deletions | additions       diff --git a/Convective Efficiency.tex b/Convective Efficiency.tex  index 6c5fda0..38c79ec 100644  --- a/Convective Efficiency.tex  +++ b/Convective Efficiency.tex 
...
\subsection{Convective Efficiency}  Energy losses (and gains) need to be accounted for during the convective element rise/fall.  When these are not negligible, $\nabla \neq \nabla\ \neq \adgrad$ \adgrad$.  In the deep interiors of stars energy loss can be due to neutrinos, while gains are produced by nuclear burning. In the outer layers however radiation can be is usually  responsible for departure from adiabaticity. These gains/losses are regarded as purely ``horizontal'' in the sense that one has to think as the average effect created by all the convective elements over a shell. Therefore no direct contribution to the net vertical heat transport is expected (on average there are as many hot and cool elements crossing the shell), but an indirect contribution can arise due to the modification to the convective properties (e.g. convective velocities). Focusing on horizontal radiative losses, the usual assumption is that the convective element is optically thick. The change in temperature of a rising convective element is due to its adiabatic expansion and the horizontal radiative losses \begin{equation}  \bigg(\frac{\D T}{\D r} \bigg) = \bigg(\frac{\D T}{\D r} \bigg)_{ad} - \frac{}  \end{equation}