# Missing Metals in DQ Stars; a Compelling Clue to their Origin††thanks:
Dedicated to the memory of Prof. G. Fontaine (1948–2019)

J. Farihi1, P. Dufour2, T. G. Wilson1,3,4
1Department of Physics and Astronomy, University College London, London WC1E
6BT, UK
2Département de Physique, Université de Montréal, Montréal, Québec H3C 3J7,
Canada
3Isaac Newton Group of Telescopes, E-38700 Santa Cruz de La Palma, Spain
4School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16
9SS, UK E-mail<EMAIL_ADDRESS>
###### Abstract

White dwarf stars frequently exhibit external pollution by heavy elements, and
yet the intrinsically carbon-enriched DQ spectral class members fail to
experience this phenomenon, representing a decades-old conundrum. This study
reports a high-resolution spectroscopic search for Ca ii in classical DQ white
dwarfs, finding that these stars are stunted both in pollution frequency and
heavy element mass fractions, relative to the wider population. Compared to
other white dwarf spectral classes, the average external accretion rate is
found to be at least three orders of magnitude lower in the DQ stars. Several
hypotheses are considered which need to simultaneously account for i) an
apparent lack of accreted metals, ii) a dearth of circumstellar planetary
material, iii) an observed deficit of unevolved companions in post-common
envelope binaries, iv) relatively low helium mass fractions, and remnant
masses that appear smaller than for other spectral classes, v) a high
incidence of strong magnetism, and vi) modestly older disk kinematics. Only
one hypothesis is consistent with all these constraints, suggesting DQ white
dwarfs are the progeny of binary evolution that altered both their stellar
structures and their circumstellar environments. A binary origin is already
suspected for the warmer and more massive DQ stars, and is proposed here as an
inclusive mechanism to expose core carbon material, in a potential
evolutionary unification for the entire DQ spectral class. In this picture, DQ
stars are not descended from DA or DB white dwarfs that commonly host
dynamically-active planetary systems.

###### keywords:

circumstellar matter— planetary systems— stars: abundances— stars: evolution—
white dwarfs

##  1 Introduction

White dwarf stars whose spectra exhibit molecular carbon features have been
known for over 60 years, when the first pair were discovered by carefully
constructed spectrophotometry (Greenstein & Matthews, 1957; Bell, 1962). Owing
to instrumentation limits on wavelength at that time, only the weaker C2 (1,0)
feature near 4670 Å was observable (Eggen & Greenstein, 1965; Greenstein,
1970), and not the stronger C2 (0,0) feature with bandhead near 5165 Å that
would later be found in many more stars (e.g. Oke 1974; Liebert 1977; Wesemael
et al. 1993). It has been understood for nearly half a century that white
dwarfs exhibiting C2 Swan bands (or other carbon features) in their spectra –
denoted as spectral class DQ (McCook & Sion, 1987) – have atmospheres that are
dominated by helium, with only trace abundances of carbon, and little or no
hydrogen (Bues, 1973; Grenfell, 1974).

Early pioneering work to understand the carbon abundances in DQ stars pointed
out that convective mixing in a gravitationally-stratified atmosphere was
problematic, because full mixing would likely result, and thus lead to a
carbon-dominated atmosphere by mass (Vauclair & Fontaine, 1979; D’Antona &
Mazzitelli, 1979), in contrast with the most successful models for the
observed spectra. There were already theoretical indications that the mass of
the helium layer in DQ stars should be significantly thinner than in their
non-carbon bearing, helium atmosphere counterparts (Muchmore & Böhm, 1977),
and this has been corroborated by successful models that dredge up carbon,
where it emerges through the diffusive (upward) tail in a stratified
atmosphere (Koester et al., 1982; Fontaine et al., 1984). The first time-
dependent modeling of this process showed that the inferred trace carbon
abundances are best reproduced with helium layer mass fractions in the range
$-4.0\lesssim\log q({\rm He})\lesssim-3.5$, and demonstrated that higher mass
fractions, such as those predicted by stellar evolution, fail to dredge
sufficient carbon to reproduce the observations (Pelletier et al., 1986).

On a similar timescale, it was already known that white dwarfs sometimes
exhibited metals such as Ca, Mg, and Fe in their spectra, and that these were
also trace abundances in helium-dominated atmospheres (e.g. vMa 2; Weidemann
1960; Strittmatter & Wickramasinghe 1971; Wegner 1972. In this way, the metal-
rich DZ spectral class was shown to share basic characteristics, such as
temperature and atmospheric composition, with the DQ stars, based on the
earliest discoveries and successful modeling. In contrast with the DQ stars,
however, there was already a scientific consensus that the metals observed in
DZ stars could not have been brought to the surface from their interiors
(Fontaine & Michaud, 1979; Vauclair et al., 1979). Over a wide range of
temperatures, white dwarfs should be gravitationally sedimented (via
diffusion), on timescales much faster than any potentially competing process
(Dupuis et al., 1992), including convective mixing – a mechanism that actually
enhances downward diffusion (Alcock & Illarionov, 1980). Thus it has been
clear for many decades that an external source is necessary for DZ stars, but
that DQ stars can be understood as polluted via their own interiors.

Fontaine et al. (1984) first mentioned the striking problem that is the
subject of this work, “… it is interesting to speculate on the reasons why
metals and carbon still appear to be mutually exclusive in the atmospheres of
cool helium-rich white dwarfs.” The DQ stars are around 9–10 per cent of the
local population as defined by the 20 pc sample, which is well constrained if
not complete (Hollands et al., 2018), and thus are not outliers or unusual by
any means. Indeed, more than 60 years after the first discovery, this
fundamental question about their nature remains unanswered, despite recent and
ongoing work with far superior data and modeling, and based on hundreds of DQ
and DZ stars (Koester & Kepler, 2019; Coutu et al., 2019; Koester et al.,
2020).

This paper reports a deep spectroscopic search for external, heavy element
pollution in DQ stars using one of the largest ground-based telescopes
available. A dramatic lack of metal pollution is revealed for DQ white dwarfs,
as compared to the similarly helium-rich DZ stars. Section 2 reports on the
high-resolution spectroscopic observations, and Section 3 quantifies the
differences between the DQ stars and DZ stars in terms of pollution
frequencies and accretion rates, using both the high-resolution spectral data
as well as available SDSS spectroscopy. Section 4 examines other threads of
available evidence on DQ stars, and in particular where there are distinct
properties as compared to DZ-type and other white dwarfs (mass and stellar
structure, circumstellar matter, duplicity, magnetism, and kinematics).
Section 5 examines several hypotheses that might account for all the available
evidence, and concludes that the DQ white dwarfs are the result of binary
evolution, where mass loss has resulted in a thinner helium layer than
otherwise expected, and which also depleted any nearby planetary material that
might pollute the star. Section 6 discusses this possibility in detail, and
provides suggestions for future work that can test this novel proposed origin
for DQ stars.

##  2 Spectroscopy and metal abundances

Table 1: Target parameter summary, Ca upper limits and detections.

WD# | $T_{\rm eff}$ | Mass | [C/He] | [Ca/He] | Notes
---|---|---|---|---|---
| (K) | ($M_{\odot}$) |  |  |
0038$-$226 | 5210 | 0.51 |  $-$8.4 |  $<-$12.3 | 1
0042$-$238 | 10500 | 0.5: |  $-$2.7 |  $<-$11.8 | 2
0312$-$084 | 9080 | 0.5: |  $-$4.2 |  $<-$12.0 | 3
0435$-$088 | 6400 | 0.55 |  $-$6.3 |  $<-$12.8 | 4
0548$-$001 | 6080 | 0.66 |  $-$6.6 |  $<-$12.5 | 1
0806$-$661 | 10210 | 0.58 |  $-$5.5 |  $<-$12.5 | 5,9
0811$+$250 | 7920 | 0.54 |  $-$5.4 |  $<-$12.2 | 4
0856$+$331 | 9490 | 0.87 |  $-$3.5 |  $<-$11.8 | 4
0913$+$103 | 8410 | 0.53 |  $-$5.2 |  $-$11.6 | 4
0935$-$371 | 9380 | 0.78 |  $-$4.2 |  $<-$12.3 | 6
1015$+$088 | 7580 | 0.53 |  $-$6.0 |  $<-$12.5 | 4
1142$-$645 | 7950 | 0.58 |  $-$5.5 |  $<-$12.9 | 4
1149$-$272 | 6440 | 0.56 |  $-$6.7 |  $<-$12.8 | 4
1708$-$147 | 9280 | 0.54 |  $-$3.9 |  $-$11.7 | 7,9
1831$+$197 | 7120 | 0.56 |  $-$6.2 |  $<-$12.8 | 4
1837$-$619 | 8500 | 0.5: |  $-$5.0 |  $<-$12.5 | 2,9
1917$-$077 | 10400 | 0.62 |  $-$5.8 |  $<-$12.8 | 5,10
2059$+$316 | 9100 | 0.66 |  $-$5.0 |  $<-$12.5 | 4
2140$+$207 | 7520 | 0.50 |  $-$6.3 |  $<-$12.8 | 4
2147$+$280 | 11000 | 0.5: |  $-$7.0 |  $<-$12.0 | 2,10
2154$-$512 | 7190 | 0.60 |  $-$4.4 |  $<-$12.8 | 5
2311$-$068 | 7350 | 0.55 |  $-$6.1 |  $<-$12.5 | 4
2317$-$173 | 10800 | 0.5: |  $-$6.5 |  $<-$12.3 | 2,10
Other DQ(Z) stars: |  |  |  |
0208$-$510 | 8180 | 0.59 |  $-$4.8 |  $<-$11.8 | 8
0736$+$053 | 7590 | 0.55 |  $-$5.9 |  $-$11.8 | 4

Notes and references: (1) Blouin & Dufour (2019); (2) Weidemann & Koester
(1995); (3) Sayres et al. (2012); (4) Coutu et al. (2019); (5) Giammichele et
al. (2012); (6) Dufour et al. (2005); (7) Subasavage et al. (2017); (8) Farihi
et al. (2013a); (9) C2 band detected in the optical for the first time; (10)
carbon detected only in ultraviolet.

