= Homologous adiabatic stellar stability
{title2=$E(x)=U_0x^{-3(\gamma-1)}+W_0x^{-1}$}
A mass-preserving <stellar homology> with scale $x$ has <mass density> proportional to $x^{-3}$. For an <adiabatic process> with constant <adiabatic index> $\gamma$, its <internal energy> scales as $x^{-3(\gamma-1)}$ and its <Newtonian gravitational potential energy> as $x^{-1}$. Writing $a=3(\gamma-1)$, stationarity at $x=1$ requires $W_0=-aU_0$, and the second scale derivative is $a(a-1)U_0$. Thus this particular displacement is restoring for $\gamma>4/3$, destabilizing for $1<\gamma<4/3$, and marginal at $\gamma=4/3$. This is a test of homologous radial stability; it does not by itself test every stellar displacement.
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