oplical aid dnd the

Luai Ripe.ess Law is .ot elisfied 'nth is logicallv valid as the Lua' Ripencss Law

was deduced onlt for mlcd eve ob*tvatiotr Tlis bas he'n the dolivadon durine thN

$ ork lo ood i fy th€ Lunft Rip.ne ss Law ro e ocompass the otricallv aided obse d'tions

The columN ofihe table contain Ob$flario' Ssial Nunber as led bv Odeh

(2004), Date of ob*tvation titilude oi lhe Place Longitud€ ol the Place Visibililv

colunN N (lor unaided visibilirv), B (wirh bimculd) and T (wirh iele*ope)' The*

visibilily colums d. €mptv io! invisible cresent md co ain v for lisible R€st of lhe

colws contalns Ase of ctesce.t in hou6 fo' the best ime (Y6llop' 1998)' Lae in

nin(es, S.padtion b€twc€n the Su ed tbe M@n At of LiShl (ARCL)' the angle v =

900 - (9 + d), that thc €clipiic mai<es wih ile hotizon on the wsted holizon for the dav

of c'lculrtioi/obsedation, OM th€ latitude ol $e Moon' l'M rh€ longtude of rhe M@n Is

100

the lonsirnde orthe sd, Arc{f-*pdtion fa.tor, Sitd by (3.3.10), Esdnat d Ripencss

Iuclion, ,{.r olcularcd eqution (3.4.2), Ar6of-epdalion ractor, Aversge Ripen.ss

Fwrion X., calcllared Ning avcds. tu = 10.5 d.gEs dd lh€ €quarion (1.4.2), Aclul

RiDene$ Fucrio. ,Rdy calculated sins equtioh (3.4.I ), AP-" the difie€ncc of AveEge

RiFn s Fwrion & th. Acrul one. Th. ribL l.l. I des nor show thc visibility @luons

s tbis table @mprises of css wher ihe cesftnt Ms r€Podedly seen withont oflical

The lsbl€ is soned on the values ofd&y. On the basis of a clo* dalrsn of the

results of conparing ARNi ihe dilleences of Av.lage Ripeness Funclion valu€s End lhe

Actual Ri!..ess Fuclion v.lucs we ob*de dEt:

The Tablc shows ft.t lh.E are only 14 pGitive sidrrines out ofa rolal nmber of

196 positive siShlings that are nor according lo lhe Lunar Ripe.ess t { (ARo, = Rn",

The€ a!. no posilire siSndne wirh A&q < -1.58 wirh or wirhour oprical aid the

Ens€ in wnich l8 aft.npb have ben nention.d in the lne6ud. We consid€r it as

Coup-A for thc Modificd Muslin Lund Ripencas Las lhal we slal. ih ihis erlq 6:

''tt th. tlill..e,ce ofAv.rase Np.ts Fanctior R-, omt the Actuot Rip.h6s

Fu"cion Rro F /Re) is l.ss nnh -J.6 it is ihDossibl. to see nE n.N luat .rcsc.nl

tor att ldtitula uith or Nilhott optical aid'.

The ncxl I27 c6es dc grcuprd 6 coup-B. Therc de la (llolo) mked cye

viibility cdes, 25 (19.?%)binoculd visibihy cesdd l6 ( 12.6%) r€lesopic visibiliry

cses for values of AFw lying betwen -1.58 dd 0.0. Oul oflhe 14 natred cye lisibiliry

ca* in thn range, threc lery low AR{yfvalue cdcs have comon chancterisdcs. Th.*

e No. 286. 2 dd 2?2 witb AR.!i valu. -3-5, -3.4? md -2.88 All lh* c4s m ner

autumnal.equiiox (Sepr.20. Oct.2l &d Ocl. t epecdvety), €r€$enl hs oldc. age

(39.1 l, 39.24 dd 4l .91 hous Elpectiv.ly) md hav€ consqueitly larger phe.

l0t

All lhe* conditions falolr rhe lisibility and it wds ncntDncd above in view of

(1.41) that aor oldcr a8e cresccnb (/.M ls ta€e) sh! er arc or scparation may be

auowed. Thisexplainsrhe very sm.ll LAG (29.s, 33.62 and 32 07 respecriv€ly) in rhese

cas. Modov€r, in all $.s cases lhe arc ofvbion is snau (ARcv 6.85, 6.8 dd 7.34

deg!€es rcspectively) buftelorile uihulhs rc taree (DAZ = 18.4,2O.j and 18.l d€grees

r$perilely) aioh the sun and tos€r widlhs (51,65 ed 5r dc seconds respccrilely). Atl

lbese lacloB suppolt thc ctaims of lisibitjry md weE anlrcrpatcd above whcn il sas

suggesred rhat smalter L,{C valnes oa!, be alosed in such cases. All d€F clains de

fon latirudejusr moE rhm 30 des*s. The shath, values ot Rj"y in conpdison to R.,,

h lnes cscs is due ro larsc lalues ofa + A (noe th& 54 deee$),lhe &gle oalhe

eclipric with venical ed large nesaivc vatu$ of tarnude of Mooo LM (tes than _

4.5degrcs) lhar rcduces Rla to hate n nuch shalter lnu /w _ 1.. Alt rhe* rhr€

obsen.rion de rncgR.nedqrrh rhe Babytonru cr.rerion

Therc is no fudhq c6e ofcesenr visibitity tjl. varue of dRq < -1,6. Fbh

dong$ otherposilile cases wilh -1.6 < AR."r < O, two de

very young ccenr' Thee

t02

m obddation no.274 (ARm = -1.19 ) drd 416(A&tr- -l_06) i{i$ age 14.8 sd t5.9

houE rcsredvely. Despile being lery yorS having FLlively l&aer LAGS (39.j md

17.7 ninulcs sperively) in both c&s rhe Moon ws very close ro the Eanh rcsuhine

in ldse visua.l dimcler bur lh€ cltsnl widrhs wcrc sb.ll (10_? &d ll.9 e. s@ndr

only). with $all the rclalivc eimurhs (4.6 a l.8 dese, the crcsent eas atnost

ve'liMlly abov. the su rhat bring it in ihe ided cordnion foi visibiliry bul lh€ snau

elative altitudes (8.5 ed 9.1 deele) nake rhe$ claims highly oplini$ic. Borh th6e

obseârio6ec in d$dgreementwirh Lh€ BdbyloniM qirenon.

Out of th€ 14 positive obseru.tioi wnh A&r < O, ihte werc very f.ini cEscenrs

i.e. obsrarion n@be6 389 (d&. = {.94), 341 (AR^, = -0.87) ed a55 c0 62) The*

crscent werc low in ottitude (?.2, ?.8 and 8.j deg,es resp€dively) ed hrd sna

erongarion (10.9. 13.3 dd 9 deg@s Esp.cively), This hakes these ct6i6s lo be higbly

opr'n'strc os well. Two otthcsc {389 and 455) are sko in djeereocot viri Babylonian

cnlenon {heec 341 is a mdSiMl6e in Babylonian qirerion (with ARCV + LAC =

How€v€r, for all $ee I I 9 c6es when $e crescnt wd daoed ro hdve been sen

wirh naked eye,lhe conmo. ieatule w6 relariv€ty hi8h l&uoes {generalty g@&r thq

S0 degrres on €ilher side of rhe €quator) except fo! obFNatbn no. 416. Th€ cbim 4l6 n

tioh latitud€ 6.5 degres nonh Apan fmm rhh lonely c6e it appe&s lhat fo, A&r < O n

is ihpossible to see lh€ cE*entatpt&es witlllatirud€s less fian 30 desEes (borhNodh

Inlhe l19caksofsroup B,lhe frequency for optica y aid€dyisibiliry (botb wnn

binocula and retescopcs) ihceses ed one ce cdily g€nerau€ lhal whcn Atqr, ties

between -3,58 and O.O theE is s hjgh rDssibility of crescenr visibilhy wilh soe oprical

sid for bod lh€ high latifude as wel s low l.rilude obefres. I hereby jn this wort the

s4ondpd of rhe Modilied t\4ujlih Lus RiFne$ LaB $$Rds

t0l

''if h. nns tizt b.nv.r -3,t otd 0.0 th. posibili9 oJ rhibilit! ollittt crcnetl

*ith and *itio optical did ln re6s vnh ihcE6iig tolu6 o!.4R-. fot hlEh.r

latitqda, gq*o ! gedt.r thon 30 d.erc$ notth a4d souk ard ZR*. b.l"g h th.

ronq. -3.5 b A0 h. postibility ol nak.tt erz visinlw ako irceqe llowvq, in t'.

tong. oI th.G wluzs oJ r'RnIor tMllo lotitad6 th. "oh.d

.tz vitibilitr h dlMt

N€xt is the Oroup-C $at mnrains 76 cases wilh AFrv in rhe dnB.0.0 ro L6, In

lhB goup rh.r. e 12 naked cye visibihy.as.s (15.8ol.),26 binocuta (14.7%) and 15

&l6copic visibilily cs4 (19.?%) e lhar vGibility with both naled cye and with opdcal

rid b.com$ ooe probablc. UnfonuMt€ty rhe dala is h.avily inclined lowads the high

htnude clesanda cl€rdenarcation for u.aided visibilit, for snalter laritudc ob*Fe6

cd nor be made. Srill, the thid pan oftbe [,todilied Muslim Lund Ripeness Law is

''fth. wtu6 oJ/R@ tk 6ztwa 0.0 an.t r.6 ttt. pEribitiiu of,i,n i, oJJint

cftsceat teith and tuithout oplicdt k ,trong fo, higlrq latitud.d,.

The Croup-D containine a ro|at nlmber of 221 c6es has 170 cases ofoprically

unaided vhibilitr oflunar cscs forr'i!' > 1.6,Il is

îl r'B@ >t.6 ttt. posnbitu, oJ ,bibititr

siorg lot boh towt aMt hiEha totitd.r".

of li61 c6c.nt fitttott opti.dt b

Finally, the 3umnarized Moditied Muslim r_umr Ripencs l-awr:

ZR-< -3.5 ihpNibL to tee th. n.v luaar c.6c."tlo, a tdrru!., ,ith o,

2,

-3.5 < /R@ < A0 ihposibt. to ee the neN crcsce"t with ot *,tthout oplicnt oit!

Jo, tnt .r latitutl$. For hiehe, t,titnds therc b n high pnsibiti,r ol gibititt

otli6t cE .nt dth oyial

104

t

0.0 < .1R,. < 1.6 p8tbiti,tr o! ,hltw o! Ji6t M@nt btrh ond withoul

optt ol h staq !o, tigt.r tdlituds. Lo.a!|rA N6t tuith opticd! a ! a".!

th.a try nelnE lt Nlth haL.d e!. h6 a eood .ranc. o! opticdlr u"tided

ne,> 1.6 th. pNtlbnrr oJ' tbi!i!, oifust cnse wtuhout optcat t st ona

Jor bo& Iow.r and highu tatttud.t,,

Another way oI lootjng inio ih€ delaih of Lue Ripene$ hodel is lo look inro

the plots ofaverage ripehess fmcrion md rhe acod ,ipenes tuncion for borh vhibh &d

invisiblc crcsei$ for single talirude, U.forluarcly, rhe dru avaitable dd considercd in

lhis work h Esrficlcd in rhc F6e lhat scieltifically re@rded ob*dalions fo, a sinetc

hnud€ a€ nor found lery fr€qucnrly €xcept tor Athens (ladrude 38 dc8,as .orrh) ed

Cape Tou (laritude 13.9 degr4s so!rh). !n panicula, for sn,I€r tatitudes. plac6 cto*

IoequarorobseruatiouarelolvcryfFquenuyav.ilable.Anunb€rofplaccsecsetecred

herc od tbeir d.h pto(cd for Alerage Ripenes Fundion &,, and the Acllal Ripen*

Fuiclion R6y for borh rhe Eponedty inlisibte dd the vilibtc oe*mr! Thesc jnctudc

plees with laritudes L8N, 33 95, 6.5N dd 38N (FiguB 3.3.6 1o 1.1.9).

Figure 3.j.6 for tadtude 3 L8N snows tbe b€sl slc

sieh,incs ou, of 6 (83.37") in asrcen€ wnh $" ,_J:'iff:ln:"11,'#:J:

obscdat'ons (6 our of6) e in agrccm€nl wilh the ta* toi this larilud€. Nexr is Cap.

'rown-(hnnde

ll.9 S) wirh succcss percenhgc 65% (ll posrlire siehings in sgenenr

od, ol 20, the needrvc obseryauons fo, Cap€ Toq are In agrehenr sirh la\ tor

9lJ% c66. This 6 ao owed by ,40r.6 (ladrlde 39N) w,rh 2 oul of3 {66.7910) posirivc

eghlingsad II o of l8 (61.l%) neSariv€ sighdnes a8rce snh fic lav. Ior laritude 6.5

deg!.es, I our of2 (50%) posilive sightings od 4 oul of6 (66 ?y.) asE. wirh rhe tan.

I05

I

4

15

't0

0 loo zlo

3b

400

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

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

---r;-+:

;b.,RF6.'"".b,.

.acrld R

Fig. No I 1.7: Rip€lB Fudid for C.p. Tow4 Sqth Anic!, Lgnbde r3.9 it€lr*

t06

f

-111+.-Ig'1.rl'1 t r.tl'::.gE::. f

Fig. No l.l.8: Ripdess Fudi@ for Lrtibde 6.j d|:gG Non\

20

1a

l6

10

20n

J.E""'g:f*,","..,v"ltr"

300 /too

iU,ar nr b. rnti"itrc ]

Fig. No 3.1.9: Ripdcas luDrid for Arhdq lrriMe t8 d€@ Non[

Loogitudc 21 7 d.grca Easr

t01

All thes. figu6 show ! lend for lhe Aveog. RiFn€s Funcrion lhal is indicaL<l

bt rh6&thal consitt ralions tud deEocr.at.d in figGs 3.3,6 b 3.3.9, c..ealy ihc

invisible ce$enr hav. Actu.l Rip€ne$ Fucrion vdu6 b.low lh. AwnC. Ripoess

"Cwe md lh. visibte cBcens have v!l@s rhal @ abov. rhc cle. Devi.tions oa

both foms ac prc*nt ald e discu$ed ,bove. A rh€orcricalty visiblc cG$en$ (aclual

Rip€n6s Funclion valu. noE rb& the avcRg€) is reponcd invisibt. $ar gendlty

.ounts s 'Positive E@lied lh.oErically i.visibte q€sc€m (acr@l Ripc.ess Function

valuc les 1ha lhc ave@g€) Eponed to be se.n, a..Negalive Edof,. The posirive ero^

dponed halc no atiecr on the modet as rh.se cm6 nay occurdue ro mdy uncontroued

fado6 ( like seather cond id ons and obseNer,s abi I ity to scn* lhe conr 160, Thc negative

cmB nay eirher by hiShty oprimislic bur inconect clains or they nsy rndicsre hcr of

authcnticity ofthe nodet both di$lsed abole ih derajlfor $e LunarRipenes Law

,a,n ovd atl @hprrisoi of the Lud Rip€ness taw wih thc Babytoni& oitcrion

shows lhat lhc Babylonie crjrelion huch noe succcssfut. .ltE succcs percenr,g€ for

pcllNe cs$ fq lh€ ipenqs law h 92.8% aeaitrsr 96.490 or Babytohid qiroion. Ior

the negatve si8]]iings lhc .iFne$ las succ*dr in 57.?ro css ,€ainn 59.9% for the

3,4 EMPIRICAL MODELS OF EARIY 2o.rr CENTURY

After lookine i o Ine deraits of tc success &d lne co.srE,nrs of rhe tupcness

Modd oftheMslinsa ctoser look in|o rhe figw3.3.1 ad tne eatysis of th€ obseryed

oara rn comparison to itc Lu.& RiFn€ss Las, d impond dp€ct is rcv.ated. The

following figee3.4 t whicb an exlhsion of fie 3.t,1, in addrrDn to Moon b€ing nonh of

$c rcfiptic. als shows rhe M@n b be sourh ofectipric. In lhis €6e though sign of

latillrde. pM oflhe M@n tats c@ ofwhcrher rhe tmgih HT is to be added or subrracted

nom ln - ,6 in equario. (3.1. r t) or (3.3.6) bur tuother qwsron becones rlever rn rhe

cae shoh in rhe tisuE for sane as tbe Moon is huch tunner away ror lhe Sun when

Moo. is south of $e Sun (in nonhem hemisphec), rhe cese n6 ldes m of tishr

108

rd t mwh thidd fd !.d FlriFM@ dir.E. So n n DosibL thlr $e d6acrr eirh

m.lld & md @.r.qudtly strdla .o mr b. vi3ibt . Ilis i! p@j$ly thc ce for rhe

obgMtaom m. 286,2.rd 272, lna @.ctn s ct.itried ro b. sl eirh nrl€d eyc

but the Lury RiFnB te i.di.d6 rh. otsuriB P@ imf.sibt. (My lo, A&,

vlluet wh€@ the Blbylonia. dilsio. atlos fi.n. So lb. quGrion lriss, for pt.as

with hign6 bdlud.q etipric mch inclin.d rorsds hdia Dd dE qegn b€inA o.

hori@n !id. oI ilE eliptic ah@ld w giv. do.! .Uolle for !c? Shoutd rhe p€.I! ol

tlE Rip.B Fundion {q be low th.. $c, e &cotding io 6glB 3.1.2 to 1.3 5 clos.

to tlE rltmr.l .quimr? TrF luF ii ddd, lcr.rt jn lh. dilosio. of thc

@nsloinrs ofrhc Lu@ &F|€3s tuturion wtEn ir i5 poi.t d od rhr for rcry low AR",

v.ru.3 rh. d*qn of bw LAc hn okt agc.td hjc DAz i, E ofl.d ro ben sd. thjs

nay b. th. po$ibl€ l6$n for rt. nrort n sirob@! tik. MDidd !.d Fothdiigh|n

tu qploriig .el.ri@ b.tqq ARCV .d DAz for the ft51 vhibitity of tu@ cresn

o,^. --.

ig

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

t09

In cae ofMoon nonh ofediflh:

DAZ ! ES = EJ J JS lassine

es I

-elr

@te

{r.4.r)

and c4c ofM@n soulhof(tipric:

'n

o,lz - o5 - p1g , 115

= o, *6s - 6n

(1.4.2)

"in, cos p

DAZ siven by (l.a.l) js much sfraller $& rhar eivcn by (1.4.2) for th. r.son tar in

(3.4.1) rhe difirEicc of dqliiatioD of rh€ Sun qd rhe M@n (\ _ 6M) is sna er

rr{ |han

in (3.4.2). wnh k4er aRcL (SM') for M, (3.4.2) as compared |o M (1.4.r) &ry j,

rar8.r bul &[ or Rd. is sdc. Ahhongh A&,i (Ra, - &ryJ r sme tdr the two ca$s bur

crc$cd al M. is older, rhick.. and brighrei mucb soaler rhan for M. Esutts into

dill.Enr ofarcs oftishr (ARCL) or older cE$eit with large! ph6e. Thus in case of in

case ofM it hst be difijcult to ke the crescqr s conp&.o ro rne c6eofM..

When€ler as is v.niet (perpodicuttr to rhc horiz!) DAZ vanish€s

optihuetud

aidnioo of I Oo (when Moon is ctosesr !o rhe t dnh) b l2v (,hen f.nhd lhe M@i

frcn is

rh€ Ea.rh) ccu_ As and whei tu i; nor v€niet lhe ol. DaZ coh€s

play

inro

lnd Oe oplimum condiroN for as ca b€ retded. For hucn ls8er valu€s of

4d DAZ

older dd wjder cesc€nr may b. visibte wirh s@ er vahca of ARCV or aD. So rhe

oodels inlolvine ARCV-DAZ rclarions comc into play. h these hodel ARCV is a

runcrron ofDAZ so as should aho be a fuction ofDAZj

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

- -[(DAZ)

(J4.3)

Il0

bdh of such obFrystios dut were povided in Monm*n, s chromtogie lt8a3), W_

This n@s rhar for coBtaor LAC (e a) ARCV decE|s

com|anr 9, t.AG ed ARcv e di@tty lotDnionat ro ech

ktnude L lrgc DAZ dd taige ARCL m@s mall€r ARCV

LAC.

Duing Elarivety ucnI lioes rhe dploorions of rhe @tist visibiliry oa nd

rutu cr€sc.nt or rh€ lat visibjtity of rhe old cFsc€nr b.gd wnh lhe obstuations oad€

in Alhens and iG viciniry by Schnidt md othes. fteoretical exploradon was iniriared

rurc du€ ro @leldtui@t E6otu ll)e ey paniculd sbononiqt qusdon

(Foth*inshm 1903).

h lhc beginning of th. trcnlietb ceirury n was eatized rhat ncrhods of verifying

dates, panicltarly tum, dotes, wcrc nor avrjtable md pspje were @ncehed 6bout the

6r6noorc.t ondniom thar gov€n dE nsted €rc visibilny of $e rs lun.r .,esccnr

(FodE nsllah, t90l). To evatuare anonohicrl condirions for rhc ktiesl risibitny of

:::::: *'*!' "-" " *.ber ofsrudies ,ppeaEd. .rhe slrk ofr. K. Forhe,iishd

(1910) hinsell4d that ofE. W. Mau.dq (l9ll) is of vjtal importance. aou rhcsc

onu'bur'ons *eE bed on fie nal€d er. obcenarions ot new luna cr€*enr nade by

Augusl Momeseo, JutiN schhidt sd Friedricb schn

@nriburo^ ee .,p,.r*, i"

"'* ";; ;;;;:;Til:,#:lTJ:,i:;;

^, . lonn n.ghao claim ro havc sueg.sEd (in his d,crc aprHed

Philqophx in rh€ roumat

1 90)

of

lhat in ord.r to catcularc thc rrue date ol phasis on ough ro have a

lable of te requisne depre$ion of the Sun betow honzon at fic 6oonkt, or of lhe

aurtude ol the M@n at ttE $tuer for dif€re o8uld dishnces of the M@r fom rhe

sun. He h6 .l$ adnitted abou his hl'en* .

Monhsn in ,his rcead. Motre.-;;;rJ,'* T,,'1:::;:;llTf

moo. hade h the laGr half of thc nine&flti century by Jutic Schnjd| Iriedrjch

Schmidr and Morn*n himslr n€ hblcs menlioned e;liq euld be @nslrEred on the

lll

69{0. Fo(brilgb.(l (t910) hs &!.rd,!od lt !. oh.{trdo.! giviq cilit drB of

ot66htio! od ir! r!$lnc Fsvit d Dy tioi.iaod ll. !E!Xind.,nd thaidltrh

of t! M@ Ebiis ro lh. Stl! r drc lim of Ms.r (or !ui!.) crrqld.d by

Fo$dDgh.n tin!.lt Tt4 ob6.'vdo$.r. ,!lrrr!8ed i! rh. T$tc No. 3.a.2 { h

sb. rddidool .rlc-rr@.. Tb culs &! .ts Fs&d oo o ARCV-DAZ chti in

Fi8@ No 3.4.2. Fo{t.iishn (t9t0) .t$ Gidna I ddrrEy r.u. lhar is

r.prottuc.d a Tibt.3.4.1 b.lo% rhd sivB rt nisinM lltiidc for lh. !.nibt. @*a|l

fot rr|ridl! valu.s ofrcldirc .?hn'h.

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

H. .!o it"EtoFd . mlrhoni.d Gt rion lo dc'qitc rtc k,E:

Mioioue Atrtr* - 12..0 _ 0r.0OrZ,

(3.a.a)

]T1,:.* y* *** ni5 cod. d.fn* a r.sioD or sry lroud rh6 poiot of

$Ds a $ow! u n6 tgu€ 3.a.2 b.!oe. If, .l 6. d4 of su!.! rL crBc..l i! .boE

d! cu.rc ir ltould tc viliuc orh€.eirc iot T[. cruE h crinlit to. Fo6qingh.n,s

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

lt2

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

- A@-on ii-

'

d66

3' 1,1 lo&!

&

ia

I|l

w. d.firc a pa@et . !r ,

.,visibihy pamerd,, accordiDg ro Forheringhao sl

rt = (aRcv - t2 + 0.00s21) | tO

(1.4.s)

rn lhc T!bl. No. 3.4.2 rhe tsl colhn @nrajd ihe v!l@ of ,. fo, €mh

oD.eryaxon Epon€d by Forherineh@ ed rhq rh.lablc is $n d in rh. inccasiag ordcr

ofyl . ln !i.w of{1.4.4) iflnc lund attitlde is lcss rba. 12,-0-0..OO8Zr lhe qcscent

should nol bc visible_ Alr€matct, in vj€w of(1.4.5) if rhe value or ,ts is rcSalive the

ccsnl shoutd nor be visible, Thc vdB of Elarivc altiiude of the Moon dd irs Gtaliv€

ainuths aE rho* calcul.&d by Fothqinghah. Ir is e.s y nobd thal out of 20

obsetuations ao. which lhe varueof,r is negatilc two obseryario$ ar€ posilivc: One. on

Oci 27, 1859 (obseru.don no.2) od tbe 0lh0 on tlE nohmg ot scpr. 14, t87t

roDsnar'oo no.4t). Hosrver he hin*lfdnilslhat his marheh.dc€l elarion (3.4.4) js

nor $ Etiabtca tE sumary labte.bla. Thc eo€ is exhibile{r i.lhe FiC@No.1.4.j.

All ir'c esl ofdE posnjvc ob$dadons e aborc FotheringtEn 3 vnibitiry cw..

hsread ofEtying on lh. ..bueb,. rylDxihale natbeharical

nay consider the su6My tabte dala and ltr a quadratc cune

approxrhation. Following rhis we oblliEd lhe tolosins @.a@n

ARCV dd DAZ jor rhe shmarJ rabt. r.4.r d,h:

/RCv = -0 0ot2g Da z' +o.o7442aDAZ +|.86429 (3.4.6)

ed dennc e dlkrure visrbitiry pamercr ya ba*d or

apProx'&afior d folto$:

Seond d€88 L@r squd

I L\ = URcl/ +a.00929D,42, '.O.O14429DAZ _|t.s642g)I |) (j.4_7)

lt4

Tabl.No.1.4.3

5

fhis nbdi6c.tio! i! rn oL by Fo$dnBltn it lppli€d to dE obseutio4

repsr€d by Forh€.ingh.n .rd e$lrs e pc5I.d in Tlble m. I 4 I TIE dat! of the

lnbL n d$ pr*ft€d in FiA No I4.1 !t ir 4ily sa th3t rh€ l@t sq@e ntins ro a

s@nd d€gE polynohisl h.5 id inprovcd $ylhin8 6ttF tN nore positive

ob*N.don have lill€n into th€ mg€ ofn g{ive vd!$ of vd Th* ee Nmb€red 6?

(ofo.t 27, 1878) a$d l7 (F.b 20, l87l).

Dunng th€ sme.6 Mdundg (l9ll) @Nid€rcd &olh€r b6io data s.i given in

the hbl. .o 3 4 4 b nr $e obwnioml drl..

20 30

hvl3bL cro.c.nt3 . villbL cr.8c€nb . Forh6rinoham tute

Fig. No 343 Fothringhan s Rule

l16

25

F-d;:

Fi8 No 144

A last sque qu.d6dc poltDnnl 6ncd b $jt d.L iGld! [E fi)llowing cl.rion:

at-v DAz'1 PAzl + tt (r.4 s,

lm 20

Usiig lbls polynoni.t ed $e @nditid $d the (,B@tt would be visiblc if:

ARC1'> !4: V1+r

(34e)

a.d lppli.d i to dd. us€d by rdtsirghd $e @h3 obrlid 4c prc&tn d in t$lc m

3 4.4 !dfig.no 3 44 In lh. tabb t t lh. 'visibility p.m€rcl dcfin.d asl

,,fl!0o20)), =f no", -l DAzl -P4.,\\

(3 4 r0)

II?

35

30

25

20

15

10

5

0

-10

. hisible Cre€cents . Visibt€ Cr€sconts

fig. No 1.4.5

Th. rrbl. 1.45 3how U[r !E @ft! br.d on Medd,s oaM &. such

inprevcd Md rhs. b dly * obqvdid j.. m 4j d le?l t4, tS7l, thd dqin6

eon rhc @rdidoa (3.4 9) IE irbt. j 4.4 and dr fg!rc ooln sr|N rh|r ood.t du€ ro

M.udq is nu.h inpotld s ohprld lo rh. Dodet (!l. b ho Ein*hrn.

rin.lly in rhis vort bolh the @ddq .trl. Forhdi|ghe .tr rh.r due |o M&i<t, &€

appljed ro thc Ft..i.d 463 obs.rio$ nm '!br

nr6d),e (Od.b 2OO4) nE @lb

e p'lsrd i! ffBod m. I 4 6 (Fo.tEngn n,r mdct) &d 1.4 ? (Mr!id6t mod.t)

arld oc rpFndix-tr (Colu@ C .d D, r6p€djidy).

