Which transition can be used in sodium mirrorless lasing for mesospheric magnetometry?
Directed mirrorless lasing from the mesospheric sodium layer has been proposed as a way to overcome the isotropy of laser-guide-star fluorescence, with demonstrated cell-scale analogues and a demonstrated stand-off magnetometry application. Several transition paths on the Na ladder compete for the same pump photons, and which of them can sustain a population inversion---and under what pumping format---is usually settled by numerical scans of a multi-level rate model. Here the same question is answered in closed form, and three design-relevant results follow from the atomic data alone. (1) A cascade inversion on $u\to l$ requires $\tau_u/\tau_l>(g_u/g_l)b_{ul}$. The inequality reproduces the full continuous-wave classification---admitting $4P_{3/2}\to4S_{1/2}$ (2.21~\textmu m), $4P_{3/2}\to3D_{5/2}$ (9.1~\textmu m) and the fine-structure companion $4S_{1/2}\to3P_{1/2}$ (1138~nm), excluding $4D_{5/2}\to4P_{3/2}$ (2.34~\textmu m) and $3D_{5/2}\to3P_{3/2}$ (819~nm)---and shows the 2.34~\textmu m line to miss by only 9\%, which is why it is transiently accessible. (2) Velocity selectivity reverses the naive scheme ranking because a Doppler-averaged treatment, exact for one-step pumping, underestimates two-step pumping by a participation factor of order the Doppler-to-natural width ratio, $\sim$$10^{2}$; equal division of power between two pump beams is exactly optimal. (3) The transient window on 2.34~\textmu m closes after $\sim$2$\tau(4P)\approx210$~ns, set by the reservoir lifetime rather than by the pump. Each conclusion is traceable to a lifetime, a branching ratio or a linewidth, and transfers to another species without rerunning a model.