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”)

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: \(\nabla_r > \nabla > \nabla' > {\nabla_{\mathrm{\!ad}}}\)