Th. fiCW m. 3.4.6 Dd dE r.bte i, r'p.ndir-n (@bm C) lhow Lt l rhq. !E a

larae tumbd (90 our of t96 po.nirc 3igtni.g) of obb ,on! wh.n rh. dcaar B

reloned vbibte fi.r dwin. AM FortEi.eho,s trFdet rd Lc betow rt. linir mvij.rt

by th€ tul€ (3 4.4) d@ to Fdtdi,Bh.r Thi. a|€g.q! tb.r rrh. ll@s of the dte is

highly relr.icted to ih€ d!r. m which Fothdingnm wdr.d on and .equind siou

ll8

E- 6i

6i

intTii:i

'E

s

na

n3

E

ll9

:i::

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

'l

Fia. No l4 7

On lh€ orher han4 figure 3 4 ? ,nd rh. tdlte i. 6ppe.dix tl

od ol 196 pGniv! nShd.Ss o,ly lO o6@tirotu d*i.1e |ion

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

t20

lh. €floft e nol s su@esstul s lh. Babylonia cdlcnon ed Lud RiFnc$ Law

coBider.d in the prelioN two articl€s 6 fd s lb. nMbq ofdeviorion fon the law in

the ob*tued @seni d€ @ren.d.

Anolher eforr ofgEar significance rnar h found in tir.6tuE is based on rh€ wft

of Schoch ( I93O) and h known d lndie Melhod give. in the Exp tnlnion ro The trdian

Astronohi.at Ephenetis , t^ ttlh n€lhod dr ba5ic dala u!€d rs givs hee il lablc no

3_4.6,

A leasr square quadralic polynoniat fincd b rhis dala yields lhc fo owing Etaionj

A nC t = l0 t74l - u Ot 3l)DAZI _ O.@aj DAZI (1.4 r r)

UsinS lhBpotynomiaI od rhc condirion dur rhecre*enr $outd be \6rble ii

/ RC v >1e it4i - o.ot lTDAzl_ o.@st DAZ. (t.4.12)

@d applied il to dara *l

Pc*nled in fis. no.1.j.8. For 6is figuE ,// Bed G

Ihrs hrl( $e Esutrr ohained aE

''visibilily peamelef defined sl

\ =(ARCV _ lt}.3:,43 _ O.o137DAzl _ o.!n97 D,12..)) (3.4.1:l)

,,,,,.i." "" -O 19 our 196 porir.\e sishrinss Dar devidrc rbn l,ll. condi on

TIs rill rhe hrer hatfofrhc 2Od c.otury rh. Indiao n.rh@ ws consid.red to be rhe besr.

Horck in vi€w of rhe analysis of Babylonian cnlcrion and the Lund Ripen s Law

exploFd 6d prsenred in this voik. dot w.d rhe Indim mehod rs s sle6sful in |.ms

l2l

of Fsnivo 3igidier dwidirB froo rbd.l. Dring rlE nodsa tis6 rh. Belylonim

cniqioo dd Ripenss tuaction h.! no( ben exploEd s rho@8fly 4 is done in rhe

wott This cxploorion hs lqd ro a sianili@1flnding rhai rh. eient md rh€ n€nievat

nE<t€ls for rn orlien visibility of @ tuD ff*rr e s u$tuI s loe of d€

{-.

ilr .;'.':i. .,

r lgrlq-;':t! *:;19;!t *j:.t l

Fig. 3 4.3

3.5

COMPARJSON AND DISCUSSION

TIE wort F6qn.d i. rhi, .h!rr6 6 b. sDtuis.d e & os:

For o obe*rioid lur c.L.d!r h is inpondr b rel,& uur rhe @ndnion

"binh of @ MM o difri@ b.fd. tqt $dc- i, d .t .|| . rclirbl.

ondition for visibi|ny of B tuE @$.,n Thij i! p.niotdy impolt nr in

€es wnen tbe M@. @ s.t b.fore $ma *n .na @juldion Thus a lunr

@lerdd bred o.onjudid ofM@ *ith rh. sr b.forc trE b..t $nst dd

e obseitio@t tue c.lcid., rh.t ..qui6.dut sighdne of rh. ew cltenr

nav€ lo De €slnirtty dif@nr.

t22

Tne ddc oa aliflic wirh lnc horizon plats d impondt olc for th€ condirioE of

edli.st visibility ofn.w tun( cese.i. For norlhd h.misrhcre if. conjunctio.

falls nes aururual equinox this ssle is snall for borh hiddle md high.r

latitudes md fi€rcfoE cBeol is eidlcr cloe to rhe horian or .vcn b€low lhe

hori&o .r rhe lin ofsu*!. Th.EfoG old.r @scenG may *ape si8hli.g.

Th€ acicnt Bobllonie crildion for the €elid visibitit, or new tunar ccscenr

h6 the hienc$ suee$ pcencge (96.4rr'o) monesr alt rhc nodets consid.Ed in

lhis cbapr€r for posiljve sighdigs coGidered in lhjs tr!,k. Ho$!ver, $c su@e$

perccnlage fof neSalive lidri.gs is nol good enouSh (59.9%). Th$ the o!e6[

succ.ss pe rcen rzge ot lhe Babylonie c rir.rion is 75.4%.

The ideas retated b lh. Luar Ripeness tuncrion thal deletoped dlrin8 $e

nedi.val ca ae lhobughly invcsligared. Wirh ftoucm rahnques of

ohpulltioB rhis hs Esultcd i o a useful ne6od for det mrnrng lhe day ofrhe

fi6t sighdlg of @w lumrc&$enr. The onl/ probteh thar sdaced againsr lhis

nethod i3 lhc sightings that deviared from rhe nod€l i. hisher tatlludes. Thee are

older c,€*ents a.d brighler crcscentr h,v€ lowq Rip.ness

11"' ly:. rh€ s@ess per.nlas. of Lunar RiFnB rarv for posnive

siel incs G 9.8% (bcrb rhan alt hod.ls coNidcEd excepl Babylonitu cril.rion)

bul rhal fo regadve sightings n only 57.?% (wose lhan all ofter crneria

consideEd in this .hapte.). The o!.hlt succe$ p.,cenbg€ is 72.l./o.

Theadw.rage ofnelhods lhar sre bakd on Etarion beNeen ac ofvision &d

!€hdve .zinulhs ed thal dc moE lhomu8hly invesric"ted durjng Eod.rn.6 is

a6o exptored. Il is fomd lh, rhe Indie Dcrhod b.sed on dr b.sic dala of Scho.h

is rhe besl aDones! lhe ARCV_DAZ ba*d Dedods

amon3st the €npidel Dodels ofte e&ly 20,r cenrur', rh€ rndid De$od be.l

uccss peEenlaec tor posnrve srghrings {eo.l"b)

rollw.d b) rhe Mlmdets herhod r84?%r and rh. me$od due ro Maundcr

t23

(54.lYo). Howv.r, in t m of th. su.6e pc@nla8e {o. eSatve sighdnes

Forhdinghm\ crilfion is rlE bcn (@ons$ slt m.thods consid@d in th.

chaptet wnh 94.7% fotlMd by Momdeas (82%) ed th€n rhe Indid ednod

/67.8r/'.

The oveEll success pcrced.ge of rhe In<tian mcthod is 79.5%, oflhc Maundeas

merho! is 81.lqodd rhat ot FohrinShm\ method a 7@.

The .uth@ ciry or succ.s of .ach is m.6uEd in rems of nhb.r of cGscnr

siShtirgs wirhour oprical aid (posi,jve sigtrings) rhal ae In aEreehent qnh $ecnl.non.

Some aurhorc have sre$ed on lesdng cdt.ria on rhc basis oI nunbcr of 6es whcn rh.

cnlqon prcdicts sighrins ed dlc qe*ent i, nor s@n (nee.rv. sghrncs) (Far@hi e1 sl,

1999) 6 wctl. How€vq odlcB hav€ indicated rhar cban

inoe.s wirh insca* in moe .,"","J;; ;;::,;:T:::"::i1",#;

1988a). -IheEfoE our eftpbdis is on .xploring @nditions uder ehich cE$enr cm be

scn od not on whcrh.r il is &rE y stu or noi e rhat lDNo.s may be 3t€lchcd for

Judeing ft€-eliabiliry ofthe cl.jG of siShtin8e A3 menrioned cadier thee c& be a

ro n€gadve sightings. rfa c terion pedicb sighilg snd

mc @rccnr E not 6crldly *en d@s nol sl dll ncm lhat 1h. $irerion is not rcliabic. TIE

:::lq condi,roro 8@,ry str6, vis,brny.!cn ,f

'nc

sry,{ no, ovft$r as sh,rr br *.n

rn rhc n€xt chapG. B.tole authcndcn, of a cdledon is ,

:jTt ":t, ::_ * *; ; ;;;;".;" *:lj:: :"ffj;j:T

Frw€cn the dimtt illmimied Ihin cecot ad lhe bignrnes ot IM&h sky,

t24

Chapter No. 4

PHYSICAL MODEIS & THEIR EVOLUTION

AI thblclh rhe twen €rh .entury and Inro $e ie€nry ftrst century a tot o,

wotk h.s been done on vanous aspects ot the probtefr of vtstli ty ot new tqnaf

ce$€.t. th€se other t$u65 In.tud6 (t) ton€th o, tunar cr6sc6hi {Danjon 1932.

1936, yas 19a3b, 1garta, McNa y, schaeisr, 199ft, McN.xy, 19a3, sotran, 2005,

Qu6hl & khan, 2006, .rc,) (ia) the mintmun o, timtthg €tonga(on of n6w vistbr.

lunar cfescenr (Danjon, 1932, ras 19a3b 6tc,), (thr $asohat varaa ons In the

€anbt vjstblliry oi .ew luna,.r.scenr (llyas, 19A5, Crrdwel & !an€y, 2OOO erc.).

On€ ot rhe mosr stgntfi.ant or these and olher sffons B rhe hrroduction of rhe

-|rrohattomt

Lmr D6te Ltne" or |LDL (contras0na rh. |nr.ma onat date ltno

(solar)) by lltas (ltyas! 19A6b). Thoogh the td6a h.s n€ve, D.en !s6d h pbctic€ of

lunar cat€.daE but rhe !€me has b€en erten.ivety usoc |n lonrare (ror insrance

luooioat by t{an4r lnd A@urat. llmo by Odeh) as a gu|d€ for the rcgtons of

vlslblllty or i.vtltbllty oi ths n6w l!..f cr€s6nt, How€ver, In thi. wort the main

emphasts ls on th6 modets rhat deat wlrh rh€ probt.m o, eaniest vlsrbi ry of new

lun.r cesco.t so that other tssues 6r€ nor constdered.

The ri6t asr@phlstc.t modet fo, ervtng thls polreh was rhat o, aruin

{8tu1n, 1977),Ihis kas ba$d on tho av.rage bdgttness hodetrorru tltoon, rh6

averaSo b.tghtnoss ot the twlighr sty and lhe theory or extlnction ((ooman, 1952,

B6mporad, 1904, Sted€ntopf, 1940). Sruh was ,bo rhe rid jn modern rimes ro

exprott lho v.rtaflons of tunrr s.miii.meter wlth rhe Eaft]vtoon dlsrance,

Afiotu€rds, app€ared lh6 .xtenstve u!6 o,lho phFlca .nd scionce ot vtslblity by

Schaetor donng rh6 t6st quarte, o, th€ twe. eth @ ury Gchaefrer, 19s6, gaaa,

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

ma€l tude or the sky. H6 al$ lhe.att2od rhe hportance ot (a) tack ot hrorm.lton

about seather predrclton st iems a.d (b) necd ot tunher qtlo.6 on ot th6

physlolos/ of huhan vi3lon c.pabthbs. Thus,3tnc6$€ rheoFltcat modet teadi€lo

Lunar Rlp€nes law by modtqvat Mus ms th6 only rheoreltcat mod€ts a€ du€ ro

Bruln and Schaeter In rhl. york Schaoter'. t$hntque! .16 .pptied ro th6 ,6cent

obs.aalion.l data and a.e toond to b6 In good aet!€metrt wtrh the obsetoalonal

The exptotts or yattop (ya[op, 199a) whlch was .gatn mor€ of enoticat tn

.aruE b ba36d on the ob$da{onat dsta and p.n o, Brutn,. mod6t bur wtrh rhe

srmprrcrt ot a 5tngl6 par.meref crito.ton tor the n.w oes.enr usibtr(v, Thls

Yallop s modet c.. b€ termed as a s.mbmpt car moder. one o, rh€ m6t

srgnlllc.nt connbutons ot y6lop ts hts concept of b€st th€ of vistbfiiy, Tne

sonware Hrtatol computes borh th. q_vatues (ya op, 199a) and rhe naCrltod€

@ntrst (rhe rerm cotned In rhts work) rhal tsthe dnf.ron@ o, rhe r{agnjtud€ ot tho

llroon and th. timtflnS hagnltud6 of rhe sky ctos6 to cr€*enr. Ihe conpartson ot

lh€ tso i3 di$ussed ..d som. ot th6 extra ordtnary oDsendtbns are cdflc.flr

analy$d. rlle mosl stgniftcant part ol thts ch6pter ts the d€vetopnenr ot a new

srnge psrsmeter crttodon for th€ fl6r vistb lty o, new tuhar crsc6nt, we have

con3ld€red the acruatbnghrne$ ot the cr6c6nt rhat b pnase dep.nd€nt(instead of

aver.€. bd€ntnBs ot ihe tul Moon ctos. to horizon usod by sruh).nd rho.ctual

b ghrne$ ot lh€ iw ght 3ky close lo rhe potnt whee rn€ cesce rs prese( For

lh€ bighhe$ of bolh (the 63conr and th€ sky) the loots devoloped Schr.fer.nd

otheE hav6 been osed, thts has ,6!utted hlo new vlrtb tty and lmtring vbtb iry

cunqr lhb te6ds to a n€w s€t of basic dat. whtch In run E.onvened hro a rcw

slnge paramoter cdredo. ba*d on . retarion het e€n aRcv and width o, crescenl.

lh6 osr mod.t ts anorh€r somfempnbat modot, Our cdterion is lo{.(t to havo

bettor succe$ percentago than any orh.r crile on devetopod dq ng rh€ 2Oh

t26

4.1 BRUIN'S PHYSICAL MODEL

Atuin bsed his wolk (BNin, 1977) on 1he obseded avedge brigbh.ss of sky

agaiBl rhe posirion of th. sun b.los ho,iD, ,ner ssel (0ut natched tnc resuls ol

K@h{ et. al. 0 952)) and &e bdshrB of rh. M@n s a tu&tio. of ddud., b6ed or

lhe 1h6ry ote inclion dw B€npoFd (Benpomd, l9O4). The fi8ures gi!e, by him, Fie.

? $d 3 (Bruin. 1977, pp.339) arc r.poduced here in Fi8 No.4.t,1. On rh. b6h ofthcse

sludies Aruin developed the Lund visibilily cuFes (relarin8 atdtude, , of qeeenr

plottcd asainst s, lhe els depE$ion b€low hoizon) 6d fie Lioiring Visibitiry curyes

{Elatine, + r,gai6r r) ed pEenled in fis. no. 9 (in€tude<t in rh. eme Fig No.4.Ll)

(Bruin., 1977pp.339).

Fis ? shows how $e .!e,a8e brighhss of the sLy ,s diminishes afre! suist s a

Iunclion oI the allitude of dE sun, lhe elar dcpG$ion or orp r, s !n. su g06 b€tow

hodrcn. Fig 8 (Btuin, I977) shos ihe @iarion of rhe av€ragc brightne$ offu M@D

ar s a fnncrion of the alritude ofthe M@n n above honzon .r h. Lnnar Visibility odes

dev€loped by Btuin shown in ng 9 @ruin, r97?) are d€vetop€d using the two functioB

Asum. a panicular brignness ofsky after rh. sunse! ey lU_ stitb, ead our lbc

corespondhg sotar deprh below horizon fbn nC 7, r = 4,a degEes m rhis casc. In o!de!

lhat lhe @en is visibh in such a bright sky lhe ctscenr should also be at lsr a

bighl (lor s!ilb). Then ftod fig 8 Ead onl rhc alftud. of ihe M@r wirh Ih.

coftrpo.ding bnghh€s rhar @Des our to be, = 1.9 d€gr*s. This pmduccs a poi on

!h. visibility curye jn Iig 9 1hal shows a rcta onb€rwentheahruderofth.cresc.nted

ne &ld dip r below hdjrcn foi a panicultu brighrness of cre*enl and oa sky. Thus a

visibilitycure isa coltectionofpoinbG,i =//r), shere oe bnenbes of cr*.nt ad

thal.of sky mrh for cooshfl widrh of cE*nl. The qucslion is rhar fig 8 giv6 rhe

brishl.ess of&c tull M@n (dound 30 d hiiur.s Mdc) and rh€ cGcenr is eeneo y

less rhan t alc minule in width. Bruin Galias this probteh bul t.av$ i! s ir is by shrinS

Oat any dikrcpalcies shau b€ accounrd for by soh. rtnd of .Gcsbh,, taclor As

121

cxnibircd in n8 9 ir is mr.d lhnl aI $le dip r = 0 d€gees lh. ninimM dliNd€s for

dif,cFnt vidrh ces@t3 @ girq

a2

2 3

Tbh hcms Aar rbe bndrh$ of lh. sky ar rhe* atrirudes (r) G we .s rhe brishrne$ of

th€ cBent of@ftsponding widrh when lhe su hdjul $r. I. ord{ dul l,rtc ce$. is

rt 16r a bright s the sty wilh dftEding ahiludc rh. c@e nusl b. wider ud sidc,.

Tnes ar€ thc sanins poinls of rh. vnibilny c!ryes fial aG aU shaqty d(rtsing

f!.crions ol the sotar dip, That heons lhar not only the brighh*s oi cc$e bul $e

brighr&s of sky (har cqul alons lbcs cwa) borh diminish sh!+t, wirh $e

inc@ine st& deFEssion.

Th€r.forc, for larger values ot r rh. ee$ent of sne brightne$ @ b. $en at

lower od loer alftudc ,. Tlw arc al$ the sianing points or rn€ cums ttat show lhc

b.h.viou of, + s {sh ofglribd. of M@n ed lh. sot{ depb) agaiGr th. $tar dept r

Ahhou8h the sm of th. atlitude /, of $e cresced dd the r me $lar dip Eoains alnost

consbrt 4 $e ce*enr gos doM dese cw6 coftsponding to fix€d crcsnt width

ed thcEby ro nx€d brighhess. As (he atrirudc of ce$.hl daca*s 0r. sky brighrness

n4l d<Eases bul rhen closer lo rhc hodzon the \isibilill sbns deo€4in8. Tnus rhese

cuNes fir$ sr.d decrsing with inc,eding !, reach d hininud ed lhcn sl,n i.cre6ing.

This in facr shows the raryjng @slrst of the brigl JEs, ol lhe crcsn md lhar of te

sky. with smatler r and td8.! , lhe contAl is ag.insl dc v6rb,tity, As , incpses dd /l

decEass $e contFsr beo66 favoudbt. fo! vhibitity of c!€$enr. However. 6 /,

funho d€qees rhc @nusl again bc@Des dfavombt. Ior v$ibility.

t28

Fig. No. 4.l - ! fic fi8u6 fron BNin s Pap.' ( t 977)

rrr, ?, rc rddb-r rr oi *l

t29

For thimer c6c.nr rh€ ,r + s aSaitur r plots Sive thc b.st time of vhibility 6 the

poinr whi.h is the midhum of rhe cwe, h rlso 3ueSens a tug. of tim. for *hich Oe

crc*dt my Ehajn visibl€- Th* de excellent id6 rhar @uld help cre$ent-hunre6

bul untdtunably Btuin has nol pesenl€d a clear cul ompuratond lrchriqu€. Brs.d on

sihil& ided Schaefer ha worled out Enorher schehe of computario. lo b€ discu$.d

latcr. The schcnc for computrtiohs is d€duccd frch the visibility cuNes olBruin sd is

bsed on ll)e dinimum point of rbe r + It plor againn s. A relarion belken rhe cE$.nl

vidth cor.sponding to d€ r + , cwe sd rhe valu. of , + r ar rhe minimum oi lh.

cufl. nay be d.duced fron abularing rheF ratucs. For lhis purpo* rhc data deducrd

fiom 6g 9 b, YaUop (Yallop,1998) is d follows:

Table 4. L2

2 3

The valu€s of ARCV ore the valucs of , + r picked fbo rh. ninimln of rhe, + r cufl es

asaisl r whi.h coqespon.ls to the b.sl -tine ofvisibihy ofcresent. Tlh dar! 6 edto

a dird degd polynonial usinS lean sq@. apptuximrion lqdr lo the follovine r.larion

ARCV =12.4023 9.487arv +3.9512W' O.5632tr'

(4rr)

Tnis ( an be tEnslomed ro lh. \ Fibitrly oaEmero . r, lunc,ion as to'toq:

v), =(ARCV -(12.4021 9.481W +3.95t2t/) _0.5632W)Dtrc

(4.1.2)

Tnis crn be le$6d on rhc obscnarional dara with the visibitiry condilio thal a cssce.l

should be visible if,, > 0 olheNi* it should E@in invhible. Using condiion (a.1.2) on

lhe data *t ed ro evatuar. n rhods in ch.prr 3 dlc plor w obhined is shoM in ligu.

no. 4.1 ,2. ln this fig@ lhe cue nmed ..Boin\ Limit,, is the plol of $e equrion 4. t .2.

|]0

Th. r€d of th. plor CD$ th. rc|t|iE.Xirtde (in d.8r€).gaiMr th. cltsdr wirhh (i.

I snd9 €l@ld€d ar dE b€s tinc of visibility for @l obsd8rioi of ttE &tl sel

coNid.r.d ii th€ peioos chl9iq. All !h. pd iE siglding cas .rd ihe @lrs of

cdlculalimr b.$d d (4. L2) @ 31Em in Trbta 4 | 3 TtE @not.r. dda sr b shom i,

R.--E-'_;"_

2

r-r*.rl.sc.G,wu.ca*

Fig. No 4 1.2

ApFidix.|u slDB rl| @hs of .rptyin8 Btuin.s [bn, yttrop,s ondid) €itdi@ (ro

be disss€d in rexr rnicte) .nd th€ w dftdim d€vetoped in rhj3 woli (ro be repon€d

in anicl. 4 4 betoe) undq rh. hQding -Mod.r tne firn @Enn otriaitu lh. vat!* of

visibilily pmmetd d.fn€d by (4.12). TrE Ddl lh cotunr, e ror the orhq rh

ltl

t32

lll

tJ4

tlq

?$I;;

135

1lz

Our ofth. 196 positiv€ sighring in the @pt. t4 cs6 d.qarc rron th. Linjr d@

!o Btuin. TheEfoE rh. nodet du! b Bruin is thc best mongsl ql ihe orhe, 20o ccntury

nod.k ad is s good $ lh. Lufu RiFn* Law. Ar nx es rhai daiar. f.oo

Brbylonid (io.rion atso dcviak froE Bruin,s nodet. A ctoer look in b ttE derails of

14 caes $at deviab fron Bruin,s mod€t sve.ls oEt rhe la cses devi.rilg frcn the

Luar Rip.ness Law and lh.e 14 caes have 8 common cascs. rn. fihl I cs.s ofvisible

cNolntable t.3.1 (ob*nation no.286,2ed 272) ofoncrec cros to auruDnal

.qunox have !alu6 ofve srlt aboE Bruin.s timi! siyen by (4.r,2). A. n @ di*ussed

'n snicle 1.3 the ce!ce.& in 6ese .6es eE older o

vi'ibihypo$ibre. rh. Bruin,slt.i,i,b.,.d..rh"b,ich.::,:;f,J:1':;T::

cases ae erisfying 1ne Btui.\ lioir despite having sEEll las.

Our otlhe six cdes lhat d€viat. fror Btuilt linit bnr not fbn LUM Ripenes

raw{obs.4o.413,434,I19,ll5,2S5 ed 290) fiEi fiG havc qalt valu6 (o.ol,

0.061 and 0.07 rcspediv€ly) so dc hsginal for Led Ripen*s

^Rm

Law. Sihitdly si, of the

Ijg that deviat fbh rhe Lu@ Rip€n $ t.w but rc! f@h Btuin,, ljnir (obwlrioa

numb6 286,2,2?2,314,6t3 ed 716) firsl lbG have b*n dscussed befoE. The n.xr

t36

two ll4 ad 613 have smaU v, v.lu6 $ dc m&gitul €ds in BNint Mod.l. Tn hsl

one (obs. No.7l6) is asain u old ag., wide ond bdClt crceoi

Ths sc ob*rve that 6c Bruio's Linit md the Lund Ripen.ss hw ee nol

exaclly supplcm.ndng eh oth.r blr thcy @ rh.rically cquiql. Howevd a

Brui. s linit is rakin8 brighhe$ into considqation theEloG Bruin's limit h moE

logicat ud noE su€6sful.

4,2 YALLOP'S SINCLE PARAMETER MODEL

To devclop his onc pdndd nod.l of firsr visibihy of luMr clt$nl Yallop

used Bruin s wort in order to .xi6ct optimum crc*enls *idth for vdious rehtive

ahnud$ o. @s of v'sion ARCV, nedion.d in ptvio6 anicle. Like Folhcdnghan

developed a lomula cllline ARCV wilh DAZ on lhe bask ofht sunnary ldble da|a,

Yallop coroider.d fie bsic dau for d€veloping a Elalion b€lren ARCV md lhe Nidth

of crescent W, Hh data (froft pase 2 ol NAO T*hrical Note No. 69 (1998)) is

Epioduc€d in the oble no. 4. | .2.

Hoselcr. d slal.d hy Yallop, f6m 1996 Mdh HM Nautical Aldanac Ofllcc

ddided ro abndon i!3 tcsl bared on the Bruin m€lhod (4.2.1) for lh. one baed on lhc

''Indi.n" melhod d thc lndie m€thod produced mor€ scnsiblc rcsults fo! old .gc

s'-shrings al high alirnd6 thai occuB at l.rsl once d y.r fo.lalnudd rtomd 55 d.sd'

Bed on fie sorli of schoch (1930) th. b6ic dara ued in the Indis. melhod h siven in

DAZ 0' I0. 15. 20.

I00 30

Tatrl.No42l

As rhe width of rh€ qesent is sivcn by (2.E.9) dd thst @ abo 6e wifien as:

tt7

rr' = ls<t - c.{A RCv).os(D,4z))

(4.2.D

a$uming ih€ seFi-dim€ld of tne M@n to be 5 con$lnt 15 dc ninutes. This basic Daia

cs b. t@sfomed inlo .Lt! rclating the width of the cesenl ed @ of lhion ARCV

likc thlt in table 4.1.2 ofYallop bed on Btui.. Yallop tr4Iomed ihal d4ta fittine it lo

r cubic polynomial6ing 16l sq(@ appoximdion 6d ohainqlIh. following eladoni

ARCy = ll -aJ7l - 6.3226u/ +0.7119tf1 -0.10)8w'

(4.2.)

While applying Brui!'s melhod (€qu.tion 4.1.2) or lhe Indim method (4 2 2) $e actual

vidrh t/ sbould be consid@d (nol a @nstonl l5 e minulet. The method says lhat if

,4,RC,/ for lhe place of obseruation is norc than $e value of lhe right nmd side oa (4 2,2)

the .Escent sholld be visible. Thus a visibilitt pameler may be dein€d s:

V| -\ARCV (rr.8J7r - 63226W +073r9w'-0tntw')Jtto

(4.2_3)

Yallop €alls lhG ,/p 6 4-value. In g€n€tal if the visibilily pemetei /p n Positivc lh.

crcsce.l.may b€ s*n. Applyins this lisibility condnio. on rhe dala et of chaPtct 3 ii is

found rhot 16 positive sighi.Ss out 196 deliate fom this lndian visibililv condilion

Thcse d.viating cases ae shoM i. 6guft 4 2 1 as vhible ciescenls below the lndian

"Limit' ed re rabulated in rable 4.2.2 .nd ihe r€sults for cosPlele dao set ats prescnled

in rpp€ndixlll io the s4ond colmn und.t lhc heading "Models" l. view of lb'sc

eluhs w find rhar rhe Bruin's Mo&l is ndgiMlly better ihm Yallop's crnction fo! this

Yallop ds @nvert d Maundd's condilion tom DAz-based lo lhe width-bsd

and oblaircd lh€ foUoving cubic reladon b.tw.et ARCV ed W:

ARC|/ lJt78) e.Olt2w - 2.O1OaW1 -Ot16OW' 142.4)

B8

*lich l6ds ro tlE visibiliiy pabr..ld:

,ip = (,{XCr. . 01.t733-9.0312W +2011J4W" 013601/ )/10

(42t

Fig No 421

. lnda Umir . htsibl.c€.e

ApplyinS Mendd's nodifi€d ondition to thc daia sr of Ch.pLr I it is fou.d

thal oul of 196 positiw sightings @ iher.27 clg th,l dwiat€ fron the 6nd;lion

d.fined by (4 2 5) rhe se e shoM in 6g@ 4 2 2 Thus Me 6s nodited

@tdition is dill mr bdd th3r th. ltdiu lr|d rn Btuit's @rditios All thc 14 {,g

rnat ddilr. iom Bruin s linit tte d6nd. lion rh. Indie linn Ttu rvo tddidonll

wr io!! m. 3r4 .rd 33) ttd ddid. fron lndis $nn & Idtd to b. tugjn.l cet

(vr = 0 ol8 and o oo2 repedively) in Y.lloP! Indiln dEd.l Thu3 thc t@ hod.ls ar€ in

clore sgr4rcot s Iney re both bas.d on sinilu lpprs.hes a.d onlv slishtlv dillq itr

i

{

Fig No 4'2 2

lD addirion 10 @Ndting .ll visibiliiy otditions from DAZ-bas.d lo th. Widlhb!*d

Y.llop achiev€d tqo orhd rdeltbl. t8ks. Oft oftheh is hh d.ducion of BBt

ine of Visibihy The otht r.Mbble ontdbllion liom Yallop is io tkaw lincs

140

b.twe.'r€giotrs of v&ids vbibility condilions on lh. b.s of viribihv pdma.. ,/}

used in lhe Indian mdlDd th|t be cals 6.8tu .

