# Magnetic dynamos in white dwarfs –III: explaining the occurrence of strong
magnetic fields in close double white dwarfs

Matthias.R. Schreiber,1,2 Diogo Belloni,1 Monica Zorotovic,3 Sarai Zapata,1
Boris T. Gänsicke,4 Steven G. Parsons5
1Departamento de Física, Universidad Técnica Federico Santa María, Av. España
1680, Valparaíso, Chile
2Millennium Nucleus for Planet Formation, NPF, Valparaíso, Chile
3Instituto de Física y Astronomía, Universidad de Valparaíso, Av. Gran Bretaña
1111, Valparaíso, Chile
4Department of Physics, University of Warwick, Coventry CV4 7AL, UK
5 Department of Physics and Astronomy, University of Sheffield, Sheffield S3
7RH, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The origin of strong
($\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$}1MG$)
magnetic fields in white dwarfs has been a puzzle for decades. Recently, a
dynamo mechanism operating in rapidly rotating and crystallizing white dwarfs
has been suggested to explain the occurrence rates of strong magnetic fields
in white dwarfs with close low-mass main sequence star companions. Here we
investigate whether the same mechanism may produce strong magnetic fields in
close double white dwarfs. The only known strongly magnetic white dwarf that
is part of a close double white dwarf system, the magnetic component of NLTT
12758, is rapidly rotating and likely crystallizing and therefore the proposed
dynamo mechanism represents an excellent scenario for the origin of its
magnetic field. Presenting a revised formation scenario for NLTT 12758, we
find a natural explanation for the rapid rotation of the magnetic component.
We furthermore show that it is not surprising that strong magnetic fields have
not been detected in all other known double white dwarfs. We therefore
conclude that the incidence of magnetic fields in close double white dwarfs
supports the idea that a rotation and crystallization driven dynamo plays a
major role in the generation of strong magnetic fields in white dwarfs.

###### keywords:

binaries: close – stars: individual: NLTT 12758, NLTT 11748, SDSS
J125733.63$+$542850.5 – stars: magnetic field – white dwarfs

††pubyear: 2021††pagerange: Magnetic dynamos in white dwarfs –III: explaining
the occurrence of strong magnetic fields in close double white dwarfs–B

##  1 Introduction

White dwarfs have been speculated to potentially have strong magnetic fields
(exceeding 1 MG) since 1947 (Blackett, 1947), but the first detection of a
magnetic field in a white dwarf was only obtained more than twenty years later
(Kemp et al., 1970). Ever since, the question why some white dwarfs become
strongly magnetic while others do not, has been one of the fundamental
unsolved issues of stellar evolution. Answering this question appears to be
very complicated largely because strongly magnetic white dwarfs are found in
different relative numbers among single white dwarfs, white dwarfs in close
detached binaries, and white dwarfs in close semi-detached binary stars.

The incidence of strongly magnetic white dwarfs in magnitude-limited surveys
of isolated white dwarfs is roughly five per cent (Kepler et al., 2013), but
the fraction is clearly higher among nearby white dwarfs which contain more
old systems (Kawka et al., 2007; Bagnulo & Landstreet, 2020, 2021) which
indicates that the fractions derived from magnitude limited surveys
potentially largely underestimate the true fraction of strongly magnetic white
dwarfs.

Among close detached white dwarf binaries the fraction of systems containing a
strongly magnetic white dwarf is very small, at most a few percent, and all
known systems are close to Roche-lobe filling and contain cold white dwarfs
with temperatures
$\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}$}10\,000$
K (Parsons et al., 2021). In contrast, the fraction of strongly magnetic white
dwarfs among their semi-detached descendants is very high, exceeding most
likely one third (Pala et al., 2020).

