= Convective instability of a uniform-density star
A uniform-density perfect-gas sphere supported by decreasing pressure has <buoyancy frequency> squared $N^2=-2\omega_d^2r^2/[\gamma(R^2-r^2)]$ 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.
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