Rotation plays a role in the generation of magnetic fields in single white dwarfs

Recent surveys of close white dwarf binaries as well as single white dwarfs have provided evidence for the late appearance of magnetic fields in white dwarfs, and a possible generation mechanism a crystallization and rotation-driven dynamo has been suggested. A key prediction of this dynamo is that magnetic white dwarfs rotate, at least on average, faster than their non-magnetic counterparts and/or that the magnetic field strength increases with rotation. Here we present rotation periods of ten white dwarfs within 40 pc measured using photometric variations. Eight of the light curves come from TESS observations and are thus not biased towards short periods, in contrast to most period estimates that have been reported previously in the literature. These TESS spin periods are indeed systematically shorter than those of non-magnetic white dwarfs. This means that the crystallization and rotation-driven dynamo could be responsible for a fraction of the magnetic fields in white dwarfs. However, the full sample of magnetic white dwarfs also contains slowly rotating strongly magnetic white dwarfs which indicates that another mechanism that leads to the late appearance of magnetic white dwarfs might be at work, either in addition to or instead of the dynamo. The fast-spinning and massive magnetic white dwarfs that appear in the literature form a small fraction of magnetic white dwarfs, and probably result from a channel related to white dwarf mergers.


INTRODUCTION
White dwarfs have been speculated to have strong (> 1 MG) magnetic fields many decades ago when Blackett (1947) postulated that the presumed fast rotation of white dwarfs can drive a dynamo, but the first detection of a magnetic white dwarf was obtained more than twenty years later (Kemp et al. 1970).Ever since this groundbreaking discovery, the question of why some white dwarfs become strongly magnetic, while others do not, represents an unsolved questions of stellar evolution.
Today we know large numbers of magnetic white dwarfs.One of the most puzzling facts is the different fractions of strongly magnetic white dwarfs among single stars and in different binary star settings.The volume-limited sample of magnetic single white dwarfs established by Bagnulo & Landstreet (2021) shows that the fraction of magnetic white dwarfs increases with age and is about 20 per cent.Studies of the (incomplete) local 40 pc sample of white dwarfs shows ★ E-mail: mercedes.hernandez@usm.cl† E-mail: matthias.schreiber@usm.clthat massive white dwarfs (which are absent in the 20 pc sample) often exhibit strong magnetic fields during the initial stages of the cooling phase (Bagnulo & Landstreet 2022).
A high incidence of magnetic white dwarfs, i.e. around 36 per cent, is seen among cataclysmic variables (CVs), semi-detached close binary stars in which a white dwarf accretes from a Roche-lobe filling main sequence star (Pala et al. 2020).
Intriguingly, among the progenitors of CVs, detached white dwarf plus main-sequence star binaries, the fraction of systems with strongly magnetic white dwarfs is negligible (Liebert et al. 2005(Liebert et al. , 2015)).Of the more than one thousand known systems (Schreiber et al. 2010;Rebassa-Mansergas et al. 2016), only about a dozen magnetic white dwarfs have been serendipitously identified (e.g.Reimers et al. 1999).In these few detached magnetic white dwarf binaries, the main sequence star companions are close to Roche-lobe filling and the white dwarfs have effective temperatures below 10 000 K (Parsons et al. 2021).The only exception to this trend is the weakly magnetic white dwarf in the young post common envelope binary CC Cet (Wilson et al. 2021).In the majority of the cold detached and strongly magnetic white dwarf binaries, the white dwarf rotation is synchro-nized with the orbital motion of the secondary star.The only clear exceptions are AR Sco, the first radio-pulsing white dwarf binary star (Marsh et al. 2016), its recently discovered analogue, J191213.72-441045.1 (Pelisoli et al. 2023), and 2MASS J0129+6715 which also shows some indications for non-synchronous rotation (Hakala et al. 2022).
Different again is the situation in close double white dwarfs.These systems must have evolved through two phases of mass transfer.Among the dozens of known detached close double white dwarf binaries, only one strongly magnetic white dwarf is known (Kawka et al. 2017;Schreiber et al. 2022), This may indicate that the fraction of systems with magnetic fields may be rather low or that detecting magnetic fields in double white dwarfs can be extremely challenging.Only very recently, the first potentially weakly magnetic white dwarfs have been detected among semi-detached double white dwarfs, socalled AM CVn binaries (Maccarone et al. 2023).
Several ideas have been put forward to explain the origin of strongly magnetic white dwarfs.The three most popular scenarios that have been suggested in the last decades are (i) the fossil field scenario in which the magnetic field of the progenitor of the white dwarf is preserved during the white dwarf formation (e.g.Angel et al. 1981;Braithwaite & Spruit 2004;Wickramasinghe & Ferrario 2005); ii) a dynamo generated during common-envelope evolution in close binaries (Regős & Tout 1995;Tout et al. 2008;Wickramasinghe et al. 2014) and (iii) coalescing double degenerate cores/objects (García-Berro et al. 2012).However, all three scenarios face serious difficulties when compared to observations.The relative numbers of strongly magnetic white dwarfs predicted by the fossil field scenario are far lower than the observed numbers if updated star formation rates and evolutionary time scales are taken into account (Kawka & Vennes 2004).The solution to this problem suggested by Wickramasinghe & Ferrario (2005), who postulated the existence of a large number of main sequence stars slightly less magnetic than Ap and Bp stars, was refuted by spectropolarimetric surveys (Aurière et al. 2007).
The common envelope dynamo scenario in its current form predicts relative numbers of magnetic systems far too large when compared to observations (Belloni & Schreiber 2020), and the biggest weakness of the double degenerate merger scenario is that it cannot explain a large number of magnetic white dwarfs among CVs.In addition, all three scenarios do not offer an explanation for the absence of young detached magnetic white dwarf binaries (e.g.Liebert et al. 2005) and the late appearance of the magnetic fields in single white dwarfs (Bagnulo & Landstreet 2021).
Based on the idea originally put forward by Isern et al. (2017), an alternative model to explain the incidence of magnetic fields in white dwarfs has been recently suggested by Schreiber et al. (2021a).This scenario has been shown to explain a large number of observations of magnetic white dwarfs in binaries: the increased occurrence rate of magnetic white dwarfs in CVs, the paucity of magnetic white dwarfs in the sample of observed double white dwarfs, the relatively large number of detached but close to Roche-lobe filling cold magnetic white dwarf plus M-dwarf binaries, the existence of radio-pulsating white dwarfs such as AR Sco (Schreiber et al. 2021a(Schreiber et al. ,b, 2022)), as well as the absence of high accretion rate polars in globular clusters (Belloni et al. 2021).One of the key predictions originally made by this scenario is that strongly magnetic crystallizing white dwarfs should rotate significantly faster than non-magnetic white dwarfs.
However, Ginzburg et al. (2022) recently suggested that the convective turnover times in crystallizing white dwarfs are orders of magnitude longer than previously thought.If this is true, white dwarfs with spin periods of several hours or even days can generate magnetic fields of the order of a MG.If super-equipartition is assumed (Augustson et al. 2016), even much stronger fields, such as those observed in many CVs, covering the range of 1-100 MG can be produced.Crucial for the context of the present paper, this model predicts a relation between spin period and field strength.However, testing this hypothesis, i.e. faster rotation in magnetic than non-magnetic white dwarfs and/or a relation between field strength and rotation requires a representative sample of spin periods of magnetic white dwarfs.
Magnetic white dwarfs can show photometric variability which allows for measuring their spin periods.This variability can have different origins.In convective atmospheres starspots can be generated by the magnetic field.As these regions are cooler and darker, starspots rotating into view, reduce the observed brightness.A strong magnetic field might however completely inhibit convection in the atmospheres, which makes the appearance of starspots unlikely (Tremblay et al. 2015).Alternatively, the Balmer lines can be split and shifted to the blue due to the presence of a strong magnetic field.The amount of this shift depends on the local field strength, which will change the spectral energy distribution locally (even if the flux remains unchanged), leading to light variation in observations in a single passband if the local field strength varies much over the stellar surface.This effect would predict larger variations (on average) in stars with stronger fields, and maybe larger amplitudes in hotter white dwarfs with stronger Balmer line blocking (Hardy et al. 2023).A third potential origin of photometric variability is that the polarized line opacities depend on the local field strength and on the angle a given region is looked at.The latter is probably a smaller effect, but might account for variations of the order of one per cent in some cases.Finally, magnetic fields can cause metals to be distributed nonhomogenously on the white dwarf surface which can cause photometric variations as well (Dupuis et al. 2000).Independent of the exact mechanism producing the variability, photometric variability of magnetic white dwarfs allows for measuring their rotation rates.
The volume-limited sample of white dwarfs within 20 pc contains 33 magnetic white dwarfs (Bagnulo & Landstreet 2021).This sample is ideal to study by measuring the rotation periods of a representative sample of magnetic white dwarfs thereby potentially constraining scenarios for the origin of magnetic fields.We found that 27 of these magnetic white dwarfs have been observed with TESS (Transiting Exoplanet Survey Satellite, Ricker et al. 2015).The resulting light curves show statistically significant and constant periodic signals (the same period in all TESS sectors) in only five cases.Given this small sample size, we included all known magnetic white dwarfs within 40 pc and identified three more periods.In addition to analyzing the TESS light curves, we followed up two additional targets (one of them part of the 20 pc sample) with SPECULOOS (Search for habitable Planets EClipsing ULtra-cOOl Star, Delrez et al. 2018;Jehin et al. 2018).We compared our period measurements to those non-magnetic and magnetic white dwarfs with previously measured spin periods, and finally, we discuss possible implications for the origin of magnetic fields in white dwarfs.

