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A&A
Volume 666, October 2022
Article Number L7
Number of page(s) 17
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202244944
Published online 07 October 2022

© S. Blouin 2022

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe-to-Open model. Subscribe to A&A to support open access publication.

1. Introduction

Cool (Teff ≲ 10 000 K) helium-atmosphere white dwarfs often have detectable traces of carbon in their atmospheres. Such carbon-polluted white dwarfs are known as DQ stars. Spectroscopically, DQ white dwarfs are recognized by the presence of C2 molecular bands from the Swan transitions in the optical region. Given the relentless action of gravitational settling, heavy elements such as carbon are a priori not expected to be present in white dwarf atmospheres. However, many physical processes can compete with gravitational settling. In the case of cool DQ stars, the presence of carbon is understood as the result of convective dredge-up from the deep interior (Pelletier et al. 1986; MacDonald et al. 1998; Althaus et al. 2005; Dufour et al. 2005; Koester et al. 2020). Under reasonable modelling assumptions, the most recent evolutionary calculations agree remarkably well with the observed carbon abundance pattern of DQ stars (Bédard et al. 2022).

The atmospheres of cool helium-rich white dwarfs are also frequently polluted by heavier elements, such as Ca, Mg, and Fe. Those are known as DZ stars. The origin of heavy elements in DZ white dwarfs is now convincingly explained by the recent or ongoing accretion of rocky debris (e.g., Jura & Young 2014; Farihi 2016). Many independent observations all converge to this explanation, including the abundance patterns observed in DZ atmospheres (Zuckerman et al. 2007; Klein et al. 2010; Gänsicke et al. 2012; Doyle et al. 2019; Harrison et al. 2021), the identification of circumstellar disks (Rocchetto et al. 2015; Wilson et al. 2019; Manser et al. 2020), the discovery of planetary debris by the transit method (Vanderburg et al. 2015; Vanderbosch et al. 2020, 2021), and the detection of X-rays from the accretion process itself (Cunningham et al. 2022).

Approximately 15% of all helium-atmosphere white dwarfs are DZs (McCleery et al. 2020; Hollands et al. 2022). Under the assumption that the DQ phenomenon is independent from the external accretion of rocky planetesimals, about 15% of DQ white dwarfs should also have atmospheres polluted by elements heavier than carbon. However, it is well established that DQZ and DZQ stars (white dwarfs showing both carbon features and absorption lines from heavier elements) are exceedingly rare. Only about 2% of DQs show the presence of heavier elements in their spectra (Coutu et al. 2019; Farihi et al. 2022). To add to this mystery, the inferred external accretion rates of the few known DQZs are orders of magnitude smaller than those typical of DZ stars. To solve this long-standing conundrum, Farihi et al. (2022) recently suggested that DQ stars are the product of a binary evolution that has altered their circumstellar environments in a way that prevents the pollution of the white dwarf.

In this Letter, we show that this hypothesis is not necessary. We demonstrate that the C2 Swan bands of typical DQ white dwarfs are efficiently suppressed if the atmosphere is polluted by an amount of metals typical of that observed in DZ stars. This naturally explains both the paucity of DQZ white dwarfs and the systematically low accretion rates inferred for known DQZ stars. In Sect. 2, we present DQZ model atmosphere calculations that show how the Swan bands can disappear if the atmosphere is moderately polluted by heavy elements following the accretion of rocky debris. We then detail in Sect. 3 the physical mechanism that leads to the suppression of the Swan bands. In Sect. 4, we compare our DQZ models to observations to demonstrate that the rarity of DQZ stars can naturally be explained by this process. Finally, our conclusions are stated in Sect. 5.

2. Metals can suppress the C2 Swan bands

The possibility of the suppression of Swan bands in DQZ stars was recently studied by Hollands et al. (2022). Their Fig. 3 shows how their models predict that the Swan bands disappear if a sufficient quantity of polluting metals is added to the atmosphere while keeping the other parameters constant (specifically, Teff = 8100 K, log g = 8.02, log H/He = −3.21, and log C/He = −4.5). However, Hollands et al. (2022) argue that this disappearance of the Swan bands cannot explain the extreme rarity of DQZ stars, since they find that the Swan bands are suppressed only at extreme levels of metal pollution. Therefore, they conclude that while the Swan bands can be inhibited to the point of becoming undetectable in some strongly polluted white dwarfs, they are generally too weakly suppressed to explain the dearth of DQZ white dwarfs.

