Let the mass of our sun be #M_s# and the distance of earth from the sun is #R#.
So by Newton's law of gravitation the force of attraction between them will be
#F = (GM_sm_e)/R^2#,
where #m_e# is the mass of earth and G is Gravitational constant.As this force of attraction provides the centrifugal force to revolve the earth around the sun with constant angular velocity #omega#, we can write
#m_eomega^2R=(GM_sm_e)/R^2#
If #T_e=(2pi)/omega# reprsents the time period of rotation of earth in its orbit.then the above relation becomes.
#m_eomega^2R=(GM_sm_e)/R^2#
#=>(4pi^2)/T_e^2=(GM_s)/R^3#
#=>T_e^2=(4pi^2R^3)/(GM_s)#
So the period of revolution of a planet #T_p# in circular orbit around a star of 4.0 solar masses #i.e.4M_s# will be given by the following relation, if the planet is at a distance from the star equal to that of the Earth from the Sun .
#=>T_p^2=(4pi^2R^3)/(4GM_s)=1/4T_e^2#
Hence
#T_p=1/2T_e#