Fis.No423

Fdthc bd rinc ofq.sit lisibililv' hc nor6 ih.t h fg 9 of Btuin tlE siniM

of viribilitv c!tues of , t r agaiin s lor different ges.ent widths, fom a slrsisl]t line

which whs Drcj.cl€d n€els tn orisin of the @rdiMt€ svst'm (" r) The 3{m h

shom h6e in fi8u.. 4 2 3 s a G{t lir. PNing ttroSli niiinun of @L @rc Th'

slop. olrhis saSl lin.i3(, trta/4d,rs 5/{ Itis dnimun @@sponds to lh'

b€d vilibility @dnio|\ thc b.d cerlln bd}6 rlE N6'ee briahtns ortlt d(v !fl|

the b.iShtness oa th. c..stl ..dding lo Bruin Yall@ inrdPr'tt it a 'poinr in tm'

rhNt rlivi.!€s "the tim€ lin€" beiv@ the Sunst ts ed lhe M@n!€t l, in a nlio 5 4 Thus

rhie "poi.t in tin€" i3 th. b.!i tine I, of c..s..ot vitibilitv Sivd bv:

t,=

5\ + 4\rs + LlG) ,

9

l4l

T. *!1,4c

(427)

In all crloldjoE in rhis rcrk rhe coiDur.rioa have b.6 dotu for this besl tim.

sivcn by (4.2.7). EsFcially the calculatiotu fo. rhe iable no. 4.2.2 iD $ltich rbe Indid

condition (4.2.4) c used lhc @mputations ac done for this best tim..

Finally, Y.llop d.duccs rhe visibihy @ndilios for dillcrcnt M8es oI q_vatu4

aic. a detailed dltysk of rhe dlta *r of Nund 256 obsdado6 avaihble in his timc.

Our Esulti in lh. sdond l$1 colmn of rlblc in app€ndix-lll, e lPpliqtion of lhc

Yalloo s condnion (bascd on bsic dala of schoch, l9l0 or lhe Indid ndhod) io! $e

doia selcted inchapter 3 wbich is taLh mostly iion Od€h (Odeh,2004) Theconditions

d.duced by Yallop d. Eprodu€ed heE in labl.4 2 4 Ihese conditio.s e iddicalo6 for

lisibility eiih or wilhout oPlica! aid.

Tlble 4.2.4r Thc t-test crileria.

{A) q> 0 216 ENily visible (,4ICl 2 I2'r {EVl

(B) 02le,l> {014 visible und( p.rfd condnioN (v U PC)

(c)

The linilins ralucs of t

(Yallop,1998):

May nedoptrtl lid'o find cerdIMNOA)

(D) { 160:4 > -0 232 w-itt neo oprielaia o frna crescent tRoet

(E) -0 2l>4'-0 29q froivsrbG viii[lcmpe,4nC. < 8 5'tlt (D

(F)

4-299>q

weE chosn for lhc aix cnbria a to F for the following reens

-tlor

'sitk,

ueto" Dujon tinir;Rcl < 8' I

A lo@ limit is rcquir€d to spdl. ob*talio6 $at d uivial 6on rhos l'ul

nave ed. .len.nt ofdifiqitv. Accotding 10 Yallop il wc foud that lhe id'al

sirrîon ARCL = l2o nd DAZ = O" prcducd a Fnsible cut-otl poin! for shich q

= +0 216, Tnere aG Ill exmDles i! Tabl. in appendixlll G'cond lai column)

wh€n q exc.eds this value, ad i. 8.Mal it should be v€rv ..sv lo se 1he new

(v)

142

c@dt in ln.e m, Providcd th.c is tro obscaing cloud in lhc sky The

rponed posilive sightings i. dr6e cascs e I 14. Tns it is @mble to consider

cscs wilh I > +0.216 10 b. lhose wbcn the cccnl is esily visible.

(B)

Fon obseryc^ rcporls it has been fou.d that, in gcneral, t = 0 is close to lhe

lowr linit for fi6t visibility ud.r p€lfecr ahosphenc co.ditio.s 3l s.. l€vel

withour Equirine op1ic.! aid. YatloP u$d his Tlble 4lo st lhis losd linn for

vhibility mot p@helyand he says thal from inspstion of Tabl. 4 ihe

siSnilidcc ofq - 0 @ b. str bul t = -{ 014 is &olhd Possible cul_of val@'

Thce @ 68 €Aes in Tabl.4 sith q in ihis Ens. in the daia u*d bv Ytllot wirh

48 positive siShtings. Th. data sed in this qotk (in app.ndixlll second lst

@lu.) thec aE lll ca*s with q > 4Ol4 bul les lhan +0216 Oll of th€*

l1l cse) rhe!. ae 6a posirive sighnngs withoul oPtical aL

(c)

Yallopusd hisia6L4lo fi.d|hecut ffpoinl *hs opti@laid isal@vs ne€ded

lo Iind the crcse.t moon by rutching lhe q_test visibilirv codc wilh scha'Iecs

.od.. The rcund€.I value of 4 = _O 160 @ cho*n for lhc cut_olT dirclion ln

Tablc 4 (YalloP, 1998), thcc \,w 26 cNs thar s Gfv rhis diterion t0'16 < q <

-0.014), only lhree ces out oflh.se wre positive unaided siSltings tbe tcst were

se.n vith hiMul6 or r.lceoFs ln Tlble i. appendixlll' *cond lat cormn

useii in this *ork therc 62 cses in rhis tuge ol q_valu€s oul of vhich I 0 s€E

unaided si8ltinSs dd 29 times lh€ cresced *as seen either vith bin@uktr oi s'lh

(D)

ln lhis ca* (-0160 > t > -0232) YaUop s Table 4 ha' roo lew enli's fton

which lo estinale a lowt limit for 4

The situalion is nade NGe bv thc fact lhat

whe€ thse is an €nl!v. in non cas.s' the Moon ss not scen even wi$ oplical

aid. ln f4t it h @€ fot lhc cc.dl to be obedcd lElo* 6 a9pae't clo'galon

ofabout7c5 (Fatoohi d al, 1998) Ydlop's Tablc 4 h6 14 cs's in lhis dg€ oul

wbicb 6 re positiv€ sightirgs thdough binoculss o! &le$ope ed one exl6

oiilinary ce of uaid.d siShing ln th' r'bl' i' app'ndixlll Gecold 16l

141

@l@) of lhb wolk lb@ e 33 .M our of which 18 @ potiliv. sidines

with binoculd or 1el.$op6 dd th@ extra otdinary ces of unaid.d sidli.g

(ob$flalion no. 189,455 add 2?4) lh.t is dillcral fren fi. @ @sidercd bv

Yallop, Al lhe tine of Yallop {1998) lhis wG thc linil helow shich n ws

NW.d that il is nol Posible 10 c thc lhin .t snl n@n .v.n vilh a lel4oF

Alloeing I' for hoiienlal panllax of lhc Moon, and ienoring th' eff'cl of

rcfa.liol\ fo. & aptent .longadon of ?q5, ,'{XC, = 8c5 lf ,'lZ = 0p rhis

coresponds to . lowr linit of I = -{ 2l2 Wilhout good fidding lgles@p* md

posidoml infomation, obsne6 aa unlikelv [o w lhc crc$"t helow this limn

(E)

(r)

TheE is a lheoEical cul{fi Point *hen th. appatcnl elonSdlion of lhe Moo'

frem the sm is ?", knom 6 lhe Ddjon linir (Ddjon l9l2 1936, llvas' l98lb

nabohi el al, l99S) This limit is obtained bv extlapolatine obedalions nade al

lager elongtions. Allowing t" for hori@nul p@lld ofth' Moon a'd i8f,olrng

rhe effect ofrefEcion, ah.ppor€nr elongarion o' 70 is eqnivalent 1o

''{'R'1 = 8'

Wirh ,,llcl, = 8'ed D,42 = O! the corespondine low€r linil on I is -O 291'

However, in Yallopt lable 4 th€E tte 2l enfiies witl onlv ] posilive sightine

with a bin@uld fot I < 4 212 ed no sighting claim wilh s_aided cve ln lable

in apFndix-lll (se@nd lst colw) th* ae l8 c*s ir the mse -0 212 2 4

>

-0 299 oulofwhich there atc 7 sishdng snh binocuhr ot tlescopes and onlv 2

claims of un_aided siShdngs (obsdation los 389 and 455) Both the*

obfdrtio.s dcviak from all the crit.ria considered up b lns po'nl

The table in .ppendixlll G4ond last cotumn) or this wod( shoss 66 cases wtrr

-0 299 > q ed therc is one.xtra ordiMry claim of sishtins

opti*l aid (obs .o. 189) Ap&t fron $is lheE is no posilive siehting silb o!

wilhoul opdcal sid. In lable 4 of Yatlop tnee is no clain of €Esce sisblins $

A@oding |o the visibilitv dNifiorion shos

divid.d into 5 tgions bv four consi&t_q val@s As lne

abovc, rhc suf4 of E nb b

aclual visibihY of lhe crescent

depends on ils width dd oi iis dltilude .bov. hori@n d th. rioc of su.sd accodi.g lo

Bruin (Bruin, l9??) dd Yallop (Y.llop, 1998) a @nsht q'valu. describes a cufre on

th. globe i.di@lin8 sinile visibiliiy @nditio.s along all poinc of$e curye' Such a

clne is a psldcr@bolic cwe wi$ v€icx on the 6l-dost loisitude Thc

lonpludi.al p6nion of lbis ven x vei* nonih 1o nonlh 6nd th' hnude of |he vqar

d.Fn& on lhe d.cliMtion of th. Moon ed lhe Sun on the ccle$itl sphere Duing

sumheG in nonhem henisph€lc lhe sun\ declimtion h extren€ nodh dd if ihe

d.climtion of ths Moon is norrh of $c su lhis ledex no!.s to exlEne nonb and lhe

c6@ visibilily is edier i. rhc nonn hitudes Duing summer of the nodnem

ladud.s if lhe Mmn is eufi of lhe s6 then ftis v.nex d@s not @ch irs

'xcm€

nonnern posilion dd still tne .e{ cE*mt vhibilitv is better in 1ne norlhen laitudcs

Thc situaion h revese for lhe $liheal henisph€te Tne palabolu opens westMd above

(nonhwardt and below Goutbwrdt fion th€ vcnex The CuR' A is rhe collection of

points on rhe 8tob. fot which rhe q_value is 0.216 All rc8io's $ithin lhc lwo b6nches ot

$. p&abola wesl of rhc vertcx e the EgioB wh.e |h. q_vdhe is s@ler

lhm 0-216 md

the cle$ent h ea3ily visible to ln. Mlcd eve in lhis rcgion' Csc B is the colleclion or

ed B lhe

all points where the q-value is -0014. In all tbe Eeions be$€e! cuftes

cicsc.nl is lisible to thc nai.d ete onlv undet pcrfed visibililv condnions

The Fgions b.(wen cuN. B and C (q_value _0-16) de $e tgio's io wbich d

obsen.r suld Gquire opdcal did 10 locatc ihe cr*c'nl dd th€n it mav be visible lo

naked cye. For reSio.s wi$ q-vdlue I€$ t\ai _0 16 rhe cresenl wolld not be visible to

the .akcd eye For a comnon. untaincd obsePer it h highl) unlikelv thal ihe cFscem

*ould bc seen in rgions ed oflhe aNe A Thc scienificallt ecoded obsaliom (oh

ehich all $c sludy of the rwnli€th cdturv i3 bNd) do oot Pohibil obedalion of

.r.sc.nl wiln mkcd €ye in region belwn cwet A 6d C In sucn t'gioN' in facr' lhc

probabilily of obsenation incFdes whh the nlmbef of keen tdined od expenenced

145

4J

SCHAEFFER'S LIMITING MAGNITUDE MODEI,

Ilruin (197?) ued only avfrgc brislhcrs ofsk, duinsrviliehl and rh' vanarion

of brishtrcs oinE full Moon ro obEin appNximlc cohlrsl lb! (csccnls of$rtuus

sidths ro develop hh crcsccnl lisibility cuncs dhcussed a6o!c ftc t'srltins nodel

fornalized in lms of , rclation berwen cRsenl width md thc Elalilc tltiludc ol

€{*dr ar "tal rime dcduccd by Yatlop (1998) pro!.d 1o b€ hiShlv succcssful'

lihough, the basic data cxtacled by Yallop frcm Druin s visibililv cuBcs- was

rcplaced bt rhc basic das due to Scboch (1930) (lhe Indiah nclhod) lo aFivc ar his q_

value coodidons (di$nsed abovc) for rh€ nN ces€ vhibilil! n h6 bccn secn rhlr

rcsulls iom Bruin s d,tt .re mrlgi.ally bctter dan thc lndian mcthod

'or

lhc ddo scl

Schacllcr (1988a. 1988b) oh lhc o$er hand has uscd $c phvsics of \isihilirv'

cxr.nsivclt ih rcsulrcd inro a lool (6at S€hacrdconvencd inlo a compubr Dmgmm) to

ddmine $c briebhcss ol sky at anv poi ol tinc dd for di{Icrcnl aftosphenc

rcmp€*tutes aDd rclarile humidil!. Ih rhis {ork Scbaeficls progrm h ep$duccd a'd

frade pad of the lundr crc$eni visibililt software dcvclopcd. Ililalo! ro clal@re

lisibilily cohdnions. Tlis pan of the sftld is rsi b conpurc thc sk! briSbhcss (or

linni.B magnirude) al points clos to $e cre$eht md the aPPatunt nagniiudc or lh'

lunar crcscenr ro sludy $c laryins conmsr dunng twilighr for vanous tcnpcrat!rcs and

olativc fiunidny. If rhe appaFnl brighmes of the crc$e.l h morc ddi tlE bighhcs of

lhc xrilishi sty lhc cre$e should b. visible o$cNne nor Schacflcr (1988a) himsell'

hasap!li.d a sinilar rcchnique lo @lyscthc visibililvor invhibilnv data avnilablc in his

riFc. -l he @hnique is ,Pplied to $e d a*lof rh. chap161dd the rcsulB obbiocd arc

pnqntcd in rabt No 4l l Bcforc a discussion 6n rhcs' Esuhs Schrcli'is

nldhodology t-or compulins sky briShhess undc! difeEnl ahosphcnc condithns 's

Aner rhc conjunclion thc ne* lumrcrescen{cd bc seen in rh. scslcm skv ck'sto

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

rwilislr sky dcp{nds on d number of f.clou. Thcsc incllde:

P6nio! of the c@nl which n.elf is significsntlv {fTctcd bv Lhc

Tbc $.nerine orthe tieii ftom cEee fld thc sunliShr dle to (a) thc

roial amount oaair-nass the ligbi tdvels-lhrcugh. (b) thc rotal amounr ol

aeosol pesnt in lhis air. and (c) lhc slratosphctic ozonc thoush shich

rhc lighl bas lo ftvel. Th6c *!lt ring souEes cauF th. inlcnsnv of liehr

sourccs of liehr rhal includc $c S!n. lhc M@h and o$r sourcG (like

anilicial light lbat G nol considcfd in this work 8lh'v m noi s

affc(i!c during lwilisht.

The amosph{ic ldp@tuE and the relaive hunidilv

Taling in|o considmrion all lh* alldls |he to|,al brighlncss of lh ikv is

conpded at lh€ point whcE cmcol is pae lf rhe bnsl i$s of $c atv is noE than

or equal to tbe briShine$ of the ccs. ihc cGcent En nor be s(n' Eten ii rhe

brishtness of the crcsocnr is marginallv moe $an the bflghtncss ot ihc skv it is vcrv

difilculi lo loc.c lhc (erc6t withour tny oplictl lid ln $e followinS thc q@'rilalivc

bols aR di*ussed bricfly for .ll dc$ @nnuaiions:

For alirudc w.ll lbo€ horian $e lpFcnt Fsnion

anamon.tR. thc angl. ofcfmdion (Snan l953.G@n 1985). givcnby:

ofrhc crc*e isEi*d bv

x=5s.{#]*' (4.i r)

whec P is the ahoq'hcic P6$rc. / is rh. |cnFatuc and z

lhe cEscenr. for ahiildcs closr ro rhs horizin rhe following

t47

i = l,.cor ,+ 1i - l

L I+441

(4 t.2)

r rnr I

x- t.ot.corlA,+ ."- | (4.ltj

I n'+s.tt I

whcre, 90'r : lnd , - 9ou : R The sollwde Hilalol wo develoi,cd in $is work

ror lh€ delcrminarion of lisual linitiie naehitude thc major nePs ol calcularions

iadoplcd lion schacf4 s pogBE) rc lised b.low wih bncf de*riPtion Rel4nccs

and dcrailed d.sc'iptions can bc folnd in Scherd (1993)

Thc prcgEn funcion /drat r'l(,) 1at6 s inpu/pE-c.lculatcd valnes listcd bclow:

. a4r. thcal(iludcofnoon abovehonzon. tdu. ftc azinnth oithc noon

. par. the alrillde oflhc place abovc sea kvelih ndcB, Pn,z. cslimlted rclalivc

hunidny of the place. p/a/. lairud€ of the plde. PratP. esrinatcd lcnD€dlrrc of

. sd/pn4. lhe right Nensioh ofthe $n at the inc of ob.cnstion.

. \eshlil (= 0.365.0.44,0.55, 0.?. 0.9) $. {av.lcngths @@spondinS lo U- B. v.

R md I bands

. b,r.r[t (* Sxl(]iJ. ?x10 i. lx10fr. I'loi3.3xlOI) p!3ncl$ !.lues in lho

nish dhc brigihss sssilled snh .rch wvelenglh $lccrcd

. a!r1t (= 0, 0.0.01!. 0.008.0) p@eler values in $e cxrinction cctlicicnr

corespondins lo lhe ozrne factor ssdialed wirh ceh wavel.neth sclccled

. rr.!.r/, (- 0.0?4- 0.045. 0.031. 0.02. 0.015) p.Bmcrer values in $c cxrinclion

ceficicnr coftspondi.e to ihe w@ficr facloa (hmidnx lcmpedurc elc )

associatcd wilh each wavelengh sel{tcd

. @vh[j] l. -lA.%. -10,45. -11.05, -11.9. _12.7) lhc nagnitude of fuu moon

coftsponding ro dififtnt slel.d warclen€lh b6nd5

. mschtil (= -25.96. -26.09. -26.74, -27.26,- -27.55) fic $lar magnilLrdc

coftspondinS to diff€rc sel*ted w6v.le41h bmds

148

cnschtl ( t.36.0.s1. 0.00, -0.?6. -1.17) @Nrion for lund masoirudcs

!! csponding ro difiere.l sleded wavelcngih b6nds

r"a.. the ler ofde obseMtion,

c?,,ap.lhc elonsalion oalhc n@n fon lhc sun al lhe line ofob*ryorion

'lhc funcrion shs Nnh $lcclinS a poinl I wnh sky position sitnhr lult mdh

I O.l.lM tum + 0.r-i-e. a poinl clo* lo th. cstrc of $e tu.e disc Thc

^.ilh

dishnce ofrhis poid.:ed,sr is scd lo calcul.k lhc 8as. aeiosl ed lhe o7..c ndss

xr = {cos(?cdig )+0.0286 icxp( to.5'cos(zsrd,ri )r-l

(4.3.4)

x4 = bo(ud'i )+o.or I 1€xp( -2a.s'cos( rd,u/,Jl ))l-r (4l.t

. L lsintf,",./,r I l' I

(4.t.6)

"'=l lr.:oro;rrl ]

comspondine ro live diff@r \Ev€l€nglhs cleted in the @v Ydc,ll/:

J, *orr'.^-,0*r!!\

' ''o' t

This is followd by rhe calculaio. of th€ cx nction @flicicnls componcnls

& =o 1066 '*e[- ?s4J'("qrrtl'r) 4 14i7)

r, =o r,(xru;'".-,(-#).(' -a##-*,) ",",,,

Ka = o.'chlil'O +0 4'lpto! '

co{satpha ) - cosl3' Ptat )))/ .

(.r.3.9)

*,, =,,,,,,r,,.. *.(,i#

).".(tf ).".(- #il )

(4.i.10)

For cqch wav€loslh bed rh.* eidnction ccmcicnts a'c ecumulalcd in

krchli)= K, + Kd + Ko + K" (4l l l)

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)

lbe.qurioft (4.3.?) to (4-3.12) aE Pleed in a loop thal rum nv€ ddes oncc for €ach i

O. 1.2,I and 4. Aner$eexccution oflhb loop lhc rir m4ss al poin r, thc tosnion of

rhe M@nandfiarof theSunacc.lculal.dusinS:

rrr@/ = [cos{'drd'Jt )+o.o2s icxp(-II'co(z"dirt.)]

(4.l.ti)

il (nofu < 0) nnposl = 40

ar nnpoeJ -lcos(go - nul )+0025 r.xp(-ll

- . (4.1.14)

il (\ah <0) xn@ld =40

, , (4.l.tt

et:. snpoel = lcosleo - ett) + 0.025 'cxp( -l t

Th. masnitude olthe M@n is lhen @lculatd NinSl

mtut =

-.12.13 + 0.026.lt 80 -.to,@ | + a. ltso -.tons )a' r 0 t (a3.16)

+ c'.rrt4

FoUo$rd by nid -am. bnghrncs. honlidl biehh.ss. twilishr bnghl.s and $c

ctree - n

4 1. Lkhlil' nq6tl

(4.3.r7)

kn = rc15t5 ftmp" | 40) +6.2.107 t(leto,stunf ,+

tot36 .(r 06 +cos'(/.io'sui )J

$hm /elo,At, is $c clonsrtion ofF fon $. ccnft oflhc luE di$

(4.r.r8)

noohh =rc o!'lo,htus D6c'il4+.r:7) '(l - ro-0 4'ri.,l'l'"*/ )' 1t3.rot

(tuh . ahree + aaoo@ . lt - c!tuae )i)

^filb = t0-0at(,*.r,14 nschlil ttz 5 -s.t' -!'dt' t(t@ I llchlnl '

i_ roo l, (r _ r o

-0r.,,.,r,r"*./ )

(4.120)

ctort = rc4 a' t'chlt l' |rydql

/g-, - ro(6 r5:/./d,N* i a0)

+6.2'to1 tlfetoresn)z +

ro$6 .(.oe *.o.'?(tt,,r* ))

.tqh - |o ort'l' ;i,l't'n6chlil'ar'').t -I0 03rrs/'I4\fuir' )'

V"' , "fat *aaoooo'(t -

"hn))

(4.1.21)

(4.1.22)

(4.12l)

150

lf thc hviligh briebhes rv/, doninats ovc. tne day lieh briehhcs dat, rhc. thc sud

.i r,grrb and .' ,1, are slored in ,r.r/t otncffi$ rhe sun of ,'s&r, md etb rR noE'd

in L.r/t Morcover. ifrhe Moon is.bov. horizln thm de,, C lho addcd to bs.r/t/.

l-iially, fic brighbes ,.\rrlt is @neened inlo .m-tanb.6. fon cquaon (4l.16)

till rhis point all conpulalion is done id a looP that €recutes Ilve timcs again, once for

sch wavel€rylh bdd *l4td fnc €aicula$on ol 6e limitins naghitudc /€, is donc .s

la (tel < l5o0) ic,,e=loat..eo lo-re)

Els { co,a - I o ! rr . .n o " I 0 r ' }

kh = tum'\t + Jctu" tb.t r

(4.1.24)

(4.3.25)

(4.1.26)

t.. = t6 i1 -2 t,IVD) d.".htzl

\ In(I0) l

(4.) 21)

Schaelcr (Scha.fer. 1988) in his drtshold co.tdsl nodel calculates /i as rhe log

of $c ratio of rhe acrual tohl brigltoBs of th. M@n ad $e tolal biSlttnss oi $c Moon

neded for visibiht fo. rhc givcn obsping condiions ln this rc|k we considr rhc

mlg.itude oflhe Moon and $e lisual limiliog naennude slculatcd frcm thc alsoinhm

givcn abovc- The diftcmc. of Moo.s naenitude z"d!A and lhc visutl limitins

nalnitude le- is consideted .s nugnitude .onxas de"or.d 4 zDzq A plol (Iig 4l I

bclow) sbows dirercncc ofscbacfcls thEshold cohtrasl I and $c nasnnudo contasl

/r43 Scries I sho*s ,1aas lor crcsmls $ wca not $cn and erics 2 shows ,'lu zg ror

crusoent ihrl \vcre sccn Sdies I and series 4 sbow ,4 cot.spondins lo dcsccnc $al

wce not sen d lhal scte sen Esp.clively.lheda1a lor this fig is talcn lrom Sch.cfc!

(1988). P$nive vales ol cont6$ fo. crcscns $al wrc nol *o ad lhc nesrtilc

vrlues lor conlrdt lhal *erc sccn snoN lhc inconsistcncics ol thc modcLs qirh lhc

ob*ndion. Th6e inconsisteoci.s may E$ll fDn .stiDatcd lalues of t€npcaorc a.d

Elative hlmidill adopt€d for lhe calcllalions.

l5r

Thdhold conExl vr| f.e.ltld. conbtt

i-

Fig No 4l l

Ta!t.4.1.1 is d.v.lopcd sina the sc prcssn Hildol i4 o!&r !o sDlv* $'

crts@t ob*frarion @rds t Ln tod ti|c E (S.h&Lr' 1988t Ydlot' l99E Odeh

2004). In @h mw of thc l'bl. obsfrrion Dmbq (s sign d bv Odch (Od'lL 2004).

dde of ob*darion. ldrlud€, longrld. &d.lcvstion tlove s l.v.l of lh' ple ftos

wh.d th. c'€l6l is ob*dc4 followEd bv $. diMr.d ldp.tat@ 3nd 6ri6aldl

Elarivc h@idny. Th€ .ext thF @luld @nr.in $c uit.6,l lim' &d th'

ldf,rdlig Ms,in4le .dr6t wh.n th. ru&ttn d" co,rr/dt b4om$ j6t f!vo@bl'

for siShrinS of c'lsdt (@lum vith hddinS .3l!n) *id n is b€sl for sidlinS

(@lwn wi$ lsdins 'bd-) ad whd i is favo@ble io' sidtina ror th' lar rin'

(6ltm wilh hddirg 'ta'r). This 8iv6 lh. tine Engc lor $e posibl' Et'd cve

visibihy ot $. c'ls6! TIE rus,,'& .otdr i3 @Nid.ied onlv fot @id'd visibilit,

oa crcsdt. Th. lan $E 6ll,!m @Duin th. i.fonMdo. E$ditg *tEths lhc

clt* M claincd b b. vi3ible widout dv opdcal ai{ wi$ a bi'@ulu ot eith a

t57

Tt. ob*Mtio6 @sidered in th. llblc 4.3.1 e ody rno* *lq il wr claim

rhar rh. cc@ 6 sn by dy n@s .s ll|e r€cordi enen rhe

de not El.ver. Moeov.r, by vrryiog thc Grinared t€npdt@ dd Glatirc huidily il

n €valual.d 10 oblain oplimm @ndilios for nr}€d eye visibihy ol $. asc.nl, If th€

ntaEnituda co"na is obtairea ro b€ i. favour of visibihy onlt dcn lhe lim. ra.8. of

naked cye crescenr lisibility Ee delemined md included in thc rzble.lt thc nagnitude

co"rf"rr is nol found b be in favou. of u.aid.d visibility the l6t posiliv. value of lhe

nagritutu conn$t is c5lculaled md th. se vatue is included in aU the thR

coGpoodins columns. 'rhe s,m positivc vahe in all rh* @tM.s it rhc indicalion

that under lhe walh.r ondnioN consid.Ed the cr€sccm ws Ev€r visible to lhc naked

]-his table shows tbal $erc d€ ll cases ofposidve sighting claihs {ilhout opiical

aid when nagnilude contmt was nclq in iavour otunaided visibility. All tsc positile

css e aho nor in aet€emenl wilh tbr conditions due io Folhqi.ghm dnd nalndeu

Lunar Ripeoes las d@s nol allow l0 of thcse dd the two of lhen e oily ntr8'hal

cses a.cordins b n. Only one 6e mlcdly diff€D fiom Lllm RiF@ss law. Onl, I of

$csc 13 ces ue not allowed by B.bylo.id cdknon. To or thesc ces m not

allowcd by lndim n€lhod ed rhe !6t of th. thrce m n&giMl c66 i. Indid method.

