Complex hydrogen chemical equilibrium and Gaia low mass problem in cool white dwarfs

Large Gaia data set shows substantial misfit between models and observation for cool white dwarfs with $T_{\rm eff}<6000,\rm K$, resulting in severe underestimation of masses of these stars. We aim to understand the underlying modelling issues. State of the art atmosphere models have been applied to analyse the Gaia DR3 sample of white dwarfs as well as quantum mechanical calculations to quantify formation and stability of different hydrogen species in the atmospheres of these stars. We reconcile the models and observations when we artificially suppress formation of $\rm H_3^+$ species, a process which substantially alters the chemical equilibrium at $T_{\rm eff}<6000,\rm K$, resulting in an overabundance of free electrons and $\rm H^-$, and strengthening of $\rm H^-$ bound-free absorption. Removing the $\rm H_3^+$ species from chemical equilibrium consideration makes ionization of hydrogen atoms the main source of free electrons, with the resulting models reproducing well the Gaia white dwarfs cooling branch. Because $\rm H_3^+$ must form under the considered conditions, likely it is the overestimation of its partition function and resulting abundance or the formation of $\rm H_3^-$ or another anionic species that suppresses the formation of $\rm H^-$ as a countercharge for $\rm H_3^+$ in current models. Chemical equilibrium in cool, hydrogen atmospheres white dwarfs must be reconsidered in respect to the abundance of $\rm H_3^+$ species and presence of unaccounted charge species.

💡 Research Summary

The paper addresses a striking discrepancy revealed by Gaia DR3: cool (T_eff < 6000 K) hydrogen‑atmosphere white dwarfs appear systematically under‑massive by up to ~0.2 M⊙ when compared with predictions from state‑of‑the‑art atmosphere models. The authors set out to identify the root cause of this “Gaia low‑mass problem.” They employ the same sophisticated white‑dwarf atmosphere code that includes Lyman‑α red‑wing opacity (Kowalski & Saumon 2006) and H⁻ bound‑free absorption (John 1988), together with a full chemical equilibrium network comprising H, H⁺, H₂, H₂⁺, H⁻, H₃⁺ and free electrons. The models have previously reproduced the spectral energy distributions of many cool DA white dwarfs down to ~4000 K, but they fail to match the Gaia G versus (G_BP–G_RP) colour‑magnitude diagram for the coolest objects.

A detailed inspection of the equilibrium abundances shows that for T_eff > 6000 K atomic hydrogen dominates, providing most free electrons via H ↔ H⁺ + e⁻, and consequently a substantial H⁻ population that yields the observed H⁻ bound‑free opacity. Below 6000 K, molecular hydrogen (H₂) becomes abundant, and the positively charged ion H₃⁺ (trihydrogen cation) emerges as the most abundant ion because of its low formation energy. The rise of H₃⁺ dramatically reduces the free‑electron density and the H⁻ abundance, weakening H⁻ bound‑free opacity and shifting the synthetic cooling track away from the Gaia sequence.

To test the impact of H₃⁺, the authors artificially suppress its formation in the equilibrium calculation. In this “no‑H₃⁺” scenario the ionisation equilibrium is governed mainly by H ↔ H⁺, with a modest contribution from H₂⁺. Free‑electron and H⁻ densities increase, strengthening H⁻ bound‑free absorption. The resulting colour‑magnitude track now overlays the Gaia data very well, reproducing both the slope and the termination near 4000 K. This demonstrates that the presence of H₃⁺, as currently modelled, is the primary source of the mismatch.

The paper then explores why the model may over‑predict H₃⁺. The abundance of H₃⁺ is directly proportional to its internal partition function. Existing partition functions (e.g., Neale & Tennyson 1995) differ by an order of magnitude among various studies, and the models adopt the highest values. More recent path‑integral Monte‑Carlo calculations (Kylänpää & Rantala 2011) give partition functions roughly a factor of 12 lower, which, when implemented, reduce H₃⁺ and H⁻ abundances and bring the models back into agreement with Gaia.

An alternative explanation is that an as‑yet‑unaccounted anionic species neutralises the positive charge of H₃⁺, preventing the depletion of free electrons. The authors discuss possible candidates: H₂⁻, H₃⁻, or a hypothetical H₂⁻‑like species. Density‑functional theory (DFT) calculations suggest H₃⁻ could be marginally more stable than a pair of H₂ + H⁻ (by ~0.06 eV). Experimental reports of H₃⁻ as a metastable ion exist, though thermochemical data are scarce. If such anions form in cool, dense white‑dwarf atmospheres, they would balance the charge budget, maintaining higher electron and H⁻ densities even in the presence of H₃⁺.

The authors also assess non‑ideal effects (density, refractive index) that could modify ionisation potentials. While the photospheric density rises modestly at the coolest temperatures, it remains far below the 0.1 g cm⁻³ threshold where non‑ideal shifts become significant (Kowalski 2010). Hence, non‑ideal corrections cannot explain the observed discrepancy.

In conclusion, the study identifies the treatment of hydrogen chemical equilibrium—specifically the abundance of H₃⁺ and the possible omission of compensating anions—as the key factor behind the Gaia low‑mass problem for cool DA white dwarfs. Correcting the H₃⁺ partition function or incorporating missing anionic species restores agreement between models and observations. The work highlights the need for high‑accuracy quantum‑chemical calculations of exotic hydrogen ions and for experimental validation of their thermodynamic properties, paving the way for more reliable white‑dwarf cooling models and age determinations.