A Photometric and Spectroscopic Investigation of the DB White Dwarf Population using SDSS and Gaia Data

C. Genest-Beaulieu; P. Bergeron

REVIEW 3 major objections 4 minor 1 cited by

A Photometric and Spectroscopic Investigation of the DB White Dwarf Population using SDSS and Gaia Data

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Most cool DBA white dwarfs carry too much hydrogen to be explained by convective dilution, so the hydrogen must have been accreted from outside the star.

desk verdict A thorough DB white dwarf census from SDSS/Gaia with a useful catalog and binary candidates; the hydrogen-origin conclusion is conditional on homogeneous-mixing models that the paper itself flags but does not test. read the letter →

arxiv 1908.01728 v1 pith:7VJ6CZX4 submitted 2019-08-05 astro-ph.SR

C. Genest-Beaulieu , P. Bergeron This is my paper · ORCID

classification astro-ph.SR

keywords whitedwarfsDBstarsDBAhydrogenabundanceconvectivedilutiondoubledegeneratebinariesphotometricfittingspectroscopic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper analyzes nearly 1,900 helium-atmosphere (DB) white dwarfs found by the Sloan Digital Sky Survey, fitting both their colors and their spectra and using Gaia parallaxes to measure temperatures, masses, and hydrogen abundances. It compares two independent measurement routes, photometry and spectroscopy, to test which parameters can be trusted and where each technique fails. Its central claim is about DBA white dwarfs (DB stars that also show hydrogen): below roughly 20,000 K the amount of hydrogen in their mixed surface layers is far too large to be the leftover of a hydrogen-rich DA progenitor, so most of that hydrogen must have been acquired later, by accretion from the interstellar medium, comets, or disrupted asteroids. Along the way the paper finds about 65 unresolved double-degenerate binaries (DB+DB and DA+DB) and finds no evidence for single low-mass DB white dwarfs. If the central claim holds, the spectral evolution of white dwarfs is messier and more interactive than simple cooling: the surfaces of many helium-atmosphere stars are being chemically resupplied from outside.

What carries the argument

The load-bearing machinery is a grid of LTE model atmospheres for helium-rich white dwarfs, with varying hydrogen and calcium abundances, computed with the ML2/$\alpha=1.25$ mixing-length prescription and a van der Waals broadening treatment chosen for this study. Two fitting techniques are paired: the photometric technique fits the $ugriz$ energy distribution plus a Gaia parallax to obtain $T_{\rm eff}$, solid angle, radius, and ultimately mass through a carbon/oxygen-core mass-radius relation; the spectroscopic technique fits normalized SDSS spectra to obtain $T_{\rm eff}$, $\log g$, and $\log(\mathrm{H/He})$ simultaneously. The decisive element is the $T_{\rm eff}$-$\log(\mathrm{H/He})$ diagram populated with constant-total-hydrogen-mass sequences from the homogeneous-mixing simulations; those sequences define a 'forbidden region' that a cooling star cannot cross with a fixed hydrogen reservoir. Comparing the observed abundances with those tracks is what turns photospheric hydrogen measurements into a statement about the total hydrogen reservoir and its origin.

What would settle it

Measure high-signal spectra of a sample of DBA white dwarfs below 20,000 K and compute each star's total hydrogen mass with the paper's homogeneous-envelope models: if a substantial fraction fall below $\log(M_H/M_\odot) \sim -15$ (the residual-dilution range), the central claim fails. A second disproof would be a DA white dwarf with a hydrogen layer near $10^{-12}\,M_\odot$ that nevertheless converts to a DB near 20,000 K.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the hydrogen seen in most DBA white dwarfs below $T_{\rm eff} \sim 20,000$ K cannot be a residual byproduct of the DA-to-DB transition. Using the measured hydrogen-to-helium ratios together with homogeneous mixing sequences at fixed total hydrogen mass, the inferred total hydrogen in the mixed envelope falls in the range $\log(M_H/M_\odot) \sim -14$ to $-10$. A DA progenitor with such a thick hydrogen layer would not have convectively mixed and become a DB in the first place; residual dilution would leave orders of magnitude less hydrogen. The paper therefore concludes that an external hydrogen source, such as interstellar accretion, comets, or disrupted asteroids, must be invoked for the bulk of cool DBA stars, while convective dilution remains the best explanation for the DA-to-DB conversion itself. It also reports a population of 55 DB+DB and 10 DA+DB unresolved binaries, identified by the discrepancy between photometric and spectroscopic masses, and shows that the DB/(DA+DB) ratio rises from about 5 percent at high temperatures to about 25 percent near 15,000 K before falling as DB stars turn into DC stars.

