Linear stability of liquid Lane-Emden stars

We establish various qualitative properties of liquid Lane-Emden stars in R d \mathbb {R}^d , including bounds for its density profile ρ \rho and radius R R . Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when γ ≥ 2 ( d − 1 ) / d \gamma \geq 2(d-1)/d ; linearly stable when γ > 2 ( d − 1 ) / d \gamma >2(d-1)/d for stars with small relative central density ρ ( 0 ) − ρ ( R ) \rho (0)-\rho (R) ; and linearly unstable when γ > 2 ( d − 1 ) / d \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.