= Solution
The functional form is not specific to <exoplanets>. With positive <pressure> normalization and physical <temperature>, a <square-root exponential atmospheric profile> can approximate a monotone interval of the <terrestrial atmosphere>. An outward-cooling interval, such as part of the <troposphere> or <mesosphere>, requires $\alpha>0$. An upward-warming interval, such as part of the <stratosphere> heated by ultraviolet absorption or the <thermosphere> heated by high-energy radiation, requires $\alpha<0$.
The local <atmospheric lapse rate> follows from the <ideal gas> relation $P=\rho k_BT/(\mu m_H)$ and <hydrostatic equilibrium>:
$$
\Gamma\equiv-\frac{dT}{dz}
=\frac{\mu m_Hg}{k_B}\frac{2\sqrt{T-T_0}}{\alpha T}
=\frac{\mu m_Hg}{k_B}\nabla.
$$
For dry terrestrial air, approximately $\gamma=1.4$ and $\mu\simeq29$ give the dry-adiabatic lapse rate $g/c_p\simeq9.8\,\mathrm{K\,km^{-1}}$. The familiar mean tropospheric value near $6.5\,\mathrm{K\,km^{-1}}$ is less steep; a local fit must satisfy $\nabla\lesssim2/7$ to be dry-convectively stable. Moist <convection> needs the moist parcel thermodynamics instead, and the <terrestrial atmosphere> is not uniformly dry or chemically homogeneous at all heights.
The squared-<logarithm> shape cannot reproduce an exactly constant nonzero lapse rate over an arbitrary thick region, all the alternating atmospheric layers, or a finite exactly isothermal region. It is a local parametrization with a fixed sign of the <temperature gradient>; it has no terrestrial universality.
$$
\boxed{\alpha>0\text{ can fit an outward-cooling interval};\quad
\alpha<0\text{ can fit an upward-warming interval}.}
$$
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