The distance between two stars of masses and is . Here is the mean distance between the centers of the Earth and the Sun, and is the mass of the Sun. [2021]

Question

The distance between two stars of masses $3 M_{s}$ and $6 M_{s}$ is $9 R$ . Here $R$ is the mean distance between the centers of the Earth and the Sun, and $M_{s}$ is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period $n T$ , where $T$ is the period of Earth's revolution around the Sun. The value of $n$ is ______. [2021]

Smart Achievers · Accepted Answer

(9)

The centre of mass lies at a distance $6 R$ from lighter mass

In circular orbit,

Time Period, $T = 2 π \sqrt{\frac{R^{3}}{G M_{s}}}$

Binary stars system

$n T = 2 π \sqrt{\frac{{(9 R)}^{3}}{G (3 M_{s} + 6 M_{s})}}$

$or, n \times 2 π \sqrt{\frac{R^{3}}{G M_{s}}} = 9 \times 2 π \sqrt{\frac{R^{3}}{G M_{s}}}$

$[∵ T = 2 π \sqrt{\frac{R^{3}}{G M_{s}}}]$

$∴$ $n = 9$

Solution

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Solution

Ans. (9) The centre of mass lies at a distance 6R from lighter mass In circular orbit, Time Period, T=2πR3GMs Binary stars system nT=2π(9R)3G(3Ms+6Ms) or, n×2πR3GMs=9×2πR3GMs [∵T=2πR3GMs] ∴ n=9

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