Stellar convective instability

Stellar convective instability occurs when an adiabatic process displaces a fluid element and buoyancy amplifies the displacement. For uniform composition the Schwarzschild criterion gives instability when the actual temperature gradient exceeds the adiabatic temperature gradient; composition gradients instead require the Ledoux criterion.

Consider a spherical uniform-density stellar model of mass density $\rho$, radius $R$, negligible surface pressure and a fixed-composition ideal gas with adiabatic index $\gamma >1$. Define ${\omega }_d^2=4\pi G\rho /3$, so the inward gravitational acceleration has magnitude $g(r)={\omega }_d^{2}r$ and $P(r)=\rho {\omega }_d^2(R^2-r^2)/2$.

For an adiabatically displaced parcel in pressure balance with its surroundings, the squared buoyancy frequency isA constant background mass density does not mean an incompressible parcel: its pressure and temperature change during the displacement. The negative $N^2$ means a rising parcel is lighter than its surroundings and continues to rise. Equivalently, the ideal-gas specific entropy decreases outward.

Thus this thermally stratified equilibrium has stellar convective instability even if a separate test of homologous adiabatic stellar stability finds restoring radial compression for $\gamma >4/3$. Mechanical equilibrium, radial compression stability and buoyant stability answer different questions. The singular near-surface expression also reflects the zero-pressure idealized boundary and should not be extrapolated to a realistic photosphere unchanged.