<figure><img src="image_1.png" /></figure>

# Analysis of Helium-rich White Dwarfs Polluted by Heavy Elements in the Gaia Era

S. Coutu¹, P. Dufour¹, P. Bergeron¹ ID, S. Blouin¹ ID, E. Loranger¹, N. F. Allard²,³, and B. H. Dunlap⁴,⁵

¹Département de Physique, Université de Montréal, Montréal, QC H3C 3J7, Canada; coutu@astro.umontreal.ca, dufourpa@astro.umontreal.ca, bergeron@astro.umontreal.ca, sblouin@astro.umontreal.ca

²GEPI, Observatoire de Paris, Université PSL, CNRS, UMR 8111, 61 avenue de l'Observatoire, F-75014 Paris, France

³Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98bis boulevard Arago, F-75014 Paris, France

⁴Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA

⁵Department of Astronomy and McDonald Observatory, University of Texas at Austin, Austin, TX 78712, USA

Received 2019 July 12; revised 2019 September 16; accepted 2019 September 18; published 2019 November 1

## Abstract

We present an analysis of 1023 DBZ/DZ(A) and 319 DQ white dwarf stars taken from the Montreal White Dwarf Database. This represents a significant increase over the previous comprehensive studies on these types of objects. We use new trigonometric parallax measurements from the Gaia second data release, together with photometry from the Sloan Digital Sky Survey, Pan-STARRS, Gaia, or BVRI from the literature, which allow the determination of the mass for the majority of the objects in our sample. We use the photometric and spectroscopic techniques with our recently improved model atmospheres code, which include high-density effects, to accurately determine the effective temperature, surface gravity, and heavy-element abundances for each object. We study the abundance of hydrogen in DBZ/DZ white dwarfs and the properties of the accreted planetesimals. We explore the nature of the second sequence of DQ stars using proper motions from Gaia and highlight evidence of crystallization in massive DQ stars. We also present mass distributions for both spectral types. Finally, we discuss the implications of our findings in the context of the spectral evolution of white dwarfs and provide the atmospheric parameters for each star.

Unified Astronomy Thesaurus concepts: White dwarf stars (1799)

Supporting material: figure set, machine-readable tables

## 1. Introduction

White dwarfs represent the final evolutionary phase of main-sequence stars with initial mass below ~8 $M_\odot$ and are characterized by their high surface gravity, typically $g = 10^8$ cm s$^{-2}$. Because of this, elements heavier than helium will sink below the photosphere in characteristic timescales that are many orders of magnitude smaller than the cooling age of the star (Paquette et al. 1986). This gravitational separation also explains why most white dwarfs—about 80%—have pure hydrogen atmospheres. The only absorption lines present in their spectra are those of hydrogen, and they are collectively known as DA stars. For a smaller fraction, practically no hydrogen survives the late phases of stellar evolution, and a thin opaque helium layer—the lightest element remaining—will float on top and form the atmosphere. Depending on the effective temperature, they are classified as DO white dwarfs if they show ionized helium lines and DB stars if only neutral helium lines can be observed. Below $T_{eff} \sim 12,000$ K, spectra of pure helium atmosphere white dwarfs become featureless, as there is not enough energy to populate the lower energy levels of He I line transitions. Such objects with continuous optical spectra are classified as DC white dwarfs. Note that a similar phenomenon also happens for hydrogen-rich white dwarfs cooler than about 5000 K, as the electrons are mostly found in the ground state, preventing Balmer line transitions. In the absence of physical mechanisms competing with gravitational settling, the optical spectra of all white dwarfs should thus show only hydrogen lines, helium lines, or pure continuum. Nevertheless, white dwarfs with traces of heavy elements do exist, indicating that gravitational settling is not acting alone. These “contaminated” white dwarfs are found mainly in two categories: (i) those with traces of heavy elements (other than

carbon) in their optical spectra, and (ii) those with primarily carbon absorption lines, either molecular or atomic.

### 1.1. Heavy-element Pollution

White dwarfs showing absorption lines from elements such as calcium, magnesium, or iron are collectively known as DZ stars if their spectra show only heavy-element lines, DBZ stars if they show helium and heavy elements, and DAZ stars if they display hydrogen and metal features. Model atmospheres show that both the DZ and DBZ stars are helium dominated while the DAZ stars are hydrogen dominated. The presence of heavy elements in these objects is now understood as external accretion of matter from a disk of debris resulting from the destruction of a rocky object (asteroid or small planets) by the white dwarf's tidal forces (see Jura & Young 2014, and references therein). These objects are thus temporarily imprinted (diffusion timescales range from a few hundred thousand to a few million years,<fnref n="6" /> much shorter than the cooling age) with the chemical composition of the polluting body, providing a unique opportunity to study the chemical composition of extrasolar bodies (Zuckerman et al. 2007; Klein et al. 2010, 2011; Dufour et al. 2012; Jura et al. 2012; Koester et al. 2014; Xu et al. 2014, 2016, 2017; Blouin et al. 2019b). High-resolution or UV spectroscopic observations of some samples have shown that 25%–50% of all white dwarfs are contaminated by heavy elements at some level (Zuckerman et al. 2003, 2010; Koester et al. 2014).

Moreover, since observational evidence indicates that these white dwarfs had (or still have) at least some sort of planetary

<footnote n="6">⁶ Based on calculations by G. Fontaine, included in the MWDD and available at http://www.montrealwhitedwarfdatabase.org/evolution.html.</footnote>
system around them, they can provide information about the correlation between stellar mass and planet occurrence, which can give insight into the planet formation process (Johnson et al. 2010). Indeed, most observational exoplanet surveys are biased toward lower-mass stars. The *Kepler* mission prioritized G-type stars (Batalha et al. 2010), which have masses around $1 M_{\odot}$. More massive stars of type O and B are observed much less frequently by *Kepler*, mostly because of their scarcity (they spend a very short time on the main sequence), and the *Kepler* field of view was chosen to avoid young stellar populations (Batalha et al. 2010). Doppler surveys also favor Sun-like stars, as their spectral properties make their detection easier (Johnson et al. 2010). Thus, very little is known about the relation between planet occurrence and stellar mass above $M = 2-3 M_{\odot}$. While massive white dwarfs—which had massive main-sequence progenitors—are fainter, the selection bias is much less important, especially with *Gaia* DR2, which is expected to be volume complete within ~70 pc (Gentile Fusillo et al. 2019). Inferences of planetary systems around white dwarfs are thus not subjected to the same limitations.

## 1.2. Carbon Pollution

White dwarfs showing mainly carbon features are collectively known as DQ stars. They represent 9% of the white dwarfs in the local sample ($D < 20$ pc; Gianniche et al. 2012). They show only atomic carbon lines when $T_{\text{eff}} \gtrsim 10,000$ K and molecular $C_2$ bands at lower effective temperatures, with a smooth transition around that temperature where both molecular bands and atomic carbon lines are simultaneously present. Model atmosphere analyses have shown that those with molecular bands are helium dominated, with carbon abundances ranging from log (C/He) = -7 to -2 (Weidemann & Koester 1995; Dufour et al. 2005; Koester & Knist 2006) and effective temperatures between ~5000 and 12,000 K. A model where the deep helium convection zone catches up with the settling carbon in the core and dredges it up to the surface can successfully account for the observed abundances in most objects (Fontaine & Brassard 2005; Dufour et al. 2005). Dufour et al. (2005) showed, however, that several DQ stars had larger-than-average carbon abundances, forming a distinct sequence about 1 dex above the bulk of the sample in a log (C/He) versus $T_{\text{eff}}$ diagram (see also Koester & Knist 2006). Since the only object with a measured trigonometric parallax belonging to this second sequence was massive, Dufour et al. proposed that they could represent the high-mass tail of the white dwarf mass distribution. However, Brassard et al. (2007) showed that an evolutionary sequence at $1 M_{\odot}$ does not correctly predict the carbon abundance pattern that is empirically observed, indicating that another explanation to account for these stars must be sought.

While the cooler end of the DQ sequence is consistent with the expectations from the dredge-up model, very few stars were known at that time to test the theory on the hotter side ($T_{\text{eff}} \ge 13,000$ K). Many hot objects showing mainly ionized or neutral carbon lines (and also oxygen in a few cases) had been identified in the Sloan Digital Sky Survey (SDSS) by Liebert et al. (2003). The interpretation then was that these objects were a hotter version of the cool DQ white dwarfs, which had helium-dominated atmospheres with traces of carbon. However, when the hottest objects were analyzed using state-of-the-art model atmospheres, it was found in fact that the main atmospheric constituent was carbon, not helium (Dufour et al. 2007b, 2008).

Dufour et al. (2013) then showed that, in a $u-g$ versus $g-r$ color-color diagram, the hot carbon-dominated atmosphere DQ stars ($T_{\text{eff}} \sim 18,000-24,000$ K with mainly ionized carbon lines) seem to form a sequence that connects with warm DQ white dwarfs with neutral atomic lines ($T_{\text{eff}} \sim 10,000-16,000$ K), followed by cooler DQ stars with strong molecular bands ($T_{\text{eff}} < 10,000$ K on the second sequence mentioned above). Unfortunately, very few trigonometric parallax measurements were available for these objects at that time, and thus the massive nature of the elusive sequence was somewhat speculative.

Since the studies of Dufour et al. (2005, 2007a), there have been several improvements in stellar atmosphere modeling (e.g., Blouin et al. 2017, 2018a). Simultaneously, several surveys have enlarged the sample of spectroscopically confirmed white dwarfs considerably, and consequently, the number of known DQ/DZ/DBZ white dwarfs has increased by more than a factor of five. Also, thanks to the second *Gaia* data release in 2018 April, distances are now available, for the first time, for most of these objects, a quantity required to obtain precise measurements of their stellar masses.

The availability of improved model atmospheres and new data motivated us to perform an updated analysis of all these metal-polluted white dwarfs. We describe the model atmospheres in Section 2, the observational data in Section 3, and the methodology in Section 4. We present the analysis for the DBZ/DZ stars in Section 5 and that for the DQ stars in Section 6. We then discuss the implications of our results on our understanding of the spectral evolution of white dwarfs in Section 7. Our conclusions follow in Section 8.

## 2. Model Atmospheres

Our DZ/DBZ/DQ model atmosphere code is similar to that outlined in Dufour et al. (2005, 2007a), but with several physical improvements described at length in Blouin et al. (2018a, 2018b). Of particular importance for the study of DZ/DBZ stars are the new line profile calculations following the unified line shape theory of Allard et al. (1999) for strong transitions, the most important being Ca II H and K, Mg I λ2852 and Mg II λ2795/λ2802, the Mg b triplet, and Ca I λ4226. Less important transitions use Lorentzian or quasi-static van der Waals broadening profiles (Walkup et al. 1984; D. Koester 2010, private communication). For the DQ model atmospheres, one of the main improvements over Dufour et al. (2005) is the replacement of the “just overlapping line approximation” (Zeidler-K.T. & Koester 1982), to describe the $C_2$ Swan band opacity, with a complete line list provided by J. O. Hornkohl (2015, private communication; see Parigger et al. 2015 for details of the methodology). We find that the use of this new line list provides a much better representation of the shape of the observed Swan bands in DQ white dwarfs, particularly in the region around 4300 Å, which was poorly fitted in Dufour et al. (2005), compared to the prescription of Zeidler-K.T. & Koester (1982), Brooke et al. (2013), or Kurucz line lists.<fnref n="7" /> The atomic line list of Kurucz has also been replaced by the compilation from the Vienna Atomic Line Database (VALD; Piskunov et al. 1995).

For the DZ/DBZ(A) white dwarfs, we generated a four-dimensional grid of model atmospheres and synthetic spectra with $T_{\text{eff}}$ varying from 4000 to 16,000 K by steps of 500 K, log g from 7.0 to 9.0 by steps of 0.5 dex, log Ca/He from

<footnote n="7">⁷ http://kurucz.harvard.edu/linelists.html</footnote>
<table><caption>**Table 1**
Object Names, Spectral Types, and Coordinates</caption><thead><tr><th>J Name</th><th>Gaia Source ID</th><th>MWDD ID</th><th>Sp. Type</th><th>R.A.</th><th>Decl.</th></tr></thead><tbody><tr><td>J0000−0026</td><td>2449884942327523456</td><td>SDSS J000052.44−002610.5</td><td>DQ</td><td>0.21853</td><td>−0.43626</td></tr><tr><td>J0000−0850</td><td>2441060055844546816</td><td>WD 2357−091</td><td>DQ</td><td>0.04817</td><td>−8.83566</td></tr><tr><td>J0002+3209</td><td>2873769975034246272</td><td>SDSS J000215.65+320914.1</td><td>DZ</td><td>0.56522</td><td>32.15387</td></tr><tr><td>J0002−0627</td><td></td><td>SDSS J000232.35−062746.0</td><td>DZ</td><td>0.63479</td><td>−6.46278</td></tr><tr><td>J0003+2240</td><td></td><td>SDSS J000341.93+224043.7</td><td>DZ</td><td>0.92471</td><td>22.67881</td></tr><tr><td>J0004+0819</td><td></td><td>SDSS J000418.68+081929.9</td><td>DZ</td><td>1.07787</td><td>8.32499</td></tr><tr><td>J0004+2432</td><td>2849976676552398976</td><td>SDSS J000426.95+243258.9</td><td>DBZ</td><td>1.11226</td><td>24.54971</td></tr><tr><td>J0004+2531</td><td>2850199499454835968</td><td>SDSS J000439.38+253150.6</td><td>DZ</td><td>1.16411</td><td>25.53071</td></tr><tr><td>J0005+0018</td><td>2546096749538393600</td><td>WD 0003+000</td><td>DZ</td><td>1.48845</td><td>0.30924</td></tr><tr><td>J0005+4003</td><td>2881925666956510848</td><td>GD 1</td><td>DZA</td><td>1.25425</td><td>40.05996</td></tr></tbody></table>

(This table is available in its entirety in machine-readable form.)

