Answer:
[tex]6.19744\times 10^{31}\ kg[/tex]
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
[tex]M=\frac{4\pi^2R^3}{GT}[/tex]
[tex]v=\frac{2\pi r}{T}\\\Rightarrow R=\frac{vT}{2\pi}[/tex]
[tex]\\\Rightarrow M=\frac{4\pi^2\left(\frac{vT}{2\pi}\right)^3}{GT}\\\Rightarrow M=\frac{Tv^3}{2\pi G}[/tex]
[tex]\\\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[/tex]
Mass of the star is [tex]6.19744\times 10^{31}\ kg[/tex]
[tex]M=\frac{T_1v_1^3}{2\pi G}=\frac{T_2v_2^3}{2\pi G}\\\Rightarrow M=T_2v_2^3[/tex]
[tex]\\\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[/tex]
Orbital period of the faster planet is 4.91 years
