Luna and lunar visibility by ALISLAAM

ON I'HE COMPARATIYE STADY OF MATHEMATICAL MODELS FOR EARLIEST VISIBILITY OF THE CRESCENT MOON AND THEIR MODIFICATION

Cdndidare:

M UI IA MMA D SITA H ID QURESH

SupeNisor (D..) N.sidddh Khan Prcfesso.. Depadmenr ofMarlEma cs Unive$ity of Karachi

Prcscntcd in

prrtiatfulfitme ofrhe

requircments for rle deerce

DOCTOR OF PHILOSOPHY ptder&, Aslrophysics

Ar Insliruk ofSpae &

unlversity of Kamchi, Kamchi September, 2007

CIiII-TIFICA'TE welcc.pllhe lhcsh

cortunrjns to the rcq0iEd $afttnrd

CERTIFICATE

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

a-w"* ,/U*sql

.',.hs la' t r- aanh,d$

Ur/

,{' \4{t 6{" iii

The

tnU'onaneofAltah

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 *hen I

and counlloss sugg6tions dsht from the bcSinning rill rhG poirrt

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

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 cnconmAcment during

$is sort. lhanls !E ats due

aor

Mr MulNflrnrad

tor reir kind shotrl Odch

n nleins rhc wcbsllc NN\.icoon)ior! dnd Mr (hitid shorkor \\\rr!r!]N4lt!!!!.9r1!rr. prpeu aDd.laln lbr rhn

stccral rhdnlis

llr

uorh

\orl

lhc*

rcnraincd

! eurcc

"ohsitc lo hrve b..o posibt€. l_asr

Dr Shlbbir Ahs.n

$hoe {aluablc nrge$iions helped

Lahorc Universny

nriNgilg

ot infomllliotr. rcscd'lh

nor rhc leasr; I subnjn

of M.meeNnr

Scienccs

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

ar an, site of observarion phenon.non

li.

!.!ieRtd alohe si$

3en ti6t

ImniB imFnel

lunar calcnd.i. In

can be

Efrained an opch pobtem since andquirr, The for beeioing ! luw mon$ in a puEly ob*rvariomt

n6! chapler

the asrrcnomicat pdamercn relare.i

lolheprobtcnr

are

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

.strcnoncrs of 2orr'

Solvins

ch.prf Eviov or rhe conlrib[ion of thc cenrurr is pesentcd $al b.8im rvirh thc etoFical toodct or

fte pnblm ol lhe ti6l visibilny

lenedy colcrlations and thc

oa lbe n.$. lumr crcsccnr inlolvcs

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* trsc

Moon. rh. risinsand seu'hgol thr Smnnd rhe M@n_ posnionsofrhc sun,nd thc Moor ar 2n! tnrc on rhc rc day aftc. conjL .rion .t.tc ch.prr cnds rith a brict

dll or

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

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 crncrion.

wilh the prcblen are.xplorcd in nrore dclaitin oder ro evalu0le the Lunlr Ripencss Las o.0 sMcess, suggc$ed in lhc medieval e6. Son nodincado.s to rh€ la$ aru ate 'ls 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 simpk Babylonian

c ldion

significanr findin8

of the

chaprer

th€

modcls in thil

borh lhc

and rhe Lunar RipeNss IEw a!€ morc successfutin tcrms

thcn co.sistnct Nnh rn. posirile sghrings dords in comparison ddllopcd in

h $al

b fie hodeh

fiBr haliof2(]ucenrury Ascroa46Sobsonationsisuscdforleslinslll

{ork. nr.r incldc obseNations collccted

ln conparison lo lh. empnical odcls

since thc

nrd to

latcrhrlaof l9'i cemtry

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

Schacfeis model calculales th. octual

lin

lull

Mdon and lhe

lriligh

slv.

ins mdgnitud. oaihc skj and $c ntgnnud.

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

of rhc cftsdrl

tcsls a

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

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

$c founn chapler a emtce} is iEmed b *di dle tulhcnticirt of a claim of siehlins or ne\ cte$e on lhe bsis of a rmi_enpnical modcl and th' -fhe sisnific.nce of such a stat€s) hN b€cn highlignred 4s Nasnitudc conlras oodel Al $e

rherc

end of

found d nuob.r of authenric ne\r

coosisienr wnh a

cG$c

*ni-cmpilical nodel tn lhcse ces

risibihv claims thlt are

nol

semi'copnical nodel des not

cltsm wilnoul oprical This happens s aemi€npirical

allow visibilil, ofthe

aid bu rhe magnitude conkasr is in f.vour

bl. i.to considcnrion lhc elsvation of lhc sne of obseoation abole sed levsl md dr wcarher conditions t-he ofvisibility.

modcl

dcs

nor

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 location

oflhc {orld

applietion in the

and lo

fiih

lerify

chim ofcre$on lhibiliry. Ap.n froh $is prirlc

chafler rhc sedi+mpi.ical modch arc applicd

lun

devetop a

cascent. The phenom..on

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

hchnique for cilculdring the leneti oan$v obsc(€d

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

p@ti$'l obsen.t,onal lun.r cdendar i,r Paki$an for lhe

r. lcril}

lasr sc\cn

thc act(dl

s. rhc nDdcls

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

and thc

''PEdicted Obscn arioml Lunar

C.leod.i forPakisan.

A sumntr' ofrhis Nhole.flon h preseired in lhe lasr ch.prcr. issues aE hiehliSlt

d $nh a di$ussion on rh. aurur.

scope

v.nols ihpoarrr

of Esearch nr the

arcd

,i tu, L Lfz/& L J \ rE -, p p,f.,/- 6 e.t ) r-l f-.g tetL 4 6)er. ( F,J) i\v1,r' 4. ?rr' 64 i, t ) :i v i v L L./t1,' (t t)I!,y.- gl = 4. â&#x201A;¬6

,fz1

r,r"

a t, r e. u

i. z-tLl- ct g a't;' lo j 161 u ! r' 6,V. -[]c|

d,j' lr )'a t )' a.'.f4)!,J. +WJ*

;" iv t " r,.{(" F\t) it4vJ r.l

!'i'; 14

r't

,p i L -r. z.t L,r l,-t tt,1,J,""4 a/iitt,t,,a.- tf ll ,iv ( tt),( J t lu e.r6t L ,J^" q': - i r- ;- l'!I' L <c*"nst*n) .

-V- v J-) 3 v { L,U{,L,' t',} i L J t,l/ z'et tt' - wt f-,ft,' -rS1 Aa4- cl )t* V:. vf i A'!-$ .,,'.,r:Lt' 5 A 4J', l-f'a le'' ra'Ya -'{,{,k L ;v'tt,,,L,f e :,,4 L,tt tu, L,f,,t -,),t *L t ts,, 'l

J'V't

J,t' r',/ h I

t4")trLiq,/t !)()-lvtQ.? d6'{p t ri. +6/,* !. 1 4. +W e c t L - u.t t) v /' r-- {,' f-,. E,e * ll I L {, * r! i{,! 4 ..- );t,- 7 ",t 'or

ilf'LLf' +

4t ?'(

,),.t

c)'i

( ),1 -t z-r z,lizt ", J,t ] t' - tfu O.L,.l' q' )a L,.a. +f- t@I., t

it+Vr{-

J!-

-ftE)6t2!-96-tt 1

d,J,

'9,!,r

G"p1c'st a x

!lr-t'

- iJr,,'v

-t

t,'1:r;i

rU,f f-

r' j-.-yry

Laffi /:t g a,"'.,1,f'S'.,st--tr-

( t)Vhftnt tunar Ripeness

:4ttr h ct ?, L

ef r, 6

t ) ),Ji;.r [ 6,",t- r-.t

L,l,t,t-'< t ! rt tfJ r',,t g,j.t,'v !- - -i i --,. y t,1 t..,-. " L v L,,a.w1 t t6,,)t,v.;,-r Lv L u il, JqJ it-l t

'U't 1

J'* ) rV, it'ts" affi {1V l,ilr 1tt',, L r t,t'. et c =V J-,. t.r L I 16 ) &.4 E q |

Jth

a)-Ii)+

1,,1

;9,f,t,,! i-st-

f1,!,+,L J+A)1 (y ) 6,, i? i,J + v L r l t,F- /" )

L 6.a,t,<

iQ,f-- d,Jr ;- *,.r,,t ,tc,tui ;"tLi..+fu&

p?,:b

*fu !xJf+:V.{-$

{,u;,t,t L |.s,cnaeter:lf E,!J "'-,-f',t4'tovi L lL/i<-G', t,t JeF it L=V i zV pV'!J iL 4r!,t i'/- t'L J'\ r" L4.r2 (4 "r,t " (L<y ano .l V,tLt)r. l'L L lL e;.,.,ri -p.- u il J =,1,'

r1,r.

" "E

+, V d t;-,/ +s,t tf,,

--,. f, â&#x201A;¬_ d6ct tP{ L ;c/- /, r

11 t!>tt t.,L c|u,e,,t,t 6,t,,,{ L l+t L.a)! i,t,,r { d' t At a : i L /}- }/l9_ tL j"<,V[ 4{ E (r;,tL1\,J, .+Jt ritt:,tJt-Ai. J !-,udJt.,J/ $ ;u-' g4vLu Lg {)*,,J..t } :f/,t }4a-,.fv ,-.r r f - u t4 aL,' + -. {L l1Lt, LV 1 tfi" s r,,L 6 jlt t e ?V ,'t.{1tr l,.t LvL.;p-,';,! -,4v,1,4, 4,tr. : t,g, 4,,,,v .8-

.1_

6tt n'L'4,L,)

Jc,.r

4lst

r, ;9

JaX4,, 662

;

1 2 7fu,k,flt 4 a i iv,L cu _ t 6 v {J*' 41 t t, v,I;- r'L j i L 4,, L t& /t, i /_t tt t * o t q 1 r+ d,!4'/, -. o,,!!j4, it' (4t,t 4 t t4 6, v!*:u- z- t /t (J t L,.t '.P-H o j.i (4 ) E,'L : Vr,!J U,\,L _ J3 i,{* ',r 1-'J 3L 1,

t:'

{' rtc 6r t 6A 1,r.

4 6 ;75-. _ 6 .1- r/svt(.hssr ';g;: r 4* *ut t',!, aa 1 v : ;,&,

fut'.r'+yr,,fuov - - j,,,,,':I j4y.q,,,,y. ;t

N lt J'C,,t djt I

q)a

vf Ll

.,u:

jy;

{U.,J,. c,,/+v, ri)

},,

/'t' f'L ))

72' A ety

{ d-- w L,t a,,} t LJ 4r1)l!1- "

-ffiS, 4 t'V rlu,l,f-,t' 6 4 /t-g,,u L s o tlt a Lyl +'180 "t{,./J tU. E { pvto{A + 1- *6,./.+ d,! d' -l 2,,!'l'J 6*t'z.t +'5. 4v t;- tt X"'"Pr. 7w J. lfoL L ig, 4v t z- {6 rfr ,1 I ;'1'4, : 5,1,u SW.,E,fJ.z i V

,1.4 f ,t i.t +r. + i/f iJ tt "'v{r {ia=v v't*,! 67) 4 {,y''!. E < / ( F,l,t i, ov,!,J od t'-r,t,, o v } lJ t' to - L; A ;it {'4a \t,ss% qu,Lr ( $,f, u !r'& Jw..Ltl

,td.

W"bf t,t) i' +v &',{s d .il-,, t

twp ? d,!ftF, r- t;4otqtl & tttt i = 7 d,S' t/a,.,, F{,-F-a o t,p.a z-''+, -at ",!

CONTENTS

Urdu R.ndenng of Abslracl

Introduction

Ll L2 t.l 1.4 L5 2.

P@nd.6

olNry

for visibiliry

sienillcmce o| ceogdphical Tbe Rut.s oaThunb: The

Crl.ndas

Lumr

Cr$chr

calion

Ani.m & fi. Medievrl

and CelesdalCycles

Aslro.on6

Astronomicsl Algorithms & Techniqucs

condbution oflh€ zod cotury

2.1

2.2

Dyne

ofMmn ed

|he Eanh

2,t

4i 2,5

2.6

2.1

2.8 2,9

Coordindtes of rhe M@n

Caenl visibihy Pard.ic'! Thc Softve Hiblol.qpp Nes Lu@

Ancicnt, Mcdicv{l & Etrly 20$ Ccntury Modcls

| 3.2 l.l 1.4 1.5 l.

'ftc ltabylonian Crilcri!

Some Sph€ncd Trigonomebc Consi.lcations

Thc

Lud

Ripehess I,3w md ils

Ehpidcal Models of Edly

ConposonahdDiscussion

Modiicdlion

CenNry

t03 122

Phyaic.l Modala & their Evolution

4.1 4.2 4,1 44

Bruid

s Physical

Y.llopl

125

Model

t27

sinslc Pamel.r Mo.lel

Srch.f.ls Umitiog Magritudc Mo&l A Nry Crirrion For vilibility of Nry

l3? ta6

Luc cEsdt

tt9 t67

Applicrtion

5.1 5-2 6.

t'|2

Lâ&#x201A;¬n$h of

l7l l9l

Cd!c.

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

Dircusion

t9E

Th. sonwm

Hihlol.cpp

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

205

Models 2ll

PhysiolModcls Lwcd.nd.r2000'20{t Futurccalendar2ooE-2ol0

Th.

24! 255 264

21t

Chapter No.

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 of lrorature that delcribed and 'oscent. 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

PARAMETERS FOR VISIlIILITY OF NEW LUNAR CRf,SCDNT

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

also aaf.clcd by toan)

gEln

ional

oflhe Eanh. bur is

rhe ncishholiing cetestial objecN (Danby, 1992).

\\,nh

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 2,425 ,Variarions periodic rems in vicw olrhe S.culair€s des Orbnes plandaiet, rhe Fiench pr.netary th€ory lnoM as VSOP87 (B&raSnon & Fl!Bo!, t988, Meus' 1998). The

rqdB

accurat€ d€terni.alio. of lhc posiiiom of the

Su dd tl'e

explorins |he circunsr.ee of rlE vGibihy of ihc

Moon is tbe

!a lud .esnt,

fisl

slep towards

Handling sdh a

laqe hnmber of pdiodi. terms mak6 this fiEt slep very ciucial. The lest of the sludy dep.nds on th. Elativc thc aunsel for

pcitio$ of

lhe

and thc

M@n

rhe

hod&i at md aner

a.y place on the &nh.

These theori€s detemio. lhe

obj*|' fid dc b6ed

c.l$i.l

ecliptic coodiiates of the sold sy$en

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

olsliptic, iound the Su, The x-dis poin6 in the diEtion ol lhat ofthe Sun. The xy'plane

the plane

shich the

the vcmal Equimr

horbi6

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 ollhe

tp. 0. 0)

lo4.

4oP 6d e-zzo?.\i\ee P ist\.

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. prcjccrion

position P

Fig.

No I I I

SuDpoe

..d

('-.r",/.)!d (.,,r",t) E

p6isc

dist nc6,

etipri. tonSnld.

hnud. of lhc Moon .nd

rh. d.y of @njucrion. For

thc

Sltr Bperiv.ty, Ff.Gd ro fi. nsr cquircx ol .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.

rinB of rh. lcat sunsa ud rhc hooEt, ihcn d. visiblc only ilT, < To, This La4 to lh. p.md.r LAG - Tr Tr. TUC) bc lhe

Using th. cclipric c@rdiNl6 ot |he sun

orho noD.ni of tiD., Oc

(a..t,) cd

thc

G$cnr h.y b.

Mon drt.ut.l.d for rh.

1..

cqq|oi.t @rdin c! of rh. |m bodiB (a.,4)!nd

oh.ircd. Vhcdd.

rhc nght

aeBion k $c

djspt@mcnr

of$. objer

fbm wdal .qdnox .long lh. cctcsli.t cqu|or (in $. q. qudtut s I ). md d delin.rion or rhe displ.ccftnl of$. 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

lo almosr all rhc nod.ls

fo..&lid

- ,- -/'.,

de sun, 6!..riv€ty aE

hoo!-nddng, rh. mj.nt

mod.m, th. difieE@ ofeinurlrs (D,,tz = l!r.

rltiluda (ARcv

and

- /r.l. can.d rlodvc

s rcll .s lh.

szimurh)

.id

thrt

$ 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. c.llcd

of vision)

lor €nnien other

frh

Elni* &idrh, rlE e ofvision !d tn. e oflighq dE dirc.it visibility of luE d.gnr r€auircr ro aat it|ro @mid.6iion a @dba of

Ap&1

rh.

pm$€rd. On. e.h p@n€tq

ihe As. ofthe Moon (AcE) =

th€ tine elaps.d sine th€ la.t onjuncli@ rill rh. doc

of(rlE sn!.r

Tt or

To d.ffned

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

ninutF io | 65

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

T-.

B€dTimeofvhibilily

AC€

M@netTr Arc'ofvision Relative AziDurh

Arcorlight(Elo!8afion) ARCL

10.

Pbseofcqc.n Widlhofcrcs@nl

Age oflhe

5. 6,

ARCV

DAZ

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

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 New Moon or conjlnctio. or

the* coodinat.s

lGal horizonbl c@rdinates. Tbereby coFPute tnc LAC,lhe Age,

to lhe

ARCL, ARCV. DAZ and the width W. The ddoih of all

t.2

$6e

computations shau bc

SIGNIFICANCE OF GEOGRAPHICAL LOCATION Fo!

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

hoizon h significdnl. They chatrgc depeod on thc latitudc

lhe sly appsr 10

scason

of lhe place. Due

hvel

lo seaon as

\tll

tlarile

to lhe

as year lo ,car a.d olso

fte axial rctation of lhc lanh cvcrv object in

along a cidular palh (The Diuftal Patb) ext nding frcm ealcm

horimn ro thc $'eslm borircn. Thn parh lics in a plae

pellcl

equalo. Tlc obj*ts in ou sty sbos. delimtion is coNLnl

(sld

lo th€ Plane of thc

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

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 olh€r. Thc Plan. of the orbn of lhc ednh

! 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& durtng

Ap.n fion th.

plr6 chc

v6y clole to tE ho.izon. Fo. pb€.

dr Gqld bd!

d$.

tt Su!d

thc

Mmmy16€l

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) .

\l.rsllldib

1l'

Figur.No.1.2I C.l.eid SdE fo. d

obffi

in hi8lj

blnud.

hid ldiod.! vhcr d'c dccli.dion oflhc Sun k eulh (wirt r in th. mnltfr h.hirph6c bctBa Sumd $lnie rld thâ&#x201A;¬ 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@ Thi! figue clorly 3howr th.! for

Hdnd FiglE No. |

\r'

2.2: Cclcadd Sph@ &|r

D obr.ffi

in low

lrlind.

$. fg@ m. 1.2.2 dbE th. ddiE dndllod ofdc ..liptic !d equ|tioi ii oonpcilor to thc hodzon for i pLc. with low hritude ai {@rd

Sihihrly

@lqlirl

.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.

wrul

e{ui@r (eliplic in ligln blw)

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

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

Tbh is lhe hain rcason bcbind @nsidedng Age of Moon

visibilily of

cF56t.

dudng th. {ullmn and

In sprinS.nd

whcn lhe M@n is $ar ofdre Sun)

wsl

not a vcry good indicabr for

Mm€r vcry t$utrg.ge

*inle6 very old clescent

hemisphed. Thc situalion is rcveB.d

s6odlly

of fte sun

cltg cd h. sn

lnd

may .scape sighling in th€ holthern

for de

(el6rial

sud@ hmisphec,

loncnudc or rh. M@n is

noc rh&

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

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*. b.

in lhe mominss befoE lhc

Whcn thc cresent is very close ro th€ sun il rcmains invisiblc

$c ohospherc close to lhe Thqefoe fi.rc n6 to

be a

sun remains highlt illnminated evcn

to the facl tbar

afi( $e

sunser'

between

fic Sln

frininum lhrcshold epaEtion or Elongarion

,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

kn lrom thc Eanh ind !l a Ne\ Mon ar apogec mat be *cn

valyine. Vhcnclossl (at apoe€e) $e Moon h tround 350000

fdhat (perig€) il k aound 400000 lo. Thus 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

lhe

brigbler ir appead in oui sky ahd wheo il k farrher il appeaB rhc

elongalion. Thts .nother

inpona crir.rion

for a combinorion ofcenain optinlm valu.s

fom

fi.

E.nh.

A cFsenr

thi! conbiMlion Esults

wi$ small €longation $al

and less biShl at

is lhar $e New Mmn cm be

ofelong ion

into tlE

saller

op(nm

and lhe

*en

distanccof$e Moon

valu€ of thc widlh

orcFv€nl

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 & 'rhe edli€st rcferehc. for

ey rireno.

TtII Mf,Dtt:vAL:

for lhc visibitir! ofncw luntr cresccnt is

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

994. Faloohi er a1. I s99 elc.). Most

'Thc

Nons4 loas

,.n 18

lunar .rescent

hihnles h.hi4d th.

lgttsa.

yas.

oflhe cxplom a&ibute ttE tollowinS rutc of $unb

s.ek

en its

aAQ

nnrc thdn 21

ho^ un1 L

tns.!

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

2006) in

lit 6turc. Sinilady. in modcm harc bccn

ftsc.nc

lagsjnS

o.tyjr

nrnucs behind $e

su.*l

wi(h aec tcaslhan 20 n6uB. Thc dala ser

duflnC lhis $ork

crcscd $nh

time

(b b. di*lssd

in ddait in chaFcd

of46j obsetuations considcar dd 4) inctldcs jt cascs whcn rhc

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

notion

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

In vdious $ud,cs lhe vrtue

rin deCrees,,S) - conninr

of$e conqant

dedlce ftum | 7 degFes lo anyrhere

2l d.8R<. Hokercr

&ound

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)

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, vaiied fiom 42 ninures for Moon ar perigee.nd 48 ninurcs for Moon at aposee ^Ebs

Thonsh lhis -nrle

of thunt. is no4

art'buled ro |hc Babylonims Arabs

sli

jr does nor

sophisricated

povidc

compNd ro $e

rhe complere prcrup of rhe cffons

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

spherical

risonodeq devetop.d b, Ambs

depends on

one

lead to the

ol rhe

L!n& Rrp€.es flnclion (bat

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

and lhe Mooh)

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

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 Flarively r€c.nr rimes, ho$v€r,lhc atiesr Hildu

appeds to have developed only abund AD 500. Morcexplicirnentiohsofrhcsederaited calcularions are

Ohe

loud

vuios plaes

in the

dly

Islmic litenle (Brui.,

ol thc ealiesl Muslin Aircnomcr eho developed lnc lables fo.

197?).

lidnC

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 the luna!crescenl s

(w) oflhe

crescenr. Thh not only shows rhat ar rhd

Moon had been rcaliud bul it aho

mdc ir

rine

possible 10

rhe varying dislance ot rhc

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

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).

ofreatiarion of the idjponance ofwid$ various Arab asllonomcrs lor celics( vjsibilil, ofcrc$.nl thc minimun equabrial $pararion offie

OD the bash

conclndcd

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) Sun

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,

sork

is o very

AtDatani,s compuradonal

elabo6re sysr€m of mrhehari€t catculalions (Bruin, | 977).

'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.

I.,1

:1.

CALf,NDARS AND THE CELESTIAL CYCLES

6 '!

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 Dogeel defines calendar

syslcm

eme aR dtccrly 4sociared snh $c celestial

/,rur.

A @nplele daily rcvolurion of lhe

lechnicallt an apparcnr Sun ar any pllcc.

r,/df.t,'

c_aclcs.

the

D,"r,at, \h. Ann@l

s\t. de DildEl norion.

is Efertsd

rhc

a day.

h the inlcpal bcrseen rwo successivc rransiN oflhc

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

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 Each calendar is dilided

iflo

dtree major typcs. Solar Calcndm. Sriclly

L@r

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

C.tcnda6 and Luni-Solar

Solsr

''ycai in

Cal.nd6 m b4d

such calendds

on the

ulrl

motion of lhe F-!nh @und the Su..

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 at

Genwich

on Janua.y

n 165.2421898 number

l. 4712

days or 365

on yer

of a

eld

heo n@n

Julid Cal6d{. Cwnlly thc loglh of t@pi€l ycar days.5 hos.48 ninurcs ed 45.2 sends. As it is nor a whote in

calodar @rsists of

crlendr cvery fourih yc& @.lained

cirher_ 365 days

166 days (i.e_ a teap

yed)

ot t66. tn old otbq

yff

Jution

conhincd

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 165 days In lnc

haBcsing rides, lhe length ofdays,lbc dmes of imir, suser, sunds. etc. I$e JLrtian calendar insrilured on Jdu.ry l. 45 Bc by Juli$ caM. wirh rhc h€lp of

Alciandlie stronon€r Sosigm6 &d

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 r nodificadon

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 October 4, I t82

ed Micb.k

196?). When

it ws implcn.nted ofljcialy.

(Thuday) in tne Jllie cal.ndar

rhe ddre

fouosEd by rhc dale Ocrcb$ I 5.

1582

(rdday)

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

Fligiou omhMiiics adaprd to rhis DodilicElior ar diff@.1 !inc. Ir is lheEtorc high hrk for hisrorifls ro tccp racl of r]E apprepriare dalcs. and

,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

ofl0

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 six

days

summ.rhalf@ ains I84

it&n for ih. sue is lie faler molio. of thc Eznh ro $; Su dar oeun on aound January 4,i. Srdnins

days. The

rhc Frihelion. thc Eanh clossr

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

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 a

days or 29 days, 12 hou6. 44

ninui4 dd 2.9 $con.l$(A$nrcmical Atnaiac. 2007). aEase is.hrging, Aaodin8 ro rh. tun5. lltery of Chaprcnl-Tou&, dd

HoEEr

ihis

chapont

lh.e

vdiadons

'lbur'ind Chapro

!E eoukd for

by the tolowing cxpEssion (Chdpon!

, 1988):

20.5305888511 + 0.o0oo0o2t62l, a_

I.o4! IO. o ! a. Da)5

(

r.4

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 29 or 10.

is eithcr

Arithmetic Lunar calcndq rheE

lllenate hontns of29 and 30 davs. tn mey 4 rhE. rcns{ulirc mon$s o, 2,

lue Rt.ndd $.8 6 bc s days qch od d mdy s folr conKud@ momh5 of 30 days e.ch { obiwrional

oys, I 994). In an

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

nol

ycs

obseNarional tunar calcndar. rhcrccdn

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 be

rule is rhal lhe tear

nunbf /is

(

l4 +

a teap

yer

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. 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 orhcr

pncnom€na rhzr dcpcnd on Ihe

eld

cycle {to

monlb numbq 9 (R nad&) in lhe Isl6mic

Thc Luni-Solar @tfldaB

follow a slmcfiy lu,ar catddsr Ttc

qtods my f.ll

wi.l{. slmmd. aurmn

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

ofHindu calcnds (th.y have bo$ $e $lar tud rhe luni-sotar calcndaB) lunarmonrh is imclcalatcdiwhcneve,irnrsinroEconpr€resotarmoilh.tncascotrhc

HcbFw

l!ni,$lr

cole.dar

addniomt honrh of

da)6 rs mrcEararcd betorc rhc

usurl 12" nonlh oflbe yce. The En ofihe dd.its of th6c calodd soc'al ahd reliSious nlrurc aod is nor in tine with tbeprcsnt*o,k.

morc

of$c

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

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

iron calenddcal excitins

fld

aspects

logirude. Apart

$c pobl.m ofsighting v.ry yolne cGsc€nt is

challenging obseFations for both

adareur

on€ of lhe mosr

md rhe pma€$ioml

asionofre6 Besides, thc pcdicion of fic vkibilily of a paniculer .erv cEsenr panicular placc is a long and inter.sling cohpuhrional exerci$ Thc sane was as early as rhe

dedieul times by

ar a

clliad

the grelr ostonomcs Al-Khwarizni, Al Batani. At

t|t under srutiny

Farghani etc (Bruin 1977) Mocovcr.

prediclion foL nak€d eye obseivarion ha lwo

noE cohplicated hsues

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

l'hdk h

morc on thc $lrcnoNical asp..rs

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

Opinions aro diided

tq thc oriSin

of$c.llton olcomlins

in rhe ARbian l,cninsula. Accordihg to At ttazsi Prcph€l Adan calehdar

nuliplicrl

and spRod

aouid

rhc

il $anln d

plnlr

yer6 in an! tofrr

as soon as

$c cni rcnof

(Ro$nlhal l9i2. l_mqi t979) A

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 months and thcii sequenc€

Mre thc snc

calcndar Ibllowed by morc rhah

tho*

namcs

ol$c

used in lhc cuiicm tslamic lunar

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 religious evenl.

exchanging bands.

ad!dced llxoush

calendar, Althoush this event was a

puelt

b$ ol

goods

was also inportlDt ior kadc and busincss wnb

ll w6

this econonic

rcrivil, $ar w6s badly anected

*&ns. PEcufnent ofcop l1

and rhe ovailabilir)

as rhe

lunarrear

or.ac f.,ul anim...

