# New insight into the magnetism of degenerate stars from the analysis of a
volume limited sample of white dwarfs

S. Bagnulo1 & J.D. Landstreet1,2
1Armagh Observatory & Planetarium, College Hill, Armagh BT61 9DG, UK
2University of Western Ontario, London, Ontario N6A 3K7, Canada.

(Accepted June 10; Received June 3; in original form: April 7 2021.)

###### Abstract

Many stars evolve into magnetic white dwarfs, and observations may help to
understand when the magnetic field appears at the stellar surface, if and how
it evolves during the cooling phase, and, above all, what are the mechanisms
that generate it. After obtaining new spectropolarimetric observations and
combining them with previous literature data, we have checked almost the
entire population of about 152 white dwarfs within 20 pc from the Sun for the
presence of magnetic fields, with a sensitivity that ranges from better than 1
kG for most of the stars of spectral class DA, to 1 MG for some of the
featureless white dwarfs. We find that 33 white dwarfs of the local 20 pc
volume are magnetic. Statistically, the data are consistent with the
possibility that the frequency of the magnetic field occurrence is similar in
stars of all spectral classes, except that in the local 20 pc volume, either
DQ stars are more frequently magnetic, or host much stronger fields than
average. The distribution of the observed field strength ranges from 40 kG to
300 MG and is uniform per decade, in striking contrast to the field frequency
distribution resulting from spectroscopic surveys. Remarkably, no fields
weaker than 40 kG are found. We confirm that magnetic fields are more frequent
in white dwarfs with higher than average mass, especially in younger stars. We
find a marked deficiency of magnetic white dwarfs younger than 0.5 Gyr, and we
find that the frequency of the occurrence of the magnetic field is
significantly higher in white dwarfs that have undergone the process of core
crystallisation than in white dwarfs with fully liquid core. There is no
obvious evidence of field strength decay with time. We discuss the
implications of our findings in relation to some of the proposals that have
been put forward to explain the origin and evolution of magnetic fields in
degenerate stars, in particular those that predict the presence of a dynamo
acting during the crystallisation phase.

###### keywords:

white dwarfs – stars: magnetic fields – polarization

††pubyear: 2021††pagerange: New insight into the magnetism of degenerate stars
from the analysis of a volume limited sample of white dwarfs–B.7.1

##  1 Introduction

That magnetic fields occur in the latest stages of stellar evolution has been
known for more than 50 years, since the discovery that pulsars are strongly
magnetised neutron stars (Woltjer, 1964; Pacini, 1967; Gold, 1968) and the
discovery that white dwarfs (WDs) host strong magnetic fields (Kemp et al.,
1970a; Angel & Landstreet, 1971a, 1972).

When present, the magnetic field is one of the main actors that determines the
observable features of a WD, or of a binary system containing a WD, by
altering the spectral lines; by changing the physical structure through
suppression of convection or by Lorentz forces; and by affecting transport of
angular momentum and mixing during accretion and mass-loss phases, and during
evolutionary changes to internal structure. However, the origin of WD
magnetism is not well understood. Schematically, at least three scenarios have
been proposed in the literature. (1) According to the classical ‘fossil field’
theory, the field is a descendant of a field that was present when the star
was in a previous evolutionary stage, perhaps during the red giant (RG) phase
or on the main sequence (MS), and its strength was amplified due to flux
conservation while the star contracted into a WD. The favoured MS candidates
progenitors of the magnetic WDs (MWDs) are the magnetic Ap and Bp stars. (2)
According to the ‘merging scenario’, a WD that formed in the merger of a
binary pair may become magnetic as a result of a dynamo generated during the
merger (Tout et al., 2008); a variant of this scenario predicts that a
magnetic field may also be formed during the accretion of rocky debris (Farihi
et al., 2011; Briggs et al., 2018a). (3) The field may also be produced by
some other internal physical mechanism during the cooling of the WD itself
(Valyavin & Fabrika, 1999), possibly a dynamo acting in an unstable WD liquid
mantle on the top of a solid core (Isern et al., 2017). The latter theory also
provides an explanation for planetary magnetic fields.

To understand the origin of the magnetic fields it is useful to know if there
are clear correlations between the incidence of magnetic fields and other
characteristics of the WDs. For instance, we know that the strength of the
magnetic fields in the chemically peculiar stars of the upper main decays
rapidly with time as soon as the star has reached the main sequence
(Landstreet et al., 2007, 2008). Does the strength of the magnetic field in
WDs also decrease with cooling age? Is there a correlation between the
magnetic field and the mass of a WD, or with the chemical composition of its
atmosphere, or with binarity?

Over the last five decades, a lot of effort have been dedicated to the search
for and characterisation of MWDs, both as a side-product of large
spectroscopic surveys (e.g. Liebert et al., 2003; Kepler et al., 2013;
Hollands et al., 2015), and due to specifically dedicated polarimetric surveys
(e.g. Angel & Landstreet, 1970a; Angel et al., 1981; Schmidt & Smith, 1995;
Putney, 1997; Aznar Cuadrado et al., 2004; Jordan et al., 2007; Kawka et al.,
2007; Landstreet et al., 2012; Bagnulo & Landstreet, 2018). A number of
proposals have been put forward regarding various kinds of trends, for example
that magnetic WDs are more massive than non magnetic WDs (Liebert et al.,
2003); that cooler WDs are more often magnetic than hotter WDs (Liebert &
Sion, 1979; Fabrika & Valyavin, 1999); that magnetic WDs are not found in
close binary systems with a WD and a main sequence companion (Liebert et al.,
2005); that magnetic fields occur more frequently in cool stars with metal
lines, whether they are H-rich DAZ WDs (Kawka et al., 2019) or He-rich DZ WDs
(Hollands et al., 2015; Bagnulo & Landstreet, 2019b; Kawka et al., 2021); and
that hot DQs, WDs with a surface composition dominated by carbon and oxygen,
show an extremely high incidence of magnetic fields (Dufour et al., 2013).
Some of these trends, however, have been challenged by the results of other
groups; for instance, Kawka et al. (2007) could not confirm a trend of the
incidence of magnetic fields with age, and Landstreet & Bagnulo (2020) showed
that MWDs in binary WDs with a main sequence companion may be rare, but do
exist.

The reality is that obtaining firm observational conclusions about
evolutionary trends of the magnetic field is strongly hampered by various
observational biases. Spectropolarimetric surveys, sensitive to the mean
longitudinal magnetic field, are photon-hungry by nature, and tend to favour
observations of bright stars. Lower mass WDs have different luminosity and
absolute magnitude than higher mass WDs of the same age, and cooler (hence
older) WDs may be much less bright than hotter and younger WDs. Both high-mass
and cool/old WDs might be underrepresented in magnitude-limited surveys
(Liebert et al., 2003). Low-resolution spectroscopic surveys, which can go
much deeper in magnitude and examine a much larger sample of stars than
spectropolarimetric surveys, are sensitive only to magnetic fields stronger
than roughly 1–2 MG, leaving weak fields in WDs undetected, and may struggle
to identify some WDs with field stronger than 100 MG, which splits spectral
lines into numerous weak components with low contrast relative to the
continuum.

To obtain the clearest possible view of the occurrence of detectable magnetic
fields in WDs, we need to look at a sample which has a relatively clear
significance and well-defined theoretical selection criteria. The most obvious
choice is a volume-limited sample. The sample of all WDs within 20 pc from the
Sun includes about 150 stars: it is small enough to allow us a careful
analysis of each individual member, but large enough to provide some
meaningful statistical results. Because the evolution of all single stars with
initial masses below about 8 $M_{\odot}$ ends with collapse to the white dwarf
state, as does the evolution of many binary pairs, the current populations of
white dwarfs in the local 20 pc volume records a sample of the results of more
than 90% of all completed stellar nuclear evolution at this distance from the
centre of the Milky Way. The magnetic fields found in the WDs of this sample
provide a relatively well-defined record of the generation and preservation of
such fields through the lifetime of the Milky Way. White dwarfs of different
ages in the local 20 pc volume provide evidence about the production rate and
evolution of both WDs and of their fields as a function of time over the past
10 Gyr.

In this paper we therefore try to characterise the frequency of occurrence of
magnetic fields of various strengths among all WDs of the local 20 pc volume,
and to correlate this with other stellar features such as mass, chemical
composition, and cooling age. This task requires first an accurate
characterisation of the physical parameters of each member of the local 20 pc
volume, which is discussed in Sect. 2. Next, we need to check each member of
the local 20 pc volume for a magnetic field. Existing observations from the
literature are reviewed in Sect. 3. Because literature data did not provide
sufficient information, we have carried out a spectropolarimetric survey aimed
at acquiring a complete dataset of all WDs within the local 20 pc volume. Some
preliminary results of our survey have been presented by Landstreet & Bagnulo
(2019b), Bagnulo & Landstreet (2019b) and Landstreet & Bagnulo (2020), mainly
to announce the discovery of new MWDs. In this paper, we publish the remaining
observations that we have carried out in the last three years, and which
complement those previously published in the literature for the sample of all
WDs within 20 pc from the Sun. These new observations are presented in Sect. 4
(observing log and fields values are given in App. A). In Sect. 5 we describe
the dataset that we have collected for the WDs of the local 20 pc volume and
in Sect. 6 we perform our analysis, in particular we discuss how the frequency
of occurrence and the strength of the magnetic field vary in WDs of different
spectral types, effective temperatures, and ages. In Sect. 7 we discuss
whether the theories that have been put forward to explain the origin of
magnetic fields in degenerate stars are consistent with the characteristics of
the MWDs of the local 20 pc volume. Our conclusions are presented in Sect. 8.
This paper contains also a second appendix (App. B) where we discuss the
magnetic nature of all individual WDs of the local 20 pc volume; for some of
the WDS, we also discuss the estimates of the stellar parameters and binarity.

##  2 Defining the 20 pc WD sample

Our sample of WDs is taken from the list proposed by Hollands et al. (2018),
mainly on the basis of the astrometry reported in the second data release of
the Gaia space astrometry mission (Gaia Collaboration, 2016, 2018), but also
including several nearby WDs missed in the Gaia survey for one reason or
another. Practically we have considered all stars of Table 1 of Hollands et
al. (2018), and all stars of their Table 4 except for WD 0454$+$620 and WD
1443$+$256: the former seems to be outside of the local 20 pc volume, and the
latter was found to be a distant high-velocity G star by Scholz et al. (2018),
an identification confirmed by a very small Gaia parallax. The third Gaia data
release became publicly available after we had performed all our observations
and carried out most of the analysis; nevertheless we checked if the revised
parallax values would alter the membership of our sample. We found for
instance that WD 0728$+$642, a star that we had incorporated in our sample
even though Gaia DR2 had put it possibly just outside of the local 20 pc
volume ($\pi=49.97\pm 0.05$), is according to DR3 most likely within 20 pc
from the Sun ($\pi=50.06\pm 0.04$), hence full member of the local 20 pc
volume. None of the stars of our sample was found according to DR3 to be
outside of the local 20 pc volume. We also checked that none of the WDs of the
catalogue by Gentile Fusillo et al. (2019) that had parallax $\pi\leq 50.0$ in
DR2 had $\pi>50$ in DR3, except for WD 0728$+$642.

###  2.1 Atmospheric parameters

Most of the WDs of the local 20 pc volume are reasonably well studied, and
atmospheric composition, mass, $T_{\rm eff}$, gravity, and age have been
calculated on the basis of detailed modelling of large data sets such as low
resolution visible and/or UV spectra, and extensive photometry with a wide
range of filters or calibrated spectrophotometry. We have selected most of
physical parameter data from the work of Giammichele et al. (2012); Limoges et
al. (2015); Subasavage et al. (2017); Blouin et al. (2019) and Coutu et al.
(2019). When data from these sources were not available (mainly for newly
discovered WDs), we have adopted the values tabulated by Hollands et al.
(2018) or by Gentile Fusillo et al. (2019), which are derived from the Gaia
three-band photometry and parallax of each WD, on the basis of calibrations
provided by fitting a large number of well-studied WDs. However, these
calibrations do not provide reliable composition choices for newly identified
WDs in the volume, and provide no ages. For WDs without age estimate in the
literature we have relied on the on-line cooling
tables111http://www.astro.umontreal.ca/~bergeron/CoolingModels of the Montreal
group (Dufour et al., 2017), which are based on computations described by
Tremblay et al. (2011), using a two-dimensional logarithmic interpolation.

To obtain an idea of the general level of precision of the adopted physical
parameters we have compared the results obtained for a number of individual
WDs by several different groups. The scatter of such computations suggests
that the uncertainties of the physical parameters are about 300 K for $T_{\rm
eff}$, 0.1 for $\log g$, $0.1\,M_{\odot}$ for the mass, and 0.3 dex for the
age (or somewhat better for the youngest WDs).

We should also note that the parameters that we have adopted come from
modelling based on the assumption that the magnetic field has no influence on
the stellar atmosphere. In fact, a magnetic field of some kG can suppress
convection in the outer layers of the MWD. This changes the stratification of
the model, and introduces a further uncertainty into the the estimate of the
stellar parameters. Radiation magneto-hydrodynamic simulations of the
atmosphere of white dwarf stars are discussed in Tremblay et al. (2015), and
applied to observational data by Gentile Fusillo et al. (2018) to the WD
2105$-$820\. For that weakly magnetic star it was found that the best pure
radiative model that reproduces the optical spectrum has $T_{\rm eff}=9982\pm
170$ K and $\log g=8.22\pm 0.08$, and is also capable to reproduce the star’s
FUV spectrum, while the best-fitting optical convective model has $T_{\rm
eff}=10389\pm 153$ K and $\log g=8.01\pm 0.05$ and fails to reproduce the FUV
spectrum. However, the zero-field atmospheric model adopted by Subasavage et
al. (2017) leads to an estimate of $T_{\rm eff}=9820\pm 240$ K and $\log
g=8.29\pm 0.04$, consistent with the best-parameters of the radiative model by
Gentile Fusillo et al. (2018). This suggests that the uncertainties introduced
by the zero-field approximation are probably of the same order as of those
deduced from the scatter seen in literature values.

###  2.2 Multiple systems

Some of the WDs occur in binary or multiple systems. In this section we
discuss these situations.

