An introduction to astronomy: designed as a text-book for the use of students in college. By Denison Olmsted ...

250'T'IfF: PLANETS.
one-eiguhtieth that of the eartl?.lAts 24,000 miles f'rom tile
center of thle mloon, early.
Plonm. 1.2.. —..t.Suppose that the only two bodies in the univerluse
w\cr a s)phere two inchcs in diameter, of thle osame density with
thle earth, for the primary, and a material poillt for thle satellite. WNlhat would be the periodic time of the satellite, at tLhe
distance of one foot, in a citcular orbit? A2s. 2, days, 10
h0ou1rs, 13 minutels.*
o The elements usedl in tihe solution of thlis problem are, for tile dlimeter of
thxe earth, 7912.4; for the distance of the moon 238,545 miles, and ftr its peiodlic time, 27.32 days. The sollution, conductedl in the ordinlary mode, will bie
found susccptible of gre\t abridglent. Bu31t tihe following ingenious method is
still shortcr. It was suggested to the author tby) onei of his putils, Mr. Samuel
Emerson, of the class of 1818.
LmJmA. t'llte periodic times qof two sattellites revolving haout pritcrie of equalt dtt-sities,
at dietnmce-s whticfh ore eqrtimulttiples of their radii, are equal.
e)cmonstrattionl.,et
hI,.:s: the masIes of the two bodies respectively.
P, p::::: the periodic times.! t...:::. tile radii of the st)lteres.
I), ( I: tlChe istances of their sattellites.
T'teCn, 5: m:'- *
But since 1) anlt d are equimultiples of R, r, by some number tn, therefore
I)II::::R'5~a,1:la' dan..' -A;Hence, hi: atX:: -:- -: -p2. But, 1.' anld.r o 1i and -t t.
at mr MXm 5tXt'l'Therefore, i T: at::. -p..-.. - -: -.*'.F p.
p' p'.
The moonl being (distant 60.290 nadlii of the earth (as would rsutlt from tho
above elements), at the distance of 60.296 incIles that of thte smtalt satellite froma
its pritnary wotld be the same: multiplle of its radius, and1 consequently, its periodic time the s.atme. What then is its period at 12 inhels?
S 27.322: p:: 60.29G': 123,. p::::: M2d. 10h. 13ml.
Ooroltiry. -.If any two spheres of the same density be taken, the periodic times
of atellites revolving abolut them close to the sufaee, will be tihe same il both; for
thile Cate becomles this whetn t:::1:.'l'htls, the material point supposetl ill tite
above p)roblleml, rwill revolve about its little globe in tie saintt tuilme that the1
moon wouldl revolve about thto arth, both being situated close to tho surfaces of
their respective prilaries.