−12 to −7 by steps of 0.5 dex, and log H/He from −7 to −3 by steps of 1 dex. We also generated a grid with no hydrogen. Metal-to-metal ratios have been fixed to that of chondrites (Lodders 2003) with respect to calcium. Calcium thus serves as a proxy for the abundances of all other heavy elements. While metal-to-metal ratios certainly differ from that of chondrites in many objects, our assumption provides a good first-order approximation of the contribution of free electrons and opacity from heavy elements (see Dufour et al. 2007a, and also discussion below).

For the DQ model grid, since no DQ white dwarfs are found at high effective temperature and low carbon abundance, or at low effective temperature and high carbon abundance, we generated two separate three-dimensional grids. The first grid is generated with $T_{eff}$ from 8000 to 16,000 K by steps of 500 K, log g from 7 to 9 by steps of 0.5 dex, and log C/He from −5 to −1 by steps of 0.5 dex. The second grid covers $T_{eff}$ from 6000 to 10,000 K by steps of 500 K, log g from 7 to 9 by steps of 0.5 dex, and log C/He from −8 to −4 by steps of 0.5 dex. No hydrogen is included in these models, but some smaller grids with hydrogen were generated to test its effects, as discussed in Section 6.3.

### 3. Observations

#### 3.1. Note on Naming Convention

Some white dwarfs can have up to 20 different names. Here we decided to adopt names based on ICRS coordinates at epoch and equinox 2000 instead of mixing names from different catalogs. Stars will be named JHHMM ± DDMM, where the first four digits correspond to the R.A. in hours and minutes and the last four digits to the decl. in degrees and minutes in sexagesimal notation. In some cases, a second relevant name will be written in parentheses. Table 1 provides cross-references for the names of the objects in our sample with Gaia source ID, MWDD ID, and the full coordinates.

#### 3.2. Sample Selection

We first selected from the Montreal White Dwarf Database<fnref n="8" /> (Dufour et al. 2017; hereafter MWDD) all objects with either a Q or a Z in their spectral classification. This includes DZ, DZA, DBZ, DBZA, DQ, etc. We did not include DAZ white dwarfs because they have hydrogen-rich atmospheres, and we are only interested here in helium-rich atmospheres. Most objects in our

sample have previously been identified in various white dwarf catalogs from the Sloan Digital Sky Survey (SDSS; Eisenstein et al. 2006; Kleinman et al. 2004, 2013; Kepler et al. 2015, 2016). Many objects have also been classified as uncertain, such as DQ: or DC-DQ. We thus visually inspected every spectrum and rejected those that did not show the appropriate spectral lines or were too noisy to make reliable classification (spectra with better signal-to-noise ratios will eventually confirm their spectral types). We rejected known unresolved binary systems, as well as magnetic white dwarfs. In order to ensure a homogeneous analysis based exclusively on optical features, we also rejected some objects classified as DQ white dwarfs but with carbon features detected only in the ultraviolet. We also excluded stars that had carbon-dominated atmospheres, the so-called Hot DQ stars ($T_{eff} \gtrsim 18,000$ K), or any object for which our preliminary analysis gave atmospheric parameters outside of our model grids (63 objects in total; these will be analyzed elsewhere). Finally, DQpec white dwarfs with strong distorted Swan bands were also left out of our analysis because there are still large uncertainties regarding their modeling (see Blouin et al. 2019c). A detailed analysis of these stars will be presented elsewhere (Blouin & Dufour 2019).

All objects in our sample were then cross-matched with Gaia DR2 to retrieve photometric and astrometric data. For one object, J0739+0513 (Procyon B), we used the parallax from Hipparcos (284.56 ± 1.26; van Leeuwen 2007). A few objects with negative parallaxes or very large uncertainties ($\sigma_{\pi}/\pi > 1$) were rejected; these are unlikely to be white dwarfs, and if they are, the data are too imprecise to allow any satisfactory analysis. Bailer-Jones (2015) demonstrated that inverting the parallax to estimate the distance might lead to unreliable distances when the uncertainty on the parallax is larger than 20%, and that it can lead to incorrect error estimates. Fortunately, 87% of the objects in our parallax sample have smaller uncertainties. In fact, the difference between a probabilistic analysis distance and the inverse of the parallax is less than 1% for the majority of objects when $\sigma_{\pi}/\pi < 0.1$.

With the availability of trigonometric parallaxes and broadband colors from Gaia for the majority of the objects in our sample, it is now possible to place them accurately in an observational Hertzsprung-Russell (H-R) diagram (Gaia Collaboration et al. 2018). This is illustrated in Figure 1, where stars that have been erroneously classified as white dwarfs can easily be identified. Examples of such stars are the giant star SDSS J174618.94+262217.0 (Green 2013) and SDSS J192013.71+383917.7, two objects that have been erroneously classified as a DQ and a DZ white dwarf,

<footnote n="8">⁸ www.montrealwhitedwarfdatabase.org</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Gaia H-R diagram for objects misclassified as white dwarfs (orange circles). SDSS J174618.94+262217.0 and SDSS J192013.71+383917.7, the two examples discussed in the text and displayed in Figure 2, are represented by a red plus sign and a blue square, respectively. Black dots are Gaia objects within 100 pc selected using the cuts proposed in Gaia Collaboration et al. (2018, Appendix B). White dwarf candidates are located in the lower left portion of the diagram.</figcaption></figure>

respectively (Kleinman et al. 2013). The similarity of the spectra of these nondegenerate stars, displayed in Figure 2, with genuine DQ and DZ white dwarfs stresses the importance of using photometric and parallax measurements in combination with spectroscopic data during the classification process. Overall, 25 objects, listed in Table 2, previously classified as white dwarfs in various catalogs were rejected this way.

To constitute our final sample, we removed 63 white dwarfs for which no optical spectroscopy was available. In the end, we are left with a sample of 1023 DZ (679 with parallax measurements) and 317 DQ (303 with parallax measurements) white dwarfs, which we analyze in a homogeneous fashion in the following sections.

### 3.3. Photometric and Spectroscopic Data

We cross-matched our sample with Pan-STARRS DR1 and SDSS DR14 to retrieve grizy and ugriz point-spread function magnitudes, respectively. Since SDSS photometry is available for 95% of our sample, while only 80% of our objects have Pan-STARRS photometry, we decided to rely primarily on SDSS photometry for the sake of homogeneity. Note that although Pan-STARRS covers a much larger part of the sky than SDSS, most spectroscopically identified white dwarfs were discovered in the SDSS. When SDSS ugriz is not available for an object, we rely on Pan-STARRS data, previously published BVRI photometry, or simply Gaia broadband photometry, in that order of priority. For J0739 +0513 (Procyon B), we rely on Hubble Space Telescope (HST) photometry (Provencal et al. 1997).

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Optical spectra of SDSS J174618.94+262217.0 (top) and SDSS J192013.71+383917.7 (bottom), two objects that were previously misclassified as DQ and DZ white dwarfs, respectively.</figcaption></figure>

<table><caption>**Table 2**
Objects Rejected Owing to Their Location in the Gaia H-R Diagram</caption><thead><tr><th>J Name</th><th>Gaia Source ID</th><th>MWDD ID</th></tr></thead><tbody><tr><td>J0205+0058</td><td>2508108382280944768</td><td>2MASS J02053812+0058354</td></tr><tr><td>J2112+0912</td><td>1743430270303257728</td><td>2MASS J21120843+0912019</td></tr><tr><td>J1642+3617</td><td>1328051961501737088</td><td>Cl* NGC 6205 KAD 606</td></tr><tr><td>J0239-0003</td><td>2498734800840646400</td><td>SDSS J023955.79-000312.0</td></tr><tr><td>J0814+3508</td><td>902942352007933568</td><td>SDSS J081424.51+350822.9</td></tr><tr><td>J0815+1537</td><td>655412730925191808</td><td>SDSS J081502.24+153709.7</td></tr><tr><td>J0914-0022</td><td>3842424562164246656</td><td>SDSS J091455.49-002219.2</td></tr><tr><td>J1136+4702</td><td>787028812551381248</td><td>SDSS J113612.16+470206.8</td></tr><tr><td>J1208-0008</td><td>3698751446481826432</td><td>SDSS J120853.36-000847.3</td></tr><tr><td>J1228+2057</td><td>3952475922233611776</td><td>SDSS J122829.50+205747.0</td></tr><tr><td>J1336+1535</td><td>3742286441879709952</td><td>SDSS J133644.21+153554.2</td></tr><tr><td>J1357+1252</td><td>3727838511198756224</td><td>SDSS J135713.37+125230.9</td></tr><tr><td>J1507+1824</td><td>1211873229279489536</td><td>SDSS J150741.36+182406.8</td></tr><tr><td>J1549+0436</td><td>4426232766560567296</td><td>SDSS J154918.87+043651.0</td></tr><tr><td>J1610+0247</td><td>4412254439017190400</td><td>SDSS J161041.62+024706.2</td></tr><tr><td>J1714+2324</td><td>4568671714800327680</td><td>SDSS J171432.25+232423.0</td></tr><tr><td>J1746+2622</td><td>4582265290590354304</td><td>SDSS J174618.94+262217.0</td></tr><tr><td>J1920+3839</td><td>2052876071205883520</td><td>SDSS J192013.71+383917.7</td></tr><tr><td>J2105+0828</td><td>1744023628624108288</td><td>SDSS J210511.38+082853.7</td></tr><tr><td>J2106+0642</td><td>1736767710874927872</td><td>SDSS J210632.52+064233.5</td></tr><tr><td>J2111+0915</td><td>1743431683347518464</td><td>SDSS J211157.26+091554.2</td></tr><tr><td>J2153+2750</td><td>1800030173963458048</td><td>SDSS J215306.03+275057.3</td></tr><tr><td>J2259-1001</td><td>2606526522781415168</td><td>SDSS J225931.60-100146.9</td></tr><tr><td>J0124+0541</td><td>2563653437678125568</td><td>USNO-B1.0 0956-00013686</td></tr><tr><td>J0257+0111</td><td>339405495907200</td><td>WD 0255+009.1</td></tr></tbody></table>

The majority of the spectra used in this study are drawn from SDSS.<fnref n="9" /> The spectra of 51 objects without SDSS spectroscopy are taken from Limoges et al. (2013, 2015), Subasavage et al. (2007, 2009), Bergeron et al. (1997, 2001), or archival data secured by the Montreal group in the past few decades. Finally, the spectroscopic data for Procyon B are from HST (Provencal et al. 2002).

### 4. Atmospheric Parameter Determination

Our fitting method is similar to that described in Dufour et al. (2005, 2007a) for DQ and DZ stars, respectively. Briefly, we first convert the observed magnitudes into average fluxes following the method described in Holberg & Bergeron (2006). The magnitudes on the AB system (SDSS, Pan-STARRS) are converted using the relation $m = -2.5 \log f_v^m - 48.60$, where

<footnote n="9">⁹ https://www.sdss.org/dr14/spectro/spectro_basics/</footnote>
<table><caption>**Table 3**
Zero-points Derived for Different Bandpasses</caption><thead><tr><th>Bandpass</th><th>Zero-point</th><th>Vega mag</th></tr></thead><tbody><tr><td>G</td><td>-21.51572</td><td>0.00</td></tr><tr><td>G<sub>BP</sub></td><td>-20.99456</td><td>0.00</td></tr><tr><td>G<sub>RP</sub></td><td>-22.22677</td><td>0.00</td></tr><tr><td>B</td><td>-20.46331</td><td>0.024<sup>a</sup></td></tr><tr><td>V</td><td>-21.06662</td><td>0.026<sup>a</sup></td></tr><tr><td>R</td><td>-21.64524</td><td>0.033<sup>a</sup></td></tr><tr><td>I</td><td>-22.37877</td><td>0.029<sup>a</sup></td></tr></tbody></table>

**Note.**

$f_{\nu}^m$ is the stellar flux averaged over the bandpass response; the corrections from Eisenstein et al. (2006) are applied to put the SDSS photometry on the AB system. The magnitudes on the Vega system (BVRI, Gaia) are converted using the relation $m = -2.5 \log f_{\lambda}^m + c_m$, where $c_m$ is the zero-point in the corresponding bandpass. These zero-points are calculated using the observed fluxes and magnitudes for Vega. For the BVRI bandpass response, we use Cohen et al. (2003), and for Gaia, we use the revised bandpasses,<fnref n="10" /> while for the Vega flux, we use alpha_lyr_stis_008.fits from the CALSPEC Calibration Database.<fnref n="11" /> Our results are summarized in Table 3.