Present versions of the models suggested for generating strong magnetic fields
in white dwarfs, the fossil field (e.g. Tout et al., 2004; Braithwaite &
Spruit, 2004), the double degenerate merger (García-Berro et al., 2012), and
the common envelope dynamo scenarios (e.g. Tout et al., 2008), cannot
reproduce these observations. Either the predicted numbers of strongly
magnetic white dwarfs are too small as in the fossil field scenario (Kawka &
Vennes, 2004; Tout et al., 2004; Wickramasinghe & Ferrario, 2005; Aurière et
al., 2007), or the number of predicted strongly magnetic white dwarfs is far
too large as in the common envelope scenario (Belloni & Schreiber, 2020), or
fails to reproduce the large number of magnetic white dwarfs in interacting
binary stars as in the double degenerate merger scenario. In addition, in
their current form, none of the above scenarios can explain the absence of
strongly magnetic white dwarfs in young detached white dwarf/main sequence
binary stars (Liebert et al., 2005; Liebert et al., 2015; Belloni & Schreiber,
2020; Schreiber et al., 2021a).

We have recently developed a new model for the generation of magnetic fields
in white dwarfs that can potentially explain the different occurrence rates of
strongly magnetic white dwarfs with main sequence star companions (Schreiber
et al., 2021a). This scenario is based on the idea first put forward by Isern
et al. (2017) who proposed that a crystallization and rotation driven dynamo
similar to those operating in planets and low-mass stars could be generating
magnetic fields in white dwarfs.

According to the scenario developed by Schreiber et al. (2021a), white dwarfs
in close binaries emerge from common envelope evolution and evolve towards
shorter orbital periods without hosting a strong magnetic field. As soon as
mass transfer starts these post common envelope binaries become cataclysmic
variables (CVs) and the white dwarf accretes mass and angular momentum. If the
spun-up white dwarf has cooled enough to contain a crystallizing core, its
structure consists of an outer carbon rich convective zone and a solid inner
oxygen rich nucleus (Isern et al., 2017; Schreiber et al., 2021a). This
situation resembles that of the interior of planets and low-mass stars and the
convective motion in the liquid mantle can generate a magnetic field.

Isern et al. (2017) estimated the field strength generated by this dynamo
based on scaling laws that are used for planets and low-mass stars
(Christensen & Aubert, 2006; Christensen et al., 2009) and derived an upper
limit of around one MG. However, existing scaling laws are most likely not
appropriate for white dwarfs as the much higher magnetic Prandtl number in
white dwarfs could imply a much larger field strength (Brandenburg, 2014;
Bovino et al., 2013). Very recently, Ginzburg et al. (2022) argued that the
convective turn-over time is much larger than assumed by Isern et al. (2017)
which would imply that white dwarfs are almost always in the fast rotation
regime. Combining this finding with a scaling law based on the balance between
the Lorentz and Coriolis forces, even permits the generation of strong
magnetic fields (reaching 100 MG) without postulating a magnetic field
enhancement due to the white dwarf’s Prandtl number.

If indeed strong magnetic fields are generated in the white dwarfs in CVs due
to this dynamo, these fields may connect with the field of the secondary star
and synchronizing torques transfer spin angular momentum from the white dwarf
to the orbital motion which can cause the system to detach. At first the
detached system will appear for a short period of time as a radio pulsing
detached white dwarf binary similar to AR Sco (Marsh et al., 2016), then
evolve into a synchronized detached pre-polar (Schwope et al., 2009; Parsons
et al., 2021) and subsequently into a semi-detached magnetic CV (polar).

In addition to offering an explanation for the origin and evolution of
magnetic white dwarfs in close binary stars with a main sequence star
companion, we could recently show that the dynamo also naturally explains the
low occurrence rate of high accretion rate magnetic CVs in globular clusters
(Belloni et al., 2021) and the increased incidence of magnetism among cold
metal polluted white dwarfs (Schreiber et al., 2021b).

Despite this success, we note that some individual systems containing magnetic
white dwarfs require an alternative mechanism for the magnetic field
generation. The recently discovered post common envelope binary CC Cet (Wilson
et al., 2021) and the polar EY Eri (Beuermann et al., 2020) both contain most
likely magnetic helium core white dwarfs and one of the main ingredients of
the dynamo scenario is a crystallizing core consisting of carbon and oxygen.
In addition, the potentially weakly magnetic white dwarf in the post common
envelope binary V471 Tau (Sion et al., 2012) is too hot to be crystallizing.

