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JEE MainPhysicsGravitation

Planet A orbits a star of mass M_A and planet B orbits a star of mass M_B . The ratio of the time periods of revolution of planet A to planet B is 3 3 . If the ratio of the masses of their respective stars is M_A M_B = 1 8 , the ratio of their orbital radii R_A R_B is:

Options

  1. A6
  2. B27 8
  3. C3 2
  4. D3

Correct answer

C. 3 2

Step-by-step solution

According to Kepler's third law, the time period of revolution T of a planet around a star of mass M in an orbit of radius R is given by T^2 = 4 ^2 R^3 GM . Rearranging this expression for the orbital radius R , we get: R^3 = G 4 ^2 T^2 M This implies that R (T^2 M)^ 1/3 . Taking the ratio for planets A and B : R_A R_B = ( ( T_A T_B )^2 M_A M_B )^ 1/3 Substitute the given values T_A T_B = 3 3 and M_A M_B = 1 8 : R_A R_B = ( (3 3 )^2 1 8 )^ 1/3 R_A R_B = ( 27 1 8 )^ 1/3 = ( 27 8 )^ 1/3 R_A R_B = 3 2 Answer: 3 2

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