Photometric data are also dereddened following the procedure described in Harris et al. (2006) using the extinction maps of Schlafly & Finkbeiner (2011). We thus consider the extinction to be negligible for stars with $D < 100$ pc, to be maximum for those located at $|z| > 250$ pc from the galactic plane (z is the distance from the galactic plane), and to vary linearly between these two regimes. Genest-Beaulieu & Bergeron (2019) compared this method to that used by Gentile Fusillo et al. (2019) and came to the conclusion that they lead to similar results for white dwarfs in the SDSS. When no parallax measurement is available, we first obtain the photometric distance assuming $log g = 8.0$ and then apply a dereddening correction using this distance, and we repeat the process until the distance and the reddening converge to a single value.

Next, we transform the monochromatic Eddington fluxes from our model grids into fluxes averaged over each bandpass. These synthetic average fluxes $H_{\lambda}^m$ (or $H_{\nu}^m$) are related to the observed fluxes by the equation

$$f_{\lambda}^{m} = 4\pi \left(\frac{R}{D}\right)^{2} H_{\lambda}^{m}, \qquad (1)$$

where R is the radius of the star and D its distance from Earth. In the above equation, the average model fluxes depend on $T_{eff}$, $log g$, and the atmospheric composition, which refers to the carbon abundance for DQ stars and to the hydrogen and calcium abundances for DBZ/DZ stars. The best fit between observed photometry and synthetic fluxes is obtained using the nonlinear least-squares steepest decent method of Levenberg-Marquardt (Press et al. 1986), with the values of $T_{eff}$ and the solid angle $\pi (R/D)^2$ left as free parameters, while $log g$ and the atmospheric composition are kept fixed during the first iteration.

From the derived value of the solid angle and the distance given by the inverse of the parallax, we determine the radius R of the star. The values of $log g$ and mass are then derived by interpolating in evolutionary models similar to those described in Fontaine et al. (2001) but with C/O cores, $log q(He) \equiv \log M_{He}/M_{\star} = -2$ (where $M_{\star}$ is the mass of the star), and $log q(H) = -10$, which are representative of helium-rich atmosphere white dwarfs (Dufour et al. 2005). We repeat the process this time using the newly determined surface gravity until convergence is reached. If the parallax is unknown, we simply assume $log g = 8$ and derive a radius R from the same evolutionary models, which yields a photometric distance when combined with the solid angle. The uncertainties on the effective temperature and the solid angle are obtained directly from the covariance matrix of the fitting procedure.

We next determine the chemical composition by fitting the spectroscopic data, again using the Levenberg-Marquardt method, keeping $T_{eff}$ and $log g$ fixed to the values obtained from the photometric fit. For cool DQ stars, we fit the Swan bands between 4000 and 6500 Å, while for hotter DQs we use the atomic absorption lines between 4500 and 5500 Å. The carbon abundance, the solid angle, and a first or second degree polynomial—to account for uncertainties in the flux calibration (see Dufour et al. 2005)—are considered free parameters during the fitting procedure. For DBZ/DZ stars, we begin by fixing the hydrogen abundance, either by fitting the $H\alpha$ spectral line if it is visible or by fixing $log H/He$ at the detection limit (we also fit stars not showing $H\alpha$ with our hydrogen-free grid; see Section 5 for a discussion of this matter). This detection limit was estimated by calculating, at each temperature and $log g$ value in our grid, the amount of hydrogen required to reach a threshold of 500 mÅ for the equivalent width of the $H\alpha$ line. We then fit the Ca II H and K absorption lines to determine $log Ca/He$. Then we repeat the photometric fit, but this time using the values of $log g$, calcium abundance (or carbon abundance in the case of DQ white dwarfs), and hydrogen abundance obtained in this last iteration. We repeat the procedure, typically three to five iterations (or more in some cases), until the parameters have converged to a stable solution.

To obtain the uncertainty on the abundances measured from the spectroscopic fit, we rely on the same method described in Bergeron et al. (1992), where we first assume an arbitrary standard deviation $\sigma = 1$ for each data point and then calculate the rms deviation of the observed spectrum from the best-fit model spectrum. This is then propagated into the covariance matrix, from which the formal uncertainties of the fitted atmospheric parameters are obtained. This gives an uncertainty estimation that depends mostly on the signal-to-noise ratio of the observed spectra.

# 5. DZ/DBZ(A) White Dwarfs

## 5.1. Photometric and Spectroscopic Fits

Following the method described in Section 4, we determined the atmospheric parameters for all DZ/DBZ(A) white dwarfs in our sample. Figure 3 shows typical fits of the energy distribution and the Ca II H and K line region (all our fits are available in the online journal), while our final parameters are given in Table 4. The fit to the $H\alpha$ line, when present, is shown as an inset. Note that for stars that do not show hydrogen, we report a solution with the hydrogen abundance fixed at the detection limit (see explanation below).

<footnote>ª Holberg & Bergeron (2006).</footnote>

<footnote n="10">¹⁰ https://www.cosmos.esa.int/web/gaia/iow_20180316</footnote>

<footnote n="11">¹¹ http://www.stsci.edu/hst/observatory/crds/calspec.html</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Examples of fits to our sample of DBZ/DZ(A) white dwarfs. Left panels: photometric fits where error bars represent the observed data, while filled circles correspond to average model fluxes. A dagger symbol indicates that log g is fixed at 8.0 (no parallax measurement available), while a star symbol indicates a value of log H/He fixed at the visibility limit. Right panels: spectroscopic fits (red) to the normalized observed spectra (black). The inset shows the fit to Hα when present. The complete figure set with all our fits (171 pages) is available in the online journal.</figcaption></figure>

(The complete figure set (171 images) is available.)
<table><caption>**Table 4**
Atmospheric Parameters for DZ Stars</caption><thead><tr><th>J Name</th><th>T<sub>eff</sub> (K)</th><th>log g</th><th>M/M<sub>☉</sub></th><th>log Ca/He</th><th>log H/He</th><th>D (pc)</th></tr></thead><tbody><tr><td>J0002+3209</td><td>6447(121)</td><td>8.30(<sup>+0.23</sup><sub>-0.22</sub>)</td><td>0.77(<sup>+0.16</sup><sub>-0.14</sub>)</td><td>-9.00(0.18)</td><td>&lt;-3.08</td><td>180.6(28.7)</td></tr><tr><td>J0002-0627</td><td>7306(428)</td><td>8.00</td><td></td><td>-9.62(0.31)</td><td>&lt;-4.05</td><td>343.6(28.3)</td></tr><tr><td>J0003+2240</td><td>5493(269)</td><td>8.00</td><td></td><td>-10.27(0.22)</td><td>&lt;-3.00</td><td>229.1(22.4)</td></tr><tr><td>J0004+0819</td><td>5896(234)</td><td>8.00</td><td></td><td>-8.86(0.16)</td><td>&lt;-3.00</td><td>337.4(27.7)</td></tr><tr><td>J0004+2432</td><td>10490(614)</td><td>8.26(<sup>+0.28</sup><sub>-0.25</sub>)</td><td>0.75(<sup>+0.18</sup><sub>-0.16</sub>)</td><td>-9.74(0.18)</td><td>&lt;-5.82</td><td>287.9(49.8)</td></tr><tr><td>J0004+2531</td><td>12132(1141)</td><td>8.32(<sup>+0.81</sup><sub>-0.62</sub>)</td><td>0.79(<sup>+0.45</sup><sub>-0.36</sub>)</td><td>-8.27(0.35)</td><td>&lt;-6.04</td><td>431.5(214.5)</td></tr><tr><td>J0005+0018</td><td>8073(208)</td><td>7.81(<sup>+0.14</sup><sub>-0.14</sub>)</td><td>0.47(<sup>+0.08</sup><sub>-0.07</sub>)</td><td>-10.05(0.09)</td><td>&lt;-5.01</td><td>156.4(11.6)</td></tr><tr><td>J0005+4003</td><td>8829(427)</td><td>8.03(<sup>+0.09</sup><sub>-0.09</sub>)</td><td>0.59(<sup>+0.06</sup><sub>-0.05</sub>)</td><td>-10.42(0.04)</td><td>-4.48(0.21)</td><td>64.3(0.3)</td></tr><tr><td>J0005+7313</td><td>12673(759)</td><td>7.95(<sup>+0.08</sup><sub>-0.08</sub>)</td><td>0.56(<sup>+0.05</sup><sub>-0.04</sub>)</td><td>-10.07(0.36)</td><td>-6.11(0.00)</td><td>34.7(0.0)</td></tr><tr><td>J0006+0520</td><td>5906(160)</td><td>7.97(<sup>+0.27</sup><sub>-0.25</sub>)</td><td>0.55(<sup>+0.17</sup><sub>-0.14</sub>)</td><td>-9.78(0.10)</td><td>&lt;-3.00</td><td>178.3(27.0)</td></tr></tbody></table>

Note. A dagger indicates that the distance is a photometric distance based on the value of log g = 8 in the absence of a parallax.

(This table is available in its entirety in machine-readable form.)

We note that other lines, mostly Mg and Fe, are sometimes observed in addition to Ca II H and K. Since in our model grids the abundances of these elements relative to calcium are fixed to the chondrite values, a visual inspection of the quality of the “fit” to these lines provides a quick assessment of the validity of this approximation, which appears to be adequate (or close enough) for most stars that show such lines in our sample. However, it is certainly possible that some elements depart significantly from the chondrite values. Since what is most important, as far as the mass and effective temperature determinations are concerned, is the amount of free electrons in the photosphere (effects due to the redistribution of the flux absorbed in the UV can also play a role in some cases), small deviations should only have a minimal impact on those parameters, even more so when hydrogen is present, because electrons from ionized hydrogen will dominate the free electron budget.

Recently, Hollands et al. (2017, 2018) analyzed a large sample of 230 cool DZ white dwarfs and determined abundances of individual elements. Their results indicate that, to first order, objects are found with chemical compositions not too far from that of bulk Earth ratios, at least for the most visible elements, which are calcium, magnesium, and iron. Although their parameters were obtained using a different model atmosphere code that did not include all the improvements included in our code, the differences between our model should not be too important for such polluted, and thus less dense, objects (differences would probably be more significant for stars cooler than ~5000 K). A comparison of their results for 197 objects in common with our sample suggests that the exact metal-to-metal ratios used in their analysis have only a modest impact on the effective temperature determination. We find that the effective temperatures obtained with our approach are, on average, only 11 K lower than theirs, but with a somewhat large 330 K standard deviation. We find no correlation between the differences in effective temperature and the deviation to the chondrite abundances (relative to Ca) that they report. Instead, we find that these differences are correlated with log g values. Since the Gaia second data release was not available at the time the analyses of Hollands et al. (2017, 2018) were published, the authors assumed log g = 8 for their whole sample, while our analysis makes use of the newly available parallax measurements to determine the surface gravities. This distinction seems to be the main explanation for the different atmospheric parameters derived in both studies. Indeed, we find a strong correlation between the differences in effective temperature and the departure from log g = 8, our temperatures being higher

(lower) for larger (smaller) log g values. Similarly, there is also a correlation between the differences in log Ca/He and log g. This result is not surprising given that the difference in the strength of the calcium absorption features resulting from a change in surface gravity will need to be compensated by an appropriate change in calcium abundance, which in turn will affect the number of free electrons, and thus the effective temperature determination. We thus conclude that the atmospheric parameters derived using our approach should be reliable and not significantly affected by our assumption of the metal-to-metal ratios.

Nevertheless, when better spectroscopic data (high-resolution or UV observations) for a given object indicate some departure from our approximation, atmospheric parameters should always be rederived in a self-consistent way for better precision. For example, for the DZA white dwarf Ross 640 analyzed by Blouin et al. (2018a), changes in the abundances of Mg and Fe relative to Ca affected the UV flux level sufficiently to warrant the calculations of a specific grid with modified abundances in order to obtain atmospheric parameters in a self-consistent way. As a result, their final effective temperature was 250 K cooler, and the corresponding mass 0.04 M<sub>☉</sub> lower, than what we obtain here with our approach. Unfortunately, since high-resolution or ultraviolet spectroscopic data are not likely to become available for every object in our sample in the foreseeable future, we are forced to adopt this approximation, in particular given the fact that most objects in our sample show only calcium lines. To conclude on this topic, while our approach provides the best atmospheric parameters that are possible to infer with the available data, the solutions for individual objects will always suffer from small intrinsic uncertainties related to our adopted metal abundances.

