 Linear adiabatic stellar oscillation equations

For a static spherical star, a fluid displacement $\mathbf{\xi }{e}^{-i\omega t}$ gives ${\rho }_1=-\nabla \cdot (\rho \mathbf{\xi })$, $-{\omega }^2\rho \mathbf{\xi }=-\nabla p_1-{\rho }_1\nabla \Phi -\rho \nabla {\varphi }_1$ and ${\nabla }^2{\varphi }_1=4\pi G{\rho }_1$. The adiabatic relation is $\Delta P=-{\Gamma }_{1}P\nabla \cdot \mathbf{\xi }$. Centre regularity and surface mechanical/gravitational boundary conditions select discrete eigenvalues in a conservative model.