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

ABSTRACT

Many stars evolve into magnetic white dwarfs (MWDs), 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 WDs 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 (WDs). We find that 33 WDs 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 WDs with higher than average mass, especially in younger stars. We find a marked deficiency of MWDs younger than 0.5 Gyr, and we find that the frequency of the occurrence of the magnetic field is significantly higher in WDs that have undergone the process of core crystallization than in WDs 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 crystallization phase.

1 INTRODUCTION

That magnetic fields occur in the latest stages of stellar evolution has been known for more than 50 yr, since the discovery that pulsars are strongly magnetized neutron stars (Woltjer 1964; Pacini 1967; Gold 1968) and the discovery that white dwarfs (WDs) host strong magnetic fields (Kemp, Swedlund & Evans 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 candidate 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 MS (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 efforts have been dedicated to the search for and characterization of MWDs, both as a side-product of large spectroscopic surveys (e.g. Liebert, Bergeron & Holberg 2003; Kepler et al. 2013; Hollands, Gänsicke & Koester 2015), and due to specifically dedicated polarimetric surveys (e.g. Angel & Landstreet 1970a; Angel, Borra & Landstreet 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 MWDs are more massive than non-MWDs (Liebert et al. 2003); that cooler WDs are more often magnetic than hotter WDs (Liebert & Sion 1979; Fabrika & Valyavin 1999); that MWDs are not found in close binary systems with a WD and an MS 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 an MS 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_⊙ ends with collapse to the WD state, as does the evolution of many binary pairs, the current populations of WDs in the local 20 pc volume records a sample of the results of more than 90 per cent 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. WDs 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 characterize 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 characterization of the physical parameters of each member of the local 20 pc volume, which is discussed in Section 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 Section 3. Because literature data did not provide sufficient information, we have carried out a spectropolarimetric survey aimed at acquiring a complete data set of all WDs within the local 20 pc volume. Some preliminary results of our survey have been presented by Landstreet & Bagnulo (2019b, 2020) and Bagnulo & Landstreet (2019b), mainly to announce the discovery of new MWDs. In this paper, we publish the remaining observations that we have carried out in the last 3 yr, 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 Section 4 (observing log and fields values are given in Appendix  A). In Section 5, we describe the data set that we have collected for the WDs of the local 20 pc volume and in Section 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 Section 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 Section 8. This paper contains also a second appendix (Appendix  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 (π = 49.97 ± 0.05), is according to DR3 most likely within 20 pc from the Sun (π = 50.06 ± 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 π ≤ 50.0 in DR2 had π > 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_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, Bergeron & Dufour (2012), Limoges, Bergeron & Lépine (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 online cooling tables¹ of the Montreal group (Dufour et al. 2017), which are based on computations described by Tremblay, Bergeron & Gianninas (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_eff, 0.1 for log g, |$0.1\, \mathrm{M}_\odot$| 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 estimate of the stellar parameters. Radiation magnetohydrodynamic simulations of the atmosphere of WD 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_eff = 9982 ± 170 K and log g = 8.22 ± 0.08, and is also capable to reproduce the star’s far-ultraviolet (FUV) spectrum, while the best-fitting optical convective model has T_eff = 10 389 ± 153 K and log g = 8.01 ± 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_eff = 9820 ± 240 K and log g = 8.29 ± 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 MS stars, with separations ranging from about 2 arcsec to several arcminutes. In such systems, the nature of all components is almost always well established. In three cases (DA WD Sirius B, 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 an MS 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 an MS companion would have had any influence on the evolution of the WD predecessor through the RG and AGB stages, although the MS 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 uDDs 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 (≲0.45 M_⊙), 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 e.g. section 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_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 Appendix  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 Appendix  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 arcsec 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 per cent 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 characterized 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_eff = 5270 K and log g = 8.0 but its spectral type (and magnetic nature) are unknown. The list is given in Table 1, which will be fully described in Section 5.

The mean mass of all WDs of the local 20 pc volume is 0.64 M_⊙ with a standard deviation of 0.13 M_⊙, fully consistent with the mean mass of non-MWDs of |$0.66 \pm 0.14\, \mathrm{M}_\odot$| estimated by Tremblay et al. (2013). The temperature distribution is highly skewed towards 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 Section 5 and in the analysis that will be carried out in Section 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. MWDs 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 〈|B|〉, i.e. the surface field strength averaged over the visible stellar disc. Low-resolution spectroscopy of WDs may detect a magnetic field when its strength is at least 1–2 MG, while high-resolution spectroscopy (say |$R \gtrsim 50\, 000$|⁠) 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 disc, or mean longitudinal field 〈B_z〉, 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, 〈B_z〉 may be measured with an uncertainty of a few hundred Gauss (see e.g. Landstreet et al. 2015; Bagnulo & Landstreet 2018). Circular polarization integrated over a spectral line is zero, therefore broad-band polarimetry is not useful to detect magnetic fields in WDs, unless the field is sufficiently strong to polarize 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 ∼0.5 MG (in absolute value) may be detected via the measurement of circular polarization in the continuum. Polarization in the continuum may be measured by spectropolarimetry or by narrow-band or broad-band circular polarimetry. Since polarization 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 broad-band polarimetric measurements. However, spectropolarimetry is often more affected by instrumental effects than simple imaging polarimetry, and our experience with FOcal Reducer and low dispersion Spectrograph (FORS2) and Intermediate-dispersion Spectrograph and Imaging System (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, multicolour narrow-band circular polarimetry may be the most suitable detection method when the polarization signal is ≲0.3 per cent 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 ≳100 MG), in which spectral lines may become very difficult to recognize, 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 characterization 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, 〈|B|〉 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 polarize 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 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 characterization of strength and morphology of a stellar magnetic field requires good sampling of the rotational cycle, for instance, 10 measurements taken about 0.1 cycles apart from each other.

3.3 Measurements of magnetic fields of WDs from previous literature

MWDs 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.

• Narrow-band and broad-band circular polarization 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 polarization was detected. These early surveys reached a degree of accuracy for V/I often of the order of 0.05 per cent 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 ∼0.2 per cent (see section 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 polarize the continuum. However, we are not aware of any systematic broad-band or narrow-band circular polarization 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.

• 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.

• In the 1990s, spectropolarimetry started to be applied more systematically to observation of WDs, mainly thanks to the survey of Schmidt & Smith (1995), Putney (1997), and later Kawka et al. (2007). These surveys allowed the measurement of 〈B_z〉 with a typical uncertainty of ∼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.

• With the use of the FORS instrument at the ESO VLT, the sensitivity of the 〈B_z〉 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, 2017). Later, Landstreet et al. (2015) demonstrated that the Echelle SpectroPolarimetric Device for the Observation of Stars (ESPaDOnS) instrument of the Canada–France–Hawaii Telescope (CFHT) could also be used to measure 〈B_z〉 in DA and DB WDs with high accuracy, and Bagnulo & Landstreet (2018) showed that the ISIS instrument of the William Herschel Telescope (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 |〈B_z〉| for 17 WDs in which no magnetic field had been detected. 10 of these non-MWDs 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 neighbourhood, finding that 9 out of the 43 WDs within 13 pc from the Sun were magnetic, for a global frequency of 21 ± 8 per cent. 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 ± 4 per cent. 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 ∼70 WDs that were never observed polarimetrically before, and another ∼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, FORS2, AND ISIS

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 data base 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 FORS2 instrument (Appenzeller et al. 1998) of the ESO Very Large Telescope (VLT), with the ISIS instrument of the WHT, and with the ESPaDOnS instrument (Donati et al. 2006) of the CFHT. 11 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 broad-band polarimetric techniques in the 70s. Only three stars could not be checked for magnetic field for the reasons explained in Section 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 ∼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 polarization, 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 ∼3700–9300 Å; we set the slit width to 1.2 arcsec for a spectral resolving power of 360 (for a justification of the choice of not using the order separating filter, see appendix A of Patat et al. 2010). For DQ stars, we used grism 600 B, covering the spectral range ∼3600–6200 Å; slit width was set to 1.0 arcsec for a spectral resolution R ∼ 780. For DA stars, we mainly used grism 1200 R + filter GG435, covering the spectral range ∼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 arcsec, for a spectral resolving power of 1400. We also report two 〈B_z〉 measurements of one star, WD 0233−242, that were obtained with FORS2 in 2013, but from which previous literature had reported only the 〈|B|〉 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 ∼3700–5200 Å with a spectral resolving power of 2600, and ∼6100–6900 Å with a spectral resolving power of 8600, respectively (for a 1 arcsec slit width). We note that for some of the observations, we erroneously adopted a 2 × 2 binning readout, which led to a slight undersampling, effectively reducing the spectral resolution in the red arm by ∼30 per cent.

4.4 Data reduction and field measurements

Bagnulo & Landstreet (2020) have shown that in FORS2 data, the background illuminated by the moon may appear circularly polarized because of cross-talk from linear to circular polarization. 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 polarization, stronger in the blue and decreasing with wavelength, being generally little noticeable at λ ≥ 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 ≃ 15.5 when not observed in dark time. We have found that spurious polarization is minimized 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 ∼2 arcsec.

The observing log and derived magnetic field values are given in Table A1. In Appendix  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

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:

• 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 an MWD via spectroscopy.

• 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 〈|B|〉.

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

• 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 (〈|B|〉 ≲ 100 kG), but it should be recalled that some DAs have fields so weak that they may be detected only via spectropolarimetry.

• 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 〈|B|〉.

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 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 is 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 give looser constraints to the magnetic field of individual members. DQ stars should be practically considered as featureless stars, in that only continuum polarization may firmly reveal the presence of a magnetic field (Berdyugina, Berdyugin & Piirola 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 ≃0.4 and 8 kG, except for one DZ star (WD 0552−106) for which the uncertainty of its non-detection was ∼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 (〈|B|〉 ≲ 1 MG).

There are four DA stars that are so cool that spectropolarimetry of H α 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 α and/or by the lack of polarization 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 |〈B_z〉| ≲ 1.5 MG, WD 0727+482A for which at best we can say that 〈|B|〉 ≲ 1 MG, and WD 0751−252 and WD 1823+116, for which we can set an upper limit of 0.5 MG to 〈|B|〉. We also note that the fainter component of the uDD system WD 0727+482B could also be a DA star, but its characterization 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 α in which spectropolarimetry is not very useful. However, the presence of a very narrow core allows us to set an upper limit for 〈|B|〉 at about 50 kG. One DA was observed with σ_z ≃ 35 kG, and five DA/DAZ WDs with σ_z = 10–15 kG (all these DAs have T_eff ≲ 5000 K except WD 0728+642). About 20 DA/DAZs have σ_z between 1 and 10 KG (but mostly ≃ 2 kG), and all the remaining non-magnetic DA and DAZ WDs, about 40 stars, were observed with sub-kG precision.

In Appendix  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.

• Based on a spectropolarimetric measurement obtained at the Mount Stromlo Observatory (〈B_z〉 = −6.1 ± 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 〈B_z〉 = −0.10 ± 0.44 kG, and with FORS2 by Bagnulo & Landstreet (2018), who found 〈B_z〉 = −0.20 ± 0.23 kG. We consider this star to be non-magnetic.

• 40 Eri B = WD 0413−077 was long considered a weakly MWD (Fabrika, Valyavin & Burlakova 2003; Valyavin et al. 2003; Ferrario, de Martino & Gänsicke 2015). However, Landstreet et al. (2015) made a number of extremely precise measurements of the longitudinal field using both the ESPaDOnS instrument of the CFHT 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.

• 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.

• Liebert & Stockman (1980) observed the cool DA star WD 1820+609 (at that time believed to be DC), and found a signal of broad-band polarization consistent with zero. Putney (1997) re-observed the same star in spectropolarimetric mode and found non-zero continuum polarization in the instrument red arm (≃−0.5 per cent, judging from her fig. 2i, and reporting this value of the polarization with our sign convention). Putney (1997) declared this signal as possibly spurious but stated that H α 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 10 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 |〈B_z〉| set by Liebert & Stockman (1980), but H α spectroscopy set for 〈|B|〉 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 summarized in Table 1, a list of physical parameters of WDs in the local 20 pc volume. This list is organized as follows. Column 1 gives the name of the star (if the star is magnetic, its name is printed in boldface); column 2 the V magnitude generally taken from the SIMBAD data base; 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. Colum 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, an 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 polarized. 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 Appendix  B. Suspected MWDs are labelled in column 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 DD or uDD. The symbol s designates 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 Section 2.2.2.

For the determination of the stellar parameters such as atmospheric composition (column 6), temperature (column 7), log g (column 8), mass (column 9), and age (column 10), we have used the references given in column 11 (see also Section 2). Column 12 gives an estimate of the average 〈|B|〉 of the MWDs. For stars in which 〈|B|〉 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, 〈|B|〉 changes as the star rotates, but usually not by very much, so that even a single 〈|B|〉 measurement allows us a good estimate of the average field strength of a star. For weaker field MWDs, however, 〈|B|〉 cannot be measured directly, and what we know with good accuracy is the average longitudinal component of the magnetic field 〈B_z〉, 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 Appendix  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 〈B_z〉 measurement and the number of measurements available, or the upper limit for |〈B_z〉| as deduced from polarimetry of the continuum (for featureless stars), or the upper limit for 〈|B|〉 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 〈|B|〉. When only spectroscopic data are available, as in the case of four WDs only, this estimate is straightforward and provided directly in column 13. For the large majority of stars, a number of 〈B_z〉 measurements are available; these measurements set a stronger constraint than spectroscopy alone, but estimating an upper limit for 〈|B|〉 is modelling-dependent. 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_〈|B|〉 the lowest value of σ_z multiplied by four, that is, |$s_{\langle \vert B \vert \rangle }= 4\, {\rm MIN}(\sigma _z)$|⁠. For example, for star WD 1202−232, the upper limit for 〈|B|〉 from direct spectroscopic measurement of its H α is 50 kG, but there are also four 〈B_z〉 measurements available, the most precise of which has σ_z ≃ 0.4 kG. We say that this star has s_〈|B|〉 = 1.6 kG. Under the approximation that the field has a dipolar structure, s_〈|B|〉 may be interpreted as an approximate estimate of the upper limit for 〈|B|〉 (this interpretation will be discussed again in Section 6.3.3). In the majority of cases of Table 1, this estimate is between 0.5 and 2 times the upper limit of 〈|B|〉 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 〈|B|〉. Finally, for featureless stars, for which only polarization of the continuum is available, s_〈|B|〉, expressed in MG, has been set equal to 15 times the upper limit of |V/I|, expressed in per cent units.

The last column of Table 1 contains a reference to the literature reporting field measurements for the star. The references to older broad-band polarimetric measurements of non-featureless WDs, superseded by modern spectropolarimetric data, are given between parenthesis. More details about individual stars are given in Appendix  B.

A general overview of the statistics that can be deduced from Table 1 is shown in Fig. 2. The top panels show the distributions of magnetic and non-MWDs with temperature, age, and mass. The bottom panels help to visualize 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_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 20–30 000 K) as H ionizes and the lines weaken, and below about 7000 K as the lines weaken because the population of the n = 2 levels of neutral H diminishes.

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 Section 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 per cent. Extrapolated to the Galactic sample, the fraction of magnetism is 22 ± 4 per cent, 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 (Section 6.1) or chemical composition of the atmosphere (Section 6.2). We also study the distribution of the field strength (Section 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 (Section 6.4) or stellar mass (Section 6.5).

6.1 The fraction of MWDs among WDs of different spectral classes

We first discuss magnetism in WDs of different spectral classes. The general statistical results are summarized in Table 2, 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 + dr.

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 1. DAZ stars, which are DA stars with a spectrum polluted by the presence of metal lines, are treated separately below. 18 DAs have a confirmed magnetic field, for a frequency of 23 ± 5 per cent. We note, however, that four DA stars in which no magnetic field was detected have in fact such weak H α lines that constraints on their field measurement come only from continuum polarimetry (s_〈|B|〉 = 0.5–1.5 MG, see Section 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 ≃25 per cent. 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 ∼200 MG (Jordan 2003) level of Grw + 70° 8247.

About 25 per cent 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 per cent (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 per cent). Two of the 12 DAZ WDs of the local 20 pc volume are magnetic, for a frequency of 17 ± 11 per cent. The incidence of the magnetic fields in DA and DAZ taken together is 22 ± 4 per cent.

Kawka et al. (2019) analysed a sample of 15 DAZ stars with T_eff between ≃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 ± 12 per cent. However, when they restrict their analysis to DAZ WDs cooler than T_eff = 6000 K, Kawka et al. (2019) find that four out of seven DAZ WDs are magnetic, and suggest that about 50 per cent of DAZ WDs cooler than T_eff = 6000 K are magnetic (the frequency is 50 ± 18 per cent). 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 ± 9 per cent, fully consistent with the frequency estimated for normal DA WDs of all ages, while 4 out of 12 DAZ cooler than 6000 K are magnetic, for a frequency of 33 ± 14 per cent. 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 K ≤ T_eff ≤ 7000 K are magnetic, and 5 out of 18 DA WDs with 5200 K ≤ T_eff ≤ 6000 K are magnetic, for a magnetic frequency of |$35\, \pm 9$| per cent and |$28\, \pm 11$| per cent, 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 broad-band or spectropolarimetry. 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 broad-band 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 ≃13 per cent. We note that the sensitivity of most of the new measurements is ≃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 per cent, for an estimated Galactic frequency of, say, 30 ± 10 per cent.