####  2.2.1 Common proper motion systems

About 30 of the WDs of the local 20 pc volume are in common proper motion
visual binary systems, usually with main sequence stars, with separations
ranging from about $2\arcsec$ to several arc-minutes. In such systems, the
nature of all components is almost always well established. In three cases (DA
WD Sirius B, and DQ WDs Procyon B and WD 0208$-$510), the components are so
close as to prevent polarimetric observations of the WD (circular polarimetric
measurements are possible only from ground-based facilities). However, HST
spectroscopy is available for all three WDs. There are two double degenerate
(DD) systems whose members are well separated and may be observed
individually. In addition there are a dozen known or suspected unresolved
double systems (uDD), all made up of two WDs (a WD plus a main sequence star
pair would usually be obvious from the composite spectrum). In only a few of
these systems is it clear what the types of WDs the two stars are, and for
half of them it is not even really clear whether one or two stars are present.

The common proper motion pairs do not present any real difficulty for our
analysis. Within the 20 pc volume, all the visual binary pairs are separated
by at least about $2{\arcsec}$, which at a distance of the order of 10 pc
corresponds to a separation in the plane of the sky of the order of 20 AU.
Therefore most of the minimum separations are hundreds or thousands of AU.
With such a large separation, it is unlikely that a main sequence companion
would have had any influence on the evolution of the WD predecessor through
the red giant and AGB stages, although the main sequence companion might
accrete some of the outflowing gas during mass loss phases preceding the
formation of the WD.

####  2.2.2 Unresolved double degenerate systems

In contrast, all the unresolved double systems consist of two WDs, and
therefore contribute not one but two WDs to the total tally of WDs within the
local 20 pc volume. Some of these uDD systems are identified simply because
the deduced mass of the object ($\la 0.45M_{\odot}$), treated as a single WD,
is too small to have formed by single star evolution. However, in some cases,
previous works provide conflicting results, in that the parameter
determination by one group may result in small deduced mass, while another
group finds a mass large enough to be the product of single star evolution
(see for example Sec. 4.2 of Hollands et al., 2018); therefore, before
accepting the system as uDD, we look for some further observational hint of a
secondary, such as discordance between computed and observed H line profiles.

For the systems that we have accepted as uDD, we have also used the available
data to make plausible proposals (often just guesses) about the
characteristics of the two stars. As these systems largely have photometry in
several bands, and the single star models of Giammichele et al. (2012) are
generally successful in accounting for the energy distributions, we have
looked for models in which both stars have similar $T_{\rm eff}$ values. We
have also started by looking for model pairs with similar radii (i.e. we have
looked for models in which the two components make similar contributions to
the total system light). Such models are of course rather uncertain,
particularly as concerns the secondary in the system. The best studied cases
are those of uDDs WD 0135-052 and of WD 0727+482 (the latter is marginally
visually resolved), for which the orbits of the WD members are known; the
orbits provide the masses of each of the two components, and light ratios, and
the other physical parameters of the stars may be reasonably estimated.

All our choices of which systems should be considered as most likely to be
composed of two stars (labelled as “uDD”), and which uDD systems are only
suspected DD (labelled as “uDD?”), are documented in App. B, where we have
also described our modelling attempts. In our statistics, we count uDD systems
as two WDs, but uDD? systems as single stars.

Finally, we note that for all established uDD systems it may not be clear how
to apportion a magnetic measurement to one of the two stars. This issue is
discussed in App. B.

A further complication of the uDD systems is that they are relatively close to
each other. For example, at a distance of 15 pc, two WDs separated by less
than 0.5″ are probably only a few AU apart. This is a small enough separation
that the evolution of each star is very likely to have affected that of the
other star. These WDs may well have evolved rather differently than isolated
single stars.

###  2.3 The list of WDs of the local 20 pc volume

Our final list of local 20 pc volume members is considered to be at least 95 %
complete and includes 152 stars, six of them in uDD systems (hence we have 146
systems), and six suspected uDD? systems that we have decided to treat as
single stars. One of the members, WD 0211$-$340 = LP 941-19, cannot be
currently well characterised because of its current proximity to a background
star. Hollands et al. (2018) commented on its high proper motion and Gaia G
and JHK magnitudes agree with a degenerate star of $T_{\rm eff}=5270$ K and
$\log g=8.0$, but its spectral type (and magnetic nature) are unknown. The
list is given in Table LABEL:Tab_Stars, which will be fully described in Sect.
5.

The mean mass of all WDs of the local 20 pc volume is 0.64 $M_{\odot}$ with a
standard deviation of 0.13 $M_{\odot}$, fully consistent with the mean mass of
non magnetic WDs of $0.66\pm 0.14\,M_{\odot}$ estimated by Tremblay et al.
(2013). The temperature distribution is highly skewed toward temperatures
lower than 10 000 K. It appears that the production of WDs was higher in the
last few Gyr – stars younger than 3 Gyr represent more than a half of the
entire sample of WDs, which have age spanning the range between 0.03 and 9.3
Gyr. Since cooler, hence older, WDs are dimmer than hotter and younger WDs, it
is possible already to anticipate that a magnitude limited survey can hardly
capture a statistically representative sample of the real population of WDs,
as will be more evident in results presented in Sect. 5 and in the analysis
that will be carried out in Sect. 6.

##  3 Field detection techniques and measurements from previous literature

We start with a general overview of the magnetic field measurements of the WDs
within 20 pc from the Sun. Magnetic WDs have been discovered with low and high
resolution spectroscopy, with narrow- and broad-band polarimetry and with
spectropolarimetry. To guide this overview, and to inform its subsequent
analysis, it is useful first to recall the sensitivity threshold of the
various methods used to detect magnetic fields.

###  3.1 Field detection techniques

####  3.1.1 Spectroscopy

Spectroscopy may reveal Zeeman splitting of spectral lines, which is sensitive
to the so-called mean magnetic field modulus $\langle|B|\rangle$, i.e., the
surface field strength averaged over the visible stellar disk. Low-resolution
spectroscopy of WDs may detect a magnetic field when its strength is at least
1–2 MG, while high-resolution spectroscopy (say $R\ga 50\,000$) may detect
fields as weak as 50 kG in DA stars with sharp Balmer line cores (e.g.,
Koester et al., 1998).

####  3.1.2 Circular spectropolarimetry of spectral lines

Circular spectropolarimetry is sensitive to the longitudinal component of the
magnetic field averaged over the stellar disk, or mean longitudinal field
$\langle B_{z}\rangle$, and may probe fields of much lower strength than
spectroscopy. The actual sensitivity of spectropolarimetric measurements
varies with spectral type. In the brightest DA stars with deep H lines,
$\langle B_{z}\rangle$ may be measured with an uncertainty of a few hundred
Gauss (see e.g. Landstreet et al., 2015; Bagnulo & Landstreet, 2018). Circular
polarisation integrated over a spectral line is zero, therefore broadband
polarimetry is not useful to detect magnetic fields in WDs, unless the field
is sufficiently strong to polarise the continuum.

####  3.1.3 Circular polarimetry of the continuum

In WDs with featureless spectra (DC spectral type), fields with a longitudinal
field of at least $\sim 0.5$ MG (in absolute value) may be detected via the
measurement of circular polarisation in the continuum. Polarisation in the
continuum may be measured by spectropolarimetry or by narrowband or broadband
circular polarimetry. Since polarisation of the continuum (or of very broad
spectral lines) may change its sign with wavelength, spectropolarimetry has
the potential to reveal signals that might cancel out in broadband
polarimetric measurements. However, spectropolarimetry is often more affected
by instrumental effects than simple imaging polarimetry, and our experience
with FORS2 and ISIS shows that only fields stronger than a few MG may be
confidently detected by spectropolarimetry of the continuum in featureless
stars (Bagnulo & Landstreet, 2020). In practice, multi-colour narrowband
circular polarimetry may be the most suitable detection method when the
polarisation signal is $\la 0.3$ % in featureless spectra. We finally note
that polarimetry of the continuum is the detection method of choice not only
for featureless cool WDs, but also for hotter DA and DB WDs in the regime of a
very strong field (say $\ga 100$ MG), in which spectral lines may become very
difficult to recognise, as well as for DQ stars, since we still do not know
how intensity profiles of C2 bands are modified by the presence of a very
strong magnetic field.

###  3.2 The Oblique Rotator Model and its implications for detection and
characterisation of a magnetic field

As the star rotates, the magnetic field, if not symmetric about the rotation
axis, changes its configuration with respect to the observer. In general,
$\langle|B|\rangle$ does not vary dramatically as the star rotates, and it is
unlikely that a spectral line may split at certain rotation phases and not at
others. Therefore, no repeated spectroscopic sampling is needed to assess
whether a star possesses a magnetic field strong enough to split its spectral
lines. In contrast, even a strong magnetic field may have a small average
component along the line of sight, and in some cases may be detectable only at
certain rotation phases. Ideally, two or three spectropolarimetric
observations per star are needed, either to rule out the presence of a
detectable field, or, in case of field detection, to confirm that a field
exists, and check if it is variable. If the magnetic field is strong enough to
polarise the continuum, it is unlikely that the entire $V/I$ spectrum is zero
at all wavelengths at a given rotational phase (or at least we have never
encountered a such a situation). Therefore we expect that even a single
polarimetric measurement of the continuum may be used to assess the presence
of a very strong magnetic field. Needless to say that the characterisation of
strength and morphology of a stellar magnetic field requires good sampling of
the rotational cycle, for instance, ten measurements taken about 0.1 cycles
apart from each other.

###  3.3 Measurements of magnetic fields of WDs from previous literature

Magnetic WDs have been discovered with low and high resolution spectroscopy,
with narrow- and broad-band polarimetry and with spectropolarimetry. Low
resolution (classification) surveys have allowed the discovery of hundreds of
MWDs in the 2–80 MG regime (Kepler et al., 2013), while the numbers of MWDs
discovered via high-resolution spectroscopy and polarimetry is an order of
magnitude smaller. Observations of MWDs may be broadly divided into four
groups.

i) Narrow band and broadband circular polarisation measurements made possible
the discovery of the first MWD by Kemp et al. (1970b), and were systematically
employed in small-scale surveys during the 1970s, leading to the discovery of
a dozen MWDs (Angel et al., 1981, and references therein). In the local 20 pc
volume, these surveys have discovered six MWDs (one DC, two DQ and three DA
WDs), and set useful upper limits to the magnetic fields of another 13 DC
stars in which no circular polarisation was detected. These early surveys
reached a degree of accuracy for $V/I$ often of the order of 0.05 % or better.
This accuracy could not be improved by modern spectropolarimetry, where
instrumental effects make it hard to reach an accuracy in the continuum better
than $\sim 0.2$ % (see Sect. 6 of Bagnulo & Landstreet, 2020), although we
must take into account that modern telescopes and detectors are able to reach
much fainter objects. Aperture polarimetry is ideal to detect WDs with a
magnetic field strong enough to polarise the continuum. However, we are not
aware of any systematic broadband or narrow band circular polarisation surveys
of WDs after the work of Liebert & Stockman (1980) and Angel et al. (1981).
The early surveys have also targetted numerous DA WDs, but for them, much
higher precision would be obtained later with spectropolarimetric techniques.

ii) MWDs were discovered with spectroscopic techniques from the 1970s on, but
mainly (because of the introduction of highly-efficient CCDs) since the late
1990s. Low-resolution spectroscopic surveys have made it possible to discover
hundreds of MWDs (Kepler et al., 2013), mostly in the range 1–80 MG. Six MWDs
have been discovered in the 20 pc volume with spectroscopic techniques.

iii) In the 1990s, spectropolarimetry started to be applied more
systematically to observation of WDs, mainly thanks to the the survey of
Schmidt & Smith (1995), Putney (1997), and later Kawka et al. (2007). These
surveys allowed the measurement of $\langle B_{z}\rangle$ with a typical
uncertainty of $\sim 10$ kG, and led to the discovery of another five MWDs in
the local 20 pc volume (including an earlier spectropolarimetric discovery by
Liebert et al., 1978), while another 32 WDs were observed with no field
detection.

iv) With the use of the FORS instrument at the ESO VLT, the sensitivity of the
$\langle B_{z}\rangle$ measurements was brought down to the 1 kG region, as
shown by Aznar Cuadrado et al. (2004), Jordan et al. (2007), Landstreet et al.
(2012) and Landstreet et al. (2017). Later, Landstreet et al. (2015)
demonstrated that the ESPaDOnS instrument of the CHFT could also be used to
measure $\langle B_{z}\rangle$ in DA and DB WDs with high accuracy, and
Bagnulo & Landstreet (2018) showed that the ISIS instrument of the WHT could
compete with the FORS instrument in terms of accuracy. (We note that ESPaDOnS
does not have capabilities in the continuum, while FORS and ISIS have.) All
these projects led to the discovery of another four MWDs in the local 20 pc
volume (one of which, WD 2047$+$372, had already been observed by Schmidt &
Smith, 1995, but with precision not sufficiently high to detect its weak
magnetic field), and set meaningful upper limits to $|\langle B_{z}\rangle|$
for 17 WDs in which no magnetic field had been detected. Ten of these non
magnetic WDs had been in fact already observed with spectropolarimetric
techniques but with a much lower precision.

###  3.4 Motivations for additional observations

None of these past surveys specifically targetted a volume-limited sample of
WDs, although Kawka et al. (2007) reported some statistics of the local
neighborhood, finding that nine out of the 43 WDs within 13 pc from the Sun
were magnetic, for a global frequency of $21\pm 8$ %. At that time, the
exploration of the 20 pc volume was still incomplete, with only 116 WDs known
members, 15 of which were known to be magnetic. For the local 20 pc volume,
Kawka et al. (2007) estimated an incidence of the magnetic field of $13\pm 4$
%. In fact, apart from the incompleteness of this sample, it is important to
note that the works by Aznar Cuadrado et al. (2004) and Jordan et al. (2007),
and later Landstreet et al. (2015) and Bagnulo & Landstreet (2018) made it
clear that all WDs with spectral lines in which Putney (1997) and Schmidt &
Smith (1995) failed to detect a magnetic field should be revisited with larger
telescopes and more efficient instruments, on the expectation that modern
measurements could reach a sensitivity up to a factor of 10 times higher than
in the past.

We finally decided to use the ESPaDOnS, FORS2 and ISIS instruments to perform
a spectropolarimetric survey of all WDs of the 20 pc volume that were not
known to have magnetic fields, but for which no highly-sensitive measurements
had been performed previously. At the beginning of our project, the target
list consisted of $\sim 70$ WDs that were never observed polarimetrically
before, and another $\sim 20$ that should be re-observed with much higher
accuracy. During the last few years we have presented the discoveries of 12
new MWDs in the 20 pc volume (Landstreet & Bagnulo, 2019a, b; Bagnulo &
Landstreet, 2019b, 2020), which is more than 1/3 of all MWDs now known in that
sample. In the next section we will present our still unpublished
spectropolarimetric observations of WDs, which include mainly non detections,
and a few marginal detections.