###  2.1 Target selection and observations

The observed DQ stars are listed in Table 1, and taken from the literature
based on favorable brightness and position in the sky. Science targets were
selected to have $T_{\rm eff}\la 11\,000$ K, and thus are primarily classical
DQ stars with C2 Swan bands, where trace atmospheric carbon is attributed to
the dredge up of core material via successful modeling in helium-rich white
dwarfs (Pelletier et al., 1986; Koester et al., 2020). These cooler DQ white
dwarfs were selected because they represent the bulk of all objects in this
spectral class, and form a modest but significant fraction of the local white
dwarf population (Hollands et al., 2018). One known DQZ star (0913+103; Wegner
et al. 1985; Kleinman et al. 2013) was included in the study in order to have
at least one confident detection of Ca ii H and K, and to search for other
metal species and thereby better assess the rare, externally polluted DQ
stars.

The overall science program was executed in service mode as priority B during
ESO periods 95, 96, and 97, where data were ultimately received for 23 out of
30 possible targets. Each white dwarf was observed with UVES (Dekker et al.,
2000) on the VLT, using the 1 arcsec slit and $2\times 2$ binning on-chip,
thereby achieving a nominal resolving power $R\approx 40\,000$. Spectroscopy
was performed using a standard dichroic configuration, with central
wavelengths of 3900 Å on the blue side, and 5640 Å on the red side. Stars were
observed using two identical integrations taken in sequence, where the
individual exposure times varied between roughly 300 s and 1800 s (typically
900 s) in order to achieve signal-to-noise (S/N) $>20$ at 4000 Å.

Raw science and calibration frames for all targets were retrieved from the ESO
archive and reduced using the standard reflex UVES pipeline. Spectra were
flat-fielded, bias- and dark-subtracted, extracted, and wavelength-calibrated
following parameter optimizations recommended in the reflex UVES
documentation. For the DA+DQ visual binary 0935$-$371, frames were reduced and
calibrated using the standard processes, however for spectral extraction the
slit length and offset parameters were adjusted to isolate each star. As
observations were taken in pairs, individual spectra were combined via a
weighted average to increase the S/N, and then normalized in each arm. For the
entire sample, S/N values of $10-70$ in the blue, and $20-120$ in the red were
found over the wavelength ranges 3500–3600 Å and 5200–5300 Å respectively. To
provide confidence in the reflex reductions, the ESO archive phase 3 spectra
for all targets were retrieved and combined in an identical manner, where it
was found that both the S/N and wavelength RMS values agree within 1 per cent.

Figure 1: Spectra of all UVES targets in the region of Ca ii H and K,
highlighted by the shaded regions in each panel. Spectra are presented at
twice the normalized flux level, so that the vertical offset between spectra
represents 50 per cent depth. For all but two targets – one previously known
(0913$+$013) and one new discovery (1708$-$147) – there are no indications of
metal pollution via this most sensitive, ground-based diagnostic (Zuckerman et
al., 2003). The spectra exhibit fringing, which is common in UVES data owing
to a light path difference between the sky and internal flat lamp; otherwise,
the spectra are featureless at these wavelengths. Data for both 0312$-$084 and
1831$+$197 are smoothed by five binned pixels so that the noise is broadly
similar to other targets. At the bottom left is the DQZ star Procyon B, whose
spectrum was obtained using HST STIS (Provencal et al., 2002) and reduced
independently (Farihi et al., 2013a). At the bottom right is the prototype DZ
star vMa 2 as observed by UVES (Napiwotzki et al., 2003), where the Ca ii H
and K features are dramatic, even at the relatively low abundance of [Ca/He]
$=-10.0$ (Dufour et al., 2007).

Figure 2: The similar yet disparate characteristics of the cool, helium
atmosphere white dwarfs of spectral classes DQ and DZ. Left: The evolutionary
tracks of the Hertzsprung-Russell diagram, based on Gaia EDR3 recommendations
(Gaia Collaboration et al., 2021), where roughly 23 200 stars are plotted as
grey points. The UVES sample of DQ target stars are plotted as open blue
circles where no Ca is detected, and as filled blue stars for Ca detections
(DQZ). Filled green circles show DZ white dwarfs (Dufour et al., 2007), where
only those stars with a $20\upsigma$ or higher detection of their parallax are
plotted. Right: Time-averaged accretion rates and upper limits for the DQ(Z)
stars from Table 1, plotted alongside the same inferences for DZ stars from
the SDSS (Farihi et al., 2010b). The population of DZ stars and DQ(Z) stars
exhibit no overlap in metal accretion rates, where $\langle\dot{M}_{\rm
DZ}\rangle/\langle\dot{M}_{\rm DQZ}\rangle\approx 800$, and
$\langle\dot{M}_{\rm DZ}\rangle/\langle\dot{M}_{\rm DQ}\rangle>3000$. While
the least metal-rich DZ stars shown here may be limited by the spectroscopic
sensitivity of the SDSS, the most metal-polluted DQ stars are nothing like
their DZ counterparts. Note that two of the plotted stars were known to be DQZ
prior to the inception of this survey; Procyon B (Provencal et al., 2002) and
0913+103 (Kleinman et al., 2013).

###  2.2 Abundances and inferred accretion rates

Abundances111Abundance notation in this paper corresponds to [Z/He] =
$\log(n_{\rm Z}/n_{\rm He})$. and upper limits for Ca ii were derived using
state-of-the-art models of helium atmosphere white dwarfs, and specifically
for stars that are enhanced in either carbon or metals typically found as
external pollutants (Coutu et al., 2019). This was achieved by the method of
line profile fitting, where, after a good fit was approximated, models were
recalculated with $\pm 0.3$ dex abundances, and visual inspection of the lines
then determined the best abundance (or upper limit). Basic stellar parameters
such as mass, effective temperature, and carbon abundance were taken from the
literature and are listed in Table 1, together with the results for calcium.

For each DQ star in the UVES sample, the [Ca/He] abundance determination or
upper limit was converted into a time-averaged accretion rate following
standard methodology. For helium atmosphere white dwarfs with temperatures in
the range considered here, metal sinking timescales are sufficiently long that
a steady state cannot be inferred with any confidence (Koester, 2009).
However, the steady-state accretion rate is equivalent to a time-averaged
value over a single sinking timescale, and is sufficiently informative in
several contexts (Farihi et al., 2009; Farihi, 2016; Hollands et al., 2018).
This is simply the mass of a particular metal in the outer, fully-mixed layers
of the star, divided by the diffusion timescale for this element (Dupuis et
al., 1993). To estimate the total mass accreted based on a single element, a
correction factor is applied; in this case Ca is assumed to be 0.016 of the
total mass as in the bulk Earth (Allègre et al., 1995; Farihi et al., 2012).
Up-to-date diffusion calculations for pure helium atmospheres with no
convective overshoot have been used (Koester et al., 2020).

##  3 Pollution inferences and implications

The following section makes use of the metal abundances and upper limits from
the UVES sample, as well as from a larger samples of DQ stars available from
the Sloan Digital Sky Survey (SDSS). While the UVES observations represent the
most sensitive search for Ca ii in DQ stars performed to date, the sample size
is only 23 objects. In contrast, there are an order of magnitude more DQ white
dwarfs with SDSS spectroscopy, and although these data are somewhat less
sensitive to Ca ii (Section 3.3), the resulting detection frequency is more
robust than for the smaller UVES sample. First, accretion rates and upper
limits are inferred for the white dwarfs with UVES spectroscopy (and for two
DQ stars observed with HST). Second, pollution frequencies are discussed in
the context of robust samples with available SDSS spectroscopy. Third, the
combined evidence is evaluated and compared with what is known for the wider
population of polluted white dwarfs.

###  3.1 Accretion rates: UVES

Figure 2 explores the inferred total (time-averaged) accretion rates for the
observed DQ stars, alongside a sample of DZ stars with similar effective
temperatures (Dufour et al., 2007; Farihi et al., 2010b). Both sets of stars
are understood to have helium-dominated atmospheres, and can be seen to occupy
the corresponding evolutionary track on the Hertzsprung-Russell diagram shown
in the Figure. Although there appear to be a few DQ stars that are outliers in
$G_{\rm BP}-G_{\rm RP}$ color, this is simply the result of objects with
strong Swan band absorption in the blue, and not from any actual difference in
the range of temperatures or cooling ages. The two DQZ stars in the UVES
sample lie directly on the helium atmosphere cooling sequence and do not
otherwise distinguish themselves.