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, Jordan & Koester (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 per cent 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 ± 14 per cent. 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 disc 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 10 000 K, have helium-dominated atmospheres (Koester, Kepler & Irwin 2020), and the presence of carbon is a result of dredge-up by the extending convection zone in the upper helium layer (Koester, Weidemann & Zeidler 1982; Pelletier et al. 1986). The 17 DQ stars of Table 1 represent more than 10 per cent of the local 20 pc WD population, a proportion much bigger than the fraction of 1.6 per cent of DQs among all WDs classified by the Sloan Digital Sky Survey (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 arcsec 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 〈|B|〉 (≃ 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, Bergeron & Fegley 1995) are magnetic. The ratio of 5 MWDs out of 16 DQ WDs results in a magnetic frequency of 31 ± 12 per cent, 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 hypothesize 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 |$T_{\rm eff}= 18\,000\!-\!24\,000$| 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 ± 4 per cent; and 13 out of 43 WDs with an He-rich atmosphere are magnetic, for a frequency of 30 ± 7 per cent. Except for the four magnetic DZ stars discussed by Bagnulo & Landstreet (2019b), all He-rich MWDs have a field strength from one 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 represent another bias for the statistics, if the frequency of field occurrence changes with cooling age (see Section 6.4). Five out of eight He-rich WDs older than 5 Gyr are magnetic, for a frequency of 62 ± 17 per cent. Such a high frequency among the oldest stars is not mirrored by H-rich WDs, as only 4 out of 25 H-rich WDs older than 5 Gyr are magnetic, for a frequency of 16 ± 7 per cent. 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

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 per cent. 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. Fig. 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 emphasize 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-infrared 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, Figs 5 and 6 show that ∼40 per cent 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 per cent of the total. Fig. 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 ≲40 kG

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 4 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 〈|B|〉 upper limits (see our considerations about s_〈|B|〉 of Section 5.3). So far, outside the local 20 pc volume we are aware of only three MWDs with 〈|B|〉 possibly smaller than 30–40 kG, namely WD 1105−048 and WD 0446−789, which were already discovered to be magnetic or suspected MWDs 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) |〈B_z〉| never rises above about 10 per cent of the value of 〈|B|〉. 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 polarize 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 recognize spectral features (if they are present at all), so that detection relies on measurements of circular polarization 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 〈B_z〉, which is the component of the magnetic field along the line of sight, averaged over the stellar disc. This average is dominated by the dipolar component of the magnetic field, while higher order multipolar components tend to contribute very little to 〈B_z〉, since their line-of-sight component averaged over the stellar disc, 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 〈B_z〉 will be about 1/10 the dipolar contribution (Schwarzschild 1950; Landolfi, Bagnulo & Landi Degl’Innocenti 1998). To make this as clear as possible, even if 〈B_z〉 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, Landi Degl’Innocenti & Landi Degl’Innocenti 1996) that produces a surface field 〈|B|〉 up to 30–40 kG, and still be undetectable either via spectroscopic or spectropolarimetric techniques. In this respect, polarimetric weak field detection techniques for WDs are less efficient than in the case of MS stars, in which often the projected surface velocity resolution of Stokes V introduced by rotation may allow one to detect a field even when 〈B_z〉 ≃ 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 characteristic 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 ≳1 MG, finding that the frequency of MWDs with MG field strength is 3.5 ± 0.5 per cent among WDs hotter than 10 000 K, and 20 ± 5 per cent among WDs cooler than 10 000 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_eff near or above 10 000 K, only 2 ± 0.8 per cent 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 K ≲ T_eff ≲ 10 000 K) modelled by Bergeron, Ruiz & Leggett (1997), and an earlier version of the 13 and 20 pc volume-limited samples, collected by Holberg, Oswalt & Sion (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 ± 2 per cent, and in the volume-limited samples the frequency of MG fields is 11 ± 5 per cent in the complete 13 pc volume, and 8 ± 3 per cent 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_eff. However, Liebert et al. (2003) also argued that this apparent difference could probably be 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-MWDs, 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 one-fourth as large as the one surveyed for lower mass, presumably less frequently MWDs. 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.

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 1 we can see that only 4 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 ≃0.36 Gyr. The frequency of magnetism of 1 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 per cent derived for the full sample of DA WDs.² These data predict the galactic frequency of young magnetic DA WDs to be 5.0 ± 4.9 per cent. 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 Fig. 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?

The key concept is that for a given T_eff, because of its smaller surface, a higher mass WD will certainly have a lower luminosity (L ∝ 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_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_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 data base, which reports T_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_eff (upper panel), luminosity (mid-panel), and the absolute B magnitudes (lower panel) of WDs of 0.6, 0.9, and 1.2M_⊙ as functions of cooling age from formation, covering the typical range of non-MWDs, fairly massive MWDs, and extremely massive MWDs. If we compare a 0.6 M_⊙ (a typical non-MWD) and a 0.9M_⊙ 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_eff values have declined to about 25 000 K (0.6 M_⊙) and 35 000 K (0.9 M_⊙), 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 |$0.9\, \mathrm{M}_\odot$| 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 ∼10 per cent greater than the fainter, which amounts to a difference in survey volume of ∼30 per cent. 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 per cent to correct for the different volumes surveyed. Note, however, that the extremely massive |$1.2\, \mathrm{M}_\odot$| WD is as much as 0.5 mag fainter in B relative to the WD of |$0.6\, \mathrm{M}_\odot$| 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 Section 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 ± 4 per cent.

From the stellar parameters of the 347 WDs identified in the P–G survey (Liebert et al. 2005), we find that about 90 per cent of these stars have cooling ages ≲0.5 Gyr. This young group includes about 312 WDs, six of which are magnetic, for a frequency of 1.9 ± 0.8 per cent 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 ± 3 per cent, in very good agreement with their estimates for cooler WDs, and also consistent with our estimate of 13 ± 4 per cent. 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 ± 1 per cent, 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 data set 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 (McCleery et al. 2020; Tremblay 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 per cent are DA stars, in which fields of an MG or more are readily identified in good classification spectra, and fields 10 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. McCleery et al. 2020; Napiwotzki et al. 2020; Tremblay et al. 2020).

Within the 40 pc volume we know of nine MWDs younger than 0.5 Gyr (see Table 3, which includes WD 2047+372, the only young MWD in the local 20 pc volume, and WD 0232+525, a new MWD 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 per cent 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 seem 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.

6.4.6 The higher incidence of magnetic fields in certain classes 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 Section 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 disc 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 Section 6.2 that the frequency of the occurrence of magnetic fields in He-rich WDs (30 ± 7 per cent) is higher than in H-rich WDs (19 ± 4 per cent), 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 0.5–5 Gyr, the frequency of the occurrence of magnetic field in H-rich and He-rich WDs is very similar: 25.9 ± 5.7 and 22.9 ± 7.1 per cent, respectively.

6.4.7 Do older MWDs have stronger fields than younger MWDs?

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, |$0.93\, \mathrm{M}_\odot$|⁠) 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. Fig. 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

Fig. 2 shows no obvious correlation between field strength and stellar mass, but it is clearly apparent that magnetic and non-MWDs 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 |$M \gt 1\, \mathrm{M}_\odot$| 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 |$0.64 \pm 0.13\, \mathrm{M}_\odot$|⁠. The mean mass of non-MWDs is |$0.62\, \mathrm{M}_\odot$|⁠, with a standard deviation of |$0.13\, \mathrm{M}_\odot$|⁠, and the mean mass of MWDs is |$0.69\, \mathrm{M}_\odot$|⁠, with a standard deviation of |$0.10\, \mathrm{M}_\odot$|⁠. Marginal evidence for correlation with age will be discussed in Section 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 |$0.69 \pm 0.10\, \mathrm{M}_\odot$| (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 Section 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 |$0.78 \pm 0.05\, \mathrm{M}_\odot$|⁠, but they found also that the average mass of more weakly MWDs is consistent with that of non-MWDs, 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 MWDs because field detections in this sample have relied heavily on spectroscopic data), McCleery et al. (2020) found that the mean mass of MWDs is |$0.75 \pm 0.05\, \mathrm{M}_\odot$|⁠.

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 |$0.67 \pm 0.08\, \mathrm{M}_\odot$|⁠, while that of MWDs younger than 2 Gyr is |$0.72 \pm 0.13\, \mathrm{M}_\odot$|⁠. The mean mass of MWDs younger than 1 Gyr is |$0.80 \pm 0.10\, \mathrm{M}_\odot$| (all standard deviations, not standard errors). Fig. 10 shows the mean and median masses of magnetic and non-MWDs 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 |$M \sim 0.82\, \mathrm{M}_\odot$|⁠. 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 |$\gtrsim 0.75\, \mathrm{M}_\odot$| and field strength ≳1 MG, with the exception of the 2.3 MG star PG 2329+267, which is listed with |$M \simeq 0.61\, \mathrm{M}_\odot$|⁠. 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 |$M \gt 0.75\, \mathrm{M}_\odot$|⁠, and PG 2329+267 has |$M = 0.91\, \mathrm{M}_\odot$|⁠. If we turn our attention to the part of the local 40 pc volume of Section 6.4.5 (see in particular Table 3), 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 3 includes WD 1105−048 (⁠|$M \simeq 0.58\, \mathrm{M}_\odot$|⁠, 〈|B|〉 ≃ 10 kG) and WD 1105−340 (⁠|$M=0.67\, \mathrm{M}_\odot$|⁠, 〈|B|〉 ≃ 150 kG). Outside the 40 pc volume we know of WD 0446−789 (⁠|$M \simeq 0.55\, \mathrm{M}_\odot$|⁠), a very young MWD already mentioned in Section 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-MWDs, 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 was found in magnitude-limited surveys, from which stronger magnetic fields seem to be associated with 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 an MS star, DD and uDD systems. Five of these systems (three CPM and two uDD systems) include an MWD, for a frequency of 14 ± 6 per cent, marginally lower than in single WDs. We note, however, that some of these systems include a DC WD and that the polarization signal of an 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 summarizing the observational constraints that we have gathered from our volume-limited survey of WDs (Section 7.1). Then (Section 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 a 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 (Section 7.3).

7.1 Observational constraints

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

7.1.1 Overall incidence of magnetic fields

About 20–25 per cent 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 Section 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-MWDs. 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 |$M \gt 1\, \mathrm{M}_\odot$| 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 Section 7.1.

7.2.1 The fossil field hypothesis

The possibility of some level of magnetic flux conservation during evolution from the MS 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 an MS structure to a WD structure two orders of magnitude smaller in radius (the fossil field hypothesis). Such a transformation could convert the MS fields of the order of 10⁴ G found in Ap and Bp MS stars (e.g. Donati & Landstreet 2009) to fields of roughly 10⁸ G = 100 MG.

It is now very clear that the strongly magnetic Ap and Bp MS 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 per cent of A and B MS 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 per cent 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 MS F stars, which require a few Gyr to end their nuclear-burning lives. These lower mass (ca. 1.2–1.6M_⊙) 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 per cent of the younger WDs, well below the observed frequency of more than 20 per cent. 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 MS 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 MS A or B stars (which are unrelated to the surface magnetic fields of the Ap and Bp stars). A series of papers summarized by Stello et al. (2016) discusses how a dynamo-driven field produced in the strongly convective core during MS evolution of an intermediate-mass star (e.g. Braithwaite & Spruit 2004; Brun, Browning & Toomre 2005) can leave a strong, stable field (Duez, Braithwaite & Mathis 2010) which can survive into the RG 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 time-scale 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 Section 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 WDs, 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 |$1\, \mathrm{M}_\odot$|⁠) and progenitor masses of around 5 or |$6\, \mathrm{M}_\odot$|⁠. 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 Section 7.2.3; instead they are almost certainly descended from single upper MS stars. These discoveries, if fully confirmed, appear to require that at least some MWDs descend from single MS 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 on to a very close WD companion. In about 25 per cent of such systems, the WD companion has an 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 RG, 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 an MS 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, b) carried out synthetic evolution calculations of binary star populations, using simple parametrized 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 MWD 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 per cent 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 crystallization

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 WD 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 crystallization 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 10³ s is sufficient to ensure saturation. A straightforward 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 dynamo efficiency on the magnetic Prandtl number leads to the expectation that much larger fields than 1 MG could be generated.

Fig. 11 shows the cooling age–mass diagram for magnetic and non-MWDs, compared to the lower limit of the age for different masses at which crystallization convection begins, as calculated by Bédard et al. (2020) for WDs with thick (q_He = 10⁻², q_H = 10⁻⁴) and thin (q_He = 10⁻², q_H = 10⁻⁴) hydrogen layer, where q_X = M_X/M_*. It appears that the frequency of the occurrence of magnetic fields after crystallization has begun, 29 ± 5 per cent, is twice the frequency in WDs with a fully liquid core, 14 ± 4 per cent (these frequencies are estimated assuming that the three MWDs between red and blue lines have not reached yet in the crystallization sequence). Fig. 11 suggests that a dynamo linked to the crystallization could be a primary mechanism that leads to the generation of magnetic fields in degenerate stars. However, we recall that the operation of this mechanism requires (1) a large increase by an unknown ‘scaling law’ factor of the fields estimated by Isern et al. (2017) to reach the observed field strengths, and (2) rapid rotation (periods of minutes or hours) to allow the crystallization dynamo to operate at peak output. It is not known at present whether either of these conditions is satisfied. Thus, although the crystallization dynamo may explain the general rise in frequency of magnetic fields already noted in Fig. 7, this possibility needs further study before being accepted as an important mechanism.

7.2.5 Mass differences between magnetic and non-MWDs

A clear feature of the MWDs of the local 20 pc volume is that their average mass is about |$0.07\, \mathrm{M}_\odot$| larger than the mean for the non-MWDs, although there is substantial overlap between the two distributions.

A straightforward interpretation of the overall excess mass of MWDs is that they may be formed preferentially from massive progenitors. This would be consistent with the hypothesis that MWD fields descend from single (magnetic and non-magnetic) A and B MS stars, or anyway from high-mass progenitors. This observation would also be consistent with the possibility that MWDs are produced in binary mergers, which will tend to lead to more massive than average WDs (Briggs et al. 2015). If most MWD fields are generated earlier in the evolution of the progenitor star by dynamo action, it is not unreasonable to suppose that the field creation process should depend fairly strongly on MS progenitor mass, and thus lead to rather different outcomes in WDs of different masses.

A particularly interesting aspect of the mass distribution of the MWDs of the local 20 pc volume is that MWDs with cooling ages below about 1–2 Gyr appear to be still significantly more massive than the average MWD mass. This constraint fits into the scenario in which at least some MWDs develop their magnetic field after crystallization: higher mass WDs crystallize sooner (even much sooner) than lower mass WDs, so there must be more MWDs among younger higher mass WDs than among younger lower mass WDs. After a certain amount of time, all WDs have gone through the process of crystallization and perhaps the residual difference in mass between MWD and non-MWDs is due to the fact that WDs with higher mass progenitors are most likely to be magnetic than WDs with lower mass progenitors and/or the result of binary mergers.

Finally, the observation that among the youngest WDs, the strongest fields are found in WDs with mass substantially higher than average, while in older stars, strong magnetic fields are found also among WD with average mass, needs to be verified with additional data. If confirmed, it would suggest again that the more massive MWDs are visibly magnetic almost immediately after formation, while the less massive MWDs require 1–2 Gyr to reveal their fields.

7.3 Evolution of the magnetic field during the WD cooling phase

The apparent deficiency of magnetic fields in the youngest WDs is consistent with different scenarios: a fossil field that emerges slowly from the interior to the surface by a diffusion process, and which can have a characteristic time-scale as long as 1–2 Gyr; or a dynamo generated field that forms when crystallization begins. At some stage, though, all field generation mechanisms discussed above, including dynamo acting during crystallization, must cease operation. The fossil theory suggests that any fields produced by earlier active mechanisms should evolve simply by diffusion and Ohmic decay. In this case, the time-scale for surface field strength growth (by field diffusing to the surface) or field decline (because of resistive losses to the supporting currents) could be as slow as the global decay time, which has been estimated to range from 0.6 to 4 Gyr for WD masses between 0.4M_⊙ and 1M_⊙ (Fontaine, Thomas & van Horn 1973).

Because we now have a record of field occurrence and strength in WDs as old as 5–8 Gyr, at which ages we find, in fact, the strongest magnetic fields in the local 20 pc volume sample, with only a modest decline in the frequency of field occurrence and no significant change in the field strength distribution (see Figs 2 and 7), it does not appear that simple Ohmic decay dominates evolution during the WD cooling phase. It is quite possible that internal field reorganization of large, invisible internal field structures (initially possibly of several thousands MG in strength?) replace surface magnetic flux that is lost by Ohmic decay, or that an internal dynamo like the one discussed in Section 7.2.4 is acting until late stages of the cooling phase. We are generally not sensitive to weak fields in older stars, but a highly sensitive survey of broad-band circular polarization aimed at probing Ohmic decay in cooler WDs could help to better understand the evolution of the magnetic fields in WDs.

8 CONCLUSIONS

We have carried out a spectropolarimetric survey of WDs that, complemented with literature data, has made it possible for us to compile an almost complete data base of highly sensitive magnetic field measurements of the local 20 pc WD population. This volume-limited sample contains approximately 152 known WDs (including secondaries from a somewhat uncertain number of uDD stars). In the course of our survey, we have observed 70 stars that have never been observed in spectropolarimetric mode before, and another 20 that had been already observed with polarimetric techniques, but with typically 1 dex lower precision than we have achieved with our new measurements. In the course of our survey of the local 20 pc volume, we have discovered 12 new MWDs, that is, more than a third of all known MWDs of the local 20 pc volume. Combining our new data with those from previous literature, field strength estimates and upper limits are now available for 149 WDs of the local 20 pc volume. This sample is the best current approximation of a statistically unbiased observational data base of MWDs, and some of its characteristics may well be representative of the entire Galactic population of WDs, or at least of WDs formed at several kpc from the Galactic Centre.

We have found that at least 33 of the observed 149 WDs have magnetic fields. We have also used the results of this survey to explore and test proposals in the literature concerning abnormally high or low frequencies of magnetic fields in various spectral classes of WDs, and to check if the frequency of the occurrence of MWDs correlates with cooling age and mass.

We found that among the commonest class, the DA WDs, 23 ± 5 per cent of the stars have detected magnetic fields. This figure serves as a reference frequency.

The incidence of magnetic fields in DC stars is 13 ± 6 per cent, approximately half that of WDs of spectral class DA, but this is likely an artefact due to the fact that in featureless WDs we can detect only magnetic fields with a strength ≳1 MG. Since we know that about half of the magnetic DAs have fields that do not produce a significant polarization in the continuum, it is quite possible that the incidence of magnetism in DCs is actually underestimated by a factor of order two, while being effectively similar to that found for DA WDs.

There are claims in the literature that magnetic fields are more frequent in WDs with metal lines, especially in the cooler ones. Indeed, combining our data with those of Kawka et al. (2019), we found that the frequency of MWDs among DAZ cooler than 6000 K is 33 ± 14 per cent, and the same result is obtained if we consider the DZ and DZAs WDs of the local 20 pc volume. It has been suggested that this could be explained by the formation of a magnetic field during the accretion of rocky debris (Farihi et al. 2011; Kawka & Vennes 2014). However, we have shown that this statistics is fully consistent with the frequency of the DA MWDs in the similar age/temperature range, therefore we suggest that the high incidence of magnetic fields observed in cooler stars with metal lines simply reflects the fact that magnetic fields are more frequent in cooler/older WDs than in hotter/younger WDs, as it will be best summarized later. In particular, the presence of metal lines in a spectrum otherwise featureless like that of DZ WDs allows us to detect weak magnetic fields that might be pretty common, but undetected, in DC stars.

The local 20 pc volume seems richer in DQ stars than what was expected from the estimate of their occurrence (a few per cent of the WDs), as more than 1 out of 10 of the local population of WDs belong to this spectral class. The frequency of fields among the 16 cool DQ and DQpec WDs is about 31 ± 12 per cent. Taken at a face value, this frequency is not significantly higher than that of magnetic DAs. However, only fields stronger than average may be detected in DQ WDs. If the field strength distribution found in DA WDs applies also to DQ WDs, then DQ stars of the local 20 pc volume could be characterized either by a much higher frequency of magnetic fields than average, or by the presence of magnetic fields much stronger than average. In either case, we could speculate that the link between magnetic field and presence of C₂ bands in an He-rich atmosphere could be due to C₂ buoyancy induced by the magnetic field. In some respect, DQ WDs could be the equivalent of chemically peculiar stars of the upper mean sequence. The immediate next step will be to check a much wider sample of DQs for the presence of magnetic field.