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

daca for thc coming

(Hahim,

yem

198?, Ahmed,

be the

fiIsl petson assisned lo derrmine thc

pilgrinage and wheth.i $e inmalarion

w6 drc or

nol

l99l).

Sincc lhe early days of

four nonths including lhe

lh.

inception of lund cakndd in the Arabim P€ninsula

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 change rh. saqed

ws abu*d i. ABbia

eEalalion

io ordq lo

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 the

lar pilgrimag€ ro Meca of

@ in pmllcl. Wi$

Propher Mubmnad in rhe lo,r ye& aner miSr.rion ro

Modina (AD 612) fequenlly abused pEctice of inicrcalation QuEnic injuiction. Th. pEcrie ofnadin8 a

abolished thbush 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

rhe cnding

of lh€ nonth of

fasting md the monrh of lhe pilgdmase.

With adoption of puel! lunar cat.nd& wirh a

fist

siehdns

ofrhe ncw IuMr

crcscehr lhrcush

luar nonrh

euMic

bcginninS $ith $e

injuncrion

Pophet Muh.mmad |he .rolution of Isl.mic lunar catedar

b.gd

lhe eyinSs or

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 aftcr Prcphet

u*d d.poch

Muhdnad mignicd to Madina

wid€ly acepGd lime ofomcial adoptjon of

in AD 6t2

Ht6 s

s ene dn6 ! mooin or tm (Iys,

! hore ofrhc Ist@ic cR n

1994). However

lhe bceinning

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

Simpl. 34heh.s or 4rolh odd appoxidlions b$cd on long b€.n dcvi*d to pcparc long

tne

d!t6

in oth.r

r.d

vid

cel.nd& in

lei-$lo ad el& olcndr

blanic legd syst h cgllcd Shdia

, is

Lm 3vs.gs h.d

ol incrconv.uion ol lslmic drGs

d.r.s.

lh. sou@ of lsldic tn+kcping

whil. le8.l pc.pls .rc efeglnd.d by seking

sisbne

ststn.

$. *i.nlilic kno*l.dge. Till th. rin. of sking of B.ddad by Hrl.su Xhu in AD 1258, Isluic law had fom

.ldr guidclires ao. elcndriol oNidcFlio.s (llya 1994). The olcndric.l sddelircs ftlv.d undcr thc Islmic lN qn bc outlin d a follows: cvolv.d

l-.nsth of

a lunar month is cnhe. 29 days

L.nA$ ol

iit

Tn rc can be

lumr tlar

.ith.r

154

dlts

or l0 days.

or 155 dayr.

mdimuh of 4 cor*cutiE monrh. of l0 datr @h or

conscutive nomhs of29 &ys each.

E4h new nonth b.sins with fiBt sightns ot ncw lu@ cGccir ovd k$em hori@n .n.r ft. l@l ss.i,

the

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.

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! puDo*ly nisled

!iii)

the

preved

dat the

Fen

pioviding wihess

pe6on ousl b. punished.

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

sky condnions.

Sighting should be caried our in sh

o.8ditd

way for €dch moflh.

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 Accunula(ion of emB @orriculally in view ofconsideling

toonth ro bc

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!

developmcnt

of $i€n€e of fie

CONTRIBUTION OF Tne

nod.h

@lid

2OIH

visibility of new

crcsnt dd

mde in

pEdicrion

CENTURY ASTRONOMERS

dcvclopmenl of lhe sciencc of erliesl noon_siehinS bc8iN vnh

fic

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

obFMtio.al work

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

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 reponed

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

1977)

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

funcion ofih€

dirnrl'ler

RCV ond DAz md a fircd Ddnh-Mmn dislaftc (in lemsolfixcd

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 due

nrodcl

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

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$ briShhess oalhe

ctrrvcs the bdehb€ss ofcrcscem (modellcd on os

$e basis of lirll Moo.) is al leasl

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

and

sl.t

depesion rhll

ldd

ds much

berwccn ARCV

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 polynomial tunction of

$e actul

cresc.nt

al$ onsideF ARCV 6 a ihi.d d.gtce 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

slctins ARCV for

a Sircn sidtn from

tlr ninin@ 2\

lbibilil, .uvs of

bnrin b!

oh lhe componding limilin8

visibility curve. So

slronohi€l 10

cri|€na

fd th.

b lhe morl oulhfltic and dep€ndable for cxper.d ali.sl visibilily ol new l$m ccenl. The modct due mod€l due to Y.llop

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

.ftiteriononthebasisofabusicd.tadeduccdfonBrui. svisibilitycurvcs.

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 Tnc mosr extctuive

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'

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

1991, Suhan.2005,

Th€ resulrinS cllendaricdl imp licariohs

.rc aho cxplorcd

hos been poinlcd our in lhe bLSinnine

determinihg $e dat when the nes

of $e chlpld thar the prublem

luar (esc€nl $onld

bc \isible has its calcndarical

inplicarion. The earli$t siShtine of new luMr calcnde is ale

maleuB

and Dlolessional

cballmging

l.st

lor bolh

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

Paranere6 on which visibiliry of new lunar cescent depends

Thc cfons ofrhe

6tro&meB

lhc lncie.l and lhe

di€ul tlnes.

Th. d.pend.nc. ofth€ dlendsB

. txB

on th. cyclic moiion

ofth. Moon sld $e

Suh.

$. Mulih! .spei.lly a lh.i. b.li.rts lticrty dphais. u tltliarsidtingorrhetwloNcr$.a fois.niosaldcldilglun&ndrha

c.l.ndar of

A! rccou ofrh€

eilons ofrhe

rsrorcmd oflhe oodc6 rim.s |o rd.lr.ss

the

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

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, Itr

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

Chapron!1943,

€91) rr.

th. lunrr

rh€ory ELP-2000 (Chapront-Tou# .nd

well sulted. A number of sotlwaE hav6

tor lho shulatlon of cel€stlal phenomona based on

ben develop€d

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.

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 Moreoler,

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

vblblllly ol luh.r cr€scent named Httatol wrttten

2.I

.natysts ot

ftst

C.t.ngs.ge dt$u$ed 6t the end

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

c6t cvery day sd iakes ovr

rv€sr ro

When de Moon is vcry

rhe Sun in

doud

every 29.5 dars on thc avcm8c.

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) on rbedayofthe conjLncton. Dqjon Linn(Danjon 1932. t9l6) has

a limn on Ihe

hininuo

lkibtc

b.cn i.$rpreted as

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

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 lhe dtoe

seen on the day

ofconjunction. TheoEricalty n is aho possibte rbar

if$e

seno.

ofconjuncrionad

ln. da, iller, orthc

the day

crekenr rs rasr

srn

rhen rhe new cEscenr

seen on

on rbe day b€to,e rhe conjurcrion .nd !h?. rh. new

crescenr

n hsl

c@dl

n *en 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 on the day

at least,

These condnions dcpehd on

obse(ets t@a on.

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

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

coNid.Ed

mwh

ths

42 hinutâ&#x201A;¬s.

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. TIE ddcmin.tion ot $c

l@l riGs

or rh.

$. 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 & ELP-2000

(di*.us.d lakr

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 rctations of th€ Eanh

ihcr

Sun ar

cEenwicb ptus 12i6. Due ro rhe in€euta ries in the

di$repancies

b.reen

the lwo rimes. rhe

TT and rhc UT.

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

AI=TT-UT Th.rcaore wh.nevcr

wanl

(2.t.2)

b dd@irc

th€ position of

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,

-4?t2 (cdlcd lhc cpoch oftne..r!lian dde,) ln rhis

r,m€ scare tlre moment described by a date (CEeorie or considercd as rhe

"nmb.r of

dayJ,, d.nor.<t

Julie)

ond rine (UT) is

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. tinc arghehl I lhat is on lhe

icguldnies in the oradon of thc Edrrb. The theories use $al. of".umber ofJulio centudcj ehps€d rinc rh. epoch r2OOO.O.

Thus usine tbn

ine

argunent md the explicii tin€ eries fomulas of the ELP

c@rdicrcs of lhe M@n d.y aAq @njudion-

th. ecliptic longitud. and

Tn*

c@rdinals

Mooisr

rhe Sun

9. the dcliplic ladtude,

i. lh ft sqt dsirely

ca*

d for

fte s.me

i6i'nl

lbe VSOP th.

of rhe day or th.

s. @ spheric.l pol& @ordimtes a th€ gtucsrdc disr.nce,

(or rhos of $e sun.ie

TIE lune

m calcrl.t

ad noosi*)

refeftd b

th. Ediprh Coordimles.

ro obrain rh. im€s of th.

Su$.i &d dE

for 1hc day in qu6rion.

(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

total dea of th€ Lund disc is called

di@tty r€laLd

nrio of rh.

of lhis

crcw

lhe.phs.. ofihc Moon, The phse ot

md lhc

th€ Moon is

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).

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 This

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

@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

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

lhen arNtomcd into Equtodal

Delination. In order lo g.r th. ihe

Su ed

the Moon, rhe

tical Hodental

ob*frers rcft$rial

(dcfin€d as lhe L@al Hou AnSle ro

fie Greensich

@dinal6 a, the RiSht A*nsion 4d 6,

oflh.

lhe

coo.dimt€si Ahitude and Azimuth, of

cooidinale

the

cal

SidftalTime

Equinox) aE rcquiEd. A simple alSorilhn lcrds

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

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

obj*t is

tical SideEolTihe

ofany moncnl is looM the Hour Anele oflhc any

oblained. The local Euatorial coordiiarcs

s, th. Hou

declinalion,lead to the Altnudc (heiehr obove lh€ holizon)

tine

ed 6. the dd Azidu$. n@ poiilsol Anele

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 co.siderinS rhe

tical

olthispointoflimc

Sidereallime

can b€

eo,

rhe

b.lhe Righr Ascension offte objccr. The *timate

inpoled uing

ireraive

pccs

(hal inmlves Eadjustihg

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

objd

offie

thisdme argumehr.

ConsiderinS Hour Angle (n€earive

io be

9Oo

or 6i" approxidare rihe of rhe rkins

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

of$e inct oflh€s denrs ac ob!.ined. This giv.s the a.d the setins orthe cenres ol$e objers (rhe sun and the Moon) thar can b€

the b.lter approximtion

risins

adjnsred |o s€t

$e actual isins (the first

app€arance

.nd acrual s.tti.s (homen! of dcappe&ance ofrh.

of the weslem limb of the objec,

cded linb

of the objsrl.

edor selting ofbodr fie Sun 6d rh€ Moon are oblained th. p.mete6 of fic import,nce for lhc atrsl (ot la0 visibilitv ot

Onc. lhe lincs of fting on€

work our att

lutr

rhe n€v (or old)

crc$.nr.

DYNAMICS OF THE MOON AND THE EARTH

2.2

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

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

Law of Cdvi|llion. Whar fouowd is a long histo.y techniques lead

describcrh€dorionofplanctsandlheirsal.lliles,dteroidsand@nebinordcrtopledicl then positions in tuturc with accey of SEater 4d g@l€r d€sG Conuibntio.s of

Cowll. Encke, Claittul. Hesn. D.launav. Hill and Brown. besides daoy olher nahemalici.ns and astdnoneb. bare ben oi grat Eul€r, Laplace, Poison. Gauss, Olber,

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

1991. 1988. Chaproni er.

rldrivu

al. I998,

etc)

Chapront.

l98l'

Einsrein's theolv of

suc(aded in dc$nbrns the molion olthe penhelion

The satllite to planet ( 1.23

Srandish I981. 1998

svst€n

larg€sl

ey o1h€r saEllite-ple.t pait (lh. nexr ldecar beins lhar of l6s ntio = 2 i lO I ). Thccforc rhe Eanh do.s not prclid. lhe dominmt

l0-r ) in compdi$n

Triton-N.Drune

nss !.tio in case of lhe Moon_E nll

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

cph€ncredF prcpded *ilhout tdtinS into

emn@u' Mey of lomJlauon of

the epheheedes of

eout Moon

lh.e contibudons b boud to be of.dlv dsvs, both b.lor' md aliet $e all

$e Neqonie ndhtuics. weE bded on

averages of various Lnds r.laled 10 lhe

dyndics of lhc Moon 30

obieBed

lompukd

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 Today it is

su(esive pa$ages of 18 oinutes

32.1

rh€ Moon throush

$cond, ed

it! perige is 27 5s4550 days (27 davs

hous

aw'age sidercal honth (inLdal b.tw€en

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

minures and I1.6 secondt. Thus on thc b6is

ofa sidcEal

h thc Moon rravel3

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

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 12.190?49

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

$lf

svsten

objsls

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 Thc smi-major arh

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

.@enrricily of the orbn is 0,t49 bul vdics

much

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

''elenentt' oi

yem

and dciltations

the orbn exhibn

deimiMdon of lund

{iih udnude of

bo$ scular

a ell

12'413 destccs

Ths all

lhc

peiodic ldiarions lhis mkes rh'

ephem€Ed€s a daunling task

dd thal of th' planets in $e nod€m *lup b€sd vith lh€ doluliodv e4loits oi Johees K'Pler $d lsc The undeslandi!8 of lhe dynanics ol the Moon

N.s,ror Kcpt.r cnpnically dcdu€ed his Lls of Pld.lary Molion on thc b6is ot o posilio' of extensile study of the obsralional .lcta @llected ovcr ce'nies of lhc pldets. Thcsc cfl be sbtcd 6 follows:

L 2.

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 Pl&erary otbiB

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

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

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

Frem rhe ddc

pl@d 6t a dista.ce

=c!+i

'r-)*s

rrr).

of publicalion ot

(2.2.r')

The force is aflnclive and G is the PmPonio.ahy constant Consranr (6.6?2x 10

dllcd Univ.dl Cavilalioml

enhs dit€ctcd ircn ,L to

tte'Priripi.l

-r

or rron

',rou|

bv NeMon in 1687 a nudbet

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

vorld think o! it no norc " lDanbv 1992) ro He had tlitficulry in describing lh. modon of Fnge (the poill in lod orbil cloesr (1749) the &nh) and could explain it ro only withi. m accurev ot 8 percenl Clanaut appli.d ml)lical merhods ad succeded in explainilg d€ molion of p€ngee bv 6'ng he.l

ache dtut kept

hih aeake

olen thut

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

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

kplM\

povid.d

e dpl.n who

sdo

a@1.61io. of the

M@! ripl@'s

mdhods

w.E

of accu!@y by sevetal malhedaicies One of then was publthcd his th@ry dd r.bl6 in 182? that Gmaircd in *id' Ne until

cdied io a high

Dmoi*e.

tion of lh. degrce

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' thc by one ahd gEdually builds up th. $lution. Autho6 cl.im thar Delaunav s vorl( n mo$

Frf{t

elulion of th.l@r problm v.t found {D&bv

Thc posilion of coordinatcs

eliflic

$e M@n dodd

th€

ElnI is desdbed

(r,,190 p) wnn r bci.s th. heli@.ntric

lonsitudc and

1992)

disr:nce or rhe tltncr' ? $e

lhe ecliFic laliode The most commonlt

lunai $bles dlrinB the nosr Pan of 20't c.n$ry \rcG dDc lo

ll]M $.ory w3 inpo!.d

bv Eckefi

&d ws tnown

bv spndical pold

edv lo handl€

BrcM (Bmm 960) This 1

ILE, sho^ tot Inprored

Lrnr

6 ELP Epr,rdr,r. Tnc lh.ory coGtruct d bv ChcPont and ChaPon(_Toul is knosn (Chapo er. al,. 1983. 1988) shon tot Ephtina des I air's Pukienhes ln ELP luDa nolion in $e simplifi.d tables have ben cxlEclqi from lhe lheorv lo rcplcsnt lh' $e fom of explicit line sries fomule. Th6e tabl's @ b€ lsed to direcllv compute ro ILE it luntr coord'nabs. ELP h Dol onlt morc Peci* md complele in conpdien For ale povides noe nodern valEs oi l(M pdmelels md olh€r Phvsical coislsts' 6000

y.4

on eacb side of J2ooo o ELP povid€s

lund c@rdiMt€s that

l4lv

have

eno$

VSOP (Vdialrons .xc.eding few arc secoo.ls Toeelher with th' d€v€lopment of t!bl6 due ro Sacnl.ifs des Olbires Ple€titt bv Bctagllon ed Fdncou (t9E8) lhe Chapront er.

destib€ the motion

sysrem. Both rh€

ofall

major bodies (€xcept Pluio) ih tbe solar

$€ori6, ELP &d rhc vsoP

dcv'lopcd

rhe Buean d's

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

BdicaUy a nhcory of pldelary

ed lund

fil o f ELP-2000-82 (Cn!po!i

cr. al. I 983) to

Propulsion Labomrory DE200/LE200 (Slstrdish

$keh iion Lasrar

For

penurbarions.

Gsklr, an

motion involvd

th. nlncrical inrcer.tion of lhe .lei

l98l). P@$ion

in this o@ry

b€en

1986).

lylic par!

ELP-2000

Fpet€s th€ nain lEblm fbm th.

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

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 paBmeted D (diffeence of ihe

ude).

ladtude and

acrons

6uhs inlo dne *.ies fomnld for Mmn\ longitudc. gocenrric disrscc @daining 2645 rcos in all. Apan flofr lh* *rics

Tlis main

problem

of all rh€ other

significant objects

solar sysren

consideed N

'Fnurbadons lo ihc mdin pbblen lhol include:

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.

Dir€ct planerary penub.tioN due lo etions of

lor

borh the direct

co6id6

th€

orbib ofthe

othr plan$

on ibe M@n

rhc indt€ct planetlry penubalions lhe ELP-2000

pldct

giv€n by BGlagnon\ VSOP82 rh@.y.

P.rurbations du. ln€ figues of $e

Edh md

thc

Mon

(Moons, 1982).

R.larivislic p.nub.rions (Lsrnd. & Chlpronl-Tou?J 1982).

5- Penurb.tuG du. to tidd afccis (villim3.t. a!

Molion of the efccnce

frde

conside&d wilh

l97E).

spect

to m in€rrigl

fime of

fomlla for geocentric of lcm !o 15:37.

consideradon ol all lhese pelturbarioB esuhs inlo rime series longitude, IatiMe

.rd disra@ of 0E M@n

An al€natc ro this

6 tinc

nds

$e numh.r

ther€icrl appro&h is !o rcpr€*nl1hc

coordinai.s explicilly

ledes fomule. This epEs.rolion of tlie tinc sed€s is dcrloped by Chap6n!

Touzi ind Chapront (Chapront-Touu & Chaprcnq

fiis work.

The

mjor fomula

used in rhh

Tbe ge@entric longitude

'/h.xpesed

l9tl,

pp. l0) und ha bcc.

!*d

vork due !o $es authob de lkrcd beloNl

as:

+0.000001856tr'

'/=218.31665416+48126?.8813424'r-0,00011268'r: 0.0000000r534.r1+,5. +(s; +/ 1si +r' xsi/1oo0o)/l0oo (2.2.t)

wh€rc

I = h in iulian

s, =tv,sr,(dj')

centuies sinceJ2000.0

+d|),,+ao *r' xro I +ao)'/r xr0' +d|,.rr (2.2.21

I )':st(o;''

-a:"'

si = tv;,s,,(d;,o

dnD.r)

2.r\

(2.2 4)

xro r)

$.t%'sr'(sfl + dI'.,)

v- r; ct, d!o8 eitb |b ot a'. G Sivto i! Chr.odCt q.od (Clg!d.To@a.nd Ci{.Gi, l9l, rp a!-56}

Tb vdB 6f

Tout.d

tb.ootM

gcoc.nttc

ldhde UL

t/-s,, +(s:/

s,,

(r25)

riE

b,:

+r'$ +i's;/10000)/100

.5,"si,'(rj". p:".,.p:!.rr

xron

+/i",r

xr0r +r'r 'r'

Q26)

x ro

Q2.?',)

e2.8)

+r4D.,)

t229'

s:,

-'u.,&4fri.t + p{t.

-tr:$4i:o

q.t!;&,(r;o +r'o.,1 Th!

y.lG

Toudsrd

Q2.tO)

th c@trt ! t - !; .iq dotE eilh lho.c of 0!.t 8lt'.! in Clqlot' Ctar@r (ct Fld-ToEald ftlnfi, 1991, I? 5?{a). of

rindly tl|c geo.6tic dllt n6 it dven by:

x-315000.57rsr +sl

+r.8i

+y'.s;/10000

Q2.rD

xron)

=t'"cdldj"'+djl

(2.2.t2)

s;=:4cd(d;'+r;,''/l s; = t,;c,rt6"4'

r,4,

(2.2.t))

(2.2.t4)

si= i"-",,(u-'' +d;o).r) -fhc

!!hEs of

'fouzi

the consr.nls rz,

and Chapronr

lnd

r; e

etc, dlong with lhose oa6s ae given in Ch.prcnl-

{Chapo.r-TouzC and Chatronr. 1991. pp 65-73),

Atlat0)s. /0)s and D{ors, ,',

arr\, y'r's, d1'\,

e.z.ts)

si,si,.';

and

desEs/6tury2,

@ in desrcB,

ac in d.s@Jcenrury, ar'\, y''?\. irlrs,

ao's,

aid d'h de in deecs/centud. kiloderrs/cenru.y. siand

s,,,sl..U,su,si./.'r.r;.,". y'r\,

ana

d's m

R, s,? and Sh

in aegeevcenruD

dc in

kilomelres,

si.si.v;

and a"rs,

5i md 4

/'rs

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

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€

enehls of thc

osc ulat

ing or

inst

aneous

orbit in lems of:

- sdimjor uis of th. dbil l, = m@ ld8itlrL ordF ddt a

l= ?,@$

sincr,i

Dsino

c= si(Xi)coso

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: whec.

is thc.@eotriciry ofthe orbn,

I"(ssii9+ I:cosp) or 1"..i€os(, +c?) $,lEE

- 0, l,

l, 4, 5. I

f= e=ia,1,, i = I

(2.2.t6J

tn. tire in lhoFnds of.,ulie

)€s fod

J2000.0, i.€.

165250

to 8,

= 9, l0

reprsent rh. m.m longnud.s of the plancc Mercury lo

6d I Lc rcpllsfl

The hsl of tlay. In thc

is the

th. Dclawy sgmen& of lh. Men D, F md

longiMe of U. Mon Bivd $nh

ah@te €xpcsion,

B= Za,^,"

6p€t

c=>d,N,

12.2.11,

(2.2.r8)

-,asinp,

lh.

md M dc Sivci in

the lable 2 of

(BdaBnon &

Flmou

l96E)

@ .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 These data series

spherical polarcoordinaEs VSOPSTBand VSOPSTDa!. used. A shonerleBionofrhese data series

hbles firsr €olumn

lheff. tr

l. the* gives.4s, fie eco.d ,s dd the lhird sives Cs. The dats file hs 6

isSilenby Meeus (M.eus, 1998)and

u$d in this@r\

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

t't i

0. l . 2.

:1.

lir

L. ll .nd

ll ne

uscd as

LA; co\B; . I jT t.

4. and 5 lor l- md I( and 0,

(L). 2(lJ) and 3{R).

x rus tolsh

I -

r2.2 rol

l,2,3,4lor B Ihc

0 lo dilfeonr

suFrrscnpt

incger lor di(feEfl

tsund for

coordinaGs and

/{i coftsponds ro lonsirudc scri.s. I = l ro lari$de sri€s and I = 3, /r,coiiesponds to dist .ce serica. ll.ch

their lssiatcd seri6- Fo. l/{iconsponds

follo\s ro obhin thc heliccDkic polar

coordin eislhcneuluatdas:

r=ltcu il' /. md

crlier

=l:, ',''

) "=lz'',')

, d. ir ndian nesuEs rhe c@.dinares of

and X is in aslrohonical unils. These

(2 2 20)

as menrioned

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:

,is = I +l8O!

ond the hcli@entric

2.3

Fs= -e

dislanft ofthe E.nI is smc

12.2.2t)

th. s€@edtic distanc' oflh€ S!n'

BIRTH OF NEW MOON

arcund 12 A3 menioned earlier the Moon in ilsjounev dound the Eanh llavels €vcry 29 5 davs whc' rhe dcgrees cvery doy in our sky and $l.s ovcr tic Su in lround

Ccoc€ntnc Lonsnude of the Sun

e.l

lhe M@n

N se

rhe

nonent is tnoM

tnc

of Nev M@ns Tine ol Bidh of Ncs Moon Tbe dudion berve€n two succe$ive Binhs petiod is n is called lhc Lun.lion Period Ho$ever dE Llnal'on from 29.2 days lo 29 8 davs This is rh. d.ys €ach or lhe con*culive

cMn

b€hind

irt tu'd

mon$s of 29

'ons For fi€ tine ofBinh ofNew

luor monlh of lO davs cacn

lonsilud's Moon one requiEs to find lhe noment when rhe seocentric

olilt

Moon and

thcn

rinc eeh of rhc sun coiNide Thus one needs to lracl lhc lo'Siludcs of lhe lonsiodes of the Sun md Coisidcrins $e najor rcms oa lhe line scrics fomulac fo' ELP-2000'8? $e $c M@tr in th€ planeurv rhcorv VSOP'2000-8? and lhc l(nw th@rv An alsori$m due b Meeus noment when thc rwo lonsiludes aG sMc can be evaluatd Ns Mmn is as rollos: (Mceus, l99E) for $e dclemiiation or $e rin' or Binh of passages ol lhe a!c€8. lhe tnpic.l Y€d (dudiion betw'en t*o conseculive (from (l I l)) Thc avedge svnodrc Sun lhroush equ ox) iscudcnrlv 16524219 davs Moonsr toten o\er a rcn'ur} ii Monrh (trre intedal bclwccn r$o cons{u'r\c t\e*

'Ixus in oft topical vear rh're aG otr a\€Esc 12 1682664 29.530589dar(fon (1.3.3)). 2O0O i e J2O0O O the nunbei of Synodic mo h5 Thqeloe since lhe $ad of lhe vcd synodic monlhs elapsd oregrveh bvl

t and lhc time in

(f,'

(2.1

- 2000) x l2 3682664

lropical ceoruries el.Psed since J2000 0 is Siven

bti

ofth.Iulid

An approximarc valu€

JDE

2451550.09166

t.2)

t236.42664

Dale ofthe Ncw M@n

ir $cn giv€n by:

+29.5JO58886t.t+0.00015437..r

-0.OOO00Ol5

j.r

+ 0.000000000?3'

(2.3.3)

$herc

t i. d

lid ces.nr 4l

*!.

inteSer Thus @ording ro this fomula rhe for

ofycar

2OOO

is 2451550.09766 lhar is

I of dynlmical lime. For

lhe p€dlrbation

horc accmle valu€

= O |h€

Jutiil

Dar.

J&ury 6.2000 ar I8n

oa ihe

of$c

r4m, md

Julia D.te of thc New Moon

tchs

due lo lhe Sun md rhc planer ar€ added, Thc pertuibafions tenrs due to thc sun d€ given byl

-0 t07 2.SlN(M )+ 0 I72J t

+0.007i9.E"'lN(M -M)-O 0 00| t

EISIN(M) +O.0l60A.SINOr M )+0

0O5

1.t.ErSINtM

t 'SlN(M -2.F)-0.000t7.StN(M

0lAj91stNe,F)

+a 0a208'E^2istN Q,M)

2*F)+O OOO|6.E.SI^-(2+M,

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.

+0.00002.s1N(1'M) wherc

-M). D -O.O0OO2.StN(r. M, + M) Q).4)

rb. mce &omaly ofth. Sd al ihc JDE = 2. I SiJ +29 I A5 3567.k-0 ,OOO0 t I -0 oaoooo I I

,r,1 =

I (2.3.5)

= thc mee anomly of de Moon d lhe JDE = 2A 1.5613+ J85 8 1693528'k+O.Ot 07j82N -0 0A0A0A0,8.

OAOA

I :38.

M@'!.4ruqtof hind. ! 6'0.? !

aE+390.670t0264.1-0.00t

0.0@w0

61 r

- 0.0U00227.

Q.3.1)

g - lncinde of lsnding md. of {!. luar orbn

!24.7716-1.5637t588.t+0.0020672'1 +

0.OOOOO2 t

(2.3.8)

Eedtricity of $. oftit of Elni 1-0.002t 16.T-0.0000074.

Th. pedurbltion |.tus

(2.1.e)

pleas eâ&#x201A;¬:

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 +stN (/tq + a 00006. stN(t7) + 0. 00u t 6.3 !N (/ E)

7.s tN (,t 9) + 0. 0000 4 2.s N (/ t q + a 0000 1. s tN(t t t ) +0.0000t7rsIN(AI2)+o0o00J5.StN(Ar i)+o W023.SlN(,| !4) (2 3 to) + 0.

