HELIOS: An Open-source, GPU-accelerated Radiative Transfer Code For Self-consistent Exoplanetary Atmospheres

1. INTRODUCTION

• Interpreting exoplanet spectra requires understanding radiative transfer and atmospheric chemistry to infer atmospheric thermal structure and molecular abundances.Atmospheric features are difficult to extract because the planet’s light is many orders of magnitude fainter than its host star.Source: §1 · p. 1
• A useful model must accommodate varied chemistry, metallicity, irradiation, and internal heat while producing customizable spectra and efficiently converging to radiative equilibrium.Such a code can form a flexible radiation package for coupling to chemical-kinetics codes or three-dimensional circulation models.Source: §1 · p. 2
• HELIOS is a customizable, built-from-scratch code for modeling exoplanetary atmospheres.Its initial version uses analytical two-stream solutions to iteratively calculate self-consistent temperature-pressure profiles and synthetic spectra.Source: §1 · p. 2
• HELIOS uses HELIOS-K opacities and correlated-k tables, with molecular abundances calculated from validated analytical equilibrium-chemistry formulae.The chemistry is parameterized by carbon and oxygen elemental abundances and includes H2O, CO, CO2, and CH4.Source: §1 · p. 2
• A calculation with 101 model layers and 300 wavelength bins takes a few minutes on a personal computer with an NVIDIA GeForce 750M GPU.The authors state that this efficiency enables parameter studies.Source: §1 · p. 2

2.1.1. Preamble

• The two-stream approximation reduces radiative transfer to an ordinary differential equation by integrating intensity over incoming and outgoing hemispheres and assuming fixed ratios of intensity moments.The formulation allows wavelength-, temperature-, and pressure-dependent single-scattering albedo and asymmetry factor for non-isotropic scattering.Source: §2.1.1 · p. 3

2.1.2. Exact solution of the radiative transfer equation in the pure absorption limit

• In pure absorption, the diffusivity factor is about 2 for very thin layers and approaches 1 as layers become optically thick.A constant D in the two-stream approximation represents a mean value across the atmosphere; the exact pure-absorption solution does not require specifying D.Source: §2.1.2 · p. 3, Figure 1 · p. 4
• The exact radiative-transfer solutions are used to compare against two-stream solutions and determine D; an analytical expression for D with scattering is unknown.The exact solution described here is in the pure-absorption limit.Source: §2.1.2 · pp. 3–4

2.1.3. Different flavors of two-stream solutions

• The model’s most sophisticated two-stream solutions include non-isotropic scattering and non-isothermal layers; pure-scattering equations recover the transparent or opaque limits.In the non-isothermal approach, a single constant Planck-function gradient is assumed within each layer.Source: §2.1.3 · p. 4
• Non-isothermal two-stream solutions generally converge to radiative equilibrium more rapidly than isothermal-layer solutions.With enough isothermal layers, the two calculations should agree.Source: §2.1.3 · p. 4

2.1.4. Rayleigh scattering

• Rayleigh scattering contributes to layer opacity and the single-scattering albedo, dominating in the optical but becoming subdominant to molecular absorption in the infrared.The model focuses on H2 Rayleigh scattering and ignores helium’s contribution, which is less than 1% of the H2 contribution.Source: §2.1.4 · p. 5

2.2.1. Model grid

• Non-isothermal layers use two sublayers so flux propagation involves both layer centers and interfaces; when Δτi < 10^-4, the isothermal prescription prevents upper-atmosphere numerical issues.In the isothermal grid, layers are separated by interfaces and spaced evenly in height or log pressure; the layer thickness follows from hydrostatic balance under isothermality and an ideal-gas equation of state.Source: §2.2.1 · p. 5

2.2.2. Boundary conditions

• The top boundary receives stellar irradiation, while the bottom can specify internal radiative heat flux; this study sets Tintern = 0 K.The adopted redistribution parameter is f = 2/3, within the stated theoretical bounds of 1/4 to 1.Source: §2.2.2 · pp. 5–6

2.2.3. Iterating for radiative equilibrium

• Radiative equilibrium is reached by iterating layer temperatures until bolometric net-flux divergence vanishes, expressing local energy conservation.Scattering couples incoming and outgoing fluxes, so HELIOS repopulates the flux arrays iteratively within each temperature step.Source: §2.2.3 · pp. 6–7

2.2.4. Numerical timestepping

• A uniform timestep converges upper layers faster than lower layers and is described as infeasible; adaptive timesteps instead scale with each layer’s radiative timescale.The algorithm reduces timesteps by 33% after detected temperature oscillations and increases them by 10% when temperature changes monotonically.Source: §2.2.4 · p. 7
• Less than 4 K at the BOA and less than 1 K in photospheric regions correspond to less than 0.5% emission-spectrum impact under the convergence criterion.The criterion tests for vanishing bolometric flux divergence, rather than imposing a threshold on temperature change.Source: §2.2.4 · p. 7