In Fig. 1 we repeat the exercise of Hollands et al. (2022) using our own state-of-the-art model atmospheres (Blouin et al. 2018a,b, 2019). We examine how the strength of the Swan bands changes for a normal-mass white dwarf with Teff = 8000 K and log C/He = −4.5 as a function of the external pollution level log Ca/He2 (no hydrogen is included). These atmospheric parameters are similar to those used in Fig. 3 of Hollands et al. (2022), but there are small differences since in that work the hydrogen and individual metal abundances were adjusted to the specific case of SDSS J095645.12+591240.7. We qualitatively replicate their finding that a high metal pollution of log Ca/He ≳ −8.5 is required to strongly suppress the Swan bands at that temperature and carbon abundance. This good agreement between independent calculations indicates that differences with respect to the constitutive physics of our atmosphere code and that used in Hollands et al. 2022 (Koester 2010) are negligible in the context of this work.

thumbnail Fig. 1.

Synthetic spectra for a white dwarf with Teff = 8000 K, log g = 8, log C/He = −4.5, and varying amounts of polluting metals. No hydrogen trace was included. The blue regions indicate the location of the two strongest C2 Swan bands.

For their analysis, Hollands et al. (2022) chose a fixed carbon abundance of log C/He = −4.5. This is not the ideal value to study the Swan bands’ suppression: DQ white dwarfs at 8000 K typically have 10 times less carbon than that. This is shown in Fig. 2, where the carbon abundances of a large sample of DQ white dwarfs are plotted as a function of Teff. At 8000 K, log C/He  =   − 4.5 (identified as a grey star) lies a full one dex above the sequence on which the vast majority of DQs are found. While some stars are in this upper region, apparently forming a continuous sequence with the ‘warm’ DQs found above 10 000 K (Coutu et al. 2019; Koester & Kepler 2019), they likely have a different origin than the classical DQs, possibly being the descendants of the hot DQ white dwarfs. In short, the analysis presented in Fig. 1 of this work and in Fig. 3 of Hollands et al. (2022) is not representative of the typical DQ white dwarf.

thumbnail Fig. 2.

Photospheric carbon abundance of DQ white dwarfs as a function of the effective temperature (data taken from Coutu et al. 2019 and Blouin & Dufour 2019). The vast majority of DQ white dwarfs follow a clear sequence in this plane as indicated by the dashed red line.

We now repeat the exercise of Hollands et al. (2022), but this time using a carbon abundance of log C/He = −5.5, a representative composition for a 8000 K DQ. Figure 3 reveals that the weaker Swan bands are more easily erased from the spectrum when metals are added. The C2 bands are very shallow even at the relatively low pollution level of log Ca/He = −10.5. This result is hardly surprising (if there is less carbon, then surely the Swan bands are weaker for any given metal pollution level), but it reopens the door to the idea that the dearth of DQZ white dwarfs could be explained by this simple effect. To test this idea, we need to compare our synthetic spectra to observations. This is the subject of Sect. 4; for now, we turn to the identification of the physical mechanism responsible for the Swan bands’ suppression.

thumbnail Fig. 3.

Same as Figure 1, but this time assuming log C/He = −5.5. A different vertical scale is used compared to Fig. 1.

3. Physical explanation

To identify the reason behind the disappearance of the C2 bands following the addition of metals, we examine in more details the log Ca/He = −11.5 and log Ca/He = −9.5 models of Fig. 3. The first thing to establish is at which Rosseland optical depths (τR) the Swan bands are formed in the vertical stratification of the atmosphere. To answer this question, we have computed a series of model spectra where the Swan bands’ opacity was artificially ignored at certain optical depths (Fig. A.1). From this exercise, we conclude that at this effective temperature and carbon abundance, the Swan bands are mostly formed in the 0.001 ≲ τR ≲ 0.03 range.

