Radial stability of a uniform-density star

For the even polynomial family of degree $n\ge 2$, the physical radial squared frequency is $[\gamma n(n+1)/2-4]{\omega }_d^2$. The fundamental $n=2$ mode is stable when $\gamma >4/3$ and marginal at $\gamma =4/3$. At angular degree zero the gradient of the regular solid harmonic vanishes, so the auxiliary displacement amplitude multiplying it is redundant.