Nonradial linear stability of liquid Lane-Emden stars

The classical model of a star is the Lane-Emden star with dynamics governed by the Euler-Poisson equations. We consider the case of a liquid star with a "stiffened gas" equation of state $p=\rho^\gamma-1$. We derive the full 3D linearised Euler-Poisson system around liquid Lane-Emden stars with no symmetry assumptions on the perturbations and show that the associated linear operator $\mathbf L$ is non-negative whenever the radial mode is non-negative. We show that $\mathbf L$ has an infinite-dimensional kernel each element of which corresponds to a linearly growing solution to the linearised system. When restricted to irrotational perturbations and modding out the three kernel elements corresponding to momentum conservation, however, we prove that $\mathbf L$ is strictly positive with coercivity bound $\langle\mathbf L\boldsymbol\theta,\boldsymbol\theta\rangle_{\bar\rho}\gtrsim\|\boldsymbol\theta\|_{L^2(B_R)}^2$. Hence we demonstrate that the liquid Lane-Emden stars are stable against non-radial irrotational perturbations whenever the purely radial mode is stable, improving upon previous results that dealt only with purely radial perturbations. However, the stability might not be as strong as one might hope, as we prove that $\|\nabla\boldsymbol\theta\|_{L^2(B_R)}^2$ cannot be controlled even in this case.