Equipped with this information, we can now compare the log Ca/He = −11.5 and log Ca/He = −9.5 models to see how they differ. Figure 4 compares their opacities as a function of τR at λ = 5160 Å (which corresponds to the wavelength where the Swan bands are strongest). It shows that the total opacity at a given τR (dashed lines) does not change much following the hundredfold increase in external metal pollution. In contrast, the C2 Swan opacity (solid lines) decreases by a factor ≃10 in the region where the Swan bands are formed (0.001 ≲ τR ≲ 0.03). Therefore, the suppression of the Swan bands seen in Fig. 3 is not caused by a masking effect from other opacity sources, but rather by a decrease in the C2 opacity itself.

thumbnail Fig. 4.

Opacity at 5160 Å as a function of the Rosseland optical depth for DQZ white dwarfs with Teff = 8000 K, log g = 8, log C/He = −5.5, and different Ca/He values (black lines correspond to log Ca/He = −11.5 and red lines to log Ca/He = −9.5). The dashed lines show the total opacity, while the solid lines show the opacity due to the Swan bands only. The wavelength chosen for this figure (5160 Å) corresponds to where the Swan bands are strongest. The grey region indicates where the Swan bands are formed.

By inspecting our model structures, we found that this decline in the Swan opacity is directly caused by a tenfold decrease in the C2 number density in the same region of the atmosphere (Fig. A.2). This decrease is in turn largely explained by a ≃2.5× decrease in the total density between the log Ca/He = −11.5 and log Ca/He = −9.5 models in the relevant atmospheric layers (0.001 ≲ τR ≲ 0.03, Fig. A.3). The reason for this decline is well documented (Provencal et al. 2002; Dufour et al. 2005; Bergeron et al. 2019). Metals act as electron donors that increase the total opacity of helium-dominated atmospheres mostly through an increase in the He− free-free opacity, which dominates the total atmospheric opacity in those stars (Saumon et al. 2022, Fig. 18). Due to this opacity increase, a given optical depth τR is attained higher up in the atmosphere, at a lower pressure.

The impact of this density decline on the molecular carbon abundance can be understood by considering the C2 dissociation equilibrium equation. Neglecting nonideal effects, it is given by

n C 2 n C 2 = Q C 2 ( T ) Q C 2 ( T ) ( 2 π m C 2 k B T m C 2 h 2 ) 3 / 2 e − D 0 / k B T ≡ f ( T ) , $$ \begin{aligned} \frac{n_{\rm \scriptscriptstyle C}^2}{n_{\mathrm{\scriptscriptstyle C}_2}} = \frac{Q_{\rm \scriptscriptstyle C}^2 (T)}{Q_{\mathrm{\scriptscriptstyle C_2}} (T)} \left( \frac{ 2\pi m_{\rm \scriptscriptstyle C}^2 k_{\rm B} T}{m_{\mathrm{\scriptscriptstyle C_2}} h^2}\right)^{3/2} e^{-D_0 / k_{\rm B} T} \equiv f(T), \end{aligned} $$(1)

where ni is a number density, Qi(T) is a partition function, mi is a mass, kB is the Boltzmann constant, h is the Planck constant, and D0 = 6.21 eV is the dissociation energy of the C2 molecule. At a fixed temperature and given C/He, we have

n C 2 ∝ n C 2 ∝ ρ 2 , $$ \begin{aligned} n_{\mathrm{\scriptscriptstyle C}_2} \propto n_{\rm \scriptscriptstyle C}^2 \propto \rho ^2, \end{aligned} $$(2)

meaning that the 2.5× decrease in mass density noted above translates into a ≃6× reduction of the C2 density.

In addition to this change in the density structure, the temperature profile is also affected (Fig. A.3). This is a second-order yet still important effect. In the region of interest (0.001 ≲ τR ≲ 0.03), the temperature rises by ≃400 K following the increase in metal pollution. This in turn translates into a twofold increase in the right-hand side of Eq. (1) (which we have plotted in Fig. A.4 for reference). Combining this effect with the density effect described in the previous paragraph, we see how moderate changes to the density and temperature stratifications explain the tenfold decrease in the C2 density in the region of the atmosphere where the Swan bands are formed. We can conclude that the suppression of the Swan bands described in Sect. 2 is due to changes to the atmospheric structure that tilt the scale in favour of dissociation in the C2/C equilibrium equation.