00001

A I -299-

2 7 + 0.

t O74o8.k-O,OO9 I 7

t. ?

t2-25t.E8+A0t6J2t.N

(2.3.1r)

12.t.t2)

,13-25 LE|+266J IEA6.t

(2.l.lr)

a4-t19,42+J64t2478.*

Q.3.t4)

/5-U.66+ ! 8 206239.k

(2.1.15)

A6-t4171+53.303771.t

(2.3.16)

/7=207-14+2.153732.t

(2.1.17) (2.3.18)

19-34_t2+27.26t2391t A ! /1

(2.3.19)

0- 207. I I +0. | 2 1 824'k

Q3.20)

64a379.t

Q.3.2tt

t I = 29 t. 34+ l.

A I 2=

6t. 7 2+24. I 98 I 54.k

(2.3.22)

t099.t

(2.3.23)

E.*

(2.3.24)

A I 3= 2 39. 5 6 + 2 t. 5 t

at 1:33

Tlus $c

,ulid

t. t5+ 2.7925

Dlle of lh.

M@n ie giv.n by

JD=JDE+X+Y Thc dm€ deeribcd by this mad€ lo get

th.

Ld.l

dat

is lhe DlaMnical Timc and lhc

r.2t

coretions for Al mcr

$e Universal Time (discu*d in 0Encxtarticle). For anr l@al coopuralion,

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-

2.4

THE TIME ARGUMEI{T

23r"

59'" t9h

thd dav.

ft w6 mentiohed earlicr thal rhedynahicsofatl soltr systcnr objec$ isdescibed

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 by

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).

.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

G@nwich on Janlary l.

oonefl

called

in edote pdsl coftsponding to rhe Noon al

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

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

tih. clapsd sine thn €pocl 6id is cxpEss.d The orhcr epoch of inpori,lcc lo rh€

J2000 0 a.d

in nmber of

€lftnl

D!i. i3. mesE

n@ eld

d.y5_

work k fie noncnt of dme €ell.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

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 .poch or

Millemia (365250 mce solrdays). BoththeseepocbsmbasedonrheinreRaloirimealted..toedn$hrdat,

qhich

a fie int rval berween lw slccessile rhsits (r6sdge rhlough rhe locll ofrhc fic ious body knoM as ft. mee Se. This ficlitious body novcs sirh

is defincd meddim)

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

of on€ lrcpical

up ro

I I minurs over a penod

(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

Th. tiDe describ€d by lh. cl@k showing

di*Ep.n i6.

d. mm

sisddd hcidia..

$lf

rinc is not wilhoul its

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)

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 on€

a r Si&F.l

Dry

ud rh.

timc

masuld @rdi.s to thb $d. is

rlrc

Sid.Flt

Timc_

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 Aglin du.lo lhc clliptird orbit

56'r" 04.09053* on

sol$ rimc).

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, In gcrcr.l rh.

lsl.m .t d. 2001.

e.wnl ir to bG p. iomcd s ed( oul.v.rging on l@l civil dh. sd dac. ror iBlhe vhcs llE rim ed darc of rh. binn or Wh.ncv.r a c.lcul.tion for

b.sd

nd M@i is compucd

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 rh.

bnsnudB and rcg.dv. for

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

gd tnc

Dylmiql Tim.

How.r s noE @nhtlon l !'pNch

@!qt loâ&#x201A;¬l <bre.rd ar rim

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

Srep-l: l8el

Local

Doaroh ercgorin

Cal.nd@

LMM, LDD

Sr.F2: INrtIBalTin. &7awTih. LHH, LMN, LSEC, ZON S4 LryY=LIY, UMM-LMM UDD.LDD UHH= LHH + ZO N, UMIN = LMIN, LIsEC = I.S EC

ln eedeEl rhe only Univcsal dme md

d.t

(considercd in inresal

Univc6d

dare

dif.rc.c. bclwen

is th. diff.cn

ho6 hcr).

. bel@n b

mar difLr by onc &d tr.!&

I[(UHH>=24)

(

Duc

......

rhis

rf (uMM> t2)

UMM-|: UW-LYY+ 1:

t)]

{

... ...

d..t @ do,

UDD=LDD.!:

{

If(uDD-o)

(

lhe

hou6 rhat cq@h zon time

ditf.rene th. t@al d.rc hd lbe

1rcft6. dar

'UMM-LMM+Ii

upHH<o)

.djcthsf.

UDD-LDD+I tf (UDD>dorqLMM)) //darr LDD=I: I

local hne tinc a.d dale

UMM.LMM.I

If(UMM-1)

nMber of days in

donrh

AYY-LW-l; J

UDD-&tr(UtlM)

olr@

l[. UEircc.l

d.rc dd

tirc

lppllFi.Lly dju{.d @ pcad! lo

{d rhc ritu. For rlis calcularion ihc nondr is Jmqry o! fcbnoy, it i! co id.Ed morh .mbq c.lcula& rhe Julih Dar. for 0E d.lc Esp@tiv€ly, ofthe

inirially 13

14.

ptwiou trd ad rh. t@ i! sle d.crcNed by l.

UW-WY-l

ofeftry yd (likc ,E | 100, 1700 .rc) d[ ln. yd lo t@6t for !mh6 of mhd rap y@.

Thc

ir Fquicd

nmbd

UYY

Stcp4: 1=INr(Wt00) 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 ln

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). cal€ndd.

d"@ledq it

casdtb' a=2-/t+rMf(1/1)

.l*1I "@Ldo

(

IdlM" B_0 J i.t

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 Now orc n6ds ro

ianing fom ys -4716 dD( thal @ balmed by subt@lion of lh. co6t ni 1524.5.

Meu

@dider this

sMbl

JD=\NTQ65.2t(UW+

17 16)+

adds addilional davs

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 16525

Notc rtEt in rhc

l6t

(242)

1155760000

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 in a

&d Millemia for vsoP)

Juli$

due to which

c€nrury ifor ELP md Julim

Mill.inia fot VSOP).

and when ttie dynmical tim. of

U.ivesal dd

Lhen into Local

il hN lo be divided by the nunbq of s6onds

cv.nt is

*rom

we need io @hv.n rl

Zon. tim.. Thc dylmical tihe is oblain d

of Julie cenluries sin@ J2000.0

that rhc Julisn dare of tbe evmt

the

nmbet

b€ calculated

15575s.(, -

TIlc int Sr.l

t|| hclion l Frt

F2:

Z-INT(JD)

d-3:

F-JDZ

usrsts =-4-), 3155?@@00J

ofdF Juliu D.!.

*ill

onEtud

ea.t, io

Cdddr Dd. .nd

th Unirwl Tim:

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 ( Ert t

t-Z ) 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 ( ( Els.

UMll-El ) UMt'r-ElJ ) a9

LMU-UMM

stcFr2: alMM>2 { Etr. t

UW-C-1716

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;

It(LHl>-2,t ... ... tt@t et) { LHR-LHR-21: LDD-UDD+,, 4t6-DD>day1vut4

LDD-|:

LMM-UMM+I:

y(LMM>t2)

{

LW-UMM+I;

)))

ste!'rs:

htn .=Aov-L| 9.60

sreFl6:

UMN-IM(nlnut.) LW-UL{N

steplT:

tu.ond-(ntr.-LMIN),60

tN-l;

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 is

Nd-M@n Algorilhn

thc our pur is thc dynmi@t

p.niculaly usfll in @nvcning Uis rinc lo

2.5

this

dgon$D

v.Mt od &y l@.t @G rin

COORI}INATES OF THE MOON For rh. dckmination of lh. c@rdinrcs ot 1h. M@n .r

dlt

lin. &d

liB| scp

lc.l

Sivcn

tim. .rd

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 is ro

lcds to lhc rim. &Bum. t eco.dinS

th. atgorjlhn dc$ib.d ,bovc,

?{51545 N 16525 1155?60000

Julon Do'. -

Using Uis rim. arsumcnr

Sr.F2r

(r 5.t)

*n6 d.*ribing rh. twr -Touza ed Chaponr t99t) s dieled in aniclc 2.2

th. @Bhcrio! ot rhc dh.

c@'dinar.s is do.c (chapo

StF||

Fo..dipdc longirudcofth. Moon us.2.2.2 ro2.2.5 ed suhdrut lh.ir csulrsi.2.2,l. For cclipdc

t.hu&

U. M@!

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

lh.nrcsulr.in2.2.1L

2.2. | 5

subsritut

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 includ. the

dimal nolion

S1.t-4r

of thc

pls.t.ry Ab.dlion sd

do.s

lol

ob6*er (W@lad & Cl.mcnce, I %6).

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_5.4)

2 5 + 177 I 96. 91r)

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 Fimlly

1he

lrue €quinox of

without th€ nul'lion in lonS itudc AV and the nubtioh inobliquityAs.

pS:

CORRECTION FOR NUTATPN

a'r =l0r

't(y, + y:..)'si,{/,I" +/d' '. +/d' '.r 'lo r) (2.s.t

Y=V+

L,t

. -22.63928-0 0ll\+0.555'lo'

12-5.6)

t.r -0,0141'l0j r tl (2.5.7)

=lo

'I(., +,' '.). cdko)

/,r' . ' +rdr

..).lo

(2.5.8) (2.5.9)

Tn€ thc

T$1.

v.lu. of F

!,'3 md

9 in Chlprcnr-Tooze

G's

in lh.

qp63iont

6d Chlpotrt (Ch.pre

ahovc

tt!

gt!€n in

-louz; &d Ch.pon!

l99l,pp.l9): Oft. rh. @retion dE to .ubrion is dre onc my c@dihal.s, the Rjght Asnsion d dd thc dcclin tion 6:

St.p6:

da Eq@tori.l

EQUATOI.ULCOORDIANIES

"=r-'( 6=

so ro 6trd

c6(tpriio")Si4v)C6(U)

-.t4

Eps i t d) Si dlu

sn't l3ih(Epsitm)Si"(v)cos(UJ

cB(Eptiron)sinei

(2.5.r0)

(2.5. I I )

Thc* re rh. nle cqubri.l c@'dinar6 of lh. Men wnh EfcEn.. b th. toe 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 to be

ofth. ob*fr.r) nay

t t.n inlo &count. Th€

be obl.iied_ In

odd rcd5

rh€

!ffftrj oftE

..paraltd" arc

Pdttd r i! givcn by:

12.s.tz)

g€catdc dist4. of dE Mer HowEr rhis q@nry paml* d.Fnds Hoq Ancle (ihe since ibe object cosad thc l@al n.ridim) tor whicn w. ncd

wh@ 4 ir on rhe thc

lad

FTl

rhe

sidc@l

tin.

LST.

Fqhtlate tiw

dgM't

tlor

UTlq !h. dat. ,tdet cwidqatio,.

I. =6r4t-50'.54841+E tot84,.s12s66I,+0'.093104.r, -0,.0000062.rr (2.5.1t

gits th. Grcawich

Mcan Sid€E5l

Ctqwich Si&@l Tio. T

To +

for &e

Ti6. .t

rirc ug@nt

oir UT of rhe drte. Th.n lh€

for ih. rimc of ob6qvation is:

(UHR + (UMLN +USEC/60y60)a0.997269j66JJ (2.5.t4)

'Iijen fic Hou anglc at ihis moncnt ofthe Moon is:

H=T

srep.sl

(2.5

tllec^ofPatala,

If p is th. goc€rntic 6dius ofrh. of irE ohsftr ih rh. dsh1 .s6io! co@lion for prhllar &€ obr.ined $:

@sd - p@sp'sin rcos

Esdh,

p,b

a' ed

ihe S.ocenkic hrirudc

rhc dEtiEtjon

d'="".I Gind -psin9 sintr)cosaa ccd-pcGg'sinr@3tt

Fin lly ro otltain rh. lhc followi.g

SrcFg:

Lad Hdianrrl C@diDtcs A2inuth.nd

t1)

(2.5.tE)

AlriMe

lissfomalio$:

Azimu$

(2.5.19)

aft.r

(2 5.16)

<2.5

haY€

.Aliitu& r=sinpUnrlcdp'cosr'costl This complcGs ihe der@inarion of dlipric, €qurqid

@rdid6

2.6

of rh.

Oe hondnIll

Mor

COORDINATES OF TI{E SUN Foi rhe d.t.mimtion of 0E c@rdiDre of rh. Sun on.

(2.5.20)

say

slcFl: lqds

for lhc

@rdi!!re

ple

Scldt to rh.

oroh6dvd (Longiru& ed tilirud.). lcal

lire

ro thc algolirhm

Dde )451545

1652t0

Using lhis

in €runy rhe

of rhc Moon.

thc egment t &coiding Julian

p@..&

d ..dd

dceiib.d lbove,

rine rhat

I | 55?600000

argment lh€ connrudion of

12.6.1)

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

2.2.20

lonsitud. of lh. Flnll' in lirc wit[ (2.2.19)

w h!rc:

2^, "e"(,, .r^,

(2.6.2)

)

2"" "."6" t r,,l

(2.6.1)

t,-'i^,-"(t"*",

c2.6.4)

L,-ZA,s\8,+c,. I

(2.6.5)

(r5.6)

=ZA,@l\tr+crt )

!,.:r,cdt

+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

-. (2.5.8)

t0' St

p2:

Fd Fliptic ldintdc ofrlF E nh;

lut Bo-Z/,.d\8,+c,. l ,r

=:/,dF"

+c"4

,, . X,{" "o.(r" +c,r

(2.6.10)

Bt-244a.+c"r I 8.

t,l,6.F" +c"'

Q.5.9)

(2.6.l|)

(2.6.r2)

(2.6.|])

Tb v.lu.t of A's, B't .rd C'! 6. (M6n, l9S, A!0.odtr-Itr,

!P. 4lS-421).

l} {oi.t nds tt giE

Fidlv lh.

lbc.otic ldlidt

For

19 l o.[8i t,,r"

c,'

(2.6.15)

(2.6.r6)

(2.6.17)

]

t,r"co't " +c"'

13 =

t0, h=>an(fr]\B,+c{

(Mc.q,

vds

199E,

(2.6.rt)

)

(2.6.19)

t,."co4p" +c"' I

(2.6rI))

lo/

TIE

ib it

lElioc.dric .lisl&e of th. E|nb

&-X,i"6!p"+c,r J zt2 t xr - I,r,6lp" +c"'J x,

of th' E

(26.r1)

l0'

$.Fl:

bv Mo.u!

4..

B's

!d C" b. lb eorlr Ei6 & tih

M6!

Aprcndix-lu, pp. 4l E-421). Fi!.lly tlE helio@tric di!l&c. of th. Elnh

ZL,r " A!" Bs lnd C!

A! i!

''id,

all in

.*!doiql

Q.62t)

hdiM for looSin|d! , a|(l ldltt|d. t uitr tu Llio.ldtc diaec X. 51

od Ct

ttdieE

The coftcctions for Abe@don md Nuradon de done in

and Slep-5 b€fore. Fi@lly

horizonlal coordiDr.s is

done lhc

2.?

als

sanc way as St

conveBion to rhc equaronat @ordinates

sme way

6 ws

done ror

then

fte Moon.

RISING AND SETTINC

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 For the

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 ofthe objed (ove.lhe locatneridian) wbich lben reads ro,nlial esrinars for de 'ransn 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. 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

ths.

rppo\imaro cdlcutduon fo.hcle appo\,dar. coordnats of thc objecr hNc to bc catcutaled

,t..,

.,.",. ,h. ."i..;;:. "t asain and wholc.atculalions

above

!!€ repealed for bekr

esrimates, The details

(he followinS p@sraphs.

Forrhe

sh.

,h.

henLion€d

of rh€e catculalions arc describ.d in

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 vdi*. Th6 Unileisat Tihe or loql tuir is simply 12 ne) is posirive for the east lons,udq ed nesarivc for Resr

wnecas for the Moon n dEily

wheE ZT

(ene

uivcrssl rime

s Ih. loql dae. Th. rime dgu. for $is rine ed date is rhen fomulatcd dd th€ lonSitudcs. At

12 -

zT wilt b.

dn€ of 0E

darc in

O@wich

offie sunar.obaincd. wlm m objdr h in rmsil irs hour dgtc (HA) ed its dsh ecdion (RA) is se s ln. Loc.l Sidqdt Tin. (LSn siNc:

coordiMtes

p.? l) As3Ming dar lhe

obj*t hs

r{A

!r da rine of t@t Fa6n e.7,I ) shos thal:

d= LSTt

If JD is

rhe

Julie ddc for $e dly

UT 0En wnn

, = eD _

mcaur.d in Julian centuri6 lh. c€enwich Mqn Sidc,@l Tide

2451545y36525

{CMST) b siven by:

=6141-50,.s4841+8640184'.812866.r+0'.093t0,t.r: _0,.000006rr,r (2.7.t)

Thch for the observer ar rhe pla.e lonenud.s a<l posirive for rhe con)

vilh eeographic lonerrude , (negative fo. 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 This rim. is rh.n converrd

b lh. UT

(2.7.5)

by:

UTU =Tlt.0Or?l?909175

(2.7 6)

west

FimUy the coordides ofthc

Ss @ Edcdar.d

for UTE

qd 0r

followins cdculadon

I,sT = a,

(2.1.7)

csnr-LST-L,

{2.7,8)

n_GSTtr-To

(2.7.9J

uTr=T,'L0027379093?5 So ihat theE is a

[or lhe

dilfer€ne of

l€ss

drs

*cond b.tween

(2.7.10)

one

v.lue

ofuTr ed ib.ext

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

de c@rdi@res of rh. Sun for UT6 (or

Urs0. The holr angl€ HA

stt'ne or nsing ovcr a l@al horjzon k

(2.7

stEE

is rh€ eeghphic laritude of rh. obsedcr lnd lhe au nd€ of a$med to b. ero, How.ver owirS to th€ phdohcnon

poinr of sky is

ofEri&tion a ,r.r, rhe Su dd s rhcy @ a€ru.lly seo sdrjng or risin8. The E@iN visibte de ir ft hs gone j4 e Dinu!€s

pld.l ae wll b.low th. hode. std

t\@g€ rFe.t of EFadioD is rhar . srat below hori^n. 'Ilis allitudc is kmh s srand.rd drilud. dcnoEo s 4, ed

6t)

is @nsid@.I

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,

lo bc -50

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. rhould be calculal.d fmo dE

object is then eloluared as:

(21.t

Thus rh€ liBr dppro\,maion for lhe

T, =UTO

And $at ofrhe

of lbe

ha\e

rhe ofrisng of$e obled h:

- E.

12.1tJ)

in8 isl

Thes.

=Wtr+

(2.7.t4\

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 borh

be catculared aglin. The Grecrwich

sid.El rinc

orrcspondjns ro

ro be obained as:

r-Ta+J6O.98564TTy'15 ,1 or

I e

rlEr l@at

12.7.\5)

lFu ogte 4d ihc &imurh of th. obj..t

12.1.t6)

Si"At = Sinpstnq +C6e,Cot6,CotH,

Th.

cordiG

for rhe

tuirg

siring

AT, =

Adding these

(21.11)

(2.7 l8)

ifto rh. appropriarc I, givcs ibpDvcd valu.s tunhq inpov€d r:lucs catcutal. $e @.dinaas of rhc Ss fd &d

A?_,

rcpeat (2.6. r 2)

lor

!ffrr

(2.6.1 8) ro obrain

Ue Moon rhc i$uc is

mt!

lh. UT for d€

complided

wau.

af€.r of panlta h signiticor.

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 Th€

ofrcfddjon h

still), Thus for fie Moon the si&dard attjtude is Civcn by:

qb.G

=E - 0-2n5.

-10134''

(2.1.te)

dc pa&Id of rb. Moon eivd byi

(2.7.20)

s $.

g€@eniric dist ncc

oflhe obsrvr &d X, rhe g@ce ric dislece otrhe Moon, p

\2.7.21'

is th€ equrorirl ndiu5 of the Eard,

th.

dd .,

The resl of thc calculatiohs for rhe irlnsil, rising

is th. pold mdids of the

lhe

stling of lhe Moon

arc

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

di*u$ed

lcaiion oo the Eanh oi ool. Tbese weE briely

ad lblcd in thc b€eimins of chaper l.

paniculd Ncw M@i or th€ conjucrion

rquned

to h€ d€t

Once

of th. Binn or

numb.r of p,@eGB

@ T". {ii) Tinc of M6Mr T..

hs ben delmined

mincd. Th€* includ. (i) Tinc of Suroet

tr tiD. a

(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

width ofcrcsent w. In vhw of the di$u$ion

P, md (x)

ofth. s&onomic.l alSorirhns ed rehniqud in tbis chaptd the* circmst nc6 de cco$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 rime ofBinh

th.s paludd ii

ofNcv hootr.

The

rh.

alsdirlm for

in anicle 2.3. U3ing $is algoddm thc obtain€d. The algorilho tates yee

tn. lh. conpuraion of (his tim. w6 pE*nl.d

coiju.tiotr of lhe M@. wnn

riic of

g@centic

binh of lhc Ns

insran@ the

bdb of lhe exp€cled dale of th.

Nry M@n in $e nodn ofApri!,2007 i. dp.ctd

,e0

m@n is

input and giles lhe Julian datc oflhe tine of the

iiy€ai is ior binn ofnew boon. Il is imporrmt ro nole $ar the i.pur is o Eal nMb.r €alculat€d on the

ihe Sm or

= 2047 +atJ0+t7)/J65

whole

iMbs, it

New M@h. For

msd

l7s d.y of thc

An |pDDxiE.r. valu. of

vholc nmba $c algoriihn ro thc

b.gimine ofthc

otuc

b..i

e G@ of f.e d.,s m*! wll. If UE "yd" it r si!4 lhc ,!lie D.rG to! $. Nd Men th.t @6 cl@n

"'*"

!d".

,ulid D.rc of$c binl ofrlE Nq M@n cl@n b tb. m. b inni.lly conwn.d lo lhc UniEEal

rhc

found

wirh

lh. .xpdrcd d.y

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

rin . B.foE this tin. only th.'bld cG*dl" ce b. lat wn b.foa on rhc dry of onjudion or a dly o! tvo b.foa-

bcfoE this

rh. sutuie

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 Ar

rimc

b. s..n ed ha ndcr ben

Thc can be

sn,

p.md4 is $c l@.1 lim. of suel ed th. M@n $t Th.* d for ey day of lhc yd Bins th€ r.chniq!.! of $c anicl. 2.7. Howvr

ndr inpondr

conput

th. dctmin ion of th. c@diEtcs of both lh. Sln &d th. $ce rehniqEs 'lquiE 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

then

trdsfomcd inlo

Ioc.i hori@ntd @ordinalcs dr. ahitud. ud &imuth, Once $e

tine of leal

o@ctcr LAG = Tn dd ncSrdv.

rhc

and

th.t of Moon sr

Tr is elculat€d. Unlc$

(I.) d

lh. LAC is posnive for

cd ncv..

for old Mmn th€ cFsent

Suppoe

sd th.

(T)

sunsct

b€

obt in.d lhe

the New Moon

sen.

(..,,r.,r.)ed t,.j,,r.) d ftc p@ie

dislanes,

aliflic lo!8it!d.

Su, Esp.ctivcly. af@d ro the ft4 .qtinox of lMtio. on $c Elnh with tftstiat coordinaies(1.r) on rh.

lalirudc of rhe M@n and 6c

dD. or sunst at any

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!

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.. TT) T", oflhc coojuction. Ncxl one requires considcrins the local rines ofsening

Using rnc .cliptic

@rdimLs

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\^

- Tdnl P)sth(E) cas6)

)c os\r

st(d) = si,,(l)co(.J

)

(2.8

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)

the

allilud. (ir) bv:

ranlAt =

si,(r)

(2.8.4)

c6(Ir)s,,{t) rz,(d)co{l)

sr,(c)si'(6)

(2.8.t

c4(d)ca(t)c,r(fl)

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'

dir@. of eim hs (DAZ = lA,-A-1. @lled El.tive @'ihuth) ard thn of or aldrudB (ARCV - ," - t.. cdl..l m of lision) d shoM in rhe lisr I, 'r the iimc

ln.

loc.l

3uEl

As ihe

$6.

sd/or

al lhe

dgulu

b$l

Tb Play a

vilil ole:

sePar.tions invol!â&#x201A;¬d

dnes @ sdaU, wi6ou1 dEh

th.

b.iwen $.

Sutr md

lh.

lunar cesenl at

of lighr (ARCL) h giEn bv:

(2.8.6)

Wh.@ for

larSct

ugl6

ARCL =

or moF

ffi.rc

Gults th.

of lieltr should b.

eldl.Ld

cdr(co(r-)co{rr,)cd(tra - s,'(r.)si(4))

Fi&28!

of lighr Uc cdlqir A@d tun lhc Ft.tiv€ .rlmuh tbc !D of villon and th. c@li<lddio t.qbcr of for dli6r ytuibrliry of t|ld qqc@t rlqoits io irL othd p!@der. One swh p€mcr{ is thc A8. of th. M@! (ACE) d!6n.d as thc tiEe

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 Gl4a.d

.iG

th. h4 cor{l|!cdo.

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 thou$nd

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

Ad tholhc c!!!6r

width is givm

bt:

(2.E.8)

(289)

"='"[ Sinc. ihc acienl birlh oI New Moon

lill

signindt p&Met€r.

thc nedieval dnes

thc sunsi of

lh. d.y in

ACE, thc time elapscd

quesdon,

siie fie

w4 eo@lly coNidecd

Howev€!, in ihe tinc of MuslinvABb $lronom€rs jl had already

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 b.en

Mooo

anat€u $e

s vcll a

€ord

ce b. ee.. Amongsl bo$ rbe $@ is aluyr a conFlilion for havinS

lhc seaens at lime a very "young' cEsanl

rhe

prcfcssioBl Grronom.u

the "youneesr"

crcsr

Amongst the early nod€ls ofnew

eiihd wilh oFical aid or wilhout

cc*.!t

i1.

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 s.pmriotr (clonsaiion) b€twn rhc

su ed $c

Mmn hd bccn

b€ ddemined once lhe

dchi'cd

usi.g (2

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 sith €nh€r horizontal coordin.les or

rh€ €quatorial

aordintle!

This elongation or Atc

des not only dcFnd on thc phe of the M@!, also 't 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

c..rral pon ol the

creot

u'dd mdgjml conditions &d ce

be @mpur€d

sine (2 E l) sisndne

i5 crirical when

THE SOFTWARf, HILAI-OI.CPP In this worl( a softwde k develop.d for the

odysh of

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 rhe

ecie.l dd Doden visibiliry nod€ls. T1| lisdng of the poelm i! ApFndix 4. Tn progm f€nrca d bri€fly de*dbed b.to*:

compur.rional and thc

Hilalo|.cpp is giyco

Th. prog@ LinAnal ucs ih&e dala filcs, rwo for input

one for oulplt. The

For.ad.rv thal conrains rhe padrcr€E 4, ,, and., (s inanicle2,6 lbove ). The pdmetcs oflhis dala file de adtu8€d in rabtes2.l

two inpul dala described

il6 e

(he files

(A) lo 2.1 (F) in appcndix 2.

'Ih.

dds.d

(D) in appendix L

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€ inbbles

Thc third

(A)

fiI. u*d

10 1.3

ol]],er

prcgrd is *..t lhal

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 by lhe

is optional

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

Latil., th. ladMe oflhc

Eld.,

ple

ftom wh€E rh.

clesnr

is ob6eN.d

ple

the .lcvarion of rh. place above

levcl,

T.mp., cnimred rcmFdturc of$e riDe of obswarion,

Hmid.. .stibaied relative humidny oflh. plac. S@.i, lhe l6al tinc of sNet

at rhe

rine of obs.Nadon

JD of Conjunct, rh€ Julie Dat of the binh

10.

As., lhc a8c ofrhc Moon.r

ll. 2

L{C,

ARCL, dc ot light or elonsetion of lhe M@n

ll,

ARCV, the eladve altitudc of fie

1,4.

D,AZ the rclaiile @ibulh ai the best time, in degrc.s,

15. 16.

Widtb, the cenlral widlh of lhc

the diff€rcncc

17,

tim. e@ding

berNcn th€ Moo.

q-val. rh. visibihy

discssd

rh€ besr

*r dd

crc*ol .t

Fsr€nt

ptue1q

of.q 10

M@n or the

lsr

Yallop (Yallop, 1998),

it minucr, su at the b.st tme,

rh€ sunsel.

fion th. the

b.st 1ioe,

al bcs1 rime, in

d.crees

dc minues.

dcfined by YalloP (Yallop, 1998)

lo h.

in d€rail in chapl.r 4.

Phi+della,

fie dglc

thal lhc.cliPlic dalcs with lh€ venical on th. wslem

hoii2on, in d.sE.s.