Boih Bruin's limit 6d Yallop s critcrio. do not allow.ine ol rh.s cases ln cae or

orher tou dlaims Ellfoutcs* are marginally lllow€d by Bruih's critelion bul Yallop s

dlsion difieB a lot in one oflhem.

Thc liBt pase of the lahh 4.1.1 shows t$o etaa odinarv claims of nakcd cte

vi.ibihy of crc$enl, ob6 No. 189 &d 455 wilh q_values -029 dd _0 216 Thc Yallop s

c.it.don d@s nor allow naked €y. visibitilt for lhc$ q Qlrcs 8d tlE naanittde

.o"r'4r, b ale nor favoMble ev.n wirh hiSlly exassenled wealh.r condnions The

Modifi€d Ripene$ Fuciion (chapb 3) !alu6 corespoldine to lh€s oberations e

aho noi favoudbte G0.95 md -0.62 in tablc 1.5.1) for cEsc€ni vkibililv Therefole th.*

obsedations, as lhey fdil to etisry evoy model, de highly melilble ahd ar. outlie*

l5l

TheE is only onc noc clain ofnajcd eye vilibility of ffiant wilh t-Elue lcs

rnd -0.16, obeMtion no. 2?4, with q-BlE -0.22 | (AR- = - l .l9)- I1tis obs.dalion it

not allowed by both the Yallop s crilerion dd th. Lunar Ripenes Law but wnh highd

.tevariotr (1524 n 16 ahove s lcvel) dd loe hmidity (6linat d to bc lo'ld lhe

tuEnitude cohftast is favourabl€ for uaided visibilily lnd $e ob*nolion is not

uNlidble. In rll the esl of lhe cB@nl oben rions *ith I-valB ls lnd -0 16 lhe

claih ofvisibilily of cressnt is with binocular or wit lelescope. Thee claiN de also

nor ucliable 6 out of25 s@h claim 8 harc favounble uas,itt.L cotun lor n*.d

ete visibility with oplimm 4tinales of lehpcdtwcs ed rclative hunidiry'

The hagnilude conrAr for sme other r.porledly Positivc cE$€nl sidrinss

{ithoul oplical aid n nol favoudble. Th6e e obeFalion nunbe6 14l, 3 19, 416, I16,

315. 2a6. 611, 314,272, 2. The q-values (and A\"J fo! $ese descents & _0 153 C

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?),

0.012 (-1.01).0.018 (-2-88),0.109 G1.47). For 341.416.Il6 rhe thrc. crireda (Yallop\.

Lumr Rrpcoess Law md ihe nagnihde €oi!r60 e. co6islenl- For 286,633,114,212

dd 2 Y.llop's citerion allows.alcd cye vGibilily ud€r "perf*l lkibility condnionJ

but bolh lhc Lumr Rip€ness Ls* and lh€ naerrilude conltdt are nnfavoulable for nk.d

eye visibilily. Thu ir apFd thll Lus RiFmss L3w is nole coNistcnl with the

m.Snilude conlrdt rcsulis.

Io rhis work limiling t€lesopic hagnitudes &e not coGidcrcd as lhe r€poned

ccsem ob$fr.tions wnh bineuls {d lclcsP.s do.ol pevid. approPriale d.iails.

ThcrefoE lhe appopiale limiling lelescop.s cm not be conpnled MoEover, out wo*

is horc conccmed wilh visibility of n w casd witboul dy otircal aid

154

t!.2131)

-'.gr""il-

,,;* I " ,.;;; I

,.!('!,"]

;;; t"

r-?tp!!

,,;p,{ I

155

t56

l5?

t58

4,4 A NEW CRITERION TOR NEW CRtrSCENT IISIBILITY

while d€vcloping rhe "vbibilitv cudcs" (it againsl 5) $d the limi$n8 visibilitv

c!ru€J (, + r against r) for.onstad brightn.ss Bruin @nsidcrcd the avedgc brightnes

of w$m horian duiog rwiligh od fi. ldistion of lh. bdgltN of lh' full M@n

with tlE alitude above hotian c me ion d @lier' Instcad of @cid'dtrg av'ase

brightness oi sky we hav€ coNideEd dual blighrnds of skv and fte cr'scent calculakd

sing the techniques developcd by S€hefet sd olh'6 (Schmier' 1988b, l99l) i' lhe

$nw@ Hilalol. W. $lslcd c@ent vi.ibilitv circUmtarcls of ldious rcw Mm6

css wh€n lhe crcscenl Ms rponed b hav. beo sn For cFs415 ofa pan'culd

width we found lhe altiludes I oi skv points with btighhess equivalent lo that ol tho

particlle crc$. at dill€tnt $l& depEsio.s r' Th. ovcdgd of the

'ltiludes

of skv

Doinrs for difl'rnl $ln dctr.$ions for plirhuld sidlh d Lbllat€d in T'blc no 4 4 l

The left most column of lhe lablc contains ihc solar dept€ssions r and ihe toP rov grv€s

rh€ {idrhs r oflhe crcsenrs sclaled ed thc n xl one giv.s its na8lnude Thc enties of

$e 6r of rhe €blc ue lhc aldtu&s , shcE lhe stv hs rh' sm€ brighhAt 4 lhc

briehtness of tnc cresknl ofthc width al th. top of lh€ @luon.

It should b. nol€d thal duins ihe twilighl 1ne widrh of the crc*dt vd'es up b a

dc seconds for very wide d.sc.nls. Se6on to eason ed for dif|ere'r lalitldes the skv

bddrlnes fot th. sdde altitu.l€ cl6e lo th. poid ldee ihe sun sts also vdi's lor qch

column of th. lable 4.4.1 a numbq of c!*s of alnosl eme cE$enl widtb wE

@nsidered md each ahitudc is aveEg€ of lh.se cdes Thc dtla of lhe tabl' 4 4 1 is $e'

plott€d on a sraph shown i. fsw 4 4 1.lb lhis figure n as a fuhclion ofr (i' =/6,) @d r

+ ,' = s 6 a fuclion ofr (rl = sat) aG bofi ploned /ft represnl the "visibilitv

cwea &d the s(, rc9dnc the liniting visibilitv cudcs" sinild lo wh'l Btuin

(Bruin. 1977) developed.

159

T$LNa43.l

tf t li rl!.

a.l a, -r.8

1l

1 tt.a

t3 4.5

95

5 2a

5_a

5 32 2.1 1

,

2,5 1.f 1,1 0,8

3.05 0a ot5

2f

255

r,e 0.t 0.55

r.t

0l o5 o3

o,7 o25

0,2

r.t o2

2 1 oi5 o.t

Fia. r!b.4.,1.1

Altdav.r 6ot Oap,Ebo

!I

5

3

2

12

t@

Alttudo v.6 Sohr Oottesalon

19

13

€l:

!9

Th. @rdin rs oarn ninid ofafi, Ghom i. figurc 4.a.t) e lhd b.sic dals for lhe

@del pe nrv€ dseloDed Sou tu dE i.ble 4 4 2l

Using tubi. l€.e sque appronn rion * oh.ir€d lh. following El.ro. ben*cn

rchtiE .ltilud. ot@rf ARCV !d tu widd r

ARCv = 435t 37tvt +2222O75O57W2 5.42264J1tt! + t0.4341159

(4.4 D

I6l

On lhe bais ofthh dlatioo wc tlcnre th. visibil,ty p.m.t€r v, 6 follows:

v

e = QRCV - ('-O.35lg$7W3 + 2.22207 5057W2 - 5.422643U t/ + 10.4341759) / l0

(4 4.2)

Our model for .adi.sl visibility of n w ls& *s6! ir that if rp > 0 (w @ll the

visibility paEmctcr v, in (4.4.2) s th. r-vtl@) lhc ceml moy be lisiblc withonl

opdcal aid olheei* noi Applting this €o.dition on thc dat set ued in chaflet I &d in

lhb chaptcr qe present thc rsulls obtained for whole data set of463 cd* in d.ltn

colunn of table in appe.dixlll. Fid fd .ss when rhe visibilily is claiocd wjlhoul

optical aid in odcr of inccGing r-ralrcs @ sho$ in 1abl. 4.4-1. Oui ol0tcs cses

only I I 66 d4iatc ton ou nodel. our of rhese I I c6es 8 de coosisteni sill the

nagnitud€ conlrast 9 arc consisle.t wirh Yallop s crirerion and 8 ats consislcnt Mth the

Lund Rip.o6s Law, Tbe oberyaiion Nnbes 3E9,455,274, 341 dd 316lhal d.viale

from lhe Lmr Ripen.$ I-aw, ihc Yallopt crilerion ed 1be Dagnnude conlrdt d also

negrdve i. our modcl. How€vcr the obseorion nwber 4161hat is negalile in orher

models is allowcd by ou!nodel. ille Een is th.t in this ce ARCV b redonably hish

(9.41 .l€g@9. Txe widdi h small (eund la @ sondt bur |hc M@. is rcry close ro

perige s closr ro $e Efih.

In fisuE 4.4.2 the minima of each visibilily cw. is joincd resulrins inro o

nFi8ht linc stich $nen .xlodcd inr.e.l tn€ oigin of rhe (r. t) c@rdjnde stseo. The

srope of rhis linc is found ro b€ (lh + sys -) 9-3t5 ot t'/s - 4.)15. This lqds ro a Dodi6.d

''b*ttine" of cesenlvisibility a:

,3 = -=-i:

s1:s+4.3(Ts+LAC)

rBin conp&ison ro l(2.25)rh panoflAc in

(4.4 )

9.3

l(2.163)rhpanofLAG

besl tih. of crcscenr visibiliry is give.

t62

l6l

In gendl l@kingd Lhe @npt.rcdsl! $ rn ubtr jn atFndrxtlt rl (m b€ nor.d

'Ihr€ t no clain of visibiliy by ey n€{s who slalue < -O.t9l. In f{r as

obrdolion nunbf 389 is not coDsin n( wjth .ny nod.l w dject n ed

ihercrore we claim $at th@ js no aulhotic obstuation (wi1h or wi$our optical

.id) ofneq lunar cEsenr for s-vatu. < 4,162. Thus cFsccn! can not be sen

wn€n vcr r-value < -0. l6 evs wirh a rel*cop€.

For -0.16 < r-value < -0,061h.rc e 26 (45olt clains ofcrcs vrs,bihy wirh

opncd aid our of58 rcporled sd coNidercd obse(arions.

^s

unaided visibilny

.laims 45J, 274 and l4l d lol coNnlenl wirh ey nodet s w @nclude lbal

fo. lhis tu8e ofr-utues the q€scenr cs be secn wirh opdcal ad only.

Ih 49 c6es wi$ -0.06 <

sighting with opricll aid

&cmne crescenr vi0r a

r{!lu. < 0.05 theG e

(41%) we conctudc rhar

or.eutd or a Glespe

u..ided sishring, Unaided sightine b nol impossibt€

ll ueided (25_5./0) ed 2l

thqe ac stonS chdces of

oii very slih chmc6 for

For 0 05 < r-vatue < o. I 5, rhce dc j5 sishtincs wirh dtuar ai d (1[r/r) and 14

$'thour oprical ad t2ra,. I hus rh. tuflr hay b€ easit) *.n qrh opucdt aid

nr h! r4se ot s-htr &d !m be seer wirhour opticaj dn under ver) Com

condnion (sarh{ coldiriotrs and heichr above *. lerct).

For r-ulue > O. t 5, oul of nexl 2 t3 obeR.rions th.

oplrcal aid I 65 timcs (77.5yo). Th@foE wc conclude

cEeent can bc esity seen.

c@kenr was s.en wirhour

th.t for r-valu > 0. 15 rh.

Thus ou oodcl $ar we @ll .eu€hi & (lh critcnon..cd be sleheiz€d 4

164

l

Calculate r-vdu. (or vp @odins ro 44.2) fo. th. c|lsdl a1 ilc besr tine

2.

Th. vbibilily condnior a girn by our 6odcl @ ihq giv€n in Tabte

vhiutc uaer pertco onairiii'lili[

May rcquirc opricar aia o iiiiFitii?MFoli

Requi( ophol ad (ROA)

Nor vis,ure wrrr opriciisia-li j

.0.16 <r.value < -O.oa

The success of our hodet in t m3 of nuber or posniv€ ob&fradons in

.gene.r wfth orc suggqrd c,itcrion is achi€ved dE ro lhe f&r rhat we halr u*d

Schrc&as brigblre$ hodet, i.e. actul brjghrness of sky rnsread of the avela8e

bnghD.$ ed acru.l brighbcs of ces@r ins&ld of 1h. b.ighh.s of ftU M@n. Th€

nudber ofposnivc ob*pations in agreeme Mrb a cnterion rs mostty int.rp,€ted 6 th€

-T--tn*h :,;--

,l-(rr%r

165

Wlen a ctirerion allows oprielty uaid.d visibitiry of rhc ncw crcsnr dd ihe

cFsnt is nol sn thcn it is a ncgativc.@t. Th.re c& be a.mbd of ree.s tor

negativ€ em6. D.Airc rhe f&r rhal s obsd€i may b€ .xFrioed ed r.ined

dlononer &d lmws the loerion offie cEsent rhe armosphdic €ondilions od the

Physioloey of dc obw.at cy6 @t stil lead lo non-vbibilny of ihe crent Thes

facloB ae sdlt nol wll exploEd rhus the high frequency ofrhc..gaive crc6 shos

Oil the probLm is srill nol eh.d conpl.Iety. A posniv. emr @cG when a nodel do*

allow visibihy oa cesc. sd thc visibitiry is nor claimed Snaltq rhc n!frb€r of

posirive ercB dd b.ttq is a visibility c.irerion.

The hbl€ 4.4.5 slmdises the posnile ed negative obseNarioB in 6greoe.t

or d'esrem. with diafcEnr oilcrion fmn ft€ dara sr w h.ve choen for $is M,k,

Tte labl€ is aEanScd wirh dercsing succe$ pcrce age in tems of vjsibiliry claihs

aruirenl wirh lhe $ircrion. Surprisingly lhe Babylonim cnrenon ha rhe besr succcs

percenh8e ti)ttow€d by our r-vatu. cril€rioo, Th. l.btc funher shows lh Btuin,s

cnrenon (4,1_2) and th. L@r Ripens law e.qudly suae$tuI, Thse @ followed

by $. q-valu. critcnon of yalop dd rnc ARCV,DAZ_bdcd Indid ncrhod (1,6.t3)

Tlle crn*ion duc ro Ma@dd (3.6.10) od lo$erinsnd,s crjterion aE dD lc6t

sucessful ofde mdhods shom in rhc |2bte, Ir sbould i, nor.d lhar tess succdstut a

modd is in d€$nbine pGilive sighdnS gncEr it is. Moreoler, strcrer a hodel jt should

be nore coosislenr wirh .cgaljve obs€ryatioDs (wl|en fte cGsc.hl ,s nor sn).

,

The number of iegarive obsrnadons in ag@mc urh lhc diterioo is at$

omrdkd 6 a resl of a cnr.non by sodc alrhos (Fatooni et ot, 1999). Howe!e!! ii

should be notcd $ar rhe exploration of Scha€fcr shows fior $e bneh|ness or nagnnude

contra$ is highty dcpendenr on rhe w.alher .ondir io.s. All rhe single pameter cn&ria

consdsed in the rable 4.4.5 do nor conlider wea$er ondiriotu of indilidu.l

obsw.'r'on.If ay ofd* qitcdon alom viribility in soh€ cae il ,. sl,U possibte lhal

lne setbq @ddirioN &c rcr f.vour.bk for vkibiliry &q oe crc*en is not acrualty

166

Tl|rcforc, if Fothcdnghd's criGdon is nosl s@4stul i. b.i4 cotui$dr fot

negalive obedadon rhG dcs not at dl htu thrl it ii h.tcr or noa d.p.dabl. thd

Ydlop'3 mod.l. Bolh Mah&E $d FolhdinSldt hod.k @

oa resative oh6d6tioB (whd ctlsdr is mt *n) but e lc.st sussful for pciliv.

ob*d.lion. Ihh is tre sinply b..!4 the criteri! a. sficlcr s @nFGd lo o$d

crireria. On th. oih6 h.nd Babylonib @ndition, i-v.lue criedon ed lhe Luar

Riperess law e highlt cotuistent with rhe positive obervalions but l4t coNislent wilh

the ncgativ. ob*Nations. The* dd other crit.ria (Bruin's limn dnd Yallop\ 4.valuc

onedon) arc noF conc.hed with condnions ud€r rh. visibihy o{ n w lutu oesenl i3

tossible .nd not *nh fte .onditions un<ld which ir is idp6siblc.

ln kms of ovmU coosislcncy M.!Mc6 mcthod atpcd lo bc best, HoBevcr,

Bruin'r lini! Y.llop s 4-valu. cnrcrion, Lufu RiD€n.s law and o$ r-vdG ciieion @

ba*d on ene Iheoelic.l co.sid.radons &d ihc orhcr hctlDds m only empiri@I. Thc

methodr bai.d on @y thcore cal @nside€rion ndy bc inproved with be d

unde6ta.dilg ol etual physical ed physiologicat dpccts of the probhn.

4.s DISG.USq!9N

In this chapter rh. tobten ofde 6El vkibilny ofnew t6d cEscenr is cxpto..d

on de basis of pbysical modeh d6cribiiS rhe bnSlttncs of c@$ot and that of rhc

lMligh sty. Tn * nod.ls ha€ @u6cy of 2oyo vhjch is erhjbned in de succs

Ftqiag. of Edin's lihir. Yallop,s 9-v.t@ crit€,ion .nd ou r-value cntedon eith

ov€all @Nisr.my of?9.q/.. ?9.5% &d ?5.4% (espelivelr) wilh Ihe obseryatiotu. AU

th* hodeb have tngh s@ss perolag. (90% ed plu) for posilive obFNarioB

(eheo Ih. crcse.r wa Epo.ledly sn). Wilh horc accudtc nodets of biehrles or

c6c.nr &d r*ilighl sty $s. merhods cd be ihproved funher.

Th. physical nodek co.lid€rcd in tht chaprer cd bc divid.d iflo rm clses

On€ th,r b ba.d diectly on ln. brighrn* nodets and includ6 onty fi€ algoljlhn du€

ro sch4fd Thc orh.. ct$ or nod.ts dcdec !tuibihty @ndirjoN o; lhe basn of

t6l

visibihy cws inii.lly corceived by BNin, Thb clss includes Bruin's lidit, Yallop\

a-EIu€ cnt€rion md th. s-value cilerion th.l is d.velop€d in dis work

w. hdc ucd Schaf€ls algo.nhn ro cxploE rhe Epoaed ob$natoN out do

not cone uder ftc "Edily visible ' @ndition duc io YolloP md when $. cEse m

r€ponedly sen, AnohSsl th€s ob$palio$ ces lhal have ufavouiablc ha8litude

cont6l are critically exoined md some sft Gjccted (esp4iallv obs.nation nlnbcts

389 od 455) s $ey de lol consislent wnh.ny of lhe visibililv ctit rion consideEd in

rbis work. Bo$ lhce rcponed cses m nol .oDsi.lercd Fliable as .ven undq niehlv

exrgeerated .hosphdic t nFEtw md Glativ. humidny rhc magiudc contrdt is

foud ro b. unfavoubL for cE$ent visibilily ve foud al lcdl onc ob*oation wh€n

ile crc*enr sd Gpon€dly *n wi$olt.ny oflhal oid (obs. no. 274) ed thal is nol

co.sistent wilh any cilcrion bul havc 6 lalourable nagnifude co 6t fo! tlalivc

humidiry less than 50% and atnospheric lefrpcrature aound l0 deglee centiSBde

Apan fron obervarion nDmb.6 389 sd 455 1nee e ll olh.r caes shen the

cEse w4 r.poncdly s@ without oPlical .id but ihe ma8lilud. .ont6l ws not

favowbt.. Howvcr,olee m ofFsitiv. ob$nado$ re consined wilh ar lcsl on€

oder crilerion consitleed in 6is sork A3 the brighhess model tE still nol pedecl

rneEfore thcs. I I obsenalio.s ce nol bc dl.d out .s umliable.

The models $at @ dedued Iioh rh€ Bruin's isibilitv cud6 ud liniling

visibility cud6, th. Bruin's linil md YalloP s single p&meter (it .ion drc found to be

consisr.d qith.{h other. Bolh hav. stmosl.quivaldt succGs Frc€ntagc br Bruin s

We ha!. developed new visibihy curves and limiting visibililv curlcs using the

brid nss mod.h due 10 schaeler md o$es O. the basis ofthese n'* cuBes a new

dara *t ad a ncw sinsle pmetei cril.rion b d.duced The .s liniting visibililv

cufl* havc lcad lo a slishdy nodifi.d "h.st lime of cE$nt visibilnv Our nw

visibiliry cril..ion, thc r-value ;ilaion, is found lo be norc succ6sful fot posnive

168

obsMriom bul lcs sue.senn in @mpdien 10 thc Bruin's I'mir dd Yallop'3 ctil.tion

for n.g.tiv. obsdaliors.

In vieq ofthe facl rh.t all th€ vkibilily oiroia de ai6ed al explorin8 conditiohs

undd which rhe new luM rcsert may b. seen the suc@$ oI a modcl lor positive

obsrv{ion is hor€ inporlrnl d i$uc as cohparcd to ils succe$ fo! ncradle

obs.F iions. None of lh€ sodel h limcd ai deducin8 condnioa und.r which lhc

vieibihy of ftw lbr cJ€sMt is Dposibl.. ThcEfo&, a nodcls see$ for bcing

consblcnl wift $c positirc obsmlion b lh€ sses ofthe nodel.

In vi.s of the nagnitude @nl@r model bed on brienb6s of c@.nt dd of

rwiligbr sky il h6 ben *en lllar rhc lbibiliiy of .ew lun& cr$enl is Sreatly aff.cted by

(i) $e eleelion above sca l€v.l oflhe obs.Nation sne, (ii) rhe almospheric lenpcdurc

and (iii) lhe relalive humidily. Higher is the.levation morc islhe maghiiudc contrasl rn

favou ofvhibilitt. Lowd is lhc lenpcEiuF or hunidily the nagnilude contdt is mote

favouBblc for cEse sighdne. Evcn if *niampidal diteri likc Yallop's g-vtlu.

cril.rion or ou s-value critdiotr <to.r rct allows cE$enl visibilny wirhoul &y oflical

aid thc mgnitlde conBl my b€ ir favour of nlcd cye visibilirt of the no ccsccnt lt

hay b. due hid* elevation ofrlE siL o. v.ry low lcdFarue or humidily.

Thus in makins decisions about rhe duthenticiy of ey clain of vnibihy of .eq

lun6! crcscant any s€mi-empi cal or a simpl..mpirical cribnon alon€ may nol pbve to

be sumcie.r, Such a qiidon nNi b. suppledcnted b, m molysis of maglitld€

contdt befoe any althdiicalion is dorc.

All th€ empirical ad the si4mpirical mod.ts N d€duced ftom $ne baic ei

ofdara lhd is d.dued vithout tating th. ahosph.ric @nditions and cldarion .bov. Fa

l.vel. This b6ic et of dala mv bc b@d on d ARCV-DAZ rclarion (lik.

Folh.rin8bm's, Maud€B or Schdh nodcb) or ARCV-WIDTII relaion (likc Ylllop\

nodd or our model dev€loped in rhh wo*) only. AlthouS}. atmospbdic conditions cu

nor be Dr.dicred ;ccnrately, rhe sesonal svelases for temp€rature md Elalivc bwidirv

169

nay be considced for an adv&k pGdicliotr. I. cdc of vdifyins a visibilig claim $.

dual ahosbh.ric conditions mv b. 6oded.

Thercforc, a possiblc slral€gy for vedliatio. day be to usc eslituted alevalio.,

t, esriMtcd tcmp.rarurc, 7, dd $dnakd Glaive huidity, l/0 md get thc rsults for

g-value or r.value. lf n allows crcsccnr visibilny for lh. eve.ing in question dd fte claim

h nade, tbe clai6 is ac€pted lf1h. s.mi-.mpiricsl cribnon d@s nor.llows ccsc.nr

visibility dd cresccnl sishtins is claim.d, .voluare mleniludc contBsl M,. Iftlo > 01he

progran Hilalol alloqs for vElidtio. i. el€valion, rempc€tur€ dd Elrtive hunidny e

onc con adjust for thse quanfii* and qiry whether a favoulable ma8nirude contmsr (M

< 0) is obtaincd or nor If rhe new mgnilude contmsr is favoldblc dcepl lhe claim

On bsis of the d!r. generaled by pDgEn Hilalol GhoM io ngu6 on the n xt

page) Elations berween magnitude coht6$ and rhe quantiries on wbich ir dep.nds {t I

ahd I, ar€ obtained 6 aollows:

M = i 8177 -1.autE + O.0Ao000ze

M= I t626e40n22)

M = -A 4 368 + 0 Ai26T _ A 0008f +A OAOA6t'

ThG for meter inc.edc in elevalion M dece66 by O.OO0OO0I4' _ O,0O4l, for €ach

pcF€nlagc ircre4 in relaive hmidity rl' vdies by incrss by O.O3l 56e(0 o,o:d) ed for

€ach degce cenrigrade inc@sc in t€npqarue / incrcdes by 0.0126 - O.OO08T +

0.000006T?. This n6y ledd to the alproprjat€ elcvalio,! rehriv€ hhidiry dd

athospheric lenperarure tequjEd for faloldbl€ nagnituo€ conrsr An applopnate

erclation my b€ tbe el€vation of . hill top o. buildirg Mf fbm wher€ m obseFalion

nay bc @de or oay hale be. @dq The apprcpiare lchlsar@ ed rclalive hMidity

t70

may 6ose that nay b. lhe ay€rlse for lhc eson or lhc valB $at wE acNally

re@rded ar rh€ $ne of obsc(ation,

ELEVATION VRS NAGXIIUOE CONTRAST

y= ?E.ri.odtr+ 5 af?

3.

E3

$,

3.

33

HUMIDIIY VRS IIAONTUOE CONTRAS1

t

TEIIIPERATURE VRS MAGNITUOE CONTRAST

8

B

2

t7l

Chapter No.5

APPLICATIONS

Durin8 rhls work lt ls obs€oed thar since th6 Babytontan eE flt recenty a

nomber ol pr.diction crltsrta, mathehatic.t .s we as obsewational. were

developed ro detemln€ whon the new tunardesc€nl woutd be flrsr seen tor a gtven

rocataon. as tho fist appea.ance ot new tunar crescent marks tho beginning ot a

new month in obseryarionat tunar catendaE thele crireria and modets are

sagnirrca.t fo, calendaicat purposes_ wh€ther an actuat obsrya ohat tuna,

carendar, like the tstamlc Lunar catendar, ui izes lh.se crireria tor arangtng its

carendar of not rhe36 c.trerta provtd6s a €uidance for both teslhg an evtdenco ot

crescenl si8lting by common peopte and tracing down ihe dares ot a catendar jr

hErory where app.opriate dates are not we recoded. Ttus the main uti ty oi the

prediction c terta tor the eartiest visibiny of new crescent ts to regutate tho

obsefr atlon.l lunar catendaf .

Although ftrst order approximations, rike Anrhmetc LunarCatenda, thar are

oased on lh6concept of LeapyeaFand rh6 averag6 mooon or rhe Moon fave been

in use, Nluslims hEve been to owing dctuat si€htng o, crescenls ar teast tor the

monrhs of fas n8 (Ramazan) and pitgnmage (z hajjah), the acruat moion oi the

Moon varies grea y due to vartous tactoB whtch causo rne obseryarionat catendar

lo be ditfered froh the arithmelc c6t6trdar.rh6 Catendars if based on a prediction

c lerlon llko thar otya op or ihe one devoloped In this work are rhe cto56t to rhe

ob*patlonat catendai In this chapter, wecohp6rethese carendaBwith th€ a.tual

obseryatronat catend6r in pracrice In paktstan tor th6 years 2ooo ro 2OOz, ft ts

found rhst on avorage 93.7% obseryadons ,16 ac@rdtng to rho yalop,s o_v6tue

t72

criterion or our s-value (or Q&K) cnterion. The disagr.ededt ls lhe resull ot eilher

the bad weather du.lo whlch the.ew lonarcr6cent could not be slghted 6nd th.

lunar nonlh b68an one day lare, or too opilmjsrlc clalms ol obse atloh and tho

Lunarmonthbeganonedayearllerthanpredlcted.