Load-bearing premise

The whole hydrogen-origin argument assumes that the envelope models used to convert photospheric hydrogen abundance into a total hydrogen mass, including the location of the forbidden region, are correct; if chemically stratified envelopes or improved convection models change that conversion, the seemingly excessive hydrogen could still be residual.

Editorial extensions

If this is right

Most cool DBA white dwarfs must have gained hydrogen after becoming DB stars, making external pollution a common stage of white dwarf evolution rather than a rare event.
The DA-to-DB transition is confined to a narrow band of hydrogen layer masses, around $\log(M_H/M_\odot)\sim -14$, setting a tight constraint on DA progenitor envelopes.
Most apparently low-mass DB white dwarfs are unresolved double degenerates; single DB white dwarfs below about 0.48 solar masses appear not to exist.
Massive DB white dwarfs below about 22,000 K may be former Hot DQ carbon-atmosphere white dwarfs, linking two otherwise separate white dwarf populations.
The DB/(DA+DB) ratio climbs from about 5 percent to about 25 percent as stars cool to 15,000 K, then drops as the coolest DB stars become DC stars, broadly matching convective dilution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if accretion supplies most cool DBA hydrogen, then hydrogen abundance should correlate with metal abundances and with infrared excess from debris disks; combining these SDSS measurements with follow-up mid-infrared photometry would test that link.
Beyond the paper: the 55 DB+DB candidates, if confirmed by radial-velocity monitoring, imply a large population of merging double-degenerate systems that could contribute to gravitational-wave sources and Type Ia supernova progenitors.
Beyond the paper: the continued existence of pure DB stars without hydrogen hints at a separate formation channel, possibly born-again post-AGB stars; searching for carbon or kinematic peculiarities in the pure DB sample could distinguish that channel.
Beyond the paper: the authors' finding that 3D corrections over-shoot $\log g$ below 20,000 K suggests hydrogen itself alters convection; computing 3D models that include trace hydrogen would test whether the forbidden region shrinks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague—

This is the paper to know if you work on helium-atmosphere white dwarfs. The authors fit SDSS photometry and spectroscopy plus Gaia parallaxes for 1,915 DB stars, publish the resulting Teff, log g, mass, H/He, and Ca/He parameters, and identify 10 DA+DB and 55 DB+DB unresolved binary candidates. That catalog and the Gaia-volume-limited DB/(DA+DB) ratio as a function of Teff are the durable results. The ratio rising from ~5% above 27,000 K to ~25% near 15,000 K is a clean confirmation of the convective dilution picture, and it is built on a larger, distance-limited sample than previous work.

The paper is careful. There is a genuine error budget, internal and external error estimates, a transparent comparison with Koester & Kepler (2015), and an honest discussion of where the 3D corrections seem too strong. The authors do not oversell the photometric mass scale, and they explicitly exclude spectra with marginal helium lines where parameters become unreliable. The mass distributions of DB and DBA stars are statistically indistinguishable, which is a useful null result.

