OREAL-H: Exoplanet Radiative Transfer

• OREAL-H is a computational framework that determines outgoing longwave radiation (OLR) and top-of-atmosphere (TOA) albedo using advanced gas absorption and continuum opacity models for rocky exoplanetary atmospheres.
• It employs GPU-accelerated, line-by-line radiative transfer methods with a high-resolution spectral grid, benchmarking its outputs against legacy models and Earth observation datasets.
• The procedure enables rapid and precise climate modeling, making it ideal for iterative exoplanet habitability studies and zonal energy budget simulations.

OREAL-H is a comprehensive computational procedure for deriving the outgoing longwave radiation (OLR) and top-of-atmosphere (TOA) albedo for rocky exoplanetary atmospheres under temperate, potentially habitable conditions. Developed as an adaptation of the HELIOS and HELIOS-K GPU-accelerated atmospheric radiative transfer codes, OREAL-H systematically incorporates high-fidelity gas absorption and continuum opacity models, advanced radiative transfer solvers, and state-of-the-art line databases. The methodology is tailored to support global and zonal climate modeling, benchmarked extensively against legacy radiative transfer codes and Earth-observational datasets, and enables rapid computation to facilitate iterative exoplanet habitability studies (Simonetti et al., 2021).

1. Radiative Transfer Framework and OLR Computation

OREAL-H operates under local thermodynamic equilibrium and employs a plane-parallel, clear-sky assumption for atmospheric columns. The core radiative transfer framework is the monochromatic two-stream solution: $$\mu\,\frac{dI_\lambda(\tau_\lambda,\mu)}{d\tau_\lambda} = I_\lambda(\tau_\lambda,\mu) - S_\lambda(\tau_\lambda)$$ with $I_\lambda$ denoting specific intensity, $\tau_\lambda$ the vertical optical depth, and $S_\lambda$ the source function. The two-stream closure follows Heng et al. (2014, 2018), iteratively updating upward and downward fluxes $F_{i,\uparrow}$ and $F_{i,\downarrow}$ through layers via: $$F_{i,\uparrow} = \frac{1}{\chi} \left[ \psi\,F_{i-1,\uparrow} - \xi\,F_{i,\downarrow} + 2\pi\,\epsilon\,\mathcal{B}_\uparrow + \frac{1}{\mu_*} \mathcal{I}_\uparrow \right]$$ where $\mu_*$ is the stellar zenith cosine, $\epsilon$ is the Eddington coefficient (fixed at 0.5), and other parameters encode scattering and closure.

The OLR is defined at TOA as the spectrally integrated upward thermal flux: $$\mathrm{OLR}(T_s) = \int_0^\infty F^\mathrm{thermal}_{\uparrow,\mathrm{TOA}}(\lambda)\; d\lambda$$ Numerical computation is performed line-by-line on a high-resolution spectral grid ($\Delta\lambda/\lambda\sim10^{-3}$ spanning 0–30,000 cm$^{-1}$).

2. Top-of-Atmosphere Albedo: Formalism and Implementation

The clear-sky TOA albedo computation is based on the two-stream solution for shortwave (stellar) radiation, encompassing both direct and diffuse (Rayleigh-scattered) components: $$F^\mathrm{ref}_{\uparrow,\mathrm{TOA}} = \int_0^\infty \left[ F_{\uparrow,\mathrm{scattered}}(\lambda) + s\,F_*(\lambda)\,e^{-\tau(\lambda)/\mu_*} \right] d\lambda$$ Here, $F_*(\lambda)$ is the incident stellar flux, $s$ is the (spectrally flat) surface albedo, and $\tau(\lambda)$ denotes the atmospheric optical depth.

The TOA albedo is then defined as: $$A_\mathrm{TOA} = \frac{F^\mathrm{ref}_{\uparrow,\mathrm{TOA}}}{\int_0^\infty \pi\,\mu_*\,F_*(\lambda)\,d\lambda}$$ Spherical geometry corrections (Toon et al. 1989 closure) adapt the path length for zenith angle effects. Clouds are excluded from the clear-sky prescription; Rayleigh scattering by gas molecules is the only shortwave scatterer.

3. Gas Opacity and Continuum Treatment in HELIOS-K

Gas-phase absorption is parameterized using line-by-line computations from HITRAN/HITEMP 2016 databases for H$_2$O, CO$_2$, CH$_4$, O$_2$, and N$_2$. The monochromatic absorption coefficient in cm$^{-1}$: $$k_\mathrm{abs}(\nu) = \sum_i S_i(T)\,f_i(\nu-\nu_i,\gamma_i)$$ where $S_i$ is the line strength, $f_i$ the Voigt profile, and $\gamma_i$ is the half-width at half maximum from combined broadening. For CO$_2$ far wings, sub-Lorentzian corrections (Perrin–Hartmann 1989; Tonkov et al. 1996 χ-factors) are applied beyond specific offsets.

