Homologous adiabatic stellar stability ($E(x)=U_0{x}^{-3(\gamma -1)}+W_0{x}^{-1}$)

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 $W_0=-a{U}_0$, 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.