The soft spot is the main interpretive claim. Section 7.2 concludes that DBA stars below ~20,000 K have too much total hydrogen (MH ~ 10^-14 to 10^-10 Msun) to have a residual origin, so external sources 'must be invoked.' That inference converts photospheric H/He to total MH using homogeneous, fully mixed envelope models from Rolland et al. (2018). The paper itself flags in Section 8 that chemically stratified atmospheres would invalidate that conversion: the same photospheric H/He would correspond to much smaller total MH if hydrogen floats near the surface. Stratified models are not tested against the cool DBA sample, so 'must' is stronger than the evidence. It should be 'is likely to require' or should be backed by a sensitivity analysis. This is a caveat on interpretation, not on the catalog.

The DB+DB binary candidates are indirect, inferred from Mspec - Mphot >= 0.2 Msun or Mphot <= 0.45 Msun. That depends on the photometric mass scale and the adopted mass-radius relation. Some could be artifacts of model or calibration errors, but the DA+DB systems are individually modeled and more convincing. The completeness correction for u-g>0 objects (weight 1.5) is standard but untested.

Citation pattern: heavy on the Bergeron group's own work, but that is because they wrote most of the relevant model atmosphere and evolutionary tracks. The parameters are fitted to public SDSS/Gaia data, so there is no problematic circularity.

Bottom line: this deserves a serious referee. The catalog alone is worth publishing, and the binary candidates are a useful target list. The hydrogen-origin conclusion needs softening or additional modeling, but that is a revision, not a rejection. I would bring it to a reading group if anyone cares about white dwarf evolution, and I would cite it for the catalog.

Referee Report

3 major / 4 minor

Summary. This paper presents a homogeneous analysis of 1915 spectroscopically identified DB white dwarfs from SDSS DR12, using LTE model atmospheres with an updated van der Waals broadening treatment, ML2/alpha = 1.25, and C/O-core evolutionary models. Atmospheric and physical parameters are measured independently from ugriz photometry plus Gaia DR2 parallaxes and from optical spectroscopy. The authors compare the two techniques, derive photometric and spectroscopic mass distributions, identify 10 DA+DB and 55 DB+DB unresolved double degenerate candidates on the basis of mass discrepancies and low photometric masses, measure hydrogen and calcium abundances, and use the hydrogen abundance versus effective temperature diagram to infer total hydrogen masses from homogeneous mixed-envelope models. They conclude that the bulk of DBA white dwarfs below roughly 20,000 K have total hydrogen masses too large to be of residual origin from the convective dilution scenario, so external or internal sources of hydrogen must be invoked. They also construct the DB/(DA+DB) ratio as a function of effective temperature and argue that the observed variation is consistent with convective dilution occurring near 20,000 K.

Significance. If the conclusions hold, this is a useful reference study: it is among the largest DB white dwarf samples analyzed with Gaia parallaxes, it provides machine-readable parameter tables, and it includes a careful discussion of internal and external errors as well as a comparison with Koester & Kepler (2015). The candidate lists of unresolved DB+DB and DA+DB systems are valuable, and the finding that the DB and DBA mass distributions are indistinguishable is a substantive empirical result under the stated model assumptions. The paper is also transparent about several known limitations, including van der Waals broadening, 3D effects, and SDSS flux calibration. However, the central hydrogen-origin claim depends on the conversion of photospheric H/He into a total hydrogen mass using homogeneous envelope models; the paper itself flags the chemically stratified alternative in Section 8 but does not test it. Because that untested assumption is load-bearing for the conclusion that non-residual hydrogen sources 'must be invoked,' the manuscript requires revision before the central claim can be accepted.

major comments (3)