##  4 New spectropolarimetry with ESPaDOnS, FORS and ISIS

Figure 1:  Distribution in apparent magnitude (either $V$ or Gaia $G$
magnitude, see Sect. 5.3) of the WDs of the local 20 pc volume. Red lines
refer to the new spectropolarimetric observations obtained in the course of
this survey, black lines to the WDs that were observed in spectropolarimetric
mode prior to our survey and that we have re-observed with higher precision;
blue lines refer to WDs that were observed in previous work and that we have
not re-observed. The three small unshaded regions correspond to the three WDs
that were not checked for magnetic field.

We present 122 new spectropolarimetric observations of 85 WDs. These new
observations add to the about 20 spectropolarimetric observations of 12 MWDs
that we have already presented, and bring the compilation of a database of
highly-sensitive magnetic field measurements of the local population of WDs
nearly to completion. As anticipated above, all our new measurements were
obtained with the FOcal Reducer and low dispersion Spectrograph (FORS2)
instrument (Appenzeller et al., 1998) of the ESO Very Large Telescope (VLT),
with the Intermediate-dispersion Spectrograph and Imaging System (ISIS)
instrument of the William Herschel Telescope (WHT), and with the Echelle
SpectroPolarimetric Device for the Observation of Stars (ESPaDOnS) instrument
(Donati et al., 2006) of the Canada France Hawaii Telescope (CFHT). Eleven WDs
were observed with two instruments, and another two (WD 0738$-$172 and WD
2336$-$079) with all three instruments. The relevance of this new survey can
be appreciated in Fig. 1, which shows the distribution in magnitude of the
observations of the WDs of the local 20 pc volume. Red lines show the
distribution in magnitude of the WDs that were observed for the first time in
spectropolarimetric mode in the course of this survey (including the 12 MWDs
presented in our previous four papers); black lines refer to the distribution
in magnitude of the WDs that were observed in spectropolarimetric mode prior
to our survey but that we have re-observed with higher precision; blue lines
refer to WDs that were checked for a magnetic field in previous works, and
that we have not re-observed because we considered that literature data
provided sufficient constraints to their magnetic field. Indeed, among the WDs
of the latter group, about 20 were already known as magnetic ones, hence did
not need to be re-observed to confirm their magnetic nature, a dozen had been
found non magnetic by high-sensitivity measurements either with the ESPaDOnS,
the FORS or the ISIS instrument in various works from 2004 to 2018; another
dozen are DC WDs that were observed with high-precision broadband polarimetric
techniques in the seventies. Only three stars could not be checked for
magnetic field for the reasons explained in Sect. 5.1. Their positions in the
histogram are shown by unshaded areas. We note that in Fig. 1, each of the six
uDD systems of the local 20 pc volume were counted as two individual stars,
even if they were observed with just one pointing (but of course it is not
trivial to estimate field values or upper limits to the individual system
members).

###  4.1 ESPaDOnS data

With ESPaDOnS we obtained 27 new observations of 24 WDs (although one star, WD
1334$+$039, turned to be featureless, and no useful field measurements could
be made, because ESPaDOnS lacks capabilities in the continuum). ESPaDOnS data
cover a useful spectral range of $\sim 3800-8900$ Å, with a spectral resolving
power of 65 000 (see Landstreet et al., 2015, for details of the use of this
instrument for WD observations). ESPaDOnS is a high-resolution
spectropolarimeter on a relatively modest telescope of 3.6-m aperture, and
lacks the ability to measure continuum polarisation, and because of this it
was used to observe only WDs brighter than $V=15$, and mainly WDs with strong
spectral features.

###  4.2 FORS2 data

We present 58 new observations of 43 WDs obtained with the FORS2 instrument
using different grisms. To observe DC stars we mainly used grism 300 V with no
order separating filter, covering the spectral range $\sim 3700-9300$ Å; we
set the slit width to 1.2″ for a spectral resolving power of 360 (for a
justification of the choice of not using the order separating filter see App.
A of Patat et al., 2010). For DQ stars we used grism 600 B, covering the
spectral range $\sim 3600-6200$ Å; slit width was set to 1.0″ for a spectral
resolution $R\sim 780$. For DA stars we mainly used grism 1200 R $+$ filter
GG435, covering the spectral range $\sim 5800-7300$ Å; with a slit width of 1
Å we obtained a spectral resolving power of 2140. For DZ stars and some DA
stars we used the grism 1200B, which covers the spectral range $3700-5100$ Å.
Slit width was set to 1.0″, for a spectral resolving power of 1400. We also
report two $\langle B_{z}\rangle$ measurements of one star, WD 0233$-$242,
that were obtained with FORS2 in 2013, but from which previous literature had
reported only the $\langle|B|\rangle$ value.

###  4.3 ISIS data

With the ISIS instrument we obtained 33 new observations of 30 WDs, using
(simultaneously) the grating R600B in the blue arm and grating R1200R in the
red arm, covering the spectral ranges of $\sim 3700-5200$ Å with a spectral
resolving power of 2600, and $\sim 6100-6900$ Å with a spectral resolving
power of 8600, respectively (for a 1″ slit width). We note that for some of
the observations we erroneously adopted a 2x2 binning readout, which led to a
slight undersampling, effectively reducing the spectral resolution in the red
arm by $\sim 30$ %.

###  4.4 Data reduction and field measurements

Data reduction for FORS and ISIS data is fully described in Bagnulo &
Landstreet (2018) and references therein. The methods used to measure the
longitudinal field from polarised spectral lines are summarised in Bagnulo &
Landstreet (2018) and Landstreet & Bagnulo (2019a). The method used to carry
out mean longitudinal field measurements with ESPaDOnS is explained by
Landstreet et al. (2015); Landstreet et al. (2017). Because of its high
resolving power, ESPaDOnS spectra of adequate S/N can be used to measure
$\langle|B|\rangle$ in MWDs with values above about 50 kG (Landstreet et al.,
2017). In case of stars where the magnetic field has to be estimated from a
weak level of polarisation in the continuum, we use the relationship

$\frac{V}{I}=\gamma\langle B_{z}\rangle$ (1)

where we adopt $\gamma=15$ MG per percent of polarisation, as discussed by
Bagnulo & Landstreet (2020). The polarisation in the continuum was estimated
from inspection of heavily rebinned spectra (up to $\sim 400$ Å spectral
bins).

Bagnulo & Landstreet (2020) have shown that in FORS2 data, the background
illuminated by the moon may appear circularly polarised because of cross-talk
from linear to circular polarisation. This cross-talk varies rapidly with the
position in the field of view, hence background subtraction may produce
artefacts in the form of a spurious signal of linear polarisation, stronger in
the blue and decreasing with wavelength, being generally little noticeable at
$\lambda\geq 7000$ Å. The effect depends on the degree of lunar illumination,
and is more prominent in fainter than in brighter stars, but normally will be
more noticeable on stars fainter than $V\simeq 15.5$ when not observed in dark
time. We have found that spurious polarisation is minimised when background is
estimated from the same Wollaston strip where the source spectrum is recorded,
which is possible to do when observations are taken under seeing conditions
better than $\sim 2\arcsec$.

The observing log and derived magnetic field values are given in Table
LABEL:Table_Log. In App. A we discuss individual observations of WDs whenever
results may be ambiguous, which happens in particular for the WDs of spectral
class DC for the reasons explained above.

##  5 Merging earlier with new field measurements: the magnetic fields of the
local population of WDs

Figure 2:  Top panels: Histogram of WDs of the local 20 pc volume. The red
stripes refers to all WDs, the blue stripes to all MWDs, black stripes to all
MWDs with $\langle|B|\rangle\geq 2$ MG. Bottom panels: Full symbols: average
$\langle|B|\rangle$ values versus temperature, cooling age $\tau$, and mass of
the MWDs. Empty symbols: sensitivity $s_{\langle|B|\rangle}$ of the field
measurements on stars in which a magnetic field was not detected, calculated
as explained in the text (Sect. 5.3). Circles refers to non metal-polluted
WDs: blue circles to WDs with an H-rich atmosphere (DA and some DC stars), and
red circles to WDs with a He-rich atmosphere (DQ and some DC stars). Triangles
refers to WDs with metal lines: blue triangles refer to stars with H-rich
atmospheres (DAZ stars) and red triangles to stars with He-rich atmosphere (DZ
and DZA stars). Crosses over symbols refer to DQ and DQpec WDs.

In the 20 pc volume there are 146 systems: 140 are presumably single WDs, or
in visually resolved systems, and six are uDD, for a total number of 152 of
currently known WDs. Only five WDs have not been observed in polarimetric
mode:

i) WD 0121$-$429B, a presumed DC member of a uDD system that, being
featureless, may be checked for magnetic field only with polarimetric
techniques; its companion WD 0121$-$429A was discovered to be a MWD via
spectroscopy.

ii) WD 0208$-$510 (= HD 13445B, spectral class DQB), which is too close to a
bright companion to be observed in polarimetric mode with ground-based
facilities. Its spectrum (observed with HST by Farihi et al., 2013) shows only
C2 absorption bands that are too broad to give any useful constraint on
$\langle|B|\rangle$.

iii) WD 0211$-$340, which cannot be currently observed because it is too close
to a background star (see Sect. 2). In fact, even its spectral type is
unknown.

iv) WD 0642$-$166 (= Sirius B), a WD of spectral class DA, is too close to its
companion to be observed with polarimetric techniques from ground-based
facilities. HST spectroscopy allows us to set a constraint on its magnetic
field ($\langle|B|\rangle\la 100$ kG), but it should be recalled that some DAs
have fields so weak that they may be detected only via spectropolarimetry.

v) WD 0736$+$053 (= Procyon B, spectral type DQZ) is in a binary system
somehow similar to that of WD 0208$-$510\. Mg i and Mg ii lines around 2800 Å
(observed with HST by Provencal et al., 2002) are not split by Zeeman effect,
setting a limit of 300 kG for $\langle|B|\rangle$.

The first three stars of this list should be considered as non observed and
will be ignored for statistical purposes, while we will keep in mind for that
for the magnetic field of Sirius B and Procyon B there are at least some
constraints from spectroscopy.

Altogether, combining data from the literature as described above with our new
observations, we have useful field measurements or upper limits for 149 WDs of
the approximately 152 WDs of the 20 pc volume. The presence of a magnetic
field is firmly established in 33 of these stars. The remaining non magnetic
(or presumed non magnetic) 116 WDs of the local 20 pc volume were observed
with spectropolarimetric techniques, with a sensitivity that varies greatly
with the object spectral type and temperature.

###  5.1 The accuracy of the observations of WDs in which magnetic field was
not detected

For the purpose of this work, together with the identification of the MWDs, it
is equally important to assess how reliably we can declare that among the
remaining 116 WDs, no magnetic fields are present, or at least, no magnetic
field detectable through the currently available techniques. Among these 116
WDs, 114 have been checked for the presence of a magnetic field with
polarimetric techniques, with a sensitivity that varies from a one or few kG
for the brightest DA WDs, up to $0.5-2.0$ MG for the featureless DC WDs.
Obviously, observations of uDDs gives looser constraints to the magnetic field
of individual members. DQ stars should be practically considered as
featureless stars, in that only continuum polarisation may firmly reveal the
presence of a magnetic field (Berdyugina et al., 2007).

The presence of metal lines allows us to measure the magnetic field in many of
the cool, and otherwise featureless stars, with a precision that depends on
the number and strength of metal lines. For the DZ targets of our survey, the
precision varies between $\simeq 0.4$ and 8 kG, except for one DZ star (WD
0552$-$106) for which the uncertainty of its non detection was $\sim 40$ kG,
and another star (WD 1743$-$545) that, because of the weakness of its metal
lines, can be checked for magnetic field only through the measurement of their
intensity profiles ($\langle|B|\rangle\la 1$ MG).

There are four DA stars that are so cool that spectropolarimetry of H$\alpha$
does not provide any useful constraint (at least at the S/N levels that were
obtained), therefore the constraint on their field is still given either by
the lack of observed Zeeman splitting in their extremely weak H$\alpha$ and/or
by the lack of polarisation in the continuum, for a precision that varies
between 0.5 and 1.5 MG. This is the case of WD 1345$+$238, for which we can
set $|\langle B_{z}\rangle|\la 1.5$ MG, WD 0727$+$482A for which at best we
can say that $\langle|B|\rangle\la 1$ MG, and WD 0751$-$252 and WD 1823$+$116,
for which we can set an upper limit of 0.5 MG to $\langle|B|\rangle$. We also
note that the fainter component of the uDD system WD 0727$+$482B could also be
a DA star, but its characterisation is uncertain and it will not be included
as member of the DA or the DC class in the next sections. WD 1820$+$609 is a
fifth case of a DA star with a very weak H$\alpha$ in which spectropolarimetry
is not very useful. However, the presence of a very narrow core allows us to
set an upper limit for $\langle|B|\rangle$ at about 50 kG. One DA was observed
with $\sigma_{z}\simeq 35$ kG, and five DA/DAZ WDs with $\sigma_{z}=10-15$ kG
(all these DAs have $T_{\rm eff}\la 5000$ K except WD 0728$+$642). About 20
DA/DAZs have $\sigma_{z}$ between 1 and 10 KG (but mostly $\simeq 2$ kG), and
all the remaining non-magnetic DA and DAZ WDs, about 40 stars, were observed
with sub-kG precision.

In App. B we comment on individual stars, grouped by spectral class, and we
justify why each star may or may not be considered magnetic, or suspected
magnetic. Whenever necessary, we make additional comments, for instance,
regarding the criteria that we have adopted to estimate of stellar parameters
of individual components of the uDD systems. Appendix B contains a detailed
compilation of the magnetic observations of WDs which may be used to guide
further spectropolarimetric observations of the local 20 pc population of WDs,
but is not needed to follow the rest of this work.

