Visibility of Luna and Lunar Crescent

kplM\ ficory of luM hotior, Publish.d in 1802. cnploy.d rrmfo@ing th.
eqution ofmolions $ tlt.t the lruc longltudc wa an independ.nt ldiable Hh wotk also
povid.d e dpl.n tion of lh. sdo a@1.61io. of the M@! ripl@'s mdhods w.E
cdied io a high degrce of accu!@y by sevetal malhedaicies One of then was
Dmoi*e. who publthcd his th@ry dd r.bl6 in 182? that Gmaircd in *id' Ne until
Ha.Fn! sork dppeaed. P. A. Hasen's wolk ext nded for over fonv veds fon 1829
dd his r.bl4 w.re publkM in lE5?. ThGe labl6 rcmain.d in s for w.ll ov'r finv
yds. Delaunay publishcd his work ih 1860lhal M basd on disturbiry tunctions $ai
in€ludcd 120 t€ms. By sn.ltti.al m.4 h. Gmov€d 0E teru of disorbins turulion on'
by one ahd gEdually builds up th. $lution. Autho6 cl.im thar Delaunav s vorl( n thc
mo$ Frf{t elulion of th.l@r problm v.t found {D&bv 1992)
Thc posilion of $e M@n dodd th€ ElnI is desdbed bv spndical pold
coordinatcs (r,,190 p) wnn r bci.s th. heli@.ntric disr:nce or rhe tltncr' ? $e
eliflic lonsitudc and I lhe ecliFic laliode The most commonlt Gd edv lo handl€
lunai $bles dlrinB the nosr Pan of 20't c.n$ry \rcG dDc lo BrcM (Bmm 1 960) This
ll]M $.ory w3 inpo!.d bv Eckefi &d ws tnown 6 ILE, sho^ tot Inprored Lrnr
Epr,rdr,r. Tnc lh.ory coGtruct d bv ChcPont and ChaPon(_Toul is knosn 6 ELP
(Chapo er. al,. 1983. 1988) shon tot Ephtina des I air's Pukienhes ln ELP
simplifi.d tables have ben cxlEclqi from lhe lheorv lo rcplcsnt lh' luDa nolion in $e
fom of explicit line sries fomule. Th6e tabl's @ b€ lsed to direcllv compute $e
luntr coord'nabs. ELP h Dol onlt morc Peci* md complele in conpdien ro ILE it
ale povides noe nodern valEs oi l(M pdmelels md olh€r Phvsical coislsts' For
6000 y.4 on eacb side of J2ooo o ELP povid€s lund c@rdiMt€s that l4lv have eno$
.xc.eding few arc secoo.ls Toeelher with th' d€v€lopment of VSOP (Vdialrons
Sacnl.ifs des Olbires Ple€titt bv Bctagllon ed Fdncou (t9E8) lhe t!bl6 due ro
Chapront er. al destib€ the motion ofall major bodies (€xcept Pluio) ih tbe solar
sysrem. Both rh€ $€ori6, ELP &d rhc vsoP w dcv'lopcd d rhe Buean d's
ll