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JEE MainPhysicsGravitation

For a system of planets orbiting a distant star, a graph is plotted with the square of the orbital time period ( T^2 ) on the y-axis and the cube of the orbital radius ( R^3 ) on the x-axis. The resulting graph is a straight line passing through the origin with a slope S . If G is the universal gravitational constant, the mass of the star is :

Options

  1. AGS 4 ^2
  2. B4 ^2 S G
  3. C4 ^2 GS
  4. DS G

Correct answer

C. 4 ^2 GS

Step-by-step solution

According to Kepler's Third Law, the time period T of a planet in a circular orbit of radius R around a star of mass M is given by: T^2 = 4 ^2 GM R^3 The equation of the given straight line is T^2 = S R^3 . Comparing the two equations, the slope S is: S = 4 ^2 GM Rearranging for the mass M of the star: M = 4 ^2 GS Answer: 4 ^2 GS

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