## 5.2. Hydrogen Abundance Measurements

While for most objects in our sample the metal-to-metal ratio assumed in our analysis has only a modest impact on our atmospheric parameters, the abundance of hydrogen, on the other hand, has a deeper impact because it can be one of the main free electron donors, even when present below the visibility limit. For instance, Bergeron et al. (2019) recently showed that adding undetectable traces of hydrogen in the models had a non-negligible effect on the mass determination of helium-rich white dwarfs (this is also discussed in Dufour et al. 2005 in a similar context, but with carbon as the main electron donor).
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Hydrogen abundance as a function of effective temperature for our sample of DBZ/DZ(A) white dwarfs. Objects with parallax measurements are represented with filled symbols, while open symbols are used for stars for which we assumed log g = 8. Orange circles are objects with hydrogen abundances determined from fitting Hα, while blue triangles correspond to upper limits. The dashed line represents the visibility limit, defined as an equivalent width of 0.5 Å for Hα at log g = 8. The black error bars represent the average uncertainties.</figcaption></figure>

Only 105 of the 1023 (10%) white dwarfs in our sample show Hα in their spectrum. This is much less than the 25% containing hydrogen reported by Dufour et al. (2007a) for two main reasons. The first one is that the presence of hydrogen for 7% of their objects was determined indirectly from the shape of the Ca II H and K absorption features, in the sense that much better fits to these lines could be achieved when hydrogen was included. However, the study of Dufour et al. was based on Lorentzian profiles, with log g fixed at 8.0, while here we use the unified line shape theory of Allard et al. (1999), with surface gravities constrained by parallax measurements. As a consequence, we no longer find objects with spectroscopic fits that are significantly improved by adding hydrogen, indicating that the need to add hydrogen in the Dufour et al. analysis was probably only a way to compensate suboptimal line profiles and/or incorrect surface gravities. The second reason is that since Dufour et al., the proportion of DZ white dwarfs too cool to show hydrogen has increased substantially, thanks to the thorough search for metal-polluted white dwarfs near the main-sequence color space (Koester et al. 2011; Hollands et al. 2017, 2018).

Figure 4 shows the abundance of hydrogen as a function of effective temperature for all the DBZ/DZ(A) white dwarfs in our sample. For stars that do not show Hα, we determine the maximum amount of hydrogen that can be added without being detected (the scatter is explained by variations in log g for each object). While the hydrogen content, and thus its impact on the free electron budget, is well constrained for T<sub>eff</sub> ≳ 9000 K, increasingly large quantities of hydrogen can be hidden as the effective temperature decreases. To examine the impact of the unknown amount of hydrogen on our atmospheric parameter determinations, we fit each star in our sample with no detectable Hα with both a hydrogen-free model grid and the abundance of hydrogen fixed at the detection limit. The results of this experiment are displayed in Figure 5.

Clearly, masses and effective temperatures are significantly reduced for the coolest stars when hydrogen is included. This reduction in both effective temperature and mass was recently

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Comparison of masses and effective temperatures obtained with hydrogen-free models (orange circles) and those obtained with models where the hydrogen abundance was set at the visibility limit (blue circles). A black line connects the two solutions for each object.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Synthetic spectra (Eddington fluxes) for models at T<sub>eff</sub> = 8000 K, log Ca/He = -9, and log g = 8, with two different hydrogen abundances indicated in the figure.</figcaption></figure>

explained by Bergeron et al. (2019, see their Figures 10 and 11). Briefly, adding hydrogen in the model increases the number of free electrons, which in turn increases the He⁻ free-free opacity. This has a quite dramatic effect on the continuum, as can be appreciated from Figure 6, where the energy distributions of two models that differ only by their hydrogen content are shown. As a result of the increased He⁻ free-free opacity, a lower temperature and a larger solid angle are required to match the observed fluxes, which translate into a larger radius and thus a smaller mass. Note that this effect practically disappears for effective temperatures above ~11,000 K, as the contribution from ionized helium starts to dominate the free electron budget.

### 5.3. Mass Distributions

As discussed above, the parameters we derive depend intimately on the assumed amount of invisible hydrogen present in the star. It is certainly reasonable to expect at least some amount of hydrogen to be present in each object. After all, not only have these white dwarfs been traveling through the
<figure><img src="image_8.png" /><figcaption>**Figure 7.** Mass distributions for DZ and DZA white dwarfs in our parallax sample. The DZA stars are shown in orange, while the DZ stars fitted with hydrogen-free models or with the hydrogen abundance set to the visibility limit are shown in black and blue, respectively.</figcaption></figure>

interstellar medium for billions of years, but potentially they also have accreted bodies that may have contained large amounts of water and ice (Klein et al. 2010; Farihi 2011; Farihi et al. 2013; Raddi et al. 2015). While it is impossible to determine exactly the hydrogen abundance in each object, we know that this abundance must lie within the two limiting cases explored here. We thus, in what follows, take a deeper look at the consequences this parameter has on the derived global properties of the sample, and most importantly on the mass distribution.

Figure 7 compares the mass distributions of our parallax sample for stars with hydrogen abundances measured directly from Hα (105 objects) with those with no detectable Hα feature (918 objects) analyzed both with hydrogen-free models and with hydrogen abundances set to the visibility limit. We find that the mean mass for the DBZA/DZA stars is 0.612 M⊙, very close to the value recently reported by Genest-Beaulieu & Bergeron (2019) for DA (0.617 M⊙) and DB (0.620 M⊙) white dwarfs, which represents the most comprehensive photometric and spectroscopic analysis of DA and DB white dwarfs identified in the SDSS. However, the mean mass is significantly higher (0.675 M⊙) for stars without Hα if no hydrogen is included in our models. This indicates that the effective temperatures and masses obtained from hydrogen-free models are probably overestimated, most likely due to the presence of invisible traces of hydrogen in these stars. Adding the maximum amount of hydrogen pushes the mass distribution toward values that now become consistent (mean mass of 0.631 M⊙) with the mean mass of DA and DB white dwarfs, although the peak is still slightly shifted toward higher masses with respect to the DBZA/DZA stars.

Since there is no reason to believe that these objects represent a distinct massive population, this probably means that our effective temperatures and masses are still slightly overestimated for many objects using our approach, most likely due to small deviations of the abundances of some elements with respect to the chondrite values. For example, if the accretion episode responsible for the metal pollution in some of these stars has stopped a long time ago, elements with different masses will start to settle with different timescales

(photospheric abundances will be very close to that of the polluting body only during the early phase and steady-state phase; Dupuis et al. 1993; Koester 2009), leading to abundances that can depart significantly from our assumed ratios. This will also affect, to a lesser extent, the relative strength of the absorption in the UV. Consequently, the uncertainties on the atmospheric parameters for these stars will be intrinsically larger than the statistical values reported here. Spectroscopic observations in the ultraviolet for every object in our sample would be required for a more accurate analysis, but unfortunately, such data will not become available anytime soon. Note that DZA white dwarfs are less affected by these uncertainties because when hydrogen is present in sufficient quantities to be detectable, it will also be the main free electron donor at the photosphere, diminishing the impact of small variations of heavy-element abundance ratios with respect to chondrites.

Even though it is not possible to derive the exact photospheric hydrogen abundance for each object, the fact that the peak of the mass distribution appears more realistic when traces of hydrogen are included suggests that most of these objects probably have abundances close to the detection limit. As will be discussed in Section 7, this corresponds also approximately to the amount of hydrogen expected if such DZ stars are the result of convectively mixed DA white dwarfs. Consequently, our adopted atmospheric parameters reported in Table 4 are obtained with the hydrogen abundance set to the visibility limit, keeping in mind that the true solution for individual objects may be off by a few hundred degrees and a few hundredths of solar mass, depending on their real hydrogen (and also metal) content.

Note finally that objects in Table 4 with $M \lesssim 0.47 M_\odot$ are most likely unresolved double-degenerate binaries, because such low-mass white dwarfs would have low-mass progenitors on the main sequence, with lifetimes longer than the age of the Milky Way. Such binaries are more luminous, resulting in a larger solid angle for a given distance, and thus a larger radius (lower mass) is inferred when analyzed under the assumption of a single object.

## 5.4. Accreted Material

Figure 8 shows the abundance of calcium as a function of effective temperature for the 1023 DBZ/DZ(A) white dwarfs in our sample. The two gaps in the distribution reported by Dufour et al. (2007a), namely, between 5000 and 6000 K and in the upper right corner of the diagram, have been mostly filled, thanks to the recent discovery of many new cool polluted white dwarfs (Koester et al. 2011; Kleinman et al. 2013; Kepler et al. 2015, 2016; Hollands et al. 2017, 2018). The absence of objects at high effective temperatures and low calcium abundances is still present and is due to the detection limit of Ca II H and K lines, as indicated in the figure. At the typical resolution of our spectroscopic observations, stars with such parameters would simply appear as DC white dwarfs below $T_{\text{eff}} \sim 12,000$ K and as DB stars above this temperature.

More interesting than the calcium-to-helium abundance ratio is the total mass of calcium contained in the convection zone of these white dwarfs. Using the derived effective temperature and surface gravity of each object, we can calculate the total mass of the helium convection zone using envelope models similar to those described in Fontaine et al. (2001; see Dufour et al. 2010 for further details) ranging from $T_{\text{eff}} = 7000$ to 30,000 K,
<figure><img src="image_9.png" /><figcaption>**Figure 8.** Calcium abundance as a function of effective temperature for all DZ white dwarfs in our sample. The black plus sign indicates the mean error bar. White dwarfs with a parallax measurement are shown with filled symbols, while $log g = 8$ was assumed for objects with open symbols. Orange circles represent DZA stars with hydrogen abundances determined by fitting $H\alpha$, and blue circles represent DZ stars with the hydrogen abundance fixed at the detection limit. The dashed line indicates the detection limit of calcium, defined as an equivalent width of 0.5 Å for the Ca II H line.</figcaption></figure>

$log g = 7.5$ to 9.0, and with the ML2/$\alpha$ = 1.0 version of the mixing-length theory (the effect of changing the mixing length is negligible below $T_{\text{eff}} \sim 16,000$ K; see Figures 9 and 10 of Rolland et al. 2018). Our results are presented in Figure 9. Note that these values represent only lower limits to the total mass accreted since we do not know how much mass has already diffused at the bottom of the convection zone. Also shown for comparison in Figure 9 is the estimated mass of calcium present in well-known asteroids in the solar system, assuming that calcium accounts for 1.6% of the mass of a typical planetesimal (see Farihi 2011). Our results are similar to those presented in Farihi (2011) and Hollands et al. (2018), with masses ranging from $10^{18}$ to $10^{22}$ g, with possibly smaller mass values at lower effective temperatures (the lack of low calcium mass values at high effective temperatures is simply a visibility limit effect; see Figure 8). Surprisingly, the total mass of calcium remains constant as a function of the white dwarf mass between 0.5 and 0.75 $M_{\odot}$, with a possible decrease in more massive white dwarfs, although we are probably dealing here with small number statistics.

### 5.5. Main-sequence Progenitors

It is also possible to estimate the mass of the white dwarf progenitors using the empirical initial–final mass relation (IFMR). While there are many different IFMRs published (see Figure 8 of Williams et al. 2009 for a comparison), most of them rely on the observations of star clusters. El-Badry et al. (2018) propose an alternative way of calculating the relation using the Gaia color–magnitude diagram. Their results agree remarkably well with those of Williams et al. (2009), which we decided to use for simplicity. One must keep in mind that this relation includes many systematic errors, one being the possible dependence on initial metallicity (Marigo & Girardi 2007), information that is lost when evolving to the white dwarf phase. Because of this, it is difficult to determine accurate initial masses of individual objects, but we can obtain a good idea of the distribution. The result is shown in Figure 10 for stars with

<figure><img src="image_10.png" /><figcaption>**Figure 9.** Total mass of calcium in the convection zone as a function of effective temperature (left) and mass (right). Orange circles are objects with hydrogen abundances determined by fitting $H\alpha$, and blue circles are objects with the hydrogen abundance fixed at the detection limit. We also show the estimated mass of calcium in Ceres, Vesta, and Minerva, assuming that calcium constitutes 1.6% of their total mass as in bulk Earth (Farihi 2011).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 10.** Mass distribution of white dwarf progenitors based on the IMFR of Williams et al. (2009) for the DBZ/DZ(A) stars in our sample with $\sigma_{\pi}/\pi < 0.1$.</figcaption></figure>

$\sigma_{\pi}/\pi < 0.1$. Note that below $M_i = 1 M_{\odot}$ the results are meaningless, since the white dwarfs in these bins have masses below 0.47 $M_{\odot}$ and are most probably unresolved double-degenerate binaries (single white dwarfs with such low mass have long main-sequence lifetimes and are not expected to have evolved into white dwarfs within the age of the galactic disk yet).

