astronomical formulae

8/22/2019 ASTRONOMICAL FORMULAE

1/51

Chapter No, 2

ASTRONOMICAL ALGORITHMS &TECHNIQUESFor lhe determlnatlon ol th prsiso locallon ol the obJsts in th6 Solar

SFtem, panbulany the Sun .nd rh6 Moon, ih6 Frnch planetary theory VSOPaT,(sretagnon & Francou, 1994) and th. lunrr rhory ELP-2000 (Chapront-Tou# .ndChapron!1943, 91) rr. well sulted. A number of sotlwaE hav6 ben developdtor lho shulatlon of celstlal phenomona based on the$ tho.16 and similar otnerworks.In the cunanl studylhe same lhori.s havs b6.n u*d lo lollowthe pGitlonsotthesun and rhe Moon.

Moreoler, to conven the theorles Into computatlonal iechnlques,malhematlcal lechnlques, tools and dlgo thms nave been dtloted. Moch ot thecomputailon.l work is ba*d on the algorithms devoloped by Meus (Meus,199a)but a subslantlal ahount or work on computatlonal aEonthms has been doneIndpendsnlly. For a ihorolgh und.Franding ol lhE compul.tlon.l toob theproblem of llme ls explor6d rnd discused in d6tatl. ln this 6Ebrd the @nblbutionof 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$. and Parry, 1995, E3*n et, al., 195a, Gurnot dndS6ldelmann, 1944, Markowiu er. Er. 1954, Mullor 6nd rappel, 1977, t9unk andlrlacoonald, 1975, Nel$n et, al., 2001, Newcomb, 1495, Sadler (Ed.), 1960,seldehann rnd Fukqshlmai 1992, Spencer, 1954, Stephenson and Moi$n, 1944,1995, Sleph6n$n, 1997, Wlls, 1963, otc.).


2/51

2.I INTRODUCTIONFor rhc derernination of lhc visibilily mnditions otNew lund Cre$hl (or theold.st lutur cresccn0 over a tocal hori2on, lhe 6Bt rask h ro deremine rhe Unive6alTine (uT) and dai. ofrbe scocenhic cobjucrion of thc Moo. &d rhe sb or thc Binhof New Mmn. In ils morion tuDnd rhe Eanh rhe Mmn tralcls doud 12 deg@s frcn

rvsr ro c6t cvery day sd iakes ovr rhe Sun in doud every 29.5 dars on thc avcm8c.When de Moon is vcry clo* lo ve$ ot (he Sun it aprears betore the sunrie ed vhen ilN elst oilhe Sui it appcmsjusr anq rhc sunset. The lunar crcscenr is vcly rarel) lkibtcon rbedayofthe conjLncton. Dqjon Linn(Danjon 1932. t9l6) has b.cn i.$rpreted asa limn on Ihe hininuo clongalion ofrhe visible lu@ cE*cnr Accordine lo rhis lihirthe ruMr ccscnr is nor visibb if lhc elonSadon is te$ lhd ? deSEes (Do88er &Sch.efer 194. Schaefe! 1991, yallop 1998). Thc hdimM elonSadon ot $e Moon arlhe dtoe of$e Eeocentdc conjuncrion is same as the inctinarion otrhe lunar orbjt fbnrhe plo. of ectipric (50 9). When rhe Moon lakes orr rh Sun ar ir ndimunelongation $c ninimun time il lals b molc foo bein8 / fmm rhe $n (on the .6remside) ro be ?o again {on &e w6om side) is eoud th,ee quans ofa day. Th6 it istheorcdcally possible lhrl rhe qesent is t6l sen on $e dly of conjunction or is tsrseen on the day ofconjunction. TheoEricalty n is aho possibte rbar if$e crescenr n hslseno. the day ofconjuncrionad rhen rhe new cEscenr 6 seen on ln. da, iller, orthccrekenr rs rasr srn on rbe day bto,e rhe conjurcrion .nd !h?. rh. new c@dl n *enon the day oflhe conjunction. In th6e cM rhe cre$e r.mDs hv$ible for.onea-half' day. Howevcr non of the$ lheoletical possibihia are rcatird in practice toofreque.dy. Mosrty rhe cEscent Enains inlisible for .two-sd.a-holt, days at least,These condnions dcpehd on tc obse(ets t@a on.

th. output ot ihesa ftort! b a .omplter progr.m for .natysts ot ftstvblblllly ol luh.r crscent named Httatol wrttten In C.t.ngs.ge dt$u$ed 6t the end

25


3/51

O@ thc rim. of thc g.cdric biil of th. Ncw MFn is d.lmin d, dE rcxlr.3l( ir b d.cdic dE locd circuhlt .c6 oflhc Su ud dE M@n tt dE tirc ofsu5.lon lhc day of rh. conjwrid or r &, .ncr thc djulction (or d lh. lirc of swi$ onlh. dly ol conjncdon or thc dly b.foE). Fd this i'lk ooc mEt d.r.miE th. l@ltim6 of thc ru$t a.d lhc noon3ct In tr. mmins in otdcr to b. vkiblc th. Su rhouldtag b.hind thc Moon in odd rh|lth. @s i3 visiblc .d in thc dcnine! thc M@nsholld bc hgAitrg b.hind rh. Su. Clsiotly thc LAC of $c M@n h!5 Em.in d onih!{ndr cosidcElion for lhc adi.sr vbibility of$. w lua ct6..nl. SiM thcrim6 of th. Babyloni.ns thtuu8h ni.tdl6 !96 .rd ill tlE 20d e uy it hs b.qrcoNid.Ed I deisiv. aerd. Blbyloni@ @Nidc!!d nininm LAG rcquiGd for tlFvisibihy or new lund ca*nt to b. 48 miNt6 wh.@ thc MBlidAFbs co.sid.td illo b. 42 to 48 minut s dcp.ndinS on th. E ih-M@n disl'ne. In Dod.m liis tlDoghthc lisibihy condnioB hlvc b.cn eFncd nainly d@ lo .ll kinds ot lnifi.illpollurios dc ncw luM cGs6t hls bc.n rcpon d lo b. tishr.d vlEn ia LAC Bmwh ls ths 42 hinuts.

TIE ddcmin.tion ot $c l@l riGs or rh. suEr ..d $. n@nsr, though statd!o b. s@nd t.st in *qu.na, i! dcp.nd.nt on rhc dct mimtion ol dt PEci*toldenkic @diMl* of th. Su ed thc M@n- h is rh.tfoE imFnliv. lhat b.foE rh.&t frinatioi of rhc LAG orc nusr find th. E.@ ric @rdiMl6 ed lh.n lhebpo@ntdc c@rdinarcs foi $c l@don on lh. globc fom sh.E ob6cNalion it to bcm&. of th.* bodic!. Th.* e dcrivcd fren th. two $.oti.s, dF VSOPET .rd th.ELP-2000 (di*.us.d lakr in $. ch!pr.o. As both rh* thod.s &*db. lh.luM.nd$. iol& c@dinatcs 6 .xplicit tin sid, mlkiig olt thc prilc "lim. usm.nt' is*nlid tor lh. appUcadon of tI6c fodul8. 'It. "ritu" co*idcrcd in thc* [email protected] rh. orh.r thcod6, is . rim ird.Dcnd.nr oflh. dillioB of thc E nl ed ir g.tudllyLfr.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 ofRGl ivily (Ch.po -To@a & Ch!!ro4 l9l, Ncl$n.t..1,2001, Guiml &

26


4/51

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

T"I=TAI+]2F.IE4 c.l.t)

AI=TT-UT (2.t.2)

Btce TAI it dgul.td &cording ro atomic rimc, In TAI $c b6ic uoir of rime is thc SIsond {defined by Bueu Inremarional des Poids.l M.su6, BIPM, in t96?, adudlion of9,192,611,770 periods of rodiarions cooesponding ro lh. tmsition brweenuo htFrfinc levcls ofd.gbund slat oflneCesinm lll abm G.t tenerat.,2OOt)).A day on rhk scale is 86400 SI seonds long {Astonomicat Atndrc, 200?). On rbeother hand rhe clet dm." is thc Univ.Ml Tin (Ul denned wirb rcf.r.nc ro nemsun and Nocialed vi1h thc Crccnwicn Md Sid@t Tim. (GMST). UT is detircd as$e hour angle ofrhe Med Sun ar cEenwicb ptus 12i6. Due ro rhe ineuta ries in therctations of th Eanh ihcr N di$repancies b.reen the lwo rimes. rhe TT and rhc UT.This dillcrcrce is ruferred ro as $c delht (At):

Th.rcaore wh.nevcr w wanl b dd@irc th position of $. SM and lhe [email protected] a panicular lime on ou! clocks w have to fomutate rhe rine argudenr using rhesecons'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 ofthe UT in question. TIE Julim Dale is th. sys!h ofconrinuols limc sle lhar begins o.Noon or crcen*i.h Jduary t, y6 -4?t2 (cdlcd lhc cpoch oftne..r!lian dde,) ln rhisr,m scare tlre moment described by a date (CEeorie or Julie) ond rine (UT) isconsidercd as rhe "nmb.r of dayJ,, d.nor.


5/51

ine argunent md the explicii tin eries fomulas of the ELP ed lbe VSOP th.c@rdicrcs of lhe M@n ad rhe Sun m calcrl.t d for fte s.me i6i'nl of rhe day or th.d.y aAq @njudion- In s. @ spheric.l pol& @ordimtes a th gtucsrdc disr.nce, X"

- th. ecliptic longitud. and 9. the dcliplic ladtude, refeftd b as th. Ediprh Coordimles.Tn* c@rdinals i. lh ft sqt dsirely ro obrain rh. ims of th. Su$.i &d dEMooisr (or rhos of $e sun.ie ad noosi*) for 1hc day in qu6rion.TIE lune ca* (or rhc cE*enrs of Mrcury &d venls) k fomcd by $e

region of $e lbn suface tow,rds fie Sun tnar fa.lls belwn rhe two plancs rhrcu8} thee E of rh. Moon, one perp.nd'.ulr ro rhe tine ol vi.w of fie obFn.. dd thc o$.rFrrEtrdicule to rhe di@rjotr of rhe Sun. Thc nrio of rh. ea of lhis crcw md lhctotal dea of th Lund disc is called lhe.phs.. ofihc Moon, The phse ot th Moon isdi@tty rlaLd ro rh. seperion bcrwen rne Sun and rhe M@n or .lorgation.

