Visibility of Luna and Lunar Crescent

s- =t'"cdldj"'+djl xron)
s;=:4cd(d;'+r;,''/l
(2.2.t2)
(2.2.t))
s; = t,;c,rt6"4' + r,4, .4 (2.2.t4)
si= +d;o).r) e.z.ts)
i"-",,(u-''
-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),
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, /'rs
aid d'h de in deecs/centud. R, s,? and Sh dc in kilomelres, 5i md 4 .t in
kiloderrs/cenru.y. siand 4 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 of 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€ el enehls of thc osc ulat ing or inst aneous orbit in lems of:
t7