An important result of our analysis is that field strength frequency distribution is approximately constant per decade over a range extending from a few tens kG up to several hundreds MGs. This is in very sharp contrast to the field frequency distribution found in the overall sample of known MWDs, which (because of the impact of SDSS on new discoveries) exhibits a very strong frequency spike between about 2 and 80 MG. Furthermore, because of the extreme sensitivity of many of our measurements which have nevertheless revealed almost no WD fields below 〈|B|〉 ≈ 40 kG, we argue that we may have detected the effective lower limit of the field strength frequency distribution. That is, we suggest that WD fields occur essentially in the range of 40 kG to several hundred MG.

We found that in the local 20 pc volume, the frequency of the occurrence of the magnetic field in young stars is significantly lower than average (only 1 out of 20 WDs younger than 0.5 Gyr is magnetic). A qualitatively similar result had been obtained in the past but called into question as possible effect of an observational bias against high-mass WDs. With the help of theoretical models, we have argued that magnitude-limited surveys are not strongly biased against hot high-mass WDs, except when the stellar mass starts to exceed |$1\, \mathrm{M}_\odot$|⁠. Therefore, the low number of MWDs found by magnitude-limited surveys reflects mainly the fact that magnetic fields are rare in younger WDs. This result is tentatively supported by a superficial analysis of the situation in the 40 pc volume-limited sample of WDs. The shortage of young MWDs suggests that some surface fields may be produced during WD evolution, or at least revealed during this evolution, rather than all being inherited from earlier evolution and being revealed as soon as the WD is born. We also note that in the local 20 pc volume, strong magnetic fields are more common in older than in younger WDs, but this result is statistically too weak to be extended outside of the local 20 pc volume, where we have no data to confirm its validity.

We have found that the mean mass of the MWDs of the local 20 pc volume is higher than average, that is, |$0.69 \pm 0.10\, \mathrm{M}_\odot$| (where the uncertainty is the variance of the distribution). This result is qualitatively in agreement with what has been previously suggested in the literature, except that the mean mass value of the MWDs of the local 20 pc volume is lower than what had been estimated from magnitude-limited spectroscopic surveys. Considering that magnitude-limited spectroscopic surveys were probably biased in favour of young MWDs, and considering that in the local 20 pc volume, MWDs younger than 2 Gyr are on average more massive than older MWDs (but at a low-significance level), it is possible that in general, younger MWDs are more massive than older MWDs. In the local 20 pc volume we did not find a correlation between the mass and the field strength, except perhaps for the younger MWDs. When we consider MWDs of all ages, the mean mass of MWDs with field weaker than 2 MG is the same as the mean mass of the MWDs with field stronger than 2 MG. Observations outside of the local 20 pc volume suggest that in young MWDs, strong fields, when found, occur only or mainly in WDs with mass higher than the average (⁠|$\gtrsim 0.75 \!-\! 0.80\, \mathrm{M}_\odot$|⁠). Young lower mass MWDs seem to host weaker fields (which as such tend to escape detection in spectroscopic surveys). However, further observations are needed to better clarify the relationship between mass, age, and field strength.

We have explored possible constraints on the mechanism(s) leading to the occurrence and evolution of magnetic fields in WDs. We have shown that the frequency of the occurrence of magnetic fields in WDs appears to rise steadily with increasing cooling age. Interestingly, this rise appears to coincide roughly with the ages at which cooling WDs start to experience core crystallization, and using the onset of this process as a divider line in a age–mass diagram, we find that the frequency of MWDs is roughly twice as large on the cool side of the crystallization boundary as on the hot side. This suggests that the presence of a dynamo as predicted by Isern et al. (2017) is a possible mechanism for the production of magnetic fields. This dynamo mechanism, which is similar to the one that produces the magnetic field of the Earth, is due to a combination of rapid rotation and core convection that is driven by sinking of solid O. While the mechanism as originally presented by Isern et al. (2017) could only explain fields up to ∼100 kG strength, Schreiber et al. (2021) have argued that it could actually produce much stronger fields. We note that the operation of this mechanism would be qualitatively consistent with the observed dearth of young MWDs, and with the higher average mass of the MWDs compared to the non-MWDs among younger stars (because higher mass WDs crystallize earlier than older WDs). However, both the suggested strong increase of the dynamo efficiency, and the necessary rapid stellar rotation are still to be demonstrated, therefore the connection of the increased frequency with the core crystallization must remain speculative at present. That fields appear more frequent in older than in younger WDs could be also explained by slow emergence of fossil fields at the stellar surface, and the fact that MWDs have a mass higher than average is consistent with other possible channels of field formation, including that MWDs are the results of merging. The new data suggest in fact that more than one channel for field formation in WDs exists.

The evolution of fields during WD cooling also provides an important constraint on the magnetic field physics. On the one hand, after formation, the field evolution of MWDs should be governed mainly by diffusion and Ohmic dissipation on time-scales of the order of 1–2 Gyr. This naively suggests that typical field strength should decline gradually with cooling age. In contrast, the distribution of field strengths is observed to remain roughly constant with age out to several Gyr. Possible solutions to this dilemma are that the declining complexity of very large internal fields tends to replace surface flux lost from the surface by Ohmic decay, and/or the presence of a dynamo acting until late in the cooling phase. Apart from improving the statistics of the occurrence of magnetic fields, the modelling of individual stars of different masses and ages may further help to identify the mechanism(s) that generate and possibly sustain the magnetic field of WDs.

Extending our conclusions beyond the local 20 pc volume, we infer that in this part of the Galaxy, and potentially on a Galactic scale, 22 ± 4 per cent of WDs have a magnetic field. This is likely an underestimate of the real frequency of the occurrence of magnetic fields because of two reasons: in DC WDs, weaker fields may well exist but are non-detectable with the current techniques; and the frequency of the occurrence of magnetic fields in WDs increases with age, hence the fraction MWDs is higher if we consider WDs older than a certain age. It is reasonable to conclude that in our region of the Milky Way, at least one star out of four will end its life as an MWD. This high frequency demonstrates clearly that magnetism cannot be considered a rare and exotic phenomenon among WDs. It appears to be even more common than metal pollution. Magnetism is a major aspect of WD physics that must be integrated with efforts to fully understand the structure of degenerate stars, and to explore the multiple evolution pathways leading to the creation of WDs and evolution through this state.

ACKNOWLEDGEMENTS

The new observations presented in this work were made with the FORS2 instrument at the ESO Telescopes at the La Silla Paranal Observatory under programmes ID 0101.D-0103, 0103.D-0029, and 0104.D-0298; with ESPaDOnS on the CFHT (operated by the National Research Council of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii), under programmes 15BC05, 16AC05, 16BC01, 17AC01, and 18AC06; and with the ISIS instrument at the William Herschel Telescope (operated on the island of La Palma by the Isaac Newton Group), under programmes P15 in 18B, P10 in 19A, and P8 in 19B. This research made use also of additional FORS2, UVES, and X-Shooter data obtained from the ESO Science Archive Facility. JDL acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number 6377-2016. The authors would like to acknowledge the great help consistently offered by the support astronomers and instrument and telescope operators at the three observatories during the entire observing campaign. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

DATA AVAILABILITY

All raw data and calibrations of FORS2, ISIS, and ESPaDOnS data are available at the observatory archives: ESO archive at archive.eso.org; Astronomical Data Centre at http://casu.ast.cam.ac.uk/casuadc/; and the Canadian Astronomical Data Centre at https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/.

Footnotes

REFERENCES

APPENDIX A: NEW SPECTROPOLARIMETRIC OBSERVATIONS OF WDS

Our new spectropolarimetric observations are presented in Table A1. In the following,we comment on those observations for which interpretation may be ambiguous. Most cases are those of DC stars observed with FORS2 in which we have tried to detect circular polarization of the continuum. Bagnulo & Landstreet (2020) have argued that, because of cross-talk from linear to circular polarization, observations of faint stars obtained in grey and bright time may be contaminated by a circularly polarized background that may be difficult to subtract correctly, because it changes rapidly with position in the field of view. We comment also on the observations of WDs of other spectral type in which interpretation or background subtraction is problematic.

A1 WD 0123−262

We observed this DC star with FORS2 and grism 300V. The observations were obtained in dark time, so they are not contaminated by polarized background.

The signal of circular polarization is consistent with zero within the error bars of less than 0.05 per cent in 100 Å bins between 4000 and 6500 Å. At longer wavelengths the uncertainty increases up to 0.1 per cent per 100 Å, but no systematic deviations from zero are noted. If we consider 400 Å spectral bins, it is safe to assume that the observed polarization is consistent with zero within 0.05 per cent, for a |〈B_z〉| ≲ 0.75 MG. For a 800 Å rebinning, the upper limit is 0.02 per cent, for |〈B_z〉| ≲ 0.3 MG. Note though that observations were obtained with detector E2V, and therefore are affected by fringing at wavelengths longer than H α. However, with heavy rebinning, fringing does not seem to leave spurious signatures in circular polarization, and this data set provides us with what is probably our most precise measurement of circular polarization in the continuum obtained with FORS2 (see Fig. A1).

A2 WD 0210−083

We observed this DA both with ISIS and with ESPaDOnS; the ESPaDOnS intensity profile of the core of H α is slightly shallower than the cores of other WDs of similar temperature in our sample but about the same width. The Stokes V/I profiles are flat, and the ISIS data in both arms suggest no field (see Fig. A2). We consider this WD to be not magnetic but the star deserves further observations.

A3 WD 0415−594

This DA WD was observed with FORS2. Background subtraction needs a special attention due to the uneven illumination from the bright K2 companion.

A4 WD 0743−336

This DC WD was observed with FORS2. We note that the magnitude of the star is G = 15.3, not V = 16.7 as found in Simbad and reported by Kunkel, Liebert & Boroson (1984). In the B band,we measure a signal of V/I = −0.15 ± 0.05 per cent. At longer wavelengths, the polarization signal is zero within noise. Since the star was observed at 60° from the Moon, which was 64 per cent illuminated, background is affected by cross-talk from linear to circular polarization. The star is bright enough that background subtraction is not particularly challenging but the origin of the signal of circular polarization at shorter wavelengths may well be spurious. We set |V/I| ≲ 0.15 per cent.

A5 WD 0806−661

This is a DQ star observed with FORS2. V/I is smaller than 0.02 per cent except at λ ≳ 8000 Å, where both V/I and N_V depart from zero. Ignoring this as a probably spurious signal, we set |V/I| ≲ 0.02 per cent.

A6 WD 0810+489

This is a DC star observed with ISIS. Data were obtained with bad seeing. There is a signal of ∼−0.2 per cent in the blue, while in the red the signal is around +0.1 per cent. We consider this as a non-detection and we set |(V/I)|_max ≤ 0.2 per cent.

A7 WD 0856−007

This star was observed with FORS2 and grism 300V because it was originally classified as DC ut our spectra revealed the presence of H α and Ca ii lines. The star is thus a DAZ. From very low resolution spectropolarimetry of H α we obtained a marginal detection (〈B_z〉 = 27 ± 12 kG) which prompts for further observations at higher resolution with grism 1200R.

A8 WD 1055−172

ISIS observations of this DC star were obtained with bad seeing and strong moon background. The blue part of the spectrum shows a very noisy signal of V/I ≃ 0.2 per cent and the red ≃−0.4 per cent, which we do not accept as detection.

A9 WD 1116−470

This DC star was observed twice with FORS2 and grism 300V, the first time at 60° from a half illuminated moon, the second time at 100° from full moon. In both observations, the background is strongly circularly polarized, and in both observations the star shows a constant signal of ∼−0.2 per cent, that goes to zero at λ ≳ 8000 Å. The interpretation of this signal is problematic, because on the one hand it is clear that background may contaminate the signal of the star, but on the other hand the signals measured in two different nights are consistent with each others (see Fig. A3), even though the background polarization had the opposite sign in the two different epochs. Furthermore, contamination from cross-talk should manifest itself more at shorter wavelength than at longer wavelength, while the observed signal of polarization seems constant with wavelength. In conclusion, while we cannot firmly confirm that the observed signal is real, the star should be considered as a strong candidate magnetic DC WD (with a field strength of the order of ∼3 MG), and should be re-observed during dark time.

A10 WD 1145−747

This DC star was observed during dark time with FORS2, both with grism 600B and with grism 300V. The latter data show no polarization (|V/I| ≲ 0.1 per cent, for |〈B_z〉| ≲ 1.5 MG), but the spectrum obtained with grism 600B has V/I = −0.2 per cent. Since the same offset is shown by the null profile, we conclude that the signal seen with grism 600 B is spurious.

A11 WD 1310−472

This DC WD was observed during dark time with FORS2. The background is not polarized and the star shows a nearly constant signal of polarization V/I ≃ 0.09 per cent. This may be explained as cross-talk from I to V, but the star certainly should be re-observed with higher accuracy.

A12 WD 1316−215

This is one of the faintest DA stars. It was observed with FORS2 twice with grism 1200R, once in bright time, with a strongly polarized background, and once during dark time. This data set may be used as a benchmark for contamination from a polarized background with grism 1200R. Because H α is present, 〈B_z〉 was accurately estimated to be zero on both occasions. Therefore, we do not expect circular polarization. From the observations obtained during bright time we measured V/I ≃ +0.1 per cent in the continuum, while during dark time we observed a signal much closer to zero, with peaks of |(V/I)|_max ≲ 0.05 per cent.

A13 WD 1334+039

This is a DA star observed with ISIS. The spectrum shows a broad H α, a bit offset to the red (by 5 Å) with respect to the absorption H α of the bright sky background.

A14 WD 1338+052

This DC star was observed during dark time and show a signal of circular polarization in the continuum (≃−0.1) per cent. Although it cannot be explained in terms of contamination from a polarized background, the signal is still too small to be deemed to be real. The star should be re-observed with higher precision.

A15 WD 1345+238

This is a very cool DA with an extremely weak H α. Because of strong background and bad seeing, the quality of our ISIS observations does not allow us to set a strong constraint, with upper limit |V/I| ≲ 0.3 per cent.

A16 WD 1821−131

This DA WD was observed both with ISIS and FORS2, and with both instruments, spectropolarimetry of the Balmer lines shows that 〈B_z〉 is consistent with zero. However, in the blue arm of ISIS we measure a signal of circular polarization of +0.3 per cent, and in the red a signal of −0.2 per cent. With FORS2 + grism 1200R, we measured a signal of circular polarization of −0.2 per cent in the continuum, consistent with what we have observed with ISIS. Because H α is not polarized, and because there is no hint that the star may have a DC companion, we consider the polarization signal to be spurious.

APPENDIX B: COMMENTS ON THE INDIVIDUAL WDS OF THE LOCAL 20 PC VOLUME

Here, we review the circular broad-band and spectropolarimetric measurements of WDs within 20 pc from the Sun that help to characterize their magnetic nature. We comment on the magnetic field measurements of individual stars, grouped by spectral types. We recall that while the definition of the sign of the mean longitudinal field has been fairly consistent in the literature, different definitions of the sign of circular polarization have been adopted – see Bagnulo & Landstreet (2020) for details. In the following, we define circular polarization to be positive when the electric field vector, in a fixed plane perpendicular to the direction of propagation of the light, is seen to rotate clockwise by an observer facing the source. This definition conforms to that adopted in textbooks such as Shurcliff (1962) and Landi Degl’Innocenti & Landolfi (2004), in publications with numerical simulations of radiative transfer (e.g. Wade et al. 2001), and in observational work such as that of Kemp et al. (1970b), Landstreet & Angel (1971), and Bagnulo & Landstreet (2020), but opposite to what was probably the more common convention in the literature of WDs (e.g. Angel et al. 1981; Schmidt & Smith 1995; Putney 1997; Vornanen et al. 2010). When we report literature values, we adapt them to this convention, hence the V/I values reported in this section may or may not have the same sign as reported in the original paper.

In general, we found that the accuracy of the older measurements in the continuum, obtained through broad-band filters, are comparable to or even better than those obtained from polarization spectra, even when highly rebinned. Therefore, for DC WDs, which have featureless spectra, and for some DQ stars, with very broad molecular lines, null detections obtained with broad-band filters still set useful constraints on the upper limit of the field strength. Conversely, for the WDs with spectral lines, low-resolution spectropolarimetry permits us to achieve a much higher sensitivity than broad-band measurements. In these stars (mainly DA WDs), null detections obtained with broad-band or narrow-band circular polarimetric measurements are no longer useful, if recent spectroscopic or spectropolarimetric measurements have been obtained. An example is represented by DA WD 40 Eri B, in which Angel & Landstreet (1970a) measured 〈B_z〉 = 20 ± 5 kG, while Landstreet et al. (2015) obtained a series with a typical uncertainty of 80–90 G, for an improvement of sensitivity better than 1.5 dex. Angel et al. (1981) tried twice to measure the magnetic field of WD 0752−676, at that time classified as DC, with an uncertainty of ∼350 kG; later the star was found to be DA, and with FORS we obtained a field measurement with a 0.4 kG uncertainty. Landstreet & Angel (1971), Liebert & Stockman (1980) and Angel et al. (1981) obtained field measurements for 32 of the WDs in the local 20 pc volume, with a typical uncertainty of hundreds of kG. About half of these stars are DC or DQ, and these measurements are still very useful for this work. The measurements for the remaining half have now been superseded by much more accurate spectropolarimetric spectral line measurements, and references are given in notes referenced in the final column of Table 1.

B1 DA stars

B1.1 WD 2359−434 = J 915 – well monitored and modelled magnetic variable with 50–100 kG field

The presence of a magnetic field in this star was first suggested by Koester et al. (1998) and confirmed with ESO FORS1 low-resolution spectropolarimetry by Aznar Cuadrado et al. (2004; we note that these data were re-reduced as described by Bagnulo et al. 2015, and the sign of 〈B_z〉 changed to conform to our usual convention). Landstreet et al. (2017) have modelled the star using 13 ESPaDOnS data, and concluded that it has a complex magnetic configuration. The stars are a photometric variable, and both light and magnetic variations agree on a rotation period of 2.7 h.

B1.2 WD 0009+501 = GJ 1004 – well monitored and modelled magnetic variable with 150–250 kG field

WD 0009+50 was discovered to be an MWD by Schmidt & Smith (1994) through spectropolarimetry. The star’s magnetic field was measured by Fabrika et al. (2003) and Valyavin et al. (2005); the latter work presented a model obtained by adopting a rotational period of 8 h. With Valyavin, we have obtained numerous ESPaDOnS measurements that we will present in a future paper.

