Question

A trinary star system has a time period, $T=3$ years, while the distance between its components is $2$ astronomical unit. If mass of the sun is represented by $M_s$, then the total mass of this multiple star system will be

Note :
(1) System of three star, orbiting around centre of mass of system, is called trinary star system. For simplicity assume all three components to be identical.
(2) 1 astronomical unit = distance between earth and sun
(3) 1 year = time period of earth to complete one full revolution around sun

• A. $3M_s$
• B. $2M_s$
• C. $\frac{7}{9}{M}_s$
• D. $\frac{8}{9}{M}_s$

Answer 1

The correct option is D $\frac{8}{9}{M}_s$

For earth and sun system
$T_0^2=\frac{4{\pi }^2}{G{M}_s}{R}^3$ .........(i)
For trinary star system
$2\frac{G{M}^2}{a^2}\frac{\sqrt{3}}{2}=M{\omega }^2\frac{a}{\sqrt{3}}\Rightarrow {\omega }^2=\frac{3GM}{a^3}{T}^2=\frac{4{\pi }^2}{{\omega }^2}=\frac{4{\pi }^2}{3GM}{a}^3...........\left(ii\right)\text{Given},T=3T_{0}anda=2R9T_0^2=\frac{4{\pi }^2}{3GM}8R^{3}Using\left(i\right)\Rightarrow 9\frac{4{\pi }^2}{G{M}_s}{R}^3=\frac{8}{3}\frac{4{\pi }^2}{GM}{R}^3\Rightarrow M=\frac{8M_s}{27}\text{Total mass}=3M=\frac{8}{9}{M}_s$