OBSERVATIONS
In this work, we combine photometric data of magnetic white dwarfs from TESS with light curves obtained using the SPECULOOS instrument.In what follows we briefly describe the data acquisition for both cases as well as the procedure we used for determining the rotational period.

TESS
For all magnetic white dwarfs within 20 pc listed in Bagnulo & Landstreet (2021) we searched for TESS light curves.Of the 33 targets on the list, we found TESS observations for 27 magnetic white dwarfs which are listed in Table A1 with their corresponding sectors.We also took a careful look at the new magnetic white dwarfs identified in the 40 pc sample (O'Brien et al. 2023;Bagnulo & Landstreet 2022) TESS data are available for 23 of the 30 new magnetic white dwarfs (O'Brien et al. 2023, their Table 4) and for eight white dwarfs from Bagnulo & Landstreet (2022), all listed in Table A2.
The TESS light curves were obtained from the Mikulski Archive for Space Telescopes (MAST1 ) web service.We extracted the Presearch Data Conditioned Simple Aperture Photometry (PDCSAP) which removes trends caused by the spacecraft, and removed all data points with a nonzero quality flag and all NaN values in each sector.
Contaminating flux from unexpected sources which occurs due to a combination of pixel size and flux integration is a known issue in TESS light curves.We therefore performed a test to identify possible contaminating flux in the light curves using the flux contamination tool2 (FluxCT, Schonhut-Stasik & Stassun 2023).We note that this tool is based on Gaia G-band magnitudes of the objects in each pixel, i.e. the tool is using a band-pass different from TESS.Therefore, the estimated levels of contamination may not be entirely accurate.How much the real values deviate from the estimates depends on the colors of the source and contaminants.However, given that the two bands overlap, our estimates should not differ from the real contamination level by more than a few per cent in most cases.
We emphasize that a critical examination of the data of each target is fundamental to avoid wrong conclusions being drawn.To that end, we slightly modified the code that provides contamination levels for TESS targets.The original version only offers the contamination level and the Gaia G-magnitude of the target for the first observed sector.More insight can be gained by providing the contamination level and G-magnitude of the target for each sector.
We analyzed each sector of each target with the least-squares spectral method based on the classical Lomb-Scargle periodogram (Lomb 1976;Scargle 1982) to obtain the main period of the photometric TESS data.For each star, we started with the least contaminated sector to make sure the signal we are picking up is coming from the white dwarf and then requested the detected period to show up with a consistent amplitude in all other sectors.We searched for periods in frequency space up to the Nyquist frequency.
To make sure the signal is significant, we also performed a false alarm probability (FAP) test.We formally requested this false alarm probability to be below 5 per cent.For all the periods we detected we found the FAP to be less than 10 −6 .The uncertainties of the periods were computed using the curfit routine from Bevington (1969), which is a Levenberg-Marquardt non-linear least-squares fitting procedure.
In case a given white dwarf light curve passed all the above tests we finally inspected adjacent TESS pixels to check whether nearby bright stars could have contaminated the white dwarf light curve.The FluxCT mentioned above only provides information on stars located within the same TESS pixel.We therefore used the lightkurve tool (Lightkurve Collaboration et al. 2018) for this exercise.If a bright source was found we downloaded its TESS light curve (in case available) and ran a period search.If the same period was found as for the white dwarf, we used the amplitude of the variation to decide if the photometric variability is indeed coming from the white dwarf.
As a first example that illustrates the importance of a careful analysis of TESS data, we show in the appendix (Figure A1) the light curve we obtained for WD 2150+591.A very strong signal is clearly present in the data with a period of 116.38 hr.However, the amplitude largely exceeds those found for other white dwarfs and the light curve resembles that of an eclipsing binary.The up-dated FluxCT provided the G-magnitude for the target of each sector which revealed that the G-magnitude of the target in the first sector was clearly different to that of WD 2150+591.In other words, the detected star in the first sector was a nearby eclipsing binary instead of the white dwarf we aimed to analyze, and we therefore eliminated this white dwarf from our sample of systems with measured period.
Taking a detailed look at adjacent pixels with the lightkurve tool turned out to be important as well.In one case (WD 1009-184), we indeed found that the period measured from the white dwarf light curve most likely corresponds to that of a nearby bright star (Figure 1) and we excluded the white dwarf from our sample of stars with reliable periods.Bagnulo & Landstreet (2019) report variations on the magnetic field strength based on two measurements for this white dwarf but more measurements are needed to constrain the spin period.
Up to this point, we have identified a total of 18 white dwarfs exhibiting significant variability: 11 within 20 pc and 7 within 40 pc.For a final decisive test, we analyzed all 18 white dwarfs with the new tool tess_localize3 (Higgins & Bell 2023).This tool is especially designed to localize the source most likely responsible for observed variations in each TESS pixel.tess_localize delivers the optimized column and row coordinates corresponding to the most probable location of the observed variability.The algorithm complements TESS using queries to the Gaia Archive4 (Gaia Collaboration et al. 2021) for star locations and offers metrics such as p-values and relative likelihoods to facilitate interpretation of the fit outcomes.Prior to the fitting process, the tess_localize tool offers the option to discern prevalent trends among pixels located outside the designated aperture.This task is accomplished through principal component analysis (PCA).The resulting PCA components can be effectively applied to and subtracted from the light curves extracted by tess_localize.Nonetheless, it is imperative to ensure that these PCA trends do not represent the signals targeted for localization; otherwise, the signals may be inadvertently removed from the data.It is important to note that tess_localize should provide a substantial detection, meaning that the 'height' parameter in the fit is significantly different from zero given its uncertainty, ensuring it is not a false positive detection.
Eighteen targets were initially considered but 10 were subsequently eliminated from the sample due to tess_localize results showing that the signals observed in the lightcurves did not originate from the white dwarf targets.These eliminated targets are as follows: WD 0810-353, WD 0816-310, WD 1036-204, WD 1829-547, WD 1900+705 and WD 2153-512 from the 20 pc sample and WD 0232+525, WD 1008-242, WD J091808.59-443724.25, and WD J094240.23-463717.68 from the 40 pc sample.For all the white dwarfs discarded with tess_localize, the origin of the detected variation was well located on the field with the exception of WD 1008-242 where we were unable to identify the source of the measured variability.
Considering all tests listed above, the final sample of white dwarf periods measured from TESS contains eight white dwarfs, five with distance less than 20 pc and three within the 40 pc.For these systems  Right: The image obtained with the lightkurve tool.Pixels with a white edge were used to extract the TESS light curve of WD 1009-184.The variation found in the light curve of this system, however, results from contamination of these pixels from a bright star in the pixel with the red border.The star located in this pixel shows cleaner variations with the same period and a larger amplitude.This type of contamination cannot be identified if TESS data is only analyzed with the FluxCT.However, tess_localize confirms that the origin of the variation corresponds to target Gaia DR3 5669427508702256896 located at the pixel marked with the red border.
the average G-magnitude in the TESS sectors is in agreement with the one reported in literature for the corresponding white dwarf and contamination from nearby sources can be excluded as the source of the identified variations.The contamination level and the exposure times of each TESS sector of the confirmed white dwarfs are shown in Table A1 and A2.The measured periods, G magnitudes and normalized amplitudes obtained from TESS light curves as well as the frequencies and PCA entries for the tess_localize tool and the best fits results (p-values and likelihood) for the eight targets with clear, significant and consistent variations (that can be assumed to reflect the rotation period) are listed in Table 1.The periodograms and the phase-folded light-curves (which include all the sectors mentioned in Table A1 and A2) for each of the eight TESS targets are shown in Figure 2.
The periodicity in the light curves of WD 0009+501, WD 0011-134, WD 0011-721, WD J075328.47-511436.98 and WD J171652.09-590636.29 is difficult/impossible to spot in the phase folded light curves.With the aim to provide a better visualization of the five mentioned light curves, we therefore calculated the mean flux for 100 bins.The corresponding folded light curves are shown in Fig. 2. As mentioned earlier, the false alarm probability for the highest peak in each periodogram is below 10 −6 .We illustrate the FAP level corresponding to a probability of 10 −3 in all periodograms in Fig. 2. We note that in two cases, WD 0009+501 and WD 0041-102, a second highly significant peak appears in the periodograms.Previous observations show that these second highest peaks correspond to the rotation period of these two white dwarfs (Valyavin et al. 2005a;Achilleos et al. 1992).The light curves folded over the true rotation period are shown in Figure A2 in the appendix.

