# The Ledoux Criterion: a neat trick

Last updated 2017-05-07 03:27:43 SGT

Let $s$ be the entropy per unit mass of a spherical configuration of matter in hydrostatic equilibrium. Then consider its variation as a function of other thermodynamic variables:

where $T$, $P$, $\mu$ are respectively the temperature, pressure, and mean molecular weight (apologies to the statistical physicists). Defining the pressure scale height as $-P/H_p = {\mathrm d P \over \mathrm d R}$ and the constants $\delta, \phi$ through a differential equation of state

we proceed to rewrite the above as

but by the identity $\left.{\partial x \over \partial y}\right|_z \left.{\partial y \over \partial z}\right|_x \left.{\partial z \over \partial x}\right|_y = -1,$ we have that $\left.{\partial T \over \partial s}\right|_P \left.{\partial s \over \partial P}\right|_T = -\left.{\partial T \over \partial P}\right|_s,$ and $\left.{\partial \log T \over \partial \log \rho}\right|_{\log \mu} \left.{\partial \log \rho \over \partial \log \mu}\right|_{\log T} = -\left.{\partial \log T \over \partial \log \mu}\right|_{\log \rho} = -{\phi \over \delta}.$ This gives us (with $c_p = T \left.{\partial s \over \partial T}\right|_P$ as the specific heat capacity at constant pressure)

Defining (per classical stellar astrophysics convention) the “nabla” quantities $\nabla = {\mathrm d \log T \over \mathrm d \log P}, \nabla_\mathrm{ad} = \left.{\partial \log T \over \partial \log P}\right|_s, \nabla_\mu = {\mathrm d \log \mu \over \mathrm d \log P}$ allows us to rewrite this last expression as

Remarkably, when we compare this with the usual statement of Ledoux's criterion for stability against convection, which can be written as

we find that this is equivalent to a statement that, for stability against convection, we require a constraint on the specific entropy gradient:

I find this a great deal more illuminating than (often quite contrived) arguments about displaced fluid packets. Rather than their usual interpretation as temperature gradients when reasoning about the various $\nabla$-quantities, it appears that a more physically natural intuition about their significance should be as contributions to the entropy gradient, instead.