Funher, In ihls work another aoollcarlon ol these models is consldered. Tnls

is lhe use ot lhese modelsto dwelop a compui.tional roolro detemlnethe lenglh

of cesce.t from cusp to cus!. flie n€w lun.. c.scent as well as crescent on next

few evenan$ b obsened io bs shorter tha. its rheoreli.ar length Le. 180 degree

IromonecusDlotheother.Anumberoialrho6haved€scribedthereasonsforthe

shortenln€ of the obsenBd crescent (Danlon, 1932, Schaefer, 1991, McNally,

1943). However, Jew have atlempted devislng a mathemallcal technlque lo

delenino the exrenr of rhis shorrening ol rength ot crescent. On the basis ol one

pafamelef dlleia ee have used cresent ot minimum visible widlh as limil on lhe

le.$h or .re$enr and devrs6d a simple rechnlque to calculate lr. The chapter

beginswilh a desc ption ofthesme.

5.1 LINCTH OF LUNAR CRESCENT

Ihe hct,lld rltc nc\v luhlrclesor appcds stroncr llEn 180. terA r.islnorn

ldt cenlrics. Il wns Denj.h wlD fi^r savc rn exptturarnr for the pbenoneion (Dor.ru_

lt32&1936)andalributcdi(o$etunarteniincloseto$ecuspsMcN! yanribuluda

dillerenl reNon snh tis pbenom€ion disadine Danjon\ hlpo$esis (McNa y, 198:t).

McNall) proved rhar lh€ tcngrh ofrhc shadoss ctose to $€ hrnar cNp ond fic d.panolc

oflunar suiface IoD beins pertecr spherc coutdnl,jimiiish rtre brignllrss oflhe regions

ofcresent close ro cusps lo bc rhe caNc ol rhe phenomenon. IIe arribured ttrc tength

shorlcdng ofcNscent to lbe,.seing afeca,due lo $e turbrlcDe of lne ,rhosplrere.

McNally has also developed a limula for calcutalnlg te knBh of rhc desceDl, Larcly

Sull (Sulta., 2005) has anribured 0E shortening of lengtn of crcscenr duc to the

Blackwell Conr.st Threshold (Btackwclt, 1946) and has d€leloped I forhula lo

c.lcurale ciesenfs lcngrh. W. have aho devctopcd a sinpte &chniqne tor catcul.ring

173

l.nglh of new lum ccscenl (Qucshi and Khrh, 2007). In the fouowins we reprcduce

this cflorr wilh slighr nodiilcation md a codection,

Schrrer cjecred McNall! s expldaion on $e bais or his Yiw thar rhc

shonening oflhe cr{ent length is situply b.ca6c oflh€ sharP d*line offie brigntn€ss

of rhe cr€scenr clo* ro rhe cusps ( SchaelTer, 199 ! ). Usins rh. dcudte dodel of Hatl@

(llapk , 1984) for calcularing rhc surface brighhcs ollhe cesnt Scbaetrer claims that

Daljon\ collected obse aions anJ his own n€* dara fic tbe nodel Howewr, nenhet

Danjon nor Schaefer have sneecsred a ddhod lor calculaling cacehl ldg$ Hapk€t

rDdcl nat be accurat for $eoreical setling rclatcd lo thc clongalion ofrhe M@n bul as

frr as the obscryed cresc€nls are conconcd dEre oughl lo be a depanur€ frod snooth

relarion berween elongation and lhe crescenl lcngth. Thc tason we co.sider n baFd on

obsenarions ofsone norning and {enihs cesccDh.

ltosl of the edly desfiption of the phononen, concenuated on rclatine it to the

plEs (or dongation) of the Moon ihal is gcnet lly th€ !e6on b.hind the Phenonenon

As rhc clongalion increass rne lengrh of rhe cesccnl fronr cusp to cusP k€eps ioc€Ning

Nhich a connon obseryalion. The nathemalicd dc*riprion ol lhe phenomenon lems 'n

ufdeficicncy arc. br Daijon Nd !ho\t to bc incoftd bv McNallv Hosev€r. e or€r

esrimrrcd limil oo rh€ ninimu \idlh of visibl. cEse (2 to 6 arc seconds) bv

VcNJlly le ro ren .mdll "dlu*

lbr DJnjotr Liilil lhe d*np'ion due lo McNallv i:

logicall, souDd so is $at of Suh dd bolh rcsulkd in$ tcchniques for calculatng

hDglh of se$oa. HoNerer bolh llcNally dnd Sullan hrve nol reponed lhe aptlicalion

of rlEn desription o. lh€ Ecorded hisloricll dria found in lii€iatuE (Yallop 1998.

Schaeffer l99l elc ). Accordinglo Danjon, d desctibed by F.l@hiet al

si,(a) = st(.) cot(@) (5'l l)

Wh.c $e @ PQ = o is fie deli.icncy at in fiS l (Fatoobi e|' Al" 198) a is rhe

elongltion dd o i5 half the cccenl l€nElh tr apP€6 rhal Dojon tr*d lhe Sine

t14

fomulE. by Nunins sphdicEl 6gL at Q to b. idr egle McNallv rejeded Dmjoo s

dSmcnt ed 6ing fou-Pafl fomuh adiv6 .l:

(5.1)

Nlserically, for small ddgtes d sd a thee is onlv . muejn't dillemce b'lsFn lhe lso

Esults. Generally thc .lonsation (4) cd be cahulaled and the cre$ent l'nrth (2o) is

obede.!. e the rw fomuls 6 b. LPd lo find fi' dclici'ncv arc' Hosder' ro

calcul.rc the c6..1 lenelh nonc of these @ be ued- McNally daetoPs a fomula fot

angula! rpdlion 9 ftoD a cusp in tems ofclescent width R d:

I

SinE

-T *t'

t Rl

(5.r r)

nininum lisibl. *id$ n.sured ih ddid dn€cdon 'wtv

lron dE

disc. Thercforc the logh of th€ ce*ent h€ oblaits is l8O0 _ 29

Usiry (he Blackqell $rcshold con(6t Sutan (sultan 2oo5) adres at tlE

ninimun vhibG *id$ (in |ems of didetcr of rh' smalld eqlival€nl Blackselr

(Bhckstll, 1946) di*) of c@enl ar Frigee dd apoge al his l@rion of ob$Parion'

Tbe fomula rhal hc deleloped for calculali'8 thc crc$61 lenglh rsi

,=f!1.'to"

l2t )

o.l4)

whde r h th€ sdldidetd of tbe Moo[ md

o5

l2t +w \

{5.1.5)

wirh w is the dim€ter of e snallsr visible .quivalent Bl&kqcll dnc and l/ h cental

widtholrhecre$e.t. (5.1 5) is$ecorsted fomof (1.5)inQureshi a Knan(Qutshi &

Kian, 2007). Sultd.onside6 minimum diameter of Blacliwell dGc lo bc 0.14 alc

minucs Nho the Moon b near peris€e md 0.16 m ninul€s wh€n the Mmn is n.or

The presenl $ork is based on rhe obseNalion or the last (old) rcsceit oD

February 26,2006. Duling this obsenation rhat sGned from lhc bcginhing ofnorning

oviliglr rill \ell past sutrie, il was noriced rhar for rhis 48 holrs Mon. noE tlrn 2?

dcFcs aMy ton lhe sun ftc ce$ent l€n8fi srancd ro derede \rith lhe rising suD

simihr obsnalion on Mdch 28'' shen lhe aee of Moon wds dound 3l tous !.d

arolnd 18.5 degrees asay lron rhe sun. rhe cdscenr leneth d*leased nrorc npidly qi(h

d(E6ing co.ra$, The l.n rine thb ciescent wJs $en lvilhou optical aid well at$

sunriF s?s less lhd 90 dege* in l€ngh. Tso dtys later n€w cc$enl with agc 27 6

hols ar 15 degees a\ey fom lhe sun Nas olsned till setling Close lo lhe lDtiab

though rhick humid a(mosphde the crcs€nt lenglh Ms aganr obscrved to bed4rcsnrg

Thee obsFations cleaily demonslrare thai lheF isnuch notc 1o b€ expload abolr lhe

phcnohenon ol shonening of cc$ent lcngth apan fom tlapke s .c€urate model thrl

shows detendoce ofcB*cnt lcnsth on clongalion tlone.

TherefoE, inrhis qorl,lo besin si$ qecoisidera simPl. seodeuical

model ior

rhe ce$e.r. fu model dFsibes the phenoncnon of shortnine of lcn8lh that depends

on th. aclual seniSiaocLr of lh. Moon d *ll a the rclaliv. alliiude (ARCV) of th€

csc.nl o\fr tftal sky. This is denved fro,n thc sinste pmnct.t (q-vatue or !!al!c)

crilerion of edlie$ visibiliry or cre$ent (chapler 4) fld is bascd on th€ facl thai

whe.cve! rhe widlh (o' btigb$e$) of cF*c clos to cusp is below lhe mininuto

vhiblc c€nrnl widrh (or b ghtn6, or rh. dcscent the vhol€ l.hsth or ccscent wirh

t76

s.lld width eould nor b. visibl.. Applying ou mod.l on lhe rc@ded vbibihv rnd

iNisibiliiy dala lvaitablc fic ldgth ofcEsenl in 4ch ce is calculaiql Thc calcnlaled

lenSrhs of c$en1 @ al$ omParcd wih thc obseded leheths dd wi$ thc lengds

catculsred uing fodlls (t.1.3) de b McNally md (s.1.4 & 5 1 5) due b Slltan. ln

cs ofusing McNally's fomuh AA is @nsidded liom lhe cilerioD used in lhis work

&d for Sul|!n's fomula the ninimM width or Bl&kwll disc (,) it 66ideEd b be in

the d.ge 0.14 alc minul€ io 0.16 arc Finut€ and dep€nds on the dislane b.tqeen Eanh

&d th. Mmn obtained using simpl. lined inieQolalion

Irvi€wofQueshi & Khan (QuEshi & Khan,2007) tlre bLishrn€s ofcfsenr ar

displacement v frcn a cusp is gvei bY

8.. = tF...Cdv tr -cos t)+

(5.r.6)

sh* t G th€ radius of Moon, F. is fie mdimun nux of sunlight

olVoon.l islhc 'ePar-rion

berqecn lhe Sunand 'te

Moon in our

the lutu suface dd X B th€ disldc. of the M@n ftom the Eanh'

lhe cescenl at angular sepaElion V fom lhe cusP is eivcn bv

lV,r, =rcosWll CosE) = tl ccatv

(5.!.7)

whc t/. is lh. width of fie cdnr i! lne diddle ln bolh the* €quttioc !' vari6 lmh

oo ro 9Oo alo.g fie lenerh of rhe cFscent frcn centE ro a cusp respeclverv

For rhc develop'nen( ofanv nodel thai dcsibes $e ni'inun Po$ibl' sidlh rhtt

cd bc visiblc lttrough naked eve one requi.es 10 s'ek guidance fron $e aclual

obedalioN. ln $e history of *idtjficatlv rcponcd obeNation of thc srv voug

cr.sent Moon, lh€ ecord is that due lo Pi.rce od February 25' 1990 (eporled bv

s.h!.fd ad Yatlop) Thc cl€*nt he claift |o have sn wirh narcd ev' B jusl 14 E

houB and ils widrh wa3 O 18 dc minut€ Amonesl

'll rfie '*oded ob*flalios lh'

t11

siglling of such a youg ed rhin cesc.nr wb n€vd rcpon.d- In lhe norlel $ar is

developed in this sork thc losct limil ofihe *idth ofvisibl. ctsent is considered ro be

o.l8 aE minut€. However this ninimum is not th€ abelule ninimm for all c6@nls for

all posible el ile altitudes (ARcv). I. fiis work *e consider rhh ninimun of 0. 1 8

. arc-ninutcs of crcscenl width shen fic Elaliw dinulh. DAZ of $c M@n is ao md

rlr r.larir. alliiudc ARcv gives lhe q-lalueof_0.22. Y.llop s ditedon (4 2 3) for $n q-

,1R(y =9 6311-6J226W +0 7319 t'i2 ol0ls,t/3 6'18)

Thc values of ARCV according lo lhis *iteion giving $e q_rdlue equal to '0 22

Nould ield ! difcre lotr€rlihiion lhc visiblc width olthe crescenr' Tbis is causcd bv

dilTriem relarivc dzinulhs DAZ FoT lh. lasl posibl' ARCV (4o) th€ *id(h ol lbe

iNisibk ae$erl Nould bearcund 108 alc{cconds $at occur dl a large valucofDAZ ln

\i.\r ol diis $ilcrion ihi nininud {idth of lisible cr€$eol for @v ARcv is Gm'd trand

s te *esent isjusi invisible for$is{idth:

(tIa)

Whqrc l'l. is tlE tlmrclical enlEl sidlh of he ces'enr and 4costl

is $e r€duced

rvidth al aogte V Irom lhe center of $e de$ent

'/-

n dre sidth r€duced bv the chnve

ahnude ARCV li ordcr thal soEe pan ol dsenl is visibl€ 4(!'rvl

> l'l' ar sonc

\alue V = Vn 'llrus thc eftstive visibte widlh of$e ce$cnl tor v nnsing fon 0" ro

(5.1.r0)

Th. brishlness of crescenl falls sha4lv s 0 approaches 900 - t th' aclul visible reidrh

of cE$.nt at any val!. of V m6r tE less tbd d€ e'onetic valu€ of vidh eiven bv

equatioh (5.1.8) Therefore rhc etr6tiv' v6ible width siven bv (5 1 10) is justiced

178

IE it ir ma dly lh koglh of.ilsc.d 6d nodo! htr lh. vilibL with of {tt*.al

b! lo diminjrl rlso. Wh@\tr lhc cr.rcdt it ilviliblc 'n

vi.w of (5.19) 4r'bs to

c*v-fr=r

(5.1.Il)

I!.ll oOE qscr. ia vi.r6e lb .tleI it vit'bL ia sftlth tl to'E 'rgh

!'n Dlrt

dilh!6l!.cc..ttti3ncv.r!d.conDldc 180!irlcrgh Tbu!'tV=V.

(5.r.r2)

Tladc i. (51.12) ||,! i! ! ddr.. of Idf l!! l6€tt of ih' cr'!c'dt $ itll lt' iol'l

lcogth oflh. ctelcai k Sivcn bY:

t;rr*"CY'*)

(t.l.lr)

D

mt

'I}lus $e ae$c len$h cd be dalualed whelever $e fieoelical sidlh lr'

€xcecds the mi.ioM width ,/, visible acording b YalloPt q'lalue ctn'rion oi our s_

value criterion, for $. palticular lalues of ARCV In $c Fis 5. l l the *8nent [D or

AC is the ninimm width lil- at dv ARCV inlisibl€ according to Yallop's cliterion

The scgnenl AB is rhe thorcdcal widlh ltl" at fie cenrE of fie cresccr! Al sngular

spaBrion vr from tbc centre olc€sent ED equh4(nrl/" 'lheefore the ponnt on

rhe ouler linb oi rlc cresceni tbal las dguhr separaion fon cerrc geal$ ihan

s. = ZDOA should noi be visibl€ TIE lisiblc cls€ then crtnds fom D o D' @d

has letrBlh 2!rn. One should nole lhal Nheneve! ttl" (nininum vhible wid$ accordnr8 b

Yallop s nitrion) is Srearer tan ttlq ($coetical widtb) {5 1 13) can nor bc sed ind the

cresceDi isnotlisible, ie iihas m lengdr'

The tr1odel developed nl this Nork lo cotrlpute tlB lenglhoflhe crcsccnl hrs been

applied lo a nuobc. ol ob*rvalions lepon'd in IneralE (Schaeffd 1984 Ylllop'

1998) Tbe resuls foi ihe crescenl length against lhe elongatio' also kno*n as irc of

lighl orARCL,ae ptscnled in lable 51 1and in Fig t'l2 Tlre loBlhs mcalculaFd

by scleciing lhc nnrinum vhiblc Nidrh Vn of dcscenl foi tlrc valLE ofARcv 'c'ordrg

b {t.1.8) in rh€ elerv ca* and ihen usnrg (5 1 13)

The colunns or rhe hble 5 l ' I sllow rhc darc ot ob*ivirion thc @dinares of

rhe lcatioD of obseryer (Ldlilude and longitud€ Fspeclilclv) tbe rclatilc umull)

PAZ). rlre dladve .hnude (ARCV) lhe clon8alion (ARCL) of $e Moon at lhc be'(

'r:d

rinc ofobstvalion. lbllowcd bv setr1i_didetr and lhc cental widh ol cr'scnl in arc

ninules. rlt 4_valu., lhe niDimum cF$enl sidlh viiihle at lbe rclatile alritude

c.lculated lom (5.1 8) and finallt the cE$e'r lensth cdlculared usiie (5 1 11) Thetablc

is aFmged in chmnotogi@t or<lr of ob$mtio's dd co't'ains onlv Pai

of th€ conPlele

data s€l lhat hs b€en used for conpaine mod'ls for carli esl crescenl siehting in pe!'ous

180

200

F 180

fi reo

3ro

Era

i roo

o80

i;@

JM

=ao

0

20 30

E mno

tig- 5. | ,2: Crlgn Lcogll! tr' vdi.Ii@ dE ARCV

Tlr! Fi8 5 1 2 slD{! lhai lh. nbdioel ctdid bdwen ct!*4t lcntlh

'd lhc

ctdgltion h not snolh s Eporr.d bv sd|Itrcr on {E b$i! of H@k''t mdcl Th'

tui! r.rlds fd tl 5 N (i) dE cr.rc.nt hngth ha to bc tff"t'd bv dE ElihMoon

dist ncc .! cl'in d abos (ii) 0F dttdgbdic n!td4' cloe ro lqizd trN|l tfrecr li'

cr$ent losrh., cl.imcd bv McN.[v ald Silrd MoEov{ v'rv wi'lc cle

'ot! ftal

c invilibte dn lo $.it d4 vicitritv qftb $t hori4o mull mt v|tilh $ddolv for s

invisiblc crcsc.nr the elongaiion mv ba ttiSc $d nult not h'r'' as lcngh (i3 msl'

dt b. i isiblc) !.dding to cLi$ of scn*&t Ttt invisibilitv of t eidt '!l*al

!t

de to ii! clo..n6 io lFdzon (sd[ ARCV) Ho*E' sftn dft$L diiod' (ARCV)

$e rnc G..nt 3lEI bc d!ibl.. Ot <ftosnlid lnd cltir b tbai for r sw' of

rrn (sfrcidt) $idlh tlt stving tud' (ARCV) mud clN lo dtc4" dt ldgft of

.le.nl fron nt mdinwn (al DAZ = 0) 10 iB minimun (terc lcnr$ vh'n itt {-valu' is

h.low limils of !tuio!) sooulv dd iot sdltolt' b nE $din'd 69 5 12 odlv dto*

obsflllloB .rc coNidctld tri.n lh' d!€c'!t M cltimcd b h!rc b'@ i"n or [E

cr.!c.d Loglt i! c!a!.bla

lEr

182

T.bl. 5.1.1 {Continued

r8l

Tabl€ 5.'1.1 (Conllnued)

Durins rhis {ork $e ?O obw.tions of Dfljon (ne ionc{ bv Scha'lTs

and Fabohi €1. al.) @uld nol he acc6sd, howvc. the pictotjal data of cdccnt ltngth ir

beins eeneraied at fte Asaolonical Obsdllory al Univdilv of Kdachi. Th! obseNed

cBcell lengh fon phologEphic Ecoids i5 given in Tabl€ 5 1 2. This inclldes sone

obsryadons made by olhe^ durirg past fe{ ycas and tbeir pictutes ate availoble liom

M.icproi.on maiDtaind by oteh,

184

Tne nodel dcvcloped in lhis e for borh Ih€ brigh$6s ad th. lenglt of

cmce.i is nainly geomerric, suppl€n .led by rhc single pmercr crilcrjon for $e

earlien visibility of ncq lunar cFsc€n1, Thc model prcvid6 a simpte method of

cdculaling lenglh of lunar ce$eot and tak€s inlo accounl lhe atnospheric afects

indiicclly through singlc pdaneler crit ria for th. vnibility of new llnar cressenr only.

whenerer t/. < n/- fi€ €r€$ent l6grh is nor cakulaled dd rh€

according lo eordcd obsedation (lable 5.1,1), TL nodel hd been Gn.d in nw $ays

I:i6r. ior so6e of $e rcccnt obeflarions whose phologlaphic Ecoids arc available $e

cilculated dd obscrvcd cescenr le.erhs tre compared and shown in jablc 5.1 2 and nr

lig 5.1 7. The cFscent l.nerhs calculared ushg ou! lornula (5.1.13) arc SEaler thm lhc

obseoed values dd rhose due ro Sultan srechnique arc genedllt closer to the obefled

!alu6. In calcul{ions using th. fomnla of McNrlly A-t = ,r- dd X = seni'diafreter ol

The colu$ns nr rable 5.12 show rhe dalc ofobseruation lhc coor\iiMlcs ol rhc

locatior ofthe obseNer (lalirode md longitudes ii degr€e resp*li!ely),lhe elongation o1'

rhe Moon fod rhe sun in d€8rees, the seni-diamcrer oflhe Moon. thc c.nlral rvidt[ ol

lh cEsce.l. ninioum visible width of cre*.nl (all in dc minul6). lollosed bv lhe

crescenl length crlculated by our nodel ,id $e obsded crc*chl l€nglh. lhc .a'nc ordrc

obseNer Th€ l.( No colunns contain cGsce|n lcngtlrs as calculaled usirr8 rlE nrodek

rlu" ro sutron onj Uctlutty .,p.ctvely. Thc du(a in rhh ubl€ has bccnlnlnucd iD ordcr

oiincre.sine elongalion ot ARCL.

To d€Gmine rh. l€ngh ol! crcsenl fom picroial record the dieital phoroemph

is opened in any sraphic $nw l@l- SeleclinS coordinales of lhe oa the poin6 (*o

ofr&n do*roeachoftu cusp md on€ closc ro supposed cenftof$ccrc*enoon the

vhible oute! liob ot lhe oe$eol an equation of circlo h developed $at leads to dre

coordinales of lhe centre of the luft disc, toining lhe cenlre of the lund disc with lh€

two yisible cusps of thc crsce.l tbe dsle made al fie centre bv thc tso cnspc is

mc6u€d. Tn€ picue of one of tlE cre$ent m.$u.d in this Mv is shoM in Fi8 5 l ?

185

Fig. No. 5. L7: Mesuemcnr of Length for Cterent of Mav 28. 2006 as pholoeraphed rr

Xffiltr UtoveElty obsefrd@ry.

'the dlta aor the obsiedcEscehl lenglhshoM in |able 5 12 and the cbart i. fig

5. L? shows inlqesrine paficn. Clain d by Schacfer (Sch@fer, l99l ) fie crc$ent lcngh

is a sn@$ fiDction of elonSatio.. but lhis drE er sho*s a tMd that cl*lv exhibns

delialion tion oy such sn@$ rel.tion. I]lc dala sdple is snall bnl lheE de lwo

subsets eech having neally sE@th relations, separatelv. b€twcc. lhc crescent length tnd

the €longation. Hovever, rhen consideed s a conbiNd dala sel lh€ obsflatios uith

elongarion ll93 de8lees (by AnFa} l44l de8@s (bv Oner) 1667 degEes (bt

Ralini) md 20.12 degrees (by Qwshi) deviarc Mk€dlv lbm $e apPaEnl snoofi

rclarion exhibiled by the rcsl oflhe dala set Cecchl l€nsths in rhcse four cNs are huch

snallcr thm lhe uend shown by the rcsl ofob$wationsas wellss thcresultsofe&h rhc

mod.ls co.sid€red abovc fot calculaline lhs le.gtb of $e ciesccnl

186

AII the* fout deviati.S cdes ae photogdphically Ecoded dd have the ledi

posibiliry o| obseNational ercs. Out of |he t€s1 of th. €ight ca*s anothcr six @

plDlosnphic. Thc rccords due lo Sch&fq de lhc only cag 0tar "ob$nation.l lcngrh

of .ew lunr cE$ent is @nMed ils Eladon wift elo.Salion mul nol bc sneln 6

shoq in flsuE 5. L2 as w€u 6 table 5. l2 (or nsurc 5. L7).

It is tunher nored $ar rhe rool mear square erot calculated foi lhe lhree

conputational frdhods (Ouls, Sultmt and McNally t it n found $at Suhan s melhod

h6 rhe led( cror (4.76 degr€et follosd by McNally's thal ha m .mr of 6 16 decrs

dd our model hd an eror of ? 22 degR' The najo. difer€nce h€lwen our model dd

rhN Sulbn's dd McNaUy s is that olr nodel 8iv* coosislmtlv hisher valucs ol the

lenefi oi crc$ent, wliets $e orner lNo nodcls havc bolh posidve lnd n.8flive emn

Th€functionalrel.tionbet*eenelongationardlhecrcscenll€nglhissinlilorinolrtuodcl

aDd $ar due to Srll.n s. bul lhe oneerhibi(ed by McNally's model is m.rkedly difitrcnl

McNally s mod.l is Sivins betre esuhs for larser elo.gaion but a3 $. clo.Sation

becond snaller the emr given by McNally s nod€l beconcs leser and ltrger

Tdhl. No t.1-2tth:.^r"la r'!h,lak.t L?ngths o[CrctehB

t87

180

160

'110

1m

100

80

60

zlo

15

ELOIGAN()|{

+OBSERVED FOURESHTS r SULTAMS o

MCMI.LYS

Fia. No 51.7

200

i60

160

E.ro

:100

380

360

20

0

15 20 25 3)

ELq{GAIOI{

Fig 5 | 8r C'lsat tdgo5 $rinc Eloqiion withdn wi.rioa dc ro aRCv

188

Tlre modcl for rhe calculation of cc*ot lenglh may be u$d d rh. .dli6t

visibiliry ctuerion 6uch in the smc say 6 Yallop\ cilcdon cm b. usd, Hos.ver, our

€nphasis *6 nor b dcv€lop d allcm e cdr€.ion for $c s€. This *ork ws i .nded

for a b€rter und.htdding ofthe grcme$y of the lunar cEscent md lo delelop a ne$od

for cllcuhting iis l€nsrh.

The scond md indi@t tesi of the 6odel is its conparison qitb the rcsuhs of

Ddjon ncnliored in Fal@hi el- al. Ifthe niiimlm visiblc eidlh ,/, for vdious ARCV

according lo Yallop's critcrioh is r.placed by rhe nininum ever visible cenralwid$ of

0.18 aa ninuks tlEt is equivalor b igoring $e atmospheric allst for lower ARCV

|hen the Eladon b.tsen cEscdr lcngrh and .lon9rioo bccon€s snoolh as pE*nrcd by

Schaefer (Schaeftr. l99l). The smc is shoM ii Fis. 5.1.8. The chart in figwc 5.1.8 aho

sho\s lhat limiling value ofelo.gation lor po$ible lisibilily oflh€ cEscsr is areund 8

d.grees. As our compurlrions do considerthe an'cctofparallaxrhis limil is equivatenr lo

te ? degrce linil very popultrly rlown $ DanjoD\ linit.

On thc b6is of $e calcuhred lenghs of {escenr and ns elonearior using ou

hodel the Ddjon deficicncy arcs aE calculrted igno.ing the af&ch ol ARCV by

aomula giv€n by Danjon and thal given by McNally (shown in Fig. 5.1.9 and s.t.lo

Bpedi!€ly) These resuhs ae in clos agreencnr wilh fte Fig I in Flroohi el al (lbar is a

^rrcduoion of Danjon\ lis 2)- 189

*.

Fi8. 5.l.9i D.fici@y Arc rg.ind Elongtiod f.dding to D6jon

;r

3.

Pig.5.1.l0: D.tr idct A!!.gd!d Eld$tLa...qdiosb LlcNdly

190

6.2 LUNAR CALENDAR FOR PAKISTAN

Mosl of the hlami. counri* follow d ob*wadonal lu.d cal€nde, ar leasl fot

ftn rcligious purposes, Although subslantial work is done to cvolle a pediction

ciiie on ro dcvclop a univerel calenda! by llys (1984b, 1988, 1991, 1994a, 199,1b,

1997) ed otheB, r truly lnilersd calendd could nor be d€velop.d. As according Io

coDmon Islmic b€liefactual sighline of$c n6! L6d crceenl is n*essaty to b€gin a

Lund monlh, such a univesl calend& $cns lo be inpo$ible. This b |fue €speci.lly

b€cau* just aner conjuncion $e new luna! crcscenr is nol visibl€ on rhe sme day

lhroughoullh€ worl,l, eveD ift*o places ofobseNatio.areoD sm€ loDgirude. According

1o recent delelopnenb (Scabefer. 1991. Yallop, I998, Qurcshi &Khao.2007)

ob*ryational l@r calendaB for each Isl.mic counlry m be d.veloped bur dc ooi

s.ncally acceptcd b)'' rarious Isl.nic conmunities.

ln Pakisian an Offrcial Commiirle (Ruer-e-Hilal Comnft(ec) decidcs about when

1o beg$ a Lunar nFdrh on tbc bdis ofpublic cvidence and obseNatioos. This connilre

sa$e6 infodaioi 3bour thc claim ofsishriig rhe csce Thc clrims arejudged oi

rhc basis oflhe diFdncsof$e Islmic Laws. Som€ FpEsenlalivesofvdousscienrific

organiatiom de cle consnhed. Oncc $c clain/s is jNritled relisiously andor

soientilicauy, $e Ennoucemenl is nDde rboul begiming ofthe nexr Lun,r m.nrh In x

way,lhe beginning ofE\v lunemond is bascd on public obscBations ' verined ii licw

ofde pnncipals laid dosn br lslamic shariaat laws (lisled inanicle 1.3).