SPECULOOS
For two white dwarfs, WD 2138-332 (part of the 20 pc sample) and LSPM J0107+2650 (a more distant magnetic DZ white dwarf), we performed observations with the SPECULOOS Southern Observatory (SSO).SPECULOOS is composed of four telescopes with primary mirrors of 1.0 m diameter, run by the European Southern Observatory (ESO) and located at Paranal, Chile.We used the SPECU-LOOS/Y4KCam CCD with the SDSS g' filter.This filter covers the range where the Balmer lines  and , and some of the strongest Helium lines, can be found when observing white dwarfs.The dates, exposure times, and the name of the telescope that was used are listed in Table B1.
Data reduction was performed using prose (Garcia et al. 2022), a Python framework to build a modular and maintainable image processing pipeline.Dark, bias and flat field corrections were executed to all the images, followed by an automated line-up in preparation for the aperture photometry.We then analyzed the resulting photometry with the least-squares spectral method used for TESS targets and found that both targets showed clear variation associated with the rotation of the white dwarf.For WD 2138-332, the periodogram shows several aliases, and the light curve fit is reasonable with a number of different periods between 4 and 12 hr, with 6.19 hr corresponding to the highest peak in the periodogram providing the best fit.However, with the data currently available, we can only constrain the period to be between 4-12 hr.Spectral type, effective temperature, mass and magnetic field strengths (Hollands et al. 2017;Bagnulo & Landstreet 2019) together with the rotational period measured with SPECULOOS are listed in Table 1, while the periodograms and the phase-folded light curves are shown in Figure 3. .09-590636.29 we binned the phase-folded light curve (using 100 bins) as the amplitude of the variation is too small to be spotted in the un-binned light curve.For WD 0009+501 and WD 0041-102 we know from previous observations that the true period is twice the period we detect in the TESS data (Valyavin et al. 2005a;Achilleos et al. 1992).We use the longer (and correct) periods in the distributions discussed in this paper.The TESS light curves folded over the rotation period can be found in the appendix.For the other seven objects, we interpret the observed periods as reflecting the rotation periods of the white dwarfs.The measured periods range from a few hours to several days.
Table 1.Rotation periods measured from TESS and SPECULOOS together with their normalized amplitudes.We provide two period approximations for WD 2138-332, i.e. the period corresponding to highest peak in the periodogram which represents the best fit and the range of possible rotational periods given the data currently available.The extra line for WD 0009+501 and WD 0041-102 provides the rotational period measured through independent observations by Bagnulo, Landstreet & Valyavin (in preparation) and Achilleos et al. (1992) which correspond to twice the period measured with TESS ( see section A3 for more details).We provide the entries for the tess_localize tool; TIC name, the frequency measured from the TESS light curves, and the number of signals removed (PCA) to obtain the best signal to noise for the signal fitting.Finally, we present the resulting p-values and relative likelihoods for the best fit indicating that the white dwarf is the variable source.

THE ROTATION PERIODS OF MAGNETIC WHITE DWARFS
We have measured periodic photometric variations for ten magnetic white dwarfs, five of which are within a distance of 20 pc.In addition, eight of these periods have been obtained from TESS light curves which cover at least one month of continuous observations and are therefore not biased towards short periods, in contrast to those determined from short (a few days) observing runs using ground-based telescopes.In what follows we compare our findings with periods of magnetic and non-magnetic white dwarfs provided in the literature.Most rotation periods of magnetic white dwarfs result from photometric measurements with ground-based telescopes such as the two periods we determined with SPECULOOS.Additional periods have been determined through variations in the polarization degree.Given the fact that it is virtually impossible to use ground-based telescopes to reach the same cadence and baseline as the Kepler surveys or TESS, the full sample of rotation periods of magnetic white dwarfs is likely biased towards shorter periods when compared to that of non-magnetic white dwarfs.
We used two different approaches towards a more representative samples of magnetic white dwarf spin periods.First, we only used periods measured from TESS light curves, except for WD 0009+501, whose period measured with TESS is half of the real period measured with spectropolarimetry.The cadence and baseline of TESS is much longer than the periods measured for non-magnetic white dwarfs.Therefore, the eight periods we measured from TESS (Table 1) should not be biased towards shorter periods, and thanks to the very short exposure time in some of the sectors (20 sec), the chances to miss an extremely short spin period are low.The obvious disadvantage of this sample is its small size.
Because of the small size of the TESS sample, we also establish a volume-limited sample consisting of spin periods of magnetic white dwarfs within 20 pc.This sample combines five period measurements from TESS, the rough period estimate we obtained for WD 2138-332, and four periods of magnetic white dwarfs within 20 pc from the literature ( Table C1).This results in a sample of ten magnetic white dwarfs with measured rotation period within 20 pc which represents 30 per cent of the 20 pc sample of magnetic white dwarfs.The disadvantage of this sample is that it is incomplete and might again be biased towards short periods.

