Dilution of accreted planetary matter in hot DA white dwarfs according to their mass

A large proportion of observed white dwarfs (WDs) show evidence of debris disks, remnants of the former planetary systems, and/or signatures of heavy elements in their atmospheres, induced by the accretion of planetary matter onto their surfaces. The observed abundances are the result of the balance between the accretion flux and the dilution of this planetary material by internal transport processes. A recent study showed that more massive DA WDs are less polluted than smaller mass ones. It was suggested that the reason could be related to the formation of planetary systems when these stars were on the main sequence. The aim of this work is to test how internal dilution processes, including thermohaline convection, change with WD masses, and whether such an effect could account for variations in the observed pollution. We computed the efficiency of atomic diffusion and thermohaline convection after the accretion of heavy elements onto WDs using static DA models with various masses, effective temperatures, and hydrogen contents. We confirm that thermohaline convection is always more efficient in diluting accreted elements than atomic diffusion, as previously shown in the literature. However, we find that element dilution by thermohaline convection is less efficient in massive WDs than in smaller mass ones, due to their larger internal density. We showed that the differences in observed heavy element pollution in WDs according to their masses cannot be explained by the dilution induced by atomic diffusion and thermohaline mixing alone. Indeed, the pollution by planetary system accretion should be more easily detectable in massive WDs than in low-mass ones. We discuss other processes that should be taken into account before drawing any conclusion about the occurrences of planetary systems according to the mass of the star on the main sequence.

💡 Research Summary

White dwarfs (WDs) represent the final evolutionary stage of the majority of stars, and a substantial fraction of them exhibit signatures of planetary debris: infrared excesses from dusty disks and heavy‑element absorption lines in their atmospheres. These metals cannot remain in the photosphere for long because gravitational settling drives them downward. Consequently, the observed atmospheric abundances are set by a balance between the accretion rate of planetary material and the internal processes that dilute or transport those elements deeper into the star.

A recent statistical study (OuId Rouis et al. 2024, hereafter OR24) examined more than 250 hot hydrogen‑rich (DA) WDs with effective temperatures between 13 kK and 30 kK. They found a striking mass dependence: roughly 44 % of low‑mass WDs (M < 0.7 M⊙) show silicon (and sometimes carbon) lines, whereas only about 11 % of high‑mass WDs (M > 0.8 M⊙) are polluted. Two explanations have been proposed. One attributes the trend to differences in planetary system formation around main‑sequence progenitors of different masses. The other suggests that internal dilution mechanisms might operate more efficiently in low‑mass WDs, thereby making pollution easier to detect. However, previous works, including OR24, considered only atomic diffusion (gravitational settling) and ignored thermohaline convection, a double‑diffusive instability that can be triggered when heavy material overlies lighter material.

The present paper addresses this gap by modelling the post‑accretion transport of heavy elements in a grid of static DA WD structures that span three masses (0.6, 0.8, and 1.0 M⊙) and three effective temperatures (20, 25, and 30 kK). The stellar structures are built with the STELUM code, using core compositions from BASTI evolutionary calculations (mass‑dependent C/O ratios), a fixed C/O/He mantle (10⁻¹·³ M⊙), and a pure hydrogen envelope with a mass fraction H/M_WD = 10⁻⁴.

Accretion is modelled as an instantaneous deposition of a total mass M_accr = 10¹⁶ g (comparable to the mass of Phobos) into a thin surface layer of mass M_mix = 10¹⁵ g (depth 39–141 km depending on the model). Above this mixed zone the accreted material is assumed to decay exponentially with radius, characterized by a scale σ that controls the steepness of the decay. Two scenarios are explored:

• Case 1 – σ is the same (0.25 % of the stellar radius) for all three masses. This leads to different mean‑molecular‑weight (μ) gradients in each model, because the same absolute mass distribution corresponds to different fractional depths.

• Case 2 – σ is tuned for each mass (0.25 %, 0.177 %, 0.132 % of the radius for 0.6, 0.8, and 1.0 M⊙ respectively) so that the resulting μ‑gradient profiles are identical across the three models. This isolates the intrinsic effect of stellar mass on the transport processes.