In stark contrast to their shared effective temperatures and luminosities, in
terms of metal accretion rates, the DQ and DZ stars show no overlap
whatsoever. It may appear that there are no DZ stars with abundances and
inferred accretion rates as low as the DQ(Z) objects, but there has not yet
been any search for weak metal lines in DC white dwarfs using large telescopes
and high-resolution spectroscopy. The conventional detection of DZ stars is
straightforward using low-resolution spectroscopy of modest quality, based on
the fact that cool helium atmospheres are largely transparent owing to a lack
of major opacity sources (Koester, 2009). Notably, the Ca ii lines in vMa 2
were detected with photographic plate technology over 100 years ago (van
Maanen, 1919), and, as can be seen in Figure 1, using an 8 m class telescope
and a high-resolution spectrograph is clearly overkill, even for [Ca/He]
$=-10.0$ ($\dot{M}_{\rm z}=1\times 10^{7}$ g s-1). For context, DZ stars with
metal abundances 30 times smaller than vMa 2 are known, and the corresponding
Ca ii line strengths are readily detected using the $R\approx 2000$ resolving
power of SDSS spectroscopy (Dufour et al., 2007; Coutu et al., 2019).

The use of the VLT and UVES, together with a sample size of 23 targets,
demonstrates conclusively that the pollution in DQZ stars never approaches the
levels typically attained in DAZ, DBZ, and DZ stars. For systems that are
likely in accretion-diffusion equilibrium (i.e. a steady state), instantaneous
accretion rates can exceed $10^{9}$ g s-1 (Farihi et al., 2016), and roughly
speaking, that is 5600 times higher than the rates inferred for polluted DQ
white dwarfs. And while steady-state accretion cannot be assumed for helium-
rich atmospheres, if the time-averaged mass accretion rates are compared
between the DQZ and the DZ stars, the top accretors in each class are
separated by a factor of 18 000(!). Furthermore, most DQ stars have only upper
limit metal abundances and accretion rates, and therefore the gulf between
these two populations is vast.

###  3.2 Pollution frequency: SDSS

Although Ca ii in the stars 0913$+$103 and 1707$-$147 was detected with UVES,
these lines should be detectable in DQ stars via SDSS spectroscopy, for the
same reasons outlined above for DZ stars; transparent atmospheres and broad
lines. Because DZ and DQ stars both have atmospheres dominated by helium, and
similar effective temperatures, the much larger SDSS datasets of DQ white
dwarfs will result in lower yet comparable sensitivity to Ca ii absorption.
For this purpose, two SDSS spectroscopic samples were analyzed, where DQZ
candidates were visually inspected, all of which are shown in Figure 3.

First, there are 164 stars designated as DQ (160) or DQZ (4) in the SDSS DR7
white dwarf catalog, ignoring those spectral types deemed uncertain (Kleinman
et al., 2013). On further scrutiny, three of these four are genuine DQZ
(J091602.83$+$101109.7, J133205.62 +274003.9, J153447.54$+$414559.4), and one
is likely a false positive with noisy data (J140256.39 +111332.3; Figure 3).
This represents 3:164 or $1.8$${}^{+1.7}_{-0.6}$ per cent of DQ stars that are
DQZ. Second, a sample of 221 DQ white dwarfs with reliable SDSS spectroscopic
identifications based on S/N, and robust Gaia parallaxes were taken from the
literature (Koester & Kepler, 2019), where each spectrum was examined for the
possible presence of Ca ii lines. Of these, there are four DQZ stars; two are
in common with the DR7 catalog, plus the objects J075230.82$+$444749.9 and
J131953.49$+$084422.0, yielding 4:221 or $1.8$${}^{+1.4}_{-0.5}$ per cent.
During these visual searches, weaker candidates with noisy spectra were also
considered, and the strongest of these are shown in Figure 3; they are not
considered further, and even if real the preceding frequency statistics would
not be significantly altered.

There is a modestly larger sample of DQ stars that has been analyzed by Coutu
et al. (2019). The catalog is comprised of all stars designated as DQ and DZ
in the Montréal white dwarf database and thus prone to selection effects, but
is largely comprised (80 per cent) of sources with SDSS spectroscopy. Despite
any possible biases within this heterogeneous sample, the numbers are
revealing. Of the 319 DQ white dwarfs studied, only four DQZ stars are noted;
Procyon B and three SDSS objects re-evaluated in Figure 3, where
J090051.91$+$033149.3 is rejected here due to insufficient signal. The
remaining three sources with metal lines represent 3:319 or
$0.9$${}^{+0.9}_{-0.3}$ per cent of DQ stars. It is also noteworthy that in
the same database, there are 1023 DZ white dwarfs with no evidence for carbon
enrichment (Coutu et al., 2019). Again keeping in mind that this catalog has
selection effects, it seems that external pollution favors helium atmospheres
without carbon at a ratio of 1026:3 or in 99.7 per cent of cases.

Figure 3: An approximate sequence of stronger to weaker Ca ii absorption lines
in DQ white dwarfs with SDSS spectroscopy. All spectral data are displayed
analogously to Figure 1, at twice the normalized flux level, so that the
vertical offset between each spectrum represents 50 per cent depth. From top
to bottom, the first five spectra are considered to have confident detections
of both the Ca ii H and K features, where these are marked with solid vertical
lines. One of these was also observed with UVES; 0913+103 = J091602.83
+101109.7 (Kleinman et al., 2013). The final three spectra towards the bottom
are considered too noisy, and these are likely false positives. Altogether,
these sources represent around 2 per cent of DQ stars with comparable
spectroscopy.  Figure 4: A side-by-side comparison of the Ca ii H & K lines
detected in 0913+103 using UVES and SDSS spectroscopy, where all spectral data
are normalized and offset by a constant of 30 per cent depth. The resolving
power of the SDSS (BOSS) spectrograph is $R=1650$ at these wavelengths, and
the spectrum is shown at the top. The UVES spectrum has resolving power
$R=41\,400$, and is shown without modification at the bottom. In between these
are shown two modified versions of the UVES data; a rebinned spectrum that has
the same number of wavelength points as the SDSS spectrum, and a smoothed
spectrum that averages over a width equal to the rebinning factor. The
rebinned UVES and SDSS data are comparable, but the intrinsically weaker, Ca
ii H line is better detected in UVES.

###  3.3 0913+103: a bridge connecting UVES and SDSS

The previous section takes advantage of large samples of DQ white dwarfs for
statistical purposes, but a priori, $R\approx 2000$ spectroscopy with the 2.5
m SDSS telescope may not be readily comparable to $R\approx 40\,000$ data from
the VLT. However, and perhaps surprisingly, the UVES target 0913+103 is weakly
but clearly detected as a DQZ in the SDSS with designation
J091602.83$+$101109.7, and was selected for the VLT survey based on this
foreknowledge (Section 2).

Figure 4 compares the Ca ii lines detected in both SDSS and UVES, where the
broad width of the weak lines – K in particular – enables their detection at
moderately low spectral resolution. The SDSS spectrum has an average S/N $=53$
(per resolution element) over its entire wavelength range from an exposure
that is close to 3600 s (4 integrations). Although the UVES exposure time is
comparable at 3960 s (2 integrations), the raw spectrum has only S/N $=23$ in
the blue arm where Ca ii is found. In the Figure, the UVES data are rebinned
to the same number of points contained in the SDSS wavelength array, yielding
a spectrum that is remarkably comparable in terms of S/N. However, in these
rebinned data, the H line appears more robustly detected, and both the H & K
features have modestly greater line depths, a likely result of the original
higher resolution.

This example demonstrates that, owing to the intrinsic width of the metal
absorption features in cool, helium atmosphere white dwarfs – exemplified by
vMa 2 in Figure 1 – the detection of DQZ stars using SDSS spectroscopy is
broadly comparable to their detection with UVES, at least to the same degree
as found in 0913+103. Using this DQZ white dwarf as a benchmark, it can be
inferred that abundances and time-averaged accretion rates as low as [Ca/He]
$=-11.6$ and $\dot{M}_{\rm z}=3\times 10^{5}$ g s-1 are likely detectable, and
thus absent in the $N>200$ DQ stars in the SDSS. This conclusion adds
considerable weight to the stark difference in pollution between the DQ and DZ
white dwarfs.

###  3.4 DQZ stars vs. other polluted white dwarfs

In Table 2, the different types of polluted white dwarfs are compared by 1)
$f_{\rm z}\equiv$ the frequency at which pollution is detected and 2)
$\langle\dot{M}_{\rm z}\rangle\equiv$ a typical mass accretion rate inferred
for a given spectral class.

  1. 1.

For the spectroscopic class of DA stars (= hydrogen Balmer lines strongest,
typically but not always hydrogen-dominated atmospheres), pollution frequency
and accretion rate statistics are taken from Zuckerman et al. (2003) and
Koester et al. (2014). Polluted spectral types include DAZ and DZA, where the
latter implies the metal lines are the stronger than the Balmer lines.

  2. 2.

For the spectroscopic class of DB stars (= He i lines strongest, helium-
dominated atmospheres), the occurrence rate of pollution and accretion
statistics are taken from Zuckerman et al. (2010). Polluted spectral types
include DBZ and DBAZ, where the latter indicates weak hydrogen lines, and even
weaker metal lines.

  3. 3.

For the spectroscopic class of DC stars (= continuum only, no lines, typically
but not always helium-dominated atmospheres), pollution statistics and
accretion rate inferences are taken from Farihi et al. (2010b) and Hollands et
al. (2018). Polluted spectral type is DZ.

Table 2: White dwarf pollution as a function of spectral class.