Comparison with previously identified periods of magnetic white dwarfs
To compare our results with previously measured rotation periods of magnetic white dwarfs, we established a list of robustly measured rotational periods from the literature, i.e. in what follows we ignore uncertain period estimates or measurements that provided a range of possible periods.Our final sample of magnetic white dwarfs with robustly measured periods from the literature is given in Table C1.
It contains seven white dwarfs from Brinkworth et al. (2013), who presented observations of 30 isolated magnetic white dwarfs performed with the Jacobus Kapteyn Telescope and compiled a list of previously published periods.We further complemented our literature search by going through the periods listed in Kawka et al. (2007) and Ferrario et al. (2015) -ignoring rather rough estimates -and by adding more recent measurements.In addition, we list three yet unpublished periods measured by Landstreet and Bagnulo through polarimetry.The observations that allowed the determination of these periods will be published elsewhere.This compilation of literature encompasses a total of 42 white dwarfs with reliably measured spin periods.Five of these 42 are in our TESS sample.Six of these 42 magnetic white dwarfs are very likely the product of the merger of two degenerate stars, as they are very massive white dwarfs (> 1.25 M ⊙ ) with rotation periods below 0.38 hr.An atypical result of a merger is SDSS J125230.93-023417.72 with low mass (0.58 M ⊙ , Reding et al. 2023).Occasionally we eliminate these likely merger products from the sample in order to constrain other possible mechanisms for generating the white dwarf magnetic fields.
We start the comparison of our samples with previously published spin periods with WD 0912+536.This white dwarf was one of the first known magnetic white dwarfs and was found to show periodic variations in circularly polarized light with a period of = 32.16hr (Angel & Landstreet 1971a).The most obvious explanation for periodic variations in polarized light is the rotation of the white dwarf in combination with a magnetic field that is not symmetrical about the spin axis.The period we determined from the TESS light curve (31.93 hr) is very close to the value measured by Angel & Landstreet (1971a).This agreement confirms that we can indeed assume periodic light curve variations to reflect the rotational periods.Other examples for spin periods measured with TESS and through polarimetry are WD 0011-134 and WD 2359-434, in both cases the periods also agree (compare Table 1 with Table C1).An interesting example is WD 0009+501.For this star Valyavin et al. (2005b) finds the magnetic field to be variable with a period of eight hours and Valeev et al. (2015) found photometric variability confirming this period but mentioned that the light curve showed two maxima per period.The TESS light curve confirms this finding but without the knowledge of the field strength variability, we would have interpreted the stronger periodic signal as the rotation period (4 hr) which is in fact half the real rotation period.The TESS light curve folded over the true rotation period is shown in Fig. A2 in the appendix.
Also the highest peak in the periodogram of WD 0041-102 corresponds to half the rotational period previously measured.If we fold the light curve over the period derived by Achilleos et al. (1992), we find two humps of different amplitudes in agreement with their results (see Fig. A2 in the appendix).Achilleos et al. (1992) showed that this variation is produced by a magnetic field that is dipolar and orientated at about 90 degrees to the spin axis.Thus, as the star spins, the abundance pattern on the surface seen from Earth varies.The strength of spectral lines, the flux they block directly, and the line blocking blueward of the Balmer jump, as seen from Earth, also change as the star rotates.This most likely produces the very strong light variability of this star.For an alternative interpretation for white dwarfs displaying two minima in their light curves see Farihi et al. (2023).
It is worth pointing out that a similar issue as identified for WD 0009+501 and WD 0041-102 could affect virtually all our other targets (i.e.we could be off by a factor of 2 for most objects).However, even if this is the case the conclusions presented in this paper would not change.
To evaluate whether the TESS and the 20 pc samples we defined are systematically different to the periods of magnetic white dwarfs in the literature we performed a Kolmogorov-Smirnov (KS) test and show the cumulative distributions in Figure 4. Assuming the usual 2-sigma significance criterion (p-values below 0.05) to reject the Null-hypothesis (both distributions come from the same parent distribution), both our samples do not show significant indications for being different to that of magnetic white dwarfs we found in the literature (Table 2).More precisely, the observed differences have an unacceptable high probability to be caused by chance (exceeding 30 per cent).We also compare the average and median values of our samples and those of previously identified periods of magnetic white dwarfs and find comparable values (Table 2).For this exercise, we also excluded magnetic white dwarfs that are very likely the product of a merger (very fast-spinning, i.e.  < 0.38 hr and very massive, i.e.  > 1.25 M ⊙ , white dwarfs) in agreement with the predictions made by Schwab (2021).These white dwarfs are WD 0316-849, WD 1859+148, WD 2254+076, WD 2209+113, Gaia DR3 4479342339285057408, and ZTF J1901+1458.While the median values in our samples are slightly longer than those of previously identified periods (even if we exclude mergers), the average periods are somewhat shorter.We conclude that the more representative samples defined here do not show significant differences to previously measured periods of magnetic white dwarfs.Hermes et al. (2017a) and Kawaler (2015) used data from the Kepler space telescope to measure the spin periods of pulsating nonmagnetic white dwarfs and found that the mean spin period of isolated non-magnetic white dwarfs with masses in the range of 0.51 − 0.72 M ⊙ is ≃ 35 hr with a standard deviation of 28 hr.In other words, the majority of non-magnetic white dwarfs have spin periods between 0.5-3 days.In the context of the crystallization and rotation-driven dynamo, and the origin of magnetic fields in white dwarfs in general, it is of fundamental importance to compare spin periods of non-magnetic and magnetic white dwarfs.

Comparison with non-magnetic white dwarfs
In order to perform the comparison of spin periods of nonmagnetic and magnetic white dwarfs, we need to consider possible observational biases.The rotation periods of non-magnetic white dwarfs have been measured through asteroseismology.The 36 nonmagnetic white dwarfs with asteroseismological measurements of their rotation period are listed for completeness in Table C2.They cover the mass range from 0.45 − 0.88 M ⊙ , which covers the peak of the mass distribution of isolated white dwarfs.Pulsating white dwarfs are located in the instability strip, and therefore cover a rather small range of temperatures (between 10 000 and 14 000 K, depending slightly on the surface gravity).However, this bias with respect to temperature (and therefore age) is unlikely to affect the distribution of spin periods given the absence of an efficient braking mechanism such as, for example, magnetic braking.
On the other hand, given that magnetic white dwarfs within 20 pc are on average older (Bagnulo & Landstreet 2022), we need to consider that they may have experienced spin-down through the same process assumed for pulsars, that is, magnetic dipole radiation.This means that the initial spin periods of magnetic white dwarfs could have been shorter than the ones we measure today.To explore this possibility, we calculated the initial spin period of the white dwarf using Equation 2 in section 3.3 from Williams et al. (2022), which requires the mass, radius, and the measured spin period (Table 3).We then integrated the equation over the cooling age and found that for our targets, the initial spin period was not significantly different (less than 5 per cent shorter) from the one currently measured, except for WD 0011-134, which has a short period of 0.74 hr and is more than 4 Gyr old.This white dwarf might have been fast spinning initially.However, there is strong evidence for the late appearance of the magnetic field in white dwarfs (Schreiber et al. 2021a;Bagnulo & Landstreet 2021), which means that white dwarfs we observe today as magnetic white dwarfs, very likely did not emit magnetic dipole radiation throughout their entire cooling age but only since the magnetic field emerged.Therefore, reconstructing initial spin periods remains impossible as long as we do not understand the mechanisms responsible for the magnetic fields in white dwarfs.
We are aware of the fact that the rotation rates from photometric variations and those inferred from asteroseismology may be measuring slightly different quantities, i.e., surface and globally averaged rotation rates, respectively (Oliveira da Rosa et al. 2022).These quantities might be slightly different but we expect them to be in general very similar.This assumption seems to be justified as Hermes et al. (2017b) could show for the helium-atmosphere white dwarf PG 0112+104 that the rotation period derived from the light curve (10.17 hr) was consistent with the rotational splittings from pulsations which indicate a period of ∼ 10 hr.We therefore decided to compare the currently measured spin periods of magnetic white dwarfs with that of non-magnetic white dwarfs.Neither the fact that the latter are systematically younger nor that the periods derived from light curves and pulsations measure slightly different quantities affects the conclusions we draw in this paper.
The corresponding cumulative distributions are shown in Figure 4. We used the Kolmogorov-Smirnov test to statistically evaluate possible differences between the samples.The results of this test are listed in Table 2.The full sample of measured periods for magnetic white dwarfs is clearly different to the rotation rates of non-magnetic white dwarfs with a high significance ( ≃ 3.76 × 10 −7 ).As the periods of the magnetic sample are potentially biased towards short periods (mostly coming from ground-based observations), this result does not allow to derive strong conclusions.
TESS provides continuous light curves for long enough time spans that in principal allow to measure periods of several days and there- fore this sample is not biased against periods as long as those measured for non-magnetic white dwarfs.The range of periods we measured with TESS still appears to be systematically shorter than that of non-magnetic white dwarfs.While the latter cover periods from 0.33 − 2.34 days, the TESS periods we present here range from 0.01 to 0.88 days.The KS tests comparing the periods of non-magnetic white dwarfs with those we measured here confirms these differences (the -value is reaching 2 significance).The median and average values further confirm the impression that magnetic white dwarfs rotate on average faster than non-magnetic white dwarfs (Table 2).Our measurements, however, do not allow us to draw strong conclusions.First, our samples are simply rather small.To provide further evidence for our findings, measuring the rotation periods of all magnetic white dwarfs in the 20 pc sample, e.g through polarimetry, therefore represents a necessary next step which, unfortunately, requires relatively large amounts of large-telescope observing time.Second, as long as we do not understand what is causing the photometric variations in magnetic white dwarfs, we cannot fully exclude that the amplitude of the variations depends on the spin period which would introduce a bias in the distribution of periods.

