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A uniform-density perfect-gas sphere supported by decreasing pressure has buoyancy frequency squared $N^2=-2{\omega }_d^2{r}^2/[\gamma (R^2-r^2)]$ in the interior. Its specific entropy decreases outward and adiabatic displacements are buoyantly unstable. For the nonradial polynomial family, the product of the two normalized squared mode frequencies is $-l(l+1)$, displaying one growing branch even when radial compression modes are stable.

For an ideal gas with constant adiabatic index $\gamma >1$, let $c_v$ and $c_p$ be specific heat capacities and $\mathcal{R}=c_p-c_v$. The ideal gas law and the internal energy of an ideal gas give $P=\rho \mathcal{R}T$ and specific internal energy $e=c_{v}T$. Since $\gamma =c_p/c_v$, the internal energy per unit volume is $\rho e=P/(\gamma -1)$. Consequently a star with spherical symmetry hasThe radial integral requires the volume measure $dV=4\pi r^{2}dr$: the first printed expression omits this factor if its differential is read literally as $dr$. The mass-coordinate expression confirms the required volume interpretation.

Define the enclosed mass $M(r)$, total mass $M_{\mathrm{tot}}$ and dimensionless shell label $m=M(r)/M_{\mathrm{tot}}$. Mass conservation gives $M_{\mathrm{tot}}dm=dM=\rho dV$, so the change of variables givesHere $m$ labels a fixed material shell rather than its radial position. In a mass-preserving stellar homology, $r(m)=x{r}_0(m)$ with $x=R/R_0$. Both a shell's radius and thickness gain a factor $x$, so its volume gains $x^3$ and its mass density becomes $\rho (m)=x^{-3}{\rho }_0(m)$. An adiabatic process keeps $P/{\rho }^{\gamma }$ fixed separately in each shell, even if that constant differs between shells. Thus $P=x^{-3\gamma }{P}_0$ and $P/\rho =x^{-3(\gamma -1)}{P}_0/{\rho }_0$. Integration at fixed mass labels provesThe spherical shell theorem gives the Newtonian gravitational potential energy by assembling shells against the mass already inside them:Every radius gains the same factor $x$, while $M$ and $dM$ remain fixed. HenceThis is the homologous adiabatic stellar stability test. The derivative with respect to the physical stellar radius is ${\partial }_R=R_0^{-1}{\partial }_x$. Stationarity at $x=1$ therefore requires $-a{U}_0-W_0=0$, orAfter imposing this equilibrium relation, $E/U_0=x^{-a}-a/x$ andSince $a>0$, a strict energy minimum under this homologous displacement requiresFor $a>1$, contraction sends $E$ to $+\infty$, expansion sends it to $0$ from below, and $x=1$ is the unique minimum, with $E(1)=(1-a)U_0<0$. For $0<a<1$, contraction sends $E$ to $-\infty$, expansion sends it to $0$ from above, and $x=1$ is the unique maximum. When $a=1$, the equilibrium relation makes $E(x)=0$ for every $x$: the scale direction is marginal, not strictly stable. The plotted curves display these three possibilities. This checks homologous radial stability; stability against every stellar displacement needs a fuller perturbation analysis.