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\begin{equation}  \Delta \T (\Delta r) = - \frac{\D \ln \P}{\D r} \Delta r \, \T \bigg[ \frac{\D \ln \T}{\D \ln \P}- \bigg(\frac{\D \ln \T'}{\D \ln \P}\bigg)\bigg] = \Delta r \frac{\T}{\hp}\, (\nabla - \nabla').  \end{equation}  Note that in general the value of $\nabla'$ depends on the rate at which the moving fluid element is exchanging heat with its surroundings. In the deep interiors of a star a good approximation is $\nabla' = \adgrad \equiv \big(\frac{\D \ln \T}{\D \ln \P}\big)_{\textrm{ad}}$ \P}\big)_{\textrm{ad}}$, where $\adgrad$ is the temperature gradient of a fluid element moving adiabatically.