Relation between rotation and white dwarf mass
As a first step to evaluate possible dependencies on the white dwarf mass, we plot the cumulative distributions of the white dwarf masses of all previously discussed samples in the right panel of Figure 5. Previously measured spin periods cover a mass range that is very different to that of typical white dwarfs.The mass distributions of this magnetic white dwarf sample (median: 0.82 M ⊙ ) and that of the non-magnetic white dwarfs (median: 0.62 M ⊙ ) are clearly different.In contrast, the 20 pc and the TESS sample represent the first ones covering spin periods of more typical white dwarf masses (medians: 0.74 M ⊙ and 0.78 M ⊙ , respectively).
However, even both these samples show a tendency towards larger white dwarf masses compared to the sample of single non-magnetic white dwarfs.This reflects the fact that magnetic white dwarfs in the 20 pc sample seem to be slightly more massive than non-magnetic white dwarfs as already mentioned by Bagnulo & Landstreet (2021).If we consider all rotation periods measured for magnetic white dwarfs (solid green line) and the magnetic white dwarf after removing white dwarfs that most likely result from mergers (dotted green line), they seem to rotate significantly faster than non-magnetic white dwarfs (solid black line).A KS-test between the full magnetic and non-magnetic sample gives a probability of only 3.76×10 −7 for the observed difference to be caused by chance.However, this is clearly caused by an observational bias.If only periods measured with TESS are considered (dashed light-blue line), a KS-test provides weak indications for the period distribution being different to those of non-magnetic white dwarfs (p-value = 0.027).The incomplete sample of magnetic white dwarfs within 20 pc (dotted dark-blue line) also contains slightly shorter periods compared to non-magnetic white dwarfs.
According to KS-tests, this difference in masses for both the 20 pc sample ( = 0.0008) and the smaller TESS sample ( = 0.0005).
In the left panel of Figure 5 we show the rotation period as a function of white dwarf mass for non-magnetic and magnetic white dwarfs.In the 20 pc and TESS sample we find white dwarfs with masses exceeding 0.75 M ⊙ to have short periods very similar to what Kawaler (2015) and Hermes et al. (2017a) found for the only three non-magnetic white dwarfs in this mass range.
However, as we shall see, a tendency towards shorter spin periods for larger masses seen in the sample of non-magnetic white dwarfs (but indicated by just three systems) is in general not obvious in magnetic white dwarfs.For white dwarfs exceeding 0.85 M ⊙ , spin periods are only known for magnetic white dwarfs.Several very massive and magnetic white dwarfs rotate fast, i.e. with spin periods of minutes.Such typically larger masses and short rotation periods are expected for white dwarf mergers (Schwab 2021) and are entirely absent in the 20 pc sample, i.e. they are rare and largely over-represented in Figure 5.There are, however, also four magnetic white dwarfs with masses above 0.85 M ⊙ and spin periods of the order of days (one of them is not shown in the figure because its period is longer than 400 hours).
In order to test for dependencies of the spin periods on white dwarf masses, we again used cumulative distributions and the KS-test.In the left panel of Figure 6 we separate the magnetic white dwarfs into samples of normal mass white dwarfs (≤ 0.85 M ⊙ ) and massive white dwarfs (> 0.85 M ⊙ ) and compare the cumulative distributions.In the right panel of Figure 6 we compare the same distributions but this time we excluded the obvious merger products which reduces the number of stars in the massive white dwarf sample and the full sample of magnetic white dwarfs by six.There is no significant difference in spin periods for different masses (p-values for all comparisons exceed  = 0.33).This result does not depend on the (admittedly arbitrary) value at which we separate high and low-mass white dwarfs.Using e.g.0.75 M ⊙ or 0.8 M ⊙ instead of 0.85 M ⊙ provides essentially the same results.However, we again advocate caution with the above results because we are dealing with low-number statistics.More spin period measurements are clearly needed before solid conclusions can be drawn.2017) discovered a large number of magnetic white dwarfs among samples of cool metal-polluted white dwarfs and claimed that there might be an increase in magnetism among these white dwarfs (see also Bagnulo & Landstreet 2019).This alleged increase in magnetism among metalpolluted white dwarfs motivated Schreiber et al. (2021b) to speculate that faster rotation, generated by the accretion of planetary debris, could trigger the crystallization and rotation-driven dynamo.Indeed, accretion can easily reduce the spin period from a few days to a few hours (Schreiber et al. 2021b, their Figure 2).
However, there are two fundamental arguments against a relation between metal pollution and magnetism.First, and most importantly, the volume-limited sample provided by Bagnulo & Landstreet (2021) shows that there most likely is no increase in magnetism for metalpolluted white dwarfs.Of the 26 metal-polluted white dwarfs within 20 pc of the Sun, only six are magnetic.This corresponds to a fraction of 0.23 ± 0.08 magnetic white dwarfs among metal-polluted white dwarfs which is identical to the overall fraction of magnetic white dwarfs among cool ( eff ≤ 15 000 K) white dwarfs in the 20 pc sample (33/139 = 0.24 ± 0.035).In other words, there are no indications for an increase in magnetism among metal-polluted white dwarfs in this volume limited sample.
Second, if a cold and old DA white dwarf is observed today, accretion of planetary debris onto this white dwarf might have occurred in the past and therefore could have increased the white dwarf's rotation rate.It is therefore unclear if metal-polluted white dwarfs should indeed rotate faster on average than white dwarfs of other spectral types.If the accretion of planetary debris occurs in relatively short episodes on a large number of white dwarfs over a long period of time, one would rather expect a relation between rotation rate and age.
We here present measurements of the rotation periods of metalpolluted white dwarfs which complement the rough estimate of 18.5 hr for WD G29-38 by Thompson et al. (2010).The two clearly variable light curves we measured with SPECULOOS show that these two metal-polluted white dwarfs have periods of the order of a few hours.Our SPECULOOS observations show that LSPM J0107+2650 has a period of 4.8 hr while for WD 2138-332 we estimate a period in the range of 4 − 12 hr.As we observed several magnetic DZ white dwarfs with SPECULOOS the two measurements we performed with SPECULOOS are potentially biased towards short periods.Given that in addition both targets have periods typical for magnetic white dwarfs, we conclude that the currently available data does not provide evidence for faster rotation among metal-polluted white dwarfs (nor against it).
While there seems to be no strong evidence for a dependence of the spin period on the spectral types we observe indications for a relation between spectral type and amplitude of the detected photo-  C2, and magnetic white dwarfs ( green circles) from Tables C1 and 1.The magnetic white dwarfs with periods measured from TESS light curves (light blue circles and triangles) and those within 20 pc (dark blue triangles) have, on average, shorter periods than non-magnetic white dwarfs measured from pulsations.Both samples share the light blue triangles, as some of the white dwarfs in the 20 pc sample were observed with TESS.Right: Cumulative distributions of the masses of magnetic and non-magnetic white dwarfs.Both, the TESS (dashed light-blue line) and the 20 pc sample (dotted dark-blue line) contains slightly more massive white dwarfs than the sample of non-magnetic white dwarfs (solid black line).This difference is statistically significant in both cases (  = 0.0005 and  = 0.0008), which is consistent with the fact that magnetic white dwarfs in the 20 pc sample are slightly more massive than the non-magnetic ones (Bagnulo & Landstreet 2021) .Figure 6.Cumulative period distributions of magnetic and non-magnetic white dwarfs.Left: non-magnetic white dwarfs (solid black line) are compared with the magnetic ones separated in three different samples: all magnetic white dwarfs (Tables C1 and 3, solid green line), magnetic white dwarfs with masses below 0.85 M ⊙ (dotted pink line), and massive magnetic white dwarfs (≥ 0.85 M ⊙ , dashed blue line).Right: The same samples as in the left panel, but removing white dwarfs that very likely formed throughout mergers (rotational periods shorter than 0.38 hr and masses larger than 1.25 M ⊙ ).No indications for a relation between spin period and white dwarf mass can be identified especially if white dwarfs that likely formed through mergers are eliminated.
metric variations.In Table 1 we list the amplitudes of the measured variations for the white dwarfs presented in this work.In Table 3, we list the white dwarfs according to their amplitudes, from smallest to largest, along with their stellar parameters from the literature.
While the four DA white dwarfs show variations with an amplitude of ∼ 0.3 − 0.8 per cent, the DZ and DC white dwarfs in our sample show variations with larger amplitudes ranging from 0.8 to 10 per cent.If confirmed by analyzing larger samples, this finding might indicate that the variations are produced by different effects in the different spectral types.

