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Uniform circular motion:

We know that centripetal acceleration for a satellite of mass m  going around around a planet of mass M, in a circular orbit of radius R , with a uniform velocity v :

      m v² / R  =  G M m / R²
         =>  v = √ [GM/R]  = R ω

        => ω =  √[ GM / R³ ]
       =>  T = 2π √[ R³/GM ]
               = K √[ R³/M ]    ,  K is a constant

  mass =     M = d 4π/3 * R³
   =>    R³/M = 3/(4πd)

   =>  T  =  K '  √ [ 1/d ]
So the time period of a satellite going around a planet is inversely proportional to the square root of the density of the planet.

     Hence, the time period remains same....  as  T.

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