A certain quaternary star system consists of three stars, each of mass m, moving in same circular orbit about a stationary central star of mass M. The three identical stars orbit in same sense and are symmetrically located with respect to each other (The centre of all stars lie in one plane). Considering gravitational force of all remaining bodies on every star, the time period of each of three stars is

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$2 π \sqrt{\frac{r^{3}}{G (M + \frac{m}{3})}}$

$2 π \sqrt{\frac{r^{3}}{G (M + \sqrt{3} m)}}$

$2 π \sqrt{\frac{r^{3}}{G (M + \frac{m}{\sqrt{3}})}}$

$2 π \sqrt{\frac{r^{3}}{G (M + 3 m)}}$

answer is B.

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Detailed Solution

The distance between the orbiting stars is $d = 2 r \cos 30^{\circ} = \sqrt{3} r$ . The net inward force on orbiting stars is
$\frac{G m^{2}}{d^{2}} \cos 30^{\circ} + \frac{G M m}{r^{2}} + \frac{G m^{2}}{d^{2}} \cos 30^{\circ} = \frac{m v^{2}}{r}$
$∴ G [\frac{m}{\sqrt{3}} + M] = \frac{4 π^{2} r^{3}}{T^{2}} o r T = 2 p \sqrt{\frac{r^{3}}{G (M + \frac{m}{\sqrt{3}})}}$
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