Given this situation, it is of utmost importance to further investigate to
which degree the dynamo scenario may solve the long standing puzzle of the
origin of strong magnetic fields in white dwarfs and in which cases an
alternative mechanism is required. Here we apply the suggested dynamo
mechanism to close double white dwarf binaries and investigate whether it can
explain the occurrence rate and characteristics of strongly magnetic white
dwarfs in these systems.

##  2 Known double white dwarfs and magnetic fields

In order to investigate whether the new scenario for magnetic field generation
may explain the observations of close (here defined as an orbital period below
35 days) double white dwarfs, we compiled a list of the currently known
systems (see Appendix A, Table 1). At present, only one system is known to
host a strongly magnetized white dwarf. NLTT 12758 consists of a magnetic
white dwarf with a field strength of about 3.1 MG and an apparently non-
magnetic white dwarf companion. The non-magnetic H-rich (DA) white dwarf is
more massive ($M=0.83\pm 0.03\mbox{M${}_{\odot}$}$) than its magnetic (DAP)
companion ($M=0.69\pm 0.05\mbox{M${}_{\odot}$}$), the orbital period of the
binary is 1.154 days, the spin period of the magnetic white dwarf has been
measured to be 23 minutes, and the cooling ages of both white dwarfs are
comparable (Kawka et al., 2017).

The recently proposed magnetic dynamo requires relatively rapid rotation and a
crystallizing core to generate strong magnetic fields in white dwarfs.
Saturation of the dynamo is assumed to occur for spin periods of just a few
seconds (see Isern et al., 2017; Schreiber et al., 2021a, for more details).
The magnetic and lower mass white dwarf in NLTT 12758 is rapidly rotating with
a spin period of 23 minutes which is much shorter than the typical spin period
of non-magnetic single white dwarfs (Hermes et al., 2017) but longer than the
short periods of the order of seconds proposed for white dwarfs in CVs
(Schreiber et al., 2021a). One would therefore expect the magnetic field
strength of the magnetic white dwarf in NLTT 12758 to be smaller than the
strongest magnetic fields observed in CVs. Indeed, in magnetic CVs the field
strengths of the white dwarfs reaches several hundred MG with an average field
strength of $\sim$30 MG (Ferrario et al., 2015; Belloni et al., 2020) and the
average field strength of pre-polars is $\sim$60-70 MG (Parsons et al., 2021)
while for the magnetic component in NLTT 12758 a field strength of just 3.1 MG
has been measured (Kawka et al., 2017).

The second criterion for the dynamo to work is that the core of the white
dwarf must have started to crystallize. The onset of crystallization depends
on the temperature and the white dwarf mass. The currently available cooling
tracks that include crystallization differ slightly in the predicted
temperatures for the onset of crystallization. In Fig.1 we compare the onset
of crystallization from the models by Salaris et al. (2010) and Bédard et al.
(2020) with the position of all C/O white dwarfs in close double white dwarfs
with known mass and effective temperature.

Figure 1: Comparison of the mass and effective temperature of the known white
dwarfs that form part of a close (period less than 35 days) double white dwarf
(from Table 1) with the onset of crystallization according to the cooling
sequences by Salaris et al. (2010, black dashed line) and Bédard et al. (2020,
black solid line). The vast majority of the white dwarfs are too hot to be
crystallizing. Among the few exceptions is the magnetic component in NLTT
12758. For this white dwarf the measured parameters are consistent with
crystallization in its core. As in addition the spin period has been measured
to be very short (23 min.) it appears plausible that the rotation and
crystallization driven dynamo generated the strong magnetic field. Three more
white dwarfs are also consistent with having a crystallizing core. This might
indicate that those white dwarfs, in contrast to the magnetic component in
NLTT 12758, did not accrete angular momentum during their evolutionary history
or that their magnetic field has not been detected yet.