Continuum absorption incorporates the MT-CKD 3.4 model for H$_2$O and Gruszka–Borysow–Baranov (GBB) continuum for CO$_2$, along with collision-induced absorption (CIA) for CO$_2$–CO$_2$ and N$_2$–N$_2$ (HITRAN CIS tables). Voigt convolutions are computed on GPUs using a Humliček-type algorithm, constructing multidimensional “ktables” in pressure–temperature–wavelength space ($\sim$50 pressure levels × 50 temperature levels × 30,000 cm$^{-1}$ at $\Delta\lambda/\lambda=10^{-3}$), with single Earth-like tables completed in minutes.

4. Atmospheric Structure, Vertical Discretization, and Sensitivities

The standard grid adopts $\sim$10 layers per pressure decade, extending from surface pressure (P$_\mathrm{surf}$) to P $\sim$ 1 μbar at TOA. Increasing grid resolution to $\sim$60 layers ($\sim$15 per decade) modifies OLR by $<1$ W m$^{-2}$ and TOA albedo by $<0.1$\%. Lapse rates for the troposphere utilize:

• Earth-like atmospheres: Moist pseudoadiabat from Pierrehumbert (2011): $$\frac{d\ln T}{d\ln P} = \frac{R}{m_nC_{p,n}} \cdot \frac{1 + \frac{L\,m_n\,r_v(T)}{R\,T}}{1 + \frac{L^2\,m_v\,r_v(T)}{C_{p,n}\,R\,T^2}}$$ where $r_v$ is the H$_2$O-to-N$_2$ mass ratio, $L$ is latent heat, $C_{p,n}$ the non-condensible heat capacity.
• CO$_2$-dominated atmospheres: Two-component adiabats per Kasting & Pollack (1991) and non-ideal EOS (Span & Wagner 1996).

Tropopause is treated as isothermal above a fixed temperature: T = 200 K (Earth-like), T = 160 K (CO$_2$-rich). Fixed-pressure tropopause conditions suppress the runaway greenhouse OLR inflection and are not preferred for habitability boundary analyses.

5. Validation Against Legacy Codes and Observational Data

OREAL-H generates OLR and TOA albedo in agreement with established radiative transfer frameworks (CAM3, SMART, SBDART, LBLRTM, LMDG, CCM3), lying within the ensemble spread for T$_s$ $<$ 280 K and diverging by up to 4% above T$_s$ = 340 K relative to CAM3 (the highest in Yang et al. 2016). For CO$_2$-rich runs, dry CO$_2$ OLR(P) for P = 0.5–5 bar falls within the range of established sub-Lorentz models (Halevy et al. 2009); moist CO$_2$ OLR matches the increased transparency of GBB-based continua versus older Pollack (1980) models.

Climate model integration (ESTM) with EOS OLR/A$_\mathrm{TOA}$ look-up tables accurately simulates the present-day Earth's zonal energy budget, matching CERES (2005–2015) zonal OLR vs. TOA albedo within $<$5 W m$^{-2}$ scatter and $<$0.02 albedo, outperforming ECM3-based radiative tables.

6. Recommendations and Practical Applications

For exoplanet habitability studies and reduced-complexity climate modeling (e.g., 1D, 2D or zonal/seasonal models), it is recommended to:

• Precompute two-dimensional tables of OLR$(T_s,P_\mathrm{nc})$ and A$_\mathrm{TOA}(T_s,\mu_*,s)$ for target gas mixtures.
• Employ $\sim$10–20 atmospheric layers per pressure decade and extend to TOA pressures $\sim$1 μbar for temperate application.
• Use fixed-temperature tropopause boundaries to preserve correct radiative features, particularly for runaway greenhouse limits.
• Partition cloud forcing externally: subtract OLR cloud effect from clear-sky OLR, and add prescribed/cloud albedo to the surface value in A$_\mathrm{TOA}$, calibrated as needed using Earth observations.
• Apply full line-by-line plus continuum (MT-CKD for H$_2$O, GBB for CO$_2$) for inner-edge/runaway greenhouse scenarios.
• Deploy HELIOS-K/HELIOS on GPU hardware for $\sim$10–100$\times$ acceleration over CPU-based radiative transfer, enabling on-the-fly recalculation in general circulation models (GCMs) or rapid energy balance model (EBM) parameter sweeps.

OREAL-H thus enables physically consistent, efficiently computed radiative flux and albedo diagnosis, supporting comprehensive exoplanetary climate investigations validated against both classical model lineups and high-quality satellite observations (Simonetti et al., 2021).