DISCUSSION
We presented periodic light curve variations of eleven magnetic white dwarfs and interpret the periodicity as reflecting the spin period of the white dwarfs.We compared these measurements with spin periods of magnetic and non-magnetic white dwarfs from the literature and obtain the following results: (i) The spin periods of magnetic white dwarfs are shorter than that of non-magnetic white dwarfs but larger unbiased samples are needed to confirm this.We note that the spin periods have been measured in an unbiased way for only six of the 33 magnetic white dwarfs within 20 pc, and therefore definite con-Table 3. Stellar parameters from TESS and SPECULOOS targets obtained from literature.The white dwarf spectral types, temperatures, masses, log g, cooling age and magnetic field magnitude have been taken from Bagnulo & Landstreet (2021) and Bagnulo & Landstreet (2022) for nine of our targets.The masses for the TESS white dwarf with distance larger tan 20 pc were obtained from Gentile Fusillo et al. (2019).For the metal-polluted white dwarf LSPM J0107+2650 whose parameters are listed in Hollands et al. (2017), and for WD 2359-434 whose parameters where obtained form Bagnulo & Landstreet (2019).The listed distances were obtained from the Gaia catalog (Gaia Collaboration et al. 2021).The targets are sorted based on the measured amplitude of their light curves.For the white dwarf WD 0009+501, the table shows the spin period measured by TESS, which is half of the real spin period previously obtained by spectropolarimetric observations in Valyavin et al. (2005a).

Name
clusions cannot be drawn.(ii) Massive ( > ∼ 0.85 M ⊙ ) magnetic white dwarfs rotate on average slightly faster than lower-mass magnetic white dwarfs.However, this difference disappears if strong merger candidates (very fast-rotating massive white dwarfs) are eliminated from the sample.A general trend for decreasing rotation periods with white dwarf mass can therefore not be established.Only the obvious merger candidates, i.e. very massive (> 1.25M ⊙ ) and fast rotating (of the orders of minutes) white dwarfs stand out.(iii) We estimated rotation rates for two metal-polluted white dwarfs and find that both of them have periods of a few hours.Given that the previously published estimate of 18.5 h for WD G29-38 (Thompson et al. 2010) can be considered a rather rough estimate, these are the first robust measurements of rotation periods of metal-polluted white dwarfs.(iv) We found the amplitudes of the variations to depend on the spectral type of the white dwarfs, with variations in DA white dwarfs showing smaller amplitudes than those of DZ or DC white dwarfs.However, a larger homogeneous sample of light curves of magnetic white dwarfs is needed to confirm this potential trend.
In what follows we discuss how robust our findings are and relate them to theories suggested for the origin of magnetic fields in white dwarfs.

Spin periods and the crystallization dynamo
Investigating the origin of magnetic fields in white dwarfs in close binary stars and based on the early work by Isern et al. (2017), Schreiber et al. (2021a) assumed fast rotation to be a necessary ingredient for the dynamo to work.The proposed scenario assumes that white dwarfs in detached post-common envelope binaries are born without a magnetic field and that only when a crystallizing white dwarf is spun-up by accretion of angular momentum in semi-detached cataclysmic variables does the crystallization and rotation-driven dynamo generates a strong magnetic field.If the field is strong enough, it can connect with the field of the secondary star and transfer spin angular momentum to the orbit.This can cause the binary to detach for a relatively short period of time.The proposed scenario is very attractive as it can explain several otherwise inexplicable observations: the absence of strongly magnetic white dwarfs in young detached post common envelope binaries (Liebert et al. 2003), the population of old and close to Roche-lobe filling post common envelope binaries (Parsons et al. 2021), the existence of the fast spinning white dwarf radio pulsar AR Sco (Marsh et al. 2016), the absence of X-ray bright polars in globular clusters (Belloni et al. 2021), and the fact all but one white dwarf in close detached double white dwarfs binaries are not magnetic (Schreiber et al. 2022).
However, as we have mentioned previously, the only magnetic white dwarfs that rotate with spin periods of the order of minutes are very massive and are consistent with being formed through stellar mergers.This implies that the fast rotation suggested by Isern et al. (2017) and Schreiber et al. (2021a) is very unlikely to generally play a role in the magnetic field generation.At the very best fast rotation may increase the strength of generated fields.
Recently, Ginzburg et al. (2022) showed that convection in crystallizing white dwarfs is slower than previously estimated by Isern et al. (2017) which translates to slower rotation rates being required for the dynamo to work.Ginzburg et al. (2022) also estimated the time scale for the generated magnetic field to appear on the surface of the white dwarf to be between ≃ 0.1 − 1 Gyr depending on the extension of the carbon enriched convection zone.The results presented in this paper might be consistent with the slow-convection dynamo playing a role in the magnetic field generation in white dwarfs.We found that seven of nine periods measured with TESS of magnetic white dwarfs rotate with periods of just a few hours and/or have only relatively weak magnetic fields which could be consistent with the slow convection dynamo.
However, it also becomes immediately clear that the dynamo cannot be responsible for all magnetic fields in white dwarfs.In particular, the DCH white dwarf WD 0912+536 has a very strong fields ( > ∼ 100 MG), a typical white dwarf mass, and a spin periods significantly longer than one day (similar to that of longer period nonmagnetic white dwarfs).Furthermore, the DAH white dwarf WD J075328.47-511436.98 rotates with a spin period exceeding 20 hours and hosts a magnetic field with a strength of 19 MG which is also difficult to explain with the dynamo (see Ginzburg et al. 2022, their Figure 4).
To further investigate to which degree the proposed dynamo may contribute to the magnetic field generation in white dwarfs, we compare the available information, white dwarf mass, rotation period, field strengths, and effective temperature of magnetic white dwarfs with the predictions of the dynamo scenario in fig. 7. If the crystallization and rotation-driven dynamo represents an important channel for magnetic field generation in white dwarfs, one would expect to see an accumulation of magnetic white dwarfs with crystallizing cores rotating relatively fast or a relation between rotation period and field strength.
Indeed, most magnetic white dwarfs have periods of a few hours or less, magnetic field strength of less than a few MG and accumulate in the lower left corner of Fig. 7, i.e. have low temperatures and typical white dwarf masses.It also seems that a significant number of magnetic white dwarfs cluster around the onset of crystallization.Although some of them rotate too slowly to generate their magnetic field through the dynamo, this clustering might imply that crystallization plays a role in the magnetic field generation of some white dwarfs.Also the magnetic white dwarfs in the TESS and 20 pc sample (stars and circles) are mostly close to the crystallization limit but some of them rotate very slowly yet host strong fields.If we include magnetic white dwarfs from the literature (triangles), it further becomes obvious that magnetic white dwarfs rotating relatively fast can be found above the temperature limit for the onset of crystallization as well as below.This suggests that the dynamo might be responsible for a fraction of the observed magnetic fields in single white dwarfs but is clearly unable to explain all of them.

Fast spinning massive white dwarfs and the merger channel
The only crystal clear tendency that can be seen in fig.7 is the existence of very fast rotators among high-mass white dwarfs, i.e. we see evidence for the merger channel (García-Berro et al. 2012) producing magnetic white dwarfs.
Our literature search provided a relatively large number of highmass white dwarfs with short spin periods.These white dwarfs are absent in the 20 pc sample (Bagnulo & Landstreet 2021) and are therefore rare (Gentile Fusillo et al. 2021).There is an increasing amount of evidence that these magnetic high-mass white dwarfs are the product of mergers as suggested early by García-Berro et al. (2012).Kilic et al. (2023) showed that 40 per cent of high-mass white dwarfs are magnetic which exceeds the 20 per cent found in volume-limited samples (Bagnulo & Landstreet 2021).This finding is in agreement with Bagnulo & Landstreet (2022) who find that in high-mass white dwarfs, magnetic fields are extremely common, very strong and appear immediately in the cooling phase while lower-mass white dwarfs become magnetic close to the onset of crystallization.Further evidence for the generation of magnetic fields through mergers is provided by Pelisoli et al. (2022) who found evidence for the three known magnetic hot sub-dwarfs to be the product of a merger.
Theoretical simulations of the merger process predict the remnant to have very short spin periods of just ≃ 10 − 20 minutes (Schwab 2021).This prediction fits with the most massive white dwarfs that are rotating with periods of this order (we used a rather arbitrary limit of 0.38 hr in fig.7).We, therefore, conclude that at least these fast-spinning, strongly magnetic, and very massive white dwarfs are the product of mergers.However, it seems that not all mergers produce strongly magnetic white dwarfs.Hollands et al. (2020) found a massive white dwarf that is likely the product of a merger and derived an upper limit on the field strength of only 50 kG.
The magnetic white dwarfs in the mass range of ∼ 0.8−1.2 that do not spin with periods of minutes, show spin periods similar to that of less massive magnetic white dwarfs.This means, the spin periods of these white dwarfs alone do not provide evidence for them emerging from a different channel than the lower mass white dwarfs.In other words, the spin periods alone do not provide indications that these white dwarfs might be mergers as well.