The measured parameters of the magnetic white dwarf component in NLTT 12758
(highlighted in blue in Fig. 1) are consistent with a crystallizing core in
the white dwarf and therefore the rotation and crystallization driven dynamo
offers a plausible explanation for its magnetic nature. Most other white
dwarfs that are members of close double white dwarfs are either He-core white
dwarfs for which the dynamo scenario does not apply and/or are too hot for
having crystallizing cores which is also consistent with the dynamo scenario
for magnetic field generation in white dwarfs. However, of the few double
white dwarfs containing C/O white dwarfs, three additional ones have
sufficiently cooled to be crystallizing.

Apart from the DAP white dwarf in NLTT 12758, components close to or beyond
the onset of crystallization are its higher mass DA companion and the more
massive white dwarfs in SDSS J125733.63+542850.5 (hereafter SDSS J1257+5428)
and NLTT 11748 (the white dwarfs highlighted with cyan and magenta in Fig. 1).
This leads to an important follow-up question: why is NLTT 12758 the only
close double white dwarf where a strongly magnetic component has been
detected?  If our hypothesis that the rotation and crystallization driven
dynamo generated the magnetic field in NLTT 12758 is correct, the other white
dwarfs with crystallizing cores must either not have sufficiently spun up
during their formation or we simply have not been able to detect their strong
magnetic fields. To understand magnetic field generation in close double white
dwarfs we therefore need to investigate the evolutionary history of NLTT 12758
and consider potential evolutionary differences to NLTT 11748 and SDSS
J1257+5482 as well as possible observational biases. We start by taking a
closer look at the formation of NLTT 12758.

##  3 Was NLTT 12758 formed through two common envelope phases?

Most close white dwarf binary stars form through common envelope evolution
(Webbink, 1984; Zorotovic et al., 2010; Ivanova et al., 2013) which occurs
when the more massive star fills its Roche-lobe on the first giant branch
(FGB) or the asymptotic giant branch (AGB). The resulting mass transfer is
typically dynamically unstable and leads to the formation of a gaseous
envelope around the secondary star and the core of the giant. This envelope is
expelled at the expense of orbital energy and orbital angular momentum.
Therefore, the emerging post common envelope binary stars have typically short
orbital periods between a few hours and a few days (Nebot Gómez-Morán et al.,
2011).

Common envelope evolution has been shown to successfully explain the observed
close white dwarf plus M dwarf binary population (Zorotovic et al., 2010) and
seems to also reproduce the short periods found among some close white dwarf
plus FGK secondary star binary systems (Hernandez et al., 2021). Two
consecutive common envelopes have been suggested as the formation scenario for
NLTT 12758 (Kawka et al., 2017) and we therefore start our discussion with a
review of this classical scenario for the formation of close double white
dwarfs.

The change in orbital separation during common envelope evolution can be
calculated using a simple energy conservation equation that relates the loss
of orbital energy of the binary and the binding energy of the envelope. This
energy conservation equation is typically parameterized with the common
envelope efficiency $\alpha_{\mathrm{CE}}$ and the binding energy parameter
$\lambda$ (e.g. Han et al., 1995; Dewi & Tauris, 2000; Zorotovic et al.,
2010).

While it is clear that the common envelope efficiency $\alpha_{\mathrm{CE}}$
should be between zero and one – there are strong indications for a relatively
low value between $0.2-0.3$ (Zorotovic et al., 2010) – there is no agreement
in the literature on the most realistic value of the binding energy parameter
$\lambda$ mostly because it remains unclear how efficient recombination energy
can contribute to the process of expelling a common envelope (Webbink, 2008;
Ivanova et al., 2015; Ivanova, 2018; Sabach et al., 2017; Grichener et al.,
2018; Soker et al., 2018).

Assuming two common envelope phases and using the bse code, Kawka et al.
(2017) found that the history of NLTT 12758 can be reproduced assuming an
initial binary with stellar masses of 3.75 M⊙ and 2.88 M⊙ and an orbital
period of 2656 d. We could reproduce their result only by assuming that $\sim
75$ per cent of the available recombination energy contributed to expelling
the envelope. This assumption can be considered unrealistic as even work that
emphasize the potential importance of recombination energy exclude such large
values (Ivanova et al., 2015). We believe that this large fraction of
recombination energy is related to a misinterpretation of the input parameters
of bse (see Appendix B, Fig.5, for more details).