Spectral Class | $f_{\rm z}$ | $\langle\dot{M}_{\rm z}\rangle$ | Ref
---|---|---|---
|  |  | (g s-1) |
DAZ / | [ DA + DAZ ] | 0.25 | $\sim 10^{8}$ | 1,2
DBZ / | [ DB + DBZ ] | 0.30 | $\sim 10^{8}$ | 3
DZ / | [ DC + DZ ] | 0.28 | $\sim 10^{8}$ | 4,5
DQZ / | [ DQ + DQZ ] | $<0.09$ | $\lesssim 10^{5}$ | 6
DQZ / | [ DQ + DQZ ] | $0.02$ | $\lesssim 10^{5.5}$ | 7

References: (1) Zuckerman et al. 2003; (2) Koester et al. 2014; (3) Zuckerman
et al. 2010; (4) Farihi et al. 2010b; (5) Hollands et al. 2018; (6) This
study, based on UVES; (7) This study, based on SDSS.

The UVES sample studied here consists of 23 targets, including a known DQZ,
and one new discovery. The nominal fraction of externally polluted DQ stars is
thus 1:22 or $4.5$${}^{+9.1}_{-1.4}$ per cent. It can be argued, however, that
the previously known DQZ is actually a part of the UVES sample, because it is
possible it could have been a selected target without prior knowledge. In that
case, the pollution frequency for DQ stars could be as high as 9.1 per cent.
However, on the basis of the much larger, unbiased SDSS samples of DQ white
dwarfs discussed above, it is far more likely the true fraction is
substantially smaller, and closer to 2 per cent.

As can be seen from the Table, there is a general coherence in the picture of
polluted white dwarfs among the spectral classes DA(Z), DB(Z), and DC/Z, where
this is now a well-understood story of dynamically active planetary systems in
the post-main sequence (Veras, 2016). Most of the aforementioned pollution
frequencies and accretion rate inferences belong to studies more than a decade
old, and in the intervening years those interpretations have been
strengthened. After cooling for 20 Myr or so (during which the sensitivity to
external pollution is poor, owing to hotter atmospheres; Koester et al. 2014;
Barstow et al. 2014), a given white dwarf may exhibit metals at the observed
frequency as a fraction of its remaining cooling lifetime (i.e. a duty cycle),
or a subset of the entire population may stay constantly polluted, while
others are never enriched by remnant planetary systems.

With a few notable exceptions, polluted white dwarfs are a population
dominated by isolated stars (Wilson et al., 2019), together with a small
fraction of wide binaries where each star has evolved in effective isolation
(Zuckerman, 2014). Although stars with helium-rich atmospheres often have
time-averaged accretion rates that exceed those inferred from DAZ stars in
accretion-diffusion equilibrium (Girven et al., 2012; Farihi et al., 2012), it
is still the case that typical rates are comparable with those of their
hydrogen-rich counterparts.

This story breaks down completely for DQ stars; not only is pollution rare,
but when it is present it is stunted by several orders of magnitude. The
frequency of external pollution in DQ stars is drastically lower than for
known polluted white dwarfs in Table 2. On top of this stark difference in the
frequency of detected metals, the discrepancy between the mass accretion rates
is also severe; in fact, there is no overlap between these populations.
Therefore, it is highly improbable that DQ stars are directly related to, or
descended from DA or DB white dwarfs that frequently retain active planetary
systems.

##  4 Additional anomalies

In the previous sections, it was shown that DQ stars do not share any
characteristics in common with either hydrogen or helium atmosphere stars that
commonly evidence their remnant planetary systems via external pollution.
However, the striking lack of metal pollution in DQ stars is just one
empirical perspective that suggests a distinct origin for these white dwarfs.
In this section, several other of their likely or potentially disparate
characteristics are explored in detail.

###  4.1 Lack of core dredge-up in DZ white dwarfs

While the statistics have been clear for decades, the present study reiterates
the fact that DZ and DQ stars are essentially two mutually exclusive
populations, and determines the size of the intervening gulf in terms of
atmospheric metals. But, importantly, there are two independent aspects that
underly this lack of common spectral features; the fact that DQ stars are
rarely DQZ (Section 3), and separately that DZ stars essentially never
manifest as DZQ (Bergeron et al. 1997; see below). Assuming both are the
result of single star evolution, then why don’t any helium-rich white dwarfs
with planetary system pollution, also dredge up their interior carbon? In a
scenario of isolated stellar evolution, the unavoidable interpretation is that
these two spectral classes have distinct interior structures. That is, there
is no plausible mechanism for a planetary system orbiting a DZ white dwarf to
prevent the host star from dredging up carbon from its core.

In fact, it is already well-established that core material dredge-up is most
efficient when the outer envelope of helium is somewhat reduced compared to
the predictions of standard stellar evolution. The earliest calculations of
carbon dredge-up indicated that efficiency peaked with helium mass fractions
$-4.0\lesssim\log q({\rm He})\lesssim-3.5$ (Pelletier et al., 1986), while a
decade later, carbon enhancements were produced with $-3\lesssim\log q({\rm
He})\lesssim-2$ (MacDonald et al., 1998). More recently, a similar range of
helium layer masses has been corroborated, and with predictions that mostly
bracket the classical, observed [C/He] sequence as a function of effective
temperature (Dufour et al., 2005), where state-of-the-art models indicate that
$-3.5\lesssim\log q({\rm He})\lesssim-2.5$ best reproduces this abundance
pattern (Koester et al., 2020). However, there are two known [C/He] sequences
at the cool end (Dufour et al., 2005; Koester & Knist, 2006), one of which
requires even smaller helium layers $\log q({\rm He})\lesssim-3$ (Coutu et
al., 2019), and the warmer DQ stars require drastically lower $\log q({\rm
He})\lesssim-5$ (Koester et al., 2020).

It is well known that some DQ white dwarfs exhibit carbon features in the
optical, while others do so only in the ultraviolet, and that this can be
ascribed to differences in helium mass fractions rather than effective
temperature (Weidemann & Koester, 1995). In fact, as noted in Table 1, several
of the DQ targets were found to have an optical C2 feature for the first time
in their UVES spectra (0806–661, 1708–147, 1837–619), while others only
exhibit carbon features in the ultraviolet (1917–077, 2147+280, 2317–173)
despite the new, highly-sensitive observations. Therefore, it is possible or
even likely that there is a continuous distribution of helium layer masses
(Koester et al., 2020), where thinner layers will lead to optical carbon
features, modestly thicker layers result in ultraviolet carbon features only,
and if sufficiently thick, no carbon will be effectively dredged.

It should be mentioned that there is one DZ star with ultraviolet carbon
features (2216–657; Weidemann & Koester 1989; Wolff et al. 2002), but the
abundance has been shown to be consistent with carbon-rich planetary material,
externally accreted, and thus not from core dredge-up (Swan et al., 2019). And
while it has been shown that the most extreme cases of metal pollution can
mask carbon features in both the optical and ultraviolet, in general this is
not the case (Hollands et al., 2022). Therefore, the number of known cases
where carbon features can be confidently ascribed to core dredge-up in DZ
stars is likely (currently) zero. That is, DZ stars have helium mass fractions
that are significantly and consistently larger than the range found for DQ
white dwarfs.

There are two related points to be made from these successful and independent
modeling efforts of [C/He] in DQ stars. The helium atmosphere, DZ and DC stars
appear to have larger $q({\rm He})$ so that core dredge-up is inefficient, and
this stellar evolutionary outcome represents the majority of all cool, helium-
rich white dwarfs (40:54 = 0.74 in the 20 pc sample; Hollands et al. 2018).
The modest to drastic reduction in $q({\rm He})$, necessary so that core
carbon is effectively dredged in DQ stars, is sufficiently common that an
explanation is needed, where possibilities are discussed in Section 5.

###  4.2 Lack of circumstellar material orbiting DQ stars

To date, all white dwarfs suspected or confirmed to have orbiting
circumstellar disks associated with planetary debris are also observed to have
atmospheres polluted by heavy elements (Farihi, 2016). There are currently
eight systems exhibiting irregular transits from their debris disks, all of
which are polluted with the exception of a few, where available data are
insufficiently sensitive (Vanderbosch et al., 2020; Guidry et al., 2021).
Confirmation of photospheric metals may require a high spectral dispersion
sufficient to resolve or rule out interstellar absorption components (Koester
et al., 2005; Zuckerman et al., 2013), or ultraviolet spectroscopy for modest
to low metal abundances in hotter stars (Farihi et al., 2013b; Xu et al.,
2015).

While Spitzer observations were ground-breaking for debris disk detections
towards white dwarfs, over its 16 year lifetime, targets were strongly biased
towards stars already known to be polluted. Owing to this bias, DQ stars were
not searched for infrared excess with comparable sensitivity, with one
possible exception (Farihi et al., 2010c). However, unbiased studies using the
WISE satellite have been able to probe much larger, unbiased samples (Debes et
al., 2011; Hoard et al., 2013; Dennihy et al., 2017), now including nearly
every sufficiently bright white dwarf identified by Gaia and utterly blind to
spectral type (Xu et al., 2020; Lai et al., 2021). Optical spectroscopic
follow-up of debris disk candidates is still ongoing, but so far there is not
a single DQ star candidate (E. Dennihy, private communication) among $N\gtrsim
100$ white dwarfs where orbiting debris is detected either by infrared excess,
optical emission or absorption lines, or transits.