Constraints on magnetic field generation in white dwarfs
Of the mechanisms suggested for magnetic field generation, the merger channel is the most convincing and the only one that can count on clear observational evidence.However, only very few white dwarfs are produced through this channel which is clearly illustrated by the complete absence of such white dwarfs in the 20 pc sample (Bagnulo & Landstreet 2021).
Concerning the dominant population of magnetic white dwarfs we know the following: the 20 pc sample established by Bagnulo & Landstreet (2021) shows that in general magnetic fields in white dwarfs appear late, i.e when the white dwarf has cooled to temperatures similar to the onset of crystallization.This is illustrated by the white dwarfs marked as stars and circles in Figure 7.In this work, we complemented this result by finding that a significant fraction of these cool nearby white dwarfs rotate fast (periods of a few hours) while others are slowly rotating (longer than 20 hours).In addition, we find that on average magnetic white dwarfs rotate faster than non-magnetic white dwarfs.These constraints seem to imply that in general rotation plays a role in the generation of white dwarf magnetic fields but not in all cases.
It could be not just one additional channel to the merger channel but several, and only one of them necessitates relatively fast rotation.
The generally late appearance of the magnetic fields could imply that crystallization plays a role or that the fields appear late for a different reason, e.g. because they were buried following the white dwarf formation.We clearly need a larger sample of well characterized magnetic white dwarfs and new theoretical ideas to finally understand the formation of magnetic fields in white dwarfs.

Magnetic white dwarfs in binaries
The absence of strongly magnetic white dwarfs in young postcommon envelope binaries was one of the main motivations for the development of the crystallization and rotation-driven dynamo (Schreiber et al. 2021a).This absence of young magnetic white dwarfs in close binaries is consistent with the absence of strongly magnetic single white dwarfs in the 20 pc sample (Bagnulo & Landstreet 2021).Whether the generally late appearance of magnetic fields in white dwarfs is caused by a dynamo that requires crystallization or if the magnetic field appears late for another reason remains an open question.
In any case, whether the magnetic field is generated late due to a dynamo (that does depend on rotation) or appears late in the evolution of a white dwarf for another reason (e.g. because it was buried), the evolutionary sequence suggested by Schreiber et al. (2021a) for white dwarf binaries (i.e.magnetic field appearance/generation during the CV phase, connection with the field of the secondary, angular momentum transfer to the orbit, cessation of mass transfer, synchronization, re-start of mass transfer) remains a promising scenario to explain the observations of magnetic white dwarfs in post common envelope binaries and CVs.
The situation for double white dwarfs is similar.As shown by Schreiber et al. (2022), only one magnetic white dwarf is known in these systems (which is close to the crystallization boundary and rapidly rotating).If rotation does not play a crucial role in the magnetic field generation, cool white dwarfs in close double white dwarfs should develop magnetic fields.In the 20 pc sample the occurrence rate of magnetism increases for effective temperatures in the range of 4 000 − 8 000 K (Bagnulo & Landstreet 2021, thier Figure 2) while the vast majority of both components in double white dwarfs are hotter.It might therefore be that in both populations, close double white dwarfs, and post common envelope binaries with main sequence star companion, the late appearance of magnetic fields has not yet been observed.
Further searches for magnetism in cool white dwarfs that are part of post-common envelope binaries or double white dwarfs would significantly help to settle open questions.If rotation is not a key ingredient for the appearance of magnetic fields, we should find magnetic white dwarfs in detached double white dwarfs and among post-common envelope binaries that contain cool white dwarfs and that are not close to Roche-lobe filling.The latter does not seem to be the case (Parsons et al. 2021), but the sample size is still small and heavily biased towards systems that are close to Roche-lobe filling as the magnetic fields are detected through cyclotron emission via wind accretion.Further observations, especially of systems containing the coolest white dwarfs are certainly required (but might be challenging).

CONCLUSIONS
The 20 pc sample provided by Bagnulo & Landstreet (2021) showed that white dwarfs become magnetic when they are older, typically with ages exceeding 2 Gyr (see also Bagnulo & Landstreet 2022), reaching effective temperatures close to the crystallization limit.While detached post-common envelope binaries are hardly ever magnetic, a large number of their descendants, semi-detached cataclysmic variables turned out to host a magnetic white dwarf.In other words, there is clear observational evidence for the late appearance of magnetic fields in white dwarfs.Based on this finding and the early work by Isern et al. (2017), a crystallization and rotation-driven dynamo has been proposed for the origin of magnetic fields in white dwarfs (Schreiber et al. 2021a;Ginzburg et al. 2022).
We presented measurements of the spin periods of eight magnetic white dwarfs using TESS light curves and found that their spin periods are similar to those of previously identified magnetic white dwarfs and shorter than that of non-magnetic white dwarfs.Four of the periods we measured agree with previously measured periods and in one of them the TESS period is half the real period.One white dwarf in our sample has magnetic field strengths of ∼ 100 MG but a long rotation rate of 32 hr.This result indicates that fast rotation plays a role in the magnetic field generation in general but is not always required.
In a sample of magnetic white dwarfs compiled from the literature, some massive white dwarfs entirely consistent with being merger products stand out.In the remaining systems, we do not find evidence for a dependence of spin period on white dwarfs mass but a larger and/or complete volume-limited sample of magnetic white dwarfs is needed to confirm this impression.
The crystallization and rotation-driven dynamo might be responsible for a fraction of the magnetic fields in white dwarfs.If it is, a third mechanism to the dynamo and mergers would probably be required.Alternatively, the proposed dynamo does not play a role but just one channel (complementing the magnetic white dwarfs resulting from mergers), that for some reason leads to the late appearance of magnetic fields and on average shorter spin periods, is responsible for the generation of most magnetic fields in white dwarfs.Independent of the reasons for the late appearance of magentic fields, the new evolutionary sequence suggested by Schreiber et al. (2021a) for white dwarfs in binaries remains plausible.

APPENDIX A: DETAILS ON THE 20 PC TESS SAMPLE A1 TESS sectors
We provide the lists of TESS sectors that we analyzed for all targets within 20 pc.This includes white dwarfs with clear photometric variations (see Table A1).

A2 An eclipsing binary contaminating the TESS light curve
During the TESS light curves analysis of WD 2150+591, we noticed that the light curves of two sectors showed features resembling that of an eclipsing binary system.We modified the flux contamination tool recently published by Schonhut-Stasik & Stassun (2023) with the aim to obtain the contamination level and G-magnitude of individual sectors in order to analyze this particular case.We compared the resulting G-magnitudes (by sector) with the white dwarf G-magnitude reported in literature.We found that the sectors showing the variations reminiscent of an eclipsing binary (Figure A1) indeed were dominated by a nearby eclipsing binary star as the average G-magnitude was significantly smaller than that of the white dwarf.Therefore the variations one can find in the TESS database for WD 2150+591 are clearly not produced by the white dwarf.