'lle advmrae. is rhal a larse numb$ of ob*Ne6" trle pan in $e exeai$ and

wilh it the pmbability of sighlins .ew cBscenl is incesed. Morcover, hon oa thcsc

observeis de iionr rural arcas sherc thde is leasl ihduriallraffic 6nd liSht pollution,

aid n k higlJy prob.blc thal lhe obseNing condilions d€ n€ar perfect. TheEfore in this

sluljy lhe* public obs.rvations e considcrcd |o bave a hki degE. of autnendciry. ln

lhis \ork all lhcsc public ob$dation-b.*d d.res ofsiart of€ach Lunu nonth frcn ye.r

2000 b 2007 ae reproduced and 4 srudi.d in conparison wirh thc Yallop\ q value and

l9l

ours-value crirerion-baed crireria. A similar wolk for the period2000-2005 has akeady

bcr reFned (Qu€shi snd Kh0, 2005)

TIE resuhs of E 95 lnDation during the Jmualy 2000 lo Scprenbd 2OO7

(Shawl, 1420 AH lo RaM, 1427 AH) .tong *nh obsRalional data 4 presenred

in yeady lables in appcndixlv All rhe conpuklio.s fo! the* tables de basd for the

city of KaEchi (Larirud. 24o 56, Longitude 6?0 l.). The fisl cotom indicales 16r

conjDct'on wjrh nonrh, day, hour, ninde and seco.d for rhe l@al rim€ of conjuictioi

in the sub-columhs. Thc calcularions are done for rhe day oftast codlncioi, fo! $e nexr

day and vherc Equird two days tater, indicaled undc. colunn hcaded .Ldt Coij

followed by rhe Dale in Oregorian catendar. Nexl $o cotlmns givc lhe r€larive ahnude

(ARCV) and reladve azimuth (DAZ) in degrecs foltowed by nooDset-sunscr tag ir

m,nules. ftc age ofM@n (in houc), rhe arc or tignt or €tonsalion {ARCL in degrt

and fie cBscenhvidrl (irr arc minlret apper $e iexr rhre. cotunns, iolhsed by

colhns indicaling the r-value, visibilily condition otr q-value. lhc .elahc and the

visibihy conditio.s on s,value. Thc visibitiry condirions are lho* de$nbed in hblc

4.2.4 (for Yallop\ oiterion) dd labt€ 4.4.4 (tor euEshi & Khan qirerioD). Under rhe

hcrding of 'Momh $c colmn givs rhe nrhc of Isl.nic mondr $at begi$ anq a

sighring reponed on $€ previous evening foltowed by $c CreBorion d.te ot slan ofrbh

lDnlh.In lhis 16l coluDn und$ Cregori& dare ofsiafling lunnnDnrh, numbsofdar-s

in dE nonrh, iE also 8nd Ttu la{ colunnr conr.ins a comnenl. If rc dccision oa

sl.nrng new lunar noDlh is in agrccment vi$ the prediotion critsioD Gratur q.ilqioi)

thc column conrains I,ROPER. tle ..ob*ryalional decision .nd tbc Nodet arc

consideed lo b. i..gEeme.t only $ne. thc s-value clirerion show Ev, i.e. crcscenr is

clsily visiblc. Tbe conrhenr in ttE tasl cotuDn cobrains ,LATE, if thc dcchion of

startng new lund nonrh is one day lare 6 indicaled by rhe dat of csy visibitiry on r_

lalue and EARLY iI the lunar nonlh has been naned ooe day edlier rhan lhar

indisted by the prediclion crn€ on. A5 staled eelier all de* calculariohs @ doDe for

KaEchi and for rhe b.sr local lime for €E$.nr lisibitily. Hower, ir should b. nored

lhal observarion claios are couected from alloverlhe counry.

192

Out of rhe 95 public obsesalioDs reponed jn rhis \o*, rher wre o.ly six

occ6ions (Jbe,2000, Juq200l, Juty 2004, ApnL 2006, Jury,2006 and Juty,2o0?)

vh.n th€ sighling war Epon€d one day latc ii compdison b rhc pcdictio, crn€ria

according ro which the New CEscent was lisible on rhe pevious day bd *s not

reponed. Tbere was oDty one oeasion {hor rhe earty sighrnrg is rcponcd Nov. D,

2004), hus,1herc weE only 6.3% enors in rhe decisio.s oflh€ moon sightingcomine

of Pakistd duing ld sven yas. llese resul$ e ba$d on verifi.d chims Tbe data

B8arding nnmberofcE*cnl siendnB d.ins $ar sre notrc ficd is nol available. The

only case of LARLY'sighring acepred by $c audonties was for rhe nonrb of

The main

'e6oi

tor lare sishlinss or rtr five indicated occdsions is th€ overBr

sky all o\€r lhe country in.lllbese caes.In case of$e onl)-' earty sigh ns for rh.Ian

scven yea6 alt€npts for cRscenr siehring \ere n,ad€ rhe Obscdatory ofu.ireBity oa

Karuchi using 6 CodC reliacror rclescopc.'the prediction crnerion based on r-value

dllowcd cescent visibility with oprical aid and rhat based on 4-vatue did hot .llowed

visibilily eleo silh lelcsope. We failed in sighinS of the crsccnt bul rhe omcials

acccpted clains of mlied eye sighli.ss fron aR. ctoe to Kamhi. Akhough,(he d\o

$n$il cosidercd did nol allow naked ete visibitny on rhisoccasion lhc misconc+tion

lhat if rhe ph4se ol rhc cresenr is noE rh.n l% n shoutd be lisible (Asrmnonicul

AhnrnJ..:l]O7r n ayha\c leJd he,te.i. or D.Jtcr rorcer rt,e.tdtrn;.

The low percenbgc ofee6 in rhis obscryalion.l elfo,1 is much noE pbnisi.g d

compaGd b lhe 6ul$ of m@n wlch, progEms cohdEted in Unncd Slales in thc tar.

1960s and rcpon€d by Dogger & Schsller ( 1994) Accordi.erorhh rcporl l5%positite

€rcs (qons clains of observing $€ cresceno and 2yo negadve eriou (cescenr was

visible bur rhe obsrveF nissd then) {qc iound monssr $c cxperienced ftooi

s"lche6. Thqe exF i.nccd h@n watcheu \v.rc geneElly consideFd b know wne,e lo

lind $c crcsce.r dd whar dc oiolatioo of rhe ..homs" wa.

l9l

Freque.dy. claims of lery dy sighiiog e made in sonc rcgioG of paknb!

(pardculely for ianins rhe dontbs of Rd.a ad Shawat 4s ienrioned beforc) $at

ad nor accefled by rhe au|hodd€s. Oie offic Dajor @!s of rh4 dly clains of

sighting ofncw se$cnt in Pakisran is lh. kenness otlh€ clainr dd lh€ nisco.epdon

.mongst trasses ftar when lhe 'tighting" hos been eponed in the Kingdom of Saodi

Arabia lhe cr$€nt nrusl be sighred the ncxt day in palistan, However. tbe oilcill

d<ision about slani.g n.w luiu Donth in lh. Kingdom ofsaudi Ambia is nor baed on

acrual signdng of lhe n w c6ent Thee d€cisions e based on cnrcria rhar have

chased tuee tnes ov€t $c pan two decad6 (odeh 2o0o).

Up ro 1999 the citria vas 6 fouows lf$e Moon.s age at drsunslis t2 hou6

or horc alter fi€ New M@q ihe pevious day is the fi6t d.y of lhe Islanic

nron6. fiis actually matu lhd lhc luDr donrh begiG on tu day ofconjuncrion

oa Moon vhich occurs on€ or two dats e.rlis than $e rime shen tbe n.N

crescent becom.s vhible ro lhe naked .ye,

Fmm 1999te crilerjon {6 chareed ro the tottowing: The tunar monrh besins on

lhe eEning *hctr the sun*1is b€forc tlt moon 9l according to Mrcca. This \as

no€ di$rtuus .s ir is Fssible lhar rhc sunsel befoe moonsd sil occur even

belbre rhe conjunction. Th€Efore. rhe hmar montn may begin three days betorc

rhe ccscent b€comes vhible to nlked cye.

in)

Ftum 2001oD\ardr rhe cirdion is:

a) Tl'e s€ocemic onjudion scurs b€foP Sb*t.

b) The Moon sck aner rhe Sun.

This is nore realistic bul it is scarccly possible thal the crcscenl becones visiblc

on dE day ofconjunction.

The Min p@mders signinc.nl in lh. .arti6r visibilily crnerion for lhe luoa

cEscenl aE LAC, AOE, ARCV, DAZ, ARCL, Phas ad lh€ widrh of cE$e.t. As

n.ntioned abov€ none ofrh€s pomelets alonc decide the visibiliiy or.on vhibilitv of

t94

lhe lmd ft*e.t. C€ncrally thc crnical valuc of thc Phsc b consid€Ed lo b. t%

(Astrononicrl AlMac, 200?). Duing rhc p€riod flom Jduary,2000 b Sepr., 200? $ar

were consid€.ed in lhis 3tudy rhe'e qs nor a sinsl. incidencc when rhe sidd.s ot

cEscenl with phN l.$ lhan l% ws eponed and am€/.d. There weE two @c6io.s

when lhe ph6e m grcarer rhan l% bur ihrr. ws no claim of sightings a.d sighting ws

not possible accoding ro bo$ lh. oitclia considqed (Oct. 23,2006 and Dec. 2l,2006).

In lhc cae ofNov, 13,2004 th. phase ws I l9% lhal may have been lhc Eason rhat rtre

siehting.lain qas accepred (as ne.don.d earlie4.

TheyounB€slcrescentof agc t9.l hou6(!r bstdheoivisibilny)eendurinClhe

period of study was that of May l'7, 2OO?. Apan from b.in8 $e youngesl cre$eht seen

duing th€ Fnod of study lhc ob$rvltion has anottrcr inleresing teatur€. Thi5 Ms $e

or y case when thc cr.eent was sccn on ft. enc cEgodm date a it N6 born

ac@rding ro Patiskn sblded line. Apari ton $is record obstualion th@ are fou.

orhd clains ofyoung Moons sighling,Nov. 11.2004(22.1hours),tan.t0.2006(21.45

ho6), June 26, 2006 (22.76 houre) dd Fcb. 17. 2007 (2 t .6 hous). ln aI $eE vee 22

(21% of Ihe studied) ascs of sishdnS daiN when $c .se of M@n @ les rhfl l0

The obsemtion of Feb. | 7, 2007 @ nadc ar fic slronomical ob*Fabry ar rh€

Instrule of Spsce & PlanetJry Astophts'cs. Unrvcrsily ol Kr',(hr *. ,pou.a U.

cre$ent nsing lhe 6' Codd Refractor T€lescoF. Both rhe prcdidion crileria alto$ed

crc*enr sighri.g wirboul oplical aid but wc could se rh. cr€scenl wi$oul tetescope. No

other claim of qesenr siShring wa ac.iv.d by $€ aulho ies on rhis day, Tte

aulho lies acceptcd ou €laim. This was $c young*r ccseft sen ar ou obseoalory

d$ng lhe period ofsludy, It is a Ko.d for ou! obs.ryarory and rhc vorld recod for rhn

part'culaf clescenfs eeli.st obervarion (sw,ftoonsiahlina,con dd

sv.icoprciecr.op). Our phoroSraphic r.cord h also posred on retevanr websnes.

!95

Tlrc'€ seE 6 c66 when thc ag. of M@n @ bciwen 24 hou's dnd 30 hous

ed rhe crescenl w4 not seen. Thus, ihis srudy also suppons th. idea that age is not a

depcndablc l&to. for any pEdiction crildion.

Anodier inrporbnt pdMeler concened is rhe noonFt-sun*l LAG. There are 22

occasions *hen the crescenr of lag 60 minures or less was Fen. Oul of rhese only two

*ft wnh lag less rhan 50 hinules, Nov. l], 2004 with lag ]5 mi.ur€s a.d Aprit t0,

2005 Nith lsg .l,l minut€s. In ihe *co.d of thcse c6es rhe prediclion c (ia a o$€d

naked ele lisibility. There wer€ 15 cases when the tag sas b€lveen 40 and 50 minules

and bolh dic pEdic on crileria did nor altov crcscent

visibility *ilhoui optic.l aid, Tbls lh. n@Mr-sN4lag alone is also not a dependabte

parametef for any prediclioh criterion. Out of 95 new Moons rhe predic on crileria difer

from $e Babylonian clnerion (Faroonhi et al, 1999) only g tines when Dabylonian

crnerion (ARCL + l"Ac(in deg.ect < 22) alto$ nalcd eye visibility but ,{luc dd

lhe s-laluc cnteria do tur Howld, on rhese nine ccas

I'r view of tn6c fads we conclude rhat bod q-vatue ed r-value cilcda arc N€I

suilcd predicrior cireria with r !.tue clireion delcloped in $is work has mdginalty

be cr success percenrage ibr posnive ob$ralions (chapter 4). porricutarly in viewofrh€

facl dar $cs fire.ia do not march acrual cr.senr visibitiry onty in one ou 95 cas

(1.057q).nd thal @casion Ns conkoversial in liew ofrhe above discussioD. Thc nve

cases Nhcn the cEscent $as seen hler tan prcdicrd aE not conside!€d as an ero! 0s thc

probl€n occun d d!. lo ore(sr skies. TtrGfoF, tor lhe tomulaion oa turue

obenarional IuMrqlendar $ese Fediction dreia de nidly dependable_

TIE € has ben only o.e occasion whd lherc mrc three co.*curivc nonrhs of

29 dars elcb (during June, July and Aususl 2ooo). This is rhe ndinum fo. Fpelnion of

29 days lund nondrs anticipoied s e&ly a loricentury AD by Muslims ( yas, 1994a).

The li6t (Rabi ul Awal, I42l) of$is lriphr ofnonthsbeema day talerrhan p!.dicled

196

oftr\i'i$ lbir tNinun rcldirio! @!H ml lLv. tc.n th.Ia llxr! vt&r D ac&noo of

frt! (diD|d) coorc.rlir.! tdt! of 30 d!y! 6d h 6b Fiod of

'lldy.

b. sighio8' ca c.r Morlly o.cN!nn8 Ep.ririon of fou 30 d.F' @d! io

f'v! o! sil Afthow\ rh. kludc shd. Lsw (cb{i.! t) dlos for con .don a! lnd

et o oh..wrrion idic:!. m m., bur ftr.dvmcc ptslirg nd dcv.toFEr 6tcdf

m ohsE|ioo l|E .j-n/- h.!G.t d Fcdidioo diEir b{' grarty bclp. Tt

AFadix-V rhosa suh r ltnr c.ldr. in qdich .oryurrioB s! bed i! Knstti d

lhe +v.lu. qilai@

tvl

Chapaer No. 6

Dtsct ssroN

'ftis rvork $is inrsd.d to qplorc fld.ompae fic mathcmaical nodels for the

crneds under which rhe new lund cresenl could be lnible at silen locdion on th€

Eanh, Mormk!, it was intendcd that acomparhoD offiese nlodcl is conducled and the

models are nodified if posible, The lask of compdien od nodiEcadon of the nodel

hs b€en succcsstuuy achielcd dd a neN model the r.vahe criterion has heen

dcveloped. A sumdary ofthis wo is presnted below sith r dkcusion on the najor

achicvedenG ofllrc Nhole eihll.

Fi6t ofalla bener undcFt.nding ol$c issues. conpubrional, aslrononrical dd

obrerarional, associated wilh rhe prcblem;f l|t firet sishri,lg of$e new lLmtr ssenr

is devcloptd. Wc have €xplodd the comPulalional tchniqucs and tlrc asrcnomical

alSornhm dd rhcir applicadon (o $e extcnt that is .ecsary aor rhe calculations

irrvolved in solving lie problem desribed in nr€ pEviors paraeraph lrritially lhc

rcchtriques wcre nnptmenled on Miciosoft Drcd work slcels bui duc lo leDglh,

calculalions and lhe use oflong fomuld Ne \cre for.ed to dcrclop a cortputfr prcgnD

tlilalol. T]le prognm has been used lo do allcrlculations lbt detemining c@ldinalcs of

$c sun sd $c Moon.s well as the padmelcs inlolved in oor pobhm lhe rc$'lts ol'

nr. applicalion of $ne of tne modcls @ obunEd $ithin ln. pro8nn and lltosc for other

modcls arc done on fic basn of $e dala generaled by pngEm thal is salcd as ao ouqul

fllc. This file is $e. kanslom.d into an MS-Dxcel rork shet md lhe resuhs of orhcr

modeh re oblained therc. Tbc hbl€s comp sing thse ts!l$ tifiin dre m.in t€xl and

thar apF{ in |hc apFndiG rc all d€velop€d fon rhes *otk sh€ets

The modcl due to Babylonies as d*dibed by Fatoohi et

$at wa modilied by the MuslintAFbs is bricfly desribcd ir

198

1..2) dd is ns applicarion is srudicd jn companson with other nod€h in cbapls ] (anicle

Ll) and chaptq .t This tule is based on lhe sun of$c elonca(ion ahd the dc oflision

-the Lunar Ripcness rule dednced by Muslins of rh. m€dieval eh in exploEd more

errensirely. h is loun'i rhal though, $e Lutur Rip.!6s function is fo Dorc sophisl,caled

a compaEd ro the Brbylonian rulc, the luo mclhods prcducc almost equivalent Esuhs

when applied to rhe re{nl obseRational ccords.

'lhe $ct|rds h3sed oD relalions bclwe$ arc ofvision (ARCV) and $e tclJrivc

azimurhs (DAZ) rhar \ere extcnsivcly develop.d durinerh€€arlypanoflhe 20- ccDrutv

ae loud noE sucesful dlrins lhis study in compdison lo U€ ancienr and Dcdielal

Derhods The suoce$ in m€asuEd in ho* nany obseNalions are in agrcened wnh lhe

models. I lorv mdny rines lh. cFsccnt is seen wlEn fi. mod€ls suggcsls ils visibihy.nd

hoN nan! tiDcs lbe c:.sceol is nol seen shen lhe nodcl aho suCgesls inrisibilny.

lhc rcnen rhl fie oodels blrd I'n ARCV-DAZ tlalions are norc $cccsslul is

irar wilh ihcrcasins rrlative azinu$ thc biCllNs corrdsl oflhc crcsccnt aalnrn rho

skybrilhhess impro\.s and the crcscentsofloseirclariveallitude ure visible. Sm.lleris

rlc DAZ lhis b e]trr$ com6l dclerionlcs lnd lhc crcscn( is only \isible al hiel'cr

,^RCv. h comprnotr to lhcse models lhc Lunlt Ripcncss tutrclior is slronglv buscd on

rc of scpdlion. lh. toblcm $nh arc ol sparaiion is th $nh l ge DAz n tan tinruch

snaller rhdn su-lgesrcd by lnc Lunft Rip$es law (10 to t2 d$recs) loi a lisibld

cr.sceit. ln rl$e cas.s \vilh large DAZ tlrc Ripene$ luncliorr ,4,r can bc llrgc so lhrl

rhe larger valLEs of R,,, arc eq Ed. Conseqlendy. aor lrse. lrtirudcs ldgc .rc of

scparalion is EqutcJ But lescr DAZ for larger laliodes auoNs snaller ARCV rtrd

con*qucnrlt smrll( rc of scpration. fhercfore, cspeci.lly for localioDs Nit largcr

hdudes Lunar Ripcn.ss law beomcs more inconsnrc s comparcd b the obscNtrions

and thc ARCV-DAZ nodels.

ll is sftn du ng $c di*usion ar rhe .nd of lh. 4ri cbapl€r lhal lhd ARCV_DAZ

ba*d nodeb, the ForherihShmr's modcl. th€ Maundeas mojel and lhe lndian nrodel aE

successiv€ly belr€r. These imprct.menls ae due to deducrion of b.ner and b.trer basic

199

dau ofnininum aE ofvision for difcrc el.tiv. uinurhs- tlowrcr. all th.* modcls

isDrc the ridrh of fi. n.w lba cas4.d ftar vdie SMrly with thc Eanh-M@n

diskrcc for rhc saec elongllion. Thk c.!s!s vdiatioB in thc elully bdshh.s or th€

.Eeent of emc donsdion. Cons.qEntly, ft. s. pln of ARCV ad DAZ $c

t'i8hh.s of casnt v&i6 for dif.rcnt Eanh-M@n disranc.s. Th€ lsk of this

considdrion is rh. Fain caus. of l.ss.r succ.ss pec a8. of $.* n.thods N

conDlEd to the later ftod.ls foi posnilc ob*nations.

Th. Glliaion of vdyiiS bid nas of crc$crl si$ fic widrh or cE*.nt ro!

se €lonSa on (and $m. pih ofARCV-DAZ €luct thc phriical d.$nPlion otth.

poblcm by Bruin lead Yallop ro dcduce bdic dala €lating widtu ol cErent to $c arc

ol visioi. Yallop d.dwed this dll! fod $c minima of lhe linning visibihy .ur$ of

Inuin. Cons.qu.ndy. Ydlop @s su.cc$ful in dcducins hh3inel. p.6nttc' r.sr. rhc 4_

ulue ciledon. This crir.ion Produed bdler Gsuhs i. compari$n to atl th. ptcvious

mdhod!. Th. dcdudion ofvadous visibiliry condirionson lh. bsis otdircBnt dngcs o!

.r.valu.s $d rh2t ol rhc besr rimc of visibility' of .een frem rh. limitins risibilirv

cunes of ENin !'e thc motr markable of Yallop\ contibution l-hc \isibili'!

condirions pro\ idc ruidclinca lbour uds *h.t condnioos rlt co*cnr \ould hL casilv

risibl€. $hcn n $ill b. visiblc undo P.rfcd st.rhct condnions whc. Ue opricrl \o(ld

b< Eqlircd .d when fte cB$ent $ould bc sinPlv nor visible $ith ot Nnhout atrv

opti.al aid. rfu* condilio* har. poKd ro b. noN and morc Eli.blc $i$ incBtsnB

Thc najor co bution ol or e

is rhc comp.ri$tr ol .ll fi. htljcls

@rclically, frathcm.ricElly, phytictlly.nd in vi.w of$eir success p.rcentagcs

s.r of ob*wations colled.d tom d. l.t. l9ri c.ntorv till dcntlv. Anorhcr 3isnillcst

contribudon ol lhh wott is lo convcn aU th. mo<l€ls ino si.8l. Padm.tcr crn€ria 1 h.*

arc listcd bclo{ in th. inccsing order ofsucc€ss pcE.nlagc:

'or

a

vp = (ARcv -

( 12.'o-0'.00s(D,{z)'z rYlo

200

", =(^* - l- # -e4,:!.,'l) f "

v, =ltncv .lo.no - o.orn1o.tz1-

o.ooot uaz2\lto

tt =(^RCV - lrt.8311.- 6.3226tv + O]llgr2 -O.tOl8V3rttO

\t, -t5 + py rxtr+,pfl

v | = tARcv - \t2.4023 - 9.4818W + 3.9512W | - 05612Wt D t rc

vr=ARCL r !rc olscpaBlion - 21"

ln rhis {ork schactels modcl of dldritc bdshrn€$ is ale erPlond in t.ms of

nasnitud€ contrrst tnd thc dsllE aF in lsRD.nr $i$ his wo* (Schelet 1988.) -n'h

has b€en ehieved by iniplcmenrine lh. rshliqus d.r.loPed bv Schel.r and o$ets |o

ev.luarc sky brighhes (in tems of linninB mgnitud.) and r brishtocss ol lhe

cE$€nL Biglrness ofsky and that otcE$cnt dcp.nd heavilv on various ihosphenc

arsrs Amongsr ths lenpeBluE and clolivc humidnv arc lcpl \2riabl. in the

posram Hilalol in ord.r to exPlor posiblc condilions undd which claims of nlted ctc

v'sibihy m.y b. Br.d for dill€Ent condilions fhis lc.ds lo wh.t Ne h!v. r€mcd 4

Magdlude Conl6r (Amag _ maenilud. of Moon - skr's linritnl8 a!!iNdr) lf

mognnrd€ .onl6{ is n.galit n is lalour of cN*tnr visibililv o'tunvie Dol [or

cxftnEly crilical n.kcd cyc obserylrion ckn$ of Nw lunlr ce*cN lhe magnitude

co 6l lE ber dallaled nrinut lv h i5 tolnd $d $he of ftcs cdcs appear doublful

6 fie oasniude conrEst @ n.v.r in Lvout or cc*ent tisibililv cv'tr with hidlv

ex.8g.ra&d wath€r coDdnioB (v.ry tow clttivc humidnv and lcmPtdur) lor the

vhole dudtion of th. ooonsd-suet hg |inc. Using th. pogim Hilatol the tin$ have

ben evaluat d of(i) \$en $c nasnitud€ contGi is ndinn (ii) whcn rh' nasnilude

conlldr jnrl bccodc ialouEble for crcsa risibililv and (iii) {hen rhc masnirudc

conm was last in falolr or ccsenr vkibilirv Th.s. esulN .r. sinild $ whar is

de*ribed by. veniccl line ovr. liditing lisibili(y cur. ofBruin

201

'fte najor achjevem€nt of this m* is th. romulation of t neq sincle pstuerer

cireion on lhe b6is of rhe Fchniqus deleloped bv Bruin and Yallop using lhe

bngnrrc$ modcls develoFd bv Sch&fer dd othe6 rhc visibilirv cuFes and thc

liniting visibilily cwes aE developed for crceenls ol diffeEnl widrhs ll'!l wcrc

actually obened tnd hav. bcen rcpodcd in lile6tue Bruin dcvelopcd lhe$ cllvcs on

|he bdis or{i) lhe ale6se bdshlnss ofthc tult M@n md thc wv i dalt,s *nh rhe

decr€sing alftude abole hoi@h and (ii) tbe avedge brighiness ot the skv dudrg

t*ilid &d fic way n depends on lhc dolld d'prc$ion b'lov horizo' On lhc olher

hand tc bave usd rhe adual brightncss for Ihe obscb-ed cEsc€nB ola fixed wid r llEn

lor tn sne ofobselvation we hare calculared th' al ude of skv poins bavins rtrc sms

bridft$ d $ar ol lh€ dcsenr for difrercht solar depcsions such compuratios m

Ep€ai.d for a nunber ofobseryed cE$e 5 of the sahe wjd$ al dilIelenl localions and

dmes. The aliudes or thc skv poitrts lhu obt'ined re ten 'vc6Bcd

our Thc wnole

prccess h rhe. rcpeaFd for cnsce.t ofdilTereni widlhs

These conpulations tesulled inlo our lisibililv curycs ed $c linnins visibihv

curyes. The linitihB lisibilitv cses we have obEined ae slishtlt diferenl

'iod

$ose

of Bruii with nini@ slightlv displ&cd The ntlighl linc joining thc ninina oi dEs

cudes has a slope of9 3/5 as compaftd to 9/5 for Bruin s cutves Tnis lcads lo slighd,

differe b.sl dme of visibililv" which is 4 l/9 1 rine th€ noo'elsuosl lag aller thc

suet (6 cohPdcd lo Yallop s be$ tine which 4/g rincs $e noonst-sunscl lae afiet

rhe suset). Th€ bdic dala ob$inod fion the ninim olor liniringlisibilnv cu8e l-rttd

b a fiird desee Polvnodial tsuls inlo the fouowine visibihv pdamslcr:

s =(.4RCV - (4351 31 /3 + 2222075057t4! - 5 422641ritf + l04l4l?tt)/l0

we have als dedeed visibilitt condi$ons in a tum€t simild io lbar of Yallop'

Our vhibihy condilions tre sliCh v difiere lron those of Yallop Howevet' ltplj_'n8

our model on the obseNatioml dal! the succcss pcrcenrage is iound lo b' thc besl

dongn all the dot nodels tll erist and lhar dc €ncd dd conpaed in lhis \orr'

Thc e6on is tbat our model is baed on $e acrual skv brighhre$ and lhe btislrhss oi

201

(escenr $idr varying wea$cr condilions whercas Bruin s nodel h based on aleragd

brishlncssofsky dd rhe FullMooo.

