H⁻ Opacity in Stellar and Exoplanet Atmospheres

Updated 26 January 2026

H⁻ opacity is defined by bound–free photodetachment and free–free absorption processes that shape the continuum in stellar and exoplanet spectra.
Radiative transfer models employ quantum-mechanical cross sections and equilibrium calculations to accurately derive the H⁻ population under varying T-P conditions.
Accurate H⁻ opacity modeling improves atmospheric retrievals by refining thermal structure, spectral features, and chemical composition estimations.

The negative hydrogen ion, H $^-$ , is the principal source of continuous opacity in the visual and near-infrared regions in a broad range of stellar and exoplanetary atmospheres. Its treatment in radiative transfer and spectral modeling is essential for accurately synthesizing stellar and exoplanet spectra, determining atmospheric structures, and retrieving precise chemical and physical parameters. The theoretical foundation for H $^-$ opacity involves detailed quantum-mechanical cross sections for bound–free (photodetachment) and free–free (inverse bremsstrahlung) processes, as well as a rigorous calculation of the H $^-$ population through chemical or statistical equilibrium, often involving non-LTE corrections. Its accurate inclusion is now recognized as vital for interpreting percent-level spectroscopic signatures in both stellar and ultra-hot exoplanetary atmospheres.

1. Fundamental Physical Processes and Cross Sections

H $^-$ contributes continuous opacity via two principal mechanisms: bound–free (photodetachment) and free–free (inverse bremsstrahlung) absorption.

Bound–free:

Photodetachment occurs as H $^-$ + $h\nu$ → H + $e^-$ at a threshold photon energy of 0.754 eV (wavelength 1.641 μm). The cross section, $\sigma_{\rm bf}(\nu, T)$ , exhibits a strong rise just above this threshold and falls off approximately as $\nu^{-3}$ at higher energies. The analytic expression, fitted to quantum calculations (e.g., John 1988; Wishart 1979), takes the form:

$\sigma_{\rm bf}(\nu, T) = \sigma_0 \left( \frac{\nu_0}{\nu} \right)^3 \left( 1 - \frac{\nu_0}{\nu} \right)^{1.5} \left[ 1 + a_1 \left( \frac{\nu_0}{\nu} \right)^{0.5} + a_2 \left( \frac{\nu_0}{\nu} \right) \right]$

for $\nu > \nu_0$ , with $\sigma_0 \sim 4 \times 10^{-17}$ cm $^2$ (Lothringer et al., 2018, Arcangeli et al., 2018).

Free–free:

The inverse bremsstrahlung process is H + $e^-$ + $h\nu$ → H + $e^-$ , contributing a smooth continuum, especially in the near- and mid-infrared. The cross section, $\sigma_{\rm ff}(\nu, T)$ , is given by

$\sigma_{\rm ff}(\nu, T) = 10^{-27} g_{\rm ff}(T,P) \nu^{-2} T^{-1/2}$

or via analytic fits (Bell & Berrington 1987; John 1988), with Gaunt factors $g_{\rm bf}$ and $g_{\rm ff}$ of order unity (Arcangeli et al., 2018, Lothringer et al., 2020).

Both contributions must be included for correct representation of the continuum at the temperatures ( $T \gtrsim 2500$ K) and pressures ( $P \lesssim 1$ bar) characteristic of hot stars and irradiated exoplanets.

2. Population of H $^-$ : Statistical and Chemical Equilibrium

The calculation of the H $^-$ number density, $n_-$ , requires knowledge of local thermodynamical and chemical conditions:

LTE Saha Equation:

Under thermochemical equilibrium,

$\frac{n_{H^-}}{n_{\rm H} n_{e^-}} = \frac{Z_{H^-}}{Z_{\rm H} Z_{e^-}} \left( \frac{h^2}{2\pi m_e kT} \right)^{3/2} \exp\left( \frac{\chi}{kT} \right)$

where $\chi = 0.754$ eV is the H $^-$ binding energy and $Z$ are partition functions. $n_{H^-} \propto n_H n_{e^-} T^{-3/2}$ at fixed $n_H$ and $n_{e^-}$ (Lothringer et al., 2018, Lothringer et al., 2020, Barklem et al., 2024).

Non-LTE Corrections:

In high-precision applications (e.g., F-K dwarfs), statistical equilibrium calculations reveal small but measurable non-LTE corrections, especially at high $T_{\mathrm{eff}}$ and low $\log g$ . Over-recombination (radiative recombination exceeding photodetachment) leads to a departure coefficient $b_- \equiv n_-/n_-^* > 1$ (up to 1–2% in the hottest models). This is quantified using rate-equation networks involving radiative and collisional processes:

$b_- = b_1\,\frac{R_{\rm rec}^\dagger + r}{R_{\rm ph} + r}$

with $r \equiv n_e k_e + n_1 k_{\rm H}(1+\Omega)$ and $\Omega = \frac{n_-^*\,k_H}{n_1^*\,n_1\,k_{3H}}$ (Barklem et al., 2024).

Reaction Network:

Key reactions include photodetachment, radiative/associative detachment, collisional detachment and three-body processes for H $_2$ formation/dissolution, with state-of-the-art rates from laboratory and quantum calculations (Barklem et al., 2024).

3. Opacity Implementation in Model Atmospheres

The quantitative treatment of H $^-$ opacity is standardized in computational frameworks for both stellar and exoplanetary applications:

Opacity Calculations:

The total H $^-$ absorption at frequency $\nu$ is

$\alpha_\nu(H^-) = n_{H^-}\:\sigma_{\rm bf}(\nu, T) + n_H n_{e^-}\:\sigma_{\rm ff}(\nu, T)$

(Lothringer et al., 2020, Lothringer et al., 2018).

