Question

Two newly discovered planets follow circular orbits around a star in a distant part of the galaxy. The orbital speeds of the planets are determined to be 43.4 km/s and 57.2 km/s. The slower planet's orbital period is 7.50 years.

(a) What is the mass of the star?

______ kg


(b) What is the orbital period of the faster planet, in years?
_______yr

Answer 1

orbital speed is v = sqrt(GM/r)

v1 = sqrt(G*M/r1)

v2 = sqrt(G*M/r2)

v1/v2 = sqrt(r2/r1)

43.4/57.2 = sqrt(r2/r1)

r2/r1 = (43.4/57.2)^2 =0.575

r2 = 0.575*r1

r1/r2 = 1/0.575

and also v1 =2*pi*r1/T1

43.4*5/18 = 2*3.142*r1/(7.5*365*24*60*60)

r1 = 4.53*10^8 m

v1 = sqrt(G*M/r1)

43.4*5/18 = sqrt(6.67*10^-11*M/(4.53*10^8))

Mass of the star is M = 9.87*10^20 Kg

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b) (T1/T2)^2 = (r1/r2)^3

(7.5/T2)^2 = (1/0.575)^3

T2 = 3.27 years