B1.3 WD 0011−721 = NLTT 681 – magnetic with 〈|B|〉 = 365 kG, observed only once

The star was spectroscopically confirmed as a nearby DA WD by Subasavage et al. (2008), and discovered to be magnetic by Landstreet & Bagnulo (2019b) using FORS2 low-resolution spectropolarimetry of H α (which shows Zeeman splitting for 〈B_z〉 ≃ 75 kG and 〈|B|〉 ≃ 365 kG). The star was observed only once, and follow-up observations are needed to check if the star is magnetically variable and whether it is a good candidate for monitoring and modelling.

B1.4 WD 0011−134 = G 158-45 – 9 MG, magnetic variable

Discovered as an MWD via spectroscopy by Bergeron et al. (1992), who from analysis of H α Zeeman splitting, estimated 〈|B|〉 ∼ 10 MG. Putney (1997) obtained six measurements and showed that 〈|B|〉 varies between 8 and 10 MG, and that 〈B_z〉 reverses sign with a rotation period that lies between some hours and a few days. It should be further monitored for modelling.

B1.5 WD 0121−429A = LP 991-16A – 5.5 MG field in DA component of uDD, flux spectroscopy only

The weakness of H α relative to the strength expected for the T_eff value deduced from the energy distribution (∼6000 K) suggests that the system is a DD with a DAH and a He-rich DC (Subasavage et al. 2007; Giammichele et al. 2012), with roughly equal contributions to the light around H α. Furthermore, the mass deduced by Giammichele et al. (2012) for a single star model is only |$0.41\, \mathrm{M}_\odot$|⁠, and both Hollands et al. (2018) and Gentile Fusillo et al. (2019) find log g = 7.57, implying |$M = 0.39\, \mathrm{M}_\odot$|⁠. Such a low mass cannot result from single star evolution. From these two lines of evidence, we conclude that this star is a DD, and we list the two components as separate WDs in our tables.

From the best fit to the single star H- and He-rich models to the photometry (Giammichele et al. 2012; Hollands et al. 2018; Gentile Fusillo et al. 2019), we assume that both the H-rich star and its its He-rich DC companion have T_eff in the range of 5800–6300 K. We take as our starting point a single star model of the system with T_eff = 6035 K, log g = 7.57, and M = 0.39M_⊙, an average of models assuming either H– or He–rich atmospheres (Giammichele et al. 2012; Gentile Fusillo et al. 2019) Assuming that each of the two stars contributes roughly equally to the observed brightness, we take this starting model and exchange the initial model for two stars with radii R_i to |$R_i/\sqrt{2}$|⁠. Keeping the same T_eff as initially, WDs with the new reduced radius have half the surface area of the initial single-star model, and together they provide the same light as the initial model. However, we need the new mass and log g parameters of the individual WDs contributing to our simple model. These are estimated from the mass–radius relationship (e.g. Giammichele et al. 2012; Bédard, Bergeron & Fontaine 2017) to be 0.69M_⊙ and 8.17 for both stars. Finally, we obtain new cooling time estimates as discussed in the text. Of course, the assumption that the two WDs contribute equally to the observed light is only a rough approximation, but it allows us to obtain plausible approximate values of parameters for both stars that are needed for our statistical study.

Subasavage et al. (2007) discovered a strong magnetic field through H α flux spectroscopy. The observed splitting corresponds to 〈|B|〉 ≈ 5.5 MG. This star has never been observed in broad-band polarimetric or spectropolarimetric mode. The clear magnetic splitting of H α and H β shown by Subasavage et al. (2007) indicates that the observed field is found in the DA. Without polarimetry, we have no information about a possible field in the DC star, so it is treated as non-observed (see Section B4.3). This system should be monitored with high-resolution spectroscopy to search for radial velocity variations of the DA, and spectropolarimetrically, to check for magnetism in the DC star and to test for possible magnetic field variability of either component.

B1.6 WD 0135−052A = GJ 64A (uDD-SB2)

The system WD 0135−052 is a clear SB2, with radial velocity semi-amplitudes of about 70 and 75 km s⁻¹, and an orbital period of 1.56 d. The SB2 system was analysed in detail by Saffer, Liebert & Olszewski (1988), and important data are summarized by Saffer, Livio & Yungelson (1998). It is composed of two nearly identical DA WDs, the masses of which have been estimated for both components from the radial velocity curves. Bergeron et al. (1989) have used mass data, distance, and photometry to estimate the values of T_eff and log g for the individual components. Ages have been estimated by Holberg et al. (2016). The resulting parameters are listed in Table 1. Three spectra obtained by Koester et al. (1998) show clearly split H α varying on a time-scale of days (fig. 5 of their paper).

Schmidt & Smith (1995) measured 〈B_z〉 = −1.6 ± 6.1 kG. Later, Aznar Cuadrado et al. (2004) and Bagnulo & Landstreet (2018) obtained much more sensitive measurements with FORS1 and FORS2 (〈B_z〉 = 0.2 ± 0.4 kG and 0.0 ± 0.2 kG respectively), without detecting any field. The single ESPaDOnS spectrum presented in this paper was fortuitously obtained at an orbital phase at which no splitting is evident (〈B_z〉 = −0.2 ± 0.3 kG). Since H α does not show any sign of Zeeman splitting in our ESPaDOnS spectrum, we can set the upper limit for 〈|B|〉 to 50 kG for each star. However, it should be noted that this upper limit refers to the combined signal of the two stars, therefore the upper limit for each star should be considered to be twice as large.

B1.7 WD 0135−052B = GJ 64B (uDD-SB2)

See section above.

B1.8 WD 0148+641 = EGGR 268 (VB: M2 at 12 arcsec)

Visual binary with an M2 companion at 12 arcsec. Schmidt & Smith (1995) measured 〈B_z〉 = 0.4 ± 4.5 kG. We observed it with ESPaDOnS (this work) and found 〈B_z〉 = 0.9 ± 0.8 kG.

B1.9 WD 0148+467 = GD 279

Schmidt & Smith (1995) measured 〈B_z〉 = −5.8 ± 7.8 kG. Bagnulo & Landstreet (2018) observed the same star with 20 times higher precision with three ISIS measurements, but did not detect a field (〈B_z〉 = −0.2 ± 0.5 kG, 0.0 ± 0.2 kG, −0.1 ± 0.4 kG). No detection was obtained in two new ESPaDOnS measurements presented in this paper (〈B_z〉 = −0.1 ± 0.4 kG, +0.1 ± 0.3 kG).

B1.10 WD 0210−083 = LP 649 − 67 (VB: dM at 3.6 arcsec and uDD?)

Recently discovered as a WD within 20 pc by Hollands et al. (2018), the star belongs to a CPM visual binary system with a 3.6 arcsec separation (Heintz 1993; Luyten 1997) to the primary of the common proper motion VB, LP 649–66, a dM star. One new ISIS (〈B_z〉 = +0.4 ± 0.8 kG, averaging the measurements in the blue and red arm) and one new ESPaDOnS measurement (〈B_z〉 = −3.1 ± 1.2 kG), all presented in this paper, show that the star is a DA but did not reveal the presence of a magnetic field (although ESPaDOnS data give a |$2.5\sigma$| detection).

The WD is possibly a uDD, based on the photometrically inferred mass of 0.43 M_⊙ (Gentile Fusillo et al. 2019), which is too low to have resulted from single star evolution. One peculiarity is the profile of the core of H α, which is clearly slightly shallower than the cores of other WDs of similar temperature in our sample, but about the same width (see Fig. A2). We treat this system as a non-magnetic single WD, and mark it as uDD? until more data are obtained.

B1.11 WD 0230−144 = LHS 1415

Observed in broad-band circular polarimetric mode by Liebert & Stockman (1980), who measured V = −0.002 ± 0.10 per cent. This star was then observed three times by Putney (1997) in spectropolarimetric mode; one of these measurement is a 2.5σ detection (σ ≃ 14 kG), and one is a 8σ detection, but described as ‘faulty’ in the text. We observed the star once with ISIS (this work), but we did not detect any field (with σ ≃ 4 kG). In our ISIS spectra, H α does not show any hint of Zeeman splitting, nor does it in the R ∼ 18 000 UVES spectra available in the ESO Archive, so we can set 50 kG as the upper limit for 〈|B|〉. This star should be probably observed again but at present we consider it as non-magnetic.

B1.12 WD 0233−242A = NLTT 8435A – 3.8 MG, variable (uDD)

This star was first observed by Vennes & Kawka (2003), who obtained a low-resolution, low S/N spectrum at Mount Stromlo and classified it as a very cool DC WD, with an estimated T_eff around 5300 K. It was modelled by Giammichele et al. (2012) (using the Mount Stromlo spectrum) as a DA on the basis of a communication from Kawka indicating that a magnetically split H α line had been detected in new data, again assuming T_eff ≈ 5300 K. A new FORS red spectrum was reported by Vennes et al. (2018), showing very clear magnetic splitting, which was modelled with a slightly decentred dipole of polar field strength about 6.1 MG. Vennes et al. (2018) also noted a radial velocity shift of at least 60 km s⁻¹ between the two spectra, which they attributed to binary motion, and reported a photometric time series that suggested a stellar rotation period of about 95 min.

We retrieved the two circularly polarized FORS spectra (taken during 2013 January for Vennes et al. 2018) from the ESO Archive. The first spectrum, from 2013-01-02, shows very clean Stokes I Zeeman components, with a rather boxy but narrow π component, and quite broad σ components. The second I spectrum, from 2013-01-09, shows a sharp π component slightly deeper than in the first spectrum, and σ components very similar to those of the first spectrum, except for rather severe terrestrial line contamination of the blue σ component. Comparing the second spectrum of WD 0233−242 to the terrestrial lines observed in a rapidly rotating MS B star indicates that one contaminating line appears in the π component and is responsible for the morphological differences between the π line core shapes observed in the two spectra.

As mentioned by Vennes et al. (2018), the sharp π components of the two spectra differ in radial velocity. We measure a velocity difference of 90 ± 10 km s⁻¹ (a shift of 1.97 ± 0.2 Å) between the two H α π components. We considered the possibility that this difference might be due to a problem with FORS wavelength calibration, but this is rejected because telluric emission lines and the O₂ atmospheric absorption bands that bracket H α appear at exactly the same apparent wavelengths in both spectra. We also considered the possibility of a wavelength shift of the π component as a function of mean magnetic field strength. However, the value of 〈|B|〉, as measured by the position of the centroid of the red σ line component (the blue component is badly distorted in the second spectrum by water vapour lines) changes by less than 0.1 MG between the two spectra, which in turn would shift the π component by only about 0.03 Å (Schimeczek & Wunner 2014).

We conclude that the velocity shift found by Vennes et al. (2018) is real and we agree with their conclusion that this WD is a member of an SB1 binary star system. As there is no sign whatever in the observed spectra of the absorption bands characteristic of MS red dwarfs, the companion is most probably a DC WD. Because the observed DA is found to have a T_eff value close to 5000 K, the expected strength of the H α line is fairly sensitive to the exact value of T_eff, and cannot be used to estimate the fractional contribution to the light of the DC without detailed modelling. Thus the only useful constraint we have on the DC companion is that its value of T_eff is probably also close to 5000 K because of the consistency of the photometric data with both H and He models having this temperature (Giammichele et al. 2012). Nevertheless, we have included the companion DC in the census of the 20 pc volume (see Section B4.5), although we cannot provide much useful information about the DC component of the system. The T_eff value found by Blouin et al. (2019), 4555 K, seems too cool to account for the clear H α line in the A component, so we will assume that both stars have the same T_eff = 4840 K, as derived by Hollands et al. (2018). We then make the assumption that both stars contribute roughly the same amount to the observed luminosity, so we reduce the radius R of the initial single star model to |$R/\sqrt{2}$| in order to divide the surface area of the star by 2. Then we recompute M/M_⊙ and log g from the mass–radius relation, as discussed in the notes for WD 0121−429 above. The resulting individual parameter values, shown in Table 1, are of course quite uncertain.

We have measured the value of 〈|B|〉 by computing the separation of the centroid of the red σ Zeeman component from the π component. We find value of 3.9 and 3.8 ± 0.1 MG for the first and second spectra, respectively (Schimeczek & Wunner 2014). However, in both spectra the large spread of the sigma components indicates a wide dispersion in local field strength |B| over the visible hemisphere, ranging between about 2.8 and 5.2 MG. (Note that the 6.1 MG field strength mentioned by Vennes et al. 2018 is the deduced polar field strength of a model decentred dipole field, not to a direct measurement of 〈|B|〉.) We have also used the circular polarization V/I data of the two available FORS spectra to measure the mean longitudinal field strength 〈B_z〉 revealed by each spectrum, using the mean separation between the centroids of the spectral line as viewed in right and left circular polarization (e.g. Landstreet et al. 2015). The measured values of the field moment 〈B_z〉 for the two spectra are +428 ± 24 and −360 ± 20 kG. The measured values thus clearly indicate that the field 〈B_z〉 is variable, and the MWD is rotating as well as orbiting around a companion. Note that these 〈B_z〉 values are both rather small compared to the 〈|B|〉 values revealed by the I spectra, and if taken at face value suggest a large inclination of the global field axis to the line of sight. However, these polarization measurements may well not provide simple snapshots of the longitudinal field of the MWD for two reasons. First, the unseen companion may significantly dilute the observed polarization, without affecting the measured value of 〈|B|〉. Secondly, the duration of each FORS observation, 2600 s, is almost 50 per cent of the rotation period of 95 min = 5700 s suggested by time series photometry of the system (Vennes et al. 2018). That is, each measurements of V/I may be smeared over nearly 1/2 rotation, probably leading to much polarization cancellation and a small residual computed longitudinal field. This smearing may also lead to the strong similarity of the Zeeman splitting as observed in the two I spectra. However, neither dilution nor rotational smearing act to reduce the measured value of 〈|B|〉 below its true value but simply average 〈|B|〉 over about 1/2 rotation of the WD.

B1.13 WD 0310−688 = GJ 127.1

Based on a spectropolarimetric measurement obtained at the Mount Stromlo Observatory, this star was identified as a suspected MWD by Kawka et al. (2007), who measured 〈B_z〉 − 6.1 ± 2.2 kG. However, the same star was observed with much higher accuracy with FORS1 by Aznar Cuadrado et al. (2004), who measured 〈B_z〉 = −0.10 ± 0.44 kG, and with FORS2 by Bagnulo & Landstreet (2018), who found 〈B_z〉 = −0.20 ± 0.23 kG. We do not consider the star to be magnetic.

The shape of H α as seen in spectra from a ESO UVES Archive with R ∼ 20 000 allow us to set an upper limit of 〈|B|〉 ≲ 50 kG.

B1.14 WD 0357+081 = G 7-16

This WD was observed once by Putney (1997), who measured 〈B_z〉 = 4.1 ± 9.5 kG using the strong H α. It was observed once by us with FORS2, a non-detection with 1.4 kG uncertainty (this work). Our FORS2 data also provide an upper limit of 300 kG for 〈|B|〉.

B1.15 WD 0413−077 = 40 Eri B (multiple system)

Observed first by Angel & Landstreet (1970a) (with a Balmer line magnetograph, in the broad-line wings of H γ, yielding 〈B_z〉 = 20 ± 5 kG), then by Landstreet & Angel (1971) (broad-band visible light, V/I = +0.04 ± 0.05), then in spectropolarimetric mode by Schmidt & Smith (1995) (〈B_z〉 = −4.5 ± 3.1 kG and 1.2 ± 0.9 kG), this star was considered for a long time as a weakly MWD (Fabrika et al. 2003; Valyavin et al. 2003). However, Landstreet et al. (2015) obtained a number of highly precise measurements of the longitudinal field using both with ESPaDOnS and with ISIS. Although six of their measurements had uncertainty as low as 85–90 G (the smallest 〈B_z〉 uncertainties ever obtained for a WD), 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 present a new ISIS field measurement, which is another high-precision non-detection.

It is interesting to note that 40 Eri was the object of a spectropolarimetric investigation by Thackeray (1947), followed by a measurement made by Babcock (1948). This star was the first WD ever observed with spectropolarimetric techniques.

B1.16 WD 0415−594 = ϵ Ret = HD 27442B (VB: K2 at 13 arcsec)

This star is the stellar companion of planet-host star HD 26442A (a K2 star at 13 arcsec) and was recognized as a WD by Chauvin et al. (2006). It was observed for the first time in spectropolarimetric mode in this work, with no field detection and σ = 0.3 kG. Since we obtained only one measurement, it may be useful to re-observe it.

B1.17 WD 0433+270 = HD 283750B (VM: K2 at 124 arcsec)

This is the secondary of a visual binary (the companion is a spectroscopic K2 binary with 124 arcsec separation). The early broad-band polarimetric observations by Landstreet & Angel (1971) (who had considered it as DC, and measured V/I = 0.03 ± 0.11 per cent) do not set a strong constraint on the star’s magnetic field. In this work, we report one FORS2 (σ = 1.5 kG) and one ISIS (σ = 5 kG) measurement, with no field detection. From ISIS spectroscopy of H α we set the upper limit for 〈|B|〉 to 100 kG.

B1.18 WD 0503−174 = LHS 1734 – 4.3 MG

A magnetic field was discovered in this star by Bergeron et al. (1992). The separation of the sigma components reveals a field of about 〈|B|〉 = 4.3 MG. Based on their mass estimate of |$0.38\, \mathrm{M}_\odot$|⁠, Giammichele et al. (2012) suggested that the star may be a unresolved DD. However, Blouin et al. (2019) derived for the stellar mass the higher value of |$0.53\, \mathrm{M}_\odot$|⁠. Both Hollands et al. (2018) and Gentile Fusillo et al. (2019) provide H-rich mass values of about 0.50 M_⊙. We conclude that the star is no longer a strong DD candidate. The star should be re-observed to check for variability and possibly monitored.

B1.19 WD 0553+053 = G 99-47 – 15 MG

The magnetic field of WD 0553+053 was discovered by Angel & Landstreet (1972) on the basis of weak broad-band circular polarization of about 0.3–0.4 per cent. A search for variability by those authors detected no statistically significant variations over any time-scale from a few tens of seconds to 1 yr. The star was re-observed by Liebert et al. (1975) and by Putney & Jordan (1995). Liebert et al. (1975) interpreted the available spectroscopy and intermediate-band spectropolarimetry as revealing a field modulus of 15 MG and a longitudinal field of 5.6 MG. Re-analysis of the data of Putney & Jordan (1995) by Bagnulo & Landstreet (2020) yields 〈B_z〉 = 6.2 ± 1 MG and 〈|B|〉 = 13.5 ± 0.5 MG, essentially the same values found 20 yr earlier. There is presently no evidence of variations on any time-scale up to decades. Gaia does not provide parallax or proper motion but these are reported by van Altena, Lee & Hoffleit (1995).

