Suppose the gravitational force varies inversely as the n th power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to

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Gravitational force, F ∝ r n 1 ​ F = r n k ​ where k is a constant. For a planet, moving in a circular orbit of radius R , F = R n k ​ But, F = m ω 2 R ​ ⇒ R n k ​ = m R ⋅ ( T 2 π ​ ) 2 ⇒ R n + 1 k ​ = T 2 m ( 2 π ) 2 ​ ⇒ T 2 ∝ R n + 1 ∴ T ∝ R 2 n + 1 ​ . ​

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