The Astronohical Alm$ac pubtished amually shtes lhar $e dew luna! crscenris sencElly nor visibtc who irs phe is te$ lhan t% (Askomniqt Almde, 2007).- This ha prcv.d ro b. nisl@dine in vicw ofihe fac| rhar the brightn$ oflhe csenrcan sr.auy vary rbr the ehe vatue ofthe phas owins lo the varyin8 dislanc. ofrheMmr froh $. Eanh. The Elrri-Mmn disunft Eics foo t5O $o@d kitomcrrs lo400 lholsdd tilomer$. Tnus when dosl lo ihe Earlh the luntr 6cshl hay bevGibl. wilh ns pnsc much ls lhan r % and in cae of fanhd ir nsy not b vkible evdnwilh 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 vdhdonii th sizc of rhe luie disc hund I OOO ye6 aeo. tn rbe Modm rimes il was nol beforeBruin lhar lhc imporrace of 1he actul visibt. widrh of lh. lutu cllsqr M @liz4Uhinately n w6 yaltop who ued the widlh of luo{ cr*cent in his ohe_psmeirnodel of tu4 cre$.nr vjsibitity etadng n b the ahnnd. of rh [email protected] 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 of2a


6/51

both rhe Sun and thc Moon. Thse corct d eclipiic coordi.ates of fie Sun and the M@n@ lhen arNtomcd into Equtodal @dinal6 a, the RiSht A*nsion 4d 6, lheDelination. In order lo g.r th. tical Hodental coo.dimtsi Ahitude and Azimuth, ofihe Su ed the Moon, rhe ob*frers rcft$rial cooidinale dd the t cal SidftalTime(dcfind as lhe L@al Hou AnSle oflh. Equinox) aE rcquiEd. A simple alSorilhn lcrdsro fie Greensich Mean Sidereal Time (GMST).r lhe Oi'Unilesal Tinre for any givendsle. AddinS lhe local longilude 1o rbis GMST erv.s 1he r,cal Md Sidereal Tihe for0i' Uhiv.sal Time for any eiv.n dale is oblained. finally to gct $ Local SidcealTinefor my nonent of the day day be obtaind keepids in mind the f6ler pace of $e

Oncc lhe tical SideEolTihe ofany moncnl is looM the Hour Anele oflhc anyobj*t is oblained. The local Euatorial coordiiarcs s, th. Hou Anele ed 6. thedeclinalion,lead to the Altnudc (heiehr obove lh holizon) dd Azidu$. n@ poiilsoltine @ imporrant lor lhis slldy. when m objecl is al lhe local meidian (i.e. TEnsi().when rhc objecl rbes ind when an object srs.

Thc time oa tEnsil may be calcularcd using rhe Hour Angle to be eo, or rheco.siderinS rhe tical Sidereallime 1o b.lhe Righr Ascension offte objccr. The *timateolthispointoflimc can b inpoled uing an ireraive pccs (hal inmlves Eadjustihgthe lime arsumena'lo be lhis approximatc tim and rccalcuhting the coordinates offieobjd at thisdme argumehr.

ConsiderinS Hour Angle t io be 9Oo or 6i" approxidare rihe of rhe rkins(nearive t, o. the se(ins (posnive 14 ae obraincd. R*rlculatins $e rimc dsunenrstor approxidale limes of thcsc eve s $c coodi.Eres ofrhe object de deremined againad the b.lter approximtion of$e inct oflhs denrs ac ob!.ined. This giv.s therisins a.d the setins orthe cenres ol$e objers (rhe sun and the Moon) thar can badjnsred |o st $e actual isins (the first apparance of the weslem limb of the objec,.nd acrual s.tti.s (homen! of dcappe&ance ofrh. cded linb of the objsrl.

29


7/51

Onc. lhe lincs of fting edor selting ofbodr fie Sun 6d rh Moon are oblainedon d work our att th. p.mete6 of fic import,nce for lhc atrsl (ot la0 visibilitv otrhe nv (or old) lutr crc$.nr.2.2 DYNAMICS OF THE MOON AND THE EARTH

T!. dcvlopnnt of nodem dy.di4l lhoics for $e sole sv$em beg& wnn$e discovry of Laws of Pleel,rf Motion by Jobannes Kepler in the 16('centurv Ataboul rhe sm.lin haac NMo. cme !p wilh his tt$ of Motion dd the UniveisalLaw of Cdvi|llion. Whar fouowd is a long histo.y ol developool ol nathemtiqltechniques lead i ng 10 the fo m! latioi o f Cd 6tial Dynam ics The e llois w.rc d ir4td todescribcrhdorionofplanctsandlheirsal.lliles,dteroidsand@nebinordcrtoplediclthen positions in tuturc with accey of SEater 4d g@lr dsG Conuibntio.s ofEulr, Laplace, Poison. Gauss, Olber, Cowll. Encke, Claittul. Hesn. D.launav. Hilland Brown. besides daoy olher nahemalici.ns and astdnoneb. bare ben oi gratsi8nificancc. A nunbei ofcldsical md nodcn books de now available thal d.scribe thcdrails ofd.F conuibutions (Sna41953. Deby, 1992, Plunmer, 1966. Pollard 1966.Woolard and Clnene 1966. Bouvct lnd Clmence, 196l elc.). Contibutions ae ohoavailabl rhat give delails ofthe lund dynamics (chaplont_Touze dd Chapront. l98l'1991. 1988. Chaproni er. al. I998, Srandish I981. 1998 etc) Einsrein's theolv ofrldrivu suc(aded in dc$nbrns the molion olthe penhelion

The satllite to planet nss !.tio in case of lhe Moon_E nll svstn s largsl( 1.23 x l0-r ) in compdi$n ro ey o1hr saEllite-ple.t pait (lh. nexr ldecar beins lhar ofTriton-N.Drune l6s ntio = 2 i lO I ). Thccforc rhe Eanh do.s not prclid. lhe dominmtefecrive fore aclins oh lhc Mooh.It is not only lhe Sun bul all msjo!pltnels and larof lh dtdoids lhat @nkibute 10 th. cffcctiv force ,cting on the Moon Thus dcphncredF prcpded *ilhout tdtinS into eout all lh.e contibudons b boud to beemn@u' Mey of the epheheedes of $. Moon of.dlv dsvs, both b.lor' md aliet $elomJlauon of $e Neqonie ndhtuics. weE bded on obieBed sd lompukdaverages of various Lnds r.laled 10 lhe dyndics of lhc Moon

30


8/51

Today it is tnoM with a g@l degK of @!6cy lhoi lhe avetagc avrodic p.riod(intepal ber\,s t$o coMurive ne* M@m) is 29.510589 davs (29 dars, 12 hom 44ninul6 dd 2.9 sondt. The avBec uomalisd. morth (intcn.l ber@n lwosu(esive pa$ages of rh Moon throush it! perige is 27 5s4550 days (27 davs 13 hous18 oinutes ed 32.1 $cond, ed 4 aw'age sidercal honth (inLdal b.twen resuccesiw p6egs of rhe M@n tnroueh ! fixed st!r) is 2? 321662 days (27 davs ? houf4l minures and I1.6 secondt. Thus on thc b6is ofa sidcEal no h thc Moon rravel3abund 12.176158 degr.et p.r.lay on the av.rage Relaiv. to lhe Su. lhc M@n traveh12.190?49 dtgc6 per day on lvmee. Howvo, rhe mininu Bt 6 b. 12 08 degesFr day and $e ndimum t2.41 deeees per day. Tnis oll happeis beau$ the orbil oirhe Moon around tbe Ennh is a hiehly "i!rcCular' .lliPe whercs lho deviadons trecau$d by p.nuibatios du. lo the S!n, lhe Planeb and olhs $lf svsten objsls

Thc smi-major arh ollhc orbn ol M@n n on alcnge 384400 kn bul has asnaU oscilldlion aound thb valn whose period is hala lhe svnodic nonth Thc.@enrricily of the orbn is 0,t49 bul vdics d much sj 0 l lT The inclinalion of theluhr orbir from the ecliplic is 500 9' bul vdi6 up tol9 Elen the nod6 (poihts ofinFdecrion ollhe lunar orbil and the ecliptic) of lhe orbn !re noi fixed 'nd go round ihcrcliptic in 18.6 yeaB with an o$illalion.bour dE $cular notion lhal moun|s to asnuch as I 6? dcsl6 Thc line ol apsd?! tle eo rcund thc echpiic complelrne on'rouod in 8.85 yem and dciltations {iih udnude of 12'413 destccs Ths all lhc''elenentt' oi the orbn exhibn bo$ scular a ell as peiodic ldiarions lhis mkes rh'deimiMdon of lund ephemEds a daunling task

The undeslandi!8 of lhe dynanics ol the Moon dd thal of th' planets in $enodm *lup bsd vith lh doluliodv e4loits oi Johees K'Pler $d lscN.s,ror Kcpt.r cnpnically dcdued his Lls of Pld.lary Molion on thc b6is ot oextensile study of the obsralional .lcta @llected ovcr ce'nies of lhc posilio' ofpldets. Thcsc cfl be sbtcd 6 follows:

3l


9/51

L2.

Pl&erary otbiB d. clliptic with lhe Sun a1 one ol lh foci.R.di6 vdtor of ! pldel (V@lor dnM liom the sun lo rhe PldDsw*ps equal &.a5 in equl lihe jorensls.Th. sqlle of rhe Friod of rcvolulion of a Plder eund rhe Su isproponionalto lhe cube ofils dea. dblance ftom die Sun.L

Nfion nol only prcentd his Law ofUnive6al Gralilaion but verified KePleasof Pleetary Motion usi.8 lh. tnw ol G6vilation. According to NMon s Larv rheof atmcion beM.en two bodi6 wilh nas*s 2' ed ,, pl@d 6t a dista.ce /

F =c!+i (2.2.r')The force is aflnclive and G is the PmPonio.ahy constant dllcd Univ.dl CavilaliomlConsranr (6.6?2x 10 'r-)*s rrr). i is enhs ditctcd ircn ,L to -r or rron ',rou|Frem rhe ddc of publicalion ot tte'Priripi.l bv NeMon in 1687 a nudbet o'astlononeB, physicisrs and na$cnalicians conribulcd sisnilicontlv in $e dcvclophenrof th. lndesl..dins of Lud sd Plmhry dy.amra.

Howele. at lh time of NeMon (and pftap3 lill lodav) the dvnanics of lhe Moonpa*nred sear dilficuhy rhar forced Ne\\'lon lo darc thal ",',e Ltmr theo'v nale hishe.l ache dtut kept hih aeake so olen thut he vorld think o! it no norc " lDanbv 1992)He had tlitficulry in describing lh. modon of Fnge (the poill in lod orbil cloesr rothe &nh) and could explain it ro only withi. m accurev ot 8 percenl Clanaut (1749)appli.d ml)lical merhods ad succeded in explainilg d molion of pngee bv 6'ngseohd o.der approximation. He published his lraorP de Ia /,,e md a se! ofnumencaltablcs ih 1752 ior computation of lhe posnion of fie Moon. the Eost sisnificmrcontribudon frcm Euler appered in l 7?2 shcn he tublishd his srcond lL@ lh'ory

32


10/51

kplM\ ficory of luM hotior, Publish.d in 1802. cnploy.d rrmfo@ing th.eqution ofmolions $ tlt.t the lruc longltudc wa an independ.nt ldiable Hh wotk alsopovid.d e dpl.n tion of lh. sdo [email protected]. of the M@! ripl@'s mdhods w.Ecdied io a high degrce of accu!@y by sevetal malhedaicies One of then wasDmoi*e. who publthcd his th@ry dd r.bl6 in 182? that Gmaircd in *id' Ne untilHa.Fn! sork dppeaed. P. A. Hasen's wolk ext nded for over fonv veds fon 1829dd his r.bl4 w.re publkM in lE5?. ThGe labl6 rcmain.d in s for w.ll ov'r finvyds. Delaunay publishcd his work ih 1860lhal M basd on disturbiry tunctions $aiinludcd 120 tms. By sn.ltti.al m.4 h. Gmovd 0E teru of disorbins turulion on'by one ahd gEdually builds up th. $lution. Autho6 cl.im thar Delaunav s vorl( n thcmo$ Frf{t elulion of th.l@r problm v.t found {D&bv 1992)