###  5.2 Four WDs that were erroneously considered magnetic in previous
literature

It is important to report that in the local 20 pc volume, four WDs were
erroneously identified as MWDs in the previous literature. Rectifying this
situation is essential in order to correctly estimate the true incidence of
magnetic fields as a function of stellar age, especially because three of them
are among the youngest WDs of our sample.

i) Based on a spectropolarimetric measurement obtained at the Mount Stromlo
Observatory ($\langle B_{z}\rangle=-6.1\pm 2.2$ kG), star WD 0310$-$688 was
identified as a suspected magnetic by Kawka et al. (2007). However, the same
star was observed with much higher accuracy with FORS1 by Aznar Cuadrado et
al. (2004), who measured $\langle B_{z}\rangle=-0.10\pm 0.44$ kG, and with
FORS2 by Bagnulo & Landstreet (2018), who found $\langle B_{z}\rangle=-0.20\pm
0.23$ kG. We consider this star to be non magnetic.

ii) 40 Eri B = WD 0413$-$077 was long considered a weakly magnetic WD (Fabrika
et al., 2003; Valyavin et al., 2003; Ferrario et al., 2015). However,
Landstreet et al. (2015) made a number of extremely precise measurements of
the longitudinal field using both the ESPaDOnS instrument of the CHFT and the
ISIS instrument at the WHT. Although most of their measurements had
uncertainty as low as 85–90 G, no magnetic field was detected. The conclusion
of Landstreet et al. (2015) was that field detections previously reported in
the literature were spurious, and the star is actually non-magnetic. In this
paper we have presented an additional measurement obtained with ISIS, that has
led again to a null detection.

iii) Holberg et al. (2016) erroneously list the 0.4 Gyr old WD 2326$+$049 as
magnetic, citing Aznar Cuadrado et al. (2004) as the source of the
measurement. In fact, this star was not observed by Aznar Cuadrado et al.
(2004), while the FORS2 measurements by Farihi et al. (2018) demonstrated that
the star is not magnetic.

iv) Liebert & Stockman (1980) observed the cool DA star WD 1820+609 (at that
time believed to be DC), and found a signal of broadband polarisation
consistent with zero. Putney (1997) re-observed the same star in
spectropolarimetric mode and found non-zero continuum polarisation in the
instrument red arm ($\simeq-0.5$ %, judging from her Fig. 2i, and reporting
this value of the polarisation with our sign convention). Putney (1997)
declared this signal as possibly spurious, but stated that H$\alpha$ revealed
a weak field, and that the star needed to be re-observed. As pointed out by
Landstreet et al. (2016), in her Table 1, Putney (1997) erroneously reported
the measurement by Liebert & Stockman (1980) with a ten times smaller error
bars, making that measurement appear as a $6\,\sigma$ detection; this error
propagated in the review paper by Ferrario et al. (2015), and the star was
considered as a magnetic one. Our ISIS measurements have a low $S/N$, and does
not improve on the upper limit of $|\langle B_{z}\rangle|$ set by Liebert &
Stockman (1980), but H$\alpha$ spectroscopy set for $\langle|B|\rangle$ the
upper limit of 50 kG. We conclude that the star should be considered as non
magnetic.

###  5.3 The WDs of the local 20 pc volume: stellar parameters and magnetic
field

The overall situation is fully summarised in Table LABEL:Tab_Stars, a list of
physical parameters of WDs in the local 20 pc volume. This list is organised
as follows. Column 1 gives the name of the star (if the star is magnetic, its
name is printed in boldface); col. 2 the $V$ magnitude generally taken from
the SIMBAD database; when this was not available (for instance, for the newly
discovered WDs) we reported the Gaia $G$ magnitude. We note that in the
context of this work, we do not need an accurate estimate of the star’s
apparent magnitude. Colums 3 reports the distance, generally as derived from
the parallax of Gaia DR2. Column 4 shows the spectral type – for MWDs we have
added the symbol "H". Traditionally, a MWD would be designated with "H" if the
field detection was made with spectroscopic techniques, and with "P" if the
detection was made with polarimetric techniques. As already discussed by
Bagnulo & Landstreet (2020), this distinction simply refers to the detection
method, and does not point to a physical stellar feature. In fact, many MWDs
can be identified as such both because they show line splitting and because
line components are polarised. Therefore we prefer to adopt the suffix H for
all MWDs, regardless the actual observing technique that was employed for
their discovery. A small number of WDs should be considered as suspected MWDs
and their cases are discussed in detail in App. B. Suspected MWDs are labelled
in col. 4 with a question mark (’H?’). Column 5 says whether the star is in a
binary system, either as a visual binary (VB) or visual multiple system (VM),
as a resolved double degenerate system (DD) or unresolved double degenerate
system (uDD). The symbol s designate stars for which there is no significant
evidence of binarity. This classification is largely based on the Tables
compiled by Toonen et al. (2017), with additional inputs as detailed in Sect.
2.2.2.

For the determination of the stellar parameters such as atmospheric
composition (col. 6), temperature (col. 7), log g (col. 8), mass (col. 9) and
age (col. 10) we have used the references given in col. 11 (see also Sect. 2).
Column 12 gives an estimate of the average $\langle|B|\rangle$ of the MWDs.
For stars in which $\langle|B|\rangle$ was estimated through the analysis of
the Zeeman splitting observed in intensity, it is possible to measure the
average field strength at the surface of the star. For rotating WDs,
$\langle|B|\rangle$ changes as the star rotates, but usually not by very much,
so that even a single $\langle|B|\rangle$ measurement allows us a good
estimate of the average field strength of a star. For weaker field MWDs,
however, $\langle|B|\rangle$ cannot be measured directly, and what we know
with good accuracy is the average longitudinal component of the magnetic field
$\langle B_{z}\rangle$, at one or several rotational phases. In these cases,
the estimate of the average field modulus at the surface of the star relies on
modelling results, or educated guesses. Again, we refer to App. B for the
discussion of the details of individual stars.

For stars in which a magnetic field was not detected, in the second last
column we have given an indication of the sensitivity of the field
measurements, noting either the uncertainty of the best $\langle B_{z}\rangle$
measurement and the number of measurements available, or the upper limit for
$|\langle B_{z}\rangle|$ as deduced from polarimetry of the continuum (for
featureless stars), or the upper limit for $\langle|B|\rangle$ from
spectroscopy, for the few cases in which polarimetric measurements are not
available. From these information, it is useful to estimate an upper limit for
the surface field $\langle|B|\rangle$. When only spectroscopic data are
available, as in the case of fours WDs only, this estimate is straightforward
and provided directly in col. 13. For the large majority of stars, a number of
$\langle B_{z}\rangle$ measurements are available; these measurements set a
stronger constraint than spectroscopy alone, but estimating an upper limit for
$\langle|B|\rangle$ is modelling-dependant. Bayesian statistics may allow one
to estimate upper limits for individual stars based on the available
measurements. However, as crude estimate we decided to define as field
sensitivity $s_{\langle|B|\rangle}$ the lowest value of $\sigma_{z}$
multiplied by four, that is, $s_{\langle|B|\rangle}=4\,{\rm MIN}(\sigma_{z})$.
For example, for star WD 1202$-$232, the upper limit for $\langle|B|\rangle$
from direct spectroscopic measurement of its H$\alpha$ is 50 kG, but there are
also four $\langle B_{z}\rangle$ measurements available, the most precise of
which has $\sigma_{z}\simeq 0.4$ kG. We say that this star has
$s_{\langle|B|\rangle}=1.6$ kG. Under the approximation that the field has a
dipolar structure, $s_{\langle|B|\rangle}$ may be interpreted as an
approximate estimate of the upper limit for $\langle|B|\rangle$ (this
interpretation will be discussed again in Sect. 6.3.3). In the majority of
cases of Table LABEL:Tab_Stars, this estimate is between 0.5 and 2 times the
upper limit of $\langle|B|\rangle$ found via a more rigorous Bayesian
approach, but there exist outlier cases, as the suspected but not confirmed
MWDs, for which our approach definitely underestimates the upper limit for
$\langle|B|\rangle$. Finally, for featureless stars, for which only
polarisation of the continuum is available, $s_{\langle|B|\rangle}$, expressed
in MG, has been set equal to 15 times the upper limit of $|V/I|$, expressed in
percent units.

Figure 3:  Left panel: age-apparent magnitude scatter plot of the local 20 pc
volume, showing all WDs cooler than 10 000 K (empty blue circles), all WDs
hotter than 10 000 K (larger empty red circles) and all MWDs (filled blue
circles). The apparent $V$ or $G$ magnitude of the $y$-axis is the same as
reported in Table LABEL:Tab_Stars (see Sect. 5.3 for more details). Right
panel: scatter plot mass-magnitude for the same sample and with the same
meaning of the symbols as in the upper panel. The horizontal dotted lines help
to visualise what the impact of a target selection based on a magnitude limit
could have on the sampling of stars with different parameters.

The last column of Table LABEL:Tab_Stars contains a reference to the
literature reporting field measurements for the star. The references to older
broadband polarimetric measurements of non-featureless WDs, superseded by
modern spectropolarimetric data, are given between parenthesis. More details
about individual stars are given in App. B.

A general overview of the statistics that can be deduced from Table
LABEL:Tab_Stars is shown in Fig. 2. The top panels show the distributions of
magnetic and non magnetic WDs with temperature, age and mass. The bottom
panels help to visualise the field strength distribution, but also show how
stellar parameters affect the sensitivity of the field measurements, in
particular that the detection threshold increases with age. In young WDs,
fields may be detected with a sensitivity of the order of a few kG, while only
fields with MG strength can be detected in most older stars. In contrast,
there is no obvious bias against detecting fields as a function of stellar
masses. Regarding temperature, field measurement sensitivity varies quite non-
linearly with $T_{\rm eff}$. For DA stars, for example, sensitivity is
affected by the depth and width of the Balmer lines, and becomes poor both at
high temperature (above 20–30000 K) as H ionises and the lines weaken, and
below about 7000 K as the lines weaken because the population of the $n=2$
levels of neutral H diminish.

It is particularly interesting to relate the physical parameters of the local
population of WDs with the scope of a magnitude limited survey. The top panel
of Fig. 3 shows that if we were to observe in the local 20 pc volume only WDs
brighter than apparent magnitude $V=14$, apart from sampling a smaller number
stars, we would miss virtually all WDs older than 3 Gyr. If in addition we
were also setting colour limits to our survey, focusing our attention for
instance to stars hotter than 10 000 K, we would look only at the very
youngest stars of the local population. No matter how deep a survey could go,
if the sample is magnitude limited, we will look at a population of WDs that
is not representative of the local situation, and all but the youngest WDs
will be vastly underrepresented. However, Fig. 3 offers no evidence that a
magnitude-limited survey of the local 20 pc volume would be biased against
higher-mass WDs. This will be discussed again in Sect. 6.4.

##  6 Analysis

Having established that a magnetic field is present in 33 out of 149 observed
WDs, we conclude that the overall frequency of MWDs in our volume-limited,
nearly complete sample is 22 %. Extrapolated to the Galactic sample, the
fraction of magnetism is $22\pm 4$ %, although we should bear in mind that we
know that weaker fields of stars with featureless spectra, if present, have
escaped detection.

In this section we analyse whether there are hints that magnetic fields are
more frequent or stronger in WDs of specific spectral classes (Sect. 6.1) or
chemical composition of the atmosphere (Sect. 6.2). We also study the
distribution of the field strength (Sect. 6.3), in particular whether the
fields are found preferably in a certain range of strength. We finally explore
whether there are correlations of field frequency with age (Sect. 6.4) or
stellar mass (Sect. 6.5).

###  6.1 The fraction of MWDs among WDs of different spectral classes

Figure 4:  Probability density function of the Galactic frequency of the
magnetic field in WDs of difference spectral classes. Table 1:  Summary of
basic data about the incidence of magnetic fields in the local population of
WDs. The frequency $f=M/N$ of col. 4 is the ratio between confirmed MWDs (col.
2) and all WDs (col. 3) for the classes of stars of col. 1; column 5 is the
corresponding expected value of $r$, $(M+1)/(N+2)$.

Spectral type | $M$ | $N$ | $f$ | $\langle rP_{r}\rangle$ | Notes
---|---|---|---|---|---
DA | 18 | 77 | 23.4$\pm$ | 4.8 % | 24.1 % | 1
DAZ | 2 | 12 | 16.7$\pm$ | 10.8 % | 21.4 % |
DA+DAZ | 20 | 89 | 22.5$\pm$ | 4.4 % | 23.1 % |
DC | 4 | 30 | 13.3$\pm$ | 6.2 % | 15.6 % |
DZ+DZA | 4 | 12 | 33.3$\pm$ | 13.6 % | 35.7 % | 2
DQ | 5 | 16 | 31.2$\pm$ | 11.6 % | 33.3 % |
Observed WDs | 33 | 149 | 22.1$\pm$ | 3.4 % | 22.5 % | 3

1 If we discard from the statistics four extremely cool DA stars for which
field measurements have very low sensitivity ($\simeq 0.5$ MG), $f\simeq 25$
%.
2 Non magnetic star WD 1743$-$545 = PM J17476$-$5436 has extremely weak
H$\alpha$ and metal lines, and the sensitivity of our field measurement is not
better what could be obtained in a DC star; therefore this star is not
considered in the statistics of DZ (nor DC) WDs.
3 Star WD 0727+482B = G 107-70B belongs to a uDD system and we were not able
to assess its spectral classification, hence it is considered only in the
global statistics.

We first discuss magnetism in WDs of different spectral classes. The general
statistical results are summarised in Table 1, while Fig. 4 shows the
probability density functions $P_{r}$ such that $P_{r}\,{\rm d}r$ is the
probability that the frequency of galactic MWDs is comprised between $r$ and
$r+{\rm d}r$.

####  6.1.1 The fraction of MWDs among DA and DAZ stars

A slight majority of WDs have spectra that show only H lines and that are
classified as DAs. There are 77 DA stars in Table LABEL:Tab_Stars. DAZ stars,
which are DA stars with a spectrum polluted by the presence of metal lines,
are treated separately below. Eighteen DAs have a confirmed magnetic field,
for a frequency of $23\pm 5$ %. We note, however, that four DA stars in which
no magnetic field was detected have in fact such weak H$\alpha$ lines that
constraints on their field measurement come only from continuum polarimetry
($s_{\langle|B|\rangle}=0.5-1.5$ MG, see Sect. 5.1). If we discard these stars
from our statistics, on the ground that their field sensitivity is much lower
than typical for this class of stars, the frequency of the occurrence of
magnetic fields in DA WDs would be $\simeq 25$ %. Field strength in magnetic
DA WDs spans the entire observed range in WDs, from kG level (for instance, WD
2150$-$820, Landstreet & Bagnulo, 2019a) to the $\sim 200$ MG (Jordan, 2003)
level of Grw +70∘ 8247.

About 25 % of DA WDs studied by Zuckerman et al. (2003) were found to have a
metal-polluted atmosphere, where the metals are believed to be accreted from
the debris of disintegrating members of a former planetary system. In the
local 20 pc volume, the ratio between DAZ and DA+DAZ stars is about 12 % (to
discuss the incidence of metal pollution in WDs it may be more interesting to
consider the ratio between all metal polluted WDs, including therefore not
only DAZ but also DZ and DZA WDs, and all WDs; in the local 20 pc volume this
ratio is about 10 %). Two of the 12 DAZ WDs of the local 20 pc volume are
magnetic, for a frequency of $17\pm 11$ %. The incidence of the magnetic
fields in DA and DAZ taken together is $22\pm 4$ %.