PHOENIX EOS/Opacity Framework:

Codes such as PHOENIX and ScCHIMERA solve for $n_{H^-}(T,P)$ using an equation-of-state (EOS) solver (e.g., ACES), merging H $^-$ opacity into detailed line-by-line or $k$ -coefficient radiative transfer (Lothringer et al., 2018, Lothringer et al., 2020, Arcangeli et al., 2018).

Data Sources and Tabulation:

Cross sections are fitted to quantum data (e.g., John 1988; Wishart 1979), tabulated at high $(\nu, T)$ resolution, and interpolated logarithmically. Partition functions for H and H $^-$ are sourced from up-to-date databases spanning $>900$ species (Lothringer et al., 2018, Lothringer et al., 2020).

Line Profile Considerations:

H $^-$ acts as a continuum source; line broadening formalism relevant for discrete transitions is not applied to its absorption. Intrinsic broadening physics (e.g., Gaunt factors) is folded into cross section tabulations (Lothringer et al., 2020).

Process	Cross Section Source	Description
Bound–free	Wishart (1979), John (1988)	Photodetachment, $\nu > \nu_0$
Free–free	John (1988), Bell & Berrington (1987)	Inverse bremsstrahlung
Population	ACES EOS, CEA2	Saha or Gibbs equilibrium

4. Impact on Opacity, Spectrum Formation, and Atmospheric Structure

The inclusion of H $^-$ opacity is fundamental for modeling radiative transfer and structure in both stellar and exoplanetary atmospheres:

Opacity Effects:

H $^-$ dominates visual and near-infrared continuum opacity for solar-type stars and irradiated gas giants at $T_{\rm eff} \gtrsim 2500$ K. In ultra-hot exoplanets (e.g., KELT-9b, WASP-18b), its continuum exceeds and suppresses molecular opacities (H $_2$ O, TiO) at $P \lesssim 0.1$ bar, 1.1–1.6 μm (Arcangeli et al., 2018, Lothringer et al., 2018).

Non-LTE Corrections to Opacity:

Non-LTE effects can alter the continuum opacity by $1$–$2$% at high $T_{\rm eff}$ and low $\log g$ ( $b_- \approx 1.01$ near $\log \tau_{5000}=1$ ), and by $0.1$–$0.2$% under solar parameters. $\Delta \kappa/\kappa_{\rm LTE} = b_- - 1$ quantifies the change (Barklem et al., 2024).

Spectral Consequences:

The increased H $^-$ opacity depresses continuum flux and weakens or erases weak-line equivalent widths, with scale $\sim 1/b_-$ . In hot Jupiters, the H $^-$ continuum mutes spectral features to the point that otherwise conspicuous H $_2$ O bands (e.g., 1.4 μm) become invisible (Arcangeli et al., 2018).

Feedback on Temperature–Pressure Profile:

H $^-$ opacity elevates the photosphere to lower pressure ( $\sim 0.01$ –$0.1$ bar) and enables thermal inversions by trapping incident starlight at intermediate depths—raising local $T$ by several hundred to $\sim$ 1000 K (Lothringer et al., 2018).

5. Numerical Techniques and Atmospheric Retrieval

State-of-the-art atmospheric modeling and retrieval incorporate the following computational strategies:

Opacity Grid Construction:

Gridded opacities in $(T,P,\lambda)$ are precalculated for both bound–free and free–free processes; codes interpolate or compute opacities "on-the-fly" for out-of-grid values (Lothringer et al., 2018, Arcangeli et al., 2018).

Radiative Transfer Integration:

H $^-$ opacity is added directly to the sum of line and CIA (collision-induced absorption) opacities. PHOENIX employs direct opacity sampling (dOS), line-by-line Feautrier solvers, and plane-parallel geometry for rapid, converged solutions on fine wavelength grids ( $\sim10^5$ – $10^6$ points) (Lothringer et al., 2018, Lothringer et al., 2020).

Atmospheric Retrieval:

Algorithms such as PETRA treat $n_e$ and $T(p)$ as free retrieval parameters, updating the H $^-$ opacity at each sampling step to jointly constrain atmospheric structure—even in the absence of molecular bands (Lothringer et al., 2020).

Validation:

PETRA/PHOENIX retrievals on synthetic data for KELT-9b produce temperature and electron density profiles matching input conditions to $\lesssim 1\sigma$ (Lothringer et al., 2020).

6. Current Limitations and Directions for Future Research

Several outstanding topics are recognized:

Completeness of Reaction Networks:

Current reaction networks, though informed by modern rates, omit vibrational/rotational H $_2$ state resolution, higher hydrogen molecules (H $_2^+$ , H $_3^+$ ), and mutual neutralization with metals. These may yield indirect effects on the H $^-$ equilibrium (Barklem et al., 2024).

Non-LTE Electron Populations:

Indirect effects via non-LTE modifications to electron density, particularly from metal ionization (e.g., Fe, Si), are not comprehensively modeled in current grids.

3D and Time-dependent Effects:

The presented analyses are 1D and steady-state; full 3D NLTE modeling and inclusion of atmospheric inhomogeneities (granulation, convection) are required for percent-level spectroscopic precision (Barklem et al., 2024).

Integration with Stellar and Exoplanetary Evolution:

Fine-grained H $^-$ opacity effects may influence interpretations of stellar parameter determination, exoplanet retrievals, and the construction of opacity tables used in evolutionary computations for substellar objects.

Future work is focused on enhancement of the reaction network, feedbacks on model atmosphere structures in full non-LTE, and the accurate tabulation of b $_-$ and $\kappa_{H^-}$ for spectrum-synthesis codes demanded by precision spectroscopy (Barklem et al., 2024).