Time period of revolution of a planet depends on radius of the orbit are mass of sun (M) and universal gravity (G) prove dimensions that T square Alpha R cube ?

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$L e t t i m e p e r i o d o f r e v o l u t i o n o f a s a t e l l i t e a r o u n d t h e s u n d e p e n d s o n a^{t h} p o w e r o f R, b^{t h} o f m a s s o f s s u n a n d c^{t h} p o w e r o f u n i v e r s a l c o n s \tan t o f g r a v i t a t i o n . S o, T α R^{a} M^{b} G^{c} B u t d i m e n s i o n a l f o r m u l a o f T, R, M a n d G a r e T^{1}, L^{1}, M^{1} a n d M^{- 1} L^{3} T^{- 2} r e s p e c t i v e l y . P u t t i n g t h e s e v a l u e s i n a b o v e e q n, w e o b t a i n, T^{1} α L^{a} M^{b} {(M^{- 1} L^{3} T^{- 2})}^{c} O R T^{1} α L^{a + 3 c} M^{b - c} T^{- 2 c} c o m p a r i n g p o w e r s o f M, L a n d T o n b o t h s i d e s o f p r o p o r t i o n a l s i g n . - 2 c = 1 o r c = - 1 / 2 . b - c = 0 o r b = c = - 1 / 2 a n d a + 3 c = 0 o r a = - 3 c = - (- 3 / 2) = 3 / 2 . p u t t i n g a l l t h e s e s u b s t i t u t i o n i n a b o v e f o r m u l a, w e g e t, T α R^{3 / 2} M^{- 1 / 2} G^{- 1 / 2} . W h e n w e c o n s i d e r M a n d G c o n s \tan t t h e n T α R^{3 / 2} O R T^{2} = R^{3} .$

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