B1.20 WD 0642−166 = Sirius B (VB: A0 at 7.5 arcsec)

In 1844, F.W.Bessell identified invisible companions of Sirius and Procyon from meridian position observation of the wobbly proper motions of these stars. An orbit for Sirius B was computed in 1850 by C. A. F. Peters, with a semimajor axis of 2.4 arcsec, and a period of 50 yr. In 1862 Sirius B was observed visually by Alvan Clark with his new 18 arcsec objective. Another 80 yr would pass before S. Chandrasekhar would clearly explain the nature of this first discovered WD in the 1930s.

H α profiles were observed at a spectral resolution of 0.56 Å per pixel with the STIS instrument of the HST by Joyce et al. (2018); the absence of significant Zeeman splitting at H α (see their Fig. 7) allows us to set an upper limit on 〈|B|〉 of about 80–100 kG. The difficulty of observing a WD of magnitude V = 8.4 only a few arcsec from a companion almost 10 mag brighter is so severe that no spectropolarimetry has been attempted.

B1.21 WD 0644+025 = G 108-26

This high-mass DA WD was observed in spectropolarimetric mode for the first time with ISIS in this work with no field detection (σ ≃ 6 kG). Because it has been observed only once, and with low precision, so it should be re-observed.

B1.22 WD 0644+375 = G 108-26

Observed in broad-band circular polarization by Angel et al. (1981) (V/I = 0.029 ± 0.057 per cent, −0.050 ± 0.038 per cent). Observed twice in spectropolarimetric mode by Schmidt & Smith (1995), who measured 〈B_z〉 = 4.5 ± 7.8 kG and 〈B_z〉 = 1.9 ± 1.6 kG, and with ISIS by Bagnulo & Landstreet (2018) (〈B_z〉 = 0.75 ± 0.91 kG). A new ESPaDOnS measurement presented in this work has σ = 0.35 kG. None of these measurements represents a field detection. ESPaDOnS data allow us to estimate the upper limit of 50 kG to 〈|B|〉.

B1.23 WD 0655−390 = GJ 2054

Identified as a WD by Subasavage et al. (2008) and observed for the first time in spectropolarimetric mode twice with FORS2 (this work). Both measurements are non-detection with σ ≃ 0.5 kG. FORS2 spectra set also the limit to 〈|B|〉 ≲ 300 kG.

B1.24 WD 0657+320 = GJ 3420

Observed only once and for the first time with ISIS (this work), with no field detection. The spectrum shows only an extremely weak H α from which we set 300 kG as the upper limit for 〈|B|〉. Due to the weakness of H α, the sensitivity of our 〈B_z〉 measurement is quite low (σ ≃ 35 kG).

B1.25 WD 0727+482A = G 107-70A (uDD)

This is a marginally resolved visual binary degenerate system with a separation of ≃ 0.6 arcsec, which is actually a member of a quadruple system – the second pair (the CPM companion system G 107-69) at about 103 arcsec separation is an unresolved MS–MS binary of spectral type M. The orbital period of the DD has been found to be 20.5 yr (Strand, Dahn & Liebert 1976; Harrington, Christy & Strand 1981; Harrington et al. 1993; Toonen et al. 2017).

Gaia DR2 reports positions but no parallax or motions of this pair, but full astrometry of the CPM dM companions. Previous parallax and proper motion studies make it clear that these four stars form a physically associated system (Toonen et al. 2017). The separation of G 107-69 from G 109-70 in 2015.5 was 115 arcsec. Bergeron, Leggett & Ruiz (2001) discuss a model of the system based on parallax, spectroscopy, and photometry, with two similar cool stars of masses around 0.6 M_⊙ and effective temperatures of about 5000 K. They observe a very weak, broad feature at H α, so at least one of the two WDs is a DA. Nelan, Bond & Schaefer (2015) provide an accurate parallax of G 107-70 and analyse the visual binary on the basis of HST fine guide sensor observations. They determine the astrometric masses of the two components: |$0.634 \pm 0.01\, \mathrm{M}_\odot$| and |$0.599 \pm 0.01\, \mathrm{M}_\odot$|⁠. Unusually, the more massive component is also the brighter star of the WD pair, which therefore must be hotter than the somewhat larger secondary star.

We adopt a hybrid model, taking the masses from Nelan et al. (2015), and the log g and radius values from the WD mass–radius relation. However, appropriate values of T_eff need to be estimated. From Bergeron et al. (2001), we adopt a mean T_eff ≈ 5000 K for the system. However, because Nelan et al. (2015) report that the more massive component is the brighter of the two stars, we cannot use the individual mass and temperature solution adopted by Bergeron et al. (2001), which have the lower mass star as the larger and brighter member. We try to identify a more consistent model as follows.

It appears widely agreed that the magnitude difference between the two components is about 0.3 mag. This allows us to estimate that the integrated fluxes f₁ and f₂ from the two stars, proportional to |$T_{\rm eff}^4 R^2$|⁠, are approximately in the ratio |$f_1/f_2 \approx 1.32 \approx T_1^4 R_1^2/T_2^4 R_2^2$|⁠. Finding radii corresponding to the masses from the mass–radius relation, finally T₁/T₂ ≈ 1.09. Starting from the system T_eff value of about 5000 K, bracketing temperatures of T₁ = 5225 K and T₂ = 4775 K provide the necessary flux ratio, and the two WDs together, with similar fluxes, would be expected to have a combined energy distribution roughly the same as that of a single WD of T_eff = 5000 K. We then estimate ages using the online Montreal cooling curves.

Fig. 18 of Bergeron et al. (2001) shows a very weak and broad H α, quite different in shape from the normal profile of this line in cool stars, and very different from the line profile computed with their adopted model. Our ISIS Stokes I spectrum is featureless in the blue but does show a weak feature at H α. Like that illustrated by Bergeron et al. (2001), it is bowl-shaped, about 40 Å wide and 2–3 per cent deep. If we assume that this feature is broadened by a very inhomogeneous magnetic field, the field would need to have a typical strength of several hundred kG. This would set a rather weak upper limit on the DA WD component of the DD. Further constraint on possible magnetic fields can be obtained with polarimetry. WD 0727+482AB was observed in polarimetric mode by Valyavin et al. (2003), but their fig. 7 shows a flat intensity spectrum around H α. Although our polarized spectra show a very weak H α line, we are not able measure 〈B_z〉, but we note that there is no evidence in our spectrum of significant circular polarization within the H α feature. Another constraint may be provided by the lack of detection of polarization in the continuum. The DD was observed also by Landstreet & Angel (1971), who measured V/I = −0.03 ± 0.10 per cent and 0.007 ± 0.022 per cent (the same measurement is also reported by Angel et al. 1981). We estimate that neither component has a longitudinal magnetic field stronger than 1 MG.

We are not able to assign a spectral classification to the companion WD 0727+482B = G 107-70B (uDD), which will be counted in the sample total but not for detailed statistical purposes.

B1.26 WD 0728+642 = G 234-4 – suspected magnetic

Putney (1997) obtained a |$3\sigma$| detection: 〈B_z〉 = 39.6 ± 11.6 kG. We observed the WD with ISIS and measured 〈B_z〉 = 14 ± 13 kG using H α (this work). We are not able to confirm the field detection, although our lack of detection could be simply due to geometry. At present we continue to consider the star as suspected magnetic but the star should be re-observed in spectropolarimetric mode.

B1.27 WD 0751−252 = SCR J0753 − 2524 (VB: M0 at 400 arcsec)

Newly discovered as a WD within 20 pc by Subasavage et al. (2008), who originally classifed as a DC. Later, Giammichele et al. (2012) obtained a spectrum that shows a weak H α. The star has a visual M0 companion at 400 arcsec. We observed it in spectropolarimetric mode for the first time with FORS2, but with a poor choice of the instrument setting: we used grism 600B, while grism 1200R would have allowed us to check if the weak H α is polarized. There is no evidence of circular polarization (this is not 100 per cent true, but we think it is spurious; otherwise we would see splitting in H α). The most useful upper limits to the strength of its magnetic field comes from low S/N UVES archive data, that show an H α with no hint of Zeeman splitting, and no broadening. This allows us to set the limit to 〈|B|〉 to ∼0.5 MG. The star could be re-observed in spectropolarimetric mode around H α, although, given the weakness of the spectral line, sensitivity will not be very high.

B1.28 WD 0752−676 = GJ 293

Observed in broad-band circular polarization by Angel et al. (1981) (V/I = −0.018 ± 0.064 per cent and 0.147 ± 0.066 per cent). We observed it in spectropolarimetric mode using FORS2 + 1200R, with much higher sensitivity, and we obtained no 〈B_z〉 detection (σ ≃ 0.4 kG). There is no hint of Zeeman splitting in archive UVES data, setting the limit of 50 kG for 〈|B|〉. Since the star was observed only once in spectropolarimetric mode, it should be re-observed.

B1.29 WD 0810−353 = UPM J0812−3529 – 30 MG

Discovered as strongly magnetic by Bagnulo & Landstreet (2020), the star should be re-observed around H α for a better characterization of its magnetic field. Note that the interpretation of the complex observed I and V/I spectrum by Bagnulo & Landstreet (2020) is very tentative, and the field strength is a rough estimate.

B1.30 WD 0821−669 = SCR J0821−6703

Spectroscopically confirmed DA WD by Subasavage et al. (2007). We observed it for the first time in polarimetric mode with FORS2 + 1200R (this work), finding that the star is non-magnetic with uncertainty of ∼4.5 kG. Our FORS2 data do not show obvious splitting in Stokes I, but both spectral resolution and S/N are low, so we adopt 〈|B|〉 ≲ 300 kG.

B1.31 WD 0839−327 = CD−32 5613 (uDD?)

Kawka et al. (2007) comment that Bragaglia et al. (1990) observed line profile variations, and suggest that this is a DD with a primary of T_eff = 9340 K and a secondary of T_eff = 7500 K. Hollands et al. (2018) and Gentile Fusillo et al. (2019) both give log g = 7.79 and M = 0.48M_⊙, which suggests that the evidence of a double WDs may be not be strong. We are therefore treating it as a suspected DD, but still as a single star for statistical purposes.

Kawka et al. (2007) attempted to detect a field obtaining a null measurement with a 2.8 kG uncertainty. A higher precision non-detection was obtained by Aznar Cuadrado et al. (2004) (〈B_z〉 = 0.35 ± 0.25 kG). From UVES archive data we suggest 〈B_z〉 ≲ 50 kG.

B1.32 WD 1019+637 = GJ 1133

Observed in broad-band circular polarization by Angel et al. (1981) (V/I = +0.02 ± 0.05 per cent). Schmidt & Smith (1995) measured 〈B_z〉 = −0.1 ± 5.1 kG. In this work, we report a new measurement with ESPaDOnS with no detection (uncertainty of 1.2 kG). For 〈|B|〉 we set the limit to 50 kG.

B1.33 WD 1121+216 = Ross 627

Schmidt & Smith (1995) measured 〈B_z〉 = 7.5 ± 3.5 kG. We obtained one measurement with ESPaDOnS (this work) with non-detection (σ ≃ 0.8 kG). For 〈|B|〉 we set an upper limit of 50 kG.

B1.34 WD 1134+300 = GD 140 (uDD?)

Possibly this is a relatively close DD (relatively close because there is an apparent velocity difference between the two WDs). We note that from its proper motion anomaly, Kervella et al. (2019) identified this star as a suspected binary system, although their results do not provide any guidance about whether the possible companion is a WD, a brown dwarf, or even a massive planet close to the primary. However, Giammichele et al. (2012) show an excellent fit to the Balmer lines of this WD, from which they deduce a distance that is very close to the value provided by Gaia. If we suppose that the WD is actually a uDD, probably to be consistent with the single star model, we would need a system of two DA WDs, both with T_eff very similar to that derived for the WD as a single star. For this system to have the same radiating surface area as the single star model, the two stars would need to be substantially smaller (and due to the mass–radius relation, more massive) than the single star model. We estimate that they would both need to have M ≈ 1.25−1.3M_⊙. This is a considerably larger mass than that of any star in the 20 pc volume, and the value is dangerously close to the Chandrasekhar limit. In addition, we would expect such high mass (and corresponding gravity) to lead to Balmer lines significantly broader than those computed by Giammichele et al. (2012) for the single star model with M = 0.97 M_⊙. Therefore, we treat this system as a single WD but note that it could be a uDD.

Schmidt & Smith (1995) measured 〈B_z〉 = −0.6 ± 2.2 kG, and Valyavin et al. (2006) measured 8.9 ± 4.5 kG and 3.5 ± 2.7 kG. Two consecutive non-detections were obtained with ISIS (one in the blue arm, one in the red arm) by Bagnulo & Landstreet (2018): averaged together, the two measurements give 〈B_z〉 = 0.3 ± 0.4 kG. In this work we report a new non-detection with ESPaDOnS (σ = 0.8 kG). The core of H α in our ESPaDOnS exposure is rounded and shallower than those of WDs with similar fundamental parameters, and has a suggestively squarish shape. A similar feature is observed in our ISIS spectra. The non-detection of a field indicates that the observed broadening is not due to Zeeman effect. Assuming that the star is not a uDD, we may ascribe the rounding of the H α line core to stellar rotation.

B1.35 WD 1148+687 = PM J11508+6831

We have only one measurement with low S/N (〈B_z〉 = −2.2 ± 2.6 kG) with ESPaDOnS (this work); 〈|B|〉 ≲ 100 kG. H α and H β have slightly rounded cores, possibly a symptom of a narrow core broadened by fast rotation.

B1.36 WD 1236−495 = V* V886 Cen

Observed in broad-band circular polarization by Angel et al. (1981) (V/I = 0.018 ± 0.070 per cent). With spectropolarimetry. Kawka et al. (2007) measured 〈B_z〉 = 2.58 ± 6.23 kG. We obtained a high-precision non-detection with FORS2 (〈B_z〉 = 0.1 ± 0.5 kG). From UVES and X-Shooter archive data we set the upper limit of 〈|B|〉 to 100 kG.

B1.37 WD 1257+037 = Wolf 457

The star was originally classified as DC. Angel & Landstreet (1970b) measured V/I = +0.07 ± 0.11, and Angel et al. (1981) report the further measurement of V/I = 0.27 ± 0.21 per cent. Putney (1997) discovered that the star has H Balmer lines; from them she measured 〈B_z〉 = 11.2 ± 18 kG and classified this as DAQZ star, but with no further comments. We could not find any reference to the presence of metal lines, and Zuckerman et al. (2003) classify it as DA. From our FORS2 polarization spectrum (with grism 1200R) we obtained a non-detection with σ = 1 kG.

B1.38 WD 1309+853 = G 256-7 – 5 MG

From splitting of H α, Putney (1995) measured an ∼5 MG field. We note that from the description of the method adopted for the field measurement, what is referred to as ‘effective field’ by Putney (1995) probably corresponds to 〈|B|〉; from the plots shown in Fig. 2 of that paper we derive 〈|B|〉 = 5.4 + ±0.5 MG. From Putney’s figure, it is clear that 〈B_z〉 is non-zero but it was not measured. Because only one measurement is available, the star should be checked for variability.

B1.39 WD 1316−215 = NLTT 33669

One of the faintest DA WDs in the local 20 pc volume (V = 16.7). Kawka & Vennes (2012) used FORS1 to measure 〈B_z〉 = −3 ± 25 kG; we obtained two more accurate measurements using FORS2 with σ ≃ 3 kG. FORS2 I spectra set the upper limit of 〈|B|〉 to about 100 kG.

B1.40 WD 1315−781 = LAWD 45 – 5.5 MG

Originally classified as DC, Bagnulo & Landstreet (2020) discovered that its low-resolution spectrum (obtained with FORS grism 300V) reveals H α and H β, and the star is magnetic, with no evidence of variability over a time-scale of ∼1 week. For a better characterization of this magnetic field, the star should be re-observed with grism 1200R.

B1.41 WD 1327−083 = BD-07 3632 (VB: M4.5 at 503 arcsec)

Observed in broad-band circular polarization by Angel & Landstreet (1970b) (V/I = +0.21 ± 0.10; the same measurement is reported also by Angel et al. 1981). Two FORS1 observations were published by Jordan et al. (2007), 〈B_z〉 = −0.5 ± 0.5 kG and 〈B_z〉 = −0.4 ± 0.5 kG; and one FORS2 measurement with grism 1200B and one ISIS measurement with grism 600B were published by Bagnulo & Landstreet (2018): 〈B_z〉 = −0.9 ± 0.4 kG and 〈B_z〉 = 0.3 ± 0.2 kG. From UVES archive spectra we estimate 〈|B|〉 ≲ 50 kG. The star is member of a visual binary system.

B1.42 WD 1334+039 = Wolf 489

The star was in the past classified as DZ in older literature, then re-classified as DC by Sion et al. (1990, see Farihi et al. 2009). Later, the star was again re-classifed, as a DA, by Giammichele et al. (2012). Angel et al. (1981) measured V/I = −0.041 ± 0.091 per cent. Most recently we obtained one measurement with ISIS (non-detection, σ = 15 kG) and one with ESPaDOnS (from which, due to low S/N, we could not obtain any useful 〈B_z〉 measurement; hence it is not reported in Table A1). Our ISIS spectrum shows indeed a (weak) H α line redshifted by 5 Å with respect to absorption H α in the night sky background. Sion et al. (1994) briefly discuss the strange motion of this star and place the star in the halo ellipse. An upper limit from our ISIS spectra for 〈|B|〉 is 300 kG.

B1.43 WD 1345+238 = LP 380-5 (VB: M5 at 199 arcsec)

Cool WD in a binary system with an M5 star at 200 arcsec. Giammichele et al. (2012) and Holberg et al. (2016) classify it as DA, although H α is extremely weak. The star was observed in broad-band circular polarimetric mode by Liebert & Stockman (1980), who measured V/I = 0.006 ± 0.08 per cent, and by us with ISIS (this work), with a non-detection (a very weak H α is possibly visible in our ISIS intensity spectrum). Due to the fact that the spectrum is nearly featureless, the strongest constraint of 〈B_z〉 comes from the broad-band circular polarization measurements by Liebert & Stockman (1980) (|〈B_z〉| ≲ 1.5 MG).

B1.44 WD 1350−090 = GJ 3814 – 460 kG, subtly variable

Discovered by Schmidt & Smith (1994), who measured 〈B_z〉 = 85 ± 9 kG. We have obtained four polarized spectra with ESPaDOnS that show 〈|B|〉 ≃ 450−465 kG. Data will be published in a forthcoming paper.

B1.45 WD 1408−591 = UCAC4 154 − 133995

Newly discovered WD by Hollands et al. (2018). Our two FORS2 observations (this work) show that it is a DA. Both measurements are 2σ detections (σ ≃ 0.8 kG). Therefore, we currently consider the star non-magnetic but further higher S/N measurements would be useful to check if the star is weakly magnetic.

B1.46 WD 1544−377 = HD 140901B (VB: G6 at 15 arcsec)

Kawka et al. (2007) presented two non-detections with FORS1 with 6–7 kG uncertainty. We have obtained a new high-precision FORS2 measurement with grism 1200R (this work) which confirms non-detection (σ = 0.6 kG). From UVES archive data we set the upper limit for 〈|B|〉 to 50 kG. The star had been observed in broad-band circular polarization by Angel et al. (1981) (V/I = −0.043 ± 0.027 per cent).