Kawka et al. (2019) analysed a sample of 15 DAZ stars with $T_{\rm eff}$
between $\simeq 5200$ and 7000 K, and noted that five of them are magnetic
(see their Table 5 – only one star is in common with our sample), for a
frequency of $33\pm 12$ %. However, when they restrict their analysis to DAZ
WDS cooler than $T_{\rm eff}$= 6000 K, Kawka et al. (2019) find that four out
of seven DAZ WDs are magnetic, and suggest that about 50 % of DAZ WDs cooler
than $T_{\rm eff}$= 6000 K are magnetic (the frequency is $50\pm 18$ %). If we
consider only the cooler DAZ WDs of the local 20 pc volume and merge them into
the sample studied by Kawka et al. (2019), we find that out of 21 DAZ cooler
than 7000 K, five are magnetic, for a frequency of $24\pm 9$ %, fully
consistent with the frequency estimated for normal DA WDs of all ages, while
four out of 12 DAZ cooler than 6000 K are magnetic, for a frequency of $33\pm
14$ %. The analysis of this extended sample does not offer statistical support
to the claim that magnetic fields are more frequent in cool DAZ than in normal
DA WDs: 11 out of 31 DA WDs with $5200\leq T_{\rm eff}\leq 7000$ K are
magnetic, and five out of 18 DA WDs with $5200\leq T_{\rm eff}\leq 6000$ K are
magnetic, for a magnetic frequency of $35\,\pm 9$ % and $28\,\pm 11$ %,
respectively.

####  6.1.2 DB stars

There are no DB stars (WDs with He-lines in the optical spectrum) in the local
20 pc volume, the closest to this class is a DBQA star (WD 1917$-$077) which
will be listed among the DQ stars.

####  6.1.3 The fraction of MWDs among DC stars

There are 31 known DC WDs in the local 20 pc volume; their magnetic fields may
be revealed only by broadband- or spectro-polarimetry. Before our survey, only
10 DC WDs of the local 20 pc volume volume had been observed in polarimetric
mode, and only one of them had been found magnetic. In the course of this
survey we have observed for the first time in polarimetric mode 20 DC WDs of
the local 20 pc volume, and re-observed one that had been already observed in
broadband imaging mode in a previous study. Only one of the DC stars of the
local 20 pc volume (WD 0121$-$429B, a member of a uDD system) remains not
observed in polarimetric mode. Our survey discovered three new DC MWDs
(Bagnulo & Landstreet, 2020). Among the observations newly published in this
paper, we report that WD 1116$-$470 is a suspected DC MWD, that could host a
field of the order of a few MG, and should be re-observed. The magnetic
frequency among DC WDs is $\simeq 13$ %. We note that the sensitivity of most
of the new measurements is $\simeq 5$ MG (in fact, the weakest field firmly
detected has a strength estimated to be about 20 MG); this situation could
certainly be improved even with current instrumentation, potentially leading
to the discovery of further magnetic fields in the DC WDs of the local 20 pc
volume.

More than a half of the magnetic DA and DAZ have a magnetic field that is not
strong enough to be detected with polarimetric measurements of the continuum,
at least with the sensitivity typical of the current measurements. If the
distribution of field strength of DC stars is similar to that observed in DA
WDs, one could hypothesize that the actual frequency of magnetic fields in the
DC WDs of the local 20 pc volume is between 27 and 33 %, for an estimated
Galactic frequency of, say, $30\pm 10$ %.

####  6.1.4 The fraction of MWDs among DZ and DZA stars

The origin of the metal lines of DZ and DZA WDs is explained in the same terms
as for DAZ WDs. All 13 known DZ and DZA WDs of the local 20 pc volume have
been observed for the first time in polarimetric mode in the course of our
survey of the local 20 pc volume, except for WD 2138$-$332 observed by Bagnulo
& Landstreet (2018), and reported as suspected magnetic, and the DZA star WD
0738$-$172, reported as suspected magnetic by Friedrich et al. (2004).

Bagnulo & Landstreet (2019b) reported the preliminary results of our survey,
confirming the magnetic nature of WD 2138$-$332, and reporting the discovery
of three additional magnetic DZs. At that time, we were also led to suggest
that the occurrence of magnetic fields in DZ stars could be particularly
frequent, having reached the conclusion that at least 40 % of DZ WDs of the
local 20 pc volume are (weakly) magnetic. However our subsequent field
measurements (presented in this paper) failed to discover any new magnetic DZ
WDs, and did not confirm the field detection by Friedrich et al. (2004) on WD
0738$-$172, a star which we now consider as an only (marginally?) suspected
MWD. Since the work by Bagnulo & Landstreet (2019b), two new DZ stars have
been identified in the local 20 pc volume, WD 0552$-$106 and WD 1743$-$545
(that were previously classified DC), and we found that neither of them is
magnetic. We note, however, the metal lines of WD 1743$-$545 are so weak that
the sensitivity of our field measurement is only at the MG level, hence this
star should not be included in the statistics. In conclusion, our new estimate
of the frequency of the occurrence of magnetic fields in the DZ stars of the
local 20 pc volume is 4/12, a fraction that is admittedly lower than the lower
limit that we had previously estimated, for a Galactic frequency of $33\pm 14$
%. This range is formally consistent with that of DA WDs, and fully consistent
with that of DA WDs within a similar age range as that of DZ and DZAs.

Metal lines allow us to detect weak magnetic fields that would be undetected
in stars that without pollution from disk-debris would have featureless
spectra. The high frequency of field detection in DZ stars is consistent with
the hypothesis that DC stars are actually magnetic at least as frequently as
DA WDs, except that the magnetic fields of DC stars are revealed only when
their strength is of the order of several MG.

####  6.1.5 The fraction of MWDs among DQ stars

DQ stars are WDs with spectroscopic traces of carbon (neutral carbon lines or
molecular C2 Swan bands). Most of them, at least below effective temperatures
of about 10000 K, have helium-dominated atmospheres (Koester et al., 2020),
and the presence of carbon is a result of dredge-up by the extending
convection zone in the upper helium layer (Koester et al., 1982; Pelletier et
al., 1986). The 17 DQ stars of Table LABEL:Tab_Stars represent more than 10 %
of the local 20 pc WD population, a proportion much bigger than the fraction
of 1.6 % of DQs among all WDs classified by the SDSS (Koester & Kepler, 2019).
This large discrepancy may be explained at least in part as a selection effect
of the SDSS. Cool WDs are probably underrepresented in the SDSS, and it is
also probable that many DQs with weak bands are mistaken for DC stars, because
many SDSS spectra have low S/N.

Five of the DQ WDs of the local 20 pc volume are confirmed magnetic, although
only 15 out of 17 could be observed in polarimetric mode. The presence of a K0
companion at 1.9$"$ prevented us from observing WD 0208$-$510 = HD 13445B with
FORS2. WD 0736$+$053 = Procyon B has a very bright F0 companion at 5.3$"$, and
could only be observed spectroscopically by Provencal et al. (2002) with the
HST. However, from these spectra we can still derive an useful upper limit for
$\langle|B|\rangle$ ($\simeq 300$ kG). All DQs in the local 20 pc volume are
cooler than 10,000 K, except for the DBQA WD 1917$-$077.

Two out of three peculiar DQ WDs (DQ stars with Swan band-like molecular band
depressions that are not at the expected wavelengths for C2 molecules, see
Schmidt et al., 1995) are magnetic. The ratio of five MWDs out of 16 DQ WDs
results in a magnetic frequency of $31\pm 12$ %, which is higher than, but
still consistent with that of DA WDs, but with an important difference: the
typical field strength that can be detected in DQ stars is much higher than in
DA WDs. Similarly to the case of DC stars, the magnetic frequency could well
be underestimated by a factor of two because of the insensitivity to fields
below MG level, in which case we could hypothesise that 2/3 of DQ stars are
magnetic. An alternative possibility is that the frequency of the occurrence
of magnetic fields in DQ stars is similar to that of DA WDs, but DQ MWDs host
much stronger fields than average (in fact, DQ stars are the WDs of the local
population that host the strongest fields). In either case, a magnetic field
and the presence of C2 in the stellar atmosphere seem correlated, and one
could speculate that a magnetic field is responsible for C2 buoyancy.

No WDs belonging to the class of rare hot DQs, WDs with a C-rich atmosphere
and little or no trace of H or He (Dufour et al., 2007), and with $T_{\rm
eff}=18000-24000$ K (Dufour et al., 2008), are present in the local 20 pc
volume, and we are not able to discuss previous claims that magnetic fields
may be nearly ubiquitous in this class of stars (Dufour et al., 2013; Dunlap,
2014).

###  6.2 The frequency of the occurrence of magnetic field versus stellar
atmospheric composition

Twenty out of 103 WDs with an H-rich atmosphere are magnetic, for a frequency
of $19\pm 4$ %; and 13 out of 43 WDs with a He-rich atmosphere are magnetic,
for a frequency of $30\pm 7$ %. Except for the four magnetic DZ stars
discussed by Bagnulo & Landstreet (2019b), all He-rich MWDs have a field
strength from 1 to hundreds MG. This field strength distribution is biased by
the fact that the local 20 pc volume does not include DB WDs, the only He-rich
stars, apart from DZ stars, in which fields weaker than 1 MG may be detected.
There are no He-rich WDs younger than 0.5 Gyr in the local 20 pc volume, which
may represents another bias for the statistics, if the frequency of field
occurrence changes with cooling age (see Sect. 6.4). Five out of eight He-rich
WDs older than 5 Gyr are magnetic, for a frequency of $62\pm 17$ %. Such a
high frequency among the oldest stars is not mirrored by H-rich WDs, as only
four out of 25 H-rich WDs older than 5 Gyr are magnetic, for a frequency of
$16\pm 7$ %. However, a proper comparison between the magnetic properties of
H-rich and He-rich WDs requires to go through a careful revision of the
chemical composition of the atmosphere of the coolest stars of the sample,
which is not so secure as for the hottest ones.

###  6.3 Distribution of magnetic field strength over the sample

Figure 5:  Blue solid line: the ratio $F$ between the number of MWDs with
${\langle|B|\rangle}$ smaller than the abscissa value $B$ and the number of
all MWDs; the dashed lines show the $\pm 1\,\sigma$ uncertainty. Blue dotted
line: the expected behaviour of a field distribution constant per decades. The
red solid line shows the ratio $S$ between the number of field measurements
with sensitivity $s_{\langle|B|\rangle}$ smaller than the abscissa value $B$
and the total number of observed WDs. Figure 6:  The red strips shows the
distribution of the sensitivity of the measurements of the magnetic field
(practically four times the uncertainty of the best $\langle B_{z}\rangle$
measurement of each star, see Sect. 5.3) and the blue strips shows the
distribution of the observed magnetic fields strengths.

Ferrario et al. (2015) have proposed that the distribution of field strength
of MWDs is peaked between 2 and 80 MG, a picture that is in conflict with the
finding by Kawka et al. (2007), who carried out a Kolmogorov-Smirnov (K-S)
test for the WDs within 13 pc, and showed that the field strength is constant
for each decade interval with a probability of 70 %. In this section we
present the results obtained from the analysis of the local 20 pc volume.

####  6.3.1 The distribution of the field strength per decades is rather
homogeneous

Compared to previous studies, we are now able to extend the analysis to a
volume more than twice as large, in which we have probed the magnetic field of
many DA WDs with sensitivity up to 1-dex better than in the past, and have
also searched for the presence of MG magnetic fields in the entire population
of DC WDs. Figure 5 shows the cumulative distribution function for the
magnetic field strength. This figure clearly demonstrates that within the
range of field strength found in the 20 pc volume, which extends between about
40 kG and 300 MG, the probability of fields occurring is roughly constant per
dex of field strength. This same behaviour is observed in Fig. 6 (which is
effectively the marginal distribution of field strengths plotted in the lower
panels of Fig 2) as well as the distribution of best measurement uncertainties
for undetected stellar fields.

It is important to emphasise that this distribution is very different from the
apparent distribution of field strengths as assembled from a census of all
known MWDs, however discovered (e.g Ferrario et al., 2015, Fig. 8). The
ensemble of currently known MWDs is dominated by discoveries made from low
resolution, low S/N optical and near IR flux spectra obtained for the SDSS
project (e.g. Schmidt et al., 2003; Kepler et al., 2013, 2016). Such spectra
are most sensitive to magnetic fields in the range of about 2–80 MG, and the
observed distribution has an excess frequency in this range about five times
higher than weaker and stronger fields outside of this range. By contrast,
Figures 5 and 6 show that $\sim 40$ % of MWDs have fields weaker than 2 MG,
and that MWDs with fields in the regime between 2 and 80 MG account for
another 40 % of the total. Figures 5 shows also that the total of fields
weaker than 500 kG contribute roughly to one third of the total observed
fields in the 20 pc volume. These weaker fields are often detected only
through highly-sensitive spectropolarimetric techniques, and strongly magnetic
DC and DQs may be detected only through polarimetry, no matter how high is the
S/N and the resolution of traditional spectroscopy. These considerations
clearly highlight how spectroscopy alone cannot provide the correct picture of
magnetism in degenerate stars.

####  6.3.2 There are no MWDs with field strength $\la 40$ kG

A still more notable, and quite surprising, feature of Fig. 5 is the complete
lack of MWDs with detected fields below about 40 kG. Extrapolating the
frequency distribution of Fig. 6 down about two more bins, a constant field
frequency suggests that there should be of the order of five or six MWDs in
our survey volume with fields between 4 and 40 kG. Instead, the distribution
comes to an abrupt halt. We have already noted that many of the available
$\langle B_{z}\rangle$ measurements are obtained with extremely high
precision. This is quantified by the red solid line in Fig. 5, that shows, for
instance, that about half of the WDs were observed with a sensitivity of 10 kG
or better, and 20 % of the WDs (that is, 33 WDs) were observed with a
sensitivity of 3 kG or better. The same situation may be appreciated also by
looking at the histograms of Fig. 6. While it is remarkable that no field
weaker than 40 kG has been detected in the 20 pc volume, we should bear in
mind that only a fraction of the WDs were checked for the presence of a
magnetic field with a sensitivity sufficient to detect the rare and ultra-weak
fields. Assuming for instance that the real frequency of MWDs with field
strength up to 40 kG is 4 %, the probability to find no magnetic field in that
range, out of 33 observed WDs, is still a non-negligible 26 %. Observations do
offer a stronger constraint, because all measurements with a sensitivity
better than 40 kG which fail to detect a field lower than 40 kG contribute to
reduce the probability that field weaker than 40 kG, if existing, would pass
undetected. Assuming that the frequency of the occurrence of field with
strength $\leq B_{\rm max}$ is constant and equal to $f_{B}$, the probability
$p$ to find no star with field weaker than $B_{\rm max}$ in a set of
observations with sensitivity $s_{\langle|B|\rangle}$ may be approximated by

$p=\prod_{i=1}^{N}\left(1-f_{B}\frac{B_{\rm
max}-s_{\langle|B|\rangle}^{i}}{B_{\rm max}}\right)$ (2)

where the index $i$ runs over all stars for which
$s_{\langle|B|\rangle}^{i}\leq B_{\rm max}$. For $B_{\rm max}=40$ kG and
$f_{B}=0.04$, we find $p=5.5$ %.

Is it possible that we have found the effective lower limit of the overall
distribution of MWD field strengths? In that case we would conclude that MWD
fields are generally limited to the range between a few tens of kG and a few
hundreds of MG, a span of about four dex.

A firm answer to this question will come after the systematic analysis of a
larger volume-limited sample of MWDs, further observations of the suspected
MWDs (such as WD 0738$-$172), and a more accurate estimate of the
$\langle|B|\rangle$ upper limits (see our considerations about
$s_{\langle|B|\rangle}$ of Sect. 5.3). So far, outside the local 20 pc volume
we are aware of only three MWDs with $\langle|B|\rangle$ possibly smaller than
30-40 kG, namely WD 1105$-$048 and WD 0446$-$789, which were already
discovered to be magnetic or suspected magnetic WDs by Aznar Cuadrado et al.
(2004), and a suspected weakly magnetic star, WD 0232$+$525, newly discovered
by us. Some or all of these stars may well have a stronger field that is seen
approximately equator-on, or for which, as in WD 2359-434, (unusually)
$|\langle B_{z}\rangle|$ never rises above about 10 % of the value of
$\langle|B|\rangle$. However their existence calls into question the general
validity of the result found in the local 20 pc volume, and suggests at the
very least that the field strength distribution may have a soft lower field
strength edge rather than a very sharp lower limit, if such an edge exists at
all.

In conclusion, although the details of the field strength distribution of the
MWDs are actually not strongly constrained by observations, the overall shape
of the cumulative distribution function of Fig. 5 appears consistent with a
distribution of the field strength constant per decades from approximately 40
kG to 300 MG, and substantially lower outside these limits. Further data are
needed to firmly extend the validity of this result outside of the local 20 pc
volume.

####  6.3.3 In the weak-field regime we are sensitive only to the dipolar
field component

We finally note that both the lower and the upper limits of the field strength
are probed mainly by polarimetry. A 30 kG dipolar field would polarise
spectral lines, but would not be strong enough to produce a noticeable
broadening of their intensity profiles, even with high resolution; at the
other extreme, magnetic fields of hundreds of MG strength make it very
difficult to recognise spectral features (if they are present at all), so that
detection relies on measurements of circular polarisation of the continuum.

At the lowest field strengths, it should be kept in mind that the Stokes $V$
profiles of spectral lines are sensitive to $\langle B_{z}\rangle$, which is
the component of the magnetic field along the line of sight, averaged over the
stellar disk. This average is dominated by the dipolar component of the
magnetic field, while higher-order multipolar components tend to contribute
very little to $\langle B_{z}\rangle$, since their line-of-sight component
averaged over the stellar disk, for a similar field strength at the magnetic
poles, is much smaller than the dipolar contribution. In particular, if we
compare a quadrupolar component and a dipolar component of similar strength,
the quadrupolar contribution to $\langle B_{z}\rangle$ will be about 1/10 the
dipolar contribution (Schwarzschild, 1950; Landolfi et al., 1998). To make
this as clear as possible, even if $\langle B_{z}\rangle$ is consistent with
zero with a sub-kG sensitivity, we cannot prove that a magnetic field is
totally absent from the surface of a WD. For instance, a WD could have a non-
linear quadrupolar magnetic field (Bagnulo et al., 1996) that produces a
surface field $\langle|B|\rangle$ up to 30–40 kG, and still be undetectable
either via spectroscopic and spectropolarimetric techniques. In this respect,
polarimetric weak field detection techniques for WDs are less efficient than
in the case of main-sequence stars, in which often the projected surface
velocity resolution of Stokes $V$ introduced by rotation may allow one to
detect a field even when $\langle B_{z}\rangle\simeq 0$ (the so-called
"crossover effect", see Babcock, 1951).

###  6.4 The frequency of the occurrence of magnetic field as a function of
cooling age

It is particularly important to understand whether the frequency of the
occurrence of a magnetic field is correlated to the star’s cooling age, as
this information may provide valuable constraints on field evolution both into
and during the WD stage. However, it is important to keep in mind that the
sensitivity of the field measurements used in this work decreases with stellar
age, except for the case of stars with metal lines.

####  6.4.1 Previous results from the literature, and their interpretations in
terms of a bias against high-mass WDs.

The question of the occurrence of fields as a function of cooling age has been
investigated in the past. Liebert & Sion (1979), Valyavin & Fabrika (1999) and
Liebert et al. (2003) suggested that the magnetic field could be more frequent
in older and cooler than in hotter and younger WDs (in fact, Greenstein et al.
1971 had already practically assumed that magnetic fields were a
characteristics of the cooler WDs only, because no hot MWDs had been found in
Greenstein’s very large collection of WD spectra at that time). In particular,
Valyavin & Fabrika (1999) suggested a threshold age around 1 Gyr for the
occurrence of magnetic fields with strength $\ga 1$ MG, finding that the
frequency of MWDs with MG field strength is $3.5\pm 0.5$ % among WDs hotter
than 10000 K, and $20\pm 5$ % among WDs cooler than 10000 K.

Liebert et al. (2003) brought in statistical considerations, by examining the
frequency of fields in three qualitatively different samples. They examined
blue spectra of the WDs found in the Palomar-Green (hereafter P-G) survey of
faint blue objects, and found that among the observed WDs, which generally
have $T_{\rm eff}$ near or above 10 000 K, only $2\pm 0.8$ % have a magnetic
field, which, owing to the selection effect introduced by the limited
sensitivity of low-resolution spectroscopy, have a strength of about 2 MG or
more. They also examined two large surveys that contain a much higher fraction
of older WDs: the sample of 110 cool WDs (mostly with $4000\la T_{\rm eff}\la
10000$ K) modelled by Bergeron et al. (1997), and an earlier version of the 13
and 20 pc volume-limited samples, collected by Holberg et al. (2002), which
are dominated by cool older WDs. In these samples the frequency of MWDs is
much higher: in the sample by Bergeron et al. (1997), the frequency of MG
fields is $7\pm 2$ %, and in the volume limited samples the frequency of MG
fields is $11\pm 5$ % in the complete 13 pc volume, and $8\pm 3$ % in the less
complete 20 pc sample. Taken at face value, this result clearly suggests that
strong magnetic fields are less common in young WDs than in old samples, and
indeed, Liebert et al. (2003) did acknowledged the possibility that some
mechanism existed that would cause the frequency of MWDs increase with
decreasing $T_{\rm eff}$. However, Liebert et al. (2003) also argued that this
apparent difference could probably due to a very strong selection effect
caused by the combination of two facts: (1) that typical masses of MWDs are
significantly higher than those of non-magnetic WDs, and (2) that higher mass
WDs are significantly smaller in radius than low-mass WDs, hence fainter. They
argued that the small radii of relatively massive MWDs led, in the Johnson B
magnitude-limited P-G survey, to sample a volume only about 1/4 as large as
the one surveyed for lower-mass, presumably less frequently magnetic WDs.
Making a correction for this effect, they concluded that the fractional
occurrence of MWDs in the P-G survey is actually very close to that of the
surveys of cooler stars.

Later, Kawka et al. (2007) analysed an incomplete sample of WDs of the local
20 pc volume, and again found no evidence of a higher incidence of magnetism
among older WDs, within uncertainties.

Figure 7:  Top panel: The ratio between the number of MWDs and the number of
all WDs that are produced during the interval of time represented by the
horizontal errorbars, as a function of cooling age $\tau$. The fraction $M/N$
printed close to the symbols represents the number of MWDs ($M$) and all WDs
($N$) in the interval of time. The vertical errorbars represent the
uncertainties associated to the frequency of the occurrence of MWDs
extrapolated to the Galactic sample. Bottom panel: The blue solid line
represents the ratio between the observed MWDs younger than the abscissa value
$\tau$ and all WDs younger than $\tau$; the blue dashed lines show the $\pm
1\,\sigma$ uncertainties of that frequency. The red dotted line shows a
constant ratio between MWDs and all WDs.

####  6.4.2 In the local 20 pc volume there is a shortage of young magnetic DA
WDs

What does the analysis of our greatly enlarged data set for the local 20 pc
volume say? From Table LABEL:Tab_Stars we can see that only four WDs out of 34
DA WDs younger than 1 Gyr are magnetic, and that there is only a single MWD
among 20 DA WDs younger than 0.5 Gyr, the very weakly magnetic star WD
2047$+$372, discovered by Landstreet et al. (2016), with age $\simeq 0.36$
Gyr. The frequency of magnetism of one out of 20 (or 19, if we consider Sirius
B as not observed) is remarkably smaller than the overall frequency of
magnetic fields of about 20–25 % derived for the full sample of DA WDs.222The
dearth of younger, hence hotter, hence more luminous magnetic WDs is reflected
also in the fact that there are no MWDs brighter than $G\sim 13$ in our sample
(see Fig. 3). These data predict the galactic frequency of young magnetic DA
WDs to be $5.0\pm 4.9$ %. The top panel of Fig. 7 shows the ratio between MWDs
and all WDs produced in various bins of interval of cooling age, and suggests
that the occurrence of magnetic fields increases with age. This behaviour may
also be appreciated by looking at the cumulative distribution functions. The
bottom panel of Figure 7 shows the ratio $r(t)$ between the number of all MWDs
younger than age $t$, $M(t)$, and the number of all WDs younger than $t$,
$N(t)$. This curve may be extrapolated to the galactic frequency of WDs with
an uncertainty $(r(t)(1-r(t))/N(t))^{1/2}$, and compared with the constant
production rate of 33/149. We can see that in the local 20 WD population,
fields are relatively rare in young stars, and more common in WDs older than 1
Gyr, in spite of the fact that detection thresholds for older stars are
definitely higher than for younger stars. The lack of young MWDs in the local
20 pc volume strongly suggests that the onset of higher magnetic frequency
occurs later than 0.5 Gyr. We note that in the local 20 pc volume there are no
WDs younger than 0.5 Gyr other than of spectral type DA. Therefore, based on
the analysis of this sample, we are not able to extrapolate our finding for
instance to DB and DQ WDs.

####  6.4.3 In fact, magnitude-limited surveys have little bias against high-
mass WDs

We now face two conflicting interpretations of the observational data. On the
one hand, the argument proposed by Liebert et al. (2003) suggests that
magnitude-limited surveys are biased against high-mass (hence more frequently
magnetic) WDs, and that this is the reason that such surveys sometimes find a
low frequency for the occurrence of MWDs. On the other hand, a volume limited
survey, which should be exempt from any bias against stellar mass,
nevertheless strongly suggests that magnetic fields are more frequent in older
than in younger WDs. How can we reconcile these views?

Figure 8:  Cooling history of $1.2\,M_{\odot}$ (black solid lines),
$0.9M_{\odot}$ (blue dotted lines) and $0.6M_{\odot}$ (tick red solid lines)
WDs. Top panel: $T_{\rm eff}$ vs. cooling age $\tau$; mid panel: luminosity
vs. cooling age; bottom panel: Johnson B absolute magnitude vs. cooling age.
Data from Montreal cooling curve database.

The key concept is that for a given $T_{\rm eff}$, because of its smaller
surface, a higher mass WD will certainly have a lower luminosity ($L\propto
M^{-2/3}$) than a lower mass WD. However, the higher mass WD, exactly because
of its smaller surface and larger internal mass and thermal energy, will also
cool more slowly than the lower mass WD, so luminosity will drop more slowly
for a higher mass WD than for a lower mass WD. While luminosity will be always
higher for a lower mass than a higher mass WD at the same $T_{\rm eff}$, the
same quantities as a function of cooling age may compare in a quite different
way. That is, the comparison we must make is not of massive and less massive
WDs of the same $T_{\rm eff}$, but of pairs of WDs of the same age.

To explore this issue in a quantitative way, we have used the Montreal cooling
curve database, which reports $T_{\rm eff}$, $\log g$ and absolute magnitude
in numerous photometric bands as functions of cooling time for a grid of WD
masses. From these data, we have plotted the values of $T_{\rm eff}$ (upper
panel), luminosity (mid panel) and the absolute B magnitudes (lower panel) of
WDs of $0.6M_{\odot}$, $0.9M_{\odot}$ and $1.2M_{\odot}$ as functions of
cooling age from formation, covering the typical range of non-magnetic WDs,
fairly massive MWDs, and extremely massive MWDs. If we compare an
$0.6M_{\odot}$ (a typical non-magnetic WD) and an $0.9M_{\odot}$ WD (a typical
MWD) that formed at the same time and at the same distance from us, the more
massive WD is more than a magnitude fainter than the less massive WD during a
short period of a few Myr after initial creation. However, after about 20 Myr,
when the $T_{\rm eff}$ values have declined to about 25 000 K ($0.6M_{\odot}$)
and 35 000 K ($0.9M_{\odot}$), the visible brightness difference has dropped
to roughly 0.1 or 0.2 mag, and after about 600 Myr the more massive star is
actually significantly brighter than the less massive one (see the bottom
panel of Fig. 8). Within the age range 0.02–0.5 Gyr, the magnitude difference
between WDs of 0.6 and $0.9\,M_{\odot}$ of the same age is only about 0.2 mag,
and the brighter star would be visible in a magnitude limited survey to a
distance $\sim 10$ % greater than the fainter, which amounts to a difference
in survey volume of $\sim 30$ %. Since it is the lower mass WD that is
slightly brighter during this cooling age interval, the volume of the higher
mass MWD should be increased by about 30 % to correct for the different
volumes surveyed. Note however that the extremely massive $1.2\,M_{\odot}$ WD
is as much as 0.5 mag fainter in B relative to the WD of $0.6\,M_{\odot}$ for
much of the first 800 Myr of cooling, then its magnitude drops rapidly below
the lower mass comparison after about 4 Gyr. Overall, it appears that the bias
that exists in magnitude-limited samples of WDs with ages between about 1 and
4 Gyr will actually lead to overestimates of numbers of the most massive WDs
compared to WDs of normal mass.

We conclude that the view that magnitude-limited surveys of WDs are strongly
biased against high-mass stars is not generally correct, although various
biases do exist for WDs of different masses at some ages. The relationships
between luminosity and cooling age change with mass in a complex way,
underscoring the fact that statistically studies of WDs must be based on
volume-limited rather than magnitude limited surveys.

####  6.4.4 Magnitude-limited surveys support the idea that there is a
shortage of young MWDs

We re-examine the results of magnitude-limited surveys in light of the
considerations of Sect. 6.4.3. We first note that even if weak (few kG to 2
MG) fields were common among young WDs, these would be still missed in the P-G
survey (and even more so by the SDSS), therefore the frequency of the
occurrence of young MWDs in the P-G survey should be compared with the
frequency of the occurrence of MWDs with field strength between 2 and 100 MG
that is deduced from the analysis of the local 20 pc volume, which is $13\pm
4$ %.

From the stellar parameters of the 347 WDs identified in the P-G survey
(Liebert et al., 2005), we find that about 90 % of these stars have cooling
ages $\la 0.5$ Gyr. This young group includes about 312 WDs, six of which are
magnetic, for a frequency of $1.9\pm 0.8$% MWDs. In their 2003 study, Liebert
et al. (2003) corrected this frequency for their estimate of the volume bias
by a factor of 3.96, bringing their estimate of actual MWD frequency in the
P-G survey to $7.9\pm 3$%, in very good agreement with their estimates for
cooler WDs, and also consistent with our estimate of $13\pm 4$ %. However, we
have argued in the preceding section that the surveyed volume should only be
corrected by a factor of roughly 1.3, leading to an estimated frequency of
$2.5\pm 1$%, substantially smaller than predicted from a volume-limited
survey. This result supports, with better statistics, our contention that WDs
with ages less than 0.5 Gyr are significantly less likely to show magnetic
fields than older WDs.

Our interpretation of the fact that magnitude-limited surveys of hot stars
find fewer MWDs than volume-limited surveys is that this is not the result of
a significant selection effect against higher-mass MWDs, but of a selection
effect in favour of younger WDs, among which we believe that frequency of MWDs
is substantially depressed from the general average. The striking difference
between the frequency of MWDs among younger and older WDs probably reflects
the action of the mechanisms that produce magnetic fields in WDs.

####  6.4.5 Future work: investigating a larger volume-limited sample

The clearest way to confirm or reject the idea that a deficiency of magnetic
fields in young WDs is a general property of WDs and not limited to the local
20 pc volume, is to extend our investigation to a larger volume limited
sample. There is no complete spectropolarimetric dataset for any volume larger
than the local 20 pc volume, but it appears that the 40 pc volume, which has
about 8 times the number of WDs as the local 20 pc volume, is likely to become
the next volume of wide interest (Tremblay et al., 2020; McCleery et al.,
2020). We look ahead to the possibilities offered by this volume for testing
our proposal.

The northern hemisphere half of the 40 pc volume (including the local 20 pc
volume) has about 81 WDs with ages of less than 500 Myr (McCleery et al.,
2020), corresponding to a total population of approximately 162 such stars.
These young stars have been fairly intensely studied because of their
brightness (the faintest WDs of this young age group in the 40 pc volume
generally $V$ or $G$ brighter than 15 mag). At least 90% are DA stars, in
which fields of a MG or more are readily identified in good classification
spectra, and fields ten times smaller may be identified with high resolution
spectroscopy. In addition, our own surveys have explored this sample rather
extensively. Our estimate is that we and others have obtained
spectropolarimetric observations on roughly half of the MWDs of ages less than
0.5 Gyr in this volume, and a very large fraction have at least been examined
with low- to mid-resolution spectroscopy (see, e.g. Tremblay et al., 2020;
McCleery et al., 2020; Napiwotzki et al., 2020).

Within the 40 pc volume we know of nine MWDs younger than 0.5 Gyr (see Table
2, which includes WD 2047$+$372, the only young MWD in the local 20 pc volume,
and WD 0232$+$525, a new MWDs recently detected by us). It would be outside of
the scope of this paper to report on a partial analysis of this sample, but as
a very preliminary result we note that at present only about 6 % of the young
sample are known to have magnetic fields. Even if further searches double the
number of young MWDs in the volume, which because of previous searches seems
unlikely to us, the magnetic fraction will still be only about half of the
average frequency of MWDs that we have clearly shown to apply to the local 20
pc volume. Thus we predict that the result of this larger future survey will
still support the conclusion that there is a dearth of MWDs among the youngest
WDs.

Table 2:  List of known MWDs younger than 0.5 Gyr within 40 pc from the Sun.

| $\langle|B|\rangle$ | $M$ |  |
---|---|---|---|---
Star | (MG) | ($M_{\odot}$) | References | Notes
WD 0041$-$102 | 20 | 1.1 | Liebert et al. (1977) |
WD 0232$+$525 | 0.005: | 0.82 | unpublished |
WD 0301$+$059 | 200 | 1.12 | Landstreet & Bagnulo (2020) |
WD 0316$-$849 | 300 | 0.86 | Barstow et al. (1995) | 1
WD 0945$+$245 | 670 | 0.80 | Liebert et al. (1993) |
WD 1105$-$048 | 0.010: | 0.58 | Aznar Cuadrado et al. (2004) |
WD 1105$-$340 | 0.150 | 0.67 | Landstreet & Bagnulo (2019a) |
WD 1658$+$440 | 2.3 | 1.32 | Schmidt et al. (1992) |
WD 2047$+$372 | 0.06 | 0.82 | Landstreet et al. (2017) |

1\. Star WD 0316$-$849 = RE J0317$-$853 = EUVE J0317$-$853 = V* CL Oct is
often confused with star WD 0325$-$857 = EQ J0317-855 = LB 9802, a hot and
young non magnetic WD (Kawka et al., 2007, Table 1).

####  6.4.6 The higher incidence of magnetic fields in certain class of stars
may just reflect a higher incidence of magnetic fields in older/cooler WDs

That DZ stars might exhibit a frequency of the occurrence of magnetic fields
higher than average would naturally be explained in the scenario of magnetic
fields that become more frequent as cooling age increases; weaker fields would
be present in many cool (hence old) H and He-rich WDs, but would be detected
only if the stellar spectrum exhibit metal lines, while the field would pass
unnoticed in featureless or nearly featureless WDs. This interpretation is
consistent also with the considerations about DAZ stars of Sect. 6.1.1, where
we noted that the frequency of magnetic DAZ WDs is actually similar to the
frequency of magnetic DA WDs, once we compare the situation in the same
temperature ranges.

Admittedly, it is possible to speculate that the magnetic field may be
originated during disk debris accretion, and its presence is longer lasting
than that of metal lines in the stellar spectrum. Some of the cooler magnetic
DA WDs could be former magnetic DAZ WDs. However, that temperature and age are
important discriminating factors is supported by the fact magnetic fields are
much more frequent in DZ and DAZ WDs cooler than $7500-8000$ K than in hotter
WDs of the same spectral class (Hollands et al., 2015; Kawka et al., 2019).

We have also seen in Sect. 6.2 that the frequency of the occurrence of
magnetic fields in He-rich WDs ($30\pm 7$ %) is higher than in H-rich WDs
($19\pm 4$ %), and we have noted that in the local 20 pc volume there are no
He-rich WDs younger than 0.5 Gyr. The higher frequency of He-rich MWDs may be
explained again as an effect of temperature/age, although we must keep in mind
that in the local 20 pc volume we have observed a specially high frequency of
MWDs among the oldest He-rich stars. In the cooling age range 0.5 to 5 Gyr,
the frequency of the occurrence of magnetic field in H-rich and He-rich WDs
are very similar: $25.9\pm 5.7$ and $22.9\pm 7.1$ %, respectively.

####  6.4.7 Do older MWDs have stronger fields than younger MWDs?

Figure 9:  Same as Fig. 7, but for MWDs with fields weaker than 2 MG (blue
symbols) and stronger than 2 MG (red symbols). Because fields weaker than
$1-2$ MG cannot be detected in stars older than a few Gyr (unless there are
metal lines in the stellar spectra), the zero frequency of weakly magnetic WDs
at $\tau=7.5$ Gyr may well be due to an observational bias.

It is very important to check next whether young MWDs have different features
than older MWDs, apart from being more rare. We note that in the local 20 pc
volume, fields with strength higher than 2-5 MG are more common in older than
in younger MWDs. The most notable exception, of a relatively young (0.92 Gyr)
WD with a very strong field (and with a higher than average mass,
$0.93\,M_{\odot}$) is Grw $+70^{\circ}\,8247$. The next youngest high-field
stars in the local 20 pc volume all have cooling ages older than 2.5 Gyr.
Figure 9 shows the same quantities as in Fig. 7, but separately for MWDs with
fields weaker than 2 MG and for MWDs with fields stronger than 2 MG, and
demonstrates in a convincing way that in the local 20 pc volume there is a
pronounced dearth of higher-field MWDs younger than 2–3 Gyr.

This issue can be further investigated only by mean of a complete survey of a
larger volume sample of WDs. With the data available we are not able to
investigate this finding beyond the 20 pc volume with unbiased statistics,
therefore we will not consider this characteristic as an observational
constraint. We also note that Wickramasinghe & Ferrario (2000) had concluded
that there is no evidence that field strength depends on temperature.

###  6.5 The frequency of the field occurrence versus stellar mass

Figure 10:  Mean (top panel) and median values (bottom panel) of the mass of
non magnetic (red empty circles) and magnetic (blue filled circles) WDs as a
function of cooling age, binned in the intervals of time represented by the
horizontal errorbars. Vertical errorbars represent the standard deviation of
the mean values, and the semi interquartile range for the median values,
calculated for each time bin. The horizontal blue dashed (red dotted) lines
represent the mean (top panel) and median (bottom panel) for the magnetic (non
magnetic) WDs. The numbers $M$, $N$ between the two panels show the number $M$
of MWDs and the number $N$ of non magnetic WDs in each time interval bin.

Figure 2 shows no obvious correlation between field strength and stellar mass,
but it is clearly apparent that magnetic and non magnetic WDs have different
mass distributions. This is not a surprise, as Liebert (1988) had already
suggested that MWDs are more massive than average, and this conjecture has
been confirmed by several subsequent studies, the most recent ones being those
of Kepler et al. (2013) and McCleery et al. (2020). We note that there are no
WDs with $M>1\,M_{\odot}$ in the local 20 pc volume, therefore we cannot
explore the full parameter space.

####  6.5.1 The average mass of MWDs is higher than the average mass of non
MWDs

We recall that the mean mass of all WDs of the local 20 pc volume is $0.64\pm
0.13\,M_{\odot}$. The mean mass of non magnetic WDs is $0.62\,M_{\odot}$, with
a standard deviation of $0.13\,M_{\odot}$, and the mean mass of MWDs is
$0.69\,M_{\odot}$, with a standard deviation of $0.10\,M_{\odot}$. Marginal
evidence for correlation with age will be discussed in Sect. 6.5.3.

####  6.5.2 Mean mass of MWDs and field strength

Formally, the field strength of the MWDs of the local 20 pc volume seems
independent of stellar mass: the mean mass is $0.69\pm 0.10\,M_{\odot}$
(standard deviation) whether we consider fields weaker than 2 MG, or fields
stronger than 2 MG. However, things might be different in certain age bins
(see Sect. 6.5.4).

Our results may be compared with the statistics gathered by Ferrario et al.
(2015), mainly on the basis of MWDs discovered via low-resolution spectroscopy
(hence highly biased in favour of MWDs with field strength between 2 and 80
MG). Ferrario et al. (2015) found that the mean mass of high-field MWDs is
$0.78\pm 0.05\,M_{\odot}$, but they found also that the average mass of more
weakly magnetic WDs is consistent with that of non-magnetic WDs, a feature
that we do not see in the local 20 pc volume. Based on a 40 pc volume-limited
sample of northern WDs (but probably still biased in favour of strongly
magnetic WDs, because field detections in this sample have relied heavily on
spectroscopic data), McCleery et al. (2020) found that the mean mass of MWDs
is $0.75\pm 0.05\,M_{\odot}$.

####  6.5.3 Does the average mass of WDs and MWDs depend on age?

In the local 20 pc volume, the average mass of the MWDs older than 2 Gyr is
$0.67\pm 0.08\,M_{\odot}$, while that of MWDs younger than 2 Gyr is $0.72\pm
0.13\,M_{\odot}$. The mean mass of MWDs younger than 1 Gyr is $0.80\pm
0.10\,M_{\odot}$ (all standard deviations, not standard errors). Figure 10
shows the mean and median masses of magnetic and non magnetic WDs as a
function of cooling age. There is marginal, not fully convincing evidence that
younger MWDs are more massive than older MWDs.

####  6.5.4 What do the observations outside the local 20 pc volume tell us
about the mass of MWDs versus their age?

Although the result that younger MWDs would appear not only rarer, but also
more massive than the average MWD, is very suggestive, it cannot be extended
with very strong statistical support, because the observational sample in the
local 20 pc volume is too small. There are only nine MWDs younger than 2 Gyr,
only four younger than 1 Gyr, and only one younger than 0.5 Gyr, the star WD
2047$+$372, which has a relatively weak field, and $M\sim 0.82\,M_{\odot}$. Is
it possible to gather information outside of the local 20 pc volume? We have
already noted that the targets of the P-G survey were mostly young WDs, and
Table 1 of Liebert et al. (2003) shows that the MWDs found in that survey have
mass $\ga 0.75\,M_{\odot}$ and field strength $\ga 1$ MG, with the exception
of the 2.3 MG star PG 2329+267, which is listed with $M\simeq
0.61\,M_{\odot}$. However, a revision based on the catalogue of Gentile
Fusillo et al. (2019) of the parameters of the same list shows that all MWDs
have $M>0.75\,M_{\odot}$, and PG 2329+267 has $M=0.91\,M_{\odot}$. If we turn
our attention to the part of the local 40 pc volume of Sect. 6.4.5 (see in
particular Table 2), we find that younger MWDs are more massive than the
average MWD. However, we note that MWDs younger than 0.5 Gyr with low mass do
exist. Table 2 includes WD 1105$-$048 ($M\simeq 0.58\,M_{\odot}$,
$\langle|B|\rangle\simeq 10$ kG) and WD 1105$-$340 ($M=0.67\,M_{\odot}$,
$\langle|B|\rangle\simeq 150$ kG). Outside the 40 pc volume we know of WD
0446$-$789 ($M\simeq 0.55\,M_{\odot}$), a very young MWDs already mentioned in
Sect. 6.3.2, with an extremely weak field. Unfortunately, there is not
sufficient information, especially on older WDs, to reach any firm conclusion.
We suggest, as a working hypothesis for further investigation, that among the
(relatively rare) young MWDs, strong fields are found mainly or only in WDs
more massive than the average MWD. Lower-mass young MWDs generally exhibit
weak, possibly extremely weak fields.

