The Ledoux Criterion: a neat trick

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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:

${\mathrm d s \over \mathrm d r} = \left.{\partial s \over \partial T}\right|_{P, \mu} {\mathrm d T \over \mathrm d r} + \left.{\partial s \over \partial P}\right|_{T,\mu} {\mathrm d P \over \mathrm d r} + \left.{\partial s \over \partial \mu}\right|_{P,T} {\mathrm d \mu \over \mathrm d r},$

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

${\mathrm d \rho \over \rho} \sim \alpha {\mathrm d P \over P} - \delta {\mathrm d T \over T} + \phi {\mathrm d \mu \over \mu},$

we proceed to rewrite the above as

$\begin{aligned}{\mathrm d s \over \mathrm d r} &= -{P \over H_p} \left(\left.{\partial s \over \partial T}\right|_{P,\mu} {\mathrm d T \over d P} + \left.{\partial s \over \partial \mu}\right|_{P,T} {\mathrm d \mu \over d P} + \left.{\partial s \over \partial P}\right|_{T,\mu}\right) \\ &= -{P \over H_p} \left.{\partial s \over \partial T}\right|_{P,\mu} \left({\mathrm d T \over d P} + \left.{\partial T \over \partial s}\right|_{P,\mu}\left.{\partial s \over \partial \mu}\right|_{P,T} {\mathrm d \mu \over d P} + \left.{\partial T \over \partial s}\right|_{P,\mu}\left.{\partial s \over \partial P}\right|_{T,\mu}\right),\end{aligned}$

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)

$\begin{aligned}{\mathrm d s \over \mathrm d r} &= -{P \over H_p} {\partial s \over \partial T} \left({\mathrm d T \over d P} - \left.{\partial T \over \partial P}\right|_s - \left.{\partial T \over \partial \mu}\right|_s {\mathrm d \mu \over \mathrm d P} \right)\\ &= -{T \over H_p} {\partial s \over \partial T} \left({P \over T}{\mathrm d T \over d P} - {P \over T}\left.{\partial T \over \partial P}\right|_s - {P \over T}{T \over \mu}\left.{\partial \log T \over \partial \log \mu}\right|_s {\mathrm d \mu \over \mathrm d P} \right) \\ &= -{c_p \over H_p} \left({P \over T}{\mathrm d T \over d P} - {P \over T}\left.{\partial T \over \partial P}\right|_s - {\phi \over \delta}{P \over \mu}{\mathrm d \mu \over \mathrm d P} \right).\end{aligned}$

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

${\mathrm d s \over \mathrm d r} = -{c_p \over H_p} \left(\nabla - \nabla_\mathrm{ad} - {\phi \over \delta}\nabla_\mu \right).$

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

$\nabla < \nabla_\mathrm{ad} + {\phi \over \delta}\nabla_\mu,$

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

$\boxed{{\mathrm d s \over \mathrm d r} > 0.}$

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.