2.3. Calculating Opacities and Transmission Functions

• The opacity set includes H2O, CO2, CO, CH4, H2-H2 CIA and H2-He CIA; omitted absorbers include NH3, HCN, C2H2 and Na/K.The authors note that the omitted species may affect atmospheric structure and identify the four-molecule set as a starting point for the code.Source: §2.3 · pp. 7–8
• A 10^-5 cm^-1 opacity-function resolution samples about 10^9 points across the wavenumber range, approaching line-by-line calculation.The authors note that coarser resolution may miss spectral lines and underestimate opacity.Source: §2.3 · p. 8
• The Voigt far-wing form is unknown; HELIOS truncates wings at 100 cm^-1, except for water at 25 cm^-1.The authors identify far-wing opacity as an uncertainty that can over- or underestimate the true contribution depending on the molecule.Source: §2.3 · p. 8
• A 20th-order Gauss-Legendre quadrature rule is reported sufficient against direct Simpson integration, and HELIOS uses 20 Gaussian points per wavelength bin.The k-distribution speeds transmission calculations by avoiding direct integration over about 10^9 opacity points.Source: §2.3 · p. 8
• Combining molecular k-distributions assumes perfectly correlated spectral lines, although real lines lie between the correlated, uncorrelated and disjoint limits.The accuracy of overlap approaches in hot atmospheres remains unquantified without a true line-by-line calculation; the authors leave that test to future work.Source: §2.3 · pp. 8–9
• 300 equilibrium wavelength bins followed by 3000-bin post-processing produce essentially identical results to running the full calculation at 3000 bins.The bins are equally spaced in wavenumber.Source: §2.3 · p. 9

2.4. Chemistry Model

• Analytical formulae calculate the mixing ratios of CO, CO2, H2O and CH4 from elemental carbon and oxygen abundances as functions of temperature and pressure.The formulae are compared with TEA Gibbs free energy minimization calculations at 1 bar and 800 K and 3000 K; the paper reports agreement.Source: §2.4 · p. 9, Figure 4 · p. 9
• Solar element abundance is defined as nC = 2.5×10^-4 and nO = 5×10^-4, giving C/O = 0.5; C/O changes vary nC while nO stays fixed.The study does not examine C/O > 1 and therefore ignores C2H2.Source: §2.4 · p. 9
• Metallicity changes scale nC and nO together, preserving C/O; for example, 3× solar metallicity retains C/O = 0.5.The model defines metallicity using these elemental abundances.Source: §2.4 · pp. 9–10

• Figure 6 presents stellar spectra used to model six hot Jupiters; the study uses PHOENIX and Kurucz models alongside the simpler Planck-function option.Source: §2.5 · p. 10
• For temperatures above 3000 K, chemistry is assumed to match that at 3000 K because tabulated Gibbs free energies extend only to 3000 K.Source: §2.5 · p. 10
• PHOENIX spectra are interpolated in stellar temperature, surface gravity, and metallicity; Kurucz spectra are interpolated in temperature and gravity.Source: §2.5 · p. 10
• Stellar spectral features imprint on secondary-emission spectra, while differences between stellar spectra also change atmospheric heating and the resulting temperature-pressure profile and synthetic spectrum.Source: §2.5 · p. 10
• Because PHOENIX and Kurucz models do not span the full 0.33 µm to 10 cm wavelength range, HELIOS patches them with a Planck function.Source: §2.5 · p. 10

• CUDA C++ parallelizes flux propagation by wavelength under coherent scattering and also parallelizes interpolation of precomputed k-distribution tables.Source: §2.6 · p. 10
• Optional Planck-function and transmission-function tables use ΔT = 10 K and Δlog10 κ = 0.1 grids, which the authors report are sufficient for converged model behavior.Source: §2.6 · p. 10
• Temperature convergence takes 2 to 15 minutes on a GeForce 750M and 0.5 to 4 minutes on a K20 GPU for a typical setup with 101 layers and 300 wavelength bins.Source: §2.6 · pp. 10–11
• After convergence, emission-spectrum post-processing at 3000 wavelength bins takes less than 30 seconds; without scattering, convergence is usually twice as fast.Source: §2.6 · p. 11

3.1.1. Comparison to GJ 1214b model of Miller-Ricci & Fortney

• HELIOS matches Miller-Ricci & Fortney (2010) around the GJ 1214b infrared photosphere, while its temperatures are about 200 K hotter at pressures above 1 bar.Adding shortwave opacity reproduces the deep atmospheric structure; the emission spectra remain similar in magnitude and trends.Source: §3.1.1 · p. 14, Figure 7 · p. 15

