A particle of mass \( m \) is under the influence of the gravitational field of a body of mass \( M \) \( ( \gg m ) \). The particle is moving in a circular orbit of radius \( r_0 \) with time period \( T_0 \) around the mass \( M \). Then, the particle is subjected to an additional central force, corresponding to the potential energy \( V_c(r) = \dfrac{m\alpha}{r^3} \), where \( \alpha \) is a positive constant of suitable dimensions and \( r \) is the distance from the centre of the orbit. If the particle moves in the same circular orbit of radius \( r_0 \) in the combined gravitational potential due to \( M \) and \( V_c(r) \), but with a new time period \( T_1 \), then \( \dfrac{(T_1^2 - T_0^2)}{T_1^2} \) is given by: \([G \text{ is the gravitational constant.}]\)