On the basis of the results displayed in Figure 10, we find that 41 white dwarfs had main-sequence progenitors with masses above $3 M_{\odot}$, indicating that the formation of rocky objects around massive stars may not be exceptional at all. Note that the Extrasolar Planets Encyclopaedia<fnref n="12" /> reports only nine exoplanets (out of 3802 exoplanets with the mass of the host star determined) around eight different stars with masses above $3 M_{\odot}$. The very small number of known planets around massive stars is mostly due to various selection effects, as the methods used to find them (transit, radial velocity, direct imaging) are all less efficient around large, bright, and massive

<footnote n="12">¹² http://exoplanet.eu/</footnote>
main-sequence stars. Nevertheless, theoretical models of planet
formation predict that planet occurrence around massive stars
should be higher (Kennedy & Kenyon 2008), and the presence
of rocky material in the photosphere of many polluted white
dwarfs with massive main-sequence progenitors seems to
confirm this hypothesis. However, to properly study planet
occurrence and its correlation with stellar mass, an analysis of a
large sample of DC and DB white dwarfs would be necessary,
preferably for a complete-volume sample, something that is
outside the scope of this work.

5.6. Discussion of Individual Objects

J0005+7313—The helium lines at 3889 and 4026 Å are deeper than those predicted by the model, indicating that the temperature determined from photometry, T<sub>eff</sub> = 12,673 K, is probably underestimated. Using Gaia photometry, we find T<sub>eff</sub> = 13,152 K, which is still too cool for a good spectroscopic fit. Bergeron et al. (2011) found T<sub>eff</sub> = 14,410 K, log H/He = -5.97, and no metals, using only spectroscopy. The absence of metals is not the explanation, since we obtain 12,715 K fitting photometry with our metal-free models. Because their technique is independent of photometry, one possible explanation is the reddening due to interstellar absorption. In this work, we use the procedure of Harris et al. (2006), and since the star is only at 34.7 pc, no correction for reddening is applied. If we use the procedure of Gentile Fusillo et al. (2019) instead, we need to apply 3.2% of the maximum absorption along the line of sight, and we now obtain T<sub>eff</sub> = 13,622 K, which provides a much better agreement with the spectra. Three-dimensional reddening maps made with Gaia should eventually allow better correction for reddening.

J0152+2418—The Mg I lines at ~3830 Å and the red wing of the Ca II H and K lines are not well reproduced by our model. One possible explanation is that the abundance ratios differ significantly from that of chondrites. A deficiency in magnesium (and possibly Fe and other elements) may explain this discrepancy. More specific adjustments of the various abundances would be necessary to obtain a good fit.

J0209+2914, J1242+0829, and J1424+5657—All three white dwarfs have “flattened” Ca II lines, similar to those observed in J1249+6514, not analyzed here, which has been identified as magnetic by Hollands et al. (2017). The spectra do not show other lines that could be used to detect line splitting. We can only consider the values we found to be approximate, but if the presence of magnetism is confirmed, J1242+0829 would be the hottest known magnetic DZ star with T<sub>eff</sub> ~ 8123 K.

J0302-0108 (GD 40)—Our effective temperature of T<sub>eff</sub> = 13,594 K is much lower than that reported by Voss et al. (2007)—T<sub>eff</sub> = 15,316 K (with log g fixed at 8)—obtained on the basis of metal-free models. They also relied on optical spectra rather than photometry to derive the temperature. Part of this large difference is probably attributable to the lack of heavy elements in their model atmosphere calculations, but since our predicted helium lines are a tiny bit too shallow compared to the observations, it is also possible that our effective temperature is slightly underestimated, possibly due to our neglect of any reddening correction given that the distance is only 64 pc.

J0555-0410 (LP 658-2) and J2201+0219—While these
objects do not show a clear Hα absorption line, setting the

hydrogen abundance at the detection limit leads to completely
spurious fits, so we decided to adopt hydrogen-free models for
these stars. J0555−0410 was also analyzed in Blouin et al.
(2018a), who also found a lower hydrogen abundance of
log H/He < −5. The difference of 61 K in T<sub>eff</sub> between both
temperature estimates can be explained by the different sets of
photometry used (BVRI + JHK instead of Pan-STARRS
grizy). Blouin et al. were also able to constrain log Mg/He to
−8.66, a value much higher than the −9.92 we used here on
the basis of the chondrite ratio.

J0801+5329, J0842−1347, and J1428+4403—These are objects similar to J0152+2418 where the magnesium lines and the shape of the Ca II H and K lines are not well reproduced. Reducing log H/He improves the fit in one case (J1428 +4403), but not to the point where the fit is satisfactory. Additional adjustments of individual metal abundances are most probably needed for these objects.

J0846+3538 and J1356+4047—Both stars seem to have an
overabundance of magnesium and sharper-than-predicted Ca II
H and K lines, as opposed to J0152+2418 and other similar
objects discussed above. Again, a fit using a tailor-made grid
with different abundance ratios would probably allow a better
fit for these objects.

J1214+7822—It was impossible to obtain a good spectroscopic fit for both the Ca II H and K lines and the Ca I line at 4226 Å simultaneously. We found the best agreement with H and K by using hydrogen-free models and ignoring the Ca I line. However, with a mass of only 0.31 M<sub>☉</sub>, this star is most likely a double-degenerate system, and a DZ+DC system could explain the unusually narrow H and K spectral lines at this temperature. Alternatively, Limoges et al. (2015) suggested that this object could have a hydrogen-rich atmosphere, with a lower atmospheric pressure and thus narrower absorption lines.

J1234+5606—The SDSS magnitudes for this object are ~0.3 mag fainter than Pan-STARRS and Gaia, which leads to two different possible solutions. We could not reach a conclusion regarding this discrepancy, but the SDSS colors lead to a much better spectroscopic fit of the helium lines at 5876 and 6678 Å, indicating a good estimate of the effective temperature. We thus decided to adopt the SDSS photometric data set.

J2253-0646 (WD 2251-070)—Blouin et al. (2019a) found for this object T<sub>eff</sub> = 4170 ± 90 K, log g = 8.06 ± 0.08, and log Ca/He = -9.8 ± 0.2 from fitting BVRI + JHK and Pan-STARRS photometry, while we find T<sub>eff</sub> = 4132 ± 53 K, log g = 8.03 ± 0.07, and log Ca/He = -10.00 ± 0.05. The values are very close, but Blouin et al. (2019a) show a much better fit in their Figure 9. This is not surprising because they relied on improved line profile calculations for the Ca I line at 4226 Å. These improved calculations are important only for objects with T<sub>eff</sub> ≲ 4500 K, and thus they have little impact on our analysis. Only four objects in our sample are in that temperature range: J1636+1619 and J0555-0410 do not show the Ca I line, and J1214+7822 has already been discussed above.

6. DQ White Dwarfs

6.1. Carbon Abundances

Following the method described in Section 4, we obtained
the atmospheric parameters for all 317 DQ stars in our sample
by fitting simultaneously the spectral energy distribution and
the carbon features (atomic and/or molecular). Figure 11 shows examples of spectroscopic and photometric fits for typical DQ white dwarfs (all our fits are available in the online journal, and our final parameters are given in Table 5). The figures show the spectral region used for the fit, i.e., either the Swan bands between 4000 and 6500 Å or the carbon lines between 4500 and 5500 Å for the hotter objects. Note that for some objects with weaker molecular absorption features, we used a smaller region centered on the observed bands to achieve a good fit.

Figure 12 shows the carbon abundance as a function of effective temperature, using a color scale to indicate the mass of each object. Two distinct populations are clearly present: one sequence with “normal”-mass (~0.6 $M_\odot$) DQ white dwarfs (bluish circles) at low effective temperature, and a second sequence (reddish circles) of massive white dwarfs ($M \ge 0.8 M_\odot$), with carbon abundances increasing with effective temperature. The first sequence follows nicely the expected evolutionary path for 0.6 $M_\odot$ DQ stars with $\log q(\text{He}) = -2.0$ (Dantona & Mazzitelli 1979; Iben & MacDonald 1985; Fontaine & Brassard 2005). Note that in Dufour et al. (2005) the bulk of the stars were following the evolutionary sequence for $\log q(\text{He})$ closer to -3.0. This is mainly due to the fact that in our improved models the band strengths with our new line list are slightly higher than with the Zeidler-K.T. & Koester (1982) prescription, resulting in a systematic downward shift in carbon abundances compared to the values published in Dufour et al. (2005). Note also that the abrupt disappearance of normal-mass DQ stars above ~9500 K is simply due to the visibility limit of carbon at these temperatures; in order to see carbon features in the optical at higher effective temperatures, the abundance must be at least $\log (\text{C/He}) = -4.5$. Unless an ultraviolet spectrum is available, objects following the predicted sequence in that range of effective temperature will simply be classified as DB white dwarfs. Also, the absence of stars below $T_{\text{eff}} \sim 6500$ K is artificial since these were not included in our sample owing to the uncertainties in modeling the pressure-shifted Swan bands (the so-called peculiar DQ, or DQpec; see Kowalski 2010; Blouin et al. 2019c).

More importantly, our analysis confirms the presence of a second sequence (Dufour et al. 2005; Koester & Knist 2006) with an abundance about 1 dex higher than the bulk of DQ white dwarfs at $M \sim 0.6 M_\odot$. It appears that this second sequence is indeed composed of massive white dwarfs, as suggested by Dufour et al. (2005), at least for the hottest ones ($T_{\text{eff}} \gtrsim 9000$ K). In fact, almost all DQ white dwarfs with an effective temperature above 10,000 K have masses higher than $0.8 M_\odot$. The coolest objects on the second sequence, however, do not appear to have larger masses. These massive DQ stars will be discussed in greater detail in Section 6.4.

## 6.2. Mass Distribution

The mass distribution of the DQ white dwarfs in our sample, displayed in Figure 13, reveals the two distinct populations very clearly, with the bulk of our sample centered around 0.55 $M_\odot$ and a second bump centered around 1 $M_\odot$, where all the hottest DQ stars in our sample are found. As stated before in the case of DBZ/DZ(A) white dwarfs, DQ stars with derived masses below ~0.45 $M_\odot$ are probably unresolved double-degenerate binaries. Such low-mass stars, if isolated, could not have been formed from single-star evolution within the lifetime

of the Galaxy. Moreover, these objects are usually interpreted as helium-core white dwarfs whose core mass was truncated by mass transfer with a companion. The presence of carbon in these apparently low-mass DQ stars indicates that at least one component has a carbon core. These binaries are thus probably composed of two non-DA stars (e.g., DQ + DC), which means that the masses reported here are most likely underestimated.

Surprisingly, the main peak of the mass distribution for the DQ white dwarfs in our sample is shifted by ~0.05 $M_\odot$ relative to that obtained for the DA and DB white dwarfs analyzed by Genest-Beaulieu & Bergeron (2019, see their Figure 21), raising suspicions about our mass determinations of DQ stars. A way to test the accuracy of our method for the mass determination is to look at the well-known DQ white dwarf Procyon B (J0739+0513), which has a very accurate dynamical mass determination of $0.592 \pm 0.006 M_\odot$ (Bond et al. 2015). Using our standard hybrid photometric/spectroscopic approach, we obtain a slightly lower mass of $0.554 \pm 0.013 M_\odot$ using HST photometry. This indicates that we are probably dealing with a systematic shift in the mass determinations of DQ white dwarfs using our models.<fnref n="13" /> Uncertainties related to our dereddening procedure for our $T_{\text{eff}}$ and mass determinations should be irrelevant here given the proximity of Procyon B. In fact, if we take only objects within 100 pc, where reddening should be minimal along the line of sight for most stars, we still find that the peak of the mass distribution is too low. In order to identify the origin of this shift in mass, we performed several tests using various model grids where we replaced our treatment of the molecular band opacity with the old just overlapping line approximation (Zeidler-K.T. & Koester 1982). We also tried various treatments for the line broadening and included undetectable traces of hydrogen or oxygen as well. In the end, similar mass distributions were always obtained, and we could not pinpoint the exact cause of this discrepancy.