[Section 7.2, Figure 24] The conclusion that the bulk of DBA white dwarfs below about 20,000 K have total hydrogen masses too large to have a residual origin is load-bearing but rests entirely on the conversion of each measured photospheric H/He ratio into a total hydrogen mass using homogeneously mixed envelope models from Rolland et al. (2018). If the atmosphere or envelope is chemically stratified, with hydrogen concentrated near the surface rather than uniformly mixed through the helium convection zone, the same photospheric H/He would correspond to a much smaller total hydrogen mass. The paper itself acknowledges this possibility in Section 8, where it states that the conclusion holds 'unless the hydrogen-to-helium abundance ratio measured using homogeneous model atmospheres is somehow overestimated, for instance, if the atmosphere is chemically inhomogeneous.' This caveat directly undercuts the strength of the claim that external sources 'must be invoked.' I ask the authors either to quantify the effect of stratification on the inferred MH values for their cool DBA sample, or to soften the conclusion to a conditional statement that depends on the homogeneous-mixing assumption.
[Section 6.1, Tables 2 and 3] The identification of 55 DB+DB unresolved double degenerate candidates relies on the assumption that low photometric masses and large photometric-spectroscopic mass discrepancies are caused by unresolved binarity. This is an indirect identification: the same signatures could in principle be produced by systematic errors in the photometric mass scale, the adopted mass-radius relation, the parallax calibration, or the spectroscopic log g scale. The paper itself documents that spectroscopic masses are affected by van der Waals broadening below about 16,000 K, by residual SDSS flux calibration above about 27,000 K, and by 3D effects near 17,000 K, while photometric masses depend on Gaia parallaxes and the evolutionary mass-radius relation. Because the double-degenerate candidate list is one of the paper's principal new empirical claims, it would strengthen the analysis to include at least one independent check, such as radial-velocity monitoring, astrometric excess noise from Gaia, or spectral decomposition for a subsample, plus a discussion of how many candidates would survive plausible systematic shifts in either mass scale. Absent such validation, the wording 'clear evidence for a large population' is stronger than the evidence supports.
[Section 7.1, Figure 23] The quantitative DB/(DA+DB) ratio as a function of effective temperature depends on a magnitude-limited SDSS sample restricted to objects within 1 kpc, a single completeness weight factor of 1.5 applied to objects with u-g > 0, and S/N cuts. The paper also reports an anomalous depletion in the DA temperature distribution near 14,000 K spectroscopically and near 12,000 K photometrically, which it attributes to temperature-scale artifacts rather than to a real change in the DA population. Since this depletion occurs in the same temperature range where the claimed rise in the DB fraction is steepest, the quantitative shape of the ratio near 15,000 K is vulnerable to the same artifacts. I request a sensitivity analysis of the ratio to the completeness weight, to the temperature scale, and to the treatment of the DA depletion, or alternatively a more cautious wording that restricts the robust claim to the qualitative rise below about 20,000 K.

minor comments (4)

[Section 5.1] The paper excludes spectra with marginal helium lines and defines a detectability limit in terms of the He I 4471 equivalent width, but it would be helpful to state explicitly how many objects are removed by this criterion and whether the remaining cool sample is sufficiently large to support the claim that part of the low-temperature log g scatter is real.
[Section 6.1, Table 3] Table 3 would be easier to interpret if each object were tagged with the specific flag that qualified it as a DB+DB candidate (Mspec - Mphot >= 0.2 Msun, Mphot <= 0.45 Msun, or both), rather than leaving the reader to recompute the thresholds from the listed parameters.
[Section 8] The reference to 'B. Rolland et al. (2019, in preparation)' for the internal dredge-up scenario should be replaced by a published reference or an arXiv identifier if one is available, since this scenario is invoked as an alternative explanation for the hydrogen excess.
[Figure captions] Several figure captions refer to 'the description of symbols is identical to that of Figure 7,' but Figure 7 has three panels with different sample definitions; a brief restatement of the symbol conventions in each caption would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper fits parameters to public SDSS/Gaia data and compares them with forward constant-MH simulations; the hydrogen conclusion is model-dependent but not derived from its own inputs.

