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For the polynomial stellar-mode coefficient reduction, write the highest coefficients asSet $x={\omega }^2/{\omega }_d^2$, $L=n(2l+n+1)$ and $k=l+n+1$. The highest powers in the equations of (b) giveThe factor $3$ follows from $4\pi G\rho =3{\omega }_d^2$; the positive $\gamma d/2$ term comes from the highest, negative $r^2$ coefficient of the equilibrium pressure.

For $l>0$, the final spectrum found below has $x\ne 0$. Eliminating $u,v$ first givesThe remaining pressure relation isA nontrivial leading coefficient has $d\ne 0$, soThusor, in dimensional form,

For the printed stability continuation, define $H=\gamma L/2-4$. The two branches for nonradial stellar oscillations areFor every $l>0$, their product is $-l(l+1)<0$. Therefore one branch oscillates and the other grows exponentially, for every positive $\gamma$. With the chosen time dependence, $x_{-}<0$ gives $\omega =i{\omega }_d\sqrt{-x_{-}}$ and growth rate ${\omega }_d\sqrt{-x_{-}}$. Increasing $\gamma$ does not remove this nonradial instability. At large $n$ or large $\gamma$, $x_{+}\sim \gamma L/2-4$ is a stiff compressive branch, while $x_{-}\sim -l(l+1)/(\gamma L/2-4)$ is a slower unstable buoyancy branch.

The buoyancy frequency supplies the physical explanation. The background density has zero gradient, but pressure decreases outward:The specific entropy of the perfect gas decreases outward, because $p/{\rho }^{\gamma }$ does. An adiabatically displaced fluid parcel therefore experiences destabilizing buoyancy. The model has convective instability of a uniform-density star, even when its radial compression modes are stable. The divergence of this expression at the zero-pressure surface also reflects the artificial equilibrium; the interior negative sign is already decisive.

For $l=0$, the formal quadratic factors asThe physical radial displacement has no $V\nabla F$ component. Using its equations directly givesIndeed $d=(n+1)u$, $j=-3u/n$, and $h=u+\gamma (n+1)u/2$ yield this result directly in the radial momentum equation. The extra identically zero factor in the two-component determinant need not represent an independent radial mode, since $V$ is redundant when $l=0$.

The lowest allowed degree, $n=2$, has ${\omega }_{\mathrm{radial}}^2=(3\gamma -4){\omega }_d^2$. Hence all these radial modes are stable for $\gamma >4/3$; the fundamental radial mode is marginal at $\gamma =4/3$ and unstable below it. Higher $n$ have their individual thresholds $\gamma =8/[n(n+1)]$. These radial stability of a uniform-density star thresholds coexist with nonradial convective instability. They are conclusions within the polynomial interior family assumed in the question; no boundary-condition quantization is being asserted.