 Weighted stellar pulsation Rayleigh quotient

For radial adiabatic stellar oscillations, the quotient is $\mathcal{R}[\xi ]=\int [\gamma p{r}^4|\xi \prime {|}^2+r^3((4-3\gamma )p)\prime |\xi {|}^2]dr/\int \rho r^4|\xi {|}^{2}dr$. Regular-centre and free-surface endpoint conditions remove the boundary term. The infimum is the fundamental squared frequency; any negative trial value proves radial instability. A variable stellar adiabatic exponent must remain inside the derivative.