Pressure-weighted radial instability criterion
= Pressure-weighted radial instability criterion
{title2=$\int_0^{R_s}r^2p(4-3\gamma)\,dr>0$}
The homologous trial function $\xi=1$ in the <weighted stellar pulsation Rayleigh quotient> has numerator $-3\int_0^{R_s}r^2p(4-3\gamma)dr$. Positivity of that integral therefore proves radial instability. It is a sufficient test: a nonpositive integral does not guarantee stability against every radial shape. For constant $\gamma$, the homologous threshold is $4/3$.