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A binary star system has two stars of masses M and 2M separated by a distance R between their centres. Both stars revolve around their common centre of mass under mutual gravitational attraction. Which of the following is/are correct?
- The heavier star revolves in an orbit of radius R/3
- Both stars revolve with the same time period equal to 2*pi*sqrt(R³ / (3*G*M))
- The kinetic energy of the lighter star is twice that of the heavier star
- The kinetic energy of the lighter star is 4 times that of the heavier star
Correct answer: The heavier star revolves in an orbit of radius R/3
Solution
The heavier star (2M) is at distance R/3 from the COM; both share the same period T = 2*pi*sqrt(R³/(3*G*M)); the KE ratio KE(M)/KE(2M) = (1/2*M*v_M²)/(1/2*2M*v₂M²) = (M*(omega*2R/3)²)/(2M*(omega*R/3)²) = (4/9)/(2/9)*1/2 = 2*1/2 = 1... let me recalculate carefully.
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