We note, however, that several model lines in the ultraviolet have wings that extend several hundreds of angstroms from the line center. This is probably not physically realistic, and the use of the impact approximation is certainly not appropriate for these lines. Unfortunately, the data required to compute line profiles with the unified line shape theory of Allard et al. (1999) are not available for these lines. For stars with available spectroscopic observations in the UV (Holberg et al. 2003),<fnref n="14" /> the use of the quasi-static van der Waals broadening (Walkup et al. 1984; D. Koester 2010, private communication) provides a good fit to the asymmetric C I λ1930 line, although the carbon abundance needed to reproduce the observations is different than that determined from the Swan bands. This problem is not new (see Dufour 2011 for a detailed discussion), and the origin of this discrepancy still remains mysterious to this day, although it is most likely related to the uncertainties of the ultraviolet opacities. In fact, for DQ stars in the intermediate-temperature regime where both atomic lines and molecular bands are present (~9500 to 11,000 K), the models also have difficulties reproducing both types of absorption features simultaneously. Examples of this problem are illustrated in Figure 14. While it is possible to obtain a good spectroscopic fit by increasing the effective temperature and carbon abundance for these objects, doing so would then

<footnote n="13">¹³ Note that a recent study by Koester & Kepler (2019) also finds a DQ mass distribution peaking near 0.55 $M_\odot$.</footnote>

<footnote n="14">¹⁴ http://vega.lpl.arizona.edu/newsips/low/</footnote>
<figure><img src="image_12.png" /><figcaption>**Figure 11.** Examples of fits to DQ white dwarfs in our sample. In the left panels, error bars represent the observed data, while filled circles correspond to our best-fit model, with the atmospheric parameters given in each panel. The photometry used in the fit is indicated at the top left of each panel. A dagger symbol indicates that the log g value has been fixed at 8.0, when no trigonometric parallax is available. A star symbol indicates that the value of log H/He has been fixed rather than fitted. The right panels show our spectroscopic fits (red) to the normalized observed spectra (black). The complete figure set with all our fits (53 pages) is available in the online journal.</figcaption></figure>

(The complete figure set (53 images) is available.)
<table><caption>**Table 5**
Atmospheric Parameters for DQ Stars</caption><thead><tr><th>J Name</th><th>T<sub>eff</sub> (K)</th><th>log g</th><th>M/M<sub>☉</sub></th><th>log C/He</th><th>log H/He</th><th>D (pc)</th></tr></thead><tbody><tr><td>J0000−0026</td><td>7420(163)</td><td>7.87<sup>+0.09</sup><sub>-0.08</sub></td><td>0.50<sup>+0.05</sup><sub>-0.04</sub></td><td>−6.11(0.12)</td><td></td><td>125.9(4.5)</td></tr><tr><td>J0000−0850</td><td>7853(226)</td><td>7.42<sup>+0.19</sup><sub>-0.17</sub></td><td>0.30<sup>+0.07</sup><sub>-0.05</sub></td><td>−5.51(0.09)</td><td></td><td>207.3(18.4)</td></tr><tr><td>J0006+1800</td><td>7422(155)</td><td>7.90<sup>+0.06</sup><sub>-0.06</sub></td><td>0.52<sup>+0.03</sup><sub>-0.03</sub></td><td>−5.98(0.03)</td><td></td><td>57.9(0.4)</td></tr><tr><td>J0007+2821</td><td>7409(195)</td><td>7.92<sup>+0.15</sup><sub>-0.15</sub></td><td>0.53<sup>+0.09</sup><sub>-0.08</sub></td><td>−5.89(0.10)</td><td></td><td>184.5(15.1)</td></tr><tr><td>J0008+2507</td><td>10801(525)</td><td>8.10<sup>+0.28</sup><sub>-0.26</sub></td><td>0.64<sup>+0.18</sup><sub>-0.15</sub></td><td>−3.37(0.21)</td><td>−3.00*</td><td>311.1(53.6)</td></tr><tr><td>J0008−1034</td><td>7587(543)</td><td>7.86<sup>+0.26</sup><sub>-0.24</sub></td><td>0.50<sup>+0.15</sup><sub>-0.12</sub></td><td>−5.85(0.14)</td><td></td><td>145.0(7.5)</td></tr><tr><td>J0015+0309</td><td>7261(217)</td><td>7.98<sup>+0.31</sup><sub>-0.26</sub></td><td>0.56<sup>+0.13</sup><sub>-0.11</sub></td><td>−6.03(0.12)</td><td></td><td>199.2(23.0)</td></tr><tr><td>J0024−1107</td><td>8330(202)</td><td>7.88<sup>+0.07</sup><sub>-0.07</sub></td><td>0.51<sup>+0.04</sup><sub>-0.04</sub></td><td>−5.09(0.06)</td><td></td><td>108.1(2.1)</td></tr><tr><td>J0033+0418</td><td>7658(214)</td><td>7.86<sup>+0.10</sup><sub>-0.10</sub></td><td>0.49<sup>+0.06</sup><sub>-0.05</sub></td><td>−6.00(0.07)</td><td></td><td>133.4(5.0)</td></tr><tr><td>J0041+7321</td><td>9433(349)</td><td>7.84<sup>+0.07</sup><sub>-0.06</sub></td><td>0.49<sup>+0.04</sup><sub>-0.03</sub></td><td>−4.38(0.12)</td><td></td><td>55.8(0.1)</td></tr></tbody></table>

Note. An asterisk indicates that the value of log H/He has been fixed rather than fitted. A dagger indicates that the distance is a photometric distance based on the value of log g = 8 in the absence of a parallax.

(This table is available in its entirety in machine-readable form.)

<figure><img src="image_13.png" /><figcaption>**Figure 12.** Carbon abundances as a function of effective temperature for DQ white dwarfs using a color scale for the mass of each object (objects without parallax measurements are shown as black diamonds). A square indicates an object fitted with models including a trace of hydrogen (log H/He = −3; see text). Solid lines represent evolutionary models from Fontaine & Brassard (2005) at 0.6 $M_{⊙}$ with, from top to bottom, log q(He) = −3 and log q(He) = −2, while the dashed line is for 1.0 $M_{⊙}$ and log q(He) = −5. The standard dredge-up scenario struggles to explain the observed abundance pattern for massive objects.</figcaption></figure>

produce an energy distribution completely at odds with the photometric observations (see right panel of Figure 14). A similar problem, based on an analysis using D. Koester’s model atmosphere code, was also reported by Gänsicke et al. (2010) for two DQ stars showing traces of oxygen. For these problematic DQ stars in our sample, we compare in Table 6 the results of our standard approach with the parameters obtained by fitting only the optical spectra, thus ignoring all photometric information. We note that even though the exact atmospheric parameters are somewhat uncertain because of this problem, the massive nature ($M > 0.8 M_{⊙}$) of these objects remains unquestionable, even when allowing for a conservative uncertainty of 0.15 $M_{⊙}$.

While the problems mentioned above become apparent only for stars that show both atomic and molecular lines, it is likely, however, that the whole temperature, abundance, and mass scales for DQ white dwarfs are affected, which could perhaps also explain the ~0.05 $M_{⊙}$ shift of the peak of the DQ mass

<figure><img src="image_14.png" /><figcaption>**Figure 13.** Mass distribution of DQ white dwarfs (blue), compared to the photometric mass distributions of DA and DB white dwarfs (orange and black lines, respectively) taken from Figure 21 of Genest-Beaulieu & Bergeron (2019).</figcaption></figure>

distribution relative to DA and DB white dwarfs. Until this problem is solved, the absolute values of the atmospheric parameters for all DQ white dwarfs should be considered uncertain, although the relative values between objects in the sample are probably reliable.

### 6.3. The Effects of Hydrogen

The presence of hydrogen is a rare phenomenon in DQ white dwarfs (see Dufour 2011). Hydrogen abundance measurements are reported only for two relatively hot and massive DQ stars in the literature, namely, G35-26 (Thejll et al. 1990) and G227-5 (Wegner & Koester 1985), while its presence is also inferred from the detection of a CH molecular band in two other objects —G99-37 and GJ 841B, both magnetic white dwarfs (Vornanen et al. 2010; Blouin et al. 2019c). Our sample of cool DQ white dwarfs with molecular carbon bands contains only six objects showing $H\alpha$, three of which have masses below 0.47 $M_{⊙}$, indicating that they are most probably unresolved double-degenerate binaries composed of a DQ and a DA white dwarf; these are listed in Table 7. Examples of other similar unresolved DQ+DA binaries can also be found in
<figure><img src="image_15.png" /><figcaption>**Figure 14.** Left panel: best solutions when the effective temperature is obtained from fits to photometric data. Middle panel: best solutions when fitting only the optical spectra. Right panel: corresponding energy distributions for the two solutions. Open circles represent the best photometric/spectroscopic solutions (our standard approach), while filled circles correspond to the solutions with the effective temperature and carbon abundance determined solely from the optical spectra.</figcaption></figure>

Vennes et al. (2012), Vennes & Kawka (2012), and Leggett et al. (2018). Except for the hotter DQ J1243+1651, the other objects have an Hα line that is too narrow to be reproduced by helium-rich models, and we believe that they are also DQ+DA binaries. Hence, of the 293 DQ white dwarfs with temperatures between ~6500 and 10,500 K in our sample, there is compelling evidence for the presence of hydrogen in only two objects, both hotter than 9000 K.

As mentioned in Section 5.1, undetectable traces of hydrogen can still have an impact on the atmospheric parameter determination of helium atmosphere white dwarfs, although in the case of DQ white dwarfs the contribution of free electrons from carbon is usually much more important than that of heavy elements in DZ stars. To test the effect of hydrogen on the determination of the atmospheric parameters of cool DQ white dwarfs, we generated a small grid of models with log H/He = -3 and refitted every object in our sample with T<sub>eff</sub> < 10,000 K. We find that by assuming such a hydrogen abundance we could produce Hα and Hβ absorption features that would be easily detected above 8000 K, as well as a CH molecular band near 4300 Å for objects cooler than ~8500 K. Even by assuming an abundance that is clearly ruled out by the spectroscopic observations, we find that the presence of hydrogen has only a marginal impact on our atmospheric parameters, increasing the derived effective temperatures by 150 K and masses by 0.035 M<sub>☉</sub>, on average. We can thus safely consider that the presence of undetectable traces of hydrogen does not affect significantly the parameters of cool DQ stars.

In the atomic line regime (T<sub>eff</sub> ≳ 10,000 K), however, spectroscopic observations tell a different story. Although Hα is blended with the carbon lines near 6588 Å, Hβ can be detected very clearly. These lines are observed in 40% of the

objects (10/25). We thus generated additional grids with log H/He = -2, -3, and -4, with T<sub>eff</sub> ranging from 10,000 to 16,000 K. We found that, surprisingly, the log H/He = -3 grid best reproduces the observations for almost every star, suggesting that the hydrogen abundance appears relatively constant in the objects where hydrogen is detected. Therefore, the solutions with log H/He = -3 are presented in Table 5 when the Balmer lines are observed. Note that it is possible that DQ stars with no detectable hydrogen features also contain some traces of hydrogen, since the Balmer lines are barely predicted for some objects with higher carbon abundance and lower effective temperature. The presence of hydrogen in these objects has a marginal impact on the measured effective temperature (265 K on average) and mass (0.032 M<sub>☉</sub> on average), but the effect can be quite significant for the carbon abundance (around 0.5 dex in some cases). Since we do not know the exact hydrogen abundance for these objects, we provide the solutions without hydrogen in Table 5.

### 6.4. Massive DQ White Dwarfs

We now turn our attention to the population of massive (M > 0.8 M<sub>☉</sub>) DQ white dwarfs clearly distinguishable in Figures 12 and 13. First, the carbon abundance pattern observed for these objects indicates that they followed a different evolutionary path than their cooler normal-mass counterparts, since the standard dredge-up prediction for massive white dwarfs completely fails to explain their chemical composition (see also Brassard et al. 2007). Second, such massive DQ stars could not have DB white dwarfs as progenitors because practically no massive helium-rich white dwarfs exist at higher effective temperatures (Beauchamp et al. 1996; Bergeron et al. 2011;
<table><caption>**Table 6**
Parameters of DQ White Dwarfs with Bad Spectroscopic Fits when the Effective Temperature Is Derived from Photometry</caption><thead><tr><th rowspan="2">Name</th><th colspan="3">Photometric Fits</th><th colspan="3">Spectroscopic Fits</th></tr><tr><th>T<sub>eff</sub> (K)</th><th>log C/He</th><th>M/M<sub>☉</sub></th><th>T<sub>eff</sub> (K)</th><th>log C/He</th><th>M/M<sub>☉</sub></th></tr></thead><tbody><tr><td>J0852+2316</td><td>11099</td><td>-3.18</td><td>0.97</td><td>12733</td><td>-1.99</td><td>1.12</td></tr><tr><td>J0859+3257</td><td>9486</td><td>-3.52</td><td>0.87</td><td>10407</td><td>-2.87</td><td>0.97</td></tr><tr><td>J0936+0607</td><td>11013</td><td>-3.07</td><td>0.97</td><td>12109</td><td>-1.99</td><td>1.09</td></tr><tr><td>J1000+1005</td><td>7869</td><td>-4.99</td><td></td><td>10395</td><td>-2.87</td><td></td></tr><tr><td>J1126+3245</td><td>9297</td><td>-4.43</td><td></td><td>11369</td><td>-2.73</td><td></td></tr><tr><td>J1140+0735</td><td>10651</td><td>-3.36</td><td>0.94</td><td>12395</td><td>-1.60</td><td>1.12</td></tr><tr><td>J1140+1824</td><td>9656</td><td>-3.54</td><td>0.81</td><td>10921</td><td>-2.62</td><td>0.94</td></tr><tr><td>J1148-0126</td><td>9680</td><td>-3.48</td><td>0.88</td><td>10868</td><td>-2.62</td><td>1.00</td></tr><tr><td>J2248+2826</td><td>9390</td><td>-4.21</td><td>0.57</td><td>11920</td><td>-2.02</td><td>0.90</td></tr></tbody></table>

<table><caption>**Table 7**
DQ White Dwarfs Showing Hydrogen Lines and Swan Bands</caption><thead><tr><th>Name</th><th>T<sub>eff</sub> (K)</th><th>M/M<sub>☉</sub></th></tr></thead><tbody><tr><td>J0836+0437</td><td>8420</td><td>0.46</td></tr><tr><td>J0928+2638</td><td>7108</td><td>0.28</td></tr><tr><td>J0950+3238</td><td>8268</td><td></td></tr><tr><td>J1243+1651</td><td>10227</td><td>0.53</td></tr><tr><td>J1406+3402</td><td>7042</td><td>0.30</td></tr><tr><td>J2310-0057</td><td>7647</td><td>0.54</td></tr></tbody></table>

Genest-Beaulieu & Bergeron 2019; Gentile Fusillo et al. 2019; Tremblay et al. 2019a). Instead, the massive sequence connects nicely with the carbon-dominated atmosphere white dwarfs at higher effective temperature, the so-called Hot DQ stars ($T_{eff} \ge 18,000$ K and $log(C/He) \ge 0$; see Dufour et al. 2007b, 2008, 2013). Note that *Gaia* trigonometric parallax measurements confirm that the carbon-dominated atmosphere white dwarfs are massive as well. We thus believe that our massive DQ stars represent cooler versions of the carbon atmosphere white dwarfs.