A3 Folding TESS data over the real rotation period
For two white dwarfs in our sample, the strongest signal in the periodogram corresponds to half the orbital period.The independently measured period appears as significant second highest peak in the periodograms.In Fig. A2 we show the TESS light curves folded over the true rotation period.In the case of WD 0041-102, this period is identical to the period indicated by the second highest peak in the periodogram.For WD 0009+501, however, the period corresponding to the second highest peak (8.007 hr) provides a poor fit to the light curve and we instead used the slightly different period (8.016 ± 0.083 hr) obtained by re-analyzing the full spectropolarimetry data set from Table A1.List of magnetic white dwarfs with TESS observations from the 20 pc sample.Targets for which we found clear periodic variations are marked with a ✓symbol.For these systems, we provide the identification number for all the analyzed TESS sectors.For those targets without clear variations (marked with ×) we only provide the total number of TESS sectors available for each white dwarf.Values in the parenthesis indicate the contamination level for the corresponding sector .Periodogram and phase-folded TESS light curve of WD 2150+591, the closest white dwarf in our sample (8.5 pc).We binned the phase-folded light curve (using 100 bins) to reduce the noise.This white dwarf was observed in five TESS sectors, by comparing the G-magnitude of each sector with the magnitude reported in the literature for this white dwarf, we found that two sectors are dominated by the light curve of a nearby eclipsing binary.Here we show the light curve corresponding to sector 56 with a 20 seconds cadence and a contamination level of 10.5 %.Valyavin et al. (2005a) supplemented with one all-night spectropolarimetry data set in the yet unpublished work of Bagnulo, Landstreet & Valyavin.The latter magnetic period provides a decent phase-folded light curve and is very close to the period corresponding to twice the period of the highest peak in the periodogram (8.017 hr).

Name
Table A2.List of magnetic white dwarfs with TESS observations from the 40 pc sample.Targets for which we found clear periodic variations are marked with a ✓symbol.For these systems, we provide the identification number for all the analyzed TESS sectors.For those targets without clear variations (marked with ×) we only provide the total number of TESS sectors available for each white dwarf.Values in the parenthesis indicate the contamination level for the corresponding sector.

Figure 1 .
Figure 1.The pixel view obtained with the two different tools used to study the contamination from nearby sources to the white dwarf observed with TESS.Left: The FluxCT provides a white area corresponding to the pixels used to calculate the contamination.The position of WD 1009-184 is indicated by a pink star while that of the only contaminating source found by the tool is shown as a white star.The numbers right next to each star indicate their Gaia G-magnitudes.Right: The image obtained with the lightkurve tool.Pixels with a white edge were used to extract the TESS light curve of WD 1009-184.The variation found in the light curve of this system, however, results from contamination of these pixels from a bright star in the pixel with the red border.The star located in this pixel shows cleaner variations with the same period and a larger amplitude.This type of contamination cannot be identified if TESS data is only analyzed with the FluxCT.However, tess_localize confirms that the origin of the variation corresponds to target Gaia DR3 5669427508702256896 located at the pixel marked with the red border.

Figure 2 .
Figure2.Periodograms (left: full Nyquist range; middle: zoomed in to the highest peak) and phase-folded light curves (right) of eight white dwarfs with a TESS light curve that shows statistically significant variations with the same period in all available sectors.The horizontal red line illustrates the false alarm probability level (FAP level) for a probability of 10 −3 per cent.In the case of WD 0009+501, WD 0011-134, WD 0011-721, WD 0041-102, WD J075328.47-511436.96 and WD J171652.09-590636.29 we binned the phase-folded light curve (using 100 bins) as the amplitude of the variation is too small to be spotted in the un-binned light curve.For WD 0009+501 and WD 0041-102 we know from previous observations that the true period is twice the period we detect in the TESS data(Valyavin et al. 2005a;Achilleos et al. 1992).We use the longer (and correct) periods in the distributions discussed in this paper.The TESS light curves folded over the rotation period can be found in the appendix.For the other seven objects, we interpret the observed periods as reflecting the rotation periods of the white dwarfs.The measured periods range from a few hours to several days.

Figure 3 .
Figure 3. Periodograms (left: full Nyquist range; middle: zoomed in to the highest peak) and phase-folded light curves (right) of the two white dwarfs observed with SPECULOOS.The obtained periods are less certain than those measured from TESS light curves.The horizontal grey line illustrates the false alarm probability level (FAP level) for a probability of 10 −6 per cent.For WD 2138-332 we can estimate only a range of periods because several periods (peaks in the periodogram) provide reasonable fits to the data.However, both targets clearly have periods of the order of a few hours.

Figure 4 .
Figure 4. Cumulative period distributions of magnetic and non-magnetic white dwarfs.If we consider all rotation periods measured for magnetic white dwarfs (solid green line) and the magnetic white dwarf after removing white dwarfs that most likely result from mergers (dotted green line), they seem to rotate significantly faster than non-magnetic white dwarfs (solid black line).A KS-test between the full magnetic and non-magnetic sample gives a probability of only 3.76×10 −7 for the observed difference to be caused by chance.However, this is clearly caused by an observational bias.If only periods measured with TESS are considered (dashed light-blue line), a KS-test provides weak indications for the period distribution being different to those of non-magnetic white dwarfs (p-value = 0.027).The incomplete sample of magnetic white dwarfs within 20 pc (dotted dark-blue line) also contains slightly shorter periods compared to non-magnetic white dwarfs.

Figure 5 .
Figure 5. Left: Rotation periods as a function of white dwarf mass for non-magnetic white dwarfs (black dots) from TableC2, and magnetic white dwarfs ( green circles) from TablesC1 and 1.The magnetic white dwarfs with periods measured from TESS light curves (light blue circles and triangles) and those within 20 pc (dark blue triangles) have, on average, shorter periods than non-magnetic white dwarfs measured from pulsations.Both samples share the light blue triangles, as some of the white dwarfs in the 20 pc sample were observed with TESS.Right: Cumulative distributions of the masses of magnetic and non-magnetic white dwarfs.Both, the TESS (dashed light-blue line) and the 20 pc sample (dotted dark-blue line) contains slightly more massive white dwarfs than the sample of non-magnetic white dwarfs (solid black line).This difference is statistically significant in both cases (  = 0.0005 and  = 0.0008), which is consistent with the fact that magnetic white dwarfs in the 20 pc sample are slightly more massive than the non-magnetic ones(Bagnulo & Landstreet 2021)

Figure 7 .
Figure 7. Mass-temperature distribution of magnetic white dwarfs.White dwarfs with periods measured from TESS light curves are represented by stars, white dwarfs with measured periods and within 20 pc from TableC1are marked as filled circles, and the white dwarfs from TableC1are represented by triangles.Colours show the magnetic field strength with a maximum of 300 MG while the size of the objects (scale on the top of the image) represents different ranges of the rotational period of the white dwarfs.The solid and dashed lines provide the temperature limits for the onset of crystallization according toBédard et al. (2020) andSalaris et al. (2010), respectively.
Figure A1.Periodogram and phase-folded TESS light curve of WD 2150+591, the closest white dwarf in our sample (8.5 pc).We binned the phase-folded light curve (using 100 bins) to reduce the noise.This white dwarf was observed in five TESS sectors, by comparing the G-magnitude of each sector with the magnitude reported in the literature for this white dwarf, we found that two sectors are dominated by the light curve of a nearby eclipsing binary.Here we show the light curve corresponding to sector 56 with a 20 seconds cadence and a contamination level of 10.5 %.

Figure A2 .
Figure A2.Periodograms (left: up to the Nyquist frequency; middle: zoomed in to the second highest peak) and phase-folded TESS light curves (right) of WD 0009+501 and WD 0041-102.The horizontal red line illustrates the false alarm probability level (FAP level) for a probability of 10 −3 .For both objects the true rotation period is close to the second highest peak in the periodogram.The data of WD 0041-102 was folded over the period indicated by the second highest peak in the periodogram (2.192 hr) and we recover the double humped light curve previously reported by Achilleos et al. (1992).In the case of WD 0009+501, the light curve is folded over a period of 8.016 ± 0.083 hr, obtained by re-analyzing the full spectropolarimetry data set from Valyavin et al. (2005a) supplemented with one all-night spectropolarimetry data set in the yet unpublished work of Bagnulo, Landstreet & Valyavin.This magnetic period is almost identical to twice the period of the highest peak in the periodogram (8.017 hr) and close to the period indicated by the second highest peak in the periodogram (8.007 hr).

Table 2 .
Two-sided Kolmogorov-Smirnov test results for the different samples: all magnetic white dwarfs (Tables C1 & and TESS targets from Table3), TESS targets from Table3, and white dwarfs within 20 pc compared to nonmagnetic white dwarfs (TableC2).The unbiased sample of periods measured with TESS confirms previous indications that magnetic white dwarfs rotate on average faster than non-magnetic ones.

Table B1 .
Observational specifications for targets observed with SPECU-LOOS.