JEE Advanced 2021 Paper 2 · Q18 · Kepler's Laws
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 centre 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 _____.
Reveal answer + step-by-step solution
Correct answer:9
Solution
Total mass $M_{\text{tot}} = 9 M_S$, separation $d = 9R$. For a binary system, Kepler's third law: $T_{\text{bin}}^2 = 4\pi^2 d^3/[G(M_1 + M_2)] = 4\pi^2 (9R)^3/[G \cdot 9 M_S] = 4\pi^2 R^3 \cdot 729/(9 G M_S) = 81 \cdot 4\pi^2 R^3/(G M_S)$. Earth-Sun: $T^2 = 4\pi^2 R^3/(G M_S)$. So $T_{\text{bin}}^2 = 81 T^2$, $T_{\text{bin}} = 9 T$. Hence $n = 9$.
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