B1.47 WD 1620−391 = CD−38 10980 (VB: G5 at 345 arcsec)

One (low-precision) measurement by Kawka et al. (2007) (〈B_z〉 = −2.96 ± 2.60 kG), one by Kawka & Vennes (2012) (〈B_z〉 = 0.5 ± 3.5 kG), and six high-precision measurements by Jordan et al. (2007) (best uncertainty of 0.3 kG), all non-detections. Angel et al. (1981) had obtained broad-band circular polarimetry (V/I = 0.016 ± 0.014 per cent). UVES archive spectra set the upper limit of 〈|B|〉 to 50 kG.

B1.48 WD 1630+089 = G 138-38

Discovered as DA WD within 20 pc by Sayres et al. (2012). Parameters for this star have been derived by Subasavage et al. (2017), Hollands et al. (2018), and Blouin et al. (2019). The gravity and mass derived by Blouin et al. (2019) are quite discrepant from the other determinations. We prefer the results of Subasavage et al. (2017), which are based on a spectrum and photometry from B to K, and are concordant with those of Hollands et al. (2018).

We obtained one ESPaDOnS measurement (this work), which is a non-detection with 2 kG uncertainty. The upper limit for 〈|B|〉 from H α spectroscopy is 50 kG. Possibly more measurements would be useful to confirm that the star is non-magnetic.

B1.49 WD 1647+591 = G 226-29

Angel et al. (1981) obtained broad-band circular polarimetry (V/I = 0.046 ± 0.022 per cent). Three non-detections with ISIS published by Bagnulo & Landstreet (2018) each with 0.5 kG uncertainty. ISIS H α spectroscopy set the upper limit for 〈|B|〉 to 100 kG.

B1.50 WD 1703−267 = UCAC4 317−104829 – 8 MG variable

Newly discovered as a WD by Hollands et al. (2018). Discovered as a strongly magnetic star by Bagnulo & Landstreet (2020), who showed that the star is variable on a time-scale of some weeks or less. Further observations are required for monitoring purpose.

B1.51 WD 1756+827 = EGGR 199

Schmidt & Smith (1995) measured 〈B_z〉 = −2.8 ± 4.2 kG. One ESPaDOnS measurement (this work) results in a non-detection with 0.8 kG uncertainty for 〈B_z〉. H α spectroscopy sets the upper limit for 〈|B|〉 to 50 kG.

B1.52 WD 1814+124 = LSR J1817+1328

Discovered as a DA WD by Lépine, Rich & Shara (2003). We observed this star once with ISIS (this work) obtaining a non-detection with large uncertainty (σ ≃ 10 kG) because only a weak H α is visible. From our ISIS spectra we also deduce a 200 kG upper limit for 〈|B|〉.

B1.53 WD 1820+609 = G 227-28

Cool DA. Liebert & Stockman (1980) measured V/I = 0.066 ± 0.11 per cent, then Putney (1997) erroneously listed it as highly polarized (see Landstreet et al. 2016, section 4). ISIS observations presented in this paper (non-detection) show that H α is too weak to measure 〈B_z〉, which is therefore better constrained by the broad-band circular polarization measurements of Liebert & Stockman (1980) (|〈B_z〉| ≲ 1.5 MG). From H α spectroscopy we deduce that 〈|B|〉 ≲ 50 kG.

B1.54 WD 1823+116 = UCAC4 508-079937

Newly discovered as a WD within 20 pc by Hollands et al. (2018). Originally classified as DC, it actually shows an extremely weak H α (Tremblay et al. 2020). We observed it with ISIS (this work) with no field detection, but based on loose constraints that come from our non-detection of polarization in the continuum we estimate |〈B_z〉| ≲ 2 MG. A stronger constraint comes from inspection of H α: 〈|B|〉 ≲ 0.5 MG.

B1.55 WD 1829+547 = G 227-35 – 120 MG

Strongly MWD. Discovered as magnetic by Angel et al. (1975), using both broad-band polarimetry and low-resolution spectropolarimetry. It was observed again in spectropolarimetric mode by Cohen et al. (1993) who suggested the star has a dipolar field seen pole-on with a dipolar field strength of 130 MG. Putney & Jordan (1995) observed the star again, and revised this estimate to a dipolar field strength of 170–180 MG. It does not appear to vary. We observed the same star with ISIS twice in circular polarization and once in linear polarization, and will report on these data on a separate paper.

B1.56 WD 1900+705 = Grw +70° 8247 – 200 MG

First WD discovered to be magnetic (by Kemp et al. 1970a), it has a magnetic field of the order of hundreds MG, and shows little to nearly no variability. See Bagnulo & Landstreet (2019a) for a review and analysis of its polarimetric characteristics.

B1.57 WD 1919+061 = UCAC4 482-095741

Discovered as WD by Hollands et al. (2018). Our new observation with ISIS (this work) confirms the star is a DA (see also Tremblay et al. 2020), and that it does not show the presence of a magnetic field. The H α core is slightly rounded. The non-detection of a mean longitudinal field (with a σ ≃ 6 kG) suggests that this broadening is not due to the Zeeman effect. The upper limit to 〈|B|〉 is about 150 kG.

B1.58 WD 1919+145 = GD 219

Observed by Schmidt & Smith (1995) (〈B_z〉 = 5.7 ± 6.6 kG), then two times by Jordan et al. (2007), who measured 〈B_z〉 = −1.5 ± 0.8 kG and 〈B_z〉 = −0.8 ± 0.8 kG. UVES archive data allow us to set the upper limit for 〈|B|〉to 50 kG.

B1.59 WD 1935+276 = G 185-32

A pulsating star (ZZ Ceti type). One ISIS measurement was presented by Bagnulo & Landstreet (2018) (〈B_z〉 = −0.3 ± 0.3 kG) and one additional ESPaDOnS measurement is reported in this work, also a non-detection with uncertainty of 0.6 kG. Previously observed also by Schmidt & Smith (1995) but with lower precision (〈B_z〉 = −8.5 ± 10.5 kG), and in broad-band circular polarization by Angel et al. (1981) (V/I = 0.00 ± 0.09 per cent). From ESPaDOnS H α spectroscopy we can estimate that the upper limit for 〈|B|〉 is 50 kG.

B1.60 WD 1953−011 = GJ 772 – 100 to 500 kG variable and modelled

Observed spectropolarimetrically by Schmidt & Smith (1995) who did not detect a field (〈B_z〉 = 15.6 ± 6.6 kG). Koester et al. (1998) observed a Zeeman pattern in a noisy high-resolution SPY H α spectrum and suggested that a field of about 93 kG was present. Maxted & Marsh (1999) obtained an H α spectrum which showed apparent Zeeman sigma components in I corresponding to a field of about 500 kG, and suggested that the star could actually be a DD system with both components magnetic. Maxted et al. (2000) described a series of I spectra that supported a model of WD 1953−011 with a global dipolar field with polar field strength of order 100 kG, together with a spot with a field of order 500 kG that is sometimes visible and sometimes not, as the star rotates. Finally, a series of polarized spectra obtained with FORS1 and on the Russian 6-m telescope revealed a rotation period of 1.448 d and confirmed the basic idea of a global dipole with a single spot of much higher field (Valyavin et al. 2008).

B1.61 WD 2007−303 = GJ 2147

Two low-precision measurements by Kawka et al. (2007) (〈B_z〉 = 5.5 ± 8.1 kG, = 3.7 ± 3. kG), one low-precision measurement with FORS1 by Landstreet et al. (2012) (〈B_z〉 = 1.1 ± 2.8 kG), and two higher precision FORS1 measurements by Jordan et al. (2007) (〈B_z〉 = 0.3 ± 0.4 kG, −0.5 ± 0.4 kG) all non-detections. From UVES archive data we set the upper limit for 〈|B|〉to 50 kG.

B1.62 WD 2032+248 = HD 340611

Low precision from narrow-band polarimetry by Angel & Landstreet (1970a) (V/I = 0.06 ± 0.08 per cent). One measurement by Schmidt & Smith (1995) (〈B_z〉 = 1 ± 2.3 kG), one ISIS measurement published by Bagnulo & Landstreet (2018) (〈B_z〉 = 0.0 ± 0.2 kG), and one ESPaDOnS measurements presented in this work (σ = 0.3 kG), all non-detections. ESPaDOnS spectroscopy of H α sets the upper limit for 〈|B|〉 to 50 kG.

B1.63 WD 2039−682 = EGGR 140

Koester et al. (1998) found that H α has a broadened core that could be explained either by a 50 kG magnetic or a |$v\, \sin \, i= 80$| km s⁻¹. Kawka et al. (2007) measured 〈B_z〉 = −6.0 ± 6.4 kG, and suggested that the reason for broadening could be rotation rather than the Zeeman effect, but also the possibility that a magnetic spot is present at the surface of the star, but not visible at the time of their observation. We obtained four more high-precision 〈B_z〉 measurements with FORS2 + 1200R (this work), all non-detections, with σ = 0.5–0.8 kG, which confirm the conclusion that the observed H α broadening is due to rotation. Formally, we still set the upper limit of 〈|B|〉 to 50 kG.

B1.64 WD 2047+372 = GJ 4165 – 60 kG variable, modelled

Discovered as magnetic by Landstreet et al. (2016), and monitored and modelled by Landstreet et al. (2017) with ESPaDOnS data. Currently the weakest WD field that has been modelled in detail, based mainly on a series of 18 ESPaDOnS spectra. The rotation period, determined from the variation of 〈B_z〉, is 0.243 d. Originally, the star had been observed by Schmidt & Smith (1995), who did not detect its weak and sign reversing field (〈B_z〉 = −42 ± 59 kG and −2.5 ± 4.6 kG).

B1.65 WD 2048+263A = G 187-8A (uDD)

The suspected binary system WD 2048+263 was earlier thought to be just outside the 20 pc limit, but Gaia has made it a clear member of the 20 pc volume. It is described by Bergeron et al. (2001) and Giammichele et al. (2012) as a probable DD on the basis of a derived mass of |$M = 0.24\, \mathrm{M}_\odot$|⁠, and listed by Toonen et al. (2017) as unconfirmed but probable DD. The low derived mass has been confirmed by Hollands et al. (2018). The derived mass is so low (i.e. the star is so overluminous compared to other WDs) that the automated fitting routine of Gentile Fusillo et al. (2019) rejects it as a WD. We consider this object to be a well-established uDD system.

The optical spectra published by Putney (1997) and Bergeron et al. (2001) show no sign of the absorption bands of a typical dM star, so the companion is almost certainly a DA or DC WD. Modelling by Bergeron et al. (2001) shows that the observed H α line is only about half as strong as it would be if both WDs are DAs of similar temperatures, so we conclude that the system probably contains one DA and one DC (the DC companion in listed in Section B4.27). The good fit of the single star model to the photometry suggests both stars have similar temperatures.

We adopt the most recent model temperature from Blouin et al. (2019) and assign it to both stars. We use their log g and mass values to derive the single star radius from the mass–radius relation, and decrease the model radius to 0.707 of its initial value (to divide the observed luminosity equally between two stars). We then return to the mass–radius relation to obtain the new, larger mass (0.53M_⊙) and gravity (log g = 7.90) of the two smaller WDs. This is of course only a rather rough first estimate of the parameters of the two stars, but is unlikely to seriously misrepresent either. Note that the two stars in the new model of the system both have masses within the normal range, rather than the very small mass estimated for the single star model of this system, so they may not have interacted very strongly during evolution.

The system was observed in broad-band continuum polarization by Angel et al. (1981) who report no detection with V/I = 0.04 ± 0.05 per cent. This pair was initially classified as a DC but reclassified by as DA9 by Putney (1997), who detected a weak but clear H α, and reported 〈B_z〉 = 13.6 ± 13.7 kG, a non-detection. It should definitely be re-observed with spectropolarimetry to be checked for magnetic field with higher precision.

B1.66 WD 2057−493 = WT 765 (VT: at 5 arcsec, 64 arcsec)

Newly identified as a nearby DA WD by Subasavage et al. (2017). For this work we obtained one FORS2 + 1200R measurement which resulted in a non-detection with a 1.8 kG uncertainty. Overall this is a triple system in which the WD is separated enough to be the only object in the slit. H α is quite visible but slightly asymmetric, with a feature at 6516 Å. The upper limit to 〈|B|〉 is 300 kG. It should be re-observed again to confirm that it is not magnetic.

B1.67 WD 2117+539 = GJ 1261

Observed twice by Schmidt & Smith (1995), who measured 〈B_z〉 = −11.5 ± 6.9 kG and 2.5 ± 2.7 kG. One ISIS measurement was reported by Bagnulo & Landstreet (2018) (〈B_z〉 = 0.0 ± 0.2 kG) and one ESPaDOnS measurement is presented in this work (〈B_z〉 = 1.2 ± 0.4 kG), all non-detections. ESPaDOnS H α spectroscopy allows us to set the upper limit for 〈|B|〉 to 50 kG.

B1.68 WD 2140−072 = PHL 1716?

Recently identified as a WD by Hollands et al. (2018). One ISIS measurement presented in this work is a non-detection with 0.8 kG uncertainty. ISIS spectroscopy of H α suggests an upper limit for 〈|B|〉 of 100 kG.

B1.69 WD 2150+591 = UCAC4 747-070768 – 800 kG, variable (VB: M2 at 15 arcsec)

This star was discovered to be a strong WD candidate at a distance of only 8.47 pc, and thus to be a probable member of the (supposedly already complete) 13 pc volume-limited WD sample, by Scholz et al. (2018). It was spectroscopically confirmed as a WD, and found to have a magnetic field, by Landstreet & Bagnulo (2019a), who reported two ISIS measurements. These ISIS spectra showed clearly that the field is variable with a period of hours or days. We have monitored the star with one ESPaDOnS observation and several more ISIS spectra. These observations and a model of the star’s magnetic field will be presented in a forthcoming paper. According to Scholz et al. (2018) the star is member of a wide binary system including an M2 star, but was inadvertently omitted from the study of Sirius-like systems with an MWD member by Landstreet & Bagnulo (2020).

B1.70 WD 2159−754 = V* CD Oct

Observed twice by Kawka et al. (2007) who obtained 〈B_z〉 = −7.8 ± 8.6 kG and −11.8 ± 7.1 kG. The one FORS2 + 1200R measurement presented in this work is another non-detection with 0.8 kG uncertainty. From UVES archive data we set the upper limit for 〈|B|〉to 50 kG.

B1.71 WD 2211−392 = LEHPM 4466 – suspected magnetic

Observed in polarimetric mode for the first time in this work, with three FORS2 measurements: one |$3\sigma$| detection and two non-detections (σ ≃ 1 kG). We cannot consider this star as a magnetic one, but it may well be a weakly and variable MWD. The situation of WD 2211−392 may be somewhat similar to that of WD 1105−084, a DA star outside of the local 20 pc volume, in which the (weak) magnetic field is detected only in a fraction of the observations (Aznar Cuadrado et al. 2004; Bagnulo & Landstreet 2018). From the FORS2 intensity spectrum we estimate an upper limit on 〈|B|〉 of 300 kG.

B1.72 WD 2246+223 = EGGR 155

Observed in polarimetic mode for the first time in this work, with two ESPaDOnS measurements (non-detections with 1.4 kG uncertainty). Upper limit from H α from ESPaDOnS spectroscopy is 〈|B|〉 ≲ 50 kG.

B1.73 WD 2248+293A = WD 2248+294A = GJ 1275A (uDD)

The system WD 2248+294 was recently classified as WD by Holberg, Bergeron & Gianninas (2008). The parameters derived by Giammichele et al. (2012), T_eff = 5592 K, log g = 7.55, and |$M = 0.35\, \mathrm{M}_\odot$|⁠, strongly suggests that this object may be a uDD. The parameters obtained by Hollands et al. (2018), Gentile Fusillo et al. (2019), and Blouin et al. (2019) are all very close to 5615 K, 7.71, and 0.43 M_⊙. As the derived mass is always found to be too small to be the result of single star evolution, it appears that this object is a uDD (a view supported by Hollands et al. 2018). Accordingly, we count this system as two objects in the statistical discussion. Because the fit obtained by Giammichele et al. (2012) to the observed H α line is essentially perfect, we conclude that the system is composed of two DAs with very similar parameters and ages. We adopt the T_eff value of Blouin et al. (2019); we assign it to both stars; we take the single star model radius and reduce it to 0.70 of its initial values, and then retrieve the new mass and log g values of the two stars using the mass–radius relation. Finally we interpolate new Montreal cooling ages as described in the text.

This system was observed polarimetric for the first time in this work, with only one ISIS measurement (non-detection with 4 kG uncertainty, but the upper limit for |〈B_z〉| should be increased by a factor of two because of binarity). From H α spectroscopy we estimated the upper limit for 〈|B|〉 to be 100 kG. We see no sign of a companion in our ISIS spectra. The star should be re-observed.

B1.74 WD 2248+293B = WD 2248+294B = GJ 1275B (uDD)

See section above.

B1.75 WD 2307+548 = LSPM J2309+5506E (VB: K3 13 arcsec)

Observed for the first time in polarimetric mode in this work, with one ISIS measurement (non-detection with 5 kG uncertainty), the spectrum shows both H α and a very weak H β. From H α spectroscopy we set our upper limit for 〈|B|〉 to 150 kG. The star is a member of a visual binary system with a K3 star at 13.1 arcsec. The star should be re-observed.

B1.76 WD 2336−079 = GD 1212

This star was observed in polarimetric mode for the first time in this work: twice with ISIS, once with FORS2 + 1200B, and once with ESPaDOnS: all these measurements are non-detections, with lowest uncertainty of 0.3 kG. 〈|B|〉 upper limit from ESPaDOnS spectroscopy is the usual 50 kG.

B1.77 WD 2341+322 = LAWD 93 (VB: M3 at 175 arcsec)

The star was observed in broad-band circular polarization by Angel & Landstreet (1970a) (V/I = 0.11 ± 0.18 per cent), and four additional observations were published by Angel et al. (1981). Two much higher precision non-detections were obtained by Bagnulo & Landstreet (2018) with 0.3 kG uncertainty, and in this work we have presented another ESPaDOnS non-detection (σ = 0.9 kG). From ESPaDOnS H α spectroscopy we set the upper limit for 〈|B|〉 to 50 kG.

B2 DAZ stars

B2.1 WD 0141−675 = LTT 934

Observed two times by Kawka et al. (2007) at the Mount Stromlo Observatory with the Steward CCD spectropolarimeter (results 〈B_z〉 = 2.93 ± 5.44 kG, and 〈B_z〉 = −1.68 ± 4.65 kG), and three times by us with FORS2 and grism 1200R (this work): one measurement was obtained with accuracy similar to those by Kawka et al. (2007), while two more recent measurements had a 10 times smaller uncertainty (0.3–0.4 kG). None of these measurements suggests the presence of a detectable field.

