Two-segment adiabat: dry above a transition pressure, moist (shallower from latent heat) below. A smooth blend region between them controls how sharply the gradient changes.
Radiative temperature: T⁴(τ) = (3/4) Tint⁴ (τ + 2/3) + (3/4) Teq⁴ [2/3 + 2/(3γ) + (γ/3 − 2/(3γ)) e−γτ].
The adiabatic gradient is a smooth blend between two regimes using a tanh: ∇ad(P) = ½ (∇dry + ∇moist) + ½ (∇dry − ∇moist) tanh[(log Ptrans − log P) / w]. Above Ptrans, ∇ad → ∇dry; below, ∇ad → ∇moist.
The adiabat curve is then integrated from the RCB outward and inward using dT/d log P = T · ∇ad(P) · ln 10. The RCB itself is found by comparing the local ∇rad from the Guillot solution to the local ∇ad(P).
The amber band on the plot marks ±w around log Ptrans, where the gradient is in transition. Real condensation transitions follow Clausius–Clapeyron and are sharper at the saturation curve; this tanh blend is a phenomenological approximation.
Gradient panel and the 0.25 asymptote. The lower panel plots ∇rad = d ln T / d ln P versus pressure, with horizontal reference lines at the slider values ∇dry and ∇moist. With a single gray thermal opacity (nT = 0, the default), τ ∝ P and the deep Guillot profile gives T⁴ ∝ τ ∝ P, so ∇rad asymptotes to exactly 1/4 = 0.250 at depth — regardless of Tint, Teq, or γv. For ∇dry > 0.25 (including ideal H2's 2/7 ≈ 0.286) no strict dry-only Schwarzschild crossing exists in this gray-opacity case.
T- and P-dependent opacity (nT, nP sliders). The thermal opacity follows κth(T, P) = κth,0 (T/1000 K)nT (P/1 bar)nP, which is the form used by Rogers et al. for H2/He atmospheres. Their fit gives κ ≈ 1.3 × 10−2 (T/1000 K)0.45 (P/1 bar)0.68 cm2/g, i.e. κth,0 ≈ 1.3 × 10−3 m2/kg (the SI form used internally here). Set log κth,0 ≈ −2.9, nT ≈ 0.45, nP ≈ 0.68 to match the Rogers scaling. The P-dependence is the more important effect physically — it reflects CIA-like absorption that scales with density of collision partners. With nP > 0, τ(P) is computed by integrating κ(T,P)·dP/g along the profile rather than the simple τ = κ·P/g of the gray case, and the Guillot solution is iterated for self-consistency. As nT or nP grows, ∇rad deep rises above 0.25 and can cross the ∇dry line — producing a real dry-Schwarzschild crossing that the gray case forbids.