Question

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 42.1 km/s and 59.1 km/s. The slower planet's orbital period is 7.50 years.

(a) What is the mass of the star?

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

Answer 1

a)

for slower planet :

V_s = speed of slower planet = 42.1 x 10³ m/s

T = time period = 7.50 years = 7.50 x 365 x 24 x 3600 sec

r = orbital radius

V_s = 2 $\pi$ r/T

(42.1 x 10³ ) = 2 x 3.14 r / (7.50 x 365 x 24 x 3600)

r = 1.6 x 10¹² m

M = mass of planet

orbital speed is given as

V_s = sqrt(GM/r)

(42.1 x 10³ ) = sqrt(6.67 x 10⁷M /(1.6 x 10¹² ))

M = 4.25 x 10¹³ kg

b)

V_f = speed of faster planet = 59.1 x 10³ m/s

R = orbital radius

using the formula

V_f = sqrt(GM/R)

(59.1 x 10³ ) = sqrt(6.67 x 10⁷(4.25 x 10¹³ ) /R)

R = 8.12 x 10¹¹ m

Time period = 2 $\pi$ R/V_f = 2 x 3.14 (8.12 x 10¹¹ ) / (59.1 x 10³ ) = 8.63 x 10⁷ sec = 2.73 years