deletions | additions       diff --git a/methods.tex b/methods.tex  index 92028f7..4a5a0ef 100644  --- a/methods.tex  +++ b/methods.tex 
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\label{eq:level_pops}  \frac{n_\mathrm{u}}{n_\mathrm{l}} = \frac{g_\mathrm{u}}{g_\mathrm{l}} e^{\frac{\Delta E}{kT_{\mathrm{ex}}}}  \end{equation}  $n_u$ and $n_l$ are the number densities in the respective upper and lower states, $g_u$ and $g_l$ are the degeneracies of those states, and $\Delta E$ is the transition energy between them.  The population levels are become  inverted whenever when  $n_\mathrm{u}/g_\mathrm{u} \gt n_\mathrm{l}/g_\mathrm{l}$, which corresponds to a negative $T_\mathrm{ex}$. Maintaining an inverted population  requires an energy source, referred to as the "pump" {\it pump}  (see XXX ADD SUBSECTION XXX). Transistions in the microwave and radio require the smallest pump action, since $n_\mathrm{u}/g_\mathrm{u} \approx n_\mathrm{l}g_\mathrm{l}$ for kinetic temperatures of a few hundred kelvin, even when in thermal equilibrium. The role of the pump is to excite molecules into the upper and lower states of the maser transition. Let transition, and is typically achieved through a collisional or radiative process (see XXX add subsection XXX). Without regard for the details of how the pump achieves this, let  $P_\mathrm{u}$ and $P_\mathrm{l}$ be the rate at which the pump adds molecules to the upper and lower states, respectively. respectively, from other states of the molecule.