full rationale

The paper's parameter determinations are fits to public SDSS photometry/spectroscopy and Gaia parallaxes using model atmospheres; the atmospheric parameters are not derived from the paper's conclusions. The central hydrogen conclusion compares measured H/He versus Teff to constant-MH sequences from Rolland et al. (2018). Those sequences are forward calculations from assumed total hydrogen masses and stated mixing-length prescriptions; they were not fitted to the SDSS points, so the inference that cool DBA stars require MH in the range -14 < log MH/Msun < -10 is a model-dependent measurement, not a tautology. The paper explicitly flags the key assumption in Section 8: 'unless the hydrogen-to-helium abundance ratio measured using homogeneous model atmospheres is somehow overestimated, for instance, if the atmosphere is chemically inhomogeneous.' That is a genuine caveat about model dependence, but it is not a circular reduction: the observed H/He values and the Rolland et al. curves share a homogeneous-mixing convention, yet the conclusion follows from the location of the data relative to curves of constant MH, not from the definition of MH. DB+DB candidates are identified through photometric versus spectroscopic mass discrepancies, which is a classification based on an assumed binary interpretation, not a derived prediction. Self-citations (GBB19, Rolland et al. 2018, Genest-Beaulieu & Bergeron 2017) are used for grids, methods, and prior simulations, but these are published, stated-assumption results with independent content; the core measurements and comparisons are externally anchored to SDSS and Gaia data. No step reduces by construction to its own input.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central results rest on a chain of model assumptions inherited from prior work by the same group (model atmospheres, mixing-length efficiency, mass-radius relation, extinction, completeness corrections). None of these are derived in this paper; they are standard but not independently validated here.

free parameters (5)

Mixing-length parameter alpha = 1.25
Adopted from prior model calibration (Bergeron et al. 2011); not fitted here, but controls the temperature and mass scales of the analysis.
Thin hydrogen layer mass fraction q(H) = 1e-10
Assumed in evolutionary models for helium-atmosphere white dwarfs (Section 3); affects mass determinations.
Photometric uncertainty floor = 0.03 mag per band
Applied in Section 4.1 to prevent a single precise magnitude from dominating the fit; changes error estimates modestly.
Completeness weight for u-g>0 objects = 1.5
Adopted in Section 7.1 to correct SDSS target selection; directly scales the DA number counts and therefore the DB fraction.
Extinction regime thresholds = D<=100 pc negligible; maximum at |z|>250 pc
Harris et al. (2006) prescription applied in Section 2.2; affects photometric temperatures and masses for most of the sample.

assumptions (5)

domain assumption LTE model atmospheres with ML2/alpha=1.25 mixing-length convection describe DB atmospheres
Adopted in Section 3; the convective efficiency is a major uncertainty, especially near 25,000 K.
domain assumption Van der Waals broadening treatment from Deridder and van Rensbergen (1976) is adequate for cool DB stars
Used throughout; Section 5.1 acknowledges it is the largest source of uncertainty below 16,000 K and may bias spectroscopic log g and mass.
domain assumption Evolutionary models with C/O cores, q(He)=1e-2, and q(H)=1e-10 provide the mass-radius relation
Used to convert photometric radii and spectroscopic log g into masses (Section 3).
domain assumption Interstellar extinction follows the Harris et al. (2006) prescription
Applied in Section 2.2 to all objects; most of the sample lies beyond 100 pc, so extinction corrections significantly affect photometric parameters.
domain assumption SDSS spectroscopic completeness for u-g>0 is 66% of that for u-g<0, corrected with a 1.5 weight
Adopted in Section 7.1 from Eisenstein et al. (2006); directly affects the DB/(DA+DB) ratio and its temperature evolution.

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Cite this review

Pith. "Pith review of A Photometric and Spectroscopic Investigation of the DB White Dwarf Population using SDSS and Gaia Data." pith.science (2026). https://pith.science/paper/7VJ6CZX4