In order to investigate whether more realistic parameter combinations allow to
reproduce the formation of NLTT 12758 through two common envelope phases we
reconstructed both potential common envelope phases following Zorotovic et al.
(2010) and Zorotovic et al. (2014). We found that without contributions from
recombination energy the system cannot have formed by two consecutive common
envelope phases. Assuming a relatively small fraction of recombination energy
($\leq 10$ per cent, but at least $2$ per cent) and a value of the common
envelope efficiency exceeding 0.5, reasonable solutions exist.

These possibilities exist because the $\lambda$ parameter is extremely
sensitive to the inclusion of recombination energy. Even for an efficiency of
recombination energy of just $2$ per cent, the $\lambda$ parameter for the
more massive white dwarf progenitor (first common envelope phase) is
$\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$}3$,
leading to a much smaller reduction of the orbital separation with respect to
simulations that do not consider recombination energy at all (where $\lambda$
is closer to 1).

This situation is similar to that of the two white dwarf plus main-sequence
post common envelope binaries known so far that require extra energy sources
in order to explain their current orbital periods (IK Peg and KOI 3278).
However, both systems can be reproduced by assuming that
$\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}$}2$ per
cent of the recombination energy contributed to the ejection process
(Zorotovic et al., 2014) and for NLTT 12758 we need a slightly larger
fraction. We therefore conclude that two consecutive common envelope phases
remain a possible scenario for the evolutionary history of NLTT 12758 but that
an (uncertain and unusual) source of additional energy (possibly recombination
energy) is required.

If two common envelope phases produced NLTT 12758, the white dwarf with the
smaller mass must have formed through the second common envelope phase and the
more massive white dwarf must have formed through the first one. This is
because the onset of common envelope evolution is defined by dynamically
unstable mass transfer which quickly reaches mass transfer time scales that
are orders of magnitude shorter than the thermal time scale of the secondary
star. The secondary star is therefore supposed to accrete only a small amount
of mass (according to Chamandy et al. 2018 the total accreted mass is roughly
in the range of $10^{-2}-10^{-4}$ M⊙). As in addition the spiral-in process
significantly reduces the orbital separation, the initially less massive star
can not evolve further up the giant branches than the first one. In other
words, in two consecutive common envelope phases the more massive white dwarf
is formed first. For NLTT 12758 this implies that the fast spinning and
strongly magnetic white dwarf formed in the second common envelope phase. If
that was indeed the evolutionary history of NLTT 12758, it remains an open
question why the magnetic white dwarf is rapidly rotating.

However, the fact that the cooling ages of both white dwarfs in NLTT 12758 are
very similar, i.e. $2.2\pm 0.2$ and $1.9\pm 0.4$ Gyr (Kawka et al., 2017)
might indicate that the formation history of the system was actually
different. If the two progenitor stars evolved off the main sequence at about
the same time, their masses must have been similar and therefore the first
mass transfer phase could have been stable.

In fact, according to Table 5 in Kawka et al. (2017), even in their solution
for two common envelopes the first mass transfer started when both stars had
virtually identical masses which, according to Ge et al. (2020), should lead
to stable mass transfer instead of common envelope evolution if the mass
transfer is not fully conservative (i.e. a fraction of the transferred mass
leaves the system).

In the following section, we therefore discuss an alternative formation
scenario for NLTT 12758 which includes stable non-conservative mass transfer
and show that it offers a natural explanation for the observed binary and
stellar parameters of the system as well as the short spin period of the
magnetic white dwarf.

##  4 An alternative formation model: Combining stable mass transfer and
common envelope evolution

As noted first by Nelemans et al. (2000) two common envelope phases using the
classical energy budget equation fail to explain several observed double white
dwarf systems. To solve this issue, they suggested the $\gamma$ algorithm
based on angular momentum conservation which allows for common envelope
evolution without spiral in during the first mass transfer phase. However, by
implicitly assuming energy conservation the $\gamma$ algorithm is rather
hiding the energy problem instead of solving it (Ivanova et al., 2013). As an
alternative, Webbink (2008) suggested that instead the first mass transfer
could be stable and non-conservative which occurs when the mass ratio of the
two progenitor stars is close to one (Ge et al., 2020). Detailed models for
double He-core white dwarfs were subsequently performed and showed that stable
non-conservative mass transfer indeed offers an explanation (Woods et al.,
2012). We here explore the evolutionary scenario in which the first white
dwarf forms through stable mass transfer for NLTT 12758 using detailed stellar
evolution calculations.