Thc posilion of $e M@n dodd th ElnI is desdbed bv spndical poldcoordinatcs (r,,190 p) wnn r bci.s th. [email protected] disr:nce or rhe tltncr' ? $eeliflic lonsitudc and I lhe ecliFic laliode The most commonlt Gd edv lo handllunai $bles dlrinB the nosr Pan of 20't c.n$ry \rcG dDc lo BrcM (Bmm 1 960) Thisll]M $.ory w3 inpo!.d bv Eckefi &d ws tnown 6 ILE, sho^ tot Inprored LrnrEpr,rdr,r. Tnc lh.ory coGtruct d bv ChcPont and ChaPon(_Toul is knosn 6 ELP(Chapo er. al,. 1983. 1988) shon tot Ephtina des I air's Pukienhes ln ELPsimplifi.d tables have ben cxlEclqi from lhe lheorv lo rcplcsnt lh' luDa nolion in $efom of explicit line sries fomule. Th6e tabl's @ b lsed to direcllv compute $eluntr coord'nabs. ELP h Dol onlt morc Peci* md complele in conpdien ro ILE itale povides noe nodern valEs oi l(M pdmelels md olhr Phvsical coislsts' For6000 y.4 on eacb side of J2ooo o ELP povids lund c@rdiMts that l4lv have eno$.xc.eding few arc secoo.ls Toeelher with th' dvlopment of VSOP (VdialronsSacnl.ifs des Olbires Pletitt bv Bctagllon ed Fdncou (t9E8) lhe t!bl6 due roChapront er. al destib the motion ofall major bodies (xcept Pluio) ih tbe solarsysrem. Both rh $ori6, ELP &d rhc vsoP w dcv'lopcd d rhe Buean d's

ll


11/51

BdicaUy a nhcory of pldelary ed lund motion involvd inies ionolasysrcmof di0eGnial eqladoi dat constin&s th. major pad of th. sfudy of elcstial n46mic'Therc & i. gddt t$o.ppoachcs for slving such dyndical systctu, ealytol udnusical, Anallril nerhods m b6ed on solutio. by Iouder sens ed thc Poiso.\Slies. Thc ELP-2000-85 (Chdprcnlr,al, 1988) is semidalttic sd has ben obla'ncdfrod a fil o f ELP-2000-82 (Cn!po!i cr. al. I 983) to th. nlncrical inrcer.tion of lhe .leiPropulsion Labomrory DE200/LE200 (Slstrdish l98l). P@$ion in this o@ry ha ben$keh iion Lasrar Gsklr, 1986).

For te an lylic par! ELP-2000 Fpets th nain lEblm fbm th.penurbarions. Th. 6ain problem tak s into accout thc &tio. of thc E nn's ce E ofm6s aid lhe acion ofrhe Sun's orbir aound lhe Eorlh-Moon baycentr. such lhat IheSun\ odit is asuned to be Keplrid cllipse. This rcrulls inlo Fouri.r series w'lhnumdical c@fiicie s snd ar$mdb th.t m suns ol mulripls of fou fundade.talpaBmeted D (diffeence of ihe ned longitudcs of the Sun ed $e M@n). / (meanmonaly of rh Moon), l(hem anohaly of $e Sun) lnd a (M@n s argumen( olh ude). Tlis main problem 6uhs inlo dne *.ies fomnld for Mmn\ longitudc.ladtude and gocenrric disrscc @daining 2645 rcos in all. Apan flofr lh* *ricsacrons of all rh other significant objects in solar sysren dt consideed N'Fnurbadons lo ihc mdin pbblen lhol include:

L tndiKr pl&c(ary perturbarions rbar re induced by lne diff.tsoc.s blwecnthe lrue orbit of the Sun aound lhe Eanh-Moo. bdycenue and tssunedKepbnm Elliplhal orbil oflhc Sun *uned in the min Pmblem.

2. Dirct planerary penub.tioN due lo etions of othr plan$ on ibe M@nlor borh the direct ad rhc indtct planetlry penubalions lhe ELP-2000co6id6 th orbib ofthe pldct givn by BGlagnon\ VSOP82 [email protected].

L P.rurbations du. ln figues of $e Edh md thc Mon (Moons, 1982).

34


12/51

4. R.larivislic p.nub.rions (Lsrnd. & Chlpronl-Tou?J 1982).5- Penurb.tuG du. to tidd afccis (villim3.t. a! l97E).6. Molion of the efccnce frde conside&d wilh spect to m inrrigl fime of

consideradon ol all lhese pelturbarioB esuhs inlo rime series fomlla for geocentriclongitude, IatiMe .rd disra@ of 0E M@n at nds $e numh.r of lcm !o 15:37.

An alnatc ro this thericrl appro&h is !o rcpr*nl1hc coordinai.s explicilly6 tinc ledes fomule. This epEs.rolion of tlie tinc seds is dcrloped by Chap6n!Touzi ind Chapront (Chapront-Touu & Chaprcnq l9tl, pp. l0) und ha bcc. !*d infiis work. The mjor fomula used in rhh vork due !o $es authob de lkrcd beloNlTbe ge@entric longitude '/h.xpesed as:

'/=218.31665416+48126?.8813424'r-0,00011268'r: +0.000001856tr'0.0000000r534.r1+,5. +(s; +/ 1si +r' xsi/1oo0o)/l0oo(2.2.t)

whrc I = h in iulian centuies sinceJ2000.0s, =tv,sr,(dj') +d|),,+ao *r' xro I +ao)'/r xr0' +d|,.rr xro r)

(2.2.21I = )':st(o;'' -a:"' '4 (2 2.r\si = tv;,s,,(d;,o + dnD.r) (2.2 4)


13/51

Tb vdB 6f tb.ootM v- r; ct, d!o8 eitb |b ot a'. G Sivto i! Chr.od-Tout.d Ct q.od ([email protected] Ci{.Gi, l9l, rp a!-56}Ib gcoc.nttc ldhde UL riE b,:

t/-s,, +(s:/ +r'$ +i's;/10000)/100 Q26)s,, .5,"si,'(rj". p:".,.p:!.rr xron +/i",r xr0r +r'r 'r' x ro ')

Q2.?',)s:, -'u.,&4fri.t + p{t. tl e2.8).$ -tr:$4i:o +r4D.,) t229'

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

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

(r25)

Q2.tO)

Th! y.lG of th c@trt ! t - !; .iq dotE eilh lho.c of 0!.t 8lt'.! in Clqlot'Toudsrd Ctar@r (ct Fld-ToEald ftlnfi, 1991, I? 5?{a).rindly tl|c geo.6tic dllt n6 it dven by:

x-315000.57rsr +sl +r.8i +y'.s;/10000 Q2.rD


14/51

s- =t'"cdldj"'+djl xron)(2.2.t2)(2.2.t));=:4cd(d;'+r;,''/l

s; = t,;c,rt6"4' + r,4, .4 (2.2.t4)

si= +d;o).r) e.z.ts)-fhc !!hEs of the consr.nls rz, /; etc, dlong with lhose oa6s ae given in Ch.prcnl-'fouzi and Chapronr {Chapo.r-TouzC and Chatronr. 1991. pp 65-73),

i"-",,(u-''

Atlat0)s. /0)s and D{ors, ,', s,,,sl..U,su,si./.'r.r;.,". md r; @ in desrcB,arr\, y'r's, d1'\, si,si,.'; and ,; ac in d.s@Jcenrury, ar'\, y''?\. irlrs, si.si.v;lnd r; e in desEs/6tury2, ao's, y'r\, ana d's m in aegeevcenruD r and a"rs, /'rsaid d'h de in deecs/centud. R, s,? and Sh dc in kilomelres, 5i md 4 .t inkiloderrs/cenru.y. siand 4 sr. in kilometftVcenruly:.

For Tbe detemimtion of plan.lary coodinstes lhe complele n-body problcn isrequied to be slvd. A. ml,,lic solution of planerary dotion wa3 pre*nted by8rct.Slon (BElaglon, | 982) of Burau d.s LonSiluds of Fnce lhat described only thc.llipric @rdinates of the pldels. Th. elution is populdly knosn d VSOP82(Variations S.culai6 d6 Orbits Pldauirct. Latcr, BEtagnon and F6Nou of th. seBur6u hodificd VSOP82 inro VSOP87 (Bcl8non & FMcoq 1988) in 3uch a uy lJlltth.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 slllionV SOP8t descibes ln el enehls of thc osc ulat ing or inst aneous orbit in lems of:

t7


15/51

a - sdimjor uis of th. dbill, = m@ ld8itlrL ordF ddtl= ?,@$p= sincr,i Dsinoc= si(Xi)coso

whec. is thc.@eotriciry ofthe orbn, r is ln.oflh. orbit fmm rh. plD. ofeliptic dd O isolbit. .cholth. rcctarsul$c@rdioat (X Lt) is u.xplicir tunctionofiime ed is inlheEvery lcd ot Ues $.i6 is in fic tod of:

bngnud. ofp.dhelion, i ir th. incliodonthe logilud. oflh e.lding nod. oa th.a or &c aph.ic.l polar coordimts (2. Iforn of p.dodic eds md Poison $ies.

(2.2.t6J

$,lEE a - 0, l, 2, l, 4, 5. I is tn. tire in lhoFnds of.,ulie )s fod J2000.0, i..I"(ssii9+ I:cosp) or 1"..ios(, +c?)

f= 165250e=ia,1,, i

c=>d,N, 12.2.11,

= I to 8, ! reprsent rh. m.m longnud.s of the plancc Mercury lo= 9, l0 6d I Lc rcpllsfl th. Dclawy sgmen& of lh. Men D, F mdThe hsl of I is the na longiMe of U. Mon Bivd $nh 6pt to lh.tlay. In thc ah@te xpcsion,

B= Za,^," + P

S - -,asinp,l8

(2.2.r8)


16/51

,l md M dc Sivci in the lable 2 of (BdaBnon & Flmou l96E)These data series @ .vailabl on CD's dd laFs For r.cta.gular c@rdiml6 of

rhe phf,ls |he dlia llles VSOPS?A. VSOPSTC dd VSOP8TE dc @d sd for thespherical polarcoordinaEs VSOPSTBand VSOPSTDa!. used. A shonerleBionofrhesedata series isSilenby Meeus (M.eus, 1998)and lheff. tr u$d in this@r\ l. the*hbles firsr olumn gives.4s, fie eco.d ,s dd the lhird sives Cs. The dats file hs 6sedes fo, dh or rhe coodi.ates L (h helioccnlric Lonsnud.) and R (lne helioccntkdislanc. ofthe Eanh) and 5 for fte [email protected] B (lhe hclioc..lric Lafiudc)

Iiiclr s$ics lir L. ll .nd ll ne uscd as follo\s ro obhin thc heliccDkic polar

t't = LA; co\B; . I jT t. r2.2 roli 0. l . 2. :1. 4. and 5 lor l- md I( and 0, l,2,3,4lor B Ihc suFrrscnpt tsund for I(L). 2(lJ) and 3{R). x rus tolsh 0 lo dilfeonr incger lor di(feEfl coordinaGs andtheir lssiatcd seri6- Fo. I - 0- /{i coftsponds ro lonsirudc scri.s. I = ll/{iconsponds ro lari$de sris and I = 3, /r,coiiesponds to dist .ce serica. ll.chcoordin eislhcneuluatdas:

(2 2 20)

/. md , d. ir ndian nesuEs and X is in aslrohonical unils. These m as menrionedcrlier rhe [email protected] of $c Eanh in heliocsnkic c@rdinare sysrcm wnce6 for {heproblem ofdckmrining posirion oflh Sun inour sky w. ctually requiE rhe Cmcentriccoodinales of lhe Sun instead. In case of rhe Eanh this uNformation is sinple:

=l:, ','' I) "=lz'',')=ltcu il'

l9


17/51

,is = I +l8O! Fs= -e 12.2.2t)

(2.1 l)

ond the hcli@entric dislanft ofthe E.nI is smc as th. s@edtic distanc' oflh S!n'

2.3 BIRTH OF NEW MOONA3 menioned earlier the Moon in ilsjounev dound the Eanh llavels arcund 12

dcgrees cvery doy in our sky and $l.s ovcr tic Su in lround vcry 29 5 davs whc' rheCcocntnc Lonsnude of the Sun e.l lhe M@n N se rhe nonent is tnoM s tncTine ol Bidh of Ncs Moon Tbe dudion berven two succe$ive Binhs of Nev M@nsis called lhc Lun.lion Period Ho$ever dE Llnal'on petiod is nfrom 29.2 days lo 29 8 davs This is rh. cMn bhind 'ons irt tu'd mon$s of 29d.ys ach or lhe con*culive luor monlh of lO davs cacn For fi tine ofBinh ofNewMoon one requiEs to find lhe noment when rhe seocentric lonsilud's olilt Moon andrhc sun coiNide Thus one needs to lracl lhc lo'Siludcs of eeh of thcn rincCoisidcrins $e najor rcms oa lhe line scrics fomulac fo' lhe lonsiodes of the Sun md$c M@tr in th planeurv rhcorv VSOP'2000-8? and lhc l(nw th@rv ELP-2000'8? $enoment when thc rwo lonsiludes aG sMc can be evaluatd An alsori$m due b Meeus(Mceus, l99E) for $e dclemiiation or $e rin' or Binh of Ns Mmn is as rollos:

On a!c8. lhe tnpic.l Yd (dudiion betw'en t*o conseculive passages ol lheSun lhroush equ ox) iscudcnrlv 16524219 davs (from (l I l)) Thc avedge svnodrcMonrh (trre intedal bclwccn r$o cons{u'r\c t\e* Moonsr toten o\er a rcn'ur} ii29.530589dar(fon (1.3.3)). 'Ixus in oft topical vear rh're aG otr a\Esc 12 1682664Synodic mo h5 Thqeloe since lhe $ad of lhe vcd 2O0O i e J2O0O O the nunbei ofsynodic monlhs elapsd oregrveh bvl

t = (f,' - 2000) x l2 3682664and lhc time in lropical ceoruries el.Psed since J2000 0 is Siven bti


18/51

(2 t.2)t236.42664An approximarc valu ofth.Iulid Dale ofthe Ncw M@n ir $cn givn by:JDE - 2451550.09166 +29.5JO58886t.t+0.00015437..r -0.OOO00Ol5 j.r

+ 0.000000000?3' r!(2.3.3)

$herc t i. d inteSer Thus @ording ro this fomula rhe for t = O |h Jutiil Dar. of$clid ces.nr ofycar 2OOO is 2451550.09766 lhar is J&ury 6.2000 ar I8n r4m, md4l *!. I of dynlmical lime. For a horc accmle valu oa ihe Julia D.te of thc New Moonlhe pdlrbation tchs due lo lhe Sun md rhc planer ar added, Thc pertuibafions tenrsdue to thc sun d given bylx= -0 t07 2.SlN(M )+ 0 I72J t T EISIN(M) +O.0l60A.SINOr M )+0 0lAj91stNe,F)+0.007i9.E"'lN(M -M)-O 0O5 1.t.ErSINtM + V +a 0a208'E^2istN Q,M)0 00| t t 'SlN(M -2.F)-0.000t7.StN(M + 2*F)+O OOO|6.E.SI^-(2+M, + M)

-0 000J2.slN(3.M )+ A OOO.|2+E.StN(M+ 2tF)+0,aoB8*E.SIN(M_2.F)-a 00021'E.stN(2. M'-!r' o 0u | 7,stN4.] 0.00007.stN(M'+ 2.io+0 0a0u.slN(2.M -2. F) +O.00001'S!N(1.M)+0 \oaol.stNi,,t-+ M-2+F)+0 0040j.SIN(2'M + 2.FrO 00O03.S|N(M +M.2.F)+0. 0000J.StN (M -M + 2.D -0. @OO2.S|N(M. -M). D -O.O0OO2.StN(r.M, + M)+0.00002.s1N(1'M) Q).4)wherc ,r,1 = rb. mce &omaly ofth. Sd al ihc JDE

= 2. I SiJ +29 I A5 3567.k-0 ,OOO0 t I t -0 oaoooo I I I(2.3.5)M = thc mee anomly of de Moon d lhe JDE

= 2A 1.5613+ J85 8 1693528'k+O.Ot 07j82N I +0 OAOA I :38. I-0 0A0A0A0,8. 11

4l


19/51

F " M@'!.4ruqtof hind.- ! 6'0.? ! aE+390.670t0264.1-0.00t 61 r 8. | - 0.0U00227. ?+ 0.0@w0 .f Q.3.1)

(2.3.8)

(2.1.e)

g - lncinde of lsnding md. of {!. luar orbn- !24.7716-1.5637t588.t+0.0020672'1 + 0.OOOOO2 t 5r IE = Eedtricity of $. oftit of Elni

= 1-0.002t 16.T-0.0000074. ?Th. pedurbltion |.tus de io pleas e:v = A00032t.StN(A I)+A000165+sNLtr+a000 t6.trSIN(/r+0.0a01 26.stN(A1)

+ 0. 000 t t.srN(,4, + a 0,f062 +stN(/tq + a 00006.stN( 7 + 0. 00ut6.3!N( E)+ 0. 00001 7.s tN (,t 9) + 0. 000042.sN (/ t q + a 00001.stN(t t t)+0.0000t7rsIN(AI2)+o0o00J5.StN(Ar i)+o W023.SlN(,| !4) (2 3 to)

A I -299- 2 7 + 0. t O74o8.k-O,OO9 I 7t. ?t2-25t.E8+A0t6J2t.N,13-25 LE|+266J IEA6.ta4-t19,42+J64t2478.*/5-U.66+ ! 8 206239.kA6-t4171+53.303771.t/7=207-14+2.153732.t

19-34_t2+27.26t2391tA ! 0- 207. I I +0. | 2 1 824'k/1 t I = 29 t. 34+ l. 64a379.t

(2.3.1r)12.t.t2)(2.l.lr)Q.3.t4)(2.1.15)(2.3.16)(2.1.17)(2.3.18)(2.3.19)Q3.20)Q.3.2tt

42


20/51

A I 2= 1 6t. 7 2+24. I 98 I 54.kA I 3= 2 39. 5 6 + 2 t. 5 t t099.tat 1:33 t. t5+ 2.7925 I E.*

Tlus $c ,ulid Dlle of lh. Na M@n ie giv.n by

(2.3.22)(2.3.23)(2.3.24)

(2 r.2tD=JDE+X+YThc dm deeribcd by this dat is lhe DlaMnical Timc and lhc coretions for Al mcr bemad lo get $e Universal Time (discu*d in 0Encxtarticle). For anr l@al coopuralion,th. Ld.l Zonc lide and d!i. n6r b. calculated ftom |he UniveMl Tim. and daleohaincd abovc on rhe bdis oatbe lonsirudc ofany place on rhe Eanh. Thc darc h $ende day of conjuncrion fo! the place and rho rime of conjuncdon (he bnrh orNcw M@n)can b dy rinc fom 0- 10 23r" 59'" t9h on thd dav.

2.4 THE TIME ARGUMEI{Tft w6 mentiohed earlicr thal rhedynahicsofatl soltr systcnr objec$ isdescibed

by lomula b6cd on 6adcs of Cl6sic.l Mcchdics and lhe Relarivisric Dyneics int.m3 or rid serics. In order b ersriv.ly ue rhcsc fomulas an apprcpriate ?rrsargMerr cotrcsponding to rhe honeht of obseiving the lunar crescenr at my place onlhe surface of the E rth h6 io be valualcd. Such a lime a.sudent h6 ro b co.inuouand mui halc a clearly dcUncd point of i's beeimine (|he aro rine). called .ph.various theorics and problems uF difercnr .pochs depending on rhs co.rext for ageneEl consideFtion in pls.lary ad luE dynmics thft ,re lwo ihporlanl epochs.

Th nr$ of lhese epochs is a oonefl in edote pdsl coftsponding to rhe Noon alG@nwich on Janlary l. 4? l2 B.C.E. on th. Julid c{l.ndar (or Nolember 24, 4? I I onCdgode cal.ndtu) (Ringold & D.Rhowila ?001), Fren $is pojnt of timc rh. rineelapsed tiu my laler point oflime in number of days md a possiblc fierion ofa day h

4l


21/51

elled th. Julid Dare abb@ialed B JD. So the ,D @Gponding lo thc 5i" 30.i. onOcrober 5, 2004 ar Grcenwich is 245t284.2708t11t.. Thls rhe Jutim D!i. i3. mesEof tih. clapsd sine thn pocl 6id is cxpEss.d in nmber of n@ eld d.y5_

The orhcr epoch of inpori,lcc lo rh lftnl work k fie noncnt of dme ell.dJ2000 0 a.d il Eprcseds lhe l2ln TDTon J.nuary l,2O0O i.. (Abononhal Almahac,2007). Tbe JD conesponding ro rhis hom.nt dl C@nwicb is 2451545 days. This is the.poch or 2eo me for borh rhe Luq Th@iy ELP,2000 (Chopo.lTouz{ & Chaponr,l9l) dd th pl&elary th@ry VSOP,87 {Brc|,gnon & Fmcou, 1988). In bo$ rhes0leodcs lih. m@!red frcn J2000.0 bo foMrd od b6ct*ards. In ELP thir rimc is inis coBidcrcd in rulian CentJies (i.c.16525 me.n sle days) md i! VSOP il is inJulieMillemia (365250 mce solrdays).

BoththeseepocbsmbasedonrheinreRaloirimealted..toedn$hrdat, qhichis defincd a fie int rval berween lw slccessile rhsits (r6sdge rhlough rhe locllmeddim) ofrhc fic ious body knoM as ft. mee Se. This ficlitious body novcs sirhuifom sF.d alory lhc cel6ial cquabr .hd is consider.d in place of lh. acrul Sun $aroovcs vith non-unifom sped (dw ro rhe .llipli orbn of tne Efin) abnS lne Ectipd.The tmsit of $c aclual Sm over a local heridi& vdes up ro I I minurs over a penodof on lrcpical ye (Astmnooical AlFanac, 2O()7), Thereby atl civil rim. reckoninghave beh a$ocialed with the hedn Sln lhot consisrl of 24 mean solar hous, Thebeeinningofa civilday, i.e. zro hou* on civil clocks occu^ at nidniehl when the nourdgle of rh. Mean Sun is rwIrc hou6 lccodin8 to the local or sisddd hcidia..

