The external gravitational potential $\Phi(r)$ due to a thin spherical shell of radius $a$ and mass per unit area $\sigma$, centred at $r=0$, will equal the gravitational potential due to a point mass $M$ at $r=0$, at any distance $r>a$, provided

$$\frac{M r \Phi(r)}{2 \pi \sigma a}+K(a) r=\int_{r-a}^{r+a} R \Phi(R) d R$$

where $K(a)$ depends on the radius of the shell. For which values of $q$ does this equation have solutions of the form $\Phi(r)=C r^{q}$, where $C$ is constant? Evaluate $K(a)$ in each case and find the relation between the mass of the shell and $M$.

Hence show that the general gravitational force

$$F(r)=\frac{A}{r^{2}}+B r$$

has a potential satisfying $(*)$. What is the cosmological significance of the constant $B$ ?