The dearth of detected circumstellar material orbiting DQ white dwarfs may
appear – superficially – redundant with the lack of atmospheric pollution, as
there is a strong connection established between these two hallmarks of
evolved planetary systems. As a rule for white dwarfs, photospheric heavy
elements require ongoing or recent accretion; that is, pollution implies
closely-orbiting circumstellar material, where it is understood that the bulk
of debris disks are simply undetectable owing to sensitivity limits (Rocchetto
et al., 2015; Bonsor et al., 2017). But the converse is not necessarily true,
and planetary matter can exist in a wide range of orbits and sizes that would
not necessarily lead to the chemical enrichment of the host star. Little is
known about the architecture of planetary systems orbiting polluted white
dwarfs outside of a few tens of R⊙ (Xu et al., 2013; Farihi et al., 2014), but
there are at least two examples of the aforementioned scenario: there is a
giant planet candidate, transiting the non-polluted white dwarf WD 1856+534
(spectral type DC; Vanderburg et al. 2020), and a 2500 AU, planetary-mass
companion orbiting WD 0806-661 (spectral type DQ, Table 1; Luhman et al.
2012).

As mentioned above, existing observations may be insufficiently sensitive to
photospheric metals, yet able to detect circumstellar material. For example,
PG 0010+280 is a $T_{\rm eff}\approx 27\,000$ K DA white dwarf with an
apparent infrared excess from a debris disk, but lacks photospheric metal
lines at optical wavelengths, presumably due to its relatively high effective
temperature and opacity (Xu et al., 2015; Walters et al., 2022). However,
current models do not predict significant opacity in DQ stars, which are
understood to have helium-dominated atmospheres, possibly with trace
abundances of hydrogen that modestly impact models at the lowest effective
temperatures (Koester et al., 2020). Atmospheric modeling of DQ white dwarfs
already includes opacities for carbon molecules and atoms (Dufour et al.,
2005; Coutu et al., 2019), and thus there is currently no other suspected
opacity source that might otherwise mask metal lines if such heavy elements
were indeed present in their atmospheres.

Another means of masking the detectable signature of external pollution, where
the detection of circumstellar matter might be more straightforward, would be
to dilute the accreted material into a much larger stellar reservoir. While
speculative, if the mass of the outer, fully mixed layers of DQ white dwarfs
were orders of magnitude larger than expected from current theory, then the
strength of any metal absorption features would be correspondingly weaker than
predicted, and the derived abundances commensurately smaller. All else being
equal, the DZ and DQ accretion rates (e.g. Figure 2, treating upper limits as
detections for heuristic purposes) could be made to agree, if the mass of the
convection zones in DQ white dwarfs are roughly $3000\times$ more massive.
While this possibility is almost certainly pathological, in this case DQ stars
could accrete from circumstellar disks that might be detected, yet the
resulting pollution would be drastically diminished.

However, no circumstellar matter has been detected towards DQ white dwarfs,
and this partly mitigates the possibly of masking any pollution as typically
observed in a DZ star. Statistics on infrared excesses indicate that robust
detections are rare below 9000 K or cooling ages significantly longer than 1
Gyr (Bergfors et al., 2014; Farihi, 2016), and thus the lack of infrared
excess is not the strongest constraint on the lack of orbiting material, but
it is suggestive.

Figure 5: The left-hand panel plots the mass distribution within the 221
classical DQ white dwarfs with reliable SDSS spectroscopy and Gaia parallaxes
(Koester & Kepler, 2019). In the histogram, masses with fractional errors less
than 10 per cent are plotted as unfilled bars (208 sources), and those less
than 5 per cent are shown as filled bars (153 sources). The low mass tail
persists in those DQ white dwarfs with high quality data, where $M\lesssim
0.45\,$M⊙ is not possible for single star evolution within a Hubble time. The
right-hand panel implements a state-of-the-art, empirical initial-to-final
mass relation (Cummings et al., 2018) to illustrate the likely distribution of
main-sequence progenitor masses for these DQ white dwarfs, assuming single
star evolution. Most of these putative parent stars would have precursor
lifetimes longer than 8 Gyr ($M_{\rm MS}\lesssim 1.1$ M⊙, based on MIST
evolutionary models; Dotter 2016), which is inconsistent with expectations for
thin disk stars (Bensby et al., 2014; Kilic et al., 2017; Sharma et al.,
2019). The addition of a typical 1.2 Gyr cooling age then amplifies the
implied total ages of the DQ white dwarfs, suggesting they should be
predominantly thick disk and halo stars if they evolved in isolation.

###  4.3 Observed lack of close DQ+dM binaries

Low-mass main-sequence stars dominate the initial mass function, and the mass
functions for companions to white dwarfs and to their A-type star progenitors
(Farihi et al., 2005; De Rosa et al., 2014). Thus, from a formation and
evolutionary point of view, it is expected that white dwarfs with M dwarf
companions are abundant, which is observed to be the case (Rebassa-Mansergas
et al., 2007; Hollands et al., 2018). Of these prevalent stellar pairs, based
on both theory and observations, it is expected that roughly two-thirds will
be in wide binaries where the original semimajor axis increased in response to
post-main sequence mass loss, and around one-third will be in short-period
orbits that resulted from common envelope evolution (Farihi et al., 2010a;
Nebot Gómez-Morán et al., 2011; Ashley et al., 2019).

Another property of DQ stars, that implies deviation from typical pathways of
isolated stellar evolution, is the fact that there are no known examples in
post-common envelope binaries with an unevolved companion (e.g. an M dwarf).
To characterize this deficit in more detail, the largest available
spectroscopic catalog of white dwarf – main sequence binaries was searched for
any DQ white dwarfs. This catalog is a compendium of all sources with
composite (white dwarf + main-sequence star) spectra within the SDSS (Rebassa-
Mansergas et al., 2012, 2016), and contains over 3200 entries, 91 per cent of
which are spatially-unresolved, plus a small fraction of partly-resolved pairs
that contribute their combined light into a single spectroscopic fiber.

The catalog was searched for all systems containing an M dwarf companion,
which represents 98 per cent of all entries, and the primary spectral type was
noted in each case. Only confident spectral type identifications were used for
both the white dwarf and the companion, and only those systems that are
spatially unresolved. Of the unresolved composites, 2263 have a white dwarf of
spectral type DA (strongest lines from H), 93 contain a DB star (strongest
lines from He i), and 70 host a DC white dwarf (no spectral lines). Not a
single DQ white dwarf is present in the catalog, including entries with
uncertain spectral type identifications.

Because DC and DQ white dwarfs share similar stellar properties – with the
likely exception of the atmospheric helium mass fraction – one would expect a
comparable number of each in the SDSS catalog, adjusted for their relative
occurrence rates as field stars. Within the 20 pc sample, there are 29 DC and
15 DQ white dwarfs (Hollands et al., 2018), and if their frequency in
unresolved (including short-period) binaries were similar, one would expect
close to 35 DQ spectral types within the composite binary catalog. Therefore,
the lack of DQ stars in this catalog is striking. While identifying a weak C2
5165 Å feature may be difficult in low S/N spectra, and in the presence of
flux from an M dwarf companion, many DQ white dwarfs exhibit deep and strong
Swan bands that are unmistakable (Dufour et al., 2005).

It is important to keep in mind that the stellar wind from a low-mass, main-
sequence star can donate hydrogen to a helium-atmosphere white dwarf via wind
capture, if the orbit is sufficiently close, as can be the case in post-common
envelope binaries. In this way, a helium atmosphere white dwarf with a
sufficiently thin convection zone (or fully mixed layer) can be transformed
into one where hydrogen is more abundant. This scenario is likely responsible
for the dominance of DA spectral types in close, white dwarf – main sequence
pairs, including the extensive SDSS catalog, and is consistent with the lack
of any known, post-common envelope systems containing a helium atmosphere
white dwarf with a thin convection zone (e.g. DB white dwarfs; van den
Besselaar et al. 2005; Rebassa-Mansergas et al. 2010).

To demonstrate the plausibility of this atmospheric transformation, consider a
$T_{\rm eff}\approx 20\,000$ K, pure helium atmosphere white dwarf, which will
manifest as a DB spectral type, and where the fully mixed outer layer has mass
$q\lesssim 10^{-10}$ M⊙(Koester, 2009; Koester et al., 2020). If this white
dwarf accretes solar composition wind from a companion at a modest rate of
$\dot{M}=10^{-14}-10^{-16}$ M⊙ yr-1 (Pyrzas et al., 2012; Parsons et al.,
2012; Ribeiro et al., 2013), hydrogen will come to dominate by number within a
fraction of a Myr. With sufficient sensitivity, a hydrogen(-enriched)
atmosphere white dwarf accreting stellar wind should appear as a DAZ spectral
type, and there are several well-documented cases among post-common envelope
binaries (Zuckerman et al., 2003; Debes, 2006; Tappert et al., 2011).

However, for the aforementioned, pure helium atmosphere case, once the
convection zone deepens due to further cooling, any superficial hydrogen layer
may be easily overwhelmed. By the time such a white dwarf has cooled to
$T_{\rm eff}\approx 10\,000$ K, the size of the fully mixed layer should be
$q\sim 10^{-5}$ M⊙, where this applies both to DQ white dwarfs and their
helium atmosphere DC star counterparts (Koester, 2009; Koester et al., 2020).
With convection zones of this depth, it would take several Gyr or longer for
the atmosphere to become dominated by hydrogen that is captured from a
companion stellar wind, with a strong dependence on the binary separation
(i.e. wind capture rate). Therefore, it is expected that some DC, DZ, and DQ
white dwarfs should be hosts to post-common envelope, main-sequence
companions. Indeed, there are a handful of such systems known to host DC stars
(Nebot Gómez-Morán et al., 2011; Parsons et al., 2013), and at least one
candidate DZ accreting metals in a stellar wind from an M dwarf (Fajardo-
Acosta et al., 2016). But there are none with a DQ white dwarf.