3.1.2. Trends associated with scattering

• Scattering shifts stellar absorption upward and cools the atmosphere; backward scattering strengthens reflected light, while forward scattering strengthens thermal emission.Scattering also mutes spectral features and raises the infrared continuum, with a stronger effect for more backward-dominated scattering.Source: §3.1.2 · p. 14, Figure 8 · p. 15

3.1.3. Isothermal versus non-isothermal layers

• 21 non-isothermal layers converge to within 3 K, whereas isothermal-layer profiles do not converge even with 1001 layers.For post-processing, 101 isothermal layers typically keep flux deviations below 1%, so HELIOS uses non-isothermal layers for radiative-equilibrium iteration and isothermal layers for spectra.Source: §3.1.3 · pp. 15–16, Figure 9 · p. 17

3.1.4. Obtaining convergence for the k-distribution tables

• 10^-3 cm^-1 opacity resolution is required for about 1% spectral error; coarser 10^-1 and 10^-2 cm^-1 resolutions yield errors above 10% and about 1–10%, respectively.The errors also appear when post-processing the reference temperature profile, so they are not solely due to radiative-equilibrium iteration.Source: §3.1.4 · p. 16, Figure 10 · p. 17

3.2. Testing the Null Hypothesis and Variations on a Theme: Benchmark 1D Models for Hot Jupiters

• Changing stellar models produces qualitatively similar spectra, and changing metallicity by a factor of three has minimal spectral effect; higher C/O cools profiles and strengthens CO absorption.The higher-C/O models continue to under-predict infrared fluxes for WASP-8b, WASP-12b, WASP-14b, and WASP-33b.Source: §3.2 · pp. 17–19

3.1.5. Using the correct value of the diffusivity factor

• D = 2 gives the smallest average spectral error for GJ 1214b and the best temperature and spectral match for WASP-12b against exact pure-absorption solutions.The authors relate this result to sufficiently fine vertical resolution, where optical-depth differences between layers are small in the photosphere.Source: §3.1.5 · p. 16, Figure 11 · p. 18

3.2. Testing the Null Hypothesis and Variations on a Theme: Benchmark 1D Models for Hot Jupiters

• The null hypothesis describes HD 189733b and WASP-43b reasonably well, but consistently under-predicts infrared fluxes for WASP-8b, WASP-12b, WASP-14b, and WASP-33b.The models assume chemical and radiative equilibrium with solar abundances; the authors identify missing opacity, disequilibrium, and non-1D effects as possible explanations for the discrepancies.Source: §3.2 · pp. 16, 18, 19

4.1. Summary

• HELIOS solves radiative transfer in 1D plane-parallel atmospheres with non-isotropic scattering, using four infrared absorbers, correlated-k opacities, and analytical equilibrium-chemistry formulae.It supports stellar-model or Planck-function irradiation and was optimized for GPU computation on a single machine.Source: §4.1 · p. 19

4.2. Comparison to previous work

• The authors position HELIOS as a novel assembly of radiative-transfer components alongside prior approaches using accelerated lambda iteration, brown-dwarf-derived methods, and two-stream or variable-Eddington-factor methods.The cited prior work includes Burrows et al. (2006, 2007, 2008), Fortney et al. (2005, 2006, 2008, 2010), Morley et al. (2013, 2015), Amundsen et al. (2014), and Molli`ere et al. (2015).Source: §4.2 · p. 19

4.3. Discussion and Opportunities for Future Work

• The current opacity set covers four main infrared absorbers and H2-H2 and H2-He collision-induced absorption; additional sources are needed for C/O > 1 and very hot planets.The authors identify C2H2, HCN, Na, K, and high-temperature continuum absorption as future opacity additions.Source: §4.3 · pp. 19, 23
• The study compares null-hypothesis models across stellar spectra, metallicities, and C/O ratios for six hot Jupiters.Figure 12 covers PHOENIX, Kurucz, and blackbody stellar inputs; Figures 13 and 14 examine metallicity and C/O variations, respectively.Source: Figure 12 · p. 20, Figure 13 · p. 21, Figure 14 · p. 22
• The authors flag correlated-k accuracy, aerosols and clouds, convective adjustment, disequilibrium chemistry, and radiative disequilibrium as areas for further work.They note that detailed case-by-case interpretation of hot Jupiters requires three-dimensional models; HELIOS is part of the open-source Exoclimes Simulation Platform.Source: §4.3 · p. 23

A. LICENSING AND PERMISSION TO USE THE TEA CODE

• The authors thank the developers of the Thermochemical Equilibrium Abundances code, initially developed at the University of Central Florida.The passage also states that the Reproducible-Research Compendium is available at github.com/exoclime/HELIOS.publications.Source: A. LICENSING AND PERMISSION TO USE THE… · p. 23