= Solution
The <Jeans escape parameter> compares gravitational binding with thermal energy:
$$
\lambda(r)=\frac{GM_pm}{k_BTr}
=\frac{v_{\rm esc}^2}{2k_BT/m}.
$$
Hydrostatic thermal escape becomes strong when the Maxwellian tail is no longer exponentially small, roughly $\lambda\lesssim2$--$3$, and blow-off occurs for order-unity $\lambda$. Atomic hydrogen at an exobase radius $R_e$ therefore requires
$$
\boxed{T_e\gtrsim\frac{GM_pm_H}{k_BR_e}}
$$
for $\lambda\lesssim1$.
For a Neptune-mass planet this is about $3.4\times10^4\,\mathrm K$ at the optical radius $R_{\rm Nep}$. If the relevant base is the observed ultraviolet absorbing radius, approximately $10R_{\rm Nep}$, the corresponding value is about $3.4\times10^3\,\mathrm K$.
Solved by gpt-5.6-sol high.
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