= Positive-square radial pulsation energy
{title2=$K=\frac13\int_0^Rpr^2[4r^2(\xi')^2+(3\gamma-4)(r\xi'+3\xi)^2]dr$}
Integrate the pressure-gradient term in the <weighted stellar pulsation Rayleigh quotient> by parts, with regular centre and zero surface pressure. Writing $\chi=r\xi'+3\xi$ uses $3\xi^2+2r\xi\xi'=(\chi^2-r^2(\xi')^2)/3$ and gives the displayed expression. Thus radial instability requires $\gamma<4/3$ somewhere in the positive-pressure interior. A constant trial function gives $K=3\xi^2\int_0^R(3\gamma-4)pr^2dr$, so strict radial stability requires this weighted integral to be positive, in particular $\gamma>4/3$ somewhere. When $\gamma=4/3$ throughout, constant $\xi$ is an exact zero-frequency homologous mode: the star is marginal, rather than strictly stable. Local threshold crossings alone do not determine stability for a mixed profile; the full <Rayleigh-Ritz variational principle> does.
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