In contrast, there are many examples of DQ stars in wider binaries. Within the
20 pc sample alone, there are several: Procyon B (F5IV+DQZ), GJ 86B (K0V+DQ),
LHS 290 (DQp+dM), 2154-512 (DQ+dM), 1633+572 (DQ+dM), and 1917-077 (DBQ+dM).
Thus, the lack of DQ white dwarfs in close or post-common envelope systems is
likely a clue to their origin, and consistent with the destruction of close
stellar companions during prior evolutionary phases.

###  4.4 Apparently low DQ white dwarf masses

The left-hand panel of Figure 5 plots the mass distribution for the sample of
221 classical DQ white dwarfs discussed in Section 3.2 (Koester & Kepler,
2019). The masses of DQ stars have been studied recently by two independent
teams, based on sources with reliable parallax data from Gaia, finding that
their average mass is roughly 0.05 M⊙ lower than those found for DA and DB
stars (Coutu et al., 2019; Bédard et al., 2022), and also lower than DZ and DC
white dwarfs (Koester & Kepler, 2019). There have been suggestions that there
may be model imperfections that prevent better agreement between DQ stars and
other white dwarfs, but to date, no other studies have considered enhanced
mass loss that may result from binary evolution.

It should be noted that two iconic DQ white dwarfs in the 20 pc sample have
masses derived independently of models, and these values are also lower than
the typical $\langle M\rangle=0.62$ M⊙ found for spectral type DA and DB white
dwarfs (Genest-Beaulieu & Bergeron, 2019). In fact, the dynamical mass of
Procyon B (0.59 M⊙) is in modest tension with the overall age estimated for
this well-studied binary system (Liebert et al., 2013; Bond et al., 2015).
Adopting a comprehensive, recent, and empirical initial-to-final mass
relation, the predicted mass of Procyon B should be closer to 0.66 M⊙
(Cummings et al., 2018). Similarly, for the planet-hosting binary GJ 86, the
DQ white dwarf has a mass of 0.54 M⊙ (Zeng et al., 2021). And while it is
likely this system is relatively old (either thick disk or somewhat younger;
Fuhrmann et al. 2014), any recent initial-to-final mass relation will yield a
progenitor that should not have evolved off the main sequence in a Hubble
time. Therefore, it is possible that the masses of DQ stars are actually low
compared to other white dwarfs, and not a result of model shortcomings, but as
a consequence of their evolutionary history.

Among the 221 DQ white dwarfs plotted in Figure 5, there are 23 objects with
derived masses $M<0.45$ M⊙, where 16 of these have
$\upvarpi/\upsigma_{\upvarpi}>20$, and thus excellent astrometric data. It is
often the case that (apparently) over-luminous white dwarfs are two unresolved
stars with comparable brightness, and therefore Gaia DR3 was searched in a 5
arcsec radius around the positions of these 23 candidate low-mass DQ stars;
none were found to be resolved into two sources. However, two were found to
have renormalized unit weight error (RUWE) suggestive of a possible or likely
non-single star: SDSS J094058.34+384350.2 (RUWE $=1.37$) and SDSS
J125245.44+194311.2 (RUWE $=2.33$). Thus, the inferred low masses of these 23
DQ white dwarfs may be partly accounted for as binary light sources, but the
bulk currently lack indications of duplicity.

Figure 6: Cumulative mass and tangential speed histograms for the DQ, DZ, and
DC stars with $T_{\rm eff}<10\,000$ K analyzed in table 3 of Koester & Kepler
(2019). The dashed lines track the distribution for stars with better than
$5\upsigma$ errors in mass or parallax (for $v_{\rm tan}$), while the solid
lines follow the distributions for sources with errors better than
$10\upsigma$.

Similarly, the argument is often made that a DQA-type spectrum must be two
unresolved white dwarfs, one a DQ and one a DA star, but currently there is
only verification of one such binary (GD 73; Vennes et al. 2012), where it is
modestly overluminous relative to single star expectations (Gentile Fusillo et
al., 2019), and has an RUWE consistent with a single astrometric source. Of
the 23 DQ white dwarfs with implied low masses, only one source shows
unambiguous (yet weak) absorption from hydrogen in its spectrum (SDSS
J150126.78+210056.9), but has an RUWE in the range considered normal for
single stars, and no other Gaia sources nearby. It can be argued that these
low-mass DQ candidates are unresolved DQ+DC white dwarfs, but again one might
expect a preponderance of high Gaia RUWE values, which is not yet observed.

Given that all DQ stars must have atmospheric carbon by definition, and thus
carbon cores, it may seem oxymoronic to speculate on the possibility of low-
mass members of this class. Owing to the finite age of the Galaxy, white
dwarfs with $M\lesssim 0.45$ M⊙ cannot yet have formed by single star
evolution. Indeed the majority of low-mass white dwarfs are found to be
single- or double-lined binaries (Marsh et al., 1995; Brown et al., 2011;
Rebassa-Mansergas et al., 2011), whose evolution is thought to be terminated
prior to helium ignition, and thus such stars would have helium cores as
opposed to carbon cores.

However, it is possible to form a low-mass carbon core in stars with $M_{\rm
MS}\gtrsim 2.3$ M⊙, where helium ignites non-degenerately (Bauer & Kupfer,
2021). Binary interactions would then be required to truncate stellar
evolution and further core growth, in order to later be observed as a low-mass
white dwarf with a carbon core. While speculative, this is a distinct
possibility for some DQ white dwarfs, as their binary properties are only
weakly constrained at present, and their molecular bands are relatively
insensitive to radial velocity changes induced by companions, especially
compared to white dwarfs with atomic spectral lines. The confirmation of any
DQ white dwarf with $M<0.45$ M⊙ would be a strong indication of a binary
origin.

The expectations for single star evolution are illustrated in the right-hand
panel of Figure 5, where main-sequence progenitor masses have been calculated
using the previously mentioned initial-to-final mass relation. Given the low
masses of the DQ white dwarfs, it is thus not surprising that the predicted
main-sequence predecessors are also of relatively low mass, but this important
detail has so far escaped attention in previous studies (Coutu et al., 2019;
Koester & Kepler, 2019; Bédard et al., 2022). The age of thin disk stars
should be less than around 8 Gyr (Bensby et al., 2014; Kilic et al., 2017;
Sharma et al., 2019), but Figure 5 indicates that most single star parents of
DQ white dwarfs would have main-sequence lifetimes longer than this. For the
DQ stars plotted in the Figure, $\langle T_{\rm eff}\rangle=7800$ K and
$\langle M\rangle=0.55$ M⊙, where white dwarf cooling would add an additional
1.2 Gyr to their inferred total ages. This picture would imply that DQ stars
are dominated by thick disk and halo members, which, as shown below, is
inconsistent with their kinematical and companion metallicity properties.

###  4.5 Contrasts with DZ and DC stars

It is useful to compare evolutionary indicators of DQ white dwarfs with their
helium atmosphere counterparts, the DZ and DC stars. While the mass
distribution of DQ stars has been discussed above, and in particular the
potential of a low-mass tail, this could simply be the result of unresolved
binaries. If that is indeed the case, and DQ stars are the product of isolated
stellar evolution, then it is expected that other white dwarf classes would
also exhibit the same basic properties in their mass distributions. In Figure
6, the cumulative mass distributions are shown for the 221 DQ white dwarfs
discussed above, together with SDSS spectroscopic DZ and DC samples from the
same study (Koester & Kepler, 2019). Only those DZ and DC stars with $T_{\rm
eff}<10\,000$ K, and which thus overlap with the DQ white dwarfs, have been
used in order to compare these three spectral classes of helium atmosphere
stars using identical criteria.

The most obvious features of the cumulative mass histogram is that DQ white
dwarfs have a larger fraction of lower masses, the DC stars have a notable
fraction of higher masses, while the planetary system hosting DZ stars are
somewhat in the middle of these two. Another feature is the concentration of
DQ white dwarf masses in the range $0.5-0.6$ M⊙, shown by a steepness in the
distribution that is not observed for DZ or DC stars.

One possible interpretation of this figure panel is as follows. Planetary
systems form and evolve commonly around the progenitors of DZ stars, which
leave remnants with a given mass distribution as observed. The DQ white dwarfs
originate in a distinct population, with lower remnant masses (reasons
explored below), and rarely formed planetary systems that lead to white dwarf
pollution. While speculative, the DC white dwarfs may descend from stars that
are, on average, somewhat more massive than DZ star progenitors, and which may
indicate that planet formation is inhibited to some degree in higher-mass
main-sequence stars.

The tangential velocity distributions tell a similar story but with less
distinction between the DZ and DC white dwarfs. The DZ stars appear to include
few stars with relatively high velocities, consistent with planetary hosts
coming from a primarily young disk population, where star formation likely
benefitted from a comparatively metal-rich environment. The DQ stars are
modest outliers compared to their helium atmosphere, DZ and DC white dwarf
counterparts, and seem to be missing stars at the lowest velocities, below
around 20 km s-1. This may imply the DQ white dwarfs are somewhat older than
the DZ and DC stars.

