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Convective overshoot carries material from an unstable convecting region into an adjacent stably stratified region because its momentum does not vanish at the stability boundary. In stars this mixes material beyond the boundary determined by a local stellar convective stability criterion, changing stellar core masses and nuclear lifetimes.

The Ledoux criterion tests local adiabatic stellar convective stability in the presence of a composition gradient. Definewhere $\mu$ is the mean molecular weight and the partial derivatives describe the equation of state. For positive $\delta$, the stratification is stable to an infinitesimal composition-preserving adiabatic process displacement whenThe parcel adjusts its pressure to its surroundings, retains its composition and changes temperature along the adiabatic temperature gradient. Comparing its resulting mass density with the ambient value determines the restoring buoyancy. For positive $\varphi /\delta$, increasing mean molecular weight inward gives ${\nabla }_{\mu }>0$ and an additional stabilizing contribution. With uniform composition the condition reduces to the Schwarzschild criterion. Thermal or compositional diffusion introduces further instability questions beyond this adiabatic parcel test.

Consider a small outward displacement $\delta r$ of a fluid element. It adjusts to the ambient pressure while retaining its specific entropy and stellar composition, so an adiabatic process gives $d\log {\rho }_{\mathrm{parcel}}=d\log P/\gamma$. Its excess mass density over the new surroundings is, to first order,Restoring buoyancy requires this bracket to be positive. With $\gamma =5/3$, the Schwarzschild criterion is thereforeEquality is neutral stability. For uniform mean molecular weight, the ideal gas law gives $d\log \rho =d\log P-d\log T$. Since hydrostatic equilibrium pressure decreases outward, the inequality is equivalent to $\nabla =d\log T/d\log P<{\nabla }_{\mathrm{ad}}=2/5$. Dividing the radiative diffusion in a star and hydrostatic equilibrium stellar structure equations givesThis is the stellar radiative temperature gradient test for stellar convective stability.

In the thin upper stellar atmosphere take $m\simeq M$, $L_r\simeq L$ and use gas pressure dominance. The prescribed opacity becomes $\kappa ={\kappa }_0\mu P{T}^{12}/\mathcal{R}$. Dividing the two structure equations then yieldsAt zero optical depth, the atmospheric boundary has $T_s^4=T_e^4/2$ and negligible gas pressure. Integrating from that boundary, with $T_s^{-8}=4T_e^{-8}$, givesTo locate the onset, put $y=T_e^8/T^8$. Logarithmic derivative gives $d\log P/d\log T=4y/(4-y)$, henceThis starts at zero at the outer boundary and increases inward. It first reaches $2/5$ atThe corresponding optical depth is ${\tau }_b=\frac{4}{3}(\sqrt{13/20}-\frac{1}{2})>0$, so the onset lies inside the stellar atmosphere. This is grey-atmosphere convection onset with thirteenth-power opacity.

Let the interior of the fully convective star have $P=K_T{T}^{5/2}$, with $K_T$ spatially constant. At its boundary $T_b/T_e$ is fixed, so the atmospheric pressure formula givesThe Stefan–Boltzmann law, $L=4\pi R^2\sigma T_e^4$ with $\sigma =ac/4$, consequently impliesThe ideal gas equation transforms the interior relation intoFor completeness, the mass-radius scaling follows by taking $\rho ={\rho }_c{\theta }^{3/2}$ and $r=\alpha \xi$ with ${\alpha }^2=5K_{\rho }{\rho }_c^{-1/3}/(8\pi G)$. Hydrostatic equilibrium and mass conservation reduce to the Lane-Emden equation $({\xi }^2{\theta }^{\prime }{)}^{\prime }=-{\xi }^2{\theta }^{3/2}$, with $\theta (0)=1$, ${\theta }^{\prime }(0)=0$. Its fixed dimensionless profile gives $R\propto \alpha$ and $M\propto {\alpha }^3{\rho }_c$. Eliminating ${\rho }_c$ gives $R\propto K_{\rho }{G}^{-1}{M}^{-1/3}$, or $K_{\rho }\propto G{M}^{1/3}R$. Thus $K_T^2\propto M^{-1}{R}^{-3}$ at fixed stellar composition. Equating this with the atmospheric expression gives $L^{17/4}\propto M^2{R}^{19/2}$, and henceThis Hayashi relation with thirteenth-power opacity concerns the interior of a fully convective star beneath a thin stellar atmosphere. The coefficient $K_T$ may vary between stars; spatially constant specific entropy does not mean equal specific entropy across the whole sequence.

Radiative transport supplies the temperature gradient used below. A purely radiative equilibrium model also needs stellar convective stability; the transport assumption alone is not a general proof of stability.

Let $g=Gm/r^2>0$, $H_P=-dr/d\log P>0$, $\nabla =d\log T/d\log P$, and ${\nabla }_{\mu }=d\log \mu /d\log P$. Consider a small radial fluid displacement $\xi$ whose sound-crossing time is short enough to keep its pressure equal to its surroundings. Its composition is frozen and its heat exchange negligible. This is the local stellar convective stability test.

For gas-pressure-dominated ideal gas matter, $\rho \propto \mu P/T$. The ambient gradient is thereforeThe displaced element conserves specific entropy and mean molecular weight, giving $(d\log \rho /dr{)}_{\mathrm{parcel}}=-(1-{\nabla }_{\mathrm{ad}})/H_P$, where ${\nabla }_{\mathrm{ad}}=({\Gamma }_2-1)/{\Gamma }_2$. The parcel-minus-environment mass density difference isIts buoyancy acceleration is minus $g$ times this difference. ThusPositive stellar buoyancy frequency squared gives a restoring force. The gas-pressure-dominated Ledoux criterion is thereforeEquality is marginal in this ideal adiabatic test. For monatomic gas ${\nabla }_{\mathrm{ad}}=2/5$. An inward increase of mean molecular weight has ${\nabla }_{\mu }>0$ and stabilizes the layer; uniform composition recovers the Schwarzschild criterion. For a radiative layer one substitutes the stellar radiative temperature gradient, ${\nabla }_{\mathrm{rad}}=3\kappa LP/(16\pi acGm{T}^4)$.

More generally define $\delta =-(\partial \log \rho /\partial \log T{)}_{P,\mu }$ and $\phi =(\partial \log \rho /\partial \log \mu {)}_{P,T}$. The same displacement argument gives $N^2=(g/H_P)[\delta ({\nabla }_{\mathrm{ad}}-\nabla )+\phi {\nabla }_{\mu }]$ and stability for $\nabla <{\nabla }_{\mathrm{ad}}+(\phi /\delta ){\nabla }_{\mu }$. Both coefficients equal one in the gas-dominated ideal-gas limit used above. This is local dynamical stability against convection; it is distinct from global radial collapse and from instabilities requiring heat or composition diffusion.

The stellar adiabatic exponents describe how pressure $P$, temperature $T$ and mass density $\rho$ respond along an isentropic process at fixed chemical composition. With specific entropy $s$,The chain rule gives ${\Gamma }_3-1={\Gamma }_1({\Gamma }_2-1)/{\Gamma }_2$. Thus the three names do not denote three independent thermodynamic responses. The adiabatic temperature gradient is ${\nabla }_{\mathrm{ad}}=({\Gamma }_2-1)/{\Gamma }_2$, while ${\Gamma }_1$ determines the squared adiabatic sound speed $c_s^2={\Gamma }_{1}P/\rho$.^[ref]

For a fixed-composition monatomic ideal gas, all three exponents equal $5/3$ and ${\nabla }_{\mathrm{ad}}=2/5$. For equilibrium blackbody radiation, all three equal $4/3$ and ${\nabla }_{\mathrm{ad}}=1/4$. In a gas-radiation mixture they generally differ; the adiabatic exponents of a monatomic gas-radiation mixture keep that distinction explicit. Partial ionization can soften the compressional response because some work changes ionization rather than immediately raising the temperature. The relevant exponent in stellar convective stability therefore need not equal a constant ideal-gas specific-heat ratio.