On the effects of clouds and hazes in the atmospheres of hot Jupiters: semi-analytical temperature-pressure profiles

Motivated by the work of Guillot (2010), we present a semi-analytical formalism for calculating the temperature-pressure profiles in hot Jovian atmospheres which includes the effects of clouds/hazes and collision-induced absorption. Using the dual-band approximation, we assume that stellar irradiation and thermal emission from the hot Jupiter occur at distinct wavelengths (“shortwave” versus “longwave”). For a purely absorbing cloud/haze, we demonstrate its dual effect of cooling and warming the upper and lower atmosphere, respectively, which modifies, in a non-trivial manner, the condition for whether a temperature inversion is present in the upper atmosphere. The warming effect becomes more pronounced as the cloud/haze deck resides at greater depths. If it sits below the shortwave photosphere, the warming effect becomes either more subdued or ceases altogether. If shortwave scattering is present, its dual effect is to warm and cool the upper and lower atmosphere, respectively, thus counteracting the effects of enhanced longwave absorption by the cloud/haze. We make a tentative comparison of a 4-parameter model to the temperature-pressure data points inferred from the observations of HD 189733b and estimate that its Bond albedo is approximately 10%. Besides their utility in developing physical intuition, our semi-analytical models are a guide for the parameter space exploration of hot Jovian atmospheres via three-dimensional simulations of atmospheric circulation.

💡 Research Summary

The paper presents a semi‑analytical framework for calculating the temperature‑pressure (T‑P) profiles of hot‑Jupiter atmospheres that explicitly incorporates the effects of clouds/hazes, collision‑induced absorption (CIA), and short‑wave scattering. Building on the dual‑band approximation introduced by Guillot (2010), the authors treat stellar irradiation (“short‑wave”) and planetary thermal emission (“long‑wave”) as occurring at distinct wavelength bands, allowing separate treatment of the short‑wave optical depth τ_S and the long‑wave optical depth τ_L.

Key ingredients of the model are: (i) a constant short‑wave absorption opacity κ_S, (ii) a scattering parameter ξ defined as the ratio of absorption to total short‑wave opacity (ξ = 1 for pure absorption, ξ = 0 for pure scattering), and (iii) a long‑wave opacity κ_L that may vary with depth. The authors derive a simple analytic relation between the Bond albedo A and ξ: A = 1 − √ξ / (1 + √ξ), which links the amount of short‑wave scattering directly to the planetary albedo.

In the short‑wave sector, the radiative‑transfer moments are solved analytically under the assumption of constant κ_S and ξ, yielding exponential attenuation of the mean intensity J_S and flux H_S with depth. The short‑wave photosphere occurs where τ_S ≈ 2/3; the photon‑deposition depth lies at τ_S ≈ 2/3 · √ξ, so stronger scattering (smaller ξ) pushes the absorption layer to higher altitudes.

For the long‑wave sector, the authors retain a general κ_L(m) and adopt constant Eddington coefficients (E₁ = 1/3, E₂ = 1/2). By integrating the moments and invoking energy conservation, they obtain a compact expression for the net radiative flux that includes an explicit term Q representing horizontal heat transport by atmospheric circulation (Q < 0 for day‑to‑night transport). The final temperature relation (equations 45 and 50) reads schematically:

T⁴ = T_int⁴ ·