4. Implications for DQZ white dwarfs

In the previous sections, we have seen how and why the accretion of metals on DQ white dwarfs can suppress their C2 Swan bands. This suggests that DQ white dwarfs may transform directly into DZs (and not DQZs) following the accretion of rocky debris, which would naturally explain the apparent dearth of DQZ stars. To test this hypothesis, we turn to the Sloan Digital Sky Survey (SDSS) DZ sample of Dufour et al. (2007). This sample contains 72 DZ white dwarfs cooler than 9000 K (a temperature range where the classical DQ sequence is well populated, see Fig. 2) that have been recently reanalysed by Coutu et al. (2019) using Gaia parallaxes and updated model atmospheres. Under the assumption that DQ white dwarfs descend from stars that hosted planetary systems similar to those hosted by the progenitors of non-DQ white dwarfs, we should expect that 11 ± 3 stars in this sample were DQs before accreting planetary material (given that ≃15% of cool helium-atmosphere white dwarfs are DQs, McCleery et al. 2020). The probability that none was a DQ is a priori ≲10−5. The fact that Swan bands are detected in none of the 72 objects can mean two things: either the progenitors of DQs actually had different planetary system architectures after all (Farihi et al. 2022) or DQs generally transform directly into DZs after the accretion of metals. Of course, those two scenarios are not mutually exclusive, but if DQs do transform directly into DZs, then the need for the first scenario lessens considerably.

To demonstrate that most DQs do transform directly into DZs, we have calculated model atmospheres for each of those 72 objects using the Teff, log g, Ca/He, and H/He values given in Coutu et al. (2019). In addition, we have computed models where we have boosted the carbon abundance so that C/He matches the expected carbon abundance for a DQ at that particular effective temperature (following the dashed red line of Fig. 2). The resulting synthetic spectra are shown in Fig. 5. For each object, we show four different models:

  • The blue model is a reference model computed assuming the same parameters as those found by Coutu et al. (2019). It has the same parameters as those given in each panel, except for C/He, which is fixed so that C/Ca is chondritic.

  • The orange model contains a trace of carbon typical of a DQ white dwarf at that temperature. It has the same parameters as those given in each panel.

  • The green model is identical to the orange model, except that no trace of hydrogen was included in the calculations. In contrast, in Coutu et al. (2019) and in the orange models, H/He was adjusted to reproduce the Balmer lines or to match the visibility limit of Hα if no Balmer line is detected (in that case, an asterisk is displayed next to the hydrogen abundance in Fig. 5).

  • The red model (dashed line) is a DQ model with no polluting metals other than carbon. It has the same parameters as those given in each panel, except that H/He = 0 and Ca/He = 0.

thumbnail Fig. 5.

Comparison between SDSS spectra (grey) and different model atmospheres (described in the text, see Sect. 4) convolved at the SDSS resolution. Panels for the 66 remaining objects are available in Fig. A.5.

In all cases, a full model structure was recalculated: the synthetic spectra and underlying model atmospheres are fully consistent. The only parameters that were adjusted to the observations are the absolute scale and the slope of the synthetic spectra. All other parameters were fixed as described above. Finally, since Coutu et al. (2019) adjusted Ca/He while keeping other abundance ratios constant (e.g., Mg/He), some objects with high Mg/Ca are not well represented by our models in the Mg I 5168/5174/5185 Å triplet region. In the context of this work, this is only a cosmetic issue which can safely be ignored.

The first thing to notice is that in most cases, the SDSS spectra have a sufficiently high signal-to-noise ratio (S/N) such that Swan bands could be detected in DQ stars at those same Teff and with typical carbon abundances (red dashed lines). In contrast, for the vast majority of objects, the same amount of carbon should not be detectable once we account for the presence of polluting metals (orange lines). For most stars, it would remain impossible to conclusively detect Swan bands even with observations with S/N = 100. This conclusion still holds if we remove any hydrogen impurities from the model calculations (green lines). Hydrogen can further suppress the Swan bands by adding more free electrons to the atmosphere, which explains the difference between the orange and green models.

