= Solution
{c}
For the <variable-latent-heat saturation curve> of a pure saturated <water> <atmosphere>, the total <pressure> equals the <saturation pressure>. The exact <Clausius-Clapeyron relation> along the <phase coexistence curve> is
$$
\frac{dp}{dT}=\frac{L(T)}{T(v_v-v_c)},
$$
where $v_v,v_c$ are the specific volumes of the <vapor> and condensed <thermodynamic phase>. Neglecting $v_c$ and using the <ideal gas law>, $v_v=R_{\rm wv}T/p$, gives
$$
\frac{d\ln p}{dT}=\frac{L(T)}{R_{\rm wv}T^2},\qquad
\ln\frac p{p_*}=\int_{T_*}^{T}\frac{L(\theta)}{R_{\rm wv}\theta^2}\,d\theta.
$$
The <triple point> provides the integration constant. Below it, the prescribed <latent heat> is constant, so
$$
\boxed{\ln\frac p{p_*}=\frac{L_s}{R_{\rm wv}}\left(\frac1{T_*}-\frac1T\right),\quad T\leq T_*.}
$$
Above it, write $K=L_*/[R_{\rm wv}(T_c-T_*)]$. Integrating $K(T_c/T^2-1/T)$ gives
$$
\boxed{\ln\frac p{p_*}=K\left[T_c\left(\frac1{T_*}-\frac1T\right)-\ln\frac{T}{T_*}\right],\quad T_*\leq T<T_c.}
$$
These are the requested implicit <temperature> profiles $T(p)$. Both branches meet at $(T_*,p_*)$, and each is monotone before the limiting <liquid-gas critical point>. The lower branch describes equilibrium with <ice>, rather than a stable <liquid> surface. A stable <liquid> <ocean> surface belongs to the upper branch; the colder upper <atmosphere> may use the lower branch. <Hydrostatic balance> is needed to convert this relation into a height profile, but not to obtain $T(p)$.
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