Suppose that in studying a particular globular cluster containing \(10^{5}\) stars, whose average mass is that of our sun, astronomers find that the total kinetic energy of the stars is 10 times that of the magnitude of their total potential energy.

(a) Estimate the amount of dark matter in the cluster, expressed in solar masses.

(b) Then assume there is no such thing as dark matter, but that the potential energy between two stars has the form \(U(r)=-\alpha r^{n}\) where \(n eq-1\). Is there a value of \(n\) such that the virial theorem would be satisfied without dark matter?