###  4.1 Simulating the evolution with MESA

We performed our simulations of the binaries with the one-dimensional stellar
and binary evolution code Modules for Experiments in Stellar Astrophysics
(mesa Paxton et al., 2011; Paxton et al., 2013, 2015, 2018, 2019, r15140).

Figure 2: The revised evolutionary scenario for NLTT 12758 consists of stable
mass transfer followed by a symbiotic phase and common envelope evolution
after which the orbital period is similar to the one we observe today.
According to this scenario, the less massive white dwarf forms first and
accretes mass and angular momentum during the second mass transfer phase. When
this fast spinning white dwarf starts to crystallize, the dynamo mechanism
suggested by Schreiber et al. (2021a) generates the strong magnetic field that
has been detected by Kawka et al. (2017).

We accounted for mass loss through winds using the standard MESA
implementations. In particular, for red giants on the FGB, we adopted the
Reimers (1975) prescription, assuming a wind efficiency of ${\eta=0.5}$, which
is consistent with metallicity-independent estimates using star clusters
(McDonald & Zijlstra, 2015). For AGB stars, we adopted the Bloecker (1995)
recipe, assuming a wind efficiency of ${\eta=0.02}$, which is consistent with
the calibration performed by Ventura et al. (2000) using the luminosity
function of lithium-rich stars in the Magellanic Clouds.

The Roche lobe radius of each star was computed using the fit of Eggleton
(1983) and mass transfer rates during Roche lobe overflow (RLOF) are
determined following the prescription of Ritter (1988). Regarding wind
accretion, we assumed two different prescriptions depending on the donor
stars. For FGB stars, we adopted the Bondi-Hoyle-Lyttleton prescription (Hoyle
& Lyttleton, 1939; Bondi & Hoyle, 1944), since in these cases the wind
velocity is much larger than the orbital velocity of the accretor. On the
other hand, winds from AGB stars can have velocities smaller or comparable to
the orbital velocity of the accretor. In such situations, hydro-dynamical
simulations have shown that efficient wind accretion is possible through the
wind Roche lobe overflow (wRLOF) mechanism (Mohamed & Podsiadlowski, 2007),
which is a regime between the Bondi-Hoyle-Lyttleton regime and RLOF. In other
words, in the wRLOF mechanism it is assumed that the slow winds fill the
donors Roche lobe, which implies that wind accretion is enhanced.

Given the importance of this mechanism on the binary evolution we investigate
here, we implemented the wRLOF model in mesa, as described in Abate et al.
(2013), Iłkiewicz et al. (2019) and Belloni et al. (2020). Regarding how much
of the donor wind is allowed to be accreted, since the accretor cannot accrete
more mass than is lost by the red giant star, we enforced that the accretion
rate efficiency in the Bondi-Hoyle-Lyttleton regime cannot be greater than
$80$ per cent. In addition, we assume that the accretion rate efficiency in
the wRLOF regime has to be $\leq 50$ per cent, which is consistent with hydro-
dynamical simulations (see Abate et al., 2013, and references therein).
Finally, during each step of the simulations, after calculating the accretion
rate efficiency in both regimes, we adopted the higher one to be used in the
wind mass transfer scheme in mesa.

The zero-age main sequence (ZAMS) stars are assumed to have no rotation.
However, as the stars evolve, their rotations are allowed to change according
to mesa standard prescriptions (Paxton et al., 2013). In addition to rotation,
we also allow the stars to eventually synchronize with the orbit, due to tidal
interaction. As usual, we distinguish the phases in which the star has a
radiative envelope from those in which it has a convective envelope since the
synchronization time-scales in the former case are generally orders of
magnitude larger than those in the latter. In particular, given the star
masses we investigate, we assume that main sequence and core-helium-burning
stars have radiative envelopes, while sub-giant, FGB and AGB stars have
convective envelopes. During each star evolution, the synchronization time-
scales are computed using the mesa standard prescription, which are based on
Hut (1981) and Hurley et al. (2002).