Th. tiDe describd by lh. cl@k showing d. mm $lf rinc is not wilhoul itsdi*Ep.n i6. In fd it is lhe Ertn, $e gtob, ilsctf rhal is ou cl@t lnd $c n.m solattine B suppos.d to b. basd on ths a!ragc rate al which lhe Eadh n spinning a@und irsdn. Howeverlhn considearion is only with rcspecl !o lh Med Sun. Du.lo the orbilalnoionoflh. Eanh beins in rhe sme direction as its dis ofmlolion (ton qesl lo EEst)on ubl totaiion coDpldes in l6s thm ihis nee sold day. So the actuat nle of dialrchtion is b.t r ralized by rhe No sw;sivc resirs of a sior. Dis in|cdal is lemed


22/51

a r Si&F.l Dry ud rh. timc masuld @rdi.s to thb $d. is rlrc Sid.Flt Timc_Aglin du.lo lhc clliptird orbit ofr[. E lrh ttris D.riod i!.le nor uifom e w h!rc ro@tuid* "M.rn Sidccal Day" rnd !@diiSly M6 Sidaql Tirc. Oc md el& dlycquls |.002?3?90915 !|16 .i&Gd dlys (or 24B 03i" 56.t553?* on m@ sid.Mttim.). Altcturcly oE md 3idcEd dly cquals 0.9?26956633 ns st.r d.ys (or 2lb56'r" 04.09053* on md sol$ rimc).

In gcrcr.l rh. o@ eld {n ir rhc rinc |.t n inro &@6r in bolh lb. civil tu.ekoning a sll a tlE $tromnic.l. Wh.Ed lh. dylmiclt rh6d* denbc |h.norions of th. objers ii solr srsr.m on th. bair of rhc coninuoBly nowins tincdc$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 @utdbc lound only bt high pcirion ob&dados of th. skics. Th. diffm. of thc lrc AT isrlbull|.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 rncqlculatioB of .cliter @ultrrion lnd 0E rin6 of th* d.nt. @rdert jn lh. tmwnhistory. Th. val@s of AT e giEn fq only |h. nan of.eh cd.nd& y.r ,nd $os forolh.r dn6 ofFd ce b. incrlDlar.


23/51

How.r s noE @nhtlon l !'pNch ir ro @!qt lol =24) ...... 1rcft6. dar( UDD-LDD+Itf (UDD>dorqLMM)) //darr is nMber of days in a donrh @yI LDD=I:'UMM-LMM+Ii

rf (uMM> t2)I UMM-|:UW-LYY+ 1:t)]upHH


24/51

AYY-LW-l;JUDD-&tr(UtlM)

olr@ l[. UEircc.l d.rc dd tirc is lppllFi.Lly dju{.d @ pcad! loc.lcula& rhe Julih Dar. for 0E d.lc {d rhc ritu. For rlis calcularion inirially ifihc nondr is Jmqry o! fcbnoy, it i! co id.Ed morh .mbq 13 or 14.Esp@tivly, ofthe ptwiou trd ad rh. t@ i! sle d.crcNed by l.

UW-WY-l

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

Stcp4: 1=INr(Wt00)ln lml of 6e $tommic.l dlculariotu ! daL d and .na Fliday.

Octoba 15, 1582 is coaid.tld io b. I dlc of Grgodd caldd.r {d a darc toniThsdly, Octob* 4, l5E2 ..d prior to rhb datc i. coGiderd s a dale i! Jutiecalndd. Ilru ifa dsl. i! fton Crc8ode cal.dde ndh.. EquiEs m accoutofnon-tlap yed tom.mong3i 0F nomal lcap yds due to th modified ruleofliad y.rinthe Grc8onlr qal.nd$(y.$divisible by 100 but rot diviribteby 400 e @r L@! yes).

d"@ledq it casdtb'I a=2-/t+rMf(1/1).l*1I "@Ldo i.t IdlM"( B_0 J

47


25/51

Now orc n6ds ro c@t 6. nub.r of diys .laps.d si@ the Julid DaLepoch (Jeulry l, 4712) iill Ihc .nd of th. PEviou yc& .rd lhe trmbr of dltrlaled flom ln. n61 d.y of rh. PEviou yd till the cnd of lhe c|lml monlhMeu @dider this sMbl ianing fom ys -4716 dD( adds addilional davsthal @ balmed by subt@lion of lh. co6t ni 1524.5.

JD=\NTQ65.2t(UW+ 17 16)+ tl]/r(30.600t (uMM+ 1))+ UDD+ B-l 524 5+ (uHH + (UMIN + USEq60)/60)/21

(2.4.r)

ln both th. lh.orics VSOP8? tnd ELP2000 lhe Poch is the J2000 0@ftsponding ro rhe Julie Dsr.2451545 theEfot one tin lly c.t5 the lin

JD 245t545 Ar (242)16525 1155760000

Notc rtEt in rhc l6t st@ lh. lid l.B on 0E lighl hfld side is the nmbd oflulie entude elap*d sine 12000.0 ad th. s@sd l@ is for At which is @allvsiven in mobd of.eonds od h.rc w. nccd to @nved n i o nmb.r of cntuies (lioELP &d Millemia for vsoP) due to which il hN lo be divided by the nunbq of s6ondsin a Juli$ cnrury ifor ELP md Julim Mill.inia fot VSOP).

lf and when ttie dynmical tim. of d cv.nt is *rom we need io @hv.n rl bU.ivesal dd Lhen into Local Zon. tim.. Thc dylmical tihe is oblain d s the nmbetof Julie cenluries sin@ J2000.0 e that rhc Julisn dare of tbe evmt cm b calculated 6:

48


26/51

1p - 15575s.(, - =-4-), usrsts ea.t,t 3155?@@00JTIlc int Sr.l Ft ofdF Juliu D.!. *ill b. onEtud io Cdddr Dd. .ndt|| hclion l Frt ro th Unirwl Tim:

St F2: Z-INT(JD)St d-3: F-JDZ

Fd &rB Fior ro ocrobcr 15, | 582 (JD = 229161) rhc inl.Sq p.rr of.IDis !.oircd ! il ii olhdtuc ldjulr.d for th. @diti@ of lhc GGgorid cdqd{

St.pa: ifz


27/51

LMU-UMMstcFr2: alMM>2 { UW-C-1716 }Etr. t UW-C-17tt )LW-UWsteFf3: hout(=drylM(dqr))r21Stepr4: UHR-tM(tbu) LHR-UHR-ZON

If (LHR-2,t ... ... tt@t et){ LHR-LHR-21: LDD-UDD+,,

4t6-DD>day1vut4'| LDD-|: LMM-UMM+I:y(LMM>t2){ LW-UMM+I; L tN-l;)))

ste!'rs: htn .=Aov-L| 9.60sreFl6: UMN-IM(nlnut.) LW-UL{NsteplT: tu.ond-(ntr.-LMIN),60

50


28/51

SGpf8: USEC=lNr(taond) LSEC-USECSr.pr9l Outpd UW. UMM IJDD, UHR, tjMIN, USECA"d LW, LMM AD LHR, LMIN, LSEC

In thc Nd-M@n Algorilhn thc our pur is thc dynmi@t lin. &d this dgon$Dis p.niculaly usfll in @nvcning Uis rinc lo U v.Mt od &y [email protected] @G rin .2.5 COORI}INATES OF THE MOON

For rh. dckmination of lh. c@rdinrcs ot 1h. M@n .r ey Sivcn lc.l tim. .rddlt liB| scp is ro fomular th. rinc a4um.nt a dieNcd in diclc 2.4. So lh. pcc$bcsins by s.l*ftg pl.c. of ob6.rq (Ljnsirude &d La ud.), te.l darc ard rim. rhrllcds to lhc rim. &Bum. t eco.dinS ro th. atgorjlhn dc$ib.d ,bovc, s:

Julon Do'. - ?{51545 N16525 1155?60000

(r 5.t)

Using Uis rim. arsumcnr th. @Bhcrio! ot rhc dh. *n6 d.*ribing rh. twrc@'dinar.s is do.c (chapo -Touza ed Chaponr t99t) s dieled in aniclc 2.2StF|| Fo..dipdc longirudcofth. Moon us.2.2.2 ro2.2.5 ed suhdrut lh.ir

csulrsi.2.2,l.Sr.F2r For cclipdc t.hu& of U. M@! E 2.2.? 10 2.2. t0 &d sub$nuE th.i

Bul{.in2.2.6,St.Fl: For gec.nr.ic dilllre of rhc M@n uc 2.2. l2 ro 2.2. | 5 ed subsritut

lh.nrcsulr.in2.2.1L

5t


29/51

DE to lh. nodotr of thc ob6*d. th. diual .nd lhe mual motior of thc Eatth.poiilior of E.y objer in th. sky is afrctcd by 0!. phcmmn of Abcmdon- Th.followinS @cid.ntion is only for rhe Eaih'Moon pls.t.ry Ab.dlion sd do.s lolinclud. the dimal nolion of thc ob6*er (W@lad & Cl.mcnce, I %6).S1.t-4r CORREC'TION FOR ABERMTION

v-y4.000 t 9521-0.0&nl059.skQ 25 1177 198 9\) 12 5.2)U- U-0.40001 754t31h(183 J +48J202't)

R- R + 0.0708.C6Q 25 + 177 I 96. 91r)

Fimlly a 1he lrue quinox of ft. d.y and dE mcm quinox of lhe dly dedifiedt dE ro thc phenom.non ofNuiation the F.is cerdinab on nol be ob|dircdwithout th nul'lion in lonS itudc AV and the nubtioh inobliquityAs.

St pS: CORRECTION FOR NUTATPN

(2.s.3)

(2_5.4)

a'r =l0r 't(y, + y:..)'si,{/,I" +/d' '. +/d' '.r 'lo r)(2.s.t

Y=V+ L,t 12-5.6). -22.63928-0 0ll\+0.555'lo' t.r -0,0141'l0j r tl

(2.5.7)

^' =lo I 'I(., +,' '.). cdko) + /,r' . ' +rdr ..).lo 1)(2.5.8)(2.5.9)


30/51

Tn v.lu. of F s, !,'3 md G's Bd in lh. qp63iont ahovc tt! gt!n inthc T$1. 9 in Chlprcnr-Tooze 6d Chlpotrt (Ch.pre -louz; &d Ch.pon!l99l,pp.l9):Oft. rh. @retion dE to .ubrion is dre onc my so ro 6trd da [email protected]

[email protected], the Rjght Asnsion d dd thc dcclin tion 6:St.p6: EQUATOI.ULCOORDIANIES

"=r-'(

b. lalfl ifto @ounr so tb.t rhc ..iopoccnt.ic" c@rdid., (coordin ts relariw lo thcposnion ofth. ob*fr.r) nay be obl.iied_ In odd rcd5 rh !ffftrj oftE ..paraltd" arcto be t t.n inlo &count. Th Pdttd r i! givcn by:

c6(tpriio")Si4v)C6(U) -.t4 Eps itd) Sidlu'l (2.5.r0)6 = sn't l3ih(Epsitm)Si"(v)cos(UJ + cB(Eptiron)sinei (2.5. I I )Thc* re rh. nle cqubri.l c@'dinar6 of lh. Men wnh EfcEn.. b th. toecqulor of $e dare ahd the trua dyunical equioox of the dar_ Th$ m stiu lheL s.oqtic c@diMl* md lhc afr.cr of lhc D6nion of lh ob*rcr on rh. glob. is ycr ro

12.s.tz)

wh@ 4 ir rhe gcatdc dist4. of dE Mer HowEr rhis q@nry paml* d.Fndson rhe Hoq Ancle (ihe since ibe object cosad thc l@al n.ridim) tor whicn w. ncdthc lad sidc@l tin. LST.St FTl Fqhtlate tiw dgM't tlor t UTlq !h. dat. ,tdet cwidqatio,.

5l


31/51

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

gits th. Grcawich Mcan SidE5l Ti6. .t oir UT of rhe drte. Th.n lhCtqwich Si&@l Tio. for &e rirc ug@nt for ih. rimc of ob6qvation is:T - To + (UHR + (UMLN +USEC/60y60)a0.997269j66JJ (2.5.t4)

'Iijen fic Hou anglc at ihis moncnt ofthe Moon is:H=T a

srep.sl tllec^ofPatala,(2.5 rt

(2 5.16)


32/51

.Aliitu& r=sinpUnrlcdp'cosr'costl (2.5.20)This complcGs ihe der@inarion of dlipric, qurqid .d Oe hondnIll@rdid6 of rh. Mor

2.6 COORDINATES OF TI{E SUNFoi rhe d.t.mimtion of 0E c@rdiDre of rh. Sun on. p@..& in runy rhese say a for lhc @rdi!!re of rhc Moon.

slcFl: Scldt ple oroh6dvd (Longiru& ed tilirud.). lcal d ..dd rine rhatlqds to rh. thc egment t &coiding ro thc algolirhm dceiib.d lbove, s:

Julian Dde )451545 Ll1652t0 I | 55?600000 12.6.1)

(2.6.2)

Using lhis lire argment lh connrudion of d. (m. eriB &enbins rh.c@rdinates of the Elnh s givd by Br.raSnon & Fdcou (1988) thc h.lie.nrriccoodinales of lhc E nb de obtained lhar ft lder trusfoftcd into fte gre.tdcc@rdimles ofth. Sun s follosl

SreFl: For heliohllic dlipric lonsitud. of lh. Flnll' in lirc wit[ (2.2.19) &d2.2.20 w h!rc:

"e"(,,

"."6"

2^,2""

.r^, )t r,,l (2.6.1)

55


33/51

t,-'i^,-"(t"*", \ c2.6.4)(2.6.5)

(r5.6)

c2.5.?)

(2.5.8)

Q.5.9)

(2.6.10)

(2.6.l|)(2.6.r2)

(2.6.|])

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

2tL,-ZA,s\8,+c,. IL. =ZA,@l\tr+crt )!,.:r,cdt r +cir ,

i, -.t0'

Fd Fliptic ldintdc ofrlF E nh;lutBo-Z/,.d\8,+c,. l,r =:/,dF" +c"4,, . X,{" "o.(r" +c,r J

St p2:

Bt-244a.+c"r I8. = t,l,6.F" +c"' J


34/51

Tb v.lu.t of A's, B't .rd C'! 6. l} {oi.t nds tt giE bv Mo.u!(M6n, l9S, A!0.odtr-Itr, !P. 4lS-421). Fidlv lh. t lbc.otic ldlidt of th' E ib it

$.Fl:l0'

For lElioc.dric .lisl&e of th. E|nb&-X,i"6!p"+c,r J

zt2 txr - I,r,6lp" +c"'J19 lx, = t,,r" o.[8i + c,' ]nt13 = t,r"co't " +c"' Jt0,h=>an(fr]\B,+c{ )lo/xr = t,."co4p" +c"' I

(2.6.15)

(2.6.r6)

(2.6.17)

(2.6.rt)

(26.r1)

(2.6.19)

(2.6rI))

Q.62t)

TIE vds of 4.. B's !d C" b. lb eorlr Ei6 & tih b, M6!(Mc.q, 199E, Aprcndix-lu, pp. 4l E-421). Fi!.lly tlE helio@tric di!l&c. of th. ElnhZL,r

" = ''id,A!" Bs lnd C! N all in hdiM for looSin|d! , a|(l ldltt|d. t B6 od Ct c in ttdieE.d A! i! .*!doiql uitr tu Llio.ldtc diaec X.

51


35/51

The coftcctions for Abe@don md Nuradon de done in dc sanc way as St p4and Slep-5 bfore. Fi@lly te conveBion to rhc equaronat @ordinates dd then bhorizonlal coordiDr.s is als done lhc sme way 6 ws done ror fte Moon.2.? RISING AND SETTINC

For the d.refti@tion ofprecise rihings of rhc setrnE or nsing of& obFcl onerequircs pEcis c.lstial coordinales of $eF objech 6i lh insht of1he occunine ofrhephehomem However! these insranh dre the points ofime that we rcqlic ro find our sorhar a process oasuccessive appoxinadon is needcd lo ahv al lhee dhes. Such aniterarive plocess is necssry b4au$ rhe objccis under con$dsaion (lbe Sun Md theMoon) significantly changc thei position relalive to the ixed celsriat sDhere durihg mi.terval around ha a ddy. The whols process shns sm m esehare ror the ritoc of'ransn ofthe objed (ove.lhe locatneridian) wbich lben reads ro,nlial esrinars for deholrmghs ar th. appoxihare line ofrhe rising orselling or thc objer The aB in facrthe csimares for rhe tocat sidereal rinres ofdc phenonena. rrco these esfnales of rtEsdceal rimcsoflhe eveors lhe unilecat hean slarrimeardlheh lhe toql lihescan becakuhted. ths. fist approximarions fo! lhe ransii rbe nsihg ed rbe sedne areobtajned usinglhc cetestirtcoordjnars ofthe obj(r evaluard ar Oh. Ur. ar any poinr orrhe glob and ao! any otrhe evenrs under considecdon lhis nomeir ((]i, Ul oay be mcail.r or a tater nomnr. This is the rson lhar the mrfial catcularions are onlvrppo\imaro cdlcutduon fo.hcle appo\,dar. ,t.., "t ,h. .,.",. ,h. ."i..;;:.coordnats of thc objecr hNc to bc catcutaled asain and wholc.atculalions henLiondabove !! repealed for bekr esrimates, The details of rhe catculalions arc describ.d in(he followinS p@sraphs.

, . Forrhe sh. rhccompnaaon. &e simptes conp@ rornore tor lhe Moon. the,(o nme or hNir ot rhe Sun can be hirra ) coturdered ar t2-,tocsl rzonc, rimewnecas for the Moon n dEily vdi*. Th6 Unileisat Tihe or loql tuir is simply 12Zl wheE ZT (ene ne) is posirive for the east lons,udq ed nesarivc for Resr58


36/51

lonSitudcs. At uivcrssl rime 12 - zT wilt b. a dn of 0E se darc in O@wich s Ih.loql dae. Th. rime dgu. for $is rine ed date is rhen fomulatcd dd thcoordiMtes offie sunar.obaincd. wlm m objdr h in rmsil irs hour dgtc (HA) iszo ed its dsh ecdion (RA) is se s ln. Loc.l Sidqdt Tin. (LSn siNc:

If JD is rhe Julie ddc for $e dly ar d' UT 0En wnn , = eD _ 2451545y36525mcaur.d in Julian centuri6 lh. cenwich Mqn Sidc,@l Tide I, {CMST) b siven by:4 =6141-50,.s4841+8640184'.812866.r+0'.093t0,t.r: _0,.000006rr,r

(2.7.t)Thch for the observer ar rhe pla.e vilh eeographic lonerrude , (negative fo. westlonenud.s a


37/51

FimUy the coordides ofthc Ss

The pbc.s bcginsUrs0. The holr angl HA

@ Edcdar.d for UTE qd 0r followins cdculadon

I,sT = a,

csnr-LST-L,n_GSTtr-TouTr=T,'L0027379093?5

So ihat theE is a dilferne of lss drs a *cond b.tween one v.lue ofuTr ed ib.ext

[or lhe locat sunrise and sunset $c base vatue $ the UTtr ahd inirialapptoxmalDns 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 rexrday csp.clively so lbar a necese,y dare adjutmeir mul b. dorc. runher ddpe.ding onthe local lairude il is funher possible thar these pbenomcna sihply donl occur.

(2.1.7)

{2.7,8)

(2.7.9J

(2.7.10)

(2.7 r)

de c@rdi@res of rh. Sun for UT6 (orstt'ne or nsing ovcr a l@al horjzon k

stEE I is rh eeghphic laritude of rh. obsedcr lnd lhe au nd of d. poinr of sky isa$med to b. ero, How.ver owirS to th phdohcnon ofEri&tion a ,r.r, rhe Su dd spld.l ae wll b.low th. hode. std rhcy @ aru.lly seo sdrjng or risin8. Thet\@g rFe.t of EFadioD is rhar . srat E@iN visibte de ir ft hs gone j4 e Dinu!sbelow hori^n. 'Ilis allitudc is kmh s srand.rd drilud. dcnoEo s 4, ed is @[email protected])


38/51

lo bc -50 N sords on aveas. f6r rlc Su, 0Ei itulud.s ihc affet of dE Ef4clionand thc sni dian.ter bo$. For a mor accwtc valuc the &rul s.mi dimet r SD,rhould be calculal.d fmo dE didea of1h. Su ed suh.actcd tioh d..veagc afi*lof Glhction, fte chag. in l.hpdl@ in ihe niddle laritDd.s my lary thjs by @ud20 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 avcEgcafects m co.nd.ed in crlculalions. Unog fte $..drd slrjM. ?01h. hou anqlc oflh.object is then eloluared as:

Thus rh liBr dppro\,maion for lhe rhe ofrisng of$e obled h:T, =UTO - E.

And $at ofrhe s. in8 isl

r-Ta+J6O.98564TTy'15,1 or I e rlEr l@at lFu ogte

(21.t 4

12.1tJ)

12.7.\5)

4d ihc &imurh of th. obj..t is

T, =Wtr+ HoThes. e only !h. n6t app,oxinarion3Esp.ctirely. Fomularing ihe lin a8unorsof lbe Su ha\e b be catculared aglin. Theborh t &d I h6 ro be obained as:

(2.7.t4\

for lh tines of swise ahd the sunselfor .ch of fien Fpaan.l, rh. [email protected] sid.El rinc orrcspondjns ro

12.1.t6)


39/51

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

Th. cordiG for rhe tuirg ! siring d:(21.11)

ao =E - 0-2n5. t -10134''qb.G r is dc pa&Id of rb. Moon eivd byi

AT, =

p s $. g@eniric dist ncc oflhe

(2.7 l8)

Adding these A?_, ifto rh. appropriarc I, givcs ibpDvcd valu.stunhq inpovd r:lucs catcutal. $e @.dinaas of rhc Ss fd&d rcpeat (2.6. r 2) ed (2.6.1 8) ro obrain lh. UT for d wau.

lor Ue Moon rhc i$uc is mt! complided s af.r of panlta h signiticor.Th !ffrr ofrcfddjon h ro dcc'le the anirh dist ne !o rhar the objed is visibl. deiif n is th@Edcat gonc dom $. horizon bln rhe aflccr of pa6 d b to inc@sc rhe zcnidd6t fte e lhe obj.cr is wlt .hovc lhc horj@n dd n a!,p.4 ro Mvc et (or rcl n*nstill), Thus for fie Moon the si&dard attjtude is Civcn by:

(2.1.te)

(2.7.20)

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

62

\2.7.21'


40/51

dd ., is th. pold mdids of theE nL The resl of thc calculatiohs for rhe irlnsil, rising ed lhe stling of lhe Moon arcth. se d 1hat for thc Sun.r is th equrorirl ndiu5 of the Eard,

