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Two stars of masses 3*Mₛ and 6*Mₛ are separated by a distance 9*R, where R is the mean Earth-Sun distance and Mₛ is the Sun's mass. They orbit their common center of mass in circular orbits with period n*T, where T is Earth's orbital period around the Sun. Find n.

1. n = 9
2. n = 18
3. n = 27
4. n = 36

Correct answer: n = 9

Solution

For a binary system separated by distance d with total mass M_total, both stars orbit the common center of mass with period T_bin = 2*pi*sqrt(d³/(G*M_total)). Here d = 9R and M_total = 3*Mₛ + 6*Mₛ = 9*Mₛ. Earth's period T = 2*pi*sqrt(R³/(G*Mₛ)). The ratio T_bin/T = sqrt[ (9R)³ / (9 Mₛ) * Mₛ / R³ ] = sqrt(729/9) = sqrt(81) = 9. So n = 9.

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