As clain€d by Sctaefd th€ brightn s modcls @v b€ in mt by a mucb a 2elo

still our Drodcl yielded be .t resul$ s comPsed 1o tho* due to Bdin ad YalloP as lar

as rhe posirive observatiohs arc conemed lt is lroped rhat wilh beller models of skv

brighhcss slill beiterclitrion ior fiGt !isibility ofnew lund descent can be delclopcd

sonE of$e lpplicarions of {he cnbda of.atliesr visibililv of ne\f llod 'rc$ctrt

havc becn considered in $is $ork The fid ond $c mosl impoddnt applicalion lre ro

d€(eaniDc Nhcn tbe new lunar cresceht would b€ visible ar soDe ldalion on lhe elobc on

rhe elennrg aicr rh€ oonjundion Anolher ar.a ol appl'calion is lo deduce d

''obscNnrioMl lunar c.lcndii- tor a 'eeion

Yct anofier aBa ol apdic'tion \e hn\c

e\ploreJ '.roJdrnrine

lhe lcnslh ofne\ ldDrr crcs(nr

t( !1ay be recalled (hal lhe liret appcardnce of new llnn crcscctrt nrarlis r

b€sinning ola new monrh in obsenalional lunor calendarq rhere

'ril'ria

and models drc

siBnillcor br calendatical Purposs whefier 'n aduul o6€n:ri'nol lunar calendai likc

rlc hliNic Lunar calendar' ulilizrs tb6e cril.da for a!6nging its cltndar ot Dol lhcsc

cLirdia ptuvidcs a guidancc Ibr bolb tesing an tvidcnce oi oescenl sightirrg bv connnon

peotlcdnd tacingdowndtdtrlesofac.l€ndar inhhrorv{herc approprirle dates arc rol

\€ll Ncordcd. rhus $c frtin ililt of the pcdicrio! crileria $e carliesr visibiln)' ol

' 'or

n€rv crc$.m is lo egntalc tnc obsefl.lional lunar calendtr dd tcsrifv lh€ clainrs ol

risibility of ne\' lunar crcsccm

Although fiIsl ordc! iPporirutions like Arnh'neLic Lunar Calendtr tlrat di'

based on lle conc€pr of Le.p Yea$ md rne a!.rage oorion o' rnc Moon havt be'n nr lse

Muslims h.vc been follosin8 acoal sighing of cresc€DLs l@l lor lhe monlhs ol

'r

i6rii8 (R.maan) $d pilsinage (zil hajjah) Thc calendars if bled o' a PEdicrion

oir*ion like tbat of Yallop or dre ohe dcveloped in this work arc $e cLosest lo rnc

obervarional calendd Conpan$n of rhese criteiia wiih lhc actual obFNsdonll

201

old! b !|dc! in ruirD frr lh. Ft! 2000 io 2005 blc Da dc (q!!thi sd

Klte, 2005). h lbir q'!rt 6. mrdy it cid.!d.d lo tb y{ 20(I|. It i! biDd lb. d

lvdsc 9J% obtlnltiol! lrc m.oding to th! Y.llop'! r-uh. €dt rion d our GEluc

(d Q&K) dilrio D. diltgt@t is lb. !!3rlt of lithd th. t d wtt|t r dM !o *hi'h

&e rw |E|r c{&. codd nol t tigln d dd tb Lud E olh b.t!! oE &v l.|c qr

bo AdrDirtic cLiu6 of otsElion tld {t L|e dot:t t g.! oE &t dlicr U|n

Tbit tt|ldttbL !|gs for th. vi.ibnft, (ritsi. for ol@t'.ridl pltrPo" hB

oorivcd |' to d.dE e "bt 6v.liodl lE[ c.lct b." for P'lii..r 0|al ts tctr

204

APPEDNIX-I

COMPUTER PROGRAM

HILALOl

205

{ cl6c{): maime.u0; cltsqo; Daiuourineo;

fclose(fptr4)t

{ gotory(]0,4hout<<"welcone io the New Moon cslculator";

goro\y(25,20):.oul<< Prs Eni.r !o $an :

void eaituoud.4void)

{ f!l.{=fop.n(schmse.hf',"a');

{ inputdalerime0i cl6cd); nonlh_chee.0i

I

ll6t_cahulatioo0;

jd=juliadare(lred,lmo trtdab,tsac);

jde=jd+denat[inl(lyeaFl620.o)]/(3600.0+24.0)i

stjd=juliedakoted,lmonrh,ldare,o.0);

$=Grjd-:15 r 545.0n6525 0j

i=(jde-2451545.0y36s2s0.0;

elonep=sin(ndeltap.Pvl 80.0f sin(sdethp+Pvl 80.0)l

elonep=eloDgp+co(mdetaprPyl80.0)rcos{sdelbp*PI/180.0)

'co((slphap-rolphap).1 5.01P1.7180.0);

elongp=ncos(€longp)r180.o/PIr

eotor)(s2,?)j pdnl('T. Elon =');eoloxy(65,7):pinr(,,%?.llr

phasep-(1,o-co(€lonSplPVl 80.0))/2.0i

solox)(5:.8);pdnt( T. Ph6e =,,);Boroxr(65,8)iprinl(,,%7.If

%'.Pbasco' 100,0);

elone=sn(nrlarrPI/l 80.0)isinGhr'Pvl 8o.o)i

c ons-elong F(on m14.Pl./ t 80 ol.@dslat ,Pt I 80.o).coq GtonSmlong.Pt/180

0li

elong=acos(€londi 180.o/Pl:

phaF=(1,oa{doie'PUl80,0)!2.0;

mseni{jia=(nBdhdsor I 80.0/Pr;

Sotor)r52.0)pn fi"Widrh -'.gotoxrr65,o'.pnntt( !i?.lltm.$rd):

Sobr)15-2.10':prinrli DAZ - ):Sotory(65.r0ripnnrt(5.7 ltl

goro\tr52.ll):prinrf( ARCV'igolorJr65.llr.prinrn o.7.1tr

qvall(oah-sall)-( I 1.837 | {,1226*wid+o,73 t 9.wid.wid-

0. l0l8'pow(qid,3)))/lo O;

ov.l=(h.lr-slrx7.r65l-6.32261wid+o.71 19.*id.wid_

0.l0l8.pov{wid,3));

,06

gotox(52,12):p.i {"Ase = '),sororr(6t,I 2ripnr'4 ./o7.llf h .0de

mjd).24.0):

Sobry(5z,11);pdn("Qvalue = ):goloxy(65,13)prinl("%?.llf',qval);

soioxy(52,1a)ipn (visib. =)rsoloxy(65,14);@ut<<'';

soroxy(68,l4);

i(qv.>o.2l6)

i(qv#-0.014)

@ul<<"VUPC r

i(qvaF-0.212)

COUK<"MNOATFC'';

i(qvaF-o.293)

cout<<'VWOAO"i

etse

cout<<,,ct ;

eoroxy(s2,15);pnnt( Lae = );

i(l|mrhs<o.0)cour<<'!"rconven bn(r'abs(ltdstrst)i

eotoxr(52,16);pinq Rd.y = ");soroxy(65,16);p.inr(,'%.51!"_rdar;

sotoxy(52.17);pinr( Rdayl = );soroxy(65,t7):pdnr(aZ.5tf',rdayl):

go'oxlr52,l8):Arnlfl tuly - l:gorory(65,t8ripnnln oo.5||.btyJ:

8o'ott(52.1c):pdnrfi S-val@ - i l:gorory(65.t9,:pnnr{l o/o.5tf .6kysoloxy(52-20);print(thilen

-

'');goloxy(65,20);pri.r(.%. jlf ,phitend.);

eoroxy(52 j21)rpiid(lhnele ho. =

")rgoroxy{65,21)ipi (,,%.51f,.ansl€_hot;

soroxr(52,21)ipdnr( Lim MaB =

"):gotoxy(65,21)iprinr("%.21f ',tindagnn0);

sotoxy(52,22);prinq Mooro Mas =

''):Eoro\t(65.22).prinr ("% 2t f .mnm.g),

sorox)(52,24).prinrt( Rel Hun =

):Boroxy(65,24);p nrr(,%..2tf .phm)i

eobxr(s2.25);pinr("T€mp@rG=

'il.sotoxy(65,25);pnnrft %.2tf ,plempl:

Sotory(5,28r.pnnr( s{s/r, nmrm/n}lou l/t, h@id{o/b, !.mptdd j

day(e/D prinr(p) rdse(q) li

{

201

rvhile(nex|sl=V');

Eolory(50,29):@ur<<"NwCalcnlaion? ;

gorox(77,29}nexts=g.rch(I clsto;

]

while(oexrsl=1r);

gorox l0,l0)r@ut<<"Allal Hanz'; gcrch0;

)

void inpurdaretiee(void)

{ goroxi20,2);coul<< "Enrcr L@dTinc & Dare,,j

eoroxy(r0,4)icoul<<'ObscpadonNo. : ;ciD>ocnr;

Soloxy(10.6)icoul<<'Day :,,lcin>>datei

goroxy( 10,8)icoul << "Mon$

:

!!;cjn >>monlhi

Soloxy(10,10);cour<<"Yeo :iiicin>>ycdj

gotoxy(10,12);cour<< ,Lo.g.(+ lor East) : icin>>plongr

goloxy(10,14);cout <<'jlkt. (+ for Nonn) :,icin >>ptari

eotoxy(I0.|6);coul<< "Alritude abovc S€a : ;cjn >>p.lli

gotoxy( 10, I 8)icout<< 'Esiim Tenperarurc : "icin >>plenp;

gotoxy( 10,20);cour<< "Estih Huidirt

)

prcs=1010.0; *0.0;

)

']

)

: ,,;cin >>ph!m;

{

i(l.apch*k(inr(year)))

nohthdayslll-29.01

clse mo hdayslll=28.0;

ld.te=dak Jmo.lh=donth JydFyear;

zonlin=plong/15.0;

i((b iDlong(anrinD0.5)

.

2ondmioubl€(tons(zondn+L0))i

entihdouble(lons(zontin));

lhorf houF (min+*d60.0y60.o)-a im:

i(lhour>=24.0)

{ lho!Flhour24.0;

ldri--ldale+1.0;

204

i(ldltohonthdayslin(lnonlhl.0)l)

{ ldaiFl.0:

lmonth=ldonth+ L0i

i(lmonrDl2.0)

{ lnonlh=I.o;

lyeFlyea$l ,0;

]])

ir0hour<0.0)

{ lhourFz4.0;

ld.re=ld.r.-l -0;

i(ldar.< | .0)

{

lhonth=lmo.th- L0i

i(lmo h<I.0)

{ tnonth=I2.o;

lyerlyed-l .0;

)

ld.tFnotrthd.rslint(lm@thr.0D;

tl

ndalFlda!.xtnonrh=lmonth:tly€clycari

tdouFlhour+d.ltltlin(lyed-1620.0)1/]500.0;

i(ttbour>=]1.0)

{ iihoui=fihou,24.0;

lldaie=ndalqfl .0;

i(ndapmonthdlyslin(rttDoiIh)l)

{ ll<ratFt.o;

tltlon!!=rnonlh+ 1.0;

i(n nonrh>12.0)

{ tlnofih=I,o;

. ttyeaFilJear+ L0;

)])

J

{

dt_new_n@n(dar..nontnrE);

soroxy(5,I rsout<<ill.w Moon ";pdnt(!D - %lfrnjd);

pdnr(' or

"od%dfld',in(mdare),ii(Mnonfi ),in(@y$r));

mdare=doublc(lo.s(mdate));

morth<oublc(long(nmonth);

myerdoubl.(lone(Myed));

pr d(" ll.'!drl6d:vdd (TD)',int(MlFu),inl(t:min),int(m*));

pint(' or eidrl6d:94d (uD",irr(ulrow),in(ain)jn(M));

6*aunou.r(uinfi 8e.r'60.0/60.0y24.Ot&lL{in(mrrd- I 620.0)l/8400.0i

IiBrcdculadonoi s.rinln0;

209

,

t

)

{

i(ho!P-24.0)

iqhoul<0.0)

ni.{hou'double(loneGou)))160.0;

s.c=(min-donble(lone(min)))r60.0;

min=double(lons(min));

*c=double(lonc(s*));

houFdolble(lonc(hou));

sun_$o; non *(1;

brim=rGs+4.0.(lr,ns-hssy9.0:

Soroxy(2,5);cout<<"(LT) olSun s.t : ";conve'l hms{ltss);

goroxr(52,5)i@ut<<", Bel Tine : ;convcn hn<bl'n):

gorox(30.5);@ul<<', Moon S€t :'iconven-hm(lrnti

goloxy(2,6)icour<< (LT) orsun Rkc : "ponven_hns(hsti

sotoxy(10,6);cout<<. M@nRi* :';.on!en hms(lrfrti

goloxy{2,3)rcour<< Lar-';

soroxy(8,3);coNert dms(fabs(pla0)r

i(plat<o.o) pinr('s'); dse

gobiy(20,3)cout<< t ns. ":eoloxy(2?,1)i@nved-dns{fabs(!long));

ifulonc>00) pnnt( E ); dF

goloxy(40,1):cout<< TZon€ ":

sobr'(48,);

i(enrim<o.o) cout<<"i; €le

sotoxy(49,1);prinq %2.01f,fab(2ontim))i

Eotoxy(2,4);p.int('LMT vo2d:%2d:%2d,int(hou).int{nin).inr(sc);

8otoxy(]5,4)jprin( (%2d l2dl%4d) UT",in(date).in(month),in(yed))i

soroxy(31,4)roNed_hns(lhout;

soroxr{42,4);print( (%2dl%2d7o4d), JD = ",in(ldal€),ii(looftn),int(lyeat);

sotoxr(60,4)iprint( %.61f jd€)i

sotoxy(6s,3)rcour<< TDr iconve _hm((lhour);

Sotoxy(22,8)rout<< s u n":soroxy(40,8 ) rcour<< Moon j

Soloxy(2,Io);cour<< Geoc. rong, ;

goroxt{2,l l )imur<< Ged- Iilir.'i

aotoxy(2,12);@ur<<"Cee. Disi.";

go!oxy(2,14);@uK<'Gc@. RA:";

sotoxy(2,15);couK< c€e. DELT ;

soroxy(2,I6):@ur<< Hou Anelc:"i

Soroxy(2,IE)Fout<< Top. RA:":

goloxy(2,19)i6ur<< Top. DEL|"i

gotoxy(2,2o);coul<<"Top- HAf ;

8otory(2.22)tcout<<"AIfi udel';

2t0

)

gobxy(2,21);cout<<"Azimut ;

gotory(2,24);cout<< Semi-Dia: ;

i{(.Y.400)l=0)

{ if(n%100)==0) k=0;

€le i((n%4F0) k=1;

)

)

double julimdate(doubl€ yrdouble mndouble dcdouble ft)

{

)

i(mn<X yr=yFlrm.=ni+12: )

a=(doubl. (long (yrl100.0)));

b=2-a+(double (ron8 (./4.0)));

jd=(doublc (lon8 ( 165.25'('r+47 I 6.0))))+(double (lo.s

{10.6001'(mn+1.0))))+d.tb-1524.5+fi i

{

l=(jde-245 I 545.0)/36525.0;

epsilon=21.0+(26.0+2 l-a48/60.0/60.0-(46.815/1600.0)'t'

(0.00059/3600.0rf 1'+(0.00t813R600-0)1.r1;

elons=29?.85016+445267.1 I 1481-0.0019142*l'l+t*t'l/I89474.0'

smon=35?.52?72+15999.05014.r-0.0001601'lir-t.r.r/300000.0

mmob=134,96298+477198.867398't+0.00869721'l+t't'VJ6250 0i

mng=91.27 I 9l +483202.01t538rr,0.0016825{1.t+l*1*r/32?2?0.0

lnod€=125.04452-l9l4.t l626l1l+0.0020708.11r+rrr*t450000 0

fo(i{;i<61;i++)

{

derbsi=delhsi+(nur obrlil[5]+nur_oblti]t6l1r).sin((nur_obltill0l.€loig+

nur,obltiltlheom+nur_obllill2l.ndom+nu obltiltll'nale+

nut_obllil[4]'lnode)1Pvl 80.0):

delra€psi+=((nur_obltill?l+nul_obrlilt8lh0{co(nui obltiltoiielone+

nul_obltiltll*smom+nulobllill2li@oE+nut_obltjl13l'nar8+

nur obllill4rhode)'PYl80-0));

)

delrasi'=(1.0(t0000.0.t600.0));

<leftaepsi'1 1.041 0000.0'3600.0));

elsilon=epsilor+dellaepsii

2l!

)

{

gmr4c6-0r{41.0+50.5484V60.0y60,0+(6640184-812866'si+0.093104'st'st-

0.0000062rst'srlsr)6600.0r

east4@=eEst Erc{€lltsi'cos(epsilon'PUl 80.0yt 5.0i

smstz€rc-lGmrzro/24.0ldoubl{long(8tui4b24-0)))'24.0r

gdu€re1Gastu o/24.0!doubL(loos(ssrz'o24.0)))'24 0;

if(smEm<o-o) 8rnst4'o+=24.0;

i(gnsldo>24.0) gott4to-=24.0i

i(s6rzerc<0.0) gaslzco+=24.0i

i(easrzerc>0.0) gastzdo-=24.0;

smstcun_Cnstrcro+frac124,01 I 002? 3 ?909I5 i

gasrcuFsdzero+firc'24.0* 1.00273?90935;

i(snslcuP24.0) snslcun-=24 0i

i(emslcun<0.0) Smslcud+=24.0i

i(sastcud>24.0) eastcuft-=24.0i

i(gastcuft<0.0) sostcud+=240;

lostcllFgnncd+plong/15.0i

l4rcuFedlcur+Plong/15.0i

lastcun-=24.0;

i(lastcuPla.0)

i(lostcud24.0) lnsrcud.=z4 0;

i(ldlcm 0O'

ii(lmsbu<0.0)

ls(c!r-240i

lhsr.!nt=24.0;

)

void convcn dmndouble c@d)

{

cdesdouble(lone(coord)):

cmin=(coo'd{d€s)i60.0;

c*clcmin'dolble(1o.8(cmio)))'60.0:

pnd( %3dooz2dn"/o2ds,in(cde8),in(cnin).in(cscc));

)

void conven bn(double cood)

{

cdes=double(loneGoord));

cmin=Good{dee)i60.0;

csc=Lcmid.double(lons(cmin)))t60 0i

prinr ( o!2dhoo)dn' o2dr".,ntrcdegl,inlr(min).rnl, cscc\):

)

void sun cood(double)

{

fplFfopen( v$pean.txt', t )i

// LO,\]GITUDE OF EARTH

fodi=o;i<6:i+)

{ tehp=o.oi

fo{=0j<lnellilj++)

2t2

{ fs@f(fpt'%lf yolf y.lf,&sx,&bx,&c);

rmr=l€mF4.cd(bx+c.1);

)

slong+{tmp.po*{!i):

)

slonrslong/l 00000000.0;

slong=jon8. | 80.0/PIi

slong=Glon g/360.Odouble(longGlons/360.0)));

slongl=360.0 slone+-180.0

i(slone<0.0)

dong+=360.0;

i(slons>350.0)

slong=slong-360.0:

// L.firurL OF EARTH

fo(i=0:i<2;i+)

ro(=oj<adlilj+)

I [sco(fpr,"%lf yolf %lr,&ax,&bx,&c);

temp-aeosrdrco(bx+c0t

)

slai+=Cemprpov{t,t);

)

slat=*IaU100000000.0j

slatl-(t80.0/PI);

fo(i=0ri<sli+)

)

to{=0i<dsrflil;j+)

{ fs.n{&rr,"o/,lf %lf.4lf'.&a,&bx3c);

r.mpaettp+u'cos{bx+cn);

)

sdst=sdsl+renp*pow(,i);

sdsi=sdst/100000000.0r

dst=sdsr. 149597E70.0:

slo.g=slong+dclt!si{20.4898/sdrry3600.0i

elph!=sin(slons.Pvl80.0).6s{.Filon+Pyl 80.0);

slph!-slph!-tdGlal'Pvl 80.0).ein(.plilon.Pvl 80.0);

salpha=slphd(@s(slong.Prl 80-0));

lalpha=aran(salPha). I 80.0/PI:

i(Glong>90.0)&&Glorg<2?0.0)) sglphaFl8o.oi

if(slong>2?0.0) salpha+=150.0i

elpha-salphe/15.0;

i(s.lph*24.0)

sha=lastcur-salph!;

salpha=elph.'24.0:

2t3

iisha<o.0)

sdelb-sinGlarrPvl80-0f cos(epsilon'PI/180 0)l

sdelta=ldclls+cos(sla1+ PV l 80.O)'sin(€ps ilon'Pv l 80 0)'si.Glons'PI/ 1 80 0) i

d€lta=6inGd€lla)' I 80.o/Pl

salphap=elpha+.lcladpha

elr- sin$laf P! I 80.o sinrsdelrap'PI/ I 80.0 ) rosulat'PL I 80 0r.on sdehtp'PV

I 80.0)rcos(sba*15.0'Pvl " 80.0):

salt=asinGalt)' I 80.0/PI;

i(sw=l)

{

;Ffra l .oz(r&(eh+l o.v(sali+5.1 l ))'Pvl80 0)fe8l 0(271 0+teoF))!(pr€s

/l0l0.o);

l

I

s.cb-nn{olarrPul80 0Is'n(!d.l!lp'PU180 0,'sinleli'Pv!80 0):

tela=$etd(cos(cdelbp'PVl 80 0/'(olleht PLI 80 0r,:

seela=acos(seela);l 80 0/Pl;

sdcltrsde liap+srcfi'cos{seela * P l/ 1 80 0);

slDr=sh!-(sefi*sin(seetai Pl/ I 80.o)ho(sdelta/u/180 0)/ I 5'0i

sdehFsdelk: shFsha, l

idh=sinLplat'PUl80O)'iirsdelt,r.Pt/lSOOFcos(pldfPLlS00rrcoil3dcltat'PI/

180.0)'cos(shd'15.0'Pvl80 0):

satFdin(s.lr)' 180 o/PIi

*lr+=r,c6ermar4erur+tnle/1000 0D' 180 o/PL/l 5 0

sa6=s,n(rd.ld. pU I 80 O Fsid ptalr PI/ I 80.0)! sintsallr Pl/ | 80 0):

sm=szn/Gos(platrPvl80 o)'co(elt'Pl/180 0))l

sad=acos(saar)1 180 o/PI;

it(shdF=o 0)

ehe

s@=180 0i

itlshaF=12o)

el* i(shd<I20)

s*nidi.=Gn(ydsf l 80.0/PIi

)

void moon c@d(do!ble)

{

fDk2=foDcnf elP2000 ur'."r):

sumv=o.o,suntp=0 0lsumvtp=0 0isunvppp-0 0:

fo(i=0ii<2l8ii+)

I r(mfl rDu2.'%[ 0/"1r 0/0lf q.lf %lf

" 0lf ,& \,&!lpo'&!lp l.&alp2 &alpl &'ly'l

sMFsuov+v.sin((atpof atpl rr+alp2r1.r/10000.0

+alp3iPo*(i,3Y1000000 0

2t4

+alp3'pow(!4)/100000000.0)rPI/180.0);

)

fo{i=oji?44;i+f)

{ fed(fpa2,"o4lf%lf%lf',&v,&rip0,&alpl);

smlT=s@vpf v'sitr((alpot.lp l'l)'Pvl 80.0);

)

fo(i=oii<154;i+r)

I fsc f(tptt2, r6lf %lfo/o1t',&v,&alp0,&alPi );

sumlpp=sumvpfi v's n((alpo+llPlil)iPI/l 80 0)i

)

fo(i=0ii<25;i++)

I

lsconi(fpu2, 9'lf ?'lf eolf'.&v.&slP0.&alpI)i

s!mwpp-smvppF ! | sin(alpo-alp I i t)t PVI 80 0li

)

mton{=218.11665+481267.88134't-13.2581'l/10000.0

+ 1 .856'po*(!3u 000000.0-1 .514'pos{tjy1000000{0 0

+sum!4Gur?+swvppl+sMvppp't'i/l 0000.0)/1000 0i

nrois=(nloD936o.o-(long (mlo.s,/360.0)))1360 0;

sumu=o,0tsumup=o.0rsunupp{,0isunuPpp=o 01

sunr=o.0;sumrFo.0isumrtp=o.0;suntppp-o.0i

lb(i=0ri<l88ri++)

r iscanfi fok2.'/'lf t'll"/.lf %lf %lf

' %lf ,&v.&.1p0,&alp l ,&alp2 &alpl.&'lp4 ):

!mu+mu+vt sin(alPo-aipl 'r falp2'r't/l0000 0'alpJ'pov'1t.J y' t000000 0

+alpl.pov(!4y100000000-0)'Ptl 80 0)i

)

fodi=0;i<64ii++)

{ f$anf( lol12,i%lf'l"lf e"lf .& v,&tlpo.&olP I )

suNup=sumufi! rsml( tlpo+alp l'r )r Pl/ l 80 0)

)

r fscanll tDk2. '/"la

%lf "/.1f ,&v,Aalpo.&alp | ):

sumuip=smupp+vtin(alPo+alpI'0'PI/l 80 0li

)

I l*anfi fDr2."%lf %ll c,6lf ,&t,&dpo.&dp | )

sunu;pp'sumuppF vrnn(.lpo'llpl'0'Pl/l 80 0):

I

;hFsumu+GuouCsunupplt+smupPp1irVl0000.0yl000 0;

fo(i-0;i<l54ii++)

I

Gcsnf(fplr2, %lf "/.lf %lr9"lf 90lt

9"lf .&v.&alD0.&alDt,&.102,&alp3,&dlp4)i

snrsmrr'cosriltpO alpt'l+;lp2'rt/lO0o0o' Jpt'po*rt l/10000000

+alp3xpowt,4yl00000000.0)'Pv180.0)r

2t5

]

fo(i=o;i<l l4ii+)

{ fs.d(fpL2,'%li %lf %lf'3v.&rlp0,&alpl );

suotFsuDr/*rcos((alpolalPl rl)'?tl 80 0).

