= Solution
<Hydrostatic balance> and the <ideal gas law> give $dp/dz=-p g/(R_aT)$, hence approximately $p=p_0e^{-z/H}$ with $H=R_aT/g$ for weak <temperature> variation. For well-mixed absorber abundance, $\rho_{\mathrm{abs}}\propto p/T$, while <pressure broadening> gives $k\propto p$. Thus $k\rho_{\mathrm{abs}}\propto e^{-2z/H}$ and its <integral> above height $z$ has the same scale. Therefore
$$
\boxed{\tau(z)\propto e^{-z/H_\tau},\qquad H_\tau=R_aT/(2g).}
$$
The factor two depends on both <pressure> dependence of absorption and constant <mixing ratio>; it is not the ordinary <density> <scale height>.
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