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 45.6 km/s and 55.1 km/s. The slower planet's orbital period is 8.68 years. (a) What is the mass of the star? (b) What is the orbital period of the faster planet, in years?

Answer 1

Answer:

$6.19744\times 10^{31}\ kg$

4.91 years

Explanation:

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

M = Mass of star

v = Oribital velocity of star

T = Oribital time period

R = Radius of orbit

$M=\frac{4\pi^2R^3}{GT}$

$v=\frac{2\pi r}{T}\\\Rightarrow R=\frac{vT}{2\pi}$

$\\\Rightarrow M=\frac{4\pi^2\left(\frac{vT}{2\pi}\right)^3}{GT}\\\Rightarrow M=\frac{Tv^3}{2\pi G}$

$\\\Rightarrow M=\frac{8.68\times 365.25\times 24\times 3600\times 45600^3}{2\pi 6.67\times 10^{-11}}\\\Rightarrow M=6.19744\times 10^{31}\ kg$

Mass of the star is $6.19744\times 10^{31}\ kg$

$M=\frac{T_1v_1^3}{2\pi G}=\frac{T_2v_2^3}{2\pi G}\\\Rightarrow M=T_2v_2^3$

$\\\Rightarrow T_2=T_1\left(\frac{v_1}{v_2}\right)^3\\\Rightarrow T_2=8.68\left(\frac{45.6}{55.1}\right)^3\\\Rightarrow T_2=4.91\ years$

Orbital period of the faster planet is 4.91 years