@misc{pith2026190801728,
  author       = {Pith},
  title        = {Pith review of: A Photometric and Spectroscopic Investigation of the DB White Dwarf Population using SDSS and Gaia Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7VJ6CZX4}},
  note         = {Machine review of arXiv:1908.01728}
}

abstract

We present a comprehensive analysis of DB white dwarfs drawn from the Sloan Digital Sky Survey, based on model fits to $ugriz$ photometry and medium resolution spectroscopy from the SDSS. We also take advantage of the exquisite trigonometric parallax measurements recently obtained by the Gaia mission. Using the so-called photometric and spectroscopic techniques, we measure the atmospheric and physical parameters of each object in our sample ($T_{\rm eff}$, $\log g$, H/He, Ca/He, $R$, $M$), and compare the values obtained from both techniques in order to assess the precision and accuracy of each method. We then explore in great detail the surface gravity, stellar mass, and hydrogen abundance distributions of DB white dwarfs as a function of effective temperature. We present some clear evidence for a large population of unresolved double degenerate binaries composed of DB+DB and even DB+DA white dwarfs. In the light of our results, we finally discuss the spectral evolution of DB white dwarfs, in particular the evolution of the DB-to-DA ratio as a function of $T_{\rm eff}$, and we revisit the question of the origin of hydrogen in DBA white dwarfs.

Figures

Figures reproduced from arXiv: 1908.01728 by the authors.

**Figure 1.** Distribution of S/N of the complete spectroscopic sample (black), the DB subsample (blue), including the DB white dwarfs showing traces of hydrogen (DBA) and/or metals (DBZ), and other subtypes (red). or parallax data. As before, we also removed all spectral types containing an M or a +. Our final photometric sample is composed of 1669 photometric data sets, of which 1350 (or 80.9%) are DB stars, including spectral … view at source ↗

**Figure 2.** Distribution of parallactic distances for the complete photometric sample (black), the DB and DBA subsample (blue), and the other subtypes (red). lowing the procedure outlined in Harris et al. (2006), where the extinction is considered negligible if D ≤ 100 pc, to be maximum for the objects located at |z| > 250 pc from the galactic plane, and to vary linearly between these two regimes. 3. THEORETICAL FRAMEWORK The g… view at source ↗

**Figure 3.** Distribution of errors on Teff (top panel) and stellar mass (bottom panel) obtained from the photometric technique, as a function of effective temperature. The open circles represent the DB and DBA stars in our sample with σπ/π < 0.25 (black) and σπ/π > 0.25 (red). The dashed lines indicate the mean errors of the DB/DBA photometric subsample with σπ/π < 0.25. All other spectral types are represented by cyan dots. Th… view at source ↗

Figures from the paper (22 more)

**Figure 4.** Figure 4: Eddington fluxes (in units of ergs cm−2 s −1 Hz−1 ) as a function of wavelength for pure helium DB models at log g = 8, and for various effective temperatures (in units of 103 K) indicated in the figure. section 4.1 of GBB19). If we restrict our sample to objects with …

**Figure 5.** Figure 5: Internal errors on Teff, log g, M, and H/He obtained from the spectroscopic technique, as a function of effective temperature. The open circles represent the DB(Z) and DBA(Z) white dwarfs with S/N > 10 (black) and S/N < 10 (red); all other subtypes are represented by c…

**Figure 6.** Figure 6: Comparison of Teff, log g, M, and log H/He for the 49 objects in our sample with multiple spectroscopic observations and with S/N > 10. is also uniform, in the sense that there are no regions with an obvious accumulation or depletion of objects. The spectroscopic log g…

**Figure 7.** Figure 7: Photometric (upper panel) and spectroscopic (middle panel) log g distributions as a function of Teff. The open circles represent the DB (black) and DBA (red), with or without metals, while the black dots correspond to other spectral types with σπ/π < 0.25 (photometry) …

**Figure 8.** Figure 8: Our best spectroscopic fits to 3 DB(A) white dwarfs near Teff = 14, 700 K, but with significantly different surface gravities. The spectra have been normalized to a continuum set to unity and the best-fit solutions are shown by the red lines. The atmospheric parameters…

**Figure 9.** Figure 9: Theoretical 3D hydrodynamical corrections in Teff (top panel) and log g (bottom panel) to be applied to 1D spectroscopic solutions, as a function of effective temperature and for various log g values, as described in [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

**Figure 10.** Figure 10: Photometric and spectroscopic masses as a function of effective temperature. The description of symbols is identical to that of [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]