####  6.5.5 Summary

While all these results are consistent with the conjecture that MWDs are more
massive than non-magnetic WDs, first proposed by Liebert (1988), it seems that
in the local 20 pc volume we have found comparatively more low-mass MWDs than
in other investigations, possibly because we have given a more appropriate
statistical weight to WDs of various ages. In contrast to what found in
magnitude-limited surveys, from which stronger magnetic fields seem to be
associated to higher mass WDs, in the local 20 pc volume, MWDs with fields
stronger than 2 MG have average mass similar to weaker field MWDs. However,
MWDs younger than 2 Gyr are marginally more massive than older MWDs. After
also looking at data available for stars outside of the local 20 pc volume, we
suggest that young MWDs are rare, and among these young MWDs, stronger fields
(of the order of 1 MG or more) are found mainly in higher-mass MWDs; lower-
mass young MWDs do exist, but seem to have weaker fields.

###  6.6 The frequency of MWDs in binary systems

There are 35 binary systems, including wide binary systems that comprise a WD
and a main-sequence (MS) star, DD and uDD systems. Five of these systems
(three CPM and two uDD systems) include a MWD, for a frequency of $14\pm 6$ %,
marginally lower than in single WDs. We note, however, that some of these
systems include a DC WD and that the polarisation signal of a MWD in an
unresolved system would be diluted by the radiation of the companion, so that
this frequency may actually be underestimated.

##  7 Discussion

In this section we begin by summarising the observational constraints that we
have gathered from our volume-limited survey of WDs (Sect. 7.1). Then (Sect.
7.2) we consider the relevance of our conclusions to two kinds of broader
questions. First, we will examine the extent to which our findings appear to
support or contradict various theories of how the magnetic fields of MWDs are
created in an previous evolutionary phase and retained in the cooling phase,
or possibly generated in close binary systems. Secondly, we address the
question of possible further field generation that may occur while WDs are in
the cooling phase. We then consider the apparent lack of evidence of Ohmic
decay (Sect. 7.3).

###  7.1 Observational constraints

The main characteristics of the WDs of the local 20 pc volume can be
summarised as follows.

####  7.1.1 Overall incidence of magnetic fields

About 20-25 % of WDs have a magnetic field. The number of WDs in the local 20
pc volume is large enough to provide strong statistical support to this
result, and we may assume it is approximately valid in the Galaxy. The
presence of a magnetic field does not seem correlated with the spectral class,
except maybe for DQ WDs, in which the magnetic field seems either stronger or
more frequent than average. Furthermore, there are hints that DZ WDs may be
more frequently magnetic than other WDs. However this result could simply be
due to the fact that magnetic fields seem more frequent in older than in
younger WDs (see Sect. 7.1.2).

####  7.1.2 Behaviour with age

There is a dearth of MWDs younger than 0.5 Gyr. This result seems to be
supported by the outcome of previous spectroscopic surveys, outside of the
local 20 pc volume. In fact, there is some evidence that the frequency of
magnetism in WDs rises with time until cooling times of the order of 5 Gyr
(upper panel of Fig. 7). Beyond that age, the frequency drops, possibly
because we are not able to detect weak fields in older stars. There is no
strong evidence that field strength decays with time.

####  7.1.3 Field strength distribution

The field strength is uniformly distributed (with roughly equal probability of
finding fields per dex of field strength) over four decades, from 40 kG to 300
MG. It appears that these field values limit approximately the range of field
strengths frequently found in WDs. Outside of the local 20 pc volume, we are
aware of only three known MWDs with a field strength marginally lower than the
lower threshold, and of a few MWDs above the general upper limit, with field
strengths of several hundreds MG.

####  7.1.4 Correlations between magnetic field and mass

MWDs are more massive than non-magnetic WDs. This result is fully consistent
with previous studies not confined to the local 20 pc volume, which are
probably biased in favour of younger and stronger field MWDs. In fact, there
is some marginal evidence that younger MWDs are more massive than older MWDs.
There are no WDs with $M>1\,M_{\odot}$ in the local 20 pc volume, which
represents a limitation of our analysis of the parameter space. Data from
outside of the local 20 pc volume show marginal evidence that young MWDs with
strong fields are more massive than young MWDs with weak fields, but data from
the local 20 pc volume show that there exist several old MWDs with low mass
and strong fields.

####  7.1.5 Magnetic fields and binarity

There are no strong indications that the overall frequency of MWDs in binary
systems is different than in single WDs. However, it should be noted that in
the local 20 pc volume there are no known close (spectroscopic) binary systems
including a dM and a WD, so this result is not inconsistent with the finding
by Liebert et al. (2005) that MWDs in close (but not interacting) systems with
a non-degenerate companion are very rare.

###  7.2 Field origin and observational constraints

In the following we discuss how various major hypotheses concerning the origin
of MWD fields stand up against the observational constraints of Sect. 7.1.

####  7.2.1 The fossil field hypothesis

The possibility of some level of magnetic flux conservation during evolution
from the main sequence to a final compact state was first discussed by Woltjer
(1964), applied to WDs by Landstreet (1967), and discussed more fully by Angel
et al. (1981). Flux conservation suggests that magnetic flux might be roughly
conserved as a star evolves from a main sequence structure to a WD structure
two orders of magnitude smaller in radius (the fossil field hypothesis). Such
a transformation could convert the main sequence fields of the order of
$10^{4}$ G found in Ap and Bp main sequence stars (e.g. Donati & Landstreet,
2009) to fields of roughly $10^{8}$ G = 100 MG.

It is now very clear that the strongly magnetic Ap and Bp main sequence stars
are not sufficiently frequent in space to account for the high frequency of
MWDs, as already pointed out by Kawka et al. (2007). Strong fields are found
in about 8% of A and B main sequence stars (Power et al., 2008). During the
first one or two Gyr of Milky Way evolution, these stars, because of their
relatively rapid evolution, provided most of the WDs. If flux conservation
from Ap and Bp stars provided all the magnetic fields of the oldest MWDs, we
would expect no more than about 8% of oldest WDs to be magnetic (assuming that
the fraction of Ap and Bp stars has been constant with time). More recently,
many WDs have also formed from main sequence F stars, which require a few Gyr
to end their nuclear-burning lives. These lower mass (ca. $1.2-1.6M_{\odot}$)
progenitors have almost no fossil magnetic fields, so WD fields formed by flux
conservation would be expected to make up a substantially smaller fraction
than 8% of the younger WDs, well below the observed frequency of more than
20%. Although this might be a minor MWD formation channel, it is clearly not
the most important process.

####  7.2.2 Origin of the fields as a result of deep-seated convective dynamo
action

Another potential source of magnetic fields inherited from pre-WD evolution
arises from the possibility of dynamo action in the interior of a star either
in the main sequence or in the AGB. This dynamo would leave behind a remnant
magnetic field that is retained in the stellar interior as a fossil field,
possibly compressed and amplified during later evolution, and finally revealed
after much mass loss in the newly formed WD.

It appears that dynamo-driven magnetic fields may originate in the convective
cores of main sequence A or B stars (which are unrelated to the surface
magnetic fields of the Ap and Bp stars). A series of papers summarised by
Stello et al. (2016) discusses how a dynamo-driven field produced in the
strongly convective core during main sequence evolution of an intermediate-
mass star (e.g. Braithwaite & Spruit, 2004; Brun et al., 2005) can leave a
strong, stable field (Duez et al., 2010) which can survive into the red giant
phase, where it could be detected by its effect on the oscillation spectrum.
Such fields would then be presumably amplified by flux compression during the
final collapse of the star into WD. It is thought that this kind of mechanism
can produce MG fields, that slowly diffuse outward to the surface of the WD
while decaying by Ohmic losses, on a timescale of order 1 Gyr.

Field generation based on a dynamo operating in the region outside the core in
a star powered by nuclear fusion in shells, possibly stimulated by the angular
momentum added by ingestion of a planet, is discussed by Kissin & Thompson
(2015). The resulting field can reach 10 MG.

Dynamos operating in single WD progenitors of various masses and rotation
rates (or in various kinds of interacting binary systems, see Sect. 7.2.3) and
leaving behind internal fields in the evolving stars, could occur over a wide
range of initial stellar parameters. Such dynamos may be capable of generating
both the range of field strengths and the high frequency of fields found in
white dwarfs, although further numerical explorations are needed to more fully
understand the capabilities of these models. The observed dearth of young MWDs
compared to the frequency of fields in older WDs may be explained by slow
emergence of such internal fields to the stellar surface.

It is worth noting the recent detection claimed by Caiazzo et al. (2020) of
magnetic fields in three young WDs that are members of open clusters, in which
the mass and age of the progenitor of each can be determined fairly
accurately. All three WDs have quite high masses (around $1\,M_{\odot}$) and
progenitor masses of around 5 or $6\,M_{\odot}$. Caiazzo et al. (2020) argue
that these new MWDs are so young that it is very unlikely that they formed
from binary merger processes discussed in Sect. 7.2.3 below; instead they are
almost certainly descended from single upper main sequence stars. These
discoveries, if fully confirmed, appear to require that at least some MWDs
descend from single main sequence stars.

####  7.2.3 Field generation during merger in a close binary

Cataclysmic variables (CVs) are close binary systems in which a non-degenerate
star is transferring mass onto a very close WD companion. In about 25% of such
systems the WD companion has a MG magnetic field. CVs form from binary systems
containing one star which has already become a WD, in which the separation is
small enough that when the non-degenerate star starts to evolve into the
giant, the WD is engulfed in the expanding envelope. If the WD survives this
common envelope event without merging with the core of the red giant, the
system becomes a CV. However, an extensive search by Liebert et al. (2005),
failed to discover any MWDs among the pre-CV systems, which are close binaries
with a main sequence star and a WD.

Following this result, Tout et al. (2008) proposed that the available orbital
energy and angular momentum could lead to the creation of a strong magnetic
field via dynamo action during the common envelope phase. The predicted
outcome of the process is that if merging occurs, the result would be a
single, high-mass high-field MWD, while those systems that finally do not
quite merge would become magnetic CVs. Later, Briggs et al. (2015, 2018a,
2018b) carried out synthetic evolution calculations of binary star
populations, using simple parameterised models of the common envelope phase
and of dynamo action during the common envelope event, to show that this
mechanism might well lead to generation of strong fields, up to the upper
field limit observed in WDs. In particular, they found parameter choices that
enabled the population synthesis field strength predictions to reproduce the
distribution of magnetic field strength depicted by Ferrario et al. (2015),
according to which most of MWDs have a magnetic fields strength between 2 and
80 MG.

A probable example of MWD creation from a binary pathway may be provided by
the magnetic WD 0316–849 = RE J0317–853 (Barstow et al., 1995), which is not
within the local 20 pc volume. This extremely hot, "young" and massive DAH WD,
with a field of about 500 MG and a rotation period of 725 s has a common
proper motion WD companion that has a much older total age. The two WDs were
presumably created together in a wide binary system. The very discordant ages
may be understood if WD 0316–849 is the result of a recent merger, which
presumably also led to the creation of the huge magnetic field (Ferrario et
al., 1997).

If we consider the possibility that some version of common envelope evolution
followed by merger is the dominant mechanism for formation of MWDs, then we
would expect a high frequency in space of initially close binary systems that
will evolve into single MWDs, a frequency close to the upper limit of systems
that will finally merge into single WDs (Toonen et al., 2017). However, it is
not yet clear that such systems are common enough to produce more than 20 % of
all WDs. Furthermore, if the mechanism operates as described by Briggs et al.
(2015, 2018a), it is predicted to produce mainly fields with a rather limited
range of field strengths; by contrast, the predicted distribution is not at
all similar to the observed roughly constant frequency per dex of field
strength in the local 20 pc volume. We also note that Belloni & Schreiber
(2020) have recently discussed the consequences of these models for a more
comprehensive range of CV and CV-like binary situations. Belloni & Schreiber
(2020) argue that the basic model predictions have important conflicts with
observations of CVs and pre-CVs, and that the original mechanism needs to be
substantially revised. However, they offer several suggestions for changes
that would bring this model of field generation into better agreement with
observations. In conclusion, it appears to us that the merging mechanism
requires further development to be shown to be a major contributor to the
observed population of MWDs.

####  7.2.4 Dynamo action during WD core crystallisation

Figure 11:  Cooling age – mass diagram for magnetic (solid symbols) and non
magnetic (empty symbols) WDs, showing the boundary at which crystallisation
convection begins as a WD cools with age, as described by Isern et al. (2017),
for stars with "thick" (blue solid lines) and "thin" (red solid lines)
hydrogen layers (from Bédard et al., 2020). Stars for which we have not
determined atmospheric composition have been arbitrarily considered as H-rich.
The same plot is shown with cooling age in logarithmic scale (left panel) and
linear scale (right panel). Circles represent DA and and DC stars, while
triangles represents stars with metal lines (DAZ, DZ, DZA). DQ stars are
represented with crosses over the symbols. Solid symbols surrounded by one
circle represent MWDs with field strength between 1 and 100 MG, and solid
symbols surrounded by two circles represent MWDs with field strength $>100$
MG.

The magnetic fields of some WDs could be slowly generated during a time of the
order of 1 Gyr by some internal physical mechanism during the cooling of the
white dwarf itself. Valyavin & Fabrika (1999) suggested that electric
conductivity in WDs could increase with time, or that magnetic field diffuses
from the inner region of the star. Isern et al. (2017) have shown that during
crystallisation in the C-O core of a typical WD, separation and sinking of the
solidifying O component leads to strong convection in the core. Provided that
the WD is rotating rapidly, a dynamo of the same type that produces the
magnetic fields of Earth, Jupiter, and M dwarfs can operate. As in those
objects, this dynamo requires rapid rotation to function in a saturated state
(i.e. to produce the largest fields of which the mechanism is capable). In
WDs, a rotation period of less than about $10^{3}$ s is sufficient to ensure
saturation. A straight-forward extrapolation of the achievable field strength
versus convective energy density suggests that this mechanism could produce
global fields only up to about 1 MG in strength. However, Schreiber et al.
(2021) have suggested that taking account of the probable dependence of the