Atomic diffusion coefficients are calculated using the prescription of Michaud et al. (2015) for trace elements in a fully ionised hydrogen plasma. Magnesium (A = 24.5, Z = 12) is taken as a representative heavy element, and the diffusion coefficient scales as D_ip ∝ T⁵⁄² / (n_p Z_i² A_i).

Thermohaline convection is treated with the classic Kippenhahn et al. (1980) formulation, later justified by 3‑D simulations (Denissenkov 2010; Traxler et al. 2011). The diffusion coefficient for thermohaline mixing is

 D_th = C_t κ_T R₀⁻¹,

where C_t = 12 (aspect‑ratio factor), κ_T is the thermal diffusivity (κ_T = 4acT³ / (3 κ C_p ρ²)), and R₀ = (∇_ad − ∇) / |∇_μ| is the density ratio that measures the competition between stabilising temperature gradients and destabilising μ‑gradients.

Results

1. Thermohaline vs. atomic diffusion – In all models and both cases, thermohaline mixing is 1–2 orders of magnitude more efficient than atomic diffusion. The heavy‑element mass fraction drops sharply within the outer 10⁻¹²–10⁻¹⁰ M_WD, creating a thin, well‑mixed surface layer followed by an exponential decline.

2. Mass dependence – Despite the higher efficiency of thermohaline mixing, its absolute effectiveness declines with increasing WD mass. The reason is the higher internal density of massive WDs, which raises the thermal diffusivity κ_T and the density ratio R₀, thereby reducing D_th. Consequently, for the same accreted mass, a 1.0 M⊙ WD retains a higher surface abundance than a 0.6 M⊙ WD after the same diffusion time.

3. Case 1 vs. Case 2 – Adjusting σ to equalise μ‑gradients (Case 2) removes the artificial enhancement of thermohaline mixing in low‑mass models caused by steeper μ‑gradients. Nevertheless, even with identical μ‑profiles, the massive models still exhibit lower D_th because the underlying physical parameters (ρ, κ_T) differ.

4. Implications for observations – If internal dilution were the sole driver, one would expect polluted signatures to be more readily detectable in massive WDs, opposite to the trend reported by OR24. Therefore, the observed decrease in pollution fraction with mass cannot be explained by atomic diffusion and thermohaline mixing alone.

Discussion

The authors argue that additional factors must be invoked. Possibilities include:

• Planetary system occurrence – High‑mass progenitors (> 3.5 M⊙) may form fewer or less massive rocky bodies, reducing the supply of accretable material.

• Dynamical delivery efficiency – The architecture of planetary systems around massive stars could lead to fewer bodies being scattered onto star‑grazing orbits during the post‑main‑sequence phase.

• Other internal processes – Rotationally induced mixing, magnetic fields, or shallow convection zones could modify the dilution timescales.

• Accretion geometry – If accretion is not globally spherical but confined to localized spots, the effective dilution volume could differ with mass.

The paper also stresses that previous works that ignored thermohaline convection (e.g., Harrison et al. 2021 for DB WDs) may still be valid because thermohaline mixing is much weaker in helium‑rich atmospheres. However, for hydrogen‑rich DA WDs, thermohaline convection is a dominant transport mechanism that must be included in any quantitative interpretation of observed metal abundances.

Conclusions

• Thermohaline convection is always more efficient than atomic diffusion in diluting accreted planetary material in hot DA WDs.

• The efficiency of thermohaline mixing declines with increasing WD mass because of higher internal densities and larger thermal diffusivities.

• The observed lower fraction of polluted high‑mass DA WDs cannot be explained by internal dilution alone; external factors related to planetary system formation, dynamical delivery, or additional internal mixing processes are required.

• Future work should combine detailed population synthesis, dynamical modeling of planetary system evolution, and more sophisticated stellar interior physics (including rotation and magnetism) to disentangle the relative contributions of these effects.

Overall, this study provides the first systematic, mass‑dependent assessment of thermohaline mixing in polluted DA white dwarfs and highlights the need for a multi‑faceted approach when using WD pollution as a probe of exoplanetary system demographics.