)

fo(i=oii<68;i+)

I

fscdf(fp$2,'/"lf o/.lf %lf'.&v.&alp0,&alpI):

vmDp=suffpPrvtosl alp0+tlp l rt l'PI/ l 80 0),

l

fo(i=oii<9:i+)

r fsdfi fDk2."!/olf./olf %lf'.& v.&tlPo.&.lp l)i

)unrppp runrtpp{'coe(dlpo'alpl't'Pl 180 oli

)

md{-li5O0O.5 7+sunr sudrsr suffpp'l_sumrPpptl't

| 0000 0:

nla{=(ml5r/l60.o{lonc (hla/160 0)))'160 0i

'"i.""',-O OOOrolla-O.O0OOt019'sin(225 O'47?lo8 o'lf PUI80 0)):

mhr+-={.d o0o0l754'sin(l83 1+481202 0r0'Pl/! 80 0))i

ndst+=(o o708rcos((225.0+a?7198 e'rl'PVl 80 0))

maipha=sin(mton8'PUlSOofcskFilon'Pvl800)i - -

nrn;ha- malDhard(mlal'Pt/l 80 0t'sin(cPsrlon'PUl80 0)'

mal;ha=ialpha(@(mlong'u/180.0));

maloha=.iu(maLlha)+ I 80 o/PI

-ri,,ir.no'go oraai.r.ng< u o o I I mdlpha- _1800:

iftmbn;27o or nalPha+-1600:

if(mh*24.0)

frba_-24 0;

inmhi<o.or dha+=24 0i

m;?lk=$n(mltr I Pv l80 ofcosl€psrlorr PV lao 0): --^ ^ -

-i"iu=na.r. -'Lmr"t'pvrtOO)tsin('psilontPVlSO O)rsininlonsrPl/1800):

nrdelh=asin(mdella)'l 80 o/Pl:

rulphlp=malpha+dell4lPhai

mha=hh.nelraatPha;

-lii 'iil""iJ" er'i do ol'"'t.aerl!pr Pr/ r 80 o,

"os(l)rar'Pv

r80 0)'co(mderbP'

Pl/l8o.o)'cos(mnar lt 0rPyl80 0l,

malr=6in(malt)1 180 o/Pli

i(sr=l)

;rcfr( 1.oz(bn(nal(+ l0 la mah s lt)fPl/l80oDft283 042730 remfl))'(tr

drcfFmefr/60 0;

216

mcela=sin(!lar*PI/180.0)'sin(mdcliaprPvl80.0)1si.(mall'PV180.0);

nft la=merdcos(ndellarPvl80.0)'@s(mlt'PUl80 0));

i(fah6(nela)<= 1.0)

I goroxy(45,20)i@ut<<"

meeta=acos(m€ela)' 180.(VPIi

mdehFnddiap+@t'1cos{mera*Pvl 80 0);

(@t'sii(6ela'rvl80.oycos(mdehe'Pl/180 0)yls 0i

malFsin(plarrPvl80.0)'sin(ndehd'Pl/180.0)+cos(plal'PVl80 0)*co(hdcllai

?v180 0)rcos{mhdll 5.0'PYl80 0)i

nralt=ain(mah)'l 80 o/PI;

nalt+=(aco(eMj(enej+hik/1000 0))x 180.0/PI);

]

)

cour<<'.eb geater lhan I i

{ hdeluFndellapi

; a,m-si1(m,lelrar'Pl lSOo)_s,n{plat'Pl l80ofs'n(moh'Pl l800r:

nDzm=mdnv(cos(olaf PVI 80 0lrcos(mah'PVI 80 0)li

nd=lcodmaa)r 180 o/PI;

i(nha-12.0)

nazm=160.0:

i(mla<12-0)

mm=1600_n@;

nrmidia=(n6<Vmdsr)' 180 o/PI;

..r,r,="."u.otrtio"o'pr rtOOttsrn(eps'lon'Pl l80 0)rpos(l 0_

po$,lsi;(epcilonrPl,I80 0)'rn(slong'Pl/l80 0' 2 0) 0 t)))

ii;i'iriio",tr.o.po",rsinrcp'it.n'Pvl80 0)rs'n{slon8'PVl 80 0) 2 0) 0 5)li

iii=.co(iii)'l 80.0/PI;

mele hor=90 o.Dhil.mdai

aii*rir.:-.*.iai*z zf oqplatrPl/l 80 0ycos{l)hil'ddt'Pvl80 0):

rda; l=l0 t'.os(plal'PVl80 0,/.osohilcmda'PVl80 0)i

217

l

i((dnon8<90.0)&&(slon8>100.0))

ddone-nlon8+360.0_slon8;

dclong=mlong'5lon8;

Bky lal'tan(phitehda'PVl 80 o)+delonsi

doubl€ modtunc(double xY)

I

xFxy/l6o:

\y=(xy{double 0on8(xY rr)f 160:

iltxy<0)

loid paBlx(double hhdoublc dd)

t ol,llliililt;lli*"""pr/r80.0!emai)rr80.0/pri

ll,l"'lili-""t.r',t, t "o',r'PVr80

0)-r'Ih'sinr'r 0'|prar'Pr/I80 0rvl600 o:

;-eniin'vntuu'pt rro or inor' hre'{nrplar'Pl 180 0) on':

yy ro\tuu'Pl/l80 o)'hit'.os(plar'Pl/180 o)/con:

rho=po*(xx's+ty'y) 0 5)'.mtji

i.rJom=aurt'pr'co"rphrPVl 80 0f sidhh'PI/l 80 0)v

"' *' - --i""ltaa'priitooFpG'cos{PlatprPtl80olrcosthh'Pl/1800)))

delrMlPhal=(l 80 0/Pl):

hp=hh_deltaalphal

i;i;i|$,ili;t4r", to'r'{sin(dd+ur

" "' _* aoirOa'er'lad Of @s'hb'Pl lso otspie'cos'pldp'Pvl80 0)l\'

dehapr=(180o/PI)i

hp/-15.0:

loid dr-new-roo.(doublcdoubledoublc)

yy=y€a.+(monLh_l O+dak/lo Oir'l2 0i

traY-tYY_20000f 12 3685,

ir(kar'0ni)aoou61.

80 or-Pie'sinuratp'Pvr 80 o))'/

06ng ltar)))- t.o;

e|s raF(dotblc oons (ksY)));

L€--l,ay/1216 85i

;;=1"' ;';93;ggiijliil['31ffi 3"'ff.i::.i:.i:::*,

-.""-."ar.""ii lSu-zq toSl5o7'kav 0 0o000l4't4'r*'

*""":l'j'#:,!';i"*::'r;;:.8 r 6ei528' kav'o o I 07582' ree' reF. ooo0 I 2r8

'_ ' .@.Le.r;co 0ooo00058r r€.re'leerrec)i

218

forg nodf6c(160.?lo8+390.67050284rtoy'0.00161I81rcc're'

0.00000227'powtle .3)-0.00000001 l'por(1e. 4)):

omes=modtun(124.??46_

I 56175588'kav'000206?2'l.e'tc. 0 000002I5'po$(tee3)l:

eFl.0-0.002516're-o.00000?414nei

msur<PV180.0);moon.-(Ptl80-olr!re'1Pvl8o.0);ones'1PU180 0);

oe--0.,r0?2.sin(mmoo.)+0. t724l.selsi!(ns!i)+o.016061sin(2.0'ndoo.)r

perF(o.ol039rsin(2.0rfaq)+o oo739tceersin(nnoon-nsui)'

o.005l4,eee,sin(msu+moon));

p€r+=(0.0020S'po{€ce,2 0)'sin(2-o'Et!.}o 0o l I t'si'(mmn'2 0'taq)

0.00057.sin(mnoon+2.01raq))i

@r+=(0.000561ee'sin(2.0xmmoon+nsun}

0.00042{sin(3.0*hm@n)+o 00042'ee\in(nsu!+2 0*fdd)r

pe*=(0.00038'e*;sin(hsun-2.orfds)_0 00024teee'sin{2 0hmoon_nsui)-

0.0001?'sin{omes));

Fr+1'0.00007'sin(mmoon+2 o'msd)+{ 00004'sin{2 o'mhon-

2.0.fa€)+0.00004.si.(3.0'osun))i

'

De.F(o.00003'si.(omoon+nsun_

20.Lre)+0.00001+!n(2.0'mmoon+20'fare))

eeft I-O.O0OO3!n nmoon-nsu 2 0'fargh0 00001"in(mmoon_

msun+2 o.fes))i

'

Fr+=Co.00002'sin(nmoon_msun 2 0rlarg)_

000002'\i l.o'mmoonrsun,'O00002'sn 40'mdoon)):

ol=(299.77+O lO?408{kay_O 009l?3rteettee);al *=(PI/1 80 0ll

a2=(25 1.88+0,016321'kay)ra2'=(PI/180.0)l

a11251.83+26.6t1336rkay)i.lr=lPul80 0):

a41140.42+16.4 I 2478'kayl.a4'1PUl 80 0)

a5{84.66+1E.206239ik.y)ra5'1Pvl 80 0)i

16=1 141.74+5l.lOl7?l rkav):a6r=lPl/180 0)l

a7=i207 l4+2 4tlTl2tkayl:r7+1Pl/180 0):

a8-l I54.84+?.106860'kov),i8'=(PUl80 0):

r9=i14.52+27 26l2torkayr.Jq.=tPVl 80.0),

slo=l207 l9+0.121824'lavl:alo'=(PUl80 0),

rl l=(291,14+l 84a379'tot) tl l'=tPl/180 0)l

al2=i16r.72+24 laSl54rkry),a12'=(PVl80 0l

,ll={219.56+25 tllo99ikav) allr=(PVl800):

ar4il3l.55+l 592518'kav);al4'=(Pl/I80 0):

addcrj Oo0l25.s,n(al ,, addq+{ 0001 65 | sin(t2)i

addc*=O.O00r64tsrn(.1), addfr=o.000126'sin(aali

iddcr+=o.oool ro'sinl15J; addc+=0.000062tsin(46)

d&cr-0000060'sin(a7)i iddcne0000056'sin(aEI

addcir=0.0OOO47rsin(ag): .ddcr=0.000042rsin(al0):

addr-0.000040'sn(.1 l)i addc*=0.000017'5in(al2)i

addcP -O oOOOJt'siral3\: addcrjo 000021'sin(514)i

mjdanjd+pe*addcr z-rdoubldlongrMjd+osDr:

2t9

)

I(24299161.0) e=z; el$

{ alph<doublc(lon8((2-186nt6.25y3652425))):,

@-z+ L0+6lph-(doubl€ (lons (ddv4.oD);

I

cldouble (lons ((bbj22.1y365.25)Di

dd=(double (lo4 (355.25'@)));

e<double (lon8 (GMdY30.60{ l)));

md.rFbbnd-(doublc (lo!s (30.600 l'c))Fr:

i(a<14.0) !)modn+l.o: clsc mmodh+13.0;

i(montF2.o) my€f<4?16.0: els mlta-rc47l50i

nnrhor-{mdar€-double(!o4(md!tc)))r24.0i

uho@houtsd.liatlin(myedl620.0)l/3600.0;

min=(mhour-doubqlone(mhout))'60.01

ms6=(ubin.doublc(lons(minD)160 0i

unin=(unoudoubl(lons(uhou)))160 0:

Na=(uin-doubqlons(unin)))'60 0;

{

i(s@>600)

i(min=60.0)

{ mi"{ 0:

hou.H;

i(houP=24 0)

{ houFo 0i

dare+;

i(daenonthdaYslint(nonthl'0)l)

{ datFloi

mo.ll|++;

i(nonth>120)

{ nonth=1o;

v6rr:

lt )))

)

{ nin=00;

if(ho!P=24.0)

{ houFo-o]

220

t ning-no;

,

void iE_tordvoio

{

i(no|t',-24.0)

dtt'+t;

iqdnl'd.ddryltin(0.dl.Ol)

( dGl.0;

noiliFli

r{rr{r,r12.0)

{ lro tsl.o;

yaftsl-;

l)))

i{d'r.>tuothdrylti!ft nornlFl,Ol)

{ dnFr.oi

mb+)i

t{montlFl2 0)

{ ftGl.oi

rttfi

))l

rcIilLno

{

)

I

rld*.ri

i{d"p@|tdry{i'(dl.0I)

. { dGr.q

n|odrh++;

i(mdb-12.0)

I moortsl.o;

.ll

l(3€l!-'d)

@no

qEodD-12-0)

( nofl.0;

l

Doll_ctee{i

2l

l

i(|enrd)

s.t6i8_n0i

l

t

{(3€c<1.0)

{ eF59.0;

lcnioLn0i

i(bll<!.0)

{ diF59.0i

hdc-;

i{hom!.0)

( hou-23.0;

dr!+_;

i{d@<I.0)

t

nto'fi_;

nin-i

i&Din<!,0)

{ DiF59.0i

i{ho8<t.0)

I bou!23.0;

i(r!.dtKt.0)

I !to#12.0i

l.

)))' uu**Il!(.*.orlt

i{dab<r.0)

I

!erl-:

i{no!.h<lo)

I dlo!tFI2.o;

dr!&io

,..

drrc!.d!dry.(in('dLl.0)1i

22

t

)

iqbou<I,0)

( !dF23r;

&l}-i

i{de<I.0)

t

mo!&-;

iftud<t.0)

t

e.dhl2.q

tqr-;

-. d.|ddrF[i!{DoodFl.0l;

,,

*!i€|{:

{

i{d.G<t.0)

{

i{Dolut<l.0)

{ mrtsl2.q

tlt.F.i

,

.

dd'nonrhdlyltidnondlt.0)l

l

{ f.c.0.0;

lo(|qFt ttqtt<Jcnn+)

i

j<Fjdr.,r4.()rr Eo.rr 4 6*r

jdFjd+ddrar[a{Ee. r62{.0)y(36{oo.z.o}

Drrrdo;

{Ciutiednq.t.r,Dodd...O.0}:

tF(dlt24s 15,.5.0yJ652J.0

8id_.te!(!tI

t 0d+245t5a5.0)865250'q

E_cod(r)

223

nmFsin({50_c/60.0).Pv180.0)-

sin(plar.PVl80.0.)rsinGdeltap.Pvls0.0):

denn<os(!lat.Ptl80.0)ico(sd.ltap'PI/180.0)i

i(rab(nmr/dcm)+ | .0)

{ h6s?c6(Ntu/d.m)'160-0/Pl;

moFsalpha!-plong/l 5.o-Cmstz€b;

mode=tuor-h8svl5.0;

hlwo-mnor+hstt 5.0i

i(onoKo.o) @or+24.0;

if(tuoF24.o) mor:24.0;

i(none<o.o) noner=24.oj

i(nonc>2a.0) none-=24.0;

i(ntwo<o.0) m$o+=24.0;

i{mtwc>24,0 dwo:24.0

sbs=smstzoo+nlqor 1.002?3790935:

brwo-nlwo+delratlin(tyer-1620.0)l/1600,0;

jd=juliandar(year,monrbdare,mr$o/24.0);

r=(jd-2451545.0y365250.0:

I

caph=ns+ptone/l 5.0-etphp.

axrs=asinhin(ptarr pt/ | 80.0f s,n(sdelrap.pt/t 80 0)

+co{ptal.Pvl80.0).cos(sdetr.p'pt/r 80,0),

cos(€ph' l5.0.Pllt 80.0)i 180.0/ptj

dexm=t(attss+50.0,60.0t1(osutar'ttl 80.0)

.codsd!lrap.Pt t80.0frn{caph. t5 Orpt l80OrJ t.0.

!cs=hlwo+detrah;

)

urss=utss-delar[in(lr€tr- 1620.0)]/3600.0;

!tss+=(acos(emdj/{cmj+hrrc/l ooo.0rr I 80.o/pl)/l 5.or:

rmcnBt24 0,

fo{lcnr=l Jcnr<4 Jcnr+)

I

jd=jutimdde{yea,,oonthdare,ner:

jde-jd+det'a(in(yeaFl 620 o, tt )ilo.o. 24 o ).,

stjd=juliddarciy.tu honoldale,o.0r,

nlsUd-2451 545.0)46525.0i

F(jd+2451545.0y365250.0;

214

)

nmFsinc(so 0/60 of?V l 80 0)-

sin(;lar'Pvl80 o)rsin(sd€l$prPVl 80 0);

dem=o(iL'Pvl80 0)t@s(sdel6P'Pvl80 0)i

iflfab(nud/d.M)<= 1.0)

I

hN=@s(.w/d@)rl800/Pl;

mot=sdphaFplonS/l 5 o-gnstzdoi

nontmot'h4Yl5 0;

ntwetuot+hlsJl5,0;

i(dot<o 0) nnot+=24 0i

i(l@1>24-0) nnor*24 0;

i(mon?<o 0) mooe+=24 01

i(oon>z4 0) mone'=24 0;

i(nt*o<0.o) mtwo+=24.0i

i(olwo>24.0)nt{o-=24 0;

slsrgmsldo+mon tl 00273?90935:

none=mone+d€ltarlin(lyed-1620.0)l/3600 0i

jd=juliedate(y.e,6odhdat,mo.e1240);

F(jd-2451545-0Y365250,0a

su._coodc):

@ph=srsr+plotrg/l 5.oelphap;

altsFbincid(Pl51Pvl80.0)isinGdellap'Pt/l 80 0)

+cos(! lat'Pl/ 1 80.0)'cos(sdellap'Pv l 80 0)

'@s(qph' 15.01PUI80 0))' I 80 o/Pli

ddlan{(all.$50.0/60.o/(cos(plat'PI/180.0)'

.os(sdehap'PI/l 80.0Xsin(€phx l5-04PU180.0))yl 5.0;

ulsFhone+delrmi

)

ulsFursFdellallin(ly.aFl 620.0)V3600.0i

uts!+=(aco(e@y(eoaj+hite/1000 0))' I 80.0/PD/15.0)i

rac=drsd24.0;

,

llsR$r+4nlrm;

i(hsr<o.o) hs.F24-0i

i(lts!>24.0) Itsr:24.0;

fo(lcnt=l Jcnt<4jcil+)

{

jd-j'iiedardyeu,nonrhdate,f.e);

jdFjd+d€llarlini(yed-1620.0)l(3600.0'24-0);

nutarion0;

stjd=juliedlle(]dJnonthdat,0.0)i

st=Grjd-245 t 545.0y3652t.0;

sid-1imc(sr)i

t=(jde,245t545.0y36525.0;

225

mooi coord(i),

llii'i--.Ji. i:s'pv r ro.or-'i"olat'Pvr 80 o

''srd

mder@P'Pr't I 80 0):

dem<os{plaf Pl/l 80 Of ds{mdelcp'Pv l 80 0):

ifffabslnw/deM)'= I o)

{ hass=a@s(num/domrrl80o/PI:

'

Dnor{alph.Fplong/15.0_gnseeroi

mone{nol-has/15 0;

mtwr.fuol+h6J15.0;

i(rndol<o 0) nnol+=24 0;

i(mtoP24 0) mnot-=24 0i

i(mone<o 0) nonar=24.0;

i(mone24 0) nonc'=24 0i

i(mrivo<o 0) ntqcF24.oi

i{mt{o>24 0)mt*o-24 0:

sts!=sstzeGrmlm' 1.00273?90935i

mlwoattu+deltarlin(lve{-1620 0)l/3600 0;

jd=jdiandat (ye&,mo hdlt€,mtwoz4 0);

r=(jd-24s1545.0)/36s25.0;

eph=qrs+plone/l 5.0-malPhoP,

dlr t -6rn(simelar'Pv180.0)'5in(ndelup'Pvl80 0)

_

c;.{pisr Pvl80 0fLo)(nddlap'Pul80 (

'cos(eph'15 o'Pvl80 0))'l80 cvPl

d.linn=(ali5e0.l25y(cos(plat'P/180 0)

'cos(ndelap'Pvl80 0)'sin(@ph'15 o'Ptl800))y15 0:

!ts=mtwo+dellaml

I

uhs+(acos(cruj(eMl+h,td l000.0J f IE0 0/Pl/l5 0)i

)

ltns=utns+aniin;

i(xms<0.o) ltmsts24.o;

i(1lme24.0) llme=24.0;

for(lcnFl Jcnr<4Jcnt++)

{

jd-juliedate(y@,monthdat€,fiac)i

jd.=jd+d.lhrlin(y.d-I620.0)l(3600 0'24 0);

stdjuliddaG8q,monthd.lc,0.0);

st=Gtjd.2451545.0Y16525.0i

t=(ide-2451545.0Y3652s.0;

nllrsin(o.125'PV180.o)-sin(llatrPVl80.0)'sin(ndctaprPVl80 0);

226

d.m=@s(ptar'PvlE00)'6(md't!prPvl80 0);

,nfabsrnutr/d.M)<=I0)

I has=aco(nln/dem)| t80 o/PI:

nnotralPhaFplong/l 5 Gghstudi

none=mnot'hddl5 0i

d{erurot+hds/15.0;

i(mot<0 0) motl_24 0;

i(moP24 0) 6nol'=24 0i

iflmoc<o 0) noneF24 0;

i(none24 0) mone'=24 0i

i(mtwo<o 0) ntsoF24.oi

i(mlso>24 0)ntsF24 0:

$mFcmsrzso+mone'1 002?3?90935i

mo;mon.rd.lollint(lved_I620 0)l/36000:

jd=juli6datdv@,hon$3ate,6rwc'/24 0)i

t=(jd-245r 545.0)/35525.0;

n@n @od(t);

caph=shr+plors/I 5.0-halPhapi

alimr-asin(<in(plaFPl/l80 0lrsinlndehap'PVl 80 0l

+.odplat'PVl8o-o). codmdeltaprPvl80 0)

tcos(eph'15.0*Pv180.0))rl80 otrli

deliaml(al$4 12t(codpkePvl80 0)

1@(4d.ltap'ryl8O.0)'sin(caph' I 5-0'PUl80 0))y15 0;

urrcmonrdcltm;

)

ulett((a@s{enaj/(cmaj+hldl000 0))r 180 o/Pl)/l 5 0);

liEcatmr,z4.0i

l

rtrutmfrzonlmi

i(xnr<o.o) ltmr+=24.0;

i(ltmr>24.0) llmr=24,01

)

void dGplay_sord(void)

{

sd=N i

Sotoxy{16,1olconv.rt dn{ilons);

eotoxy(16,I I);conven_dos(fabs(sla0)r

il(slar<o) sd= S'i ele

soroxy(16,12)iprini(' %.21{Xm dn);

i("dclb<o.o) sdrs, ele

gotox(l 6,1 4);@nv..t_bn{salpha);

sotoxy(16, I 5)iconve't_dms(fab(sdelra))i

ifid.lllp<o.ol d S: €le

soloxy(l 6,1 6)i@nven_hft s(sho);

gotoxy(l 6, I 8);mnrcn_hnB(sdph,p):

sd'N r

sd=N;

227

gobxy(l 6,1 9);conven-dns(fabs(sdelap));

iotoxy(16,20);co.v.d-hds(sha)i

coroxv{ 16,21)i@nren dmstsla)i

;oblt(I6.22 ):@ven d6l rabslsalt Di

i(elr<0.oJ sdrBi.ls. $:Ai

lobxr(I6,2axprinrn'/..a1rd nin .sFidia'4 0)i

I

loid dhplay-mcooid(void)

{

i(mtat<o.o)

sobx,{l6,l0r.on(n dns(TlongI

sobry(16.1 1),.onven-dns(labs{darr)l

!dnd( ',6c".sd),

eoroxv16.12,:Dnntt( % 8lf ,mdst/em!j) &Fmdsr:

;tmd;lia.o.or sd=s', cl*

.oloxytl6,l4),on!cn hms(nralphal,

;otoxy(16.1 5,.cDnven dms{fabs(mdelta))

ed=N i

soro\y(36,1 6)lconven-hos(rnht)

gorory(16,18).on!en hm(m.lphapli

,fidelkD<oo) sd=s: else

e;ro\ril6.l9 r;mnven dms(fabnmdeltaPr',

sd=N i

sobxy(36,20r:@Ncn hm({abs(mha)):

Robxv(16.22);conEndd{rabi mdlU:

;ftmah<O.oj sdrB ebe

sd= Al

SotoxY( ]6.21}:@nven dms(m@':

goto\y16.24,:pnn't "" 4lfdc mrn" m*midrd2 ur'

J

iDnmniDu4. \n';d1""dl"idr""d l ocnl iniidate'Jnimon'h' rn(vean)

brinttflt4. %5 IlAt%t Il^f/"d\".plal plon&in(pall)):

'ta'irii"-l."s.s.

r rn'v,s. r trr," a."d\".flmp.phu.,n(urst in((urssin(ubr f60 0)l:

frintfif;J4,'%15.t^f/'6 2llL",mjd Cd._nojor2a 0):

r.rinrntok4. %6.2lnlo/o6 2lAt',tlim$ltst'60 0'elongp)i

bnntn6t4. 0/.6 2tlt'/"6 21fli .mah_salr.s.4_m@)

ilaiirirld..as.rrê,"4.:tNz8..lllt.wid'60 'ri."iiiiiuo.""

0.qvalohlri

pac ort""s 4lNToa 4lrrt 9 5lft"phrtemda-mla(mlon&srong N

"s.alr

[i,,iiro"n."vo.zrn.z*.1 tfl *,,8 ] lt\r./,8.11{\ro^8. 1ll\e.6. 1U\".25.5-

msFidiat.2,day,d.vl Bkv ul(v_rdavrstv_davl )i

fDrinr(fpir4,"o/o8,lllvo8 3lAf '.nma8 len)i

22E

)

{

ip nt(iprr4,""/od:%d{'/o8.3lo\t",in(thou),in((lbou'in(lhou))'60),nma8-len)i

'

{

falt=malt+.1 ; fam=hem+.1:

kndisl=9o.o-fali:

Glomonlaco(sin(matl'PUl 80 0)rsin(fal1'Ptl80-0)+co(malt'Pv1 80 0)'co(i

ah.P/180.0.)r@s((n'a-faa)rPvl80.)))r l80.o/lli

relonsui=(acos(si nGan' P V 1 80.0)* sin(lalt'P V l E 0.0)+cos(salr' PI/ 1 80 0)'cos(ral(

iPl/180.0)'co(G@-ran)'Pvl80.)))1 180.o/PI;

sacon=1.0/(cos(zendist'P/180.0)10.0286rq(-10-51@s(andist1Pvl80.0)));

aacom-1.o/(cos(rndisl.Ptl80.0)+0.0l2lrexpc24.5xcos(4ndistiPI/180.0)));

oacon=pow(1.0-posGin(Endisl'PVl 80.0)/(1.0+20.0/6378.0),2.0),-0.5)i

fo(i=ori<5;i++)

{

kFo.1066'ex(-palr8200 0)rpow(Mchlil/0.55.4.0);

lc=o l'pus(*6chlil/0.55.-l.r)'dptpalL/1<00 0\

ka=kaipow(1.0-0.12llog(phunVl 00.0),1.i3)

'(1.0+0.13'si.Galpha'Pl/180 0).(pla/aab(plat)))i

ko{2$htir(3.or0.4'(plarr(PV180.0)1@s(elph.tPUl 80.0)-

@s(3.orplaCPvl 80.0))Y3.0;

kw:Brehlil*0.94.(ph$!i 100.0)*cxp(pteDp/15.0)iexpcparr/E200.0)j

li$h[i]=kd.ka+ko+kw:

dnshlil=krSeom+k .dcon+ko.olcom+kw.gacom;

)

npo$i=1.o/Go(EndistxPVl80.0)+0.025rexp(-t

ii(mah<=o.o) mhpos€f=4o.o;

l.0icos(zbdist*Pv180.0))j

nnposeF I -0lcos(90.Gtu1r). PVI 80.0)+0.025rexpt I L0tco((90.0-

mlrxPyl80.0)));

ir(elt<=o.o) snposef=4o.0i

snposcf=1.0(cos((90.onatrPr/l 80.0)r{.025.ex(-t 1.01@(90.0-

ex)1PVl 80.0)));

lb(i=oj<s:i+)

{

.idrb=boschtil.(1.or0.3'co<6.283.(ye&j992.0y1 1.0));

nigb|b=nisntb*(0.4+o.6/pow(1.G0.961po{sir(andistrpvl8o.o),2.0),-o.s));

nishrbrightb.pow(l0.0,c0_41t$htiirslpost)i

oMas=-12.?3+0.0261rabs(l 80.o-etonsp)+4.0.pow(180.0-

elonsp),4.0).(!ow(t 0.0,-9.0))i

mmag=nnmas+.nshlil;

co'@=pow(t 0.0,c0.4.k$blil.@pceo)

229

fem-Doq(IO0,(o I5'telomorv400)|'62'pov]l100 ?O'/po$rlelomor20l:

I80 0) 2 oi)i

ii'-i..-po*tro.o.s xr, r

'oo+Po$kos(felomon'Pv

m@nFrou4lo 0,(_0 4'(Fmal-noscnlrl4r z/ r"i ..

doonb=;oonb 1l O pova 10 0'( '0 4+[shtr I'stpos'r I ))i

';j,f"nn",li.ilk##ir#niti#rr'niffi ;".r"''

,fouF owlI0 0,t0 4'k$h[i]rsnpo*n):

f.s=6 i r mw( 10 0,7 0)/po*1|.loNu Z 0)

lcs+=rov'( I O 0.(6.Itf.lodu 400)rl

ili=,J"-'oi" rid 6.i:e"

' ' oe' pow;cosf'ro^nqun'Pv

da\bno$( |0 0,( 0 4'(mshtrl_o6(nlrl*) zr,n

davb=<t vb'( I G pow( 10 0 { 4rk$blil'slpo*r) rl

d.;b=lr;s.ctoud440000.011.0-crourl)'davbi

,r,aout.t*itul b$hlil=nighlb+dalb,

cle' b*hl'l=nielb+Nilb:

ifihalPoo) b(hL'l=bshlil+m@nb:

bllrlil=bschlilipo*(10 0.12 0)i

I 80 0 r'2 0)

L'=o*n,ruo.oo,,rr',*oo,, t',

;il"fi;iil;i" -jili.TrJ,,gf*,,'; :l:::SXll3:::i:l l

;il=on;.po*( l .o+;oa{cr*o'ber.0 5r) 20'r

!cn=-16 57-2 5'log(khy'log( 10 0)4nscnlzl'

2i0

APPENDI'(.U

ANCIENT, MEDIEVAI, AND EARLY 2OM

CENTTTRYMODELS

23r

2!2

23J

234

215

236

2]E

2t9

240

24r

I

2A

APPENDIX.III

PHYSICAL MODELS

211

244

!.I9

245

26

241

244

249

250

9!

!!!

!.39

!4

4!C

94

-9!_

q4!!

,-!!:

,-9!l

9lJ1

9-41

25t

2r2

253

NB

91'!

254

APPENDIX-IV

OBSERVATIONAL LIJNAR CALENDAR OT'

. PAKISTAN

200G2007

AND ITS COMPARISON WTIE

TSE VISIDILNY CBITERIA

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263

APPENDIX-V

F'UTURECALENDAR

DATA FOR TFE OESERVATIONAL LUNAR CAIEIIDAR FOR

PAXTSTIN

BASED FOR COORDINATES OT KARA'CEI, PAKISTAN

LATITUDE2"5I, I,()NGITUDE6/3'

PREDICTEI' OBSERVATTONAI, LUNAR CAI.ENI,AR

FOR

YEARS 1429 AII TO r€1 AII

(200t AL _ 2txl9 AIt)

2A

H

t

i 3 t I :

3 ?

t

I I i

6

t

5

t

9

3 t f

3

s

a

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2

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:

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E f; e p a g

N s. a s n

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3

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p

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f € ts

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c I R c E E

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n

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5

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n & s 3

q R a I B

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a

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s

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272

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2a1

Visibility of Luna and Lunar Crescent

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