# MLT

These notes are mostly inspired from reading Cox & Giuli (“Principles of stellar structure”); some insights are from A.Maeder (“Physics, formation and evolution of rotating stars”) and Kippenhahn & Weigert (“Stellar Structure and Evolution”)

## Pressure Scale Height

In hydrostatic equilibrium we can define the total pressure scale height, $${H_{\mathrm{P}}}$$, as $-\frac{{{\mathrm d}}\ln {\mathrm{P}}}{{{\mathrm d}}r} \equiv \frac{1}{{H_{\mathrm{P}}}} = \frac{\rho g}{{\mathrm{P}}}$ where $${\mathrm{P}}$$ is the total pressure ($${\mathrm{P}}_{gas} + {\mathrm{P}}_{rad}$$). The pressure scale height is a measure of the distance over which the pressure changes by an appreciable fraction of itself.

We can define the gradients $$\nabla \equiv \frac{{{\mathrm d}}\ln {T}}{{{\mathrm d}}\ln {\mathrm{P}}}$$ and $$\nabla' \equiv \frac{{{\mathrm d}}\ln {T}'}{{{\mathrm d}}\ln {\mathrm{P}}}$$. Here $$\nabla$$ represents the average temperature gradient with respect to pressure of all matter at a given level, while $$\nabla'$$ is the temperature gradient with respect to pressure of a rising/falling fluid element. At first order the temperature excess of such fluid elements can therefore be written as $\Delta {T}(\Delta r) = {T}'(r + \Delta r) - {T}(r + \delta r) \simeq \Delta r \bigg[ \frac{{{\mathrm d}}{T}'}{{{\mathrm d}}r}-\frac{{{\mathrm d}}{T}}{{{\mathrm d}}r}\bigg].$ Assuming $${T}' \simeq {T}$$, i.e. the temperature is not changing drastically within the distance $$\Delta r$$, one can write $\Delta {T}(\Delta r) = \Delta r \, {T}\bigg[ -\frac{{{\mathrm d}}\ln {T}}{{{\mathrm d}}r}- \bigg(-\frac{{{\mathrm d}}\ln {T}'}{{{\mathrm d}}r}\bigg)\bigg],$ and using the assumption of pressure equilibrium, the definitions of pressure scale height $${H_{\mathrm{P}}}$$ and the gradients $$\nabla$$ and $$\nabla'$$ we obtain $\label{eq:deltat} \Delta {T}(\Delta r) = - \frac{{{\mathrm d}}\ln {\mathrm{P}}}{{{\mathrm d}}r} \Delta r \, {T}\bigg[ \frac{{{\mathrm d}}\ln {T}}{{{\mathrm d}}\ln {\mathrm{P}}}- \bigg(\frac{{{\mathrm d}}\ln {T}'}{{{\mathrm d}}\ln {\mathrm{P}}}\bigg)\bigg] = \Delta r \frac{{T}}{{H_{\mathrm{P}}}}\, (\nabla - \nabla').$ Note that in general the value of $$\nabla'$$ depends on the rate at which the moving fluid element is exchanging heat with its surroundings. However in the deep interiors of a star a good approximation is $$\nabla' = {\nabla_{\mathrm{\!ad}}}\equiv \big(\frac{{{\mathrm d}}\ln {T}}{{{\mathrm d}}\ln {\mathrm{P}}}\big)_{\textrm{ad}}$$, where $${\nabla_{\mathrm{\!ad}}}$$ is the temperature gradient of a fluid element moving adiabatically.
Since a non-zero temperature gradient always implies a radiative flux, in a convective region part of the energy is still going to be transported by radiation. This flux can be written as $\label{eq:radtransfer} F_{r} = -\frac{4ac}{3}\frac{T^3}{\kappa \rho}\frac{{{\mathrm d}}\ln T}{{{\mathrm d}}\ln {\mathrm{P}}} = \frac{4ac}{3}\frac{T^4}{\kappa \rho}\frac{\nabla}{{H_{\mathrm{P}}}}.$ Finally the fictitious (but computable) radiative gradient can be defined as the gradient of temperature required by radiation to carry the total stellar flux: $\label{eq:totalflux} F = F_{c} + F_{r} \equiv \frac{4ac}{3}\frac{T^4}{\kappa \rho}\frac{\nabla_r}{{H_{\mathrm{P}}}}.$
In a stellar convection zone where no energy is produced by nuclear reactions, the following inequalities are satisfied: $$\nabla_r > \nabla > \nabla' > {\nabla_{\mathrm{\!ad}}}$$