The many unusual properties of Hot DQ white dwarfs— unique chemical composition, high mass, and high incidence of magnetism—recently prompted Dunlap & Clemens (2015) to propose that these hot carbon atmosphere white dwarfs represent a population of merged white dwarfs (failed Type Ia supernovae). The key piece of evidence for this proposed scenario comes from the kinematic properties of the sample, which provide an independent age indicator (Dunlap & Clemens 2015). Indeed, one interesting characteristic that stands out about the Hot DQ population is their very high tangential velocity, as a group, compared to other white dwarfs with the same age and mass. Over time, a population of stars is kinematically heated through gravitational interactions (Wegg & Phinney 2012). Hence, as a population gets older, its velocity dispersion increases. In particular, Dunlap & Clemens showed that there is a discrepancy between the young age inferred from the derived atmospheric parameters of carbon atmosphere white dwarfs<fnref n="15" /> and the old age derived from the velocity dispersion. The merger scenario provides an elegant solution to this dilemma because the reheating induced by the merging event resets the cooling age clock, and consequently, the cooling age derived from the effective temperature and mass thus becomes meaningless. If the massive DQ white

<figure><img src="image_16.png" /><figcaption>**Figure 15.** Cumulative distribution of transverse velocities for massive DQ, DBZ/DZ(A) (this paper), and DA (Genest-Beaulieu & Bergeron 2019) white dwarfs with $T_{eff}$ between 10,000 and 16,000 K.</figcaption></figure>

dwarfs observed in Figures 12 and 13 are indeed cooled-down versions of the Hot DQ stars, a kinematic analysis of the sample should then show the same discrepancy between the two age indicators.

We show in Figure 15 the cumulative distribution of transverse velocities, $V_{trans}$, for our sample of massive DQ white dwarfs, calculated using *Gaia* distances and proper motions. We selected DQ white dwarfs with $T_{eff} > 10,000$ K and $M > 0.8 M_{☉}$, a point where there appears to be a clean separation between the two DQ populations. Since there is no DQ white dwarf with “normal” masses in this temperature range, we compare the properties of the massive DQ stars with those of our sample of DBZ/DZ(A) and DA white dwarfs with signal-to-noise ratio higher than 25 taken from Genest-Beaulieu & Bergeron (2019), a large and clean sample perfectly suited for this comparison. Also, in order to make a meaningful comparison, we selected only DBZ/DZ(A) and DA white dwarfs in the same range of effective temperature where the massive DQ stars are found, i.e., between 10,000 and 16,000 K, and we compare the distributions of transverse velocities for stars with masses greater than $0.8 M_{☉}$ and in the $0.5-0.8 M_{☉}$ mass range.

First, we notice that the transverse velocity distributions for DBZ/DZ(A) and DA white dwarfs are very similar, and that those with $M > 0.8 M_{☉}$ are much less dispersed than those with $0.5 M_{☉} < M < 0.8 M_{☉}$. This is exactly what we expect, since massive white dwarfs have massive progenitors with

<footnote n="15">¹⁵ Such hot massive white dwarfs should descend from massive, short-lived main-sequence progenitors, and they should thus have a small total age if they evolved as single stars.</footnote>
shorter main-sequence lifetimes, corresponding to a shorter total age, and thus a smaller velocity dispersion (Wegg & Phinney 2012). However, the velocity dispersion for the massive DQ stars displayed in Figure 15 is extremely broad, in sharp contrast with what is expected for a population of massive white dwarfs. We find indeed that 45% (10 out of 22) of the massive ($M > 0.8 M_\odot$) DQ population has $V_{\text{trans}} > 50$ km s$^{-1}$, while the fractions are only 5% (3 out of 61) and 7% (1 out of 14) for DA and DBZ/DZ(A) with similar atmospheric parameters, respectively. The fact that the distribution is even larger than that of stars in the 0.5–0.8 $M_\odot$ mass range strongly supports the idea that these objects are the descendants of the carbon-dominated atmosphere white dwarfs (the Hot DQ stars), and consequently that they represent the outcome of the merging of two white dwarfs.

It is interesting to note, however, that very few massive DQ stars are found with a large magnetic field (J1040+0635 and J1036+6522 are the only two examples; Williams et al. 2013), in contrast with the very large fraction of magnetic Hot DQ white dwarfs (at least 70% are magnetic at some level; Dufour et al. 2013). Magnetic fields of a few MG would be easily detectable through line splitting in our spectra. It thus seems that as helium finds its way to the top of the photosphere, the strength of the magnetic field is also reduced significantly, as these stars cool down. If our hypothesis that the Hot DQ stars represent the progenitors of the massive DQ in our sample is correct, then the latter are probably still magnetic at some level. Unfortunately, the SDSS spectra used in our analysis lack the signal-to-noise ratio and spectral resolution required to properly identify weaker magnetic fields of a few $\times 10^2$ kG. We predict that a large fraction of these objects will eventually show magnetism when high-resolution spectroscopic observations become available. Also, at least one-third of the Hot DQ stars are variable (Barlow et al. 2008; Montgomery et al. 2008; Dunlap et al. 2010; Dufour et al. 2011; Lawrie et al. 2013), presumably due to the presence of magnetic spots at the surface of a rapidly rotating star. If magnetism is indeed still present at some level, a large fraction of the massive DQ stars should also be variable.

Finally, we notice that there appears to be a correlation between the mass and the effective temperature, the hottest objects near 16,000 K being significantly more massive ($\sim 0.3 M_\odot$) than their cooler counterparts near 10,000 K. As it is expected that white dwarfs should evolve at constant masses, we first believed that some missing ingredient in our models was responsible for either an overestimation of the masses of the hottest stars or an underestimation for the cooler ones. We experimented with many test grids incorporating traces of oxygen, hydrogen, and different treatments of line broadening, but none of these experiments affected the relative masses of our object in a significant way. As mentioned in Section 6.2, we know that there are some uncertainties concerning the absolute values of the atmospheric parameters of DQ white dwarfs that may explain the shifted position of the peak of the mass distribution and the difficulties in reproducing simultaneously the atomic and molecular features. However, it is unlikely that the shortcomings of our models translate into uncertainties on the masses much larger than 0.1 $M_\odot$ (the radius of the star is tightly constrained from the measured energy distribution and distance), not enough to explain the clear correlation with effective temperature that we observe in our sample. We thus believe that the observed correlation is real

<figure><img src="image_17.png" /><figcaption>**Figure 16.** Absolute *G* magnitudes as a function of $G_{BP} - G_{RP}$. Filled circles represent DQ stars with $\sigma_{\pi}/\pi < 0.1$, using a color scale for the mass of each object. Black dots are white dwarf candidates within 100 pc selected from Gaia DR2 using the cuts proposed in Appendix B of Gaia Collaboration et al. (2018).</figcaption></figure>

and that it is most probably the manifestation of an observational bias due to the crystallization of the core of these white dwarfs (see next section).

## 6.5. The Crystallization Sequence

As a white dwarf cools off, thermal energy is gradually lost from the star in the form of radiation, until the kinetic motions of the ions lose amplitude and eventually become correlated. The ionic state then evolves from a gas to a fluid to a solid, a process referred to as crystallization (Fontaine et al. 2001). This liquid-to-solid transition, which begins in the center of the stellar core and slowly moves outward, represents a first-order phase transition and is thus accompanied by a release of latent heat, which contributes to slowing down the cooling process significantly. Over 50 yr ago, Van Horn (1968) predicted this crystallization sequence, a process that decreases the cooling rate during the transition, leading to a pileup of objects that could be detected in H-R diagrams. The first direct observational evidence of this crystallization process was reported by Tremblay et al. (2019b), in the form of a characteristic pileup of white dwarfs forming a tight sequence in the Gaia $M_G$ versus $G_{BP} - G_{RP}$ H-R diagram, the so-called “Q branch” (see their Figure 2).

Our second, more massive sequence of DQ white dwarfs is particularly interesting in the context of crystallization, because for $\sim 0.6 M_\odot$ white dwarfs crystallization occurs at the same time as another physical process referred to as convective coupling, which takes place when the base of the superficial convection zone reaches into the degenerate interior (Fontaine et al. 2001). Convective coupling also decreases the cooling process temporarily, and in $\sim 0.6 M_\odot$ white dwarfs the effects of crystallization and convective coupling cannot be disentangled. Fortunately, crystallization occurs at much higher effective temperature for more massive white dwarfs, making the massive sequence of DQ white dwarfs ideal objects for examining the crystallization process.

Figure 16 shows the color–magnitude diagram for our sample of DQ white dwarfs with measured parallaxes and $\sigma_{\pi}/\pi < 0.1$. The lowest red point in this diagram is J0841+3329, for which the Gaia phot_bp RP_excess_factor
<figure><img src="image_18.png" /><figcaption>**Figure 17.** Mass as a function of $T_{\text{eff}}$ for the DQ white dwarfs in our sample (red circles), and DA white dwarfs spectroscopically identified in the MWDD with $\sigma_{\pi}/\pi < 0.1$ (white circles), taken from Figure 14 of Bergeron et al. (2019). Also reproduced are their theoretical isochrones obtained from cooling sequences with C/O-core compositions, $q(\text{He}) = 10^{-2}$, and $q(\text{H}) = 10^{-4}$, equally spaced by $\Delta \log \tau_{\text{cool}} = 0.02$ (in yr). The lower black solid curve indicates the onset of crystallization at the center of evolving models, while the upper one indicates the location where 80% of the total mass has solidified. The dashed curve indicates the onset of convective coupling. The black plus sign corresponds to the mean errors of each fitted parameter.</figcaption></figure>

value is 1.34, indicating possibly a bad quality of magnitudes according to Gaia Collaboration et al. (2018). Our best fit based on SDSS photometry predicts a lower magnitude in the G bandpass. By removing this object, all our massive DQ stars fall in the region of the crystallization sequence reported by Tremblay et al. (2019b).

Figure 17 shows the masses as a function of effective temperature for the same sample of DQ stars, together with the sample of DA white dwarfs analyzed by Bergeron et al. (2019, see their Figure 14). The black solid lines are also described at length in Bergeron et al. Briefly, the left line corresponds to the onset of crystallization at the center of an evolving model (the same as in Tremblay et al. 2019b), while the right line indicates the location where 80% of the total mass of the star has solidified. Upon crystallization, latent heat is slowly released, and the white dwarf cooling is slowed down, a process that is well illustrated by the tightening of the isochrones (shown in red) between the two black lines. Figure 17 reveals that most of our massive DQ stars fall within these lines, indicating a possible pileup due to crystallization.

Figure 18 shows the masses of DBZ/DZ(A) white dwarfs in our sample as a function of effective temperature, together with the same isochrones and crystallization sequences as before. The signature of crystallization cannot be seen, but this could be due to the fact that there are much less massive stars above $T_{\text{eff}} = 12,000$ K, which is consistent with the idea that DBZ (A) white dwarfs belong to the same population as DB(A) stars. At high temperatures, it is also more difficult to see metal absorption lines (see the detection limit in Figure 8), and the diagram is thus much less populated in that range.