When the initially more massive star becomes a white dwarf, we ignore the
contribution of the white dwarf to the tidal interaction, as its
synchronization time-scale should be much longer than that of the initially
less massive star. However, since the white dwarf can accrete mass and angular
momentum from the winds of its companion, especially in the case of a
symbiotic binary with the giant donor being on the AGB, we allowed the WD to
spin up following King et al. (1991) and Schreiber et al. (2021a). Measuring
short-time variability in symbiotic stars is very difficult, due to the fact
that steady nuclear burning on the white dwarf surface usually reduces the
amplitude of the variation, reducing in turn its detection probability.
Despite that, there is some evidence that the white dwarf can spin up during
the symbiotic star phase from the studies of the stable periodic oscillations
in two systems. In BF Cyg, the white dwarf’s spin period is ${\approx 108}$
min (Formiggini & Leibowitz, 2009), while Z And hosts a magnetic white dwarf
with spin period of ${\approx 28}$ min (Sokoloski & Bildsten, 1999). In both
cases, the white dwarf most likely spun up due to wind mass and angular
momentum accretion.

Depending on the accretion rate onto the white dwarf, hydrogen shell burning
could be stable, which results in an increase of its mass (e.g. Shen &
Bildsten, 2007; Wolf et al., 2013). We implemented the critical accretion rate
derived by Nomoto et al. (2007, their eq. 5), above which white dwarfs are
thermally stable, i.e. hydrogen burns steadily in a shell. For accretion rates
lower than this critical value, the white dwarf is considered thermally
unstable, which means that the hydrogen shell burning is unstable to flashes
and, for sufficiently strong flashes, nova eruptions are triggered, causing
that most of the accreted mass will be expelled. In addition, we assume that
there is a maximum possible accretion rate (Nomoto et al., 2007, their eq. 6)
such that white dwarfs accreting at rates above it will burn stably at this
maximum rate, and the remaining non-accreted matter will be piled up to form a
red-giant-like envelope. Since we expect such high rates when the TP-AGB
companion of the first white dwarf is close to filling its Roche lobe, i.e.
the binary is close to the onset of common envelope evolution, we assume that
this piled-up matter will be eventually lost during common envelope evolution.

Finally, we assume solar metallicity and set all other parameters in mesa as
the default values in version 15140.

###  4.2 Searching for a model that can explain the properties of NLTT 12758

Figure 3: Evolution with the DA progenitor (star 2) mass of some properties of
the best-fitting model for NLTT 12758, prior the onset of the common envelope
phase, namely star core and 1st WD masses (top left-hand panel), orbital
period (top right-hand panel), mass transfer rate during stable non-
conservative mass transfer and wind mass loss/accretion after the 1st WD is
formed (bottom left-hand panel), and spin period of the DAP (1st WD)
progenitor as well as of the DAP WD (bottom right-hand panel). The range of DA
and DAP WD masses, as inferred from observations, are indicated as gray areas
in the top left-hand panel, and the spin period of the DAP WD as a dotted
horizontal line in the bottom right-hand panel. We reasonably well reproduce
those properties in our modeling with the mesa code.

In order to search for a reasonable model that could reproduce the
observational properties of NLTT 12758 we carried out several sparse grids
covering different regions of the parameter space, which include the ZAMS
masses, the initial orbital period, the accretion efficiency during stable
non-conservative mass transfer, and the orbital energy conversion efficiency
during common envelope evolution ($\alpha_{\rm{CE}}$).

We found out that, if the magnetic WD in NLTT 12758 was formed through stable
non-conservative mass transfer, then its progenitor ZAMS mass cannot be less
massive than ${\simeq 3.60}$ M⊙. This is because the core mass growth during
the first white dwarf progenitor evolution is virtually entirely interrupted
when it becomes a Roche-lