Visibility of Luna and Lunar Crescent

Today it is tnoM with a g@l degK of @!6cy lhoi lhe avetagc avrodic p.riod
(intepal ber\,s t$o coMurive ne* M@m) is 29.510589 davs (29 dars, 12 hom 44
ninul6 dd 2.9 sondt. The av€Bec uomalisd. morth (intcn.l ber@n lwo
su(esive pa$ages of rh€ Moon throush it! perige is 27 5s4550 days (27 davs 13 hous
18 oinutes ed 32.1 $cond, ed 4 aw'age sidercal honth (inLdal b.tw€en re
succesiw p6eg€s of rhe M@n tnroueh ! fixed st!r) is 2? 321662 days (27 davs ? houf
4l minures and I1.6 secondt. Thus on thc b6is ofa sidcEal no h thc Moon rravel3
abund 12.176158 degr.et p.r.lay on the av.rage Relaiv. to lhe Su. lhc M@n traveh
12.190?49 dtgc6 per day on lv€mee. Howvo, rhe mininu Bt 6 b. 12 08 deges
Fr
day and $e ndimum t2.41 deeees per day. Tnis oll happeis beau$ the orbil oi
rhe Moon around tbe Ennh is a hiehly "i!rcCular' .lliPe whercs lho deviadons tre
cau$d by p.nuibatios du. lo the S!n, lhe Planeb
and olhs $lf svsten objsls
Thc smi-major arh ollhc orbn ol M@n n on alcnge 384400 kn bul has a
snaU oscilldlion aound thb valn€ whose period is hala lhe svnodic nonth Thc
.@enrricily of the orbn is 0,t49 bul vdics d much sj 0 l lT The inclinalion of the
luhr orbir from the ecliplic is 500 9' bul vdi6 up tol9 Elen the nod6 (poihts of
inFdecrion ollhe lunar orbil and the ecliptic) of lhe orbn !re noi fixed 'nd
go round ihc
rcliptic in 18.6 yeaB with an o$illalion.bour dE $cular notion lhal moun|s to as
nuch as I 6? dcsl6 Thc line ol apsd?! tle eo rcund thc echpiic complelrne on'
rouod in 8.85 yem and dciltations {iih udnude of 12'413 destccs Ths all lhc
''elenentt' oi the orbn exhibn bo$ scular a ell as peiodic ldiarions lhis mkes rh'
deimiMdon of lund ephem€Ed€s a daunling task
The undeslandi!8 of lhe dynanics ol the Moon dd thal of th' planets in $e
nod€m *lup b€sd vith lh€ doluliodv e4loits oi Johees K'Pler $d lsc
N.s,ror Kcpt.r cnpnically dcdu€ed his Lls of Pld.lary Molion on thc b6is ot o
extensile study of the obsralional .lcta @llected ovcr ce'nies of lhc posilio' of
pldets. Thcsc cfl be sbtcd 6 follows:
3l