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 42.6 km/s and 53.6 km/s. The slower planet's orbital period is 6.30 years. (a) What is the mass of the star? (b) What is the orbital period of the faster planet, in years?

Answer 1

Gravitational constant = G = 6.67 x 10^-11 N.m²/kg²

Mass of the star = M

Mass of the first planet = m₁

Mass of the second planet = m₂

Speed of the first planet = V₁ = 42.6 km/s = 42600 m/s

Speed of the second planet = V₂ = 53.6 km/s = 53600 m/s

Radius of orbit of the first planet = R₁

Radius of orbit of the second planet = R₂

Orbital time period of the first planet = T₁ = 6.3 years = 6.3 x (365x24x60x60) sec = 1.987 x 10⁸ sec

Orbital time period of the second planet = T₂

For the first planet,

V₁T₁ = 2R₁

(42600)(1.987x10⁸) = 2R₁

R₁ = 1.347 x 10¹² m

The gravitational force between the first planet and the star provides the centripetal force for the circular motion of the first planet.

M = 3.665 x 10³¹ kg

For the second planet,

The gravitational force between the second planet and the star provides the centripetal force for the circular motion of the second planet.

R₂ = 8.509 x 10¹¹ m

V₂T₂ = 2R₂

(53600)T₂ = 2(8.509x10¹¹)

T₂ = 9.975 x 10⁷ sec

Converting to years,

T₂ = (9.975x10⁷) / (365x24x60x60)

T₂ = 3.163 years

a) Mass of the star = 3.665 x 10³¹ kg

b) Orbital period of the faster planet = 3.163 years