= Solution
Downward-from-space <optical depth> at wavelength $\lambda$ is
$$
\boxed{\tau_\lambda(z)=\int_z^\infty\kappa_\lambda(z')\rho(z')\,dz',}
$$
where $\kappa_\lambda$ is the mass <absorption coefficient>. A nearly isothermal, <hydrostatic equilibrium>, compositionally well-mixed atmosphere with height-independent $\kappa_\lambda$ has $\rho\propto e^{-z/H}$ and $\tau\propto e^{-z/H_\tau}$ with $H_\tau=H=R_aT/g$. Thus <temperature>, mean molecular <mass> and <gravity> set the basic <scale height>. Changes in the absorber mixing ratio, <pressure broadening>, temperature dependence of line strengths, collision-induced absorption and departures from <hydrostatic equilibrium> modify it. For example if local absorption scales as $\rho^2$ in an isothermal atmosphere, $H_\tau=H/2$.
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