## 6.6. Discussion of Individual Objects

J0901+5751, J0922+2928, and J1423+5729.—These stars show many O I lines, especially at 6156 and 7772 Å, and at 5330 and 5435 Å in the case of J1423+5729. For J0922+2928,

Gänsicke et al. (2010) found $T_{\text{eff}} = 8270$ K and $\log C/\text{He} = -2.6$ using oxygen-rich models, while we found $T_{\text{eff}} = 8022$ K and $\log C/\text{He} = -4.93$. They also found $\log \text{H/He} < -5.0$ and $\log \text{O/He} = -2.0$, which implies that oxygen is more abundant than carbon. White dwarfs with such a high amount of oxygen in their atmosphere must have followed a different evolutionary path, and the authors suggest that they could be O/Ne-core white dwarfs surrounded by a layer of carbon and oxygen. Such white dwarfs would be massive. According to our results, only one of these can be qualified as massive, but we determined the mass using evolutionary models with C/O cores and did not take atmospheric oxygen into account in our analysis. However, if we take the values of Gänsicke et al. (2010) for J0922+2928, all these objects fall above the first sequence in Figure 12, suggesting indeed a different evolutionary path. We also note that these objects are rare and have little impact on the conclusions of this work.

J1040+0635.—This star shows magnetic splitting, suggesting that our solution may be uncertain. To evaluate the influence of the magnetic field on our result, we fitted the magnetic white dwarf J1036+6522 (not included in this analysis), which shows similar line splitting. We found $T_{\text{eff}} = 15,637$ K, $\log g = 8.8$, and $\log C/\text{He} = -1.87$, the carbon abundance being very approximate since we do not include splitting in this exercise. Williams et al. (2013) used magnetic synthetic spectra to fit this object and estimated $T_{\text{eff}} \sim 15,500$ K, $\log g \sim 9$, and $\log C/\text{He} = -1$, which is consistent with our solution.

J2101+3148.—This star shows a discrepancy for the Swan band at 4700 Å. Dufour et al. (2005) also reported this problem without further explanation. With a larger sample, we can now compare this object with other stars with similar properties (e.g., J1424+0833 and J1118−0314), and this discrepancy is not observed anywhere else. A change in temperature of $\pm 500$ K does not improve the situation. However, the finding
<figure><img src="image_19.png" /><figcaption>**Figure 18.** Same as Figure 17, but for DZ/DBZ white dwarfs. Blue circles correspond to DZ/DBZ stars, while orange circles represent DZA/DBZA stars.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 19.** Comparison of the mass distributions of DBZ/DZ(A) white dwarfs for effective temperatures above and below 12,000 K.</figcaption></figure>

chart on SIMBAD clearly shows a red object very near the white dwarf (potentially a companion or an object along the line of sight) that probably contaminates the spectra.

## 7. On the Spectral Evolution of White Dwarfs

Thanks to the advent of Gaia, we now have, for the first time, detailed mass distributions for a large sample of both DQ and DBZ/DZ(A) white dwarfs. Although there are still uncertainties regarding the individual mass determinations (see Sections 5.1 and 6.2), a lot of information can be extracted from the overall shape of these mass distributions, and in particular with respect to the spectral evolution of these white dwarfs.

The mass distribution of DQ white dwarfs shows two distinct peaks (see Figure 13), one centered around 0.55 $M_\odot$ and another one centered around 1 $M_\odot$. As discussed above, the location of the first peak is slightly shifted relative to that of DA and DB white dwarfs (see Figure 21 of Genest-Beaulieu & Bergeron 2019), most probably indicating a systematic error

due to some unknown opacity source in the ultraviolet. Putting aside this problem, it is interesting to compare the overall shape of these mass distributions. The main difference between the mass distribution of DB stars and that of DA stars is the absence of a high-mass tail in the former. This difference was first noted by Beauchamp et al. (1996) and later confirmed by Bergeron et al. (2011) and Genest-Beaulieu & Bergeron (2019). The mass distribution of our DQ sample does not show an extended tail extending to masses above 1 $M_\odot$, but rather exhibits two distinct peaks with a clear separation near 0.8 $M_\odot$. As discussed in Section 6.4, there is compelling evidence that the second peak represents a population of merged white dwarfs. Hence, the mass distribution of DQ stars with $T_{\text{eff}} < 10,000$ K—effectively removing practically all the massive objects—resembles much more that of DB white dwarfs than that of DA stars, reinforcing the belief that the progenitors of DQ white dwarfs are helium-rich DB stars.

In contrast, the mass distribution of DBZ/DZ(A) white dwarfs (see Figure 7) reveals a fair number of massive objects forming some sort of a tail similar to that observed in DA white dwarfs. The similitude is even more striking in Figure 19, where we show the mass distributions of our sample for effective temperatures above and below 12,000 K. We see that at low effective temperatures the mass distribution—composed essentially of DZ(A) stars—clearly shows a high-mass tail, which is barely present in the mass distribution of hotter white dwarfs, containing essentially only DBZ(A) white dwarfs. We note that the differences in the morphology of these mass distributions cannot be explained by the presence of a significant number of statistical outliers owing to poor precision on parallax measurements (and thus mass determination) since our conclusions remain the same if we look only at stars with $\sigma_\pi/\pi < 0.1$.

The most logical explanation for this phenomenon is that the progenitors of a significant fraction of DZ(A) white dwarfs are DA stars that were convectively mixed and transformed into helium atmosphere white dwarfs. Indeed, it is well known that the ratio of non-DA to DA white dwarfs increases considerably with decreasing effective temperatures. This has been interpreted as the result of convective mixing, a process that
becomes important below 12,000 K, as discussed at length in Rolland et al. (2018) and references therein. This is a process where the thin hydrogen layer is mixed with the deeper and more massive helium layer through convection, effectively turning a hydrogen-rich white dwarf into a helium-rich one. Our results thus suggest that many DZ(A) stars, and certainly most of the massive ones, represent the outcome of mixed DA white dwarfs that have also accreted planetary material. In fact, Rolland et al. (2018) also demonstrated that most DZA stars below $T_{\text{eff}} \sim 12,000$ K could not be the descendants of DBA (or DBZA) stars, and that these objects must be the result of hydrogen-rich stars that turned helium-rich as a result of convective mixing. For the DBZ(A) white dwarfs, it is reasonable to consider these objects as being simply the metal-polluted version of DB(A) stars. These polluted DBZ(A) stars, when they cool off, will eventually become helium-rich white dwarfs, or DZ(A) stars if they also accrete material (they would not become DQ stars, however; see below).

Finally, the DQ white dwarfs with masses near the peak of the mass distribution—which rarely show any trace of hydrogen (see Section 6.3)—are most likely the descendants of the so-called “pure helium” DB stars that show no hydrogen feature. In these nearly hydrogen-free DB stars, hydrogen has probably been largely depleted during the earlier born-again post-AGB evolutionary phase (Rolland et al. 2018). This interpretation could simultaneously explain the rarity of DQZ objects<fnref n="16" /> (i.e., DQ stars showing traces of metals), if the surroundings of the progenitor have been cleared of rocky debris during the active earlier post-AGB phases.

## 8. Summary and Conclusions

We presented an analysis of 1023 DBZ/DZ(A) and 319 DQ white dwarfs based on state-of-the-art model atmospheres using new parallaxes, proper motions, and photometry from Gaia DR2, as well as photometry from SDSS, Pan-STARRS, previously published BVRI, and spectroscopy from various sources. This represents a significant increase over the previous comprehensive studies on these types of objects, namely, those of Dufour et al. (2007b; 159 DZs) and Dufour et al. (2005; 56 DQs). Calcium abundance measurements for our large sample of DBZ/DZ(A) white dwarfs indicate that the rocky objects that polluted their photosphere had masses similar to those of large asteroids found in our solar system. We found several polluted white dwarfs with progenitor masses well above $3 M_{\odot}$, confirming that the formation of rocky material is also common for early-type stars. The availability of parallax measurements for nearly three-quarters of our sample allowed us to determine, for the first time, meaningful mass distributions for these types of objects (the mass distributions for DQ and DZ stars of Dufour et al. 2005, 2007a contained only 11 and 16 objects, respectively). These mass distributions revealed several interesting aspects about the properties of our samples that we summarize here:

1. The mean mass for the DBZ/DZ white dwarfs (i.e., objects not showing $H\alpha$) is significantly higher than that of DBZA/DZA stars. The two distributions are in much better agreement when undetectable traces of hydrogen are included in the model fits.

2. The mass distribution of DZ(A) white dwarfs cooler than $T_{\text{eff}} = 12,000$ K shows a high-mass tail similar to that observed for DA stars. This high-mass tail is absent for objects in our sample hotter than 12,000 K. We interpret this as a signature that a significant fraction of the DZ(A) stars are convectively mixed DA white dwarfs that have accreted rocky material.

3. The mass distribution of DQ white dwarfs shows two distinct peaks, one centered at $0.55 M_{\odot}$, whose carbon abundances are well explained by the standard carbon dredge-up scenario, and another one centered at $\sim 1 M_{\odot}$, whose high kinematic properties are consistent with the idea that they represent a population of merged white dwarfs. We note that the location of the $0.55 M_{\odot}$ peak is slightly shifted toward smaller masses relative to that of DB white dwarfs, most probably due to some unknown opacity source in the ultraviolet in our DQ models.

While traces of hydrogen are detected (or needed) in nearly all DBZ/DZ(A) white dwarfs, its presence is extremely rare in cool DQ stars. This indicates that the natures of these two populations of helium atmosphere white dwarfs are clearly distinct. The most logical way to explain the abundance pattern and mass distributions of these objects is to interpret hydrogen-free DB white dwarfs as progenitors of cool DQ stars, while the other types of cool helium atmosphere white dwarfs, namely, the DC and DZ(A) stars, would originate from both convectively mixed DA and cooled-down DB(A) white dwarfs. Within this scenario, the rarity of both hydrogen and heavy elements in DQ white dwarfs is also naturally explained by invoking particularly active post-AGB phases that would eliminate practically all the remaining hydrogen, as well as most nearby rocky objects in orbit.

The presence of hydrogen in 40% of the hotter DQ white dwarfs is, however, somewhat mysterious. We showed that this population of massive stars has much larger space velocities than what is expected for a relatively young star population. As originally proposed by Dunlap & Clemens (2015) in the case of Hot DQ white dwarfs, this indicates that the massive DQ stars in our sample, which are most likely a cooled-down version of the Hot DQs, would also be the result of the merging of two C/O white dwarfs. Dufour et al. (2007b, 2008) proposed that the Hot DQ white dwarfs would disguise themselves as massive DB white dwarfs until the underlying convection zone completely dilutes the thin helium layer, effectively transforming the object into a C/O-dominated atmosphere when the star cools to about 24,000 K. However, such massive, and most probably magnetic, DB white dwarfs are very scarce. Another possibility is that these merger remnants could instead hide as DA stars with very thin hydrogen layers, or possibly as some of the massive magnetic white dwarfs that are often found at higher effective temperatures. We predict that spectropolarimetric or high-resolution spectroscopic observations of massive DQ white dwarfs should reveal that most of them are magnetic at some level.

Interestingly, we find that there is a correlation between the stellar mass and the effective temperature for these massive DQ white dwarfs. Despite the uncertainties associated with the temperature, abundance, and mass scales for these objects (due most probably to some missing opacity in the ultraviolet), we believe the observed trend to be real, and that it represents a manifestation of an accumulation of stars at certain effective temperatures due to the slowing of the cooling process, when

<footnote n="16">¹⁶ There are only four stars with Ca II H and K absorption lines in our sample: J0739+0513 (Procyon B), J0900+0331, J1332+2740, and J1534+4145.</footnote>
the stellar core eventually crystallizes. Future work should address the shortcomings in the modeling of DQ white dwarfs in order to reduce the uncertainties on the atmospheric parameters of these objects.

This work was supported in part by NSERC (Canada) and the FRQNT (Québec). B.H.D. acknowledges support from the Wootton Center for Astrophysical Plasma Properties under the United States Department of Energy collaborative agreement DE-FOA-0001634. This work has made use of the Montreal White Dwarf Database (Dufour et al. 2017) and also the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna; the SIMBAD database, operated at CDS, Strasbourg, France (Wenger et al. 2000); and data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement (Gaia Collaboration et al. 2016, 2018). Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS website is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University (Abolfathi et al. 2018). The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg, and the Max Planck Institute for Extraterrestrial Physics, Garching, Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope

Science Institute, the National Aeronautics and Space Administration under grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation (Chambers et al. 2016).

*Software: Astropy (Astropy Collaboration et al. 2013, 2018), Matplotlib (Hunter 2007), NumPy (Oliphant 2006; van der Walt et al. 2011), PySpecKit (Ginsburg & Mirocha 2011).*

## ORCID iDs

P. Bergeron https://orcid.org/0000-0003-2368-345X

S. Blouin https://orcid.org/0000-0002-9632-1436

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