The key takeaway from Fig. 5 is that if, as expected, some objects in this sample are stars that have dredged up carbon from their deep interiors, they could not have been classified as DQZ because their Swan bands are too strongly suppressed to be detectable. The rarity of DQZ white dwarfs is not a surprise and is naturally explained by the well-known effect that electron donors have on the atmospheric structures of helium-dominated atmospheres. Figure 5 also shows why it is unsurprising that the few known DQZ white dwarfs have very small accretion rates compared to the DZ population (Farihi et al. 2022). The few objects where the Swan bands could have a chance of being detected with higher S/N observations are precisely those with the lowest levels of external pollution. For instance, we estimate that the Swan bands in the orange models of Figs. 5, A.5 for the three most weakly polluted objects could be conclusively detected with a S/N ≳ 40 (WD 1005+030, log Ca/He = −11.32) and ≳150 (LSPM J1341+4253, log Ca/He = −11.16; USNO−B1.0 0937−00210798, log Ca/He = −10.95).

So far, the hunt for more DQZs has been mostly focussed on a search for metal lines (chiefly Ca II H & K) in known DQ white dwarfs (Farihi et al. 2022). By exclusively selecting known DQs, this survey strategy is effectively selecting objects that are very unlikely to be DQZs. In fact, since most known DQ white dwarfs have easily detectable Swan bands, they are objects that must have very low (if any) external pollution, otherwise they would not have strong Swan bands in the first place. A more promising survey strategy might be to look for very shallow Swan bands in known DZ white dwarfs with very low Ca/He, since a low external pollution implies a weaker suppression of the Swan bands.

5. Conclusion

Using state-of-the-art model atmospheres, we have shown that a classical DQ star with a typical carbon abundance directly transforms into a DZ white dwarf following the accretion of a typical amount of metals from planetary debris. The accreted material decreases the density of the atmosphere, which results in a smaller C2 abundance and a strong suppression of the Swan bands. This naturally explains the observed paucity of DQZ stars as well as the very small accretion rates inferred for known DQZs.

Those findings nullify the main argument put forward by Farihi et al. (2022) to support the idea that all DQ stars are the product of binary evolution. We cannot definitely rule out this scenario, but our conclusions at least lessen the need for this hypothesis. That being said, we recognize that there are still other properties of DQ stars that remain hard to explain (notably the observed deficit of unevolved companions in post-common-envelope binaries) and each of those should be further scrutinized.

1

In this work, log X/He corresponds to the logarithm of the number abundance ratio, log10nX/nHe.

2

We only quote the Ca abundance, but all metals from N to Cu are included with chondritic abundance ratios with respect to Ca. This applies to all models presented in this work.

Acknowledgments

S.B. thanks Pierre Bergeron, Patrick Dufour and Antoine Bédard for useful discussions. S.B. is a Banting Postdoctoral Fellow and a CITA National Fellow, supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). This work has made use of the Montreal White Dwarf Database (Dufour et al. 2017).

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Appendix A: Supplementary figures

thumbnail Fig. A.1.

Synthetic spectra of a DQZ white dwarf with Teff = 8000 K, log g = 8, log C/He = −5.5, and log Ca/He = −11.5. Each spectrum was computed using the same unperturbed model structure, but the Swan bands’ opacity was omitted from the spectrum calculation for Rosseland optical depths smaller than the value given in the legend. For example, for the grey spectrum, the Swan bands’ opacity was only included for atmospheric layers deeper than τR = 0.01.

thumbnail Fig. A.2.

Molecular carbon number density as a function of the Rosseland mean optical depth for DQZ white dwarfs with Teff = 8000 K, log g = 8, log C/He = −5.5, and different Ca/He values (the black line corresponds to log Ca/He = −11.5 and the red line to log Ca/He = −9.5).

thumbnail Fig. A.3.

Mass density (top panel) and temperature (bottom panel) as a function of the Rosseland mean optical depth for DQZ white dwarfs with Teff = 8000 K, log g = 8, log C/He = −5.5, and different Ca/He values (black lines correspond to log Ca/He = −11.5 and red lines to log Ca/He = −9.5).

thumbnail Fig. A.4.

Right-hand side of Equation (1). The partition functions given in Irwin (1981) were used for this exercise.

thumbnail Fig. A.5.