Another possibility is that DQ white dwarfs have received significant velocity
kicks from asymmetric mass loss, and which was not experienced by the DZ and
DC stars. Only modest white dwarf recoil, no larger than 1 km s-1, has been
inferred using a large and pristine catalog of wide Gaia binaries, via
comparison of those pairs with two main-sequence stars, versus those
containing at least one white dwarf (El-Badry & Rix, 2018). However, based on
the large separation between binary components in that study, this modest
recoil may apply only to stars that evolved in effective isolation. In
contrast, close binary evolution may induce larger kicks that may account for
the lack of low space velocities in the DQ stars.

The space motions of DQ white dwarfs are far from that expected for Population
II stars, and neither are there any indications of a Galactic thick disk or
halo origin. Under the assumption of zero radial velocity, the spectral
classes plotted in Figure 6 have calculated mean space motions in the
direction of Galactic rotation ($V$) given in Table 3. The three samples are
without any kinematical biases, and were selected based on spectral
classification via SDSS observations (Koester & Kepler, 2019). As can be seen,
the DQ stars may exhibit a modest fraction of older – or kicked – thin disk
stars within their population, but for the most part, the DQ, DZ, and DC white
dwarfs are members of the relatively young Galactic disk (Bensby et al.,
2014). Thus, the low masses of the DQ white dwarfs, and the older ages implied
by single star evolution, are not supported by their space velocities.

###  4.6 Magnetism and rotation among DQ stars

Magnetism has long been associated with DQ white dwarfs, including detections
of strong fields (Schmidt et al., 1995; Schmidt et al., 1999; Vornanen et al.,
2010), as well as spectral indications of magnetism via modeling (Hall &
Maxwell, 2008; Blouin et al., 2019). In a pioneering effort to characterize
magnetism among all white dwarfs within 20 pc (Bagnulo & Landstreet, 2021), it
has been recently found that DQ stars are likely outliers. First, the nominal
frequency of magnetism in DQ white dwarfs is higher than in other spectral
classes. Second, the actual frequency is likely to be higher, because the
sensitivity is restricted to magnetic fields on the order of MG or larger,
while spectropolarimetry can readily detect fields of 100 kG or somewhat
smaller in white dwarfs with atomic lines (i.e. DA, DB, or DZ stars). Third,
it is noteworthy that, in fact, the DQ stars have the highest magnetic fields
of white dwarfs within 20 pc (Bagnulo & Landstreet, 2021).

Rapid rotation has been associated with some magnetic white dwarfs, especially
in the scenario where close stellar binaries or mergers may generate the
observed fields (Liebert et al., 2005; Tout et al., 2008). Thus, relatively
rapid rotation might be expected in some DQ white dwarfs, and appears to be
well documented for the warmer examples of this spectral class (Dufour et al.,
2008; Lawrie et al., 2013; Williams et al., 2016). Based on these modest
correlations, a more thorough investigation of the entire DQ spectroscopic
class would likely identify additional magnetic examples, as well as stars
rotating more rapidly than the tens to hundreds of hour spin periods typical
for isolated white dwarfs (Hermes et al., 2017).

Table 3: Space velocities in the direction of Galactic rotation.

Spectral | $\langle V\rangle$ | $\upsigma_{V}$
---|---|---
Class | (km s-1) | (km s-1)
DQ | $-13$ | 30
DZ | $-5$ | 22
DC | $-7$ | 25
Thin disk | $-15$ | 20
Thick disk | $-46$ | 38

Notes: Values for the thin and thick disk are taken from Bensby et al. (2014),
and $v_{\rm rad}=0$ is assumed for all white dwarf calculations.

##  5 Possible origins of DQ stars

The previous section forms a list of independent yet circumstantial evidence
that may help to constrain the origin of DQ white dwarfs. And with the
exception of the dramatic lack of metal pollution, the empirical data are
suggestive but far from conclusive. In this section, the various threads are
combined and addressed with concrete hypotheses, which can be tested against
the existing evidence, now and in the near future, and will hopefully set the
stage to reveal the origin of all DQ white dwarfs; classical examples, warmer
stars with atomic lines, spectral oddballs, as well as peculiar and magnetic
examples.

###  5.1 Summary of observations

Below is a concise list of observational constraints, which can directly be
compared to any hypothetical evolutionary possibilities. Any successful
hypothesis for the origin of DQ stars should be consistent with these data and
indications, where Table 4 tracks the various scenarios and their potential
success in addressing each.

(1) DQ stars and DZ planetary hosts are essentially mutually exclusive
populations, with hundreds of known examples of each.

(2) There are no known DQ stars with infrared, optical, or transit indications
of circumstellar, planetary material.

(3) DQ stars are not found in post-common envelope binaries with unevolved
companions.

(4) The DQ spectral class is characterized by low $q({\rm He})$, where the
bulk of members may have relatively low masses.

(5) DQ white dwarfs are commonly magnetic, and likely more frequently or with
stronger fields than other spectral classes.

(6) The kinematics of DQ white dwarfs reveal they are only slightly older, or
kicked, relative to DZ and DC stars.

###  5.2 Metals are masked by unknown stellar opacity

This hypothesis states that the heavy element pollution in DQ white dwarfs is
present, and similar to that observed e.g. in DZ stars, but is masked by an
unknown stellar opacity source. This possibility can in principle account for
the fact that DQ stars rarely exhibit detectable metal lines, but does not
address the bulk of relevant issues discussed in the previous section. The
derived masses of metals in the fully mixed outer layers of the star, as well
as the inferred accretion rates, are tied to line strengths. Then, if metal
lines were reduced in strength within DQ stars and not in DZ stars, owing to a
previously unaccounted for opacity source, there would be an expected trend
with effective temperature or carbon abundance (or both). Neither the deep
UVES observations, nor the hundreds of stars observed with the SDSS, show any
trends in detected Ca ii line strengths with the DQ stellar parameters. This
hypothesis fails to address most of the other outstanding characteristics of
the DQ stars, and would not, for example, prevent the detection of
circumstellar material in the infrared, nor account for their distinctive
stellar properties.

###  5.3 Metals are diluted in deeper convective layers

This idea is similar to that discussed immediately above, but instead dilutes
atmospheric metals within significantly deeper convection zones. A scenario
where the relative sizes of the fully mixed outer layers of DQ and DZ stars
are orders of magnitude different can, in principle, account for the offset
seen in the right hand panel of Figure 2. This could be achieved by increasing
the size of convection zones in DQ stars, or decreasing those in DZ stars.
However, any order of magnitude changes in the depth of these layers will also
directly effect the heavy element diffusion timescale, as these two are
strongly correlated (Koester, 2009). For example, if DQ convection zones were
$1000\times$ larger than currently predicted, and crudely extrapolating
linearly from current models, one might expect diluted metals in DQ stars to
be seen for timescales of Gyr, and thus more frequently than in DZ stars, all
else being equal. Unfortunately, this hypothesis faces all the same problems
as the previous, and would have to be reconciled with the current underlying
framework of diffusion in helium atmosphere white dwarfs.

###  5.4 Progenitors of DQ stars have high-intermediate masses

A plausible theory for the lack of planetary material in and around DQ white
dwarfs is inhibited planet formation, and one possibility is via higher-mass
main-sequence stars. If DQ white dwarfs are the progeny of stars with masses
greater than $3-4$ M⊙, then it could be argued that this is consistent with
their lack of circumstellar matter, and pollution from planetary debris.
Around such luminous stars, protoplanetary disk lifetimes may be insufficient
to form planetary precursors and full-fledged planets (Kennedy & Kenyon, 2009;
Kunitomo et al., 2021), and there are currently few confirmed cases of planets
detected towards stars with masses above 3 M⊙ (e.g. Reffert et al. 2015).
However, the expectations of this hypothesis would be that the remnant masses
of DQ white dwarfs would be higher than the field, not lower as observed and
modeled. And while such progeny may be prone to detectable magnetism (Caiazzo
et al., 2020), they should be a relatively young kinematical population,
having spent little time on the main sequence. These predictions fail to match
the properties of DQ stars as shown in Figure 6. Furthermore, higher-mass
parent stars would not address the lack of post-common envelope binaries,
unless it were shown that this leads more commonly to companion destruction
during the post-main sequence.

Table 4: Consistency of DQ hypotheses vs. observational constraints. Constraint | Metals | Metals | High | Metal | Binary
---|---|---|---|---|---
| Masked | Diluted | Mass | Poor | Evol
Pollution | + | + | + | + | +
Circumstellar | - | - | + | + | +
Companions | - | - | - | - | +
$q({\rm He})$ / Mass | - | - | - | + | +
Magnetism | - | - | + | - | +
Kinematics | + | + | - | - | +

###  5.5 Progenitors of DQ stars are (very) metal poor

Similar to the hypothesis above, a lack of planetary matter orbiting and
polluting DQ stars could be attributed to an origin as Population II stars
that are truly metal poor. There are numerous planetary occurrence rate
studies that focus on stellar metallicity for hosts of transiting or Doppler
planets, but in general these are restricted to metallicities higher than 0.1
$Z_{\odot}$. The most ancient planetary systems confirmed to date orbit thick
disk stars that are only modestly metal poor (Mortier et al., 2012; Campante
et al., 2015), and similarly debris disk masses are thought to be strongly
linked to metallicities comparable to solar (Gáspár et al., 2016). Truly metal
poor stars have not yet revealed planetary systems, and theoretical models
suggest there is a critical metallicity, below which planet formation becomes
challenging, with values in the range $0.01<Z/Z_{\odot}<0.1$ (Johnson & Li,
2012; Hasegawa & Hirashita, 2014). If DQ white dwarfs descend from such metal-
poor stars, that could account for the lack of planetary material as pollution
and circumstellar debris, and might also be consistent with their low masses,
as the remains of stars that spent the bulk of their lifetime on the main
sequence. However, as shown above, their kinematics utterly fail to support
this interpretation. Furthermore, the known widely-bound companions to DQ
stars are not significantly metal poor, and one would expect some fraction to
host e.g. M subdwarf companions (Monteiro et al., 2006; Zhang et al., 2013).

###  5.6 Progenitors of DQ stars are binaries

A binary origin for DQ white dwarfs has several advantages over other
hypotheses. This scenario can immediately be invoked to account for the
apparently low masses, and more acutely, their relatively thin helium layer
mass fractions, which are arguably their defining characteristics according to
current models (Coutu et al., 2019; Koester & Kepler, 2019; Bédard et al.,
2022). Furthermore, it is already generally accepted that binary evolution is
responsible for the warmer examples of DQ stars (Dunlap & Clemens, 2015; Cheng
et al., 2020), and while precise evolutionary modeling has yet to reproduce
this inference based on observations, at face value there is a clear pathway
to reduce $q({\rm He})$ in DQ star progenitors via binary interactions.

During close binary evolution, a post-main sequence precursor star will often
lose mass via unstable Roche-lobe overflow, initiating a common envelope that
shrinks the semimajor axis, and which is known to lead to lower remnant masses
(Marsh et al., 1995; Rebassa-Mansergas et al., 2011). This scenario relieves
the tension between the relatively young space motions of the DQ stars, and
the long single-star lifetimes implied by their low masses. A high incidence
of (strong) magnetism might also be expected for a binary population, as is
observed for DQ white dwarfs.

Of all the theories discussed above, only a binary population might account
for the apparent lack of unevolved, post-common envelope companions, and it
requires that such companions are either consumed, or remain hidden. The major
challenge for this hypothesis is to show that binary evolution does not favor
the formation or retention of planetary architectures that commonly pollute
the surfaces of white dwarfs, and theoretical work is needed to investigate
this aspect in particular. It can be speculated that asymmetric mass loss,
especially that which might occur during the cannibalization of low-mass
stellar or substellar companions, might assist in the removal or
destabilization of planetary material within the system.

##  6 A DQ binary population outlook

This study has shown that DQ white dwarfs originate in a stellar and
circumstellar population that is distinct from the DAZ, DBZ, and DZ hosts of
remnant planetary systems, and highlighted several other, potentially
disparate characteristics of these carbon-enriched stars that point to a
binary origin. It is noteworthy that, for the most part, this work has
focussed on the classical and cool DQ stars that are defined spectroscopically
by the presence of Swan bands. However, there is a miniature zoo of DQ
spectral subtypes, and in particular a binary origin has already been
suggested for those DQ stars that have warmer temperatures and even thinner,
or absent, helium layers (Dunlap & Clemens, 2015; Cheng et al., 2020), as well
as unusual spectral types such as DAQ (Hollands et al., 2020), all of which
appear to have higher remnant masses, and thus superficially amenable to
merger hypotheses. These deductions have yet to be supported by dedicated
evolutionary modeling efforts, but if correct, then one clear outcome is that
binary interactions can result in reduced $q({\rm He})$ in white dwarf
remnants.

Therefore, if binary evolution can reduce the helium mass fraction in white
dwarfs, the outlook is immediately positive for the entire class of DQ stars.
The only major difference for the cool and classical DQ stars with Swan bands,
those that form the bulk of the known population, is their lower or more
typical remnant masses. For this reason, the merger of two white dwarfs is
unlikely to be the dominant, binary formation channel for DQ stars. Moreover,
the classical DQ white dwarfs do not have kinematics of a blue straggler
population, with an added delay in cooling for the time it takes to merge.
Binary evolution does not necessarily imply older ages, as two stars can begin
to interact as soon as one has evolved sufficiently for mass transfer to
begin, and thus a binary DQ population can have ages similar to their helium
atmosphere, DZ and DC white dwarf counterparts, as observed. The common
envelope is thought to be a rapid process, and should not add significantly to
the inferred total ages of DQ stars in the case that unevolved companions are
immediately consumed; the age bottleneck is still the evolutionary timescale
of the white dwarf progenitor.

Planetary companions are also a possibility in a scenario of fatal engulfment
and binary evolution for DQ white dwarfs. Two basic outcomes are required, one
is that the white dwarf experiences enhanced mass loss compared to single star
evolution (possibly in terms of total mass and the outer helium envelope), and
that any substellar companions prohibit or diminish planetary material that
might later precipitate atmospheric pollution. There is some evidence that
giant planets in close orbits (i.e. hot Jupiters) are rarely found with
additional planetary companions, and may have cleared away planet-forming disk
material during migration, or ejected other planets dynamically (Latham et
al., 2011; Mustill et al., 2015). While it is unclear if a giant planet (or
two) is sufficient to amplify mass loss during stellar evolution, the absence
of hot Jupiters around subgiant hosts argues that giant planet ingestion does
indeed occur (Schlaufman & Winn, 2013).

A key question is exactly how binary evolution might prevent the eventual
pollution of a white dwarf, where theoretical modeling is clearly needed. From
observations, it is noteworthy that for known white dwarfs in short-period
binaries with low-mass stellar and substellar companions, there is only a
single case of circumbinary material and pollution (SDSS J155720.77+091624.6;
Farihi et al. 2017). Despite this apparent rarity, there is a general lack of
sensitivity to such circumbinary dust systems, because the cool companion can
itself cause a large infrared excess, and thus additional emission from dust
would be challenging to identify.

Binary evolution modeling is needed to test the basic premise raised here for
DQ white dwarfs. The lack of unevolved companions in close orbits is a key
observable; on the one hand, if they are destroyed then the remnant might be
expected to be somewhat more massive, and on the other hand if they survive,
it is unclear why they are not detected. It is theoretically possible for a
companion to merge and eject more material than it provides to the merged
stellar core, but such outcomes are best addressed via dedicated modeling. It
is important to note that both Procyon B and GJ 86B may have transferred mass
during their evolution, as the originally more massive stars within their
binary systems. Current data and modeling do not require such mass transfer in
these iconic systems, but it may account for remaining inconsistencies,
especially from an evolutionary perspective (Bond et al., 2015; Zeng et al.,
2021). Binary or white dwarf – main-sequence merger candidates may be
challenging to detect given the lack of atomic lines in many DQ stars, but
relatively rapid stellar rotation, photometric or astrometric variation owing
to unseen companions are likely one way forward, observationally.

Lastly, what is the nature of the rare and weak pollution in DQ white dwarfs?
The default hypothesis should be remnant planetary systems222Appealing to the
interstellar medium for the weak DQZ pollution may be tempting, but there are
myriad issues such as their utter lack of hydrogen., and there is no a priori
reason to expect other mechanisms, even if the stars are currently or formerly
binary (Rafikov, 2013; Martin et al., 2013; Bromley & Kenyon, 2015). There are
dynamical issues that are unique to binary systems, and circumbinary planet
hosts in particular, with additional instabilities as compared to single
stars, during formation and on the main sequence, where planetary ejection is
one of the likely outcomes (Smullen et al., 2016; Sutherland & Fabrycky, 2016;
Sutherland & Kratter, 2019). It might be speculated that such instabilities
might be catastrophically amplified during a common envelope or merger event
where mass loss may be asymmetric or impulsive, thus quickly reducing or
removing planets and associated planetesimal belts.

For context, the solar zodiacal cloud is replenished at a rate in the range
$10^{7}-10^{8}$ g s-1 (Nesvorný et al., 2011; Rigley & Wyatt, 2022), and the
rare DQZ exhibit accretion rates $100\times$ lower, not to mention the typical
DQ upper limits at least $1000\times$ lower, which approaches the accretion
rate of extraterrestrial material onto the Earth (Love & Brownlee, 1993; Yada
et al., 2004). It may be possible to detect multiple heavy elements in the
rare DQZ stars, and if successful, their ratios can be compared to the
planetary abundances associated with DAZ, DBZ, and DZ white dwarfs. While
challenging owing to the low abundances and line strengths, it may provide an
important clue to the nature of these stars, which are now, hopefully, less of
an astrophysical enigma.

## Acknowledgements

J. Farihi thanks D. Koester and G. Fontaine for key discussions over the years
that helped to form the foundation of this work, E. Dennihy for confirming the
fact that no DQ star has a suspected infrared excess from circumstellar dust,
A. Rebassa-Mansergas for sharing his full white dwarf-main sequence binary
catalog, and valuable exchanges with E. B. Bauer, R. R. Rafikov, and J. J.
Eldridge. Several colleagues provided feedback on an earlier version of the
manuscript, including those mentioned above, as well as T. von Hippel, A. J.
Mustill, A. Swan, and B. Zuckerman. The authors acknowledge the European
Southern Observatory for the award of telescope time via programs 095.D-0706,
096.D-0076, and 097.D-0063. This work has made use of the ESA Gaia mission,
processed by the Gaia DPAC. J. Farihi acknowledges support from STFC grant
ST/R000476/1. T. G. Wilson acknowledges support from STFC consolidated grants
ST/R000824/1, ST/V000861/1, and UKSA grant ST/R003203/1.

## Data Availability

Data acquired at ESO facilities for the PI programs listed above are available
through their archive.

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