B2.2 WD 0208+396 = EGGR 168

Observed by us once with ISIS and once with ESPaDOnS (this work), with uncertainties of ∼0.5 and 1 kG, respectively. Both 〈B_z〉 measurements are non-detections. Sharp Ca ii H and K lines apparent in Espadons spectrum, Mg not visible.

B2.3 WD 0245+541 = G 174-74

Observed twice with ISIS (this work), no field detection with σ ≃ 12 kG. It is classified as DAZ by Zuckerman et al. (2003), but Ca ii lines are very weak in our ISIS spectra. It was originally measured in broad-band circular polarization by Angel et al. (1981) (V/I = −0.14 ± 0.09 per cent).

B2.4 WD 0322−019 = G77-50 – 120 kG

The star was discovered to be an MWD by Farihi et al. (2011) from the analysis of UVES spectroscopy. The mean field modulus 〈|B|〉 has been measured from Zeeman splitting of weak metal lines using co-added spectra and is found to be 〈|B|〉 ≈ 120 kG. From the UVES times series, Farihi et al. (2011) suggest that the rotation period could be either ∼1 or ∼30 d, the longer period being more likely than the former. Two FORS2 measurements by Farihi et al. (2018) obtained during two consecutive nights (〈B_z〉 = −5.4 ± 3.0 kG and −16.5 ± 2.3 kG) suggest that the longitudinal field may actually change at a time-scale much shorter than a month.

B2.5 WD 0856−007 = LP 606-32 (uDD?)

This star could be considered to be a uDD? because the mass derived by Blouin et al. (2019) is 0.41M_⊙ with T_eff = 4655 K. (We note that this WD appears in table 3 of Blouin et al. 2019 with the wrong sign for δ.) However, Subasavage et al. (2017) find T_eff = 5240 K, log g = 8.10, and M = 0.64M_⊙; Hollands et al. (2018) find T_eff = 4963 K and log g = 7.92; and Gentile Fusillo et al. (2019) find T_eff = 4967 K, log g = 7.93, and M = 0.54M_⊙. We consider this WD as a possible but weakly supported uDD? candidate, and count it as a single star for the statistics. The star was observed for the first time in spectropolarimetric mode using FORS2 with grism 300V. This star was originally classified as DC by Subasavage et al. (2008), but our spectra clearly show the presence of H α and Ca II H&K lines. Our 〈B_z〉 measurement from H α has low precision (12 kG uncertainty) but is a |$3\sigma$| detection; the star should be re-observed with higher resolution around H α, for instance using FORS2 grism 1200R.

B2.6 WD 1202−232 = LP 852-7

One ISIS measurement published by Bagnulo & Landstreet (2018) (〈B_z〉 = 0.43 ± 0.33 kG) and three FORS1 measurements by Jordan et al. (2007) (〈B_z〉 = 0.69 ± 0.52 kG; −0.19 ± 0.35 kG, −0.06 ± 0.37 kG, as revised by Bagnulo et al. 2015). From UVES SPY spectra we estimate 〈|B|〉 ≲ 50 kG.

B2.7 WD 1208+576 = G 197-47

Classified as DAZ by Zuckerman & Reid (1998) (see also Zuckerman et al. 2003). In this work, we report one 〈B_z〉 measurement with the ISIS instrument that turned to be a non-detection with σ ≃ 2.5 kG. From non-split weak H Balmer lines we deduce that the upper limit for 〈|B|〉 is 200 kG. To confirm that is not magnetic, the star should be re-observed.

B2.8 WD 1223−659 = WG 21

Wegner (1973) found weak Ca ii H&K lines in the spectrum of this star. Later, Kawka et al. (2007) could not see them in their spectra, and classified the star as DA. However, ESO Archive X-Shooter spectra clearly show the presence of Ca ii lines, therefore we consider this as a DAZ WD. In this work, we report three measurements obtained with FORS2 + grism 1200R. They are all non-detections with a typical 0.3 kG uncertainty. From the X-Shooter archive spectra we set 200 kG as the upper limit of 〈|B|〉.

B2.9 WD 1633+433 = GJ 3965

Schmidt & Smith (1995) measured 〈B_z〉 = 3.6 ± 2.9 kG. We obtained a new ESPaDOnS 〈B_z〉 measurement (this work), which is a non-detection with σ = 1.4 kG. From ESPaDOnS H α spectroscopy we obtain 〈|B|〉 ≲ 50 kG.

B2.10 WD 1821−131 = EGGR 176

A DAZ star with very weak metal lines: Zuckerman et al. (2003) have measured a Ca ii K line with an equivalent width of 46 mÅ. We obtained one measurement with ISIS and one measurement with FORS2 + 1200R, both non-detections, with lowest σ ≃ 1.2 kG. From ISIS spectra we estimate that the upper limit for 〈|B|〉 is 200 kG. From UVES spectra archive we set 〈|B|〉 ≲ 50 kG.

B2.11 WD 2105−820 = LAWD 83 – 40 kG

This star was suspected to have a weak magnetic field by Koester et al. (1998), who observed that the core of H α was abnormally broad, but they could not decide whether this was due to rapid rotation of vsin i ≈ 65 km s⁻¹ or to a magnetic field of about 43 kG. Five FORS1 polarized spectra of the star by Landstreet et al. (2012) revealed an apparently nearly constant magnetic field of 〈B_z〉 ≈ 10 kG. Later FORS2 polarized spectra by Bagnulo & Landstreet (2018) and Farihi et al. (2018) reveal that the field sometimes decreases to 〈B_z〉 ≈ 4 kG. No period has been established yet. Our FORS2 1200B spectra show a very weak Ca ii K line, so the star really is a DAZ.

B2.12 WD 2326+049 = V* ZZ Psc

Holberg et al. (2016) erroneously lists this 0.4 Gyr old WD as magnetic, but actually Schmidt & Smith (1995) had measured 〈B_z〉 = 2.8 ± 12.8 kG. The FORS2 measurements of Farihi et al. (2018) (〈B_z〉 = −0.7 ± 0.5 kG) and our ESPaDOnS measurement (this work, 〈B_z〉 = 0.0 ± 0.5 kG) confirm that this star is not magnetic. The ESPaDOnS spectrum provides 50 kG as the upper limit for 〈|B|〉.

B3 DB stars

There are no DB stars in the local 20 pc volume, except for a DBQA star (WD 1917−077) which is listed in Section B6 among the DQ stars.

B4 DC stars

B4.1 WD 0000−345 = LAWD 1

Based on the detection of a broad absorption feature between 4500 and 4700 Å via low-resolution spectroscopy, Reimers et al. (1996) claimed that the star has a magnetic field with dipolar strength of 86 MG. However, Schmidt et al. (2001) observed the same star in circular spectropolarimetric mode, and failed to detect the polarization signal that would be expected from a magnetic star with an 86 MG field. Furthermore, Schmidt et al. (2001) failed to detect the feature around 4500-4700 Å, (there are UVES and FORS1 spectra in the archive to check this). The spectrum by Giammichele et al. (2012) (available through the MDWD) is clearly featureless, confirming the hypothesis by Schmidt et al. (2001) that the feature seen by Reimers et al. (1996) was spurious. For these reasons, this star is considered to be non-magnetic. Integrated over their optical spectra, Schmidt et al. (2001) measured once −0.05 per cent and once +0.14 per cent, from which we obtain an upper limit of 〈B_z〉 ≲ 2 MG.

B4.2 WD 0004+122 = NLTT 287 – 100 MG

Identified as a DC star by Kawka & Vennes (2006), WD 0004+122 was found to exhibit a strong signal of circular polarization by Bagnulo & Landstreet (2020), who estimated 〈B_z〉 ∼ +30 MG, suggesting that the 〈|B|〉 value may be between 60 and 200 MG. (In Table 1 we adopt 〈|B|〉 = 100 kG.) The star was observed only once and should be re-observed to check for variability.

B4.3 WD 0121−429B = LP 991-16B

This star is a member of a uDD system composed of a magnetic DA and a DC WD. Since the system was never observed in polarimetric mode, we cannot reach any conclusion about the magnetic nature of the DC component, and is not to be considered in our statistical analysis. See Section B1.5 for more details about the system and the DA component.

B4.4 WD 0123−262 = EGGR 307

The star was never observed in spectropolarimetric mode before our FORS2 measurement that shows |(V/I)|_max ≲ 0.03 per cent, for |〈B_z〉| ≲ 0.5 MG (this work).

B4.5 WD 0233−242B = NLTT 8435B (uDD)

This star is a member of a uDD system composed of a magnetic DA and a DC WD. No circular polarization is detected in the continuum away from H α (which is polarized by the magnetic DA companion), with |V/I| ≲ 0.05 per cent. We conclude that the DC component has |$\vert \langle B_z \rangle \vert \lesssim \ 1.5$| MG. See Section B1.12 for more details about the system and the DA component.

B4.6 WD 0423+120 = G 83-10 (uDD?)

Holberg et al. (2008) consider the star as too bright for its parallax and classify it as an unresolved DD. Giammichele et al. (2012) discussed WD 0423+120 as an object without H α but their spectral energy distribution is better fit by an H-rich model with T_eff ≃ 6100 K. They do not mention DD models or assign a very low mass to WD, but suggest that a large magnetic field may be present that washes out the H α profile. They note, however, that Putney (1997) did not detect any polarization signal from the star. Hollands et al. (2018) give log g = 8.20 and Gentile Fusillo et al. (2019) give the WD mass as 0.76M_⊙ (H) or 0.70M_⊙ (He). We note the star as a possible uDD system, but we consider it as a single star.

The star was observed by Landstreet & Angel (1971), who measured V/I = 0.03 ± 0.11 per cent and Putney (1997), who measured V/I = 0.038 ± 0.034 per cent in the region 3800–4600 Å, and V/I = −0.060 ± 0.020 per cent in the region 6000–8000 Å. Putney (1997) considers the star as having a magnetic field ≲20 MG, but if we transform the measurement by Landstreet & Angel (1971) and the average of Putney (1997) measurements in the two bands into 〈B_z〉 = 0.45 ± 1.65 MG and 〈B_z〉 = −0.33 ± 0.45 MG, respectively, we obtain |〈B_z〉| ≲ 0.6 MG. If this is a uDD system, a polarization signal coming from one of the stars would be diluted by a factor of roughly two; therefore, we double our upper limit and set |〈B_z〉| ≲ 1.2 MG. Perhaps it would be worthwhile to observe the star again in spectropolarimetric mode with higher precision.

B4.7 WD 0426+588 = G 175-34B (VB: dM at 9 arcsec)

This bright DC WD was observed by Landstreet & Angel (1971), who measured V/I = −0.12 ± 0.13 per cent and by Angel et al. (1981), who measured V/I = 0.006 ± 0.013 G. The star was observed in spectropolarimetric mode with ISIS by Bagnulo & Landstreet (2018); in the B filter we report V/I = 0.00 ± 0.01 per cent, and in the red 0.01 ± 0.01 per cent. Altogether, these measurements set the upper limit for |〈B_z〉| to 0.17 MG.

B4.8 WD 0708−670 = SCR J0708−6706 – At least 200 MG

Identified as a DC WD by Subasavage et al. (2008), it was discovered to be magnetic by Bagnulo & Landstreet (2020), who estimated the star to have 〈B_z〉 ∼ 200 MG.

B4.9 WD 0743−336 = GJ 288 B (VM: F9 at 870 arcsec)

Identified as a cool WD by Kunkel et al. (1984). Observed in polarimetric mode only once (with FORS2, this paper, using grism 300V), the star shows no evidence for circular polarization. We set 1.5 MG as the upper limits of the star’s longitudinal field. The star is member of a triple system. We note that the star has G = 15.3, not V = 16.7 as tabulated in Simbad and reported by Kunkel et al. (1984).

B4.10 WD 0743+073.2 = GJ 1102 A

Observed only once in polarimetric mode (with FORS2, this paper, using grism 300V), no signal was detected. We set the upper limit of |〈B_z〉| to 1.5 MG. This is a member of a resolved DD system with WD 0743 + 073.1 at 16 arcsec (see below).

B4.11 WD 0743+073.1 = GJ 1102 B

Observed only once in polarimetric mode with FORS2 (this paper), using grism 300V. No evidence of circular polarization. We set the upper of |〈B_z〉| to 0.45 MG. This is a member of a resolved DD system with WD 0743 + 073.2 at 16 arcsec (see above).

B4.12 WD 0810+489 = G 111-64

Listed as probable WD by Luyten (1970), this star was spectroscopically confirmed as a DC WD by Kawka, Vennes & Thorstensen (2004). Observed with ISIS (this paper) with bad seeing conditions, this WD shows V/I ≃ 0.25 per cent in the blue, but this signal may well be spurious. We set the upper limit for |〈B_z〉| to 3 MG.

B4.13 WD 0912+536 = G 195-19 – 100 MG, variable

This is the second WD ever discovered magnetic (Angel & Landstreet 1971a), and the first one to be discovered variable (Angel & Landstreet 1971b). An improved ephemeris was provided by Angel et al. (1972b). With circular polarization reaching 2 per cent we infer 〈B_z〉 ∼ 30 MG and 〈|B|〉 ∼ 100 MG. There has been almost no further interest in this star since the 1970s.

B4.14 WD 0959+149 = G 42-33

Observed by Landstreet & Angel (1971) in broad-band (V = 0.02 ± 0.11 per cent), and by Putney (1997) (V = 0.080 ± 0.033 per cent). Neither detect circular polarization. We set an upper limit of 2 MG for the mean longitudinal field.

B4.15 WD 1033+714 = LP 37-186

Observed by Liebert & Stockman (1980), who measured V/I = −0.039 ± 0.16 per cent. For |〈B_z〉| we set an upper limit of 1.5 MG, but the star should be re-observed in circular spectropolarimetric mode.

B4.16 WD 1055−072 = LAWD 34

Observed with ISIS (this paper) with gratings R600B and R1200R in less than optimal sky conditions, we did not detect a convincing signal of circular polarization, with upper limits of 0.2 per cent in the blue and 0.4 per cent in the red (corresponding to |〈B_z〉| ≲ 6 MG).

B4.17 WD 1116−470 = SCR J1118−4721 – suspected magnetic

Discovered as a DC WD by Subasavage et al. (2008), we have observed it twice with FORS2 (this work) with grism 300V. Both observations show a similar signal of circular polarization at −0.2 per cent, close to the instrumental detection limit. Because of the consistency between the two observations, we suspect that this small signal is real. In this case the star would have a longitudinal field of about 3 MG.

B4.18 WD 1132−325 = HD 100623B (VB: K0 at 16 arcsec)

A Sirius like system with a K0 at 16 arcsec, firmly classified as DC by Holberg et al. (2016). We observed it with FORS2 (this work) with grism 300V. The result was a non-detection, and we set the upper limit to |〈B_z〉| as 1.5 MG.

B4.19 WD 1145−747 = SSSPM J1148−7458 (uDD?)

This is the coolest and faintest WD of our sample, discovered to be a DC WD by Scholz et al. (2004), who estimated its distance to 36 ± 5 pc, but according to the Gaia parallax (π50.10 ± 0.08 mas) it is just within the local 20 pc volume. Assuming an H-rich atmosphere, Hollands et al. (2018) give T_eff = 3711 K, log g = 7.67; Gentile Fusillo et al. (2019) give 3761 K, 7.70, and M = 0.42m_⊙. Since this mass is well below the lower limit for production by single star evolution, this result suggests that this object may be a uDD. However, assuming an He-rich atmosphere leads to the parameters 4109 K, 7.81, and M = 0.47M_⊙ (Gentile Fusillo et al. 2019). If this is the case, this object could be produced by single-star evolution. Because of the ambiguity of the mass of WD 1145−747, we classify it as uDD?

We observed the object with FORS2 (this work) with grism 300V and (quasi-simultaneously) with grism 600B, with no detection of circular polarization. We set |〈B_z〉| ≲ 1.5 MG, but if the system is shown to be a uDD, this upper limit should be multiplied by two.

B4.20 WD 1310−472 = GJ 3770

One of the faintest WDs in the local 20 pc volume. We observed it with FORS2 + 300V (this work), detecting V/I ≃ 0.1 per cent. This signal is consistent with instrumental polarization, but the star should be re-observed. We set |〈B_z〉| ≲ 1.5 MG .

B4.21 WD 1338+052 = LSPM J1341+0500

Identified as a new DC WD of the local 20 pc volume by Sayres et al. (2012), this star was observed in polarimetric mode only once with FORS2 + 300V (this work), which resulted in the upper limit |〈B_z〉| ≲ 1.5 MG.

B4.22 WD 1444−174 = LP 801-9

Observed by Liebert & Stockman (1980) who measured V/I = −0.233 ± 0.09 per cent (but note that in their paper, the Vilanova identification was erroneously given as WD 1414−175). Putney (1997) detected circular polarization in the blue (0.36 ± 0.002 per cent), but not in the red (V = 0.044 ± 0.042 per cent); her conclusion was that the detection in the blue arm was spurious, and data were consistent with null polarization. This star should be re-observed. We set the upper limit for |〈B_z〉| to 5 MG.

B4.23 WD 1917+386 = G 125-3

Observed once by Angel et al. (1981) (V/I = −0.02 ± 0.03) per cent and twice by Putney (1997). In one of her measurements Putney (1997) detected a small amount of polarization in the continuum (V/I = −0.49 ± 0.03 per cent in the blue and +0.062 ± 0.001 per cent in the red), while a second observation was consistent with zero (V/I = −0.07 ± 0.02 per cent in the blue and V/I = −0.05 ± 0.03 in the red). The star was not reported as magnetic. If we assume 0.1 per cent as the upper limit for |V/I|, we obtain |〈B_z〉| ≲ 1.5 MG.

B4.24 WD 2002−110 = EGGR 498

Observed quasi-simultaneously with FORS2 with grisms 300V and 600B (this work). We obtained |(V/I)|_max ≲ 0.1 per cent and |(V/I)|_max ≲ 0.2 per cent, respectively, leading to |〈B_z〉| ≲ 3 MG.

B4.25 WD 2008−600 = SCR J2012−5956

The star was spectroscopically confirmed to be a DC WD by Subasavage et al. (2007). Observed using FORS2 with grism 300V (this work) for an upper limit of |〈B_z〉| ≲ 1.5 MG.

B4.26 WD 2017−306 = EC 20173−3036

Discovered to be a WD by Hollands et al. (2018), who assumed that it is H-rich. One measurement using FORS2 (this work) with grism 1200 B shows no features and |V/I| ≲ 0.02 per cent, for |〈B_z〉| ≲ 0.3 MG. Since it has T_eff ≃ 10 500, we conclude that the star is clearly an He-dominated rather than H-dominated DC WD.

B4.27 WD 2048+263B = G 187-8B (uDD)

This is the probable DC companion of WD 2048+263A, with which we believe it forms a uDD system. See Section B1.65 for more details. From the broad-band polarization measurement by Angel et al. (1981) (V/I = 0.04 ± 0.05 per cent), assuming a dilution by a factor of two, we estimate |〈B_z〉| ≲ 1.5 MG

B4.28 WD 2049−253 = UCAC4 325-215293 – 20 MG

Identified as a WD by Hollands et al. (2018). Observed with FORS2 with grism 1200B by Bagnulo & Landstreet (2020), who found that the star exhibits continuum circular polarization of ∼0.5 per cent. Since this discovery observation was obtained during dark time, when the background was essentially unpolarized, we believe that the star is very probably magnetic. However, this should be confirmed by further observations.

B4.29 WD 2054−050 = VB 11 (VB + uDD?)

Giammichele et al. (2012) classify this WD as DC, and provide a mass estimate of 0.37M_⊙. Both they and Toonen et al. (2017) consider it likely that the star is actually a uDD. However, Hollands et al. (2018) and Gentile Fusillo et al. (2019) agree on log g values near 7.87, implying M = 0.50M_⊙, which could result from single star evolution, and Hollands et al. (2018) do not even mention it as a possible uDD. We consider that this object is not a strong candidate uDD, and we count it as a single WD in the statistics. It is also a VB with an M3 companion at 15 arcsec.

Liebert & Stockman (1980) measured V/I = 0.23 ± 0.13 per cent, Angel et al. (1981) V/I = −0.15 ± 0.12 per cent. This allows us to set an upper limit to |〈B_z〉| of 2 MG.

B4.30 WD 2226−754 = SCR J2230−7513 (DD)

Reported as visual (resolved VB) DD by Scholz et al. (2002) (see WD 2226−755). Observed with FORS2 + 300V (this work), no detection, which sets the upper limit of |〈B_z〉| to 0.75 MG.

B4.31 WD 2226−755 = SCR J2230−7515 (DD)

Reported as visual DD by Scholz et al. (2002) (see WD 2226−754 above). Observed with FORS2 + 300V (this work), no detection, which gives us an upper limit to |〈B_z〉| of 1.2 MG.

B5 DZ and DZA stars

B5.1 WD 0046 + 051 = vMA 2

This is the prototype of metal-enriched WDs – its strong Ca ii H + K lines were observed more than one century ago by van Maanen (1917). With ISIS, Bagnulo & Landstreet (2018) obtained a |$2.4\sigma$| detection (with σ ∼ 1 kG), but a new more accurate FORS2 measurement presented in this work, obtained with (−0.5 ± 0.3 kG), suggests that the star is not magnetic. UVES archive data obtained with |$R\sim 20\, 000$| show no hint of Zeeman splitting in the Ca ii H + K lines, which allows us to set the upper limit of 200 kG for 〈|B|〉. The star is very bright, and perhaps one or two additional high-precision field measurements could be useful to check if the star has a very weak field.

B5.2 WD 0552−106 = UCAC4 398 − 010797

Newly identified as a WD of the local 20 pc volume by Hollands et al. (2018), our two ISIS spectra (this work) show weak Ca II lines H&K and no Balmer lines, consistent with the classification as a DZ star by Tremblay et al. (2020). We obtained no field detection, although with low precision (σ ≃ 40 kG). From intensity spectra we set an upper limit of 300 kG for 〈|B|〉.

B5.3 WD 0552−041 = EGGR 45

According to Giammichele et al. (2012) and Subasavage et al. (2017), WD 0552−041 is the only known DZ star in the local 20 pc volume with an H-rich atmosphere, however, Coutu et al. (2019) prefer to describe it as He-rich, and suggest that there is no H at all. We observed it with ISIS and FORS2 + 1200B (this work), and we do not see H α in our ISIS spectrum, but at T_eff ≈ 4500, H α would not be expected, and selecting the dominant element in the atmosphere must be done using the energy distribution. We found no magnetic field (our most accurate measurement is 〈B_z〉 = 5.9 ± 4.6 kG). From ISIS intensity spectra we set the upper limit of 〈|B|〉 to 300 kG.

B5.4 WD 0738−172 = L745-46A (VB: M6.5 at 21 arcsec)

The stellar spectrum shows no sign of higher Balmer lines, in spite of T_eff = 7600 K, but H α is clearly visible with very shallow broad wings and deep line core. Bagnulo & Landstreet (2019b) considered this star as a candidate (weakly) magnetic DZ star on the basis of FORS1 measurements around H α by Friedrich et al. (2004), who found that the star is magnetic, although at a low significance level (〈B_z〉 = −6.9 ± 2.1 kG). In the FORS1 archive, we have retrieved the original polarization spectra obtained on 2000-12-03 and on 2000-12-19 with grism 1200R, and using H α we measure 〈B_z〉 = +5.1 ± 1.3 kG and 〈B_z〉 = −3.8 ± 2.2 kG, respectively (note that we obtained the opposite sign for 〈B_z〉 than that found by Friedrich et al. 2004). None of our new ISIS (Bagnulo & Landstreet 2019b, and this work), ESPaDOnS, and FORS2 + 1200B observations confirms the presence of a magnetic field. Because of the very sharp core of its H α (FWHM ≲ 1 Å wide), the upper limit on 〈|B|〉 from ESPaDOnS data is as small as 20 kG. The star should still be considered as a candidate magnetic star but at present we treat it as non-magnetic.

B5.5 WD 0816−310 = SCR J0818−3110 – 90 kG, variable

Discovered as an MWD by Bagnulo & Landstreet (2019b) who obtained one FORS2 and one ISIS measurement. The field is too weak to split the star’s metal lines observed at low resolution, but using X-Shooter spectra with resolution ∼10 000, Kawka et al. (2021) measured 〈|B|〉92 ± 1 kG.

B5.6 WD 0840−136 = LP 726-1

The star was discovered as a DZ WDs in the local 20 pc volume by Subasavage et al. (2007). Two FORS2 + 1200B observations presented in this work (but see also Bagnulo & Landstreet 2019b) did not lead to any field detection with uncertainties of ∼2.5 kG. From FORS2 Stokes I spectra we estimate 〈|B|〉 ≲ 200 kG.

B5.7 WD 1009−184 = WT 1759 – 150 kG, variable (VB: K7 at 400 arcsec)

The star was classified as DZ by Subasavage et al. (2009), and was discovered to be a MWD by Bagnulo & Landstreet (2019b), who published one measurement obtained with FORS2 and one obtained with ISIS, showing that the star is variable. (The most extreme value was measured with FORS2, 〈B_z〉 = −47 ± 1.5 kG.) As is the case for WD 0816−310, the field is too weak to split the star’s metal lines, hence 〈|B|〉 ≲ 200 kG.

B5.8 WD 1532+129 = G 137-24 – 50 kG, variable

Originally classified as DZ WD by Kawka et al. (2004), the star was discovered to be an MWD by Bagnulo & Landstreet (2019b), who published two FORS2 + 1200B measurements and one ISIS measurement. The star is variable, with 〈B_z〉 values from FORS2 measurements of −21 ± 1 kG and −4 ± 1 kG, while 〈|B|〉 is not strong enough to split spectral lines, leading to 〈|B|〉 ≲ 300 kG. In revising our data we have realized that the uncertainty of ≃2 kG in our ISIS measurement reported by Bagnulo & Landstreet (2019b) was actually seriously underestimated, and should have been σ ≃ 5.5 kG.

B5.9 WD 1626+368 = Ross 640

The star is a DZA. An early broad-band circular polarization measurement was reported by Angel et al. (1981) of V/I = −0.16 ± 0.11 per cent. In this work, we present one measurement with ESPaDOnS and two measurements with ISIS. None of which show evidence of a magnetic field (with typical 1–2 kG uncertainty). ESPaDOnS data (this work) give us an upper limit for 〈|B|〉 of 50 kG.

B5.10 WD 1705+030 = EGGR 494

The only 〈B_z〉 measurement available, obtained with ISIS (this work) reveals no field, with a ≃8 kG uncertainty. The width of the core of the Ca ii H line at 3968 Å gives us a much weaker limit of 〈|B|〉 ≲ 300 kG.

B5.11 WD 1743−545 = PM J17476−5436

This star was identified as a DC WD by Subasavage et al. (2017), but the star shows very weak Ca ii H&K lines, and an extremely weak H α. Observed using FORS2 with grism 300V (this work), we obtain an upper limit |(V/I)|_max ≲ 0.2 per cent that corresponds to |〈B_z〉| ≲ 3 MG; from the 20 Å full width at half-maximum of the weak H-line we deduce 〈|B|〉 ≲ 1 MG.

B5.12 WD 2138−332 = L 570-26 – 50 kG variable

Magnetic according to two FORS2 measurements and one ISIS measurements published by Bagnulo & Landstreet (2018, 2019b). A new ISIS and a new ESPaDOnS field measurement are presented in this work. The longitudinal field of this star is certainly variable and the star should be monitored for modelling.

Very close examination of our single ESPaDOnS spectrum and of our 1200R ISIS spectrum strongly suggests that a very weak H α line is present in both I spectra. This feature is so weak (less than 100 mÅ) that its reality is not certain. It should be confirmed with a spectrum at higher S/N. In any case, if real, this feature indicates that the He-dominated atmosphere also contains a very small amount of H.

B5.13 WD 2251−070 = EGGR 453

This is the second coolest WD of the local 20 pc volume. We have only one FORS2 + 1200B observation (this work), which shows strong Ca i (4226 Å) and Ca ii (3933, 3968 Å) resonance lines. The polarization spectrum of these lines shows no magnetic field but with large uncertainty (∼5 kG).

B6 DQ stars

B6.1 WD 0038−226 = LHS 1126

Peculiar DQ. Liebert & Stockman (1980) measured V/I = −0.15 ± 0.20 per cent with no filter (integrating over the interval 3200–8600 Å). Angel et al. (1981) obtained three BBCP measurements, including one with a marginal detection (V/I = 0.120 ± 0.041). Spectropolarimetric data by Schmidt et al. (1995) show that V/I = −0.04 ± 0.03 per cent. Schmidt et al. (1995) estimated 〈|B|〉 ≲ 3 MG.

B6.2 WD 0115+159 = GJ 1037

Cool DQ WD. Angel & Landstreet (1970b) obtained one measurement (V/I = −0.12 ± 0.1 per cent) and Angel et al. (1981) reported a best measurement of V/I = 0.003 ± 0.019 per cent. Vornanen et al. (2013) published circular spectropolarimetry that they obtained in 2008 with FORS1 using grism 600B. These new data show no signal of non-zero circular polarization at about the 0.1–0.2 per cent level, and we assume |〈B_z〉| ≲ 0.5 MG.

B6.3 WD 0208−510 = HD 13445B (VB: K0 at 1.9 arcsec)

The star has a K0 companion at 1.9 arcsec. Optical STIS spectroscopy by Farihi et al. (2013) shows that the star is a helium-rich WD with C2 absorption bands and T_eff = 8180 K, thus making the binary system rather similar to Procyon. Its spectral features in intensity are too broad to estimate any useful upper limit to 〈|B|〉, and we cannot observe it in polarimetric mode with FORS2 because of its proximity to a much brighter object. For our statistical analysis, the star is considered as not observed, hence it is ignored.

B6.4 WD 0341+182 = GJ 151

Landstreet & Angel (1971) measured V/I = 0.06 ± 0.10 per cent, (the same observations were later reported by Angel et al. 1981), and by Vornanen et al. (2013) with FORS1 using grism 600B, with no field detection, for |〈B_z〉| ≲ 2 MG.

B6.5 WD 0435−088 = GJ 3306

Angel et al. (1981) obtained two measurements: V/I = −0.038 ± 0.057 per cent and V/I = −0.017 ± 0.021 per cent. Observed by Vornanen et al. (2013) with FORS2 using grism 600B with no detection. We assume an upper limit of 0.5 MG for the magnetic field 〈|B|〉.

B6.6 WD 0548−001 = GJ 1086 – 5 MG, constant

Magnetic field discovered by Landstreet & Angel (1971), and re-observed and modelled by Angel & Landstreet (1974) who deduced the value of 〈B_z〉 to be about 3.6 MG from the polarization signature of the CH G band. Newer, higher resolution FORS1 polarized spectra were obtained by Berdyugina et al. (2007) and Vornanen et al. (2010) from which 〈B_z〉 ∼ 2.5 MG is deduced. The signal of circular polarization seems constant between 2005 and 2008, and overall there is no evidence of change compared with the earlier broad-band polarization measurements by Angel & Landstreet (1974).

B6.7 WD 0736+053 = Procyon B (VB: F5 at 5.3 arcsec)

Provencal et al. (2002) observed the star with STIS of the HST, revealing a DQZ spectrum with carbon Swan bands and Mg i and Mg ii) UV lines. Their fig. 9 shows that Mg i and Mg ii lines around 2800 Å do not appear split by Zeeman effect, setting a limit to 300 kG for 〈|B|〉. Dues to its proximity (∼5 arcsec) to a much brighter F-type companion, the star has never been observed in polarimetric mode.

B6.8 WD 0806−661 = L97-3

Our FORS2 measurements with grism 300V (this work) do not show any polarization signal, with a signal smaller than 0.02 per cent. Our optical spectrum is featureless, but the star is classified as DQ on the basis of IUE spectra (Bergeron et al. 1997; Subasavage et al. 2009). The star was observed twice by Angel et al. (1981) in BBCP, for 0.060 ± 0.058 per cent and 0.031 ± 0.055 per cent. There is no evidence that the star is magnetic, and we estimate |〈B_z〉| ≲ 0.3 MG.

B6.9 WD 1008+290 = LHS 2229 – 100 MG

Cool peculiar DQ star discovered to be magnetic by Schmidt et al. (1999), who from a V/I ∼ 10 per cent estimate a field of the order of |〈B_z〉| ∼ 100 MG. We have not found other polarimetric observations since its discovery, so the star should be re-observed and checked for variability and possibly monitored.

B6.10 WD 1036−204 = LP 790-29 – 200 MG, constant

Magnetic field discovered via spectropolarimetry by Liebert et al. (1978) and observed also by Schmidt et al. (1995) (see also Schmidt et al. 1999, for further analysis). Beuermann & Reinsch (2002) have monitored the star with EFOSC in spectropolarimetric mode to search for short-term variability, without finding any significant variations. From the measured signal of circular polarization of ∼10 per cent we estimate a longitudinal field of the order of 100 MG.

B6.11 WD 1043−188 = BD−18 3019B (VB: M3 at 8 arcsec)

Visual binary with an M3 at 8 arcsec. Observed by us using FORS2 with grism 600B (this work): we did not detect any signal, and we set the upper limit for |〈B_z〉| to 0.3 MG.

B6.12 WD 1142−645 = GJ 440

Observed only by Angel et al. (1981), who did not detect any signal of circular polarimetry (V/I = 0.010 ± 0.026 per cent and V/I = −0.014 ± 0.016 per cent). From these measurements we set an upper limit of |〈B_z〉| ≲ 0.5 MG. The star is quite bright (V = 11.5) but there are no spectropolarimetric measurements available in the literature, and not having observed it with FORS was an oversight on our part.

B6.13 WD 1633+572 = EGGR 258 (MS: M-type eclipsing binary at 26 arcsec)

Observed twice by Angel et al. (1981) (V/I = −0.08 ± 0.14 per cent and −0.09 ± 0.12 per cent), and by Schmidt et al. (1995) (V/I ≲ 0.03 per cent). From these measurements we deduce |〈B_z〉| ≲ 0.3 MG. We note that this star is also called WD 1633+571/2/3, as well as G 226-17 and G 225-58.

B6.14 WD 1748+708 = G 240-72 – 300 MG, very slowly variable

Intrinsic broad-band circular and linear polarization of this MWD were discovered by Angel et al. (1974). It was later observed in linear polarization by West (1989) and by Berdyugin & Piirola (1999). Berdyugin & Piirola (1999) found evidence of a variation of linear polarization in a time-scale of 20 yr. We have observed the star with ISIS both in circular (twice) and in linear polarization; our observations will be presented in a later paper.

The intensity spectrum of this star is quite unique, with a huge ‘sag’ in energy distribution between 4300 and 6700 Å. Bergeron et al. (1997) relate this feature to a group of C₂H stars which have molecular bands in the same general region and similar T_eff (see their fig. 30), and sugggest that the feature observed in WD 1748+708 is C₂H band distorted by the presence of a strong magnetic field, so they classify the star as a DQ. Kowalski (2010) suggests that the same feature is actually an extremely pressure shifted Swan band, implicitly agreeing on the star’s classification as DQ. Nevertheless it is still somewhat uncertain whether this classification is correct, and if correct, exactly what it means.

B6.15 WD 1917−077 = LAWD 74 (VB: M6 at 27.3 arcsec)

This star is classified as DBQA. It has an extremely weak H α (see fig. 8 of Giammichele et al. 2012) but T_eff ≃ 11 000 K, which leads to the conclusion that the atmosphere is probably He dominated. The UV shows a single line of C i. There is no sign of the C₂ Swan bands in the optical Oswalt et al. (1991). This DBQA star was observed by Angel et al. (1981), who obtained V/I = 0.05 ± 0.05 per cent, and with FORS2 by Bagnulo & Landstreet (2018), who found |V/I| ≲ 0.03 per cent. We have also re-observed the star with ISIS both in the blue and in the red, but because of a instrument set-up error, we cannot measure and subtract the sky contribution (hence there is no corresponding entry in Table A1). However, from ISIS spectroscopy of the broad, shallow H α we can still set the upper limit for 〈|B|〉 to 1 MG, while from circular polarimetry we deduce |〈B_z〉| ≲ 0.5 MG. In our FORS2 spectrum we do not see evidence for He lines.

B6.16 WD 2140+207 = EGGR 148

Observed in BBCP mode by Angel & Landstreet (1970b) (V/I = −0.01 ± 0.04 per cent), then observed again in BBCP two times by Angel et al. (1981) (V/I = 0.025 ± 0.017 per cent, −0.032 ± 0.077 per cent, and once in spectropolarimetric mode by Bagnulo & Landstreet (2018) (V/I ≲ 0.02 per cent). No lines are visible in the optical spectrum. The DQ classification comes from Koester et al. (1998), who analysed IUE spectra of luminous WDs. The non-detection of circular polarization allows us to place the |$1\sigma$| uncertainty for field detection at 200 kG.

B6.17 WD 2153−512 = WG 39 – 1.3 MG (VB: M2 at 28 arcsec)

Magnetic field discovered by Vornanen et al. (2010), who derived the 〈B_z〉 = 1.3 ± 0.5 MG value from modelling their polarization observations. These data do not constrain 〈|B|〉. The star is a member of a wide binary system (see Landstreet & Bagnulo 2020). The star should be checked for variability in spectropolarimetric mode. Note that this star has also been named WD 2154−511 and WD 2154−512.

B7 DX

B7.1 WD 0211−340

This star was too close to a background star, and has not been observed spectroscopically.