E. 19. 20. 21. 22. 21. ?4, 25. 26 27. 28. 29. I

Mlalit. I,loo.

4liptic latiiude in

d.8e'

Mlongi! Moon's.cliptrc lon8itudc. in d.gtus-

*liptic longitud., in dccres M-SD, dgule s.minid.tcr of th. M@|L in @ oinut Slongil., Sun\

As-Fac!

of sep@lioh fsctor d€fined in l.ler chap1e6, in dcgrc6

R-En, cslimat€d Rip.n $ Fucrion vatue dclin€d h chapler

R-A!er, averas€ Ripeness funcrion valu€ dellned in chapl{ 3 R-actual, acbal Rip.ns Fsction vtlue dcnled in chapls 3

DR+s! $e diflecnc€ of R-rclualand R-Enimaled DRnc! thc difrccne of R-a.tu4l ed RnErage MoonS M.g, Moon\ mogniiude al the besl

lne

of Yallop,

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

m.gnitude of the Moon

dd $c linning

m.gnitutle of rhe sky ai thal monen!

B6(Dmag), fic divdsd rimc *h.o U. @ntrasl of rhe sry bnSh,ES th. Moon\ bdghret is b6r in favou of ha9irude of

rhe M@D

rhe Moon

thc lim'ring m.gn'nrd.

ald thc diflcn@ of $e

ofih. sly

al rhar monen

Lsr(Dmae), ihe univdal tme wh€n rh€ conrmst orrhe sky bnshhess

32.

tbc Moon\

biehd

nagoirude ofthe Moon

lal io &vou ad

rhc

of th. Moon

rbc difcrcn@ of

liniring magnitude of$e sly

ed $e

th.r momenr.

A3 thc .x€cution oflhe proghh b.gins ir prompts for ob*Nadon nMb.r, dale,

nonlh, y.&, longirude, Iaritude

cl.vation abole sea l€vel of th€ phcc and $e

estimrd r.larilc humidiiy of lhe place. This plompr by rh. lunction tzrdd,r.rtnd callcd by the tufuri,on nt i n n u t i

esrinated ttmperaturc ahd iniliared

"..

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

rdttirgt.le

b6ed on lh€ alsorithn dhcus$d in aaicle 2.7 abov.. Dr functio. calculars rhe b6r rimc for casent visibitiiy aeording io @ndnion du€

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 to Yallop (1998)

rh*.@rdin 10

@ c.bdat.d th. etr€c1oftuhio. dd

the

sidd.d tihc coftspondins

zdo hou uiversal tide, for the dat€ consideed. are calculted using the tuncdois

,ttttid od

l€s

ad srl_tiw,

*'€.n

E#iv.ly,

t$aiqs ouinlo,

usins rhc

thu fe Scn nted is displayed scootd (.@fitn^.s of 1he su.) ed

The infomatioi/d.tA

rtbpt

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

orher functions

O$d pffi.r.c (l7lo 20 dd 22 1o 12) re @mpui.d in (r? to 20 in tunction r@r_cMtt a\d 22 to 32 tt liM'si\ brl anicle.

display€d alo.g with these pdmet.rs. All thc the

outp

dda

(p@ncle6

Ddmd.s lisbd as I lo 32 d€ wiren lo file J.r"8ear in nmloulrro4, (pam€rs I to 29) ed tim._nnse

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

The Esl

of the functions

/.ar.r..l

used in lhc

pregrm aE lisLd

dd briety

point

dcscibed

y.e u$d is lcap yed (ac@rdi.8 ro crcsorid or rcr? lfrhe yd is lap lhc tuncdon rerums I orh€fli$ O.

chects lhe1hd the rule)

.orwt_r'ru

convens an

.,,rzt_rtu

convens lime in hours into

hodlw

Btms rh. @jnd.r .ff4 dividing

th€ inpur

h 6ed ohain 6e mst. in

pamt

eelc

inro dcgres, arc minutes

arc

*cond3.

hoE, ninuIs &d s*onds. tuB€

160.

6is fucdon

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 the

deercs

on step 8 of alsori$n in anicte 2.5.

itr._r.. rhrcud ,rr.-do, ,nd d"c_sc thrcugh chdging tine (al (at lev.ls ofdale

d.._r,,:

Ttcs. tundions altows

ld.b ofhourr ninure ad ssnds)

dd honrh).

and dare

sihin lhc tuldion di.i@{tt. lhc prcgr&n Hil.lo! ha do-|9kL l@p ^ (pl*in8 dld le,) fion thc Br. rhar remiDrd *tEtr rhe iddrifid za,r rcccivB V' If my oths key is prcsed lh. pregm rmdN in wait ndq PEsing Panicdd k ls a In fer

is obvioN in the

er.l{ar.

cohbilsriotr vdio{s&lions

iniii.l€d likc inc@ing or

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 deoeasiry tim€

datc,

s.r'"a4.t

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). ro fie out file

in one lin€ a.d prc$ins 4

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€* In view of lhc

prcbls

IE sm!.ndylicd

dyn.Dic.l lhddcs VSOP87 and ELP2000 d'at dryib. th.

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 motion

ephendb oflhc

Ite

Sun

dd th.

Mootr.

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

isues. Wiihout having a @mpl.l. kno* bow of lhse isues apprclrjalc

argmcnt for the d.lminadon of

lud ed dE $l&

cmrdimres

tbe

not be

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.

lb pcr@i !rd&d tflt t Fb&o le or*dt ci rdt!. da-!.d. dD.r

di-

dShdrS o{

rtr

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

plobLa ofib. d.olloilirS

tl.

dry

.ll

tt cqdrilrl ,trrit r@id.d

of6rr jihli!8 ofew

lurt.!..c.d

eith

it.

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

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

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 ln the

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

wilh the

is done

Babyronlan and the medievat mod6ts.

BABYLONIAN CRITERIA No lrisonobery or spheriqt rrigonomelly existed till rhe developnenb in,\rabia in lhe 9u descnbed

halheharical

rhc lorh cenlu.y AD; the probteh

rems ofcertain obseNable paranercs shown in rhc Fieure No

w6 o.ly

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

gon€

deerees

bebw norizon

(the sole deprc$ion) and rhe crcsenr at M isjusr visible. AB is rhe equaror lhat nakes

agle q, $e geogBphical latitude oflhc ptace, wirh rhe nomat rhe sepeadon b€teen lhe Sun tud $e tvl@n comhonly knoM an

ro the horizon. SM

s .@ of lishl. and

abbelialed

ARCL but shoM

difidce b.isFn the q "dc of dcs@nr" d€noi€d 6 ao (ako call.d

in th. figue. SD is the

su dd rhc Moon lnow .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 alrirudes of the

Sivcn lalitude

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

modem tines the earliesi Ffee.cc of

sigbring of nev

lud

sighling ofnew lunar crescenl

syslemalic study

cEs@nr is thrt du. rolhc.inShm (Fothainghd.

of the earliesi

l9l0). He refe6

ro l2s enlury Jryish phildopher Mcinoni<16. Asordine to lorhcringhm. the work .ppearcd in the

TEti$

by Mlimonid.s (Moshc bcn

th. Ntu Mo@ 0",islre''.'rotalt: Scfq z.hnim

Md@n) o\ \\.

SahcliJi.otion

Hilhot Kiddush H'Hodcsh, I l?8

fial

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

hgk ofrision

dirlaccs then

tle aplbre.t ahgle of tision - F6\.noghe

slarcs rhal

m.es the diffeEncc of the Sun md the Moon in rc.irh th. nle of Mainoid6 .nd his om rule for hinimlm visibt. pb4 of Mai1onides

Moon ae n€dly the

enc

Howevcr he f!fther says lhat Maimonides rute sivcs

slidrly

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 lower

undedand Maimodis' dithmdiol nelhod same (Foihciinsham,

solving

has not giecn a dchited

@ounl of rb.

l9l0).

Lder, Bruin(Bruin. 1977)gives

moe debiled

fie probl€n of findin8 sronomi€al

eounrof

any medievalatenpr

€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

peeds

a conplerc soludon. He aho nenlions lh€

AlBamni

lato accounl ofMo$s ber Madoon

lnd clains rhst ibn Mamoon larsely follos Al Balloi. One oflhe nost inponanl dp.cl

Anbs^rulins orthis ea had olEady rcaliz€d rhe sienitlcece of cE$cnr widrh. This inpoiant aFcl enaincd

ofBruin's accounr ofrhc eflods oftbe bedicval

era is rhar

hissins frem all th. najor cooiribudons of lw.ntietn century bcloE Bruin. d€scription

thc modein loowl.dge in ihis work lhe sme

.xlecively od is pEsnt

latu id

Bed

on this

been explocd

dor.

the chaPler.

lilcatue that app.ared duing 2od c.nrury lhe gencral rule of Babylods for first visibilily ofnew lunecresc€nlis:

lual su6.t shottd be g.ater than 2't hows the su6el ohd the no6et thDuld be g.ot* than 12 tim at the

titu

(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€ to lhe histotiel cflon5 of lhc Blbylonians dlsiv.lv &d have epon€d rnal eording h Eords of re* c.Mnl siSlftings ol PG Christie m lhe cderion attibured oi Babylonis is d ov€r simplificalioi (Fabohi.t al 1999) Ac@tding lo $en $udv 209 recoids of Posi{ve new cr€sceol 3ig} ing

extdctd fom Babvloni&s Astronon'car

mathem'lical lunar lheory di&ies. the Babylonitus had succceded in ioduladng a lrulv wrlhoul $hich ihey 6ed to pEdicl !&ious par@ele$ of lumr notion Unfortumtelv aboul be presnline my iheorerical d€laih of Babvlonian eE lh'v appear 10

"ilicdl

above Bluin\ slgscsion thar Babvlonia criterion was whar is mentioncd as Follo*s: lnst€Ed.1hey clain that Babvlonian ctilerion wEs

Tnc

ies cEsce.r

is een

'' Elongutio, (ARCL)

(l l l)'

+ nooe|st6't

laE

tine

(Uc) (3.12)

Even, rhis cdrerion is

apon d on

(Neue'ba@r' b6sis of the suee€stios ol Ncusebaud of a madehalcal lh'orv Ttev have

d'$rip$on *bich th€ lt@ rhsrv ol Babvlonids las nehtioned two svst€ns of sol& moliors on 1955)

&d nol

a rcsult of

wD drived al $ nol hoq lh€ new rul€ they hav. attdbded to the Babvlonims h8d on lhe giv6, They have gone on lo plesd .lifleant valucs fo! th' @nslul 'kh lalEs Engc fton 5 DiniDum of2loro a maximM ol23" lnlhe

based, but

;ideof(3.1.2) Th€se

p!€ent work we adofl fte followins slss.sted by Fatoohi el

as a bener

.l:

critcrion alfiibutcd lo de Babvlo'ians

(r. I

The

Mon

divi.s

d6dibes how bnghr

ctcse

@uld

this qitcda

paluelea moE

aurhos is thar (D ihc

of lisht

th. cFac.nl &d (ii) ihe moon*i*utuel lag de*db. hov lons the

coaii

6 d.sc'ib.d by th*.

abovc

is the

horian.n

r th€

$n sl. MoE

is rhe vql@

ofqch ofdrqc

po$ibilily ofsishdng offie ce*cni. Howver, drcr. h6 to b.5

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 conbined

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

lerion was

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

succestul in 98.7% of

orsdialios haYe smnged for coll.cdon of €r.*ent Th*e inclu& Isldic Ctffint Oberyalion Prcjdl lh

In r€ccnt lin€s a number of sighting or non{ighline @ords.

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 of lhes dords ofdening

Usins

(l.l.l)

is lhal by Od€h (OdGh

cEsnts el.dcd

appli€d

the dar.

for

2004) we hav' *lecred 463

tE compdison ofrhe nodels

9l clet€d

for this

Ll.l.

wlk

the

studied in

'6!lts

u€

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

pEs

ed in the lable no.

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 ? claims of

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 table

i. App.nditll

obseffation No. (odeh, 2OO4), dare ofobsryatio!, latitude of

lonsitude of rhe location

ob6dtr, visibihy N' for uaided visibilitv' 'B' for visibilitv thrcugl binmld, T 8l

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

for visibilny

@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

@ 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. cresccnl there

is foutrd lob€ successlul in 96.4% c.ses

css

in 59.92%

cal

success

dd !frongsl

lhe hegalive sigbdn$ ii su@ssful

of€ model in desciibihg posilile sighlings alonc is tul a ofa nodel. Thc model is not suned seu if n is nol able to dc$tibe the

negalilc sishfng

The sucess

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).

is inportant

b rrisil

*lipric nlkes wilh

Thecfoc, il

lhe ne|hods thai lead lo lhe ddelhimion oflhe mgle lhal the

the horircn al the

tinc of snel on the *$em hotian (il

panicularly significer in view oflhe fact that the new c!e*.nt appeds close lo n). This

ofthe ecliPtic slowly fislE no 3 2,I on lhe nexl pag. shovs lhe wcsrcm pan of

angle lalies as agaiNi rhe lixed celesrial equalor lhe orientation

laries lhroueh the year.

lis

$e cclcnial spheE wilh impodanl points &d angles

w' t,

m d de$nM

the west cardinal point,

rh€

P, the

v€ml cquinox, inl€eclion of.cliptic dd cqulor. Nonh

c.l.sti.l

pol€.

below:

qv - r

8}'r1

Z ZSr

rb ouiq'iry of6..dinic,

- 900 + 9, 9 bdte lb.lditd. od6. d...' - 9 + 4 lb dSb ofrb.cliP(ic xdft 6. v..dc.l,

rs-d-G5tlirst

tu,

d.clhnioofrn

- 90'- o, St - ?e 6. lonSili. ofd! PZ

$'!

FrsNo

FMn

3phicrt

1! 6su!

ti8oiodt

32l

3.2.1 in lhc tPhdic.l

tt.rdA non &. !id.

ttrtgL oe

SPZ

gBs:

ioo

b..e..o d..lidlor

1.l{ of ctsiE of

(Jr.r)

tlne=cort.o.PSz

'ft. g@d |!l

'pplvi'g

lonSind'

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

i3:

(3.2.2) (3.2.1)

FDm

ridcle sh,

st nir8 !r Pr:

@s(PSz +@+

usins (3.2.2)

A)=rer.@r,1.

(3.2.4)

(3.2.3):

0.2.5)

6d (32.t) tosethd *irh

(1.2.1) leds lo:

(1.2.6)

(3.2.5)

(l.2.6) thd sivc:

Ju"' elipti. nEt s wirh thc v.nicsl is 5.e! 6 n d.peo& on the lorgitud. ofthc su only for a fix.d place {or latirude 9).

This show that e + A or the .ngl. thlr ihc

dcp.rdcnt

ln th. Esr of rh€ di$usion in rtis snicl. rhc anglc ncxl

.!ticl. etr@ cv.r In

? + A is

L*d

it h

olol&Ld

900 -

on ibe

q + A is d€nor€d

V. In dE

6is of (3.2.7).

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

ir ir simply

cr.!ent, TLse circmshnc.s aF shom

impossibtc io 3a thc

vdal eqlinor. Sm thar is just *riirg

in the 68ure 3.2.2 on the

Cl, lh. cclsli.l .q@lor, 1 i! th.

al s.

TS is the diuFal path of th.

iii)

DE b thc diumal palh of the Moon th.l s€t before thc

6s,

decliMtion

Ds =

the Moon

zcvs = NS

vi)

whee

DM =

3t E.

letpendi.ula. to the c€lesial equlo. is lh€ dif.rence of lhe of ihe M@n &d the deli.ation 6s of rhc sun. Declinalion of

south of lh€ sun md rhe upper limn of Ds is 50 9'. lhe inclihalion

oflund orbit

sUNl

to lhc

9Oo

-4

trliplic. is the

agle

bctween

$e c.l.srial equ.tor lnd $e horihn

lhe lotitude oflh€

Pl&. TheztSD

teprenls

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

vii)

sF = pM

0M is

c.l6ial taliluds rhelinilof5'9'.

lhe

viii)

aftr co.junction

p€,Fndicultr ro

fi.

of the Men and lhc

E lifljc

is $e difiercnce

As for DS

bctw*n

never €xcads

ptrallel lo FE (E' .ol showo in rhc fisure 6 it almost coincides wilh E) h

the

ix) lislhe Vemal Equinox ed Z)6t = v h !h. angle of lhe Ecliptic with $e lary ftum zeto horizon. Dep.ndniS on laliude C and lh€ se@n, 'Y naY is alons the botizo') lo 90 deB€6 (when €cliptic is (vhctr ecliptic

pc+endicula! to lhe holzon)

cas' when

is small

thal has

paEllel lo the 4liplic is nuch ltrsei tha' DE (ftol is paEllcl io lh€ cqualo!) of$is oSle md and hdc. E sd E aE mrch s.panted Tlle det€mi'arion

sienifidc. shall be di$sed in the n'xt drcL I'lt figurc shos rh€ SM is jN| about o sel in a pl&e oI sdall b n€diud *l (dd M@n havins ddlinarion eulh of il' sun ha al@dv

its

laftude d , $e

set at

poinl E)

th.l lh€ LAG = TiG of

Mdnct Tim of suNer is

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

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: tha

lun|r

(328)

sioiLdyrt rrtdSL

AEFS b

dlo.ndl

r!d. t codrt.ld|plEdilati|lgbqi6

tilr,.= Pcpl$iry

SD by 5M

tN Ag.in in ISDE tm

(3.2.e)

E!, SF by

illn

-ts-$N -6d,H

(3.2.8) ,nd (3.2.9)

02.r0)

t!rc:

.DE

DA-@,

-r=S, ln

!! lid .llrdrudi8

-6,161

(3.2.U)

b frrr! !23 $,ldd It dlr d. Dlc ntDglc Ft sDE of6. Flvior6 68rta

rEql'itld.ortun.8 dvcLAOi.i.i.EDM<aEhr: DM

(l.2.ll)-

(d{-!.)

-(au-asr<Qu -,s)r$a

Pig.3.2.3

4.2.t2,

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.

csjud olcul.€

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) ln rh. hod.m s.tup oE

rh. Elarivc ,ltirudc ARCV

$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. .q@to'ial

Mdidle.

of lh. Su

Ne8{iv. LAG ces

on thc da, olconjun tiotr

Frlly oeu tom smdl

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 tsrl

b.low:

dllc of coijuiction.

tncd Timc of Conjmtion,

su!a, As. of M@tr .t le.l suet in ho6, Elongadon or Moon rroo tlF 56 in dcrFs,

Zorc (PST) tihe of locd

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

rable No

3.3

{u6el

in oinut

3.2.1 : posirive Ase

Nesaiifr&B-;s-i66l2oliT;R;;iifr;;

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

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 @

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 Figua (l.1.

)

bclow

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

horiar

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). Ddrall€l Io

ih. .qudor. TIE lin. HS ir

\:\.

Fig No

l.l. | : Trigmrdic

ddiFin

of @rdni@ of

luE d6ern

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 MT i3 p.rDodoio to

!cvl.{HddT-9+A.Itrb.ioplyk &b6crdtio: ra-Er +rS-t,6lp+D+L |hir

Fldih

.e.

.r .$|d@ O)

of B.uh O9ZD.

Irtia8b

HgB b

ABS

rlcd s . ph.

(3J:)

codr+ a)

{d ti.lgL

(3.3.1)

lad! lor (3.33) (33.4)

(3.3.t

'Il&

eb &uii

tih in hi! .qurid (a) |b h cbiD. nr b..o obi!.d by ld4 lhe tigoffiy. Ittt smc 6ru. i, 'slr

(3.3,5)

t'.d.d ||3toglpt ric.l

i! dimalot

Aom

!i!@.tty, turiotbADs

f|(!o .phqi6l trt|tEL tlTMr

O9,D t&

8it&.1

sin

rr'r _ rh&

sn(e+a) erhlr I.

(3.1.t)

@s(e+a)

rs = sili(sin r!.r..(e + 6r+,L

-,rr

0.l_6)

sphoical iridglc ABS:

stnaD

and in

4r @eP

(1.1.7)

sinfls=-l]1gll +

l].l.8l

stn

spherial lridgle HCS:

co(@

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

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 11'e sienificdce

sUM

for

day of lhe

yw !trd psrti.uldly

for rh.

dlt

or rt y

.Rc

UE bnh ot ncq

Moon.

wlee6,

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

oncc rhc

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 u$ or (3.1.9) dep.nds on trE vjsibifty @ndiiioG, @cly lhe

o$( hahd rhe mininu del. of

s@ml'on tu, thar.& b.

kmh

only on lb. bdis ofa

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

48 ninules of timc.

-r1li5

ks seen Nhen rhe m ot seplFrion dJ B t2 d€8res or teads b a siople Eladon b€lwn /s md lhe aduat semi-

^-Using rhis etalioo thc

*p@tion felor

@ of *pdlion

necded for ce$enr

is .alcutated in ihe sonwar.

Hilalol.

r0)

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

for d {ider but bmly vbible cresent mininm as w6 only

l0line

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 degl6

(40 minulcr. This

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) 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

as henrioned by

exccds lhar @lcolared by (1.!.6) rhe work this rule has ben named

M6lin

c..s

would be visibl. orhcNi* nor. In thn

Lunar tup€ness Law or

According ro Bruin (19?7) $e valu6 ol HS de.ored (3 3.6)

seE pGsenled in

rhc

sin

R()., 9)

lled Lunat Ripenss Tables sDd

Lunr cmcent ws walulcd by Mulin BltonomcB HS a3 deived tom (3.3.6) h d€nokds,l4 and:

sifrply Lumr

Ripen€ss

obkin.d fron

fisr visibility ofne!

usiog rne sboE rul€. In lhis work

r[sin/, r.nre+^t+lM ts

(l.r.r l)

^d.,

ofl,, sin ly is also smatl &d lko n ahosl sme as rv ,j. Bui when the *liplic is wll inclihel io$&ds lhe Fo, a fixed pl&e (e constal0 it is noted lhat forsDallvalues

hoien (q

+ A is large bu lcas rhan 9oo) 1r'4, nay becode noE rhan

snd ecliptic is roweds nonh) or renain le$ rh&

1! ri(iflv

1, - .ls (ir pr, <

0 and etipric is

iowdds nonh). Ho@er,{da is endely independenl of Oe se@. aod dep€nds only on

to ihc ACE

rhc rcladve coordinates of rhc sun md rhc Mood. As rhe major componen in (3.1.t

lhc diferenc. ol longnudes of rhe

and lhc Mooh ,taa

h closely linked

Mftovd. ir sign

0.3.1l)

shows

differcnl signs

x&" * l,{ - k. Bur whcn boil 0M 6d I + A hav. the s4e &4 *ould b. mor. thd l, - & and if pM dd e + have o,

would bc

l.$ the ,L -

The vahes for HS ohained

.ts.

Bul thcF v&iatioos u€ nor

^ sMnal

fom (3,3,9) dc denolcd s X,b and:

12)

^,,=.'""( For a

fi&d

ber@n the

place (9 consrd.t) R,, dcPcnds on ae sun

shoM larer. Using

the Moon).nd thc

(he

s.en

spa6tio. a,

+ A is

*son

(rhe equabrial

fiat shall

b€

fo! $e dav or

lh.

dependeot

looh sod le hniqu.s di*ussd in.hapter 2

ie,

t'Ac

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 day alier conjunction aor s placc of ob*ryarion

i3i

'"'(

a;t+,ry

$i"

-rsj'o (3.1.r)

Tlus Lunar Ripeness Law that piolidc

a solution of the probleh

rhe first day of visibility of n€w lunar cre*enl

oldelemining

based on the Angle lhat the Ecliptic

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

nat6

wilh

the

suns for &y leation.rc applopnab value

oi4 d

calcuhted. (3,3.1l) allo*s onc lo calculale,e,4

sivenby(3.3.10), in (3 3 12) the valu. 95

ofx"i ce

ed sine be

tound ltr

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.

dep€ndent on the rime of

el.v

val@ of

degc.s

an appopdate value for the day

bdis of

the

lS6. foi

dill@lt

of,,

vary from

sell

l0 degcca lo

oi day after conjunctioh is obtained only on the

ule disraice o. sminimd.r of the Moon. Still tor

compdie.

values

,, = loo dd

calculaled lor both $e cxrene valucs

curves,t,r aeainsr the longitude oflhe Suo (fiom March 2l) aft ploned in

FiSuGs (3 3 2) lo (3.1.5) These

nd, ats not

&d study ns veialioG scenally

$irh chdging lalilude of pl.cc. As possiblc values

120 and the

hence

ye.!. However, *ith posiblc lalues of4, one cd calculate

for @h day ofa

ladrudes,

cw€s

snow inleFsting featres lbat

from fi8.1.1.2 il is cl€ar rhar forg = sinusoid.l wirh

mdina.t

i... for

a placc on

March 21, Septcmber

D4ember 21. Snalle! v.lues of rounger cEsent m.y b€

rn.

iG nems

b€

sunmdized

$eequabr&i!

2l dd hinima

at Ju.e 21.

shaller lalues ]?d4 or !€lariv€ly

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

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

d.crcae funner making ir esier

a younger

cwc ahaFs clos. ro ll€ alhlnn l m6d thc solsti.6 flltt ns.

F g. No.

Ll.2 X* forg=00

Kg.

ffi

.r.

BcyoDd tropic of

hsimu

al lhc

.alF

rl,

;

210

180

Fi8 No.l

iiD

.,.-

rt'L forg= l0o

two of the infl@don Poids

aotmnal €quinor

simPlv gone

&d $e

tunner malir,g n FoE difficuh lo

(f8; 3J.1). Tb nioidsn oftL qttc Ed,. tt !t .quinox mrli!8 it .did io r.c yomg.r @3@l! n tr ro â&#x201A;¬ml .qu$ox . yomg.r

iv)

ct!.c.

ttl

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 rG

g6.r.lty .si6 ro @ younScr cr!.c.nt3 cl@ clqe b aull|nn l cquilor.

vlrnll equimx .ltd

diffrct lr

'17

t6 13

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!

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 For the Arctic Circle .nd to it3 lorth (latitu<tc|

c@!d!d

!o thor. clde

lutunn l cquiaox

!.d

Fi&No. 1.1.5.