2.8 I{EW LUNAR CRESCENT VISIBILITY PARAMETERSA nunber ofpdrameleB have been corsidered imponanl for deteminin8 *belber

the new cr$ent would bc lisible al a lcaiion oo the Eanh oi ool. Tbese weE brielydi*u$ed ad lblcd in thc beimins of chaper l. Once tr tiD. of th. Binn or apaniculd Ncw M@i or th conjucrion hs ben delmined a numb.r of p,@eGB @rquned to h dt mincd. Th* includ. (i) Tinc of Suroet T". {ii) Tinc of M6Mr T..(iii) LAcTn - Ti (i!) Bcs| Time orvisibiliry Tr,(!) Ase ofrh Moon ar Tb. AcE. (vi)Arc of visior ARCV, (vii) Reladle Azimulh DAZ, (viii) Arc of Lishr (Elonsatioi)ARCL. (ix) Ph@ ofcrs.nt P, md (x) width ofcrcsent w. In vhw of the di$u$ionofth. s&onomic.l alSorirhns ed rehniqud in tbis chaptd the* circmst nc6 de c-co$iderEd ro .xploE compurdiom of vdbN psmre6 $al e inponmr ao. $elnolysis of l@al visibility oflh ncs lunr cr.$nt on $e day ofconjuhdion or d. doy

The fiBl of th.s paludd ii rh. coiju.tiotr of lhe M@. wnn ihe Sm or tn.rime ofBinh ofNcv hootr. The alsdirlm for lh. conpuraion of (his tim. w6 pE*nl.din anicle 2.3. U3ing $is algoddm thc riic of g@centic binh of lhc Ns m@n isobtaind. The algorilho tates yee s input and giles lhe Julian datc oflhe tine of thebinn ofnew boon. Il is imporrmt ro nole $ar the i.pur iiyai is ior a whole iMbs, itis o Eal nMb.r alculatd on the bdb of lhe expcled dale of th. New M@h. Forinsran@ the Nry M@n in $e nodn ofApri!,2007 i. dp.ctd msd l7s d.y of thc

,e0 = 2047 +atJ0+t7)/J6563


41/51

An |pDDxiE.r. valu. of "'*" wirh e G@ of f.e d.,s m*! wll. If UE "yd" it rvholc nmba $c algoriihn si!4 lhc ,!lie D.rG to! $. Nd Men th.t @6 cl@nro thc b.gimine ofthc !d".

otuc rhc ,ulid D.rc of$c binl ofrlE Nq M@n cl@n b lh. .xpdrcd d.y hab..i found tb. m. b inni.lly conwn.d lo lhc UniEEal rift dd darc edconeqE ly rhc le.l lih. and ddc. For ln.s convffirioE ftom Julis Dat. to rh.le.l rim. ed drlc th. tehniqu6 of rhc .rricl. 2.4 6 u$d, Bcfoc thb im. lh. Ew'lun.rion" h6 nor b.!un s no qNnid ofrlr visibiliry ofrhc Bw l|!w cEghl on rh.A6ins bcfoE this rin . B.foE this tin. only th.'bld cG*dl" ce b. lat wn b.foarh. sutuie on rhc dry of onjudion or a dly o! tvo b.foa-

Ar dr rimc ot@dudion dc M@n n.y b. &rwhcrc wi$in . strip ot widd loo18 eud dG elipric, i-.. wirhin 50 9 of rhc Sun. ap.n fom |h. ecume of . slardlips $G "@.nl'cxisl5 bul dE lo irs.xlrEh. clo*ncs ro lh. slai!8 su ir co norb. s..n ed ha ndcr ben sn,

Thc ndr inpondr p.md4 is $c [email protected] lim. of suel ed th. M@n $t Th.*can be conput d for ey day of lhc yd Bins th r.chniq!.! of $c anicl. 2.7. Howvr$ce rehniqEs 'lquiE th. dctmin ion of th. c@diEtcs of both lh. Sln &d th.M@n for D dFckd rihe of lrsi! dsing dd s.ltinS of dh of thd. Thc c@diDL3of th. su !rc obl.incd usi.s dE VSOPE? fi6ry of Brdagnon ad fEncou {or asihplificd v6ion sivcn by MeE) &d bricfly prc*ntcd in aniclc 2.2 Dd 2.6. Sihildlyrh. D@iF @rdidtcs of thc M@n c oh.in d aing th. ELP2000 of ChrFo -Toted Ch.pont pr*nrcd in dticL 2.2 and 2.5. Th. .oddinal6 ofrh. Sun &d lh. M@nlhis wr e th. SMtric sph.dcd pold [email protected] (dis1@ P. alatial lonsilud. rdd d c.l.nial l! rud. n. And @n cdon for .bdnlion thc sc e lMtfomd loth. 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


42/51

d[dl! of pdallq. Using lhe local siddnat timc tb. @rdinas e then trdsfomcd inloIoc.i hori@ntd @ordinalcs dr. ahitud. ud &imuth,

Once $e tine of leal sunsct (T) and th.t of Moon sr (I.) d obt in.d lheo@ctcr LAG = Tn - Tr is elculatd. Unlc$ lh. LAC is posnive for the New Moondd ncSrdv. for old Mmn th cFsent cd ncv.. b sen.

Using rnc .cliptic @rdimLs oa lh. $n ad rh. Men E alculared for th. T6 thcegutodal @rdimrs of rhe two bodics (d.,4) dd (d,,t,) e c.lolat d usins(Me$,1998):

s t h\^ )c os\r ) - Tdnl P)sth(E)

Suppoe (..,,r.,r.)ed t,.j,,r.) d ftc p@ie dislanes, aliflic lo!8it!d.sd th. lalirudc of rhe M@n and 6c Su, Esp.ctivcly. af@d ro the ft4 .qtinox ofrhc dD. or sunst at any lMtio. on $c Elnh with tftstiat coordinaies(1.r) on rh.day, or day aftcr. lhe binh ofNew Msn (al$ oll.d the Csenlric Coijliction oi th.Moon).Thefi'slsteDinthedelemimtionofih.visibililyoflhnevlunarcresce,onthc day of conjuncrion or the day affer, is io dcternihe lh actual dy.anical ihe (TD o!TT) T", oflhc coojuction. Ncxl one requires considcrins the local rines ofsening ofth.Sun &d tb. Moon. Le! Tt dd Tn (Coordimi.d Univdsal Tine TUC) b th. tihes of [email protected] sunsel dd the nooi.eq vi|h Tj < T..

(2.8 )o.8.3)

cas6)st(d) = si,,(l)co(.J + codB)st()st(r)

wl@d, fi. nd emid is in 0r s. qudtul6 t, ed ', the obliquily orrh.dlipri. is d$ adjugd for lh. d, . ed thc b.9 rine Th. tsal Hou Atgl. H is th.noblaincd tion rhc difreren@ ofth. l@l Sidftal Tiru (Zsl) 4d th. rishl 6c.nsionTnis finally gjves lhe l@d nonzontal c@dinal6 amulh (,1) $d the allilud. (ir) bv:

65


43/51

ranlAt = (2.8.4)c6(Ir)s,,{t) rz,(d)co{l)(2.8.t

And adjusi.g for lhc rciacdon 6d $e heigh of $e obFNd's l@aiio. above *. Id'lrbc topoentic @oilinares (,r,,r")ed (,i,,7t,) of the Moon dd fte Sun. esFclivtrv

In alnost all lhc nodels for cdlicst m@mid ins $c ancidl d w'll ts ibe nod'm'ln. dir@. of eim hs (DAZ = lA,-A-1. @lled El.tive @'ihuth) ard thn ofaldrudB (ARCV - ," - t.. cdl..l m of lision) d shoM in rhe lisr I, 'r the iimc orloc.l 3uEl T, sd/or al lhe b$l Tb Play a vilil ole:

As ihe dgulu sePar.tions invol!d b.iwen $. Sutr md lh. lunar cesenl at$6. dnes @ sdaU, wi6ou1 dEh mt th. e of lighr (ARCL) h giEn bv:

(2.8.6)

si,(r) = sr,(c)si'(6) + c4(d)ca(t)c,r(fl)

Wh.@ for larSct ugl6 or moF ffi.rc Gults th. d of lieltr should b. eldl.Ld

ARCL = cdr(co(r-)co{rr,)cd(tra - s,'(r.)si(4)) (2 E ?)

66


44/51

Fi&28!A@d tun lhc Ft.tiv .rlmuh tbc !D of villon and th. e of lighr Uc cdlqir

for dli6r ytuibrliry of t|ld qqc@t rlqoits io irL ie c@li


45/51

"='"[ (289)

Sinc. ihc acienl dd thc nedieval dnes {E ACE, thc time elapscd siie fiebirlh oI New Moon lill thc sunsi of lh. d.y in quesdon, w4 eo@lly coNidecd asignindt p&Metr. Howev!, in ihe tinc of MuslinvABb $lronomrs jl had alreadyb.en c.lied tal 0E ACE is nor a nD&hc al padndcr, stll thc pardet* isinporlrnl 1o calculat. This is bccaue oflhe fact that depending on rhe aordinats of theMooo dd lhc seaens at lime a very "young' cEsanl ce b. ee.. Amongsl bo$ rbeanatu s vcll a rhe prcfcssioBl Grronom.u $@ is aluyr a conFlilion for havinS$e ord lo s the "youneesr" crcsr eiihd wilh oFical aid or wilhout i1.

Amongst the early nodls ofnew cc*.!t vnibility asltunomeB sed $c !.larionbetren rh. rcldrive altitudc (ARCV = alriMc of rhc Mmn - ahilude of lhl Sun) dd rherclrriv. uimurh rDA-/ - e'murh of Lh. Su - uimdh of de M@n' Th. rsoparmeleG arc slill consideFd inporl&l as if DAZ = 0. the crcscenl is venically abole$e point of 3uEel dd und.r such a cncM$dce the youg.st a @ll as lhc thimeslcrc$qt cd bc ecn. Wirh lugd DAZ wles only oldtr md thc thictd ccs @ be

How lhick or lhin is lh cEsenl al any lime cs b ddemined once lhes.pmriotr (clonsaiion) btwn rhc su ed $c Mmn hd bccn dchi'cd usi.g (2 8 ?)sith nhr horizontal coordin.les or rh quatorial aordintle! This elongation or Atc orLiSht (ARCL) hads to lh. Phde (P fiaction of the illuninatcd lund disc facing fieobsdr) of lb. Mftn using (2.8-E). Ho*ev.a the lhicloN or $e widlh (w) or rhec..rral pon ol the creot des not only dcFnd on thc phe of the M@!, 't alsodepclds on rhc Elnh'Moon disrece, rh. sMc cm be ohaitcd usins (2 8 9) Dte besrinc of lbc crcsdt visibihy sqggned by Yauop is criical fot siShting lhc crcsn1u'dd mdgjml conditions &d ce be @mpurd sine (2 E l) sisndne i5 crirical when

68


46/51

THE SOFTWARf, HILAI-OI.CPPIn this worl( a softwde k develop.d for the odysh of rhe fisr visibiliry of newlue 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 rhecompur.rional and thc ecie.l dd Doden visibiliry nodls. T1| lisdng of the poelmHilalo|.cpp is giyco i! ApFndix 4. Tn progm fnrca d brifly de*dbed b.to*:

Th. prog@ LinAnal ucs ih&e dala filcs, rwo for input ed one for oulplt. Thetwo inpul dala il6 e (he files For.ad.rv thal conrains rhe padrcrE 4, ,, and., (sdescribed inanicle2,6 lbove ). The pdmetcs oflhis dala file de adtu8d in rabtes2.l(A) lo 2.1 (F) in appcndix 2. 'Ih. ol]],er itpnt tie is tlc .tp2MA& rhat conlaiE thepaluercs as de$rib.d in rhc anicle 2.5 above. The pdmeles of rhis dara file ddds.d inbbles Ll (A) 10 1.3 (D) in appendix L

