Visibility of Luna and Lunar Crescent

accurat€ d€terni.alio. of lhc posiiiom of the Su dd tl'e Moon is tbe fisl slep towards
explorins |he circunsr.ee of rlE vGibihy of ihc !a lud .esnt, Handling sdh a
laqe hnmber of pdiodi. terms mak6 this fiEt slep very ciucial. The lest of the sludy
dep.nds on th. Elativc pcitio$ of lhe Ss and thc M@n ed rhe hod&i at md aner
thc aunsel for a.y place on the &nh.
These theori€s detemio. lhe c.l$i.l ecliptic coodiiates of the sold sy$en
obj*|' fid dc b6ed on sphencal polar c@rdinats. The fig Ll.l sho\s the spheiical
pol.r coordiMle sys&n, The origin of lh. syslen O is €nher lhe ccntrt of lh€ E nh ot
lhat ofthe Sun. The xy'plane k the plane olsliptic, the plane i. shich the E horbi6
iound the Su, The x-dis poin6 in the diEtion ol the vcmal Equimr l which is the
poinr of intersection of the Ecliptic (palh of rhe Eanh arcund rhe Sun) and lhd celesrial
equaror (rvhos plde coimides wnh rhe pl.ne oa rhc lereslrial equalor). P is $e objsl
llhc Moon in gcocemric sysem or thc Earth in llelioccn(ric sy(em) $hosc stheical
porq @dimrs tp. 0. 0) e p = lo4. 0 = 4oP 6d e-zzo?.\i\ee P ist\.
prcjccrion ollhe position P ollhe objccl on ro the xy-phne. In the celestial ecliptic
cmdinaB the cetesial loicitudc I = o dd rhe cel.$ial l6(ludeP= 9f - el= zPoP ).
whcn fie hcli@enilic eclipric coordinates ofthe E nh aa elaluared usins VSoP ft.y
aE $en tdslomed inlo lh€ gc@nric eliplic coodimles of lhe Sun The corjuncdon
or $e Bidh ofNe\ Men occu6 wben }{1 = Is.
Fig. No I I I