Howev€r lhe narcr i! lrot

i^fore=61

sinplc beca6' tuc olc of latitude of

rhe

cKdt'

plavs e impona role lhai i-., whcth€t thc cc$eni is !o!ih or north of ihc Sun dcFndst Now lh' stDns nuimum dercmin6 rhe v.luc ofrtdo lhal is lronglv btnude vbibl' lttitud6 indicat's thtt e old'r M@n rov rc1b' at lhe

v.oit

.qrinox for hi8h.r

ThtebE n m'v b' nisleding

b s the AGE offt. Mo@ iNrrsd $ do6 iG elo'sdi@ ad brighlress

valucs posibL ftat invisibilitv $g86l.d bv $' Rrpcnas tun.lid vd6 drd sm lqr€lEof x'L h rlnt Thh inly l.!d lo ltlollN for sdalld LAG dab avlilablc in lilettrurt e $!d( th. RiD.G$ tulcrion vtl@s for dE ob!'dltioml dx' s't th' c!56 whd thc cscenl .dculacd sd pr6.ntcd in ApFndirlt Out oflhis RipGs ltw rE also s||os nor h lg.mdt *nn dE L|M is

Eporr.dly

bui .rc

(Odeh' obae liontl dah k $lccLd AoF lh4 tPo :d bv Od'h re cosidercd onlv Ttblc ;0{a) fron *hich c.*s or cvdi!8 '6cnr obs'naiioN (Od'h' 20Oa))' dtt' of obseNttion ldnud€ 3.1.1 shows o. No. (th. oberysti@ nunb'r LAG' snele v of ecliPlic sitt' ,ftt longin/. of ih. t@dron of obaa$s' M@n\ !8e and ed th' 3d' Rd giv'n bv (13 12) rrorLo, t"lio,a" or Uo.,, toogirudes oI the Moon d'8r*s' Ri' sircn bv *ill !6 siv@ bv (3.310), Re siv€n bv (3 3 12) *ith 's = 10 5 O.3.I l) and a&. = Rdq - &"

in nblc rc. 3J.1. TtG

'Ihe table in App€n.tix'U shos S. No , the obseNation No (Od€h 2004)' date or visibilitv 'N' for lbaided obseNalio., latiiude dd longilude of the locaiion ol obseru€r, lhouel binGuld''T for visibihv thbush lelescope The

lisibility,'B' fot visibiliiy visibility coluds d€ empty

iV' in an the.res.ent wd nol s*n sd conlaitu oflhe t8ble contarn .ppropriltc colmn ifth€ q€sftnt is sen The rcsl ollhe coluons file columns de fot the live age ofMoon , LAG. ARCL, ARCV dd DAZ The ld the Babvlonid model' models consideted in this chrpter The colu. headed A is for

B and A for the Lunar Ripeness model The colnmn headed

cod'ins

AR"

,Appendixll shows lhat oul of 196 cdcs in shicb $e crese has thee tle onlv 14 cases hat do not b€en reponed to hove been een wilhoul oplicdl aid rhcs obscrydlions de tinther obey thc LuMr Ripencss La* stalcd above The debils or The nodificalion is below *ltn a Modi6ed L@r Ripeness Law is sugeest€d The lobl€

exploEd

needcd in ordet 10 sepdarc tbc used for clescent

caes of dked eve visibilitv and rhe caes when optical is

visibilitv

wnh the helP of A tot.l number of l2 posiliv€ obsedations ih tbis dala hav€ b'en vilhout oplical aid dnd the bin@ules ed klcscopes when $e cr.$enl ws nor visible

Luai

was deduced $ ork

.ot elisfied 'nth is logicallv valid as the Lua' Ripencss Law durine thN onlt for mlcd eve ob*tvatiotr Tlis bas he'n the dolivadon

Ripe.ess Law is

lo ood i fy th€ Lunft Rip.ne

Law ro e ocompass the otricallv aided

obse

d'tions

ofihe table contain Ob$flario' Ssial Nunber as led bv Odeh ol the Place Visibililv (2004), Date of ob*tvation titilude oi lhe Place Longitud€ (wirh iele*ope)' The* colunN N (lor unaided visibilirv), B (wirh bimculd) and T lisible R€st of lhe colums d. €mptv io! invisible cresent md co ain v for The columN

visibilily

fo' the best ime (Y6llop' 1998)' Lae in (ARCL)' the angle v = nin(es, S.padtion b€twc€n the Su ed tbe M@n At of LiShl hotizon on the wsted holizon for the dav 900 - (9 + d), that thc €clipiic mai<es wih ile Is th€ latitude ol $e Moon' l'M rh€ longtude of rhe M@n

colws

contalns Ase of ctesce.t in hou6

of c'lculrtioi/obsedation,

100

the lonsirnde orthe

sd, Arc{f-*pdtion

Iuclion, ,{.r olcularcd eqution

fa.tor,

Sitd

by (3.3.10), Esdnat d Ripencss

(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

196 positive siShlings that are nor according

pGitive sidrrines out ofa rolal nmber

lo lhe Lunar Ripe.ess

(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

Coup-A for thc Modificd Muslin Lund

''tt th. tlill..e,ce ofAv.rase

Ripencas

Np.ts

Las lhal we slal. ih ihis

viibility cdes,

grcuprd

see

nE n.N

erlq 6:

luat .rcsc.nl

coup-B. Therc de la (llolo) mked cye

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

ca*

Fanctior R-, omt the Actuot Rip.h6s

Fu"cion Rro F /Re) is l.ss nnh -J.6 it is ihDossibl. to tor att ldtitula uith or Nilhott optical aid'. The ncxl I27 c6es

consid€r

14 natred cye

lisibiliry

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. in thn range, threc lery low AR{yfvalue cdcs have comon chancterisdcs.

l0t

All lhe* conditions falolr (1.41) that

aor

lisibility and it wds ncntDncd above in view of oldcr a8e cresccnb (/.M ls ta€e) sh! er arc or scparation may be rhe

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

deg!€es rcspectively)

buftelorile uihulhs

7.34

(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!,

taree

alosed in such

All d€F clains de 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 _ cases.

fon

1.. Alt rhe* rhr€

obsen.rion de rncgR.nedqrrh rhe Babytonru cr.rerion Therc is no fudhq c6e

dong$ otherposilile

cases

ofcesenr visibitity

wilh -1.6 < AR."r <

t02

tjl.

varue of

dRq < -1,6. Fbh

two de very young

ccenr'

Thee

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

in ldse

sperively)

visua.l

only). with

in both

dimcler bur lh€

$all

c&s

rhe Moon

cltsnl

widrhs wcrc

the rclalivc eimurhs (4.6

l.8

very close ro the Eanh rcsuhine

sb.ll

(10_?

dese,

&d ll.9 e. s@ndr

the crcsent eas atnost

su rhat bring it in ihe ided cordnion foi visibiliry bul lh€ snau (8.5 ed 9.1 deele) nake rhe$ claims highly oplini$ic. Borh th6e

ve'liMlly abov. the elative altitudes

obse^ario6ec in d$dgreementwirh

Lh€

BdbyloniM qirenon.

Out of th€ 14 positive obseru.tioi wnh

A&r

< O,

ihte

werc very

f.ini

cEscenrs

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 i.e.

opr'n'strc cnlenon

well. Two otthcsc {389 and 455) are sko in djeereocot

{heec

341 is a

How€v€r, for all

mdSiMl6e

$ee

Babylonian

in Babylonian qirerion (with ARCV + LAC =

I I 9 c6es when $e

wirh naked eye,lhe conmo. ieatule

viri

crescnt wd daoed

ro hdve been

sen

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

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. .tz vitibilitr h dlMt tizt

"oh.d

N€xt is the Oroup-C $at mnrains 76 cases wilh AFrv in rhe dnB.0.0 ro L6, In lhB goup rh.r. 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

nor be made. Srill, the

''fth. wtu6

thid pan oftbe [,todilied Muslim Lund

oJ/R@ tk 6ztwa 0.0 an.t r.6 ttt. pEribitiiu

Ripeness Law is

of,i,n i,

oJJint

cftsceat teith and tuithout oplicdt k ,trong fo, higlrq latitud.d,. The Croup-D containine a ro|at nlmber unaided

vhibilitr oflunar cscs

forr'i!'

of 221 c6es

has 170 cases ofoprically

> 1.6,Il is

^il r'B@ >t.6 ttt. posnbitu, oJ ,bibititr siorg lot boh towt aMt hiEha totitd.r".

li61 c6c.nt fitttott opti.dt b

Finally, the 3umnarized Moditied Muslim r_umr Ripencs

ZR-< -3.5 ihpNibL 2,

to tee

l-awr:

th. n.v luaar c.6c."tlo,

a tdrru!., ,ith o,

/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 -3.5 <

104

0.0 < .1R,. < 1.6 p8tbiti,tr o!

,hltw

Ji6t M@nt btrh ond withoul

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 optt ol

ne,>

1.6 th.

pNtlbnrr

oJ'

tbi!i!, oifust

cnse

wtuhout optcat

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

tbeir

d.h pto(cd for Alerage Ripenes Fundion &,,

Fuiclion R6y for borh rhe Eponedty inlisibte plees with laritudes L8N, 33 95, 6.5N

Figure 3.j.6 for tadtude

38N

the vilibtc

(FiguB

and the Acllal

oe*mr!

Thesc jnctudc

3.3.6 1o 1.1.9).

L8N snows tbe b€sl slc

,_J:'iff:ln:"11,'#:J:

sieh,incs ou,

of 6 (83.37") in asrcen€ wnh $" obscdat'ons (6 our of6) e in agrccm€nl wilh the ta* toi this 'rown-(hnnde

Ripen*

ll.9

S) wirh succcss percenhgc 65%

(ll

larilud€. Nexr is Cap.

posrlire siehings in sgenenr

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 od, ol

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

4 15

't0

loo

Fia No.

3 3.6:

zlo

actual RF

400

br in\isibte . e.lrrr nr Or ri"tOre

Rjp.@r Fundion for r-ditute 3 L8 d€g.c Nonrr

---r;-+: ;b.,RF6.'"".b,. Fig. No

1.7:

Rip€lB Fudid

.acrld

for C.p. Tow4

t06

Sqth Anic!, Lgnbde r3.9

it€lr*

-111+.-Ig'1.rl'1 Fig. No

r.tl'::.gE::.

l.l.8: Ripdess Fudi@ for Lrtibde 6.j d|:gG Non\

20 1a

300

20n

J.E""'g:f*,","..,v"ltr" iU,ar nr b. rnti"itrc Fig. No 3.1.9: Ripdcas

luDrid

for Arhdq

]

lrriMe t8 dâ&#x201A;¬@ Non[

Loogitudc 21 7 d.grca Easr

t01

/too

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

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 invisible

both foms ac prc*nt ald

discu$ed ,bove. A rh€orcricalty visiblc cG$en$ (aclual

Rip€n6s Funclion valu. .ounts

noE rb& the avcRg€) is reponcd invisibt. $ar gendlty '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

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 ,a,n

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

Modd

o Ine deraits of

&d lne co.srE,nrs of rhe tupcness ctoser look in|o rhe figw3.3.1 ad tne eatysis of th€ obseryed to itc Lu.& RiFn€ss Las, d impond dp€ct success

oftheMslinsa

oara rn comparison

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

!.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 mwh

thidd fd

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 with hign6 bdlud.q etipric mch inclin.d rorsds tlE

Rip.B

Fundion

low th.. $c, e

&cotding io

rltmr.l

6glB 3.1.2 to 1.3 5 clos. jn lh. dilosio. of thc

.quimr? TrF luF ii ddd, lcr.rt @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 qploriig tu .el.ri@ b.tqq ARCV .d DAz for the ft51 vhibitity of tu@ cresn to tlE

o,^.

--. ig

Fig No 3

4.t: Spncriql Trigonomfiic ih..iFion

t09

6ndiiid,

lu@

de*nt

In cae ofMoon nonh

DAZ

and

ofediflh:

ES = EJ

J JS

lassine I es

-elr

@te

{r.4.r)

*6s - 6n

(1.4.2)

c4c ofM@n soulhof(tipric:

o,lz - o5 -

p1g

115

DAZ siven by (l.a.l) js much sfraller

"in,

(3.4.1) rhe

cos

rhar eivcn by (1.4.2) for th.

difirEicc of dqliiatioD of rh€ Sun qd rhe M@n (\ _ 6M) rr{ in (3.4.2). wnh k4er aRcL (SM') for M, (3.4.2) as compared rar8.r bul

or Rd. is

sdc.

Ahhongh

A&,i (Ra,

&ryJ

r.son tar in is sna er |han

|o M (1.4.r) &ry

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 optihue- aidnioo of I Oo (when

f.nhd frcn rh€ Ea.rh) ccu_

to rhc horiz!) DAZ

Moon is ctosesr !o rhe t dnh)

As and whei tu

vanish€s tud

l2v (,hen lhe

nor v€niet lhe

M@i

ol. DaZ coh€s inro play lnd Oe oplimum condiroN for as ca b€ retded. For hucn ls8er valu€s of DAZ 4d 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. these hodel ARCV is a runcrron ofDAZ so as should aho be a fuction ofDAZj

4,. = 4.. cosa

ad ARcy = ftDAz)=aL

- -[(DAZ)

Il0

(J4.3)

n@s

This

rhar for coBtaor LAC

(e a)

ARCV

decE|s

ed ARcv e di@tty lotDnionat ro ech L lrgc DAZ dd taige ARCL m@s mall€r ARCV

com|anr 9, t.AG

ktnude LAC.

Duing Elarivety ucnI lioes rhe dploorions of rhe @tist visibiliry

rutu cr€sc.nt or

rh€

rurc du€ ro

@leldtui@t

lat

visibjtity of rhe old cFsc€nr b.gd wnh lhe obstuations oad€ in Alhens and iG viciniry by Schnidt md othes. fteoretical

(Foth*inshm

exploradon was iniriared

E6otu ll)e

paniculd sbononiqt qusdon

1903).

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 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 in rh€ roumat of Philqophx ^, . 90) 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 1

aurtude

sun.

betow honzon at

.l$

fic 6oonkt,

or of lhe $tuer for dif€re o8uld dishnces of the M@r fom rhe adnitted abou his hl'en* .

the M@n at ttE

Motre.-;;;rJ,'* T,,'1:::;:;llTf moo. hade h the laGr Monhsn

in ,his rcead.

Schmidr and

bdh of

half of thc nine&flti century by Jutic Schnjd| Iriedrjch

Morn*n himslr n€

such

hblcs menlioned

e;liq euld be @nslrEred

on the

obFrystios dut were povided in Monm*n, s chromtogie lt8a3),

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: H.

Fortsiryhrn,.""-r,rffi

.!o it"EtoFd . mlrhoni.d Gt

rion lo

Mioioue Atrtr*

dc'qitc rtc k,E:

12..0 _

0r.0OrZ,

(3.a.a)

*** ni5 cod. d.fn* a r.sioD or sry lroud rh6 poiot of y* ]T1,:.* $Ds a $ow! u n6 tgu€ 3.a.2 b.!oe.

.l 6. d4 of su!.! rL crBc..l i! .boE iot T[. cruE h crinlit to. Fo6qingh.n,s If,

cu.rc ir ltould tc viliuc orh€.eirc

Fit. No. 3.al Fortdbgh!$!

lt2

clrl!

T.ble No. 1.4.2

Forh.dng[d\

Rutc

d66

A@-on

ii-

3' 1,1 lo&!

& ia

I|l

w. d.firc a pa@et . !r , .,visibihy pamerd,,

accordiDg ro Forheringhao

(aRcv - t2 + 0.00s21)

rn lhc T!bl. No. 3.4.2 rhe oD.eryaxon Epon€d by Forherineh@

tsl colhn

| tO

(1.4.s)

@nrajd ihe v!l@

ed rhq rh.lablc

fo, €mh

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

ccsnl

the

shoutd nor be visible, Thc

vdB

of Elarivc altiiude of the Moon

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 nor

hin*lfdnilslhat

$ Etiabtca tE sumary

his marheh.dc€l elarion (3.4.4) js

labte.bla. Thc eo€ is exhibile{r All ir'c esl ofdE posnjvc ob$dadons aborc FotheringtEn

i.lhe FiC@No.1.4.j.

3 vnibitiry

cw..

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: hsread

/RCv

dennc

= -0 0ot2g Da

+o.o7442aDAZ

dlkrure visrbitiry pamercr ya ba*d apProx'&afior d folto$: I L\ =

URcl/ +a.00929D,42,

+|.86429 or Seond

'.O.O14429DAZ _|t.s642g)I

lt4

(3.4.6)

d€88 L@r squd

(j.4_7)

Tabl.No.1.4.3

fhis nbdi6c.tio!

i! rn oL

by Fo$dnBltn

repsr€d by Forh€.ingh.n

.rd e$lrs

e pc5I.d in Tlble m. I 4 I

lnbL n

d$ pr*ft€d

in FiA No

s@nd d€gE polynohisl h.5 ob*N.don

(ofo.t

have

Dunng th€ the

hbl. .o

3 4 4

!t ir

TIE dat! of the

th3t rh€

lill€n into th€ mg€ ofn g{ive vd!$ of

27, 1878) a$d

lppli€d to dE obseutio4

l@t sq@e ntins ro a inprovcd $ylhin8 6ttF tN nore positive

I4.1

4ily sa

vd Th*

ee Nmb€red 6?

l7 (F.b 20, l87l).

sme.6

Mdundg

(l9ll)

@Nid€rcd &olh€r b6io data s.i given in

b nr $e obwnioml drl..

hvl3bL cro.c.nt3

. villbL cr.8c€nb .

Fig. No 343 Fothringhan

l16

Forh6rinoham

Rule

tute

F-d;:

Fi8 No 144 A last

sque qu.d6dc poltDnnl 6ncd b $jt d.L iGld! [E fi)llowing cl.rion:

DAz'1 PAzl

at-v

Usiig lbls polynoni.t ed $e @nditid

ARC1'>

a.d lppli.d 3

4.4

tt the

(r.4 s,

(,B@tt

!4: V1+r

to dd. us€d by

!dfig.no 3 44

-l

DAzl

, ,fl!0o20)) , =f no",

(34e)

obrlid 4c prc&tn d in t$lc m t t lh. 'visibility p.m€rcl dcfin.d asl

rdtsirghd

lh. tabb

would be visiblc if:

$e @h3

-P4.,\\

II?

(3 4 r0)

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 appljed

vort bolh the @ddq .trl. Forhdi|ghe .tr rh.r due |o M&i<t, ro thc Ft..i.d 463 obs.rio$ nm nr6d),e (Od.b 2OO4) rhis

e p'lsrd i! ffBod m. I 4 6 (Fo.tEngn '!br n,r mdct) &d 1.4 ? (Mr!id6t arld

&€

@lb

mod.t)

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 ,on! wh.n rh. dcaar reloned vbibte fi.r dwin. AM FortEi.eho,s

obb

trFdet

rt. linir mvij.rt Fdtdi,Bh.r Thi. a|€g.q! tb.r rrh. ll@s of the dte is to ih€ d!r. m which Fothdingnm wdr.d on and .equind siou

by th€ tul€ (3 4.4) d@ to

highly relr.icted

ll8

Lc betow

E- 6i

intTii:i

s E

ll9

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

{-:

Fia. No

On lhâ&#x201A;Ź orher han4 figure 3 4 ? ,nd rh.

tdlte

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 6ppe.dix

Forb.gnant

t20

lh. €floft

nol

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. su@esstul

Anolher eforr ofgEar significance rnar h found in tir.6tuE is based on rh€ of Schoch ( I93O) and h known d lndie Melhod give. in the Exp tnlnion ro The trdian

wft

Astronohi.at Ephenetis , t^ ttlh n€lhod

givs hee

ba5ic dala u!€d rs

lablc no

3_4.6,

leasr square quadralic

UsinS lhBpotynomiaI

polynoniat fincd b rhis dala yields lhc fo owing Etaionj

= l0

v >1e

@d applied il to dara

t74l -

u Ot 3l)DAZI _ O.@aj

rhc condirion dur rhecre*enr $outd be

it4i

Ihrs G

hrl( $e

ii (t.4.12)

Esutrr ohained aE

''visibilily peamelef defined

19 our 196 porir.\e sishrinss Dar devidrc

TIs rill rhe hrer hatfofrhc 2Od c.otury rh. Indiao n.rh@ ws Horck in vi€w of rhe analysis of Babylonian 6d

\6rble

=(ARCV _ lt}.3:,43 _ O.o137DAzl _ o.!n97 D,12..)) (3.4.1:l)

,,,,,.i." "" -O

exploFd

(1.4 r r)

- o.ot lTDAzl_ o.@st DAZ.

Pc*nled in fis. no.1.j.8. For 6is figuE ,// Bed

DAZI

rbn l,ll.

condi on

consid.red to be rhe besr.

cnlcrion and the Lund Ripen s Law 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

hs lqd ro a sianili@1flnding rhai rh. eient md rh€ n€nievat for rn orlien visibility of @ tuD ff*rr e s u$tuI s loe of d€

This cxploorion

nE<t€ls

{-.

.;'.':i. .,

ilr

lgrlq-;':t! *:;19;!t *j:.t

Fig. 3 4.3

3.5

COMPARJSON AND DISCUSSION TIE wort

F6qn.d i. rhi, .h!rr6

sDtuis.d e & os:

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. For

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

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 @lerdd bred

nav€ lo

€slnirtty dif@nr.

t22

Tne

ddc

aliflic

wirh lnc horizon plats

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

Th€ acicnt Bobllonie crildion for the

the hienc$

suee$ pcencge

*ape si8hli.g.

€elid

(96.4rr'o)

visibitit, or new tunar ccscenr 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%.

b lh. Luar Ripeness tuncrion thal deletoped dlrin8 $e nedi.val ca ae lhobughly invcsligared. Wirh ftoucm rahnques The ideas retated

ohpulltioB rhis hs Esultcd i o a useful ne6od for det mrnrng fi6t sighdlg of @w lumrc&$enr. The probteh

lhe day ofrhe

onl/

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 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.

11"' ly:.

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

rollw.d b) rhe Mlmdets

ofte

e&ly 20,r cenrur', rh€ rndid De$od be.l uccss peEenlaec tor posnrve srghrings {eo.l"b) herhod r84?%r and rh. me$od due ro Maundcr

t23

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. (54.lYo).

chaptet wnh 94.7% fotlMd by Momdeas (82%) /67.8r/'.

th€n rhe

The oveEll success pcrced.ge of rhe In<tian mcthod is 79.5%, merho! is 81.lqodd rhat ot FohrinShm\ method

Indid ednod

oflhc Maundeas

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 1988a). -IheEfoE

.,"","J;; ;;::,;:T:::"::i1",#;

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. $irerion is not rcliabic. TIE condi,roro 8@,ry str6, vis,brny.!cn ,f sry,{ no, ovft$r as sh,rr br *.n 'nc rn rhc n€xt chapG. B.tole authcndcn, of a cdledon is ,

:::lq

*; ; ;;;;".;" *:lj:: :"ffj;j:T

":t, ::_ :jTt Frw€cn the dimtt illmimied Ihin

cecot ad

t24

lhe bignrnes ot

IM&h

sky,

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. yas 1936, 19a3b, 1garta, McNa y, schaeisr, 199ft, McN.xy, 19a3, sotran, 2005, Qu6hl & khan, 2006, .rc,) (ia) the mintmun o, timtthg €tonga(on lunar cfescenr (Danjon,

of n6w vistbr.

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 Yallop

s modet

usibtr(v, Thls

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

lhe 1h6ry

bdshrB

of rh. M@n

s a tu&tio. of ddud.,

b6ed or

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 .r above honzon 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 a bighl (lor s!ilb). Then ftod fig 8 Ead onl rhc

lsr

coftrpo.ding bnghh€s

alftud. of ihe M@r wirh Ih.

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 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

dif,cFnt vidrh ces@t3 @

$le

dip

r = 0 d€gees lh. ninimM dliNd€s for

girq 2

Tbh hcms Aar rbe th€

bndrh$

cBent of@ftsponding

rt 16r

of lh. sky ar rhe* atrirudes (r) G widrh when lhe

.s rhe brishrne$ of

su hdjul $r. I. ord{ dul l,rtc ce$.

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 bright

the

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

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 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. funho d€qees rhc @nusl again bc@Des dfavombt. Ior v$ibility.

t28

Fig. No. 4.l - !

fic fi8u6 fron BNin s Pap.'

rrr, ?, rc rddb-r rr oi *l

t29

(

977)

For thimer

c6c.nr

rh€ ,r + s aSaitur r plots Sive thc b.st time of

poinr whi.h is the midhum of rhe

cwe, h rlso

Th*

de excellent

crc*dt my Ehajn

visibl€-

3ueSens a

id6

tug.

vhibility 6 the

of tim. for *hich Oe

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

follows:

Table 4. L2 2

, + 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 The valu€s of ARCV ore the valucs of

dird degd polynonial

usinS lean

sq@. apptuximrion lqdr lo

ARCV =12.4023 9.487arv Tnis ( an be tEnslomed ro

v),

lh. \ Fibitrly

the

follovine r.larion

+3.9512W' O.5632tr'

oaEmero

r, lunc,ion

(4rr)

to'toq:

=(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

|]0

.2.

Th. r€d of th. plor CD$ th. rc|t|iE.Xirtde (in d.8r€).gaiMr th.

I snd9

cltsdr

wirhh

(i.

@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, €l@ld€d

dE b€s tinc of visibility for

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 in rhj3 woli (ro be repon€d

disss€d in rexr rnicte) .nd th€ w dftdim d€vetoped 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 be

ltl

t32

lll

tJ4

tlq

?$I;;

135

1lz

Our

ofth.

196 positiv€ sighring in the

!o Btuin. TheEfoE rh. nodet du!

nod.k ad is

good

@pt.

cs6

d.qarc rron th. Linjr d@ Bruin is thc best mongsl ql ihe orhe, 20o ccntury

b $ lh. Lufu RiFn*

Law.

Ar nx

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 snicle 1.3 the ce!ce.& in 6ese .6es

vi'ibihypo$ibre. rh. cases

ae erisfying

1ne

older

Bruin,slt.i,i,b.,.d..rh"b,ich.::,:;f,J:1':;T:: 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

^Rm dc hsginal for Led Ripen*s 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

ll4 ad 613 have smaU v, v.lu6 $ dc m&gitul €ds in BNint one (obs. No.7l6) is asain u old ag., wide ond bdClt crceoi two

Ths sc ob*rve exaclly supplcm.ndng

Brui. s

linit

that

Bruio's

Linit md the Lund

oth.r blr thcy @ rh.rically

Mod.l.

hw ee

Ripen.ss

cquiql.

ud noE su€6sful.

4,2

YALLOP'S SINCLE PARAMETER MODEL

nol

Howevd

is rakin8 brighhe$ into considqation theEloG Bruin's limit

logicat

Tn hsl

moE

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 To devclop his onc

ptvio6

ahnud$ o. @s of v'sion ARCV, nedion.d in developed a lomula

cllline ARCV wilh DAZ

on lhe bask

anicle. Like Folhcdnghan

ofht

sunnary ldble da|a,

b€lren ARCV md lhe Nidth W, Hh data (froft pase 2 ol NAO T*hrical Note No. 69 (1998)) is

Yallop coroider.d fie bsic dau for d€veloping a Elalion

crescent

Epioduc€d in the oble no. 4. | .2.

Hoselcr.

Mdh

slal.d hy Yallop, f6m 1996

HM Nautical Aldanac Ofllcc

ddided ro abndon i!3 tcsl bared on the Bruin m€lhod (4.2.1) for lh.

''Indi.n" melhod s'-shrings al high

Bed

baed on lhc

thc lndie m€thod produced mor€ scnsiblc rcsults fo! old .gc

alirnd6 thai occuB

l.rsl once

fie sorli of schoch (1930) th. b6ic

DAZ

one

y.r

dara

15.

20.

I00

Tatrl.No42l As rhe width of rh€

qesent

is sivcn by (2.E.9)

tt7

d.sd'

ued in the Indis. melhod h siven in

I0.

fo.lalnudd rtomd

dd thst @ abo 6e wifien

as:

rr' = ls<t

- c.{A RCv).os(D,4z))

a$uming ih€ seFi-dim€ld of tne M@n to

cs b. t@sfomed

be 5

(4.2.D

con$lnt

dc ninutes. This basic Daia

inlo .Lt! rclating the width of the cesenl

likc thlt in table 4.1.2 ofYallop

r cubic polynomial6ing

ARCy =

bed

16l sq(@

-aJ7l

on Btui.. Yallop

ed @

tr4Iomed

of lhion ARCV

ihal d4ta fittine it lo

appoximdion 6d ohainqlIh. following eladoni

6.3226u/ +0.7119tf1

(4.2.)

-0.10)8w'

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 ,4,RC,/ for lhe place the

minulet. The method

says lhat

of obseruation is norc than $e value of lhe right nmd side

.Escent sholld be visible. Thus

a @nstonl

-\ARCV

visibilitt pameler may

(rr.8J7r - 63226W

dein€d

oa (4

2,2)

+073r9w'-0tntw')Jtto (4.2_3)

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 Yallop €alls lhG ,/p

Thcse d.viating cases

4-value. In g€n€tal

ae shoM

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 and

@nvert d Maundd's condilion

tom DAz-based lo lhe width-bsd

oblaircd lh€ foUoving cubic reladon b.tw.et ARCV

ARC|/ lJt78) e.Olt2w -

2.O1OaW1

-Ot16OW'

142.4)

*lich l6ds

tlE visibiliiy

pabr..ld:

,ip = (,{XCr. . 01.t733-9.0312W +2011J4W" 013601/

)/10

(42t Fig No 421

. lnda Umir .

ApplyinS

thal oul

196

htsibl.c€.e

Mendd's nodifi€d ondition to thc daia sr of Ch.pLr I it is fou.d positiw sightings @ iher.27 clg th,l dwiat€ fron the 6nd;lion

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 d.fined by

(4 2

(vr = 0 ol8

and o oo2

clore

sgr4rcot

repedively) in Y.lloP! Indiln dEd.l Thu3 thc t@ hod.ls ar€ in

Iney

both bas.d on

sinilu lpprs.hes a.d onlv slishtlv dillq

itr

i {

Fig No 4'2

@Ndting .ll visibiliiy otditions from DAZ-bas.d lo th. Widlhb!*d Y.llop achievâ&#x201A;¬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 lD addirion 10

140

condilions on lh.

b.twe.'r€giotrs of v&ids vbibility used in lhe Indian

mdlDd th|t

6.8tu

cals

b.s

of viribihv

pdma.. ,/}

Fis.No423 Fdthc bd rinc ofq.sit lisibililv' hc nor6 ih.t h fg 9 of Btuin tlE siniM ges.ent widths, fom a slrsisl]t line of viribilitv c!tues of , t r agaiin s lor different (" r) The 3{m h which whs Drcj.cl€d n€els tn orisin of the @rdiMt€ svst'm Th' shom h6e in fi8u.. 4 2 3 s a G{t lir. PNing ttroSli niiinun of @L @rc Itis dnimun @@sponds to lh' ortlt d(v !fl| b€d vilibility @dnio|\ thc b.d cerlln bd}6 rlE N6'ee briahtns it a 'poinr in tm' the b.iShtness oa th. c..stl ..dding lo Bruin Yall@ inrdPr'tt l, in a nlio 5 4 Thus rhNt rlivi.!€s "the tim€ lin€" beiv@ the Sunst ts ed lhe M@n!€t slop. olrhis

rhie

"poi.t

saSl lin.i3(, trta/4d,rs 5/{

tin€"

th. b.!i tine

of c..s..ot vitibilitv Sivd bv:

t,= 5\

4\rs

LlG)

l4l

, T. *!1,4c

(427)

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.. In all

crloldjoE

in rhis

rcrk

rhe

Finally, Y.llop d.duccs rhe visibihy @ndilios for dillcrcnt M8es oI

q_vatu4

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

aic.

detailed

d.duced by Yallop

lisibility eiih

Eprodu€ed heE in

labl.4

4 Ihese conditio.s

iddicalo6 for

or wilhout oPlica! aid.

Tlble 4.2.4r Thc t-test crileria.

{A) q> (B)

ENily visible (,4ICl 2 I2'r {EVl

0 216

02le,l> {014

p.rfd

condnioN (v

160:4 > -0 232

nedoptrtl lid'o w-itt neo oprielaia o

2l>4'-0

froivsrbG

(c)

May

(D)

{

(E)

-0

(F)

4-299>q

The

visible und(

29q

'sitk,

linilins ralucs of t weE chosn

find

PC)

cerdIMNOA)

frna crescent

tRoet

viii[lcmpe,4nC. < 8 5'tlt ueto" Dujon tinir;Rcl < 8' for lhc aix cnbria

a to

(v) (D I

F for the following

reens

(Yallop,1998):

-tlor A lo@ limit is rcquir€d to

spdl. ob*talio6

$at

uivial 6on rhos

l'ul

.len.nt ofdifiqitv. Accotding 10 Yallop il wc foud that lhe id'al sirr^ion ARCL = l2o nd DAZ = O" prcducd a Fnsible cut-otl poin! for shich q

nave

ed.

= +0 216, Tnere aG

Ill

exmDles

wh€n q exc.eds this value,

ad i.

Tabl. in appendixlll G'cond lai column) 8.Mal it should be v€rv ..sv lo se 1he new

142

c@dt

in ln.e

rponed posilive sightings i. dr6e cscs wilh (B)

tro obscaing cloud in lhc sky The

th.c is

Providcd

cascs

> +0.216 10 b. lhose wbcn the

I 14.

Tns

cccnl

it is

@mble

to consider

is esily visible.

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

014 is

&olhd

Possible cul_of val@'

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€*

(c)

Thce @

68 €Aes in

l1l cse)

rhe!. ae 6a posirive sighnngs withoul oPtical aL

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

cNs

thar

Gfv rhis diterion

t0'16 < q <

wre positive unaided siSltings tbe tcst were *cond lat cormn se.n vith hiMul6 or r.lceoFs ln Tlble i. appendixlll' q_valu€s oul of vhich I 0 s€E useii in this *ork therc 62 cses in rhis tuge ol -0.014), only lhree

unaided si8ltinSs

(D)

ces

out oflh.se

29 times lh€ cresced *as seen either vith bin@uktr oi

s'lh

> -0232) YaUop s Table 4 ha' roo lew enli's fton thc fact lhat which lo estinale a lowt limit for 4 The situalion is nade NGe bv oplical whe€ thse is an €nl!v. in non cas.s' the Moon ss not scen even wi$ a9pae't clo'galon aid. ln f4t it h @€ fot lhc cc.dl to be obedcd lElo* 6 lhis dg€ oul ofabout7c5 (Fatoohi d al, 1998) Ydlop's Tablc 4 h6 14 cs's in exl6 wbicb 6 re positiv€ sightirgs thdough binoculss o! &le$ope ed one ln lhis ca* (-0160 >

oiilinary

of uaid.d siShing ln th' r'bl'

141

app'ndixlll Gecold 16l

@l@) of lhb wolk lb@

with binoculd or 1el.$op6

dd th@

our of which 18

extra otdinary

@ potiliv. sidines

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

is nol Posible 10

thc lhin

.t snl

n@n .v.n vilh

lel4oF

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

Alloeing

coresponds to

. lowr linit

I = -{ 2l2

Wilhout good fidding lgles@p* md

posidoml infomation, obsne6 aa unlikelv [o

(E)

lhc

crc$"t

helow this limn

cul{fi Point *hen th. appatcnl elonSdlion of lhe Moo' l98lb frem the sm is ?", knom 6 lhe Ddjon linir (Ddjon l9l2 1936, llvas'

TheE is a lheoEical

al nabohi el al, l99S) This limit is obtained bv extlapolatine obedalions nade i8f,olrng lager elongtions. Allowing t" for hori@nul p@lld ofth' Moon a'd 1o ''{'R'1 = 8' rhe effect ofrefEcion, ah.ppor€nr elongarion o' 70 is eqnivalent Wirh

,,llcl, = 8'ed

However, in

D,42 = O! the corespondine low€r

linil

Yallopt lable 4 th€E tte 2l enfiies witl onlv

]

is -O 291'

posilive sightine

I < 4 212 ed no sighting claim wilh s_aided cve ln lable ir the mse -0 212 2 4 > in apFndix-lll (se@nd lst colw) th* ae l8 c*s

with

bin@uld fot

snh binocuhr ot tlescopes and onlv 2 455) Both the* un_aided siShdngs (obsdation los 389 and po'nl dcviak from all the crit.ria considered up b lns

-0 299 oulofwhich there atc claims

obfdrtio.s (r)

7 sishdng

wod( shoss 66 cases wtrr The table in .ppendixlll G4ond last cotumn) or this claim of sishtins -0 299 > q ed therc is one.xtra ordiMry silb o! opti*l aid (obs .o. 189) Ap&t fron $is lheE is no posilive siehting of €Esce sisblins $ wilhoul opdcal sid. In lable 4 of Yatlop tnee is no clain

A@oding |o the visibilitv divid.d into

dNifiorion shos

abovc, rhc

suf4

of E nb b

lhe tgions bv four consi&t_q val@s As lne aclual visibihY of

crescent

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 depends on ils width

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

d.Fn&

on lhe d.cliMtion of th. Moon

c6@

visibilily is edier

ladud.s if lhe Mmn is nonnern posilion

eufi

dd still

tne

Thc situaion h revese for lhe

rhc nonn hitudes Duing summer of the nodnem of lhe s6 then ftis v.nex d@s not @ch irs

.e{ cE*mt vhibilitv

is better in

$liheal henisph€te Tne

(nonhwardt and below Goutbwrdt fion

'xcm€

1ne

norlhen laitudcs

palabolu opens westMd above

vcnex The CuR' A is

th€

rhe collection

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 the EgioB wh.e |h. q_vdhe is s@ler lhm 0-216 md

ln. Mlcd eve in lhis rcgion' Csc B is the colleclion or all points where the q-value is -0014. In all tbe Eeions be$€e! cuftes ed B lhe cicsc.nl is lisible to thc nai.d ete onlv undet pcrfed visibililv condnions

the cle$ent h ea3ily visible to

_0-16) de $e tgio's io wbich The Fgions b.(wen cuN. B and C (q_value visible lo obsen.r suld Gquire opdcal did 10 locatc ihe cr*c'nl dd th€n it mav be

naked cye. For reSio.s

wi$

q-vdlue I€$

t\ai

_0 16 rhe

the .akcd eye For a comnon. untaincd obsePer

*ould

bc seen in

rgions

cresenl wolld not be visible to

it h highl) unlikelv thal ihe cFscem

oflhe aNe A Thc scienificallt ecoded

obsaliom

(oh

bNd) do oot Pohibil obedalion of in facr' lhc .r.sc.nl wiln mkcd €ye in region belwn cwet A 6d C In sucn t'gioN' probabilily of obsenation incFdes whh the nlmbef of keen tdined od expenenced ehich all $c sludy of the rwnli€th cdturv i3

145

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

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' fornalized in

lihough, the basic data cxtacled by Yallop frcm Druin rcplaced

bt rhc

iom Bruin

was

d,tt .re mrlgi.ally

ces€

vhibilil! n h6

bccn secn rhlr

bctter dan thc lndian mcthod

lhc ddo scl

Schacllcr (1988a. 1988b) oh lhc o$er hand has uscd

cxr.nsivclt

visibililv cuBcs-

basic das due to Scboch (1930) (lhe Indiah nclhod) lo aFivc ar his q_

value coodidons (di$nsed abovc) for rh€

rcsulls

'or

phvsics

of \isihilirv'

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

{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'

rcmp€*tutes aDd rclarile humidil!. Ih rhis

$c laryins conmsr dunng twilighr for vanous tcnpcrat!rcs and olativc fiunidny. If rhe appaFnl brighmes of the crc$e.l h morc ddi tlE bighhcs of (1988a) himsell' lhc xrilishi sty lhc cre$e should b. visible o$cNne nor Schacflcr lunar crcscenr ro sludy

sinilar rcchnique lo @lyscthc visibililvor invhibilnv data avnilablc in his arc @hnique is ,Pplied to $e d a*lof rh. chap161dd the rcsulB obbiocd

hasap!li.d riFc.

-l

pnqntcd

in rabt No 4l l

Bcforc

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!

forc sunsd. l hc

rwilislr sky dcp{nds on

conlEi

d number

bc(wce n lhe briebtness

f crcsccni

Nd thai

r thc

of f.clou. Thcsc incllde:

P6nio! of the c@nl which n.elf is significsntlv {fTctcd bv Tbc $.nerine orthe

tieii

cEee fld

ftom

Lhc

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

$c S!n. lhc M@h and o$r sourcG (like anilicial light lbat G nol considcfd in this work 8lh'v m noi s affc(i!c during lwilisht.

sourccs of liehr rhal includc

The amosph{ic

ldp@tuE

and the

relaive hunidilv

lh* alldls |he to|,al brighlncss of lh ikv is conpded at lh€ point whcE cmcol is pae lf rhe bnsl i$s of $c atv is noE than rhe or equal to tbe briShine$ of the ccs. ihc cGcent En nor be s(n' Eten ii Taling in|o considmrion all

it is vcrv brishtness of the crcsocnr is marginallv moe $an the bflghtncss ot ihc skv 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:

w.ll lbo€ horian $e lpFcnt Fsnion ofrhc crc*e isEi*d anamon.tR. thc angl. ofcfmdion (Snan l953.G@n 1985). givcnby: For alirudc

x=5s.{#]*' whec P is the ahoq'hcic

P6$rc. /

is rh.

|cnFatuc

(4.i r)

and z

lhe cEscenr. for ahiildcs closr ro rhs horizin rhe following

t47

,+ 1i - l L I+441 r ."rnr I x- t.ot.corlA,+n'+s.tt | i

whcre, 90'r : lnd

- 9ou

lion schacf4

and dcrailed

pogBE)

linitiie

Hilalol wo develoi,cd in $is work

naehitude thc major nePs ol calcularions

lised b.low wih bncf de*riPtion Rel4nccs

d.sc'iptions can bc folnd in Scherd (1993)

Thc prcgEn funcion

. .

The sollwde

t.2)

(4.ltj

ror lh€ delcrminarion of lisual iadoplcd

l,.cor

/drat

r'l(,)

1at6

inpu/pE-c.lculatcd valnes listcd bclow:

a4r. thcal(iludcofnoon abovehonzon.

ftc azinnth oithc noon

kvelih ndcB, Pn,z. cslimlted rclalivc place. p/a/. lairud€ of the plde. PratP. esrinatcd lcnD€dlrrc of

par. the alrillde oflhc place abovc hunidny of the

tdu.

sea

. 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

associatcd

wilh each wavelengh sel{tcd

@vh[j]

coftsponding

elc )

-lA.%. -10,45. -11.05, -11.9. _12.7) lhc nagnitude of fuu moon ro

dififtnt slel.d

warclen€lh b6nd5

mschtil (= -25.96. -26.09. -26.74, -27.26,- -27.55) coftspondinS to diff€rc sel*ted w6v.le41h bmds 148

fic $lar

magnilLrdc

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

'lhc funcrion

shs

I O.l.lM tum dishnce ofrhis

+ 0.r-i-e. a poinl

wnh sky position

sitnhr lult

clo* lo th. cstrc of $e tu.e disc

Thc

mdh

^.ilh

poid.:ed,sr is scd lo calcul.k lhc 8as. aeiosl ed lhe o7..c

= {cos(?cdig )+0.0286

bo(ud'i

"'=l

lhc sun al lhe line ofob*ryorion

Nnh $lcclinS a poinl

. L

fon

)+o.or

icxp( to.5'cos(zsrd,ri 1€xp(

)r-l

(4.3.4)

-2a.s'cos( rd,u/,Jl ))l-r

(4l.t

l' lr.:oro;rrl ] lsintf,",./,r I

ndss

(4.t.6)

This is followd by rhe calculaio. of th€ cx nction @flicicnls componcnls comspondine ro live

diff@r

\Ev€l€nglhs

& =o 1066 '*e[- ?s4J'("qrrtl'r)

in the

@v Ydc,ll/:

14i7)

r,(xru;'".-,(-#).(' -a##-*,) ",",,,

J, Ka =

cleted

*orr'.^-,0*r!!\

' +0 4'lpto! ' o.'chlil'O

*,, =,,,,,,r,,..

''o'

co{satpha

)

- cosl3'

Ptat )))/

(.r.3.9)

*.(,i# ).".(tf ).".(- #il )

For cqch wav€loslh

(4.i.10)

bed rh.* eidnction ccmcicnts a'c ecumulalcd in

(4l l l)

krchli)= K, + Kd + Ko + K" And rheir linearconbinalio. wiih mas componcnls

gathcrcd inro

dnvhJl:

,.tdchltl

lbe.qurioft O.

1.2,I

rhe

K,' x,

+ Ko' x o +

X-'

(4I

12)

(4.3.?) to (4-3.12) aE Pleed in a loop thal rum nv€ ddes oncc for €ach i

Aner$eexccution oflhb loop lhc rir

and 4.

m4ss al

poin

thc tosnion

M@nandfiarof theSunacc.lculal.dusinS:

rrr@/

= [cos{'drd'Jt )+o.o2s

(nofu < 0)

nnposl

icxp(-II'co(z"dirt.)]

(4.l.ti)

= 40

nnpoeJ -lcos(go -

nul )+0025 r.xp(-ll

il (\ah <0) xn@ld =40

et:.

snpoel = lcosleo

- ett) + 0.025 'cxp( -l

Th. masnitude olthe M@n

mtut

-.12.13 + +

FoUo$rd by

is lhen

. (4.1.14)

, , (4.l.tt

@lculatd NinSl

0.026.lt 80 -.to,@

a. ltso -.tons )a' r 0 t

c'.rrt4

nid -am. bnghrncs.

honlidl

biehh.ss. twilishr bnghl.s and $c

nq6tl ctree - n 4 1. Lkhlil'

= rc15t5

ftmp"

noohh

^filb

=rc

+cos'(/.io'sui

$c clonsrtion ofF

(tuh . ahree + aaoo@

roo

rc4 a'

t'chlt l'

'(l -

ro-0

4'ri.,l'l'"*/ )'

lt - c!tuae )i)

nschlil ttz 5 -s.t'

(r _

(4.r.r8)

fon $. ccnft oflhc luE di$

o!'lo,htus D6c'il4+.r:7)

= t0-0at(,*.r,14

i_ ctort

(4.3.r7)

| 40) +6.2.107 t(leto,stunf

tot36 .(r

$hm /elo,At,

-!'dt'

-0r.,,.,r,r"*./ o

t(t@ I llchlnl

ro(6

|rydql

r5:/./d,N* i

1t3.rot

(4.120)

) (4.1.21)

a0)

+6.2'to1 tlfetoresn)z + ro$6 .(.oe *.o.'?(tt,,r* )) 03rrs/'I4\fuir' )' .tqh - |o ort'l' ;i,l't'n6chlil'ar'').t -I0 , *aaoooo'(t V"'

/g-,

(a3.16)

"fat

"hn))

150

(4.1.22)

(4.12l)

lf thc hviligh briebhes

r,grrb

L.r/t

and

rv/, doninats

,1, are slored in

,r.r/t

Morcover. ifrhe Moon

l-iially, fic brighbes

,.\rrlt

ovc. tne day lieh briehhcs dat, rhc. thc sud

otncffi$ rhe sun of ,'s&r, md etb rR noE'd is.bov. horizln thm de,, C lho addcd to bs.r/t/.

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 (4.1.24)

l5o0)

la (tel <

Els

{ co,a

kh = tum'\t

t.. =

ic,,e=loat..eo

i1 -2

Jctu" tb.t

. .n

lo-re)

(4.3.25)

}

t,IVD) \ In(I0) l

(4.1.26)

d.".htzl

Schaelcr (Scha.fer. 1988) in his drtshold

(4.) 21)

co.tdsl nodel calculates /i

of $c ratio of rhe acrual tohl brigltoBs of th. M@n

ad $e

as rhe log

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

bclow) sbows dirercncc ofscbacfcls thEshold cohtrasl

/r43

Scries I sho*s ,1aas lor crcsmls

crusoent ihrl \vcre sccn Sdies

wce not

sen

d lhal

4 zDzq and

A plol (Iig

$c nasnnudo contasl

wca not $cn and erics 2

shows ,'lu zg ror

and series 4 sbow ,4 cot.spondins lo dcsccnc

$al

scte sen Esp.clively.lheda1a lor this fig is talcn lrom Sch.cfc!

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 (1988). P$nive

Elative hlmidill adopt€d for lhe calcllalions.

l5r

Thdhold conExl vr|

f.e.ltld. conbtt

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 Fig No

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' .3l!n) *id n is b€sl for sidlinS for siShrinS of c'lsdt (@lum vith hddinS (@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 2004). In @h mw of thc

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 visibihy ot $.

t57

Tt. ob*Mtio6 rhar

rh.

de not

@sidered in th. llblc 4.3.1

cc@ 6 sn El.ver. Moeov.r,

n €valual.d ntaEnituda

dy n@s .s

ody rno*

*lq il wr claim

ll|e r€cordi enen rhe

t€npdt@ dd Glatirc huidily il nr}€d eye visibihy ol $. asc.nl, If th€

by vrryiog thc Grinared

oblain oplimm @ndilios for

co"na

is obtairea ro b€

favour of visibihy onlt dcn lhe

lim. ra.8. of

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 naked cye crescenr

nagritutu conn$t is c5lculaled md th.

coGpoodins columns. 'rhe s,m positivc vahe in all that under lhe

walh.r ondnioN consid.Ed

]-his table shows tbal $erc d€ aid when nagnilude contmt was

css

rh*

the cr€sccm

cases

in aU the thR @tM.s it rhc indicalion

vatue is included

ws Ev€r visible

to lhc naked

ofposidve sighting claihs {ilhout opiical

nclq in iavour otunaided visibility.

All

tsc positile

aho nor in aet€emenl wilh tbr conditions due io Folhqi.ghm dnd nalndeu

oily ntr8'hal 6e mlcdly diff€D fiom Lllm RiF@ss law. Onl, I of

Lunar Ripeoes las d@s nol allow

l0 of

thcse

the two of lhen

b n. Only one $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 cses a.cordins

orher

tou

dlaims

dlsion difieB

Ellfoutcs*

are marginally

lllow€d by Bruih's critelion bul Yallop

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

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

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 TheE is only onc

claih ofvisibilily of cressnt is with binocular or nor

ucliable 6 out of25

s@h

claim

wit

lelescope. Thee

8 harc favounble

ete visibility with oplimm 4tinales of lehpcdtwcs

claiN de

also

uas,itt.L cotun lor n*.d

ed rclative hunidiry'

The hagnilude conrAr for sme other r.porledly Positivc cE$€nl sidrinss {ithoul oplical aid n nol favoudble. Th6e obeFalion nunbe6 14l, 3 19, 416, I16,

315. 2a6. 611, 314,272, 2. The q-values (and 0.8?), -0.1r (0.07),.0.101 c1.06). -0.047

(-1.t.

0.012 (-1.01).0.018 (-2-88),0.109 G1.47). For

A\"J

fo! $ese descents

_0 153 C

-0.02?(0.167), 0.00? (-3-5),0.01 c0.7?),

341.416.Il6

Lumr Rrpcoess Law md ihe nagnihde €oi!r60

rhe

thrc. crireda (Yallop\.

co6islenl- For 286,633,114,212

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

dd 2 Y.llop's citerion allows.alcd

cye vGibilily

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 is horc conccmed wilh

visibility of n w

casd

154

witboul

dy otircal

aid

wo*

t!.2131)

-'.gr""il,,;* I

,.;;;

;;;

,.!('!,"]

r-?tp!!

,,;p,{ 155

t56

l5?

t58

4,4

A NEW CRITERION TOR NEW CRtrSCENT

while d€vcloping rhe "vbibilitv cudcs" (it againsl

IISIBILITY

$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. $lslcd c@ent vi.ibilitv circUmtarcls of ldious rcw Mm6 css wh€n lhe crcscenl Ms rponed b hav. beo sn For cFs415 ofa pan'culd tho width we found lhe altiludes I oi skv points with btighhess equivalent lo that ol of skv particlle crc$. at dill€tnt $l& depEsio.s r' Th. ovcdgd of the 'ltiludes in T'blc no 4 4 l Doinrs for difl'rnl $ln dctr.$ions for plirhuld sidlh d Lbllat€d grv€s The left most column of lhe lablc contains ihc solar dept€ssions r and ihe toP rov enties of rh€ {idrhs r oflhe crcsenrs sclaled ed thc n xl one giv.s its na8lnude Thc $e

rhe

€blc ue lhc aldtu&s

briehtness of tnc

shcE lhe stv hs rh' sm€ brighhAt

lhc

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

d.sc.nls. Se6on to eason ed for dif|ere'r lalitldes the skv qch bddrlnes fot th. sdde altitu.l€ cl6e lo th. poid ldee ihe sun sts also vdi's lor column of th. lable 4.4.1 a numbq of c!*s of alnosl eme cE$enl widtb wE

seconds for very wide

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

@nsidered md each ahitudc is aveEg€ of lh.se

(Bruin. 1977) developed.

159

T$LNa43.l

a.l

rl!.

-r.8 4.5

tt.a

95 5

2a 5_a

2.1

2,5

1.f

3.05

r,e

r.t

255

1,1

ot5

0.t

0.55

o3 o25

o,7

0,2

r.t 2

oi5

Fia. r!b.4.,1.1

Altdav.r 6ot Oap,Ebo

I! 5

3 2

0,8

o.t

Alttudo v.6 Sohr Oottesalon 19 13

€l:

Th. @rdin rs oarn ninid ofafi, Ghom i. figurc 4.a.t) @del pe nrv€ dseloDed Sou tu dE i.ble 4 4 2l

Using

tubi. l€.e sque appronn rion

rchtiE .ltilud.

ot@rf

ARCv =

435t

ARCV

lhd

b.sic dals for lhe

oh.ir€d lh. following El.ro. ben*cn

tu widd

37tvt +2222O75O57W2 5.42264J1tt! + t0.4341159 (4.4 D

I6l

On lhe

bais ofthh dlatioo wc tlcnre th. visibil,ty

p.m.t€r

follows:

t/ + 10.4341759) e = QRCV - ('-O.35lg$7W3 + 2.22207 5057W2 - 5.422643U

(4 4.2)

Our model for .adi.sl visibility of n w

visibility paEmctcr v, in (4.4.2)

ls& *s6!

th. r-vtl@) lhc

ir that if

ceml

rp > 0

@ll the

moy be lisiblc withonl

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

opdcal aid

olheei* noi

colunn of table in appe.dixlll.

Fid

fd .ss

optical aid in odcr of inccGing r-ralrcs @

only

d4iatc ton ou nodel. our of

cd*

d.ltn

when rhe visibilily is claiocd wjlhoul

sho$

in 1abl. 4.4-1. Oui

rhese I

ol0tcs cses

sill

c6es 8 de coosisteni

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 from lhe

Lmr Ripen.$

negrdve

I-aw, ihc

Yallopt crilerion ed

1be Dagnnude

316lhal d.viale

conlrdt

also

our modcl. How€vcr the obseorion nwber 4161hat is negalile in orher

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. models is allowcd by ou!nodel.

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€

''b*ttine"

of cesenlvisibility

,3 rB

(lh + sys

9-3t5 ot t'/s

4.)15. This

lqds

ro a

-=-i: s1:s+4.3(Ts+LAC)

(4.4

9.3

besl

tih. of crcscenr visibiliry

is give.

l(2.163)rhpanofLAG in conp&ison ro l(2.25)rh panoflAc in

t62

)

Dodi6.d

l6l

gendl l@kingd

'Ihr€

Lhe

@npt.rcdsl!

ubtr jn atFndrxtlt

b€ nor.d

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 no

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

opncd aid our of58 rcporled

coNidercd obse(arions.

.laims 45J, 274 and

l4l d lol coNnlenl

fo. lhis tu8e ofr-utues the q€scenr

Ih 49 c6es

wi$

26 (45olt clains

-0.06 <

r{!lu.

wirh

ofcrcs

nodet

< 0.05 theG

e ll

unaided visibilny

be secn wirh opdcal

vrs,bihy wirh

@nclude lbal

ad only.

ueided

(25_5./0)

ed 2l

sighting with opricll aid (41%) we conctudc rhar thqe ac stonS chdces of &cmne crescenr vi0r a or.eutd or a Glespe oii very slih chmc6 for u..ided sishring, Unaided sightine b nol impossibt€ For 0 05 < r-vatue < o. I 5, rhce $'thour oprical ad t2ra,. I hus

nr h! r4se condnion

For

(sarh{

r-ulue >

O. t

dc j5 sishtincs wirh dtuar rh.

tuflr

(1[r/r) and 14 hay b€ easit) *.n qrh opucdt aid

s-htr &d !m be seer wirhour opticaj dn coldiriotrs and heichr above

5, oul of nexl

under ver) Com

lerct).

t3 obeR.rions th. c@kenr was s.en wirhour

oplrcal aid I 65 timcs (77.5yo). Th@foE wc conclude th.t for cEeent can bc esity seen.

Thus

ai d

ou oodcl $ar we @ll .eu€hi &

164

(lh

r-valu >

0. 15

rh.

critcnon..cd be sleheiz€d

Calculate

Th. vbibilily condnior

r-vdu. (or vp @odins

vhiutc uaer pertco

a girn

44.2) fo. th.

c|lsdl

by our 6odcl @

ihq

ilc

besr

tine

giv€n in Tabte

onairiii'lili[

May rcquirc opricar aia

iiiiFitii?MFoli

Requi( ophol ad (ROA) Nor vis,ure wrrr opriciisia-li

.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

dlononer &d lmws Physioloey

f&r

rhal

obsd€i may

b€

.xFrioed ed r.ined

offie cEsent rhe armosphdic €ondilions od the of dc obw.at cy6 @t stil lead lo non-vbibilny of ihe crent Thes

facloB ae sdlt nol

Oil

rhe

the loerion

wll

exploEd rhus the high frequency

the probLm is srill nol

eh.d

ofrhc..gaive crc6 shos

conpl.Iety. A posniv. emr @cG when a nodel do* thc visibitiry is nor claimed Snaltq rhc n!frb€r of

allow visibihy oa cesc. sd posirive ercB dd b.ttq is a visibility c.irerion. The hbl€ 4.4.5 or

d'esrem.

slmdises

the posnile

negative obseNarioB in

6greoe.t

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 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 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 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

ofd*

166

Tl|rcforc, if Fothcdnghd's criGdon is nosl s@4stul

i. b.i4 cotui$dr fot

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 negalive obedadon rhG

Riperess law

dcs not

highlt cotuistent with rhe positive obervalions but

the ncgativ. ob*Nations. The*

l4t

coNislent wilh

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 Bruin'r

lini!

Y.llop

bc best, HoBevcr,

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

4.s

the

probhn.

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

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

t6l

@ndirjoN

lhe basn of

cws inii.lly

visibihy

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

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 not cone

rbis work. Bo$ lhce rcponed cses

nol .oDsi.lercd Fliable as .ven undq niehlv

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

exrgeerated .hosphdic

humidiry less than 50% and atnospheric lefrpcrature aound l0 deglee centiSBde

sd 455 1nee e ll olh.r caes shen the cEse w4 r.poncdly s@ without oPlical .id but ihe ma8lilud. .ont6l ws not Apan fron obervarion nDmb.6 389

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. favowbt.. Howvcr,olee

The models $at

@ dedued Iioh

visibility cud6, th. Bruin's consisr.d

qith.{h

rh€ Bruin's

isibilitv cud6 ud liniling

linil md YalloP s single p&meter

(it

.ion

drc found to be

other. Bolh hav. stmosl.quivaldt succGs Frc€ntagc

Bruin s

We ha!. developed new visibihy curves and limiting visibililv curlcs using the

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 brid nss mod.h

due 10 schaeler md

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

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

rwiligbr sky il h6 ben

*en

lllar rhc

bed

on brienb6s of

lbibiliiy of .ew lun& cr$enl

c@.nt dd of

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

cEse

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 favouBblc for

hay b.

due

hid*

sighdne. Evcn

elevation

ofrlE siL o. v.ry low lcdFarue

Thus in makins decisions about rhe duthenticiy of

or humidily.

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

contdt befoe All

any

qiidon nNi b. suppledcnted b, m molysis of maglitld€

althdiicalion

th€ empirical

the

is dorc.

si4mpirical

mod.ts

d€duced ftom

$ne baic

Fa ARCV-DAZ rclarion (lik.

ofdara lhd is d.dued vithout tating th. ahosph.ric @nditions and cldarion .bov.

l.vel. This b6ic

et of

dala

mv bc b@d on d

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.

dual ahosbh.ric

conditions

esriMtcd tcmp.rarurc, 7,

cdc of vdifyins

visibilig claim

mv b. 6oded.

Thercforc, a possiblc slral€gy for

vedliatio. day

be to usc eslituted alevalio.,

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

i. el€valion, rempc€tur€ dd Elrtive hunidny e onc con adjust for thse quanfii* and qiry whether a favoulable ma8nirude contmsr (M progran Hilalol alloqs for vElidtio.

< 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 ahd

ar€ obtained

M= I

{t I

aollows:

8177

-1.autE

+ O.0Ao000ze

t626e40n22)

M = -A 4 368 + 0 Ai26T

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

- 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

incrcdes by 0.0126

may 6ose that nay

b. lhe ay€rlse for lhc eson or lhc valB $at wE

re@rded ar rh€ $ne of obsc(ation, ELEVATION VRS NAGXIIUOE CONTRAST

?E.ri.odtr+

af?

HUMIDIIY VRS IIAONTUOE CONTRAS1

$, 3.

TEIIIPERATURE VRS MAGNITUOE CONTRAST

8 B

t7l

acNally

Chapter No.5

APPLICATIONS

Durin8 rhls work lt ls obs€oed thar since th6 Babytontan eE

nomber

pr.diction crltsrta, mathehatic.t

flt

recenty a

.s we as obsewational. were

developed ro detemln€ whon the new tunardesc€nl woutd be flrsr seen tor a gtven rocataon. as tho

new month in sagnirrca.t

fist

appea.ance ot new tunar crescent marks tho beginning ot a obseryarionat tunar catendaE thele crireria and modets are

fo, calendaicat

purposes_ wh€ther

carendar, like the tstamlc Lunar catendar, carendar of not rhe36 c.trerta provtd6s

an actuat obsrya ohat

tuna,

lh.se crireria tor arangtng its a €uidance for both teslhg an evtdenco ot izes

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 yalop,s

to rho

t72

o_v6tue

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

a simple rechnlque to calculate lr. beginswilh a desc ption ofthesme.

5.1

devrs6d

The chapter

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

McNall) proved

tis

pbenom€ion disadine Danjon\ hlpo$esis (McNa y, 198:t).

rhar lh€ tcngrh

ofrhc shadoss ctose to $€

oflunar suiface IoD beins pertecr

spherc

coutdnl,jimiiish

hrnar rtre

cNp ond fic d.panolc

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 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

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

plEs (or dongation) of the Moon ihal is gcnet lly

the

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

!ho\t to bc incoftd bv McNallv Hosev€r. e or€r rh€ ninimu \idlh of visibl. cEse (2 to 6 arc seconds) bv

ufdeficicncy arc. br Daijon Nd

limil oo 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

esrimrrcd

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 Schaeffer l99l elc ). Accordinglo Danjon, d desctibed by F.l@hiet al

si,(a)

st(.)

cot(@) at

fiS l

(5'l l)

Al" 198) a is rhe dd o i5 half the cccenl l€nElh tr apP€6 rhal Dojon tr*d lhe Sine

Wh.c $e @ elongltion

1998.

PQ =

o is fie deli.icncy

t14

(Fatoobi e|'

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) . muejn't dillemce b'lsFn lhe lso the cre$ent l'nrth (2o) is Esults. Generally thc .lonsation (4) cd be cahulaled and ro obede.!. e the rw fomuls 6 b. LPd lo find fi' dclici'ncv arc' Hosder'

Nlserically, for small ddgtes

calcul.rc the angula!

c6..1

rpdlion

a thee is onlv

lenelh nonc of these

9 ftoD

a cusp in

I SinE

ued- McNally daetoPs a fomula fot

tems ofclescent width R

-T *t' t Rl

(5.r

lron dE nininum lisibl. *id$ n.sured ih ddid dn€cdon 'wtv is l8O0 _ 29 disc. Thercforc the logh of th€ ce*ent h€ oblaits (sultan 2oo5) adres at tlE Usiry (he Blackqell $rcshold con(6t Sutan rh' smalld eqlival€nl Blackselr ninimun vhibG *id$ (in |ems of didetcr of l@rion of ob$Parion' (Bhckstll, 1946) di*) of c@enl ar Frigee dd apoge al his crc$61 lenglh rsi Tbe fomula rhal hc deleloped for calculali'8 thc

o.l4)

,=f!1.'to" l2t )

whde r h th€

sdldidetd

of tbe

Moo[ md

l2t +w \

wirh w is the dim€ter

{5.1.5)

snallsr visible .quivalent Bl&kqcll dnc and l/ h cental

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

widtholrhecre$e.t. (5.1

The presenl $ork is based on rhe obseNalion or the last (old) rcsceit

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

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 phcnohenon shows

shonening of

expload abolr lhe

cc$ent lcngth apan fom tlapke s .c€urate model thrl

detendoce ofcB*cnt lcnsth on clongalion tlone.

TherefoE, inrhis rhe

1o b€

ce$e.r.

on th. aclual

qorl,lo

besin

si$ qecoisidera

simPl. seodeuical model ior

model dFsibes the phenoncnon of shortnine of lcn8lh that depends

seniSiaocLr of lh. Moon

csc.nl o\fr tftal

*ll a the rclaliv. alliiude (ARCV) of th€

sky. This is denved fro,n thc sinste

crilerion of edlie$ visibiliry or cre$ent (chapler 4) whe.cve! rhe widlh (o' btigb$e$) of

vhiblc c€nrnl widrh (or

b ghtn6,

cF*c

pmnct.t

fld

clos to

(q-vatue or

!!al!c)

is bascd on th€ facl thai

cusp is below lhe mininuto

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

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 iNisibiliiy

dala lvaitablc

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

8..

sh*

G th€ radius

tF...Cdv tr -cos

ofcfsenr

tlre bLishrn€s

Irvi€wofQueshi & Khan (QuEshi & Khan,2007) displacement v frcn a cusp is gvei bY

t)+

(5.r.6)

of Moon, F. is fie mdimun nux of sunlight

olVoon.l islhc 'ePar-rion berqecn lhe Sunand 'te Moon in our the Eanh' the lutu suface dd X B th€ disldc. of the M@n ftom lhe cescenl at angular sepaElion V fom lhe cusP is eivcn bv

lV,r,

whc t/. is lh. oo ro 9Oo

=rcosWll

width of fie

(5.!.7)

CosE) = tl ccatv

cdnr

lne diddle ln bolh the* €quttioc

!' vari6 lmh

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

$e aclual bc visiblc lttrough naked eve one requi.es 10 s'ek guidance fron srv voug obedalioN. ln $e history of *idtjficatlv rcponcd obeNation of thc 1990 (eporled bv cr.sent Moon, lh€ ecord is that due lo Pi.rce od February 25' jusl 14 E he claift |o have sn wirh narcd ev' B

cl€*nt ils widrh wa3 O 18 dc minut€

s.h!.fd ad houB and

Yatlop) Thc

Amonesl

t11

'll

rfie

'*oded

ob*flalios lh'

siglling of such a youg

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. 8 md . arc-ninutcs of crcscenl width shen fic Elaliw dinulh. DAZ of $c M@n is ao rhin cesc.nr

rlr r.larir.

alliiudc

ARcv

gives lhe

q-lalueof_0.22. Y.llop

,1R(y =9 6311-6J226W +0 7319 Thc values of ARCV according lo lhis Nould

ield ! difcre lotr€rlihiion

dilTriem relarivc dzinulhs DAZ FoT

ditedon (4

t'i2 ol0ls,t/3

2 3)

for

$n q-

6'18)

giving $e q_rdlue equal to '0 22

*iteion

lhc visiblc width olthe crescenr' Tbis is causcd bv

lh. lasl posibl' ARCV (4o) th€ *id(h ol

lbe

ln iNisibk ae$erl Nould bearcund 108 alc{cconds $at occur dl a large valucofDAZ tr\i.\r ol diis $ilcrion ihi nininud {idth of lisible cr€$eol for @v ARcv is Gm'd and s te *esent isjusi invisible for$is{idth:

(tIa) Whqrc l'l. is tlE tlmrclical

enlEl sidlh of he ces'enr and 4costl

rvidth al aogte V Irom lhe center of $e ahnude ARCV li

\alue V =

de$ent

ordcr thal soEe pan ol

'/dsenl

dre

is $e r€duced

sidth r€duced bv the chnve

is visibl€

4(!'rvl > l'l'

'llrus thc eftstive visibte widlh of$e ce$cnl tor v nnsing

ar sonc

fon

0" ro

(5.1.r0)

900 - t th' aclul visible Th. brishlness of crescenl falls sha4lv s 0 approaches valu€ of vidh eiven bv of cE$.nt at any val!. of V m6r tE less tbd d€ e'onetic (5 1 10) is justiced (5.1.8) Therefore rhc etr6tiv' v6ible width siven bv

reidrh

equatioh

178

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 it ir ma

c*v-fr=r

(5.1.Il)

ia vi.r6e lb .tleI it vit'bL ia sftlth tl to'E 'rgh dilh!6l!.cc..ttti3ncv.r!d.conDldc 180!irlcrgh Tbu!'tV=V.

I!.ll

oOE qscr.

!'n Dlrt

(5.r.r2)

Tladc i. (51.12) ||,! i! ! ddr.. lcogth

oflh. ctelcai k

Idf l!! l6â&#x201A;¬tt

ih' cr'!c'dt $ itll

lt' iol'l

Sivcn bY:

(t.l.lr)

t;rr*"CY'*)

'I}lus $e €xcecds the

ae$c

mi.ioM

width

value criterion, for

len$h

,/,

be dalualed whelever $e fieoelical

visible acording

b YalloPt

palticular lalues of ARCV In

sidlh lr'

q'lalue ctn'rion oi our

$c Fis

the

*8nent [D or

ARCV inlisibl€ according to Yallop's cliterion The scgnenl AB is rhe thorcdcal widlh ltl" at fie cenrE of fie cresccr! Al sngular 'lheefore the ponnt on spaBrion vr from tbc centre olc€sent ED equh4(nrl/" AC is the ninimm width lil- at

rhe ouler

linb oi rlc

cresceni tbal

las dguhr separaion fon cerrc geal$

noi be visibl€ TIE lisiblc = ZDOA should

cls€

then

crtnds fom D o D'

ihan @d

(nininum vhible wid$ accordnr8 b tan ttlq ($coetical widtb) {5 1 13) can nor bc sed ind the

has letrBlh 2!rn. One should nole lhal Nheneve! ttl"

Yallop

cresceDi

nitrion)

is Srearer

isnotlisible, ie iihas m lengdr'

crcsccnl hrs been The tr1odel developed nl this Nork lo cotrlpute tlB lenglhoflhe

applied

1984 Ylllop' lo a nuobc. ol ob*rvalions lepon'd in IneralE (Schaeffd irc of foi ihe crescenl length against lhe elongatio' also kno*n as

1998) Tbe resuls

lighl orARCL,ae ptscnled

in lable 51 1and in

Fig

t'l2

Tlre loBlhs mcalculaFd

foi tlrc valLE by scleciing lhc nnrinum vhiblc Nidrh Vn of dcscenl in rh€ elerv ca* and ihen usnrg (5 1 13) b

ofARcv

'c'ordrg

{t.1.8)

ob*ivirion thc @dinares of Fspeclilclv) tbe rclatilc umull) rhe lcatioD of obseryer (Ldlilude and longitud€ The colunns or rhe hble 5

sllow rhc darc ot

clon8alion (ARCL) of $e Moon at lhc be'( PAZ). rlre dladve .hnude (ARCV) 'r:d lhe and lhc cental widh ol cr'scnl in arc rinc ofobstvalion. lbllowcd bv setr1i_didetr sidlh viiihle at lbe rclatile alritude ninules. rlt 4_valu., lhe niDimum cF$enl lensth cdlculared usiie (5 1 11) Thetablc c.lculated lom (5.1 8) and finallt the cE$e'r is aFmged in chmnotogi@t or<lr of data s€l lhat

b€en used for

ob$mtio's dd

conpaine mod'ls for

180

co't'ains onlv Pai of th€ conPlele

carli esl crescenl siehting in

pe!'ous

200

180

reo

3ro

Era

io80

roo

i;@ JM

=ao0

30 20 E mno

tig-

5. | ,2:

Crlgn

Lcogll!

tr' vdi.Ii@ dE

Tlr! Fi8 5 1 2 slD{! lhai lh. nbdioel

ctdid

ARCV

bdwen

ct!*4t

lcntlh

lhc

mdcl Th' ctdgltion h not snolh s Eporr.d bv sd|Itrcr on {E b$i! of H@k''t tui! r.rlds fd tl 5 N (i) dE cr.rc.nt hngth ha to bc tff"t'd bv dE ElihMoon ro lqizd trN|l tfrecr li' dist ncc .! cl'in d abos (ii) 0F dttdgbdic n!td4' cloe ftal v'rv wi'lc cle cr$ent losrh., cl.imcd bv McN.[v ald Silrd MoEov{ 'ot! d4 vicitritv qftb $t hori4o mull mt v|tilh $ddolv for s

invilibte

$.it

$d nult not h'r'' as lcngh (i3 msl' !t eidt dt b. i isiblc) !.dding to cLi$ of scn*&t Ttt invisibilitv of t '!l*al sftn dft$L diiod' (ARCV) de to ii! clo..n6 io lFdzon (sd[ ARCV) Ho*E' b tbai for r sw' of $e rnc G..nt 3lEI bc d!ibl.. Ot <ftosnlid lnd cltir dt ldgft of $idlh tlt stving tud' (ARCV) mud clN lo dtc4" invisiblc crcsc.nr the elongaiion

rrn (sfrcidt) .le.nl fron nt mdinwn

ba ttiSc

lcnr$ vh'n itt {-valu' is 69 5 12 odlv dto* h.low limils of !tuio!) sooulv dd iot sdltolt' b nE $din'd b'@ i"n or [E obsflllloB .rc coNidctld tri.n lh' d!â&#x201A;¬c'!t M cltimcd b h!rc

cr.!c.d Loglt i!

(al DAZ = 0) 10 iB minimun (terc

c!a!.bla

lEr

182

T.bl.

5.1.1 {Continued

r8l

Tabl€ 5.'1.1 (Conllnued)

Durins rhis

{ork $e

and Fabohi €1. al.) @uld nol he

acc6sd, howvc.

beins eeneraied at fte Asaolonical

cBcell lengh fon

obw.tions of Dfljon (ne ionc{

Obsdllory

fe{ ycas

maiDtaind by oteh,

184

cdccnt ltngth ir

Univdilv of Kdachi. Th!

phologEphic Ecoids i5 given in Tabl€ 5

obsryadons made by olhe^ durirg past

M.icproi.on

the pictotjal data of

bv Scha'lTs

obseNed

2. This inclldes sone

and tbeir pictutes ate availoble liom

Tne nodel dcvcloped in lhis

for borh Ih€ brigh$6s

ad th.

lenglt of

.led by rhc single pmercr crilcrjon for $e earlien visibility of ncq lunar cFsc€n1, Thc model prcvid6 a simpte method of

cmce.i is nainly

geomerric, suppl€n

cdculaling lenglh

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

according lo eordcd obsedation (lable 5.1,1),

rh€

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

rhose due ro Sultan srechnique arc genedllt closer to the

!alu6. In calcul{ions

using th. fomnla of McNrlly A-t

obefled

= ,r- dd X = seni'diafreter ol

The colu$ns nr rable 5.12 show rhe dalc ofobseruation lhc coor\iiMlcs ol rhc

rhe Moon

fod

rhe sun

resp*li!ely),lhe elongation

o1'

in d€8rees, the seni-diamcrer oflhe Moon. thc c.nlral rvidt[

locatior ofthe obseNer (lalirode md longitudes

degr€e

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

rlu" ro sutron onj Uctlutty

.,p.ctvely.

as calculaled usirr8

rlE nrodek

Thc du(a in rhh ubl€ has bccnlnlnucd iD ordcr

oiincre.sine elongalion ot ARCL. To d€Gmine rh. l€ngh is opened in any sraphic

ol!

$nw

ofr&n do*roeachoftu

crcsenl fom picroial record the dieital phoroemph

l@l- SeleclinS coordinales of

coordinales of lhe centre of the

luft

oa the

poin6

(*o

cenftof$ccrc*enoon of circlo h developed $at leads to

cusp md on€ closc ro supposed

vhible oute! liob ot lhe oe$eol an equation

lhe

the dre

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

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 5.

delialion tion

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 the €longation. Hovever,

elongarion

ll93

s a conbiNd dala sel lh€ obsflatios uith AnFa} l44l de8@s (bv Oner) 1667 degEes (bt

rhen consideed

de8lees (by

Ralini) md 20.12

tnd

Qwshi) deviarc Mk€dlv lbm $e apPaEnl snoofi rcsl oflhe dala set Cecchl l€nsths in rhcse four cNs are huch

degrees (by

rclarion exhibiled by the

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 @

"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). plDlosnphic. Thc rccords due lo Sch&fq de lhc only

It is rhe

0tar

erot

tunher nored $ar rhe rool mear square

conputational frdhods (Ouls, Sultmt and McNally

cag

calculated

foi lhe lhree

it n found $at Suhan

led( cror (4.76 degr€et follosd by McNally's thal ha m

.mr

of 6

s melhod

decrs

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

our model hd an eror of ? 22

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

t87

L?ngths

o[CrctehB

180 160 '110

1m 100

80 60 zlo

ELOIGAN()|{

+OBSERVED FOURESHTS Fia.

SULTAMS o MCMI.LYS

No 51.7

200

i60 160

E.ro :100

380 360 20 0

15 20 25

ELq{GAIOI{ Fig 5 |

C'lsat tdgo5 $rinc Eloqiion

188

withdn

wi.rioa dc

aRCv

Tlre modcl for rhe calculation of

cc*ot

lenglh may be u$d

rh.

.dli6t

visibiliry ctuerion 6uch in the smc say €nphasis

nor

dcv€lop

6 Yallop\ cilcdon cm b. usd, Hos.ver, our 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

Schaefer (Schaeftr.

l99l).

The

lcngrh and

smc

.lon9rioo bccon€s snoolh

shoM

pE*nrcd by

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

? 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) ^rrcduoion

These resuhs

ae in clos agreencnr wilh fte Fig I in Flroohi el al (lbar is a

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

LUNAR CALENDAR FOR PAKISTAN

6.2

Mosl of the hlami. counri* follow

ftn

ob*wadonal lu.d cal€nde,

ar leasl

fot

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

sa$e6

infodaioi

rhc basis oflhe

3bour thc

claim ofsishriig

diFdncsof$e Islmic

organiatiom de

cle

rhe

csce

connilre

Thc clrims arejudged

Laws. Som€ FpEsenlalivesofvdousscienrific

$c clain/s is jNritled

consnhed. Oncc

relisiously andor

soientilicauy, $e Ennoucemenl is nDde rboul begiming ofthe nexr Lun,r m.nrh In x way,lhe beginning

ofde pnncipals

'lle

ofE\v lunemond

is bascd

public obscBations ' verined

ii licw

laid dosn br lslamic shariaat laws (lisled inanicle 1.3).

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 observeis de iionr rural arcas sherc thde is leasl

ihduriallraffic

oa thcsc

6nd liSht pollution,

aid n k higlJy prob.blc thal lhe obseNing condilions d€ n€ar perfect. TheEfore in this

lhe* public obs.rvations e considcrcd |o bave a hki degE. of autnendciry. ln \ork all lhcsc public ob$dation-b.*d d.res ofsiart of€ach Lunu nonth frcn ye.r

sluljy lhis

2000

2007 ae reproduced and

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)

of E 95 lnDation during the Jmualy 2000 lo Scprenbd 2OO7 1420 AH lo RaM, 1427 AH) .tong *nh obsRalional data 4 presenred

TIE resuhs

(Shawl,

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

day and vherc

Equird two days

codlncioi,

tater, indicaled undc. colunn hcaded

followed by rhe Dale in Oregorian catendar. Nexl

fo! $e nexr

.Ldt Coij

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

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 and

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

Isl.nic mondr $at begi$ anq a foltowed by $c CreBorion d.te ot slan ofrbh

rhe nrhc of

sighring reponed on $€ previous evening

lDnlh.In lhis 16l coluDn und$ Cregori& in dE nonrh,

also

8nd

dare

ofsiafling lunnnDnrh, numbsofdar-s

Ttu la{ colunnr conr.ins a comnenl.

sl.nrng new lunar noDlh is in agrccment

thc column conrains I,ROPER.

tle

vi$

the prediotion

..ob*ryalional

rc dccision oa

critsioD Gratur q.ilqioi)

decision .nd tbc Nodet

arc

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 consideed lo b.

startng new lund nonrh is one day lare

lalue and EARLY

indicaled by rhe dat of

csy visibitiry on r_

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

lhal observarion claios are couected from alloverlhe counry.

192

should

nored

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 reponed. Tbere was oDty one oeasion {hor rhe earty sighrnrg is rcponcd 2004), hus,1herc weE only 6.3% enors in rhe decisio.s oflh€ moon

not

Nov. D, sightingcomine

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 of Pakistd duing

only case

LARLY'sighring acepred by $c audonties was for rhe nonrb of

The main

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

'e6oi

tor lare sishlinss or

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

compaGd

lhe

ofee6

6ul$ of

m@n

1960s and rcpon€d by Dogger

€rcs (qons clains of

wlch,

progEms cohdEted in Unncd Slales in thc tar.

& Schsller ( 1994) Accordi.erorhh rcporl l5%positite

observing

visible bur rhe obsrveF

in rhis obscryalion.l elfo,1 is much noE pbnisi.g

$€

cresceno and 2yo negadve

nissd then) {qc

iound monssr

eriou (cescenr

cxperienced

was

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

sighiiog

(pardculely for ianins rhe dontbs of

Rd.a

Freque.dy. claims of lery

made

in sonc rcgioG of paknb!

ad Shawat

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 acrual signdng

luiu

Donth in lh. Kingdom ofsaudi Ambia is nor baed on

c6ent

of lhe n w

Thee d€cisions

based on cnrcria rhar have

chased tuee tnes ov€t $c pan two decad6 (odeh 2o0o). Up ro 1999 the

or horc alter nron6.

fiis

oa Moon

citria

vas

fouows

lf$e

fi€ New M@q ihe pevious

actually

drsunslis t2 hou6 is the fi6t d.y of lhe Islanic

Moon.s age at

day

matu lhd lhc luDr donrh begiG

day ofconjuncrion

vhich occurs on€ or two dats e.rlis than $e rime shen tbe n.N

crescent becom.s

vhible

Fmm 1999te crilerjon

ro lhe naked

.ye,

chareed ro the tottowing: The tunar monrh besins on

lhe eEning *hctr the sun*1is b€forc

no€ di$rtuus .s ir is Fssible

tlt

moon

according to Mrcca. This

lhar rhc sunsel befoe moonsd

sil

\as

occur even

belbre rhe conjunction. Th€Efore. rhe hmar montn may begin three days betorc rhe

in)

ccscent b€comes vhible to nlked cye.

Ftum 2001oD\ardr rhe

a) b)

Tl'e s€ocemic

cirdion

is:

onjudion scurs

b€foP

Sb*t.

The Moon sck aner rhe Sun.

This is nore realistic bul it is scarccly possible thal the crcscenl becones visiblc on dE day

The

ofconjunction.

Min p@mders signinc.nl in lh. .arti6r visibilily

cEscenl aE LAC, AOE, ARCV, DAZ, ARCL, Phas

crnerion for lhe luoa

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

(Astrononicrl AlMac, 200?). Duing were consid€.ed in lhis 3tudy rhe'e cEscenl with

phN l.$

when lhe ph6e

lhan l%

rhc p€riod flom

Jduary,2000

(as

b. t%

Sepr., 200?

nor a sinsl. incidencc when rhe

ihrr. ws

no claim of sightings

lh. oitclia considqed (Oct. 23,2006

In lhc cae ofNov, 13,2004 th. phase

siehting.lain qas accepred

consid€Ed lo

$ar

sidd.s ot

eponed and am€/.d. There weE two @c6io.s

grcarer rhan l% bur

not possible accoding ro bo$

ws I

a.d sighting

and Dec.

2l,2006).

l9% lhal may have been lhc Eason rhat rtre

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)

(21% of Ihe studied) ascs of sishdnS

Fcb. 17. 2007 (2 t .6 hous). ln

daiN

The obsemtion of Feb. | 7, 2007

when

nadc ar

$c .se of M@n

aI $eE vee 22

les rhfl l0

fic slronomical ob*Fabry

ar rh€

& PlanetJry Astophts'cs. Unrvcrsily ol Kr',(hr *. ,pou.a U. lhe 6' Codd Refractor T€lescoF. Both rhe prcdidion crileria alto$ed

Instrule of Spsce cre$ent nsing

crc*enr sighri.g wirboul oplical aid but wc could se rh. cr€scenl wi$oul

tetescope. No

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 other claim

sv.icoprciecr.op). Our

phoroSraphic r.cord h also posred on retevanr websnes.

!95

Tlrc'€ seE 6

w4 not

rhe crescenl

depcndablc

c66

l&to.

when thc ag. of M@n

seen. Thus,

bciwen 24 hou's dnd 30 hous

ihis srudy also suppons th. idea that age is not a

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 (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

from

Babylonian clnerion (Faroonhi et al, 1999) only g

tines when

Dabylonian

22) alto$ nalcd eye visibility but Howld, on rhese nine ccas

,{luc dd

crnerion (ARCL + l"Ac(in deg.ect <

lhe s-laluc cnteria do

tur

difer

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

(1.057q).nd thal @casion cases Nhcn the cEscent

probl€n occun

cas

conkoversial in liew ofrhe above discussioD. Thc nve

$as seen hler tan prcdicrd aE not conside!€d as an ero!

d d!. lo ore(sr

skies.

obenarional IuMrqlendar $ese Fediction

0s thc

TtrGfoF, tor lhe tomulaion oa turue dreia de nidly dependable_

TIE € has ben only o.e occasion whd lherc mrc three co.*curivc nonrhs 29 dars elcb (during June, July and Aususl 2ooo). This is rhe ndinum fo. Fpelnion 29 days lund nondrs anticipoied

The li6t (Rabi ul

e&ly

loricentury AD by Muslims

Awal, I42l) of$is lriphr ofnonthsbeema

196

(

of of

yas, 1994a).

day talerrhan p!.dicled

oftr\i'i$ lbir tNinun rcldirio! @!H ml lLv. tc.n th.Ia llxr! vt&r D frt! (diD|d) coorc.rlir.! tdt! of 30 d!y! 6d h 6b Fiod of

ac&noo

'lldy.

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

$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

'ftis

rvork

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 models are nodified if posible, The lask of

achicvedenG ofllrc Nhole

eihll.

Fi6t ofalla bener undcFt.nding

ol$c

obrerarional, associated wilh rhe prcblem;f

dd sishri,lg of$e new lLmtr ssenr

issues.

l|t

firet

conpubrional, aslrononrical

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

solving

lie

problem desribed

in nr€ pEviors paraeraph lrritially

lhc

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' rcchtriques wcre nnptmenled on Miciosoft

$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

nr. applicalion of

re thar apF{ modeh

oblained therc. Tbc hbl€s comp sing thse in |hc

apFndiG rc all

d€velop€d

The modcl due to Babylonies as

fon

ts!l$ tifiin

rhes *otk sh€ets

d*dibed by Fatoohi

$at wa modilied by the MuslintAFbs is bricfly desribcd 198

dre m.in t€xl and

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 1..2)

errensirely. h is loun'i rhal though, $e Lutur Rip.!6s function is

Dorc sophisl,caled

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 skybrilhhess impro\.s

and the

biCllNs corrdsl oflhc

crcsccnt aalnrn rho

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

rhe larger valLEs

cas.s \vilh large DAZ tlrc Ripene$ luncliorr ,4,r can bc llrgc so lhrl

R,,, arc

Ed. Conseqlendy. aor lrse. lrtirudcs ldgc .rc of

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

scparalion is

and thc

ARCV-DAZ nodels.

sftn du

$c di*usion

ar rhe

.nd of lh. 4ri cbapl€r lhal

ba*d nodeb, the ForherihShmr's modcl. th€ Maundeas mojel

lhd

ARCV_DAZ

and lhe lndian nrodel

successiv€ly belr€r. These imprct.menls ae due to deducrion of b.ner and b.trer basic

199

dau ofnininum aE ofvision for

fi.

difcrc el.tiv.

uinurhs- tlowrcr. all

th.*

modcls

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 isDrc the ridrh of

n.w

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!