Thc third fiI. u*d by lhe prcgrd is *..t lhal is optional dd is n*d only vhenfie Esults of fi. cobput2rions in Oc preetm m EquiEd to be sbEd. Tl!. vcGion oflh progr.n Hilalol givn in appendix 4 hs the lile nanc scrrrgaA, rhar sbres lheonpulalional Esulrs of a sinelc .x@urior of $e prcgrm. fte i.fomations stoBt in

o. No., rhc obsenarion nhber thar n scncally digned by Odch (odeh,2004).Date. datc of obwatioo,Long., th. longitudc of 0E ple ftom whE rh. clesnr is ob6eN.dLatil., th. ladMe oflhc pleEld., the .lcvarion of rh. place above s! levcl,T.mp., cnimred rcmFdturc of$e riDe of obswarion,Hmid.. .stibaied relative humidny oflh. plac. at rhe rine of [email protected], lhe l6al tinc of sNet

69


47/51

9. JD of Conjunct, rh Julie Dat of the binh of.q M@n or the lsr10. As., lhc a8c ofrhc Moon.r rh besr tim. e@ding 10 Yallop (Yallop, 1998),ll. L{C, the diffrcncc berNcn th Moo. *r dd rh sunsel. it minucr,I 2 ARCL, dc ot light or elonsetion of lhe M@n fion th. su at the b.st tme, inll, ARCV, the eladve altitudc of fie crc*ol .t the b.st 1ioe, it d.crees1,4. D,AZ the rclaiile @ibulh ai the best time, in degrc.s,15. Widtb, the cenlral widlh of lhc Fsrnt al bcs1 rime, in dc minues.16. q-val. rh. visibihy ptue1q dcfined by YalloP (Yallop, 1998) lo h.

discssd in drail in chapl.r 4.17, Phi+della, fie dglc thal lhc.cliPlic dalcs with lh venical on th. wslemhoii2on, in d.sE.s.

I E. Mlalit. I,loo. s 4liptic latiiude in d.8e'19. Mlongi! Moon's.cliptrc lon8itudc. in d.gtus-20. Slongil., Sun\ *liptic longitud., in dccres21. M-SD, dgule s.minid.tcr of th. M@|L in @ oinut s.22. As-Fac! e of sep@lioh fsctor dfined in l.ler chap1e6, in dcgrc621. R-En, cslimatd Rip.n $ Fucrion vatue dclind h chapler I?4, R-A!er, averas Ripeness funcrion valu dellned in chapl{ 325. R-actual, acbal Rip.ns Fsction vtlue dcnled in chapls 326 DR+s! $e diflecnc of R-rclualand R-Enimaled27. DRnc! thc difrccne of R-a.tu4l ed RnErage28. MoonS M.g, Moon\ mogniiude al the besl lne of Yallop,29. Lio-Mae(lin), lh. LiDiting Mt8nitud. of the stv ned |ne cs6t s

defioed by Schaff.r (Schaefd, 1988b) aloDs wnh ih Yallop's besl dne inudveEal dne dis.ts.d in cha .r 4.30. SI'(DME8), tbe univ.ssl tine wh.n the conrrdt ofthe stv bdelhes ddrh. Mmn\ bljehBt just tum in favou of thc Moon dd 0E ditrGnc' of dE

70


48/51

I_

32.

A3 thc .xcution oflhe proghh b.ginsnonlh, y.&, longirude, Iaritude od cl.vationesrinated ttmperaturc ahd estimrd r.larilciniliared by rh. lunction tzrdd,r.rtnd callcd by

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 edth. Moon\ bdghret is b6r in favou of rhe Moon ald thc diflcn@ of $eha9irude of rhe M@D ed thc lim'ring m.gn'nrd. ofih. sly al rhar monenLsr(Dmae), ihe univdal tme whn rh conrmst orrhe sky bnshhess edtbc Moon\ biehd is lal io &vou of th. Moon ad rbc difcrcn@ of $enagoirude ofthe Moon ad rhc liniring magnitude of$e sly ar th.r momenr.

ir prompts for ob*Nadon nMb.r, dale,abole sea lvel of th phcc and $e

humidiiy of lhe place. This plompr isthe tufuri,on nt inn u t "..

Ticn rh. prcg@ c,lls for $e tundi@ math .h4tg.. This tunc on n6rddmins lh. Julia Date of dE dDc of n.esr @njwdon or rhe bini of ncw M@ncrlling thc fucrion /t ]@_D@r. Th. tu dion r'r_r@_r@r is bdcd on .tgorithn dueioMeeus(1998)discqsedinadicl2.3abovc(fomu162.3.1.ro2.3.25).'Iletuncrionrorr, cna,a. $en calh the tu.ctior renir&rr lhal dekmines the tihcs of lhc sunseldd the Moon set tttroud fiD.rioB tun-q.t 6d mon_se.Ih tunciions rzu_r.r sdtutu_sa t. b6ed on lh alsorithn dhcus$d in aaicle 2.7 abov.. Dr functio.rdttirgt.le calculars rhe b6r rimc for casent visibitiiy aeording io @ndnion duto Yallop (1998) di*ussd h a larq cha .r Tnis is followd by @hpul!fion of theJulie Dat 6d dB Exlcndd Juli@ dat corcsponding ro rhe best rim. 6ing rhtunctiolirri.ndrl. rhat t bs*d on rl| algorirhn pEdred above in anicl. 2.4. ThisIesds ro fomulation orlhe rime {sMn1GIs dipused in adiclc 2.1) fo' rhc Elp2ooi)md VSOPS? lheories for the c.lculation of coordimres of lhe Moon and ihc Surespectively. Th.e coordinates e calcul.r.d according b the algorithDs in anictes 2.5dd 2.6 iopl.m.n&d in fucdoB a@r_.ou.t ed t\. su"_.erd Bpeilely. B.fore

7l


49/51

rh*.@rdin ls @ c.bdat.d th. etrc1oftuhio. dd the sidd.d tihc coftspondins10 zdo hou uiversal tide, for the dat consideed. are calculted using the tuncdois,ttttid ad srl_tiw, E#iv.ly, The infomatioi/d.tA thu fe Scn nted is displayedod *'.n usins rhc t$aiqs ouinlo, rtbpt t scootd (.@fitn^.s of 1he su.) eddbrldr_n rord (coodinatcs of tbe Moon).

Finally, fhc tunction nainrcutir. crLtl^r.s md dbplaF on $en aU rhevisibiliiy parmetcrs dircused in pFiou.rricle (2.8) &d lisred ss l0 to 16 dd 2labov. in lnc clm anicle. O$d pffi.r.c (l7lo 20 dd 22 1o 12) re @mpui.d inorher functions (r? to 20 in tunction r@r_cMtt a\d 22 to 32 tt liM'si\ brldisplayd alo.g with these pdmet.rs. All thc Ddmd.s lisbd as I lo 32 d wiren lothe outp dda file J.r"8ear in nmloulrro4, (pamrs I to 29) ed tim._nnse(p@ncle6 30 to 32), Tle tundion /ltr,r,,? is ih reproducrion ot Scba.fer's pog,aD(Shs.fer, 1998, Bogd, 2004) lor delediiirg th liniting hagnitude of dy point of

The Esl of the functions used in lhc pregrm aE lisLd dd briety dcscibed

/.ar.r..l chects lhe1hd the y.e u$d is lcap yed ([email protected] ro crcsoridrule) or rcr? lfrhe yd is lap lhc tuncdon rerums I orhfli$ O..orwt_r'ru convens an eelc inro dcgres, arc minutes do arc *cond3..,,rzt_rtu convens lime in hours into hoE, ninuIs &d s*onds.

hodlw Btms rh. @jnd.r .ff4 dividing th inpur b, 160. 6is fucdonh 6ed ohain 6e mst. in the tuB 0 deercs b 360 degE6.pamt calculat s rh. afr*tl ot psrlts in i8hl acEion ad decu.adonto cct th. ropcdFic ddr as1fuion 6d d.ctinarion and is bas.don step 8 of alsori$n in anicte 2.5.

itr._r.. rhrcud ,rr.-do, ,nd d"c_sc thrcugh d.._r,,: Ttcs. tundions altowschdging tine (al ld.b ofhourr ninure ad ssnds) and dare(at lev.ls ofdale dd honrh).

72


50/51

In fer sihin lhc tuldion di.i@{tt. lhc prcgr&n Hil.lo! ha ^ do-|9kL l@prhar remiDrd *tEtr rhe iddrifid za,r rcccivB V' (pl*in8 dld le,) fion thc Br.If my oths key is prcsed lh. pregm rmdN in wait ndq PEsing Panicdd k ls ais obvioN in the er.l{ar. cohbilsriotr vdio{s&lions e iniii.ld likc inc@ing ordeoeasiry tim dd datc, chegilg tmpdal@ or hlmidity o. wiling to th output nbr.rngafir. bitiad.g any ofthFc.ctioro Gp..G th. *ftution ofall lhe conpulalionswilb new timc, dale,lcmp.dture or huDidily. In cd. ofpre$ingP all lh data is winenro fie out file s.r'"a4.t in one lin a.d prc$ins 4 Mit s only rhe $lected !alu$ oItine and conesponding value of thc diffelcnce of doo.\ magnilude and $e liniling skymasnitude. Tlese fatues of thc progm Hihlol (varyin8 dne, dale and !@lhcrconditio6) nEle tbe progrm norc dynoiq .s compdd lo Mooncalc of M&ruiMdd, 2001 ) ad A@uai. TiF. of Odeh (Od.h, 2006).

In view of lhc prcbls of d.rminiig th. dlr of the fi6l sier ing of new [email protected] in this cnaFd w cxploEd.ll thc Mjor 6pet of conp arional cForls. Th*

IE sm!.ndylicd dyn.Dic.l lhddcs VSOP87 and ELP2000 d'at dryib. th.motion ofplseB loud thc Su sd oflh. M@n reund the Eanh. The* e m.mosl rent dd nost accunt. avlilabl. looh for the delmi.ation otephendb oflhc Sun dd th. Mootr.Ite ateonlhm lhat lelds lo lbe dct min.lion oftb. dynmical lime ofde lud@njucdor or lh. binh ot ncw Moon.Tn pmblms ssociarcd wi|b th. dynamical &d uivEal lim ed relatedisues. Wiihout having a @mpl.l. kno* bow of lhse isues apprclrjalc tbeargmcnt for the d.lminadon of lud ed dE $l& cmrdimres co not be

1)


51/51

TL dl irpde dgsitor tu d. ddroni4 m. tr..r,r.l od locd rocdn . o{ th. @ rd 6. md* lllth! s* d6itr of ti.dD.r lb pcr@i !rd&d tflt t Fb&o of di- dShdrS o{ rtrle or*dt ci rdt!. da-!.d.lrib I tdrwh ||I&tbt r of d| 6..c tt{itr d .|6mi.l G.hiS..erl .ltdl6or | .orio!.. F!!im ir redd b &r .U t ,.w lull

Tto 1b ct# &ir_.r .ll tt cqdrilrl ,trrit [email protected] eith it.plobLa ofib. d.olloilirS tl. dry of6rr jihli!8 ofew lurt.!..c.d

astronomical formulae

Documents