Linear Stability of liquid Lane-Emden stars

We establish various qualitative properties of liquid Lane-Emden stars in $\mathbb{R}^d$, including bounds for its density profile $\rho$ and radius $R$. Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when $\gamma\geq 2(d-1)/d$; linearly stable when $\gamma<2(d-1)/d$ for stars with small relative central density $\rho(0)-\rho(R)$; and linearly unstable when $\gamma<2(d-1)/d$ for stars with large central density. Such dependence on central density is not seen in the gaseous Lane-Emden stars.