$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 .r.valu.s

risibl€. $hcn n $ill b. visiblc undo P.rfcd st.rhct condnions whc. Ue opricrl \o(ld when fte cB$ent $ould bc sinPlv nor visible $ith ot Nnhout atrv b< Eqlircd

opti.al aid.

rfu*

condilio* har. poKd ro b. noN and morc Eli.blc

$i$

incBtsnB

is rhc comp.ri$tr ol .ll fi. htljcls a @rclically, frathcm.ricElly, phytictlly.nd in vi.w of$eir success p.rcentagcs 'or s.r of ob*wations colled.d tom d. l.t. l9ri c.ntorv till dcntlv. Anorhcr 3isnillcst Thc najor

bution

or e

contribudon ol lhh wott is lo convcn aU th. mo<l€ls ino arc listcd

si.8l. Padm.tcr crn€ria

bclo{ in th. inccsing order ofsucc€ss pcE.nlagc:

vp = (ARcv

- ( 12.'o-0'.00s(D,{z)'z rYlo

200

h.*

=(^*

-e4,:!.,'l) " # f l-

v, =ltncv .lo.no - o.orn1o.tz1tt =(^RCV

uaz2\lto

lrt.8311.- 6.3226tv + O]llgr2 -O.tOl8V3rttO

\t, v

o.ooot

-t5 + py rxtr+,pfl

| | = tARcv - \t2.4023 - 9.4818W + 3.9512W - 05612Wt vr=ARCL r !rc olscpaBlion -

21"

ln rhis {ork schactels modcl of dldritc bdshrn€$ is ale erPlond in t.ms of -n'h nasnitud€ contrrst tnd thc dsllE aF in lsRD.nr $i$ his wo* (Schelet 1988.) ehieved by iniplcmenrine lh. rshliqus d.r.loPed bv Schel.r and o$ets |o brishtocss ol lhe ev.luarc sky brighhes (in tems of linninB mgnitud.) and cE$€nL Biglrness ofsky and that otcE$cnt dcp.nd heavilv on various ihosphenc has b€en

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

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

Magdlude Conl6r (Amag

maenilud.

of Moon

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

ben evaluat

de*ribed by. veniccl line

ovr.

liditing lisibili(y

201

cur. ofBruin

'fte najor

achjevem€nt of this

is th. romulation of

neq sincle pstuerer

using lhe cireion on lhe b6is of rhe Fchniqus deleloped bv Bruin and Yallop cuFes and thc bngnrrc$ modcls develoFd bv Sch&fer dd othe6 rhc visibilirv

liniting visibilily cwes aE

developed for crceenls

actually obened tnd hav. bcen rcpodcd in

lile6tue

ol diffeEnl

widrhs

ll'!l

wcrc

Bruin dcvelopcd lhe$ cllvcs on

dalt,s *nh rhe bdis or{i) lhe ale6se bdshlnss ofthc tult M@n md thc wv i ot the skv dudrg decr€sing alftude abole hoi@h and (ii) tbe avedge brighiness On lhc olher way n depends on lhc dolld d'prc$ion b'lov horizo' |he

t*ilid

&d fic

wid

lor

llEn

tc bave usd rhe adual brightncss for Ihe obscb-ed cEsc€nB ola fixed rtrc sms tn sne ofobselvation we hare calculared th' al ude of skv poins bavins

hand

compuratios m $ar ol lh€ dcsenr for difrercht solar depcsions such sahe wjd$ al dilIelenl localions and Ep€ai.d for a nunber ofobseryed cE$e 5 of the our Thc wnole re ten dmes. The aliudes or thc skv poitrts lhu obt'ined 'vc6Bcd widlhs prccess h rhe. rcpeaFd for cnsce.t ofdilTereni

bridft$ d

These conpulations tesulled inlo our

lisibililv

curycs

ed $c linnins visibihv

$ose ae slishtlt diferenl 'iod linc joining thc ninina oi dEs ntlighl The displ&cd slightlv with nini@ of Bruii for Bruin s cutves Tnis lcads lo slighd, cudes has a slope of9 3/5 as compaftd to 9/5 noo'elsuosl lag aller thc b.sl dme of visibililv" which is 4 l/9 1 rine th€

curyes. The linitihB

lisibilitv cses we

have obEined

differe

lae afiet cohPdcd lo Yallop s be$ tine which 4/g rincs $e noonst-sunscl l-rttd the ninim olor liniringlisibilnv cu8e rhe suset). Th€ bdic dala ob$inod fion visibihv pdamslcr: b a fiird desee Polvnodial tsuls inlo the fouowine

suet (6

=(.4RCV

(4351

/3 + 2222075057t4! - 5 422641ritf +

l04l4l?tt)/l0

io lbar of Yallop' we have als dedeed visibilitt condi$ons in a tum€t simild Yallop Howevet' ltplj_'n8 condilions tre sliCh v difiere lron those of

Our vhibihy

pcrcenrage is iound lo b' thc besl our model on the obseNatioml dal! the succcss \orr' erist and lhar dc €ncd dd conpaed in lhis dongn all the dot nodels brighhre$ and lhe btislrhss oi is tbat our model is baed on $e acrual skv

tll

Thc

e6on

201

(escenr $idr varying wea$cr condilions whercas Bruin s nodel

based on aleragd

brishlncssofsky dd rhe FullMooo. As clain€d by Sctaefd th€ brightn

still our Drodcl yielded be

resul$

comPsed

a mucb a 2elo 1o tho* due to Bdin ad YalloP as lar

modcls

b€ in

conemed lt is lroped rhat wilh beller models of skv brighhcss slill beiterclitrion ior fiGt !isibility ofnew lund descent can be delclopcd as rhe posirive observatiohs arc

sonE of$e lpplicarions of {he cnbda of.atliesr visibililv of ne\f llod 'rc$ctrt ro havc becn considered in $is $ork The fid ond $c mosl impoddnt applicalion lre on d€(eaniDc Nhcn tbe new lunar cresceht would b€ visible ar soDe ldalion on lhe elobc

rhe elennrg aicr rh€ oonjundion Anolher ar.a ol appl'calion is lo deduce d ''obscNnrioMl lunar c.lcndii- tor a Yct anofier aBa ol apdic'tion \e hn\c 'eeion lhe lcnslh ofne\ ldDrr crcs(nr e\ploreJ

'.roJdrnrine

t( !1ay be recalled (hal lhe liret

appcardnce

of new llnn crcscctrt nrarlis

ola new monrh in obsenalional lunor calendarq rhere 'ril'ria and models drc calendai likc siBnillcor br calendatical Purposs whefier 'n aduul o6€n:ri'nol lunar Dol lhcsc rlc hliNic Lunar calendar' ulilizrs tb6e cril.da for a!6nging its cltndar ot b€sinning

cLirdia ptuvidcs

guidancc Ibr bolb tesing an tvidcnce oi oescenl sightirrg bv connnon

dates arc rol peotlcdnd tacingdowndtdtrlesofac.l€ndar inhhrorv{herc approprirle \€ll Ncordcd. rhus $c frtin ' ililt of the pcdicrio! crileria $e carliesr visibiln)' ol 'or dd tcsrifv lh€ clainrs ol calendtr lunar tnc obsefl.lional is lo egntalc crc$.m n€rv

risibility of ne\' lunar crcsccm tlrat di' Although fiIsl ordc! iPporirutions like Arnh'neLic Lunar Calendtr o' rnc Moon havt be'n nr lse based on lle conc€pr of Le.p Yea$ md rne a!.rage oorion lor lhe monlhs ol Muslims h.vc been follosin8 acoal sighing of cresc€DLs 'r l@l i6rii8 (R.maan) $d pilsinage (zil hajjah) Thc calendars if bled o' a PEdicrion

oir*ion like obervarional

lo rnc tbat of Yallop or dre ohe dcveloped in this work arc $e cLosest calendd Conpan$n of rhese criteiia wiih lhc actual obFNsdonll

201

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. â&#x201A;¬dt rion d our GEluc

old!

!|dc!

ruirD

frr lh.

Ft!

2000 io 2005

dilrio D. diltgt@t is lb. !!3rlt of lithd th. t d wtt|t r dM !o *hi'h qr &e rw |E|r c{&. codd nol t tigln d dd tb Lud E olh b.t!! oE &v l.|c bo AdrDirtic cLiu6 of otsElion tld {t L|e dot:t t g.! oE &t dlicr U|n (d Q&K)

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 Tbit

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');

{ I

inputdalerime0i

cl6cd);

nonlh_chee.0i

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 -

goro\tr52.ll):prinrf(

ARCV'igolorJr65.llr.prinrn

):Sotory(65.r0ripnnrt(5.7

qvall(oah-sall)-( I 1.837 | {,1226*wid+o,73 t 9.wid.wid0. 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

ltl

o.7.1tr

./o7.llf h .0de gotox(52,12):p.i {"Ase = '),sororr(6t,I 2ripnr'4 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

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

eotoxr(52,16);pinq Rd.y

bn(r'abs(ltdstrst)i

= ");soroxy(65,16);p.inr(,'%.51!"_rdar;

sotoxy(52.17);pinr( Rdayl = );soroxy(65,t7):pdnr(aZ.5tf',rdayl): go'oxlr52,l8):Arnlfl l:gorory(65,t8ripnnln oo.5||.btyJ: i 8o'ott(52.1c):pdnrfi S-val@ - l:gorory(65.t9,:pnnr{l o/o.5tf .6ky-

tuly -

soloxy(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)

{

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. :

Soloxy(10.6)icoul<<'Day goroxy( 10,8)icoul << "Mon$ Soloxy(10,10);cour<<"Yeo ,Lo.g.(+

;ciD>ocnr; :,,lcin>>datei !!;cjn >>monlhi :

:iiicin>>ycdj

gotoxy(10,12);cour<< 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; ,,;cin >>ph!m; gotoxy( 10,20);cour<< "Estih : prcs=1010.0;

*0.0;

Huidirt

) {

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 i(lhour>=24.0)

{

lho!Flhour24.0; ldri--ldale+1.0;

204

im:

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;

ndalFlda!.xtnonrh=lmonth:tlyâ&#x201A;Źclycari 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

pdnr('

- %lfrnjd); "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(mrrdIiBrcdculadonoi s.rinln0;

209

620.0)l/8400.0i

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<< ns. ":eoloxy(2?,1)i@nved-dns{fabs(!long)); pnnt( E ); dF ifulonc>00) 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 {10.6001'(mn+1.0))))+d.tb-1524.5+fi i

(lo.s

) {

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'st0.0000062rst'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) i(gnsldo>24.0) i(s6rzerc<0.0) i(easrzerc>0.0)

8rnst4'o+=24.0;

gott4to-=24.0i gaslzco+=24.0i gastzdo-=24.0;

smstcun_Cnstrcro+frac124,01 I 002? 3 ?909I5 i gasrcuFsdzero+firc'24.0* 1.00273?90935;

i(snslcuP24.0) i(emslcun<0.0) i(sastcud>24.0) i(gastcuft<0.0)

snslcun-=24 0i Smslcud+=24.0i eastcuft-=24.0i sostcud+=240;

lostcllFgnncd+plong/15.0i l4rcuFedlcur+Plong/15.0i

i(lastcuPla.0) i(lostcud24.0) i(ldlcm 0O' ii(lmsbu<0.0)

lastcun-=24.0; lnsrcud.=z4 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 o!2dhoo)dn' o2dr".,ntrcdegl,inlr(min).rnl, cscc\): prinr

(

) void sun cood(double) {

fplFfopen( v$pean.txt',

// LO,\]GITUDE OF EARTH fodi=o;i<6:i+) { tehp=o.oi fo{=0j<lnellilj++)

2t2

{

y.lf,&sx,&bx,&c); rmr=lâ&#x201A;¬mF4.cd(bx+c.1);

fs@f(fpt'%lf

yolf

)

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+)

[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) elpha-salphe/15.0; i(s.lph*24.0)

salpha+=150.0i

salpha=elph.'24.0:

sha=lastcur-salph!;

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 d€lta=6inGd€lla)' I 80.o/Pl

salphap=elpha+.lcladpha

elr- sin$laf P!

sinrsdelrap'PI/ I 80.0 80.0): I 80.0)rcos(sba*15.0'Pvl " salt=asinGalt)' I 80.0/PI; I

80.o

)

rosulat'PL

0r.on sdehtp'PV

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); 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

l I

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 sm=szn/Gos(platrPvl80 o)'co(elt'Pl/180 0))l

Pl/ | 80

sad=acos(saar)1 180 o/PI; s@=180 0i it(shdF=o

0) itlshaF=12o) ehe 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.'%[

q.lf %lf &alpl &'ly'l " 0lf ,& \,&!lpo'&!lp l.&alp2 rr+alp2r1.r/10000.0 sMFsuov+v.sin((atpof atpl 0/"1r 0/0lf

+alp3iPo*(i,3Y1000000 0

2t4

0):

+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++) lsconi(fpu2, 9'lf ?'lf eolf'.&v.&slP0.&alpI)i I s!mwpp-smvppF ! | sin(alpo-alp I i t)t PVI )

0li

mton{=218.11665+481267.88134't-13.2581'l/10000.0

.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++)

fok2.'/'lf ' %lf t'll"/.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 iscanfi

+alpl.pov(!4y100000000-0)'Ptl

t000000 0

0)i

) fodi=0;i<64ii++) e"lf .& v,&tlpo.&olP I ) { f$anf( lol12,i%lf'l"lfrsml( tlpo+alp l'r )r Pl/ l 80 0) suNup=sumufi! )

fscanll tDk2. '/"la %lf "/.1f ,&v,Aalpo.&alp | ):

sumuip=smupp+vtin(alPo+alpI'0'PI/l

0li

)

l*anfi fDr2."%lf %ll

c,6lf ,&t,&dpo.&dp

sunu;pp'sumuppF

)

vrnn(.lpo'llpl'0'Pl/l

0):

I ;hFsumu+GuouCsunupplt+smupPp1irVl0000.0yl000

fo(i-0;i<l54ii++)

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

0).

)

fo(i=oii<68;i+) I fscdf(fp$2,'/"lf o/.lf %lf'.&v.&alp0,&alpI): rt vmDp=suffpPrvtosl alp0+tlp l 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 o'lf PUI80 0)): OOOrolla-O.O0OOt019'sin(225 O'47?lo8 '"i.""',-O 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<

iftmbn;27o

mdlphanalPha+-1600: uo o I

if(mh*24.0) inmhi<o.or

frba_-24 0; dha+=24 0i I Pv l80 ofcoslâ&#x201A;¬psrlorr m;?lk=$n(mltr

-i"iu=na.r.

lao 0):

_1800:

--^ ^O)rsininlonsrPl/1800): -

-'Lmr"t'pvrtOO)tsin('psilontPVlSO

nrdelh=asin(mdella)'l 80 o/Pl:

rulphlp=malpha+dell4lPhai mha=hh.nelraatPha; 'iil""iJ" r80 0)'co(mderbP' er'i do ol'"'t.aerl!pr Pr/ r 80 o, "os(l)rar'Pv Pl/l8o.o)'cos(mnar lt 0rPyl80 0l, malr=6in(malt)1 180 o/Pli

-lii

i(sr=l) ;rcfr(

042730 remfl))'(tr 1.oz(bn(nal(+ l0 la mah s lt)fPl/l80oDft283 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)

goroxy(45,20)i@ut<<" meeta=acos(mâ&#x201A;¬ela)' 180.(VPIi

mdehFnddiap+@t'1cos{mera*Pvl

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

)

{

hdeluFndellapi

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;

; a,m-si1(m,lelrar'Pl

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 po$,lsi;(epcilonrPl,I80 0)'rn(slong'Pl/l80 0' 2 0) 0

ii;i'iriio",tr.o.po",rsinrcp'it.n'Pvl80 0)rs'n{slon8'PVl

iii=.co(iii)'l

t)))

80 0) 2 0) 0

80.0/PI;

mele hor=90 o.Dhil.mdai

aii*rir.:-.*.iai*z zf oqplatrPl/l 80 0ycos{l)hil'ddt'Pvl80 rda;

5)li

l=l0 t'.os(plal'PVl80 0,/.osohilcmda'PVl80 0)i

217

0):

i((dnon8<90.0)&&(slon8>100.0)) ddone-nlon8+360.0_slon8;

Bky

dclong=mlong'5lon8; lal'tan(phitehda'PVl 80 o)+delonsi

doubl€ modtunc(double xY)

xFxy/l6o: \y=(xy{double 0on8(xY rr)f 160: iltxy<0)

loid paBlx(double hhdoublc dd)

ol,llliililt;lli*"""pr/r80.0!emai)rr80.0/pri 0)-r'Ih'sinr'r 0'|prar'Pr/I80 0rvl600 o: inor' hre'{nrplar'Pl 180 0) on': ;-eniin'vntuu'pt rro or"o',r'PVr80 yy ro\tuu'Pl/l80 o)'hit'.os(plar'Pl/180 o)/con:

ll,l"'lili-""t.r',t,

rho=po*(xx's+ty'y) 0 5)'.mtji

80 0f sidhh'PI/l 80 0)v i.rJom=aurt'pr'co"rphrPVl "' *' - --i""ltaa'priitooFpG'cos{PlatprPtl80olrcosthh'Pl/1800)))

delrMlPhal=(l 80 0/Pl): hp=hh_deltaalphal

or-Pie'sinuratp'Pvr 80 o))'/ 0)l\' aoirOa'er'lad Of @s'hb'Pl lso otspie'cos'pldp'Pvl80 dehapr=(180o/PI)i hp/-15.0:

i;i;i|$,ili;t4r", " "' _*

to'r'{sin(dd+ur

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.

06ng ltar)))- t.o; oons (ksY)));

e|s raF(dotblc L€--l,ay/1216 85i ;;=1"' ;';93;ggiijliil['31ffi 3"'ff.i::.i:.i:::*, -.""-."ar.""ii lSu-zq toSl5o7'kav 0 0o000l4't4'r*' 6ei528' kav'o o *""":l'j'#:,!';i"*::'r;;:.8 '_ ' .@.Le.r;co 0ooo00058r r€.re'leerrec)i r

218

07582' ree'

reF.

ooo0 I 2r8

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.

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)); o'mhonFr+1'0.00007'sin(mmoon+2 o'msd)+{ 00004'sin{2

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

0rlarg)_

000002'\i l.o'mmoonrsun,'O00002'sn 40'mdoon)):

ol=(299.77+O lO?408{kay_O 009l?3rteettee);al a2=(25 1.88+0,016321'kay)ra2'=(PI/180.0)l a11251.83+26.6t1336rkay)i.lr=lPul80 0):

2478'kayl.a4'1PUl 80 0) a5{84.66+1E.206239ik.y)ra5'1Pvl 80 0)i rkav):a6r=lPl/180 0)l 16=1 141.74+5l.lOl7?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): a41140.42+16.4

*=(PI/1 80

| 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): addc*=0.000017'5in(al2)i addr-0.000040'sn(.1 addcP -O oOOOJt'siral3\: addcrjo 000021'sin(514)i mjdanjd+pe*addcr z-rdoubldlongrMjd+osDr:

addcrj Oo0l25.s,n(al

addq+{

l)i

2t9

0ll

I(24299161.0) e=z; el$

{

alph<doublc(lon8((2-186nt6.25y3652425))):, @-z+ L0+6lph-(doubl€ (lons (ddv4.oD);

I cldouble dd=(double

(lons ((bbj22.1y365.25)Di

(lo4 (355.25'@)));

e<double (lon8 (GMdY30.60{ l))); md.rFbbnd-(doublc (lo!s (30.600 l'c))Fr:

!)modn+l.o: clsc mmodh+13.0; i(montF2.o) my€f<4?16.0: els mlta-rc47l50i i(a<14.0)

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"{

hou.H; i(houP=24 0) { houFo 0i dare+;

i(daenonthdaYslint(nonthl'0)l)

{

datFloi mo.ll|++;

i(nonth>120)

{

lt )))

nonth=1o;

v6rr:

)

{

nin=00; if(ho!P=24.0) houFo-o]

{

220

dtt'+t;

iqdnl'd.ddryltin(0.dl.Ol) ( dGl.0; noiliFli r{rr{r,r12.0)

{

t , void

lro tsl.o; yaftsl-;

l)))

ning-no;

iE_tordvoio

{

i(no|t',-24.0)

i{d'r.>tuothdrylti!ft nornlFl,Ol) { dnFr.oi

mb+)i

t{montlFl2 0)

{

rcIilLno {

))l

ftGl.oi rttfi

rld*.ri

i{d"p@|tdry{i'(dl.0I)

. {

dGr.q

n|odrh++;

i(mdb-12.0)

moortsl.o;

.ll

l(3â&#x201A;¬l!-'d)

@no

)

I qEodD-12-0)

(

nofl.0;

Doll_ctee{i

i(|enrd) s.t6i8_n0i

{(3â&#x201A;¬c<1.0)

{

eF59.0;

i(bll<!.0)

{

diF59.0i

hdc-; i{hom!.0)

(

hou-23.0; dr!+_;

i{d@<I.0)

nto'fi_;

i(r!.dtKt.0)

l t

)))'

lcnioLn0i

!to#12.0i

l. uu**Il!(.*.orlt

nin-i i&Din<!,0) DiF59.0i

{

i{ho8<t.0)

bou!23.0;

i{dab<r.0)

dr!&io

!erl-: i{no!.h<lo) I dlo!tFI2.o;

,.. drrc!.d!dry.(in('dLl.0)1i

t iqbou<I,0)

(

!dF23r; &l}-i

i{de<I.0)

mo!&-;

iftud<t.0) t e.dhl2.q

-. ,,

tqr-; d.|ddrF[i!{DoodFl.0l;

*!iâ&#x201A;¬|{: ) {

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:

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 !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+)

jd=jutimdde{yea,,oonthdare,ner: jde-jd+det'a(in(yeaFl 620 o, tt

)ilo.o.

stjd=juliddarciy.tu honoldale,o.0r, nlsUd-2451 545.0)46525.0i F(jd+2451545.0y365250.0;

214

24 o ).,

t.0.

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)

hN=@s(.w/d@)rl800/Pl; mot=sdphaFplonS/l

nontmot'h4Yl5

o-gnstzdoi

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

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/ 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 1

) 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

ro.or-'i"olat'Pvr

mder@P'Pr't

80 o

80 0):

''srd 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;

nnol+=24 0;

i(mtoP24 0)

mnot-=24 0i

i(rndol<o

nonar=24.0; i(mone24 0) nonc'=24 0i i(mrivo<o 0) ntqcF24.oi i(mone<o

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 ( cvPl 'cos(eph'15 o'Pvl80 0))'l80 0) d.linn=(ali5e0.l25y(cos(plat'P/180 o'Ptl800))y15 0: 'cos(ndelap'Pvl80 0)'sin(@ph'15

!ts=mtwo+dellaml

uhs+(acos(cruj(eMl+h,td

l000.0J

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

226

0);

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

0) noneF24 0; i(none24 0) mone'=24 0i

iflmoc<o

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 liEcatmr,z4.0i

180

rtrutmfrzonlmi i(xnr<o.o) ltmr+=24.0; i(ltmr>24.0) llmr=24,01 )

void dGplay_sord(void) {

il(slar<o) sd=

S'i ele

sd=N i

Sotoxy{16,1olconv.rt dn{ilons); eotoxy(16,I I);conven_dos(fabs(sla0)r soroxy(16,12)iprini(' %.21{Xm dn);

i("dclb<o.o)

gotox(l

sdrs,

ele

sd'N

4);@nv..t_bn{salpha); sotoxy(16, I 5)iconve't_dms(fab(sdelra))i 6,1

ifid.lllp<o.ol d

â&#x201A;¬le

sd=N;

6)i@nven_hft s(sho); soloxy(l gotoxy(l 6, I 8);mnrcn_hnB(sdph,p): 6,1

227

o/Pl)/l

0);

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

sdrBi.ls.

i(elr<0.oJ

$: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: ed=N i sd=s',

cl*

;tmd;lia.o.or

hms(nralphal, ;otoxy(16.1 5,.cDnven dms{fabs(mdelta))

.oloxytl6,l4),on!cn

soro\y(36,1 6)lconven-hos(rnht)

gorory(16,18).on!en hm(m.lphapli sd=N i ,fidelkD<oo) sd=s: else e;ro\ril6.l9 r;mnven dms(fabnmdeltaPr', sobxy(36,20r:@Ncn hm({abs(mha)): Robxv(16.22);conEndd{rabi mdlU:

;ftmah<O.oj SotoxY(

sdrB

ebe

sd=

]6.21}:@nven dms(m@':

goto\y16.24,:pnn't "" 4lfdc mrn" m*midrd2 ur'

iDnmniDu4. \n';d1""dl"idr""d ocnl iniidate'Jnimon'h' rn(vean) %5 IlAt%t Il^f/"d\".plal plon&in(pall)): brinttflt4. 'ta'irii"-l."s.s. in((urssin(ubr f60 0)l: r rn'v,s. r trr," a."d\".flmp.phu.,n(urst 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.rr^e,"4.:tNz8..lllt.wid'60 0.qvalohlri

'ri."iiiiiuo.""

pac

"s.alr

ort""s

4lNToa

4lrrt

9 5lft"phrtemda-mla(mlon&srong

[i,,iiro"n."vo.zrn.z*.1 tfl *,,8 ] lt\r./,8.11{\ro^8. 1ll\e.6. 1U\".25.5msFidiat.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 ah.P/180.0.)r@s((n'a-faa)rPvl80.)))r l80.o/lli relonsui=(acos(si nGan'

P V

80.0)* sin(lalt'P

iPl/180.0)'co(G@-ran)'Pvl80.)))1

0)'co(i

0.0)+cos(salr' PI/ 1 80 0)'cos(ral( 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)

l.0icos(zbdist*Pv180.0))j

mhposâ&#x201A;¬f=4o.o;

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@(90.0ex)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(180.0elonsp),4.0).(!ow(t 0.0,-9.0))i

mmag=nnmas+.nshlil; co'@=pow(t 0.0,c0.4.k$blil.@pceo)

229

?O'/po$rlelomor20l: fem-Doq(IO0,(o I5'telomorv400)|'62'pov]l100 I80 0) 2 oi)i ii'-i..-po*tro.o.s xr, r 'oo+Po$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 I 80 ili=,J"-'oi" rid 6.i:e" ' oe' pow;cosf'ro^nqun'Pv zr,n ' da\bno$( |0 0,( 0 4'(mshtrl_o6(nlrl*) 4rk$blil'slpo*r) rl davb=<t vb'( I G pow( 10 0

0 r'2

{

d.;b=lr;s.ctoud440000.011.0-crourl)'davbi ,r,aout.t*itul b$hlil=nighlb+dalb,

b*hl'l=nielb+Nilb: cle' ifihalPoo) b(hL'l=bshlil+m@nb: bllrlil=bschlilipo*(10

0.12 0)i

t', -jili.TrJ,,gf*,,'; ;il"fi;iil;i"

L'=o*n,ruo.oo,,rr',*oo,,

;il=on;.po*( l .o+;oa{cr*o'ber.0 !cn=-16 57-2 5'log(khy'log(

5r) 20'r

0)4nscnlzl'

2i0

:l:::SXll3:::i:l

APPENDI'(.U 2OM ANCIENT, MEDIEVAI, AND EARLY CENTTTRYMODELS

23r

2!2

23J

234

215

236

2]E

2t9

240

24r

APPENDIX.III PHYSICAL MODELS

211

!.I9

244

245

241

244

249

250

!!! 4!C q4!! 9! !.39 94 ,-!!: !4 -9!_ ,-9!l 9lJ1 9-41

25t

2r2

253

91'!

254

APPENDIX-IV OBSERVATIONAL LIJNAR CALENDAR OT' PAKISTAN

200G2007

AND ITS COMPARISON WTIE

TSE VISIDILNY CBITERIA

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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 râ&#x201A;¬1 (200t AL _ 2txl9 AIt)

AII

i t i !

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267

;Ci{{

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264

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l:--r*s j: ;-ig;ii.EiE *ii€ I€ ;si*i i:=s;;; 269

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;l:;

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!; i;3- :;.r Ei* n S:;.:.!-:= _.rF .-+ia 3F-=:i "*.. ;:;; ig ;l!i l;;r::i'{s g:s; 3::rq;; ii ;le; {: lp l;e; 3; :i:; i: ::;; 271

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272

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