Comparison between SDSS spectra (grey) and different model atmospheres (described in the text, see Section 4) convolved at the SDSS resolution. Continued from Figure 5.

thumbnail Fig. A.5.

continued.

thumbnail Fig. A.5.

continued.

thumbnail Fig. A.5.

continued.

thumbnail Fig. A.5.

continued.

thumbnail Fig. A.5.

continued.

thumbnail Fig. A.5.

continued.

thumbnail Fig. A.5.

continued.

thumbnail Fig. A.5.

continued.

thumbnail Fig. A.5.

continued.

thumbnail Fig. A.5.

continued.

All Figures

thumbnail Fig. 1.

Synthetic spectra for a white dwarf with Teff = 8000 K, log g = 8, log C/He = −4.5, and varying amounts of polluting metals. No hydrogen trace was included. The blue regions indicate the location of the two strongest C2 Swan bands.
In the text
thumbnail Fig. 2.

Photospheric carbon abundance of DQ white dwarfs as a function of the effective temperature (data taken from Coutu et al. 2019 and Blouin & Dufour 2019). The vast majority of DQ white dwarfs follow a clear sequence in this plane as indicated by the dashed red line.
In the text
thumbnail Fig. 3.

Same as Figure 1, but this time assuming log C/He = −5.5. A different vertical scale is used compared to Fig. 1.
In the text
thumbnail Fig. 4.

Opacity at 5160 Å as a function of the Rosseland optical depth for DQZ white dwarfs with Teff = 8000 K, log g = 8, log C/He = −5.5, and different Ca/He values (black lines correspond to log Ca/He = −11.5 and red lines to log Ca/He = −9.5). The dashed lines show the total opacity, while the solid lines show the opacity due to the Swan bands only. The wavelength chosen for this figure (5160 Å) corresponds to where the Swan bands are strongest. The grey region indicates where the Swan bands are formed.
In the text
thumbnail Fig. 5.

Comparison between SDSS spectra (grey) and different model atmospheres (described in the text, see Sect. 4) convolved at the SDSS resolution. Panels for the 66 remaining objects are available in Fig. A.5.
In the text
thumbnail Fig. A.1.

Synthetic spectra of a DQZ white dwarf with Teff = 8000 K, log g = 8, log C/He = −5.5, and log Ca/He = −11.5. Each spectrum was computed using the same unperturbed model structure, but the Swan bands’ opacity was omitted from the spectrum calculation for Rosseland optical depths smaller than the value given in the legend. For example, for the grey spectrum, the Swan bands’ opacity was only included for atmospheric layers deeper than τR = 0.01.
In the text
thumbnail Fig. A.2.

Molecular carbon number density as a function of the Rosseland mean optical depth for DQZ white dwarfs with Teff = 8000 K, log g = 8, log C/He = −5.5, and different Ca/He values (the black line corresponds to log Ca/He = −11.5 and the red line to log Ca/He = −9.5).
In the text
thumbnail Fig. A.3.

Mass density (top panel) and temperature (bottom panel) as a function of the Rosseland mean optical depth for DQZ white dwarfs with Teff = 8000 K, log g = 8, log C/He = −5.5, and different Ca/He values (black lines correspond to log Ca/He = −11.5 and red lines to log Ca/He = −9.5).
In the text
thumbnail Fig. A.4.

Right-hand side of Equation (1). The partition functions given in Irwin (1981) were used for this exercise.
In the text
thumbnail Fig. A.5.

Comparison between SDSS spectra (grey) and different model atmospheres (described in the text, see Section 4) convolved at the SDSS resolution. Continued from Figure 5.
In the text
thumbnail Fig. A.5.

continued.
In the text
thumbnail Fig. A.5.

continued.
In the text
thumbnail Fig. A.5.

continued.
In the text
thumbnail Fig. A.5.

continued.
In the text
thumbnail Fig. A.5.

continued.
In the text
thumbnail Fig. A.5.

continued.
In the text
thumbnail Fig. A.5.

continued.
In the text
thumbnail Fig. A.5.

continued.
In the text
thumbnail Fig. A.5.

continued.
In the text
thumbnail Fig. A.5.

continued.
In the text

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