**Figure 11.** Figure 11: Left: Photometric mass distribution for the objects with σπ/π < 0.25. Right: Spectroscopic mass distribution for the DB spectra with S/N > 10. The objects with very marginal helium lines have been excluded. Also shown are the corresponding mass distributions for the D…

**Figure 12.** Figure 12: Left: Best photometric fits to four massive DB white dwarfs. The error bars represent the observed ugriz magnitudes and associated uncertainties, while the filled circles represent the best-fit model. Right: Corresponding best spectroscopic fits. The best-fit model (r…

**Figure 13.** Figure 13: Relative mass distributions obtained from photometry (black) and spectroscopy (blue: uncorrected; red: 3D-corrected), for the DB white dwarfs in common between the photometric and spectroscopic samples. The objects with S/N < 10, σπ/π > 0.25, or marginal helium lines…

**Figure 14.** Figure 14: Hydrogen abundances as a function of effective temperature. The description of symbols is identical to that of [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]

**Figure 15.** Figure 15: Best spectroscopic fit to four cool DB white dwarfs with no detectable Hα feature. The best-fit model (red) is plotted over the normalized observed spectrum (black). The inset shows the region near Hα (indicated by the tick mark) used to determine the hydrogen abundan…

**Figure 16.** Figure 16: Differences in effective temperatures, surface gravities, and hydrogen abundances as a function of effective temperature between our analysis and that of Koester & Kepler (2015). The cyan dots represent the objects for which Koester & Kepler assumed log g = 8.0. to th…

**Figure 17.** Figure 17: Top panel: Best photometric fit to SDSS J011356.38+301514.62 under the assumption of a single DBA white dwarf. Bottom panel: (top) best spectroscopic fit obtained by Manseau et al. (2016); (middle) our best spectroscopic fit at lower temperature; (bottom) our best fit…

**Figure 18.** Figure 18: Photometric (top) and spectroscopic (bottom) masses as a function of effective temperature. DB+DB and DA+DB double degenerate candidates are shown as red and cyan circles, respectively. The horizontal dotted and dashed lines are located at M = 0.48 M and M = 0.6 M , r…

**Figure 19.** Figure 19: Best photometric (left) and spectroscopic (right) fits to four DBA white dwarfs in our sample with extremely large hydrogen abundances. The display is similar to that described in [PITH_FULL_IMAGE:figures/full_fig_p042_19.png]

**Figure 20.** Figure 20: Best spectroscopic fits to four DBZ white dwarfs in our sample with strong Ca ii H & K lines, without (left) and with (right) detectable hydrogen features (Hα region shown in the inset). 6.4. Magnetic White Dwarfs In section 5.2, we discussed the presence of massive D…

**Figure 21.** Figure 21: Best photometric (left) and spectroscopic (right) fits to four magnetic DB white dwarfs in our sample. The display is similar to that described in [PITH_FULL_IMAGE:figures/full_fig_p045_21.png]

**Figure 22.** Figure 22: Number of DA and DB white dwarfs in our SDSS sample as a function of effective temperature. The solid and dotted distributions are based on spectroscopic and photometric temperatures, respectively. The objects with S/N < 10 and D > 1 kpc have been excluded, and a weig…

**Figure 23.** Figure 23: Ratio of the number of DB stars to the total number of DA+DB white dwarfs, as a function of (spectroscopic) effective temperature. The error bars represent the Poisson statistics of each bin. The objects with S/N < 10 and D > 1 kpc have been excluded. stars with a con…

**Figure 24.** Figure 24: Predicted hydrogen abundances as a function of effective temperature (solid and dotted lines) from the simulations of Rolland et al. (2018) for homogeneously mixed models at 0.6 M and for both the ML2/α = 0.6 (upper panel) and α = 2 (lower panel) versions of the mixin…

**Figure 25.** Figure 25: SDSS J011356.38+301514.62. The complete figure set (10 images) is available in the online journal [PITH_FULL_IMAGE:figures/full_fig_p058_25.png]

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