A particle of mass m is under the influence of the gravitational...

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 $V_{\mathrm{c}}(r)=m \alpha / r^3$, where $\alpha$ is a positive constant of suitable dimensions and $r is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius$r_0$in the combined gravitational potential due to$M and V_{\mathrm{c}}(r)$, but with a new time period$T_1$, then \left(T_1^2-T_0^2\right) / T_1^2$ is given by [ $G$ is the gravitational constant.]