Astronomy and astrophysics Stellar structure Proposition

Ledoux criterion

Builds on Schwarzschild criterion for convection Hydrostatic equilibrium Brunt–Väisälä frequency

Statement

Consider a layer of a star in hydrostatic equilibrium, in which the mean molecular weight $\mu$ may vary with depth. Let $P$ , $T$ and $\rho$ be the pressure, the temperature and the density, and let $\nabla = d \ln T / d \ln P$ and $\nabla_\mu = d \ln \mu / d \ln P$ be the actual gradients of the layer. Let $\nabla_{\mathrm{ad}}$ be the adiabatic gradient, and let $\delta = -(\partial \ln \rho / \partial \ln T)_{P, \mu}$ and $\varphi = (\partial \ln \rho / \partial \ln \mu)_{P, T}$ . Suppose that a small displaced parcel moves adiabatically, keeps its composition, and stays in pressure balance with its surroundings. Then the layer is stable against convection if $\nabla < \nabla_{\mathrm{ad}} + \frac{\varphi}{\delta} \nabla_\mu,$ and unstable if the opposite strict inequality holds.

Remarks

The argument is that of the Schwarzschild criterion, but now a rising parcel carries its own composition. So the densities of the parcel and of its surroundings also differ because of $\mu$ . For an ideal gas, $\delta = \varphi = 1$ , and the condition of stability reads $\nabla < \nabla_{\mathrm{ad}} + \nabla_\mu$ . Example: around a core that has burnt part of its hydrogen, the heavier helium-rich gas lies deeper, so $\mu$ increases inward, $\nabla_\mu > 0$ , and the composition gradient stabilizes the layer. When $\nabla_\mu = 0$ the criterion reduces to the Schwarzschild criterion. Layers with $\nabla_{\mathrm{ad}} < \nabla < \nabla_{\mathrm{ad}} + (\varphi/\delta) \nabla_\mu$ are stable by Ledoux but unstable by Schwarzschild; in them slow semiconvective mixing can occur. The criterion is equivalent to the sign condition $N^2 > 0$ on the Brunt–Väisälä frequency including the composition term.

Going deeper

For a student

The difference between the two criteria matters in the regions left behind by a shrinking or growing convective core, where hydrogen and helium are layered. Which criterion to use, and how fast semiconvection mixes, changes the computed lifetimes and tracks of massive stars. For a uniform composition both criteria agree.

Reading path

To understand this page from the ground up, read these 88 pages in order. The foundations come first.

Brunt–Väisälä frequency concept
Derivative concept
Differential equation concept
Euclidean space object
Euclidean vector concept
Euler equations concept
Function concept
Interval concept
Lane–Emden equation concept
Limit point of a set definition
Local maximum definition
Ordinary differential equation concept
Partition of an interval definition
Poisson's equation concept
Sequence definition
Velocity concept
Fréchet derivative definition
Cross product definition
Ordinary differential equation definition
Function of several variables concept
Vector-valued function concept
Lipschitz condition definition
Classical mechanics concept
Limit of a function definition
Local minimum definition
Mesh of a partition definition
Riemann sum definition
Real sequence definition
Subsequence definition
Limit of a function of several variables concept
Momentum concept
Partial derivative concept
Solution of an ordinary differential equation definition
Inertial frame of reference concept
Continuity at a point definition
Derivative definition
Riemann integrable function definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Initial value problem definition
Newton's laws of motion statement
Continuous function definition
Antiderivative definition
Riemann integral definition
Fermat's theorem on stationary points statement
Convergent sequence definition
Monotone convergence theorem for sequences statement
Every sequence has a monotone subsequence statement
Differentiable on an open set definition
Gradient definition
Higher partial derivatives definition
Picard–Lindelöf hypotheses definition
Intermediate value theorem statement
Definite integral definition
Bolzano–Weierstrass theorem statement
Iterated derivative definition
Picard operator definition
Additivity of the integral over intervals statement
Boundedness theorem statement
Monotonicity of the integral statement
C^k function characterised by continuous derivatives statement
Integral equation form of an initial value problem statement
An iterate of the Picard operator is a contraction statement
Extreme value theorem statement
Vector field definition
Picard–Lindelöf theorem statement
Mean value theorem for integrals statement
Rolle's theorem statement
Conservative vector field definition
Mean value theorem statement
First fundamental theorem of calculus statement
Classical mechanics definition
Potential energy definition
Momentum definition
Second fundamental theorem of calculus statement
Kinetic energy definition
Angular momentum definition
Fundamental theorem of calculus statement
Central force definition
Virial theorem statement
Newton's law of universal gravitation statement
Hydrostatic equilibrium statement
Schwarzschild criterion for convection statement
Ledoux criterion statement

Tag	`CVZBK60C26`
URI	`m2://CVZBK60C26`
Kind	statement, proposition
Writing	W0 stub
MSC	85A15, 76E06, 76E20
Status	This copy of the site has no build data. Run `m2 status CVZBK60C26`.
About	Schwarzschild criterion for convection
Uses	Schwarzschild criterion for convection, Hydrostatic equilibrium, Brunt–Väisälä frequency
Changed	3 November 2026