Mirrorless lasing from the mesospheric sodium layer: a quantitative path to threshold and its radiative limit

<jats:p>&lt;p dir="ltr"&gt;Mirrorless lasing from the mesospheric sodium layer could transform laser-guide-star emission into a directional, gain-driven source. Here we convert this possibility into quantitative requirements, using a ten-level velocity-resolved rate-equation model and an exact compression--irradiance equivalence. Continuous-wave operation remains below threshold across the accessible parameter space. The transient 4D&lt;sub&gt;5/2&lt;/sub&gt;-4P&lt;sub&gt;3/2&lt;/sub&gt; channel at 2339 nm reaches threshold at pulsed irradiances of 10&lt;sup&gt;4&lt;/sup&gt;-10&lt;sup&gt;5&lt;/sup&gt; W/m&lt;sup&gt;-2&lt;/sup&gt;, given six- to ten-fold velocity compression. The obstacle is therefore not laser power but momentum transfer: spontaneous scattering delivers at most one recoil per event at a bounded rate, leaving chirp-driven compression an order of magnitude short. This gap defines the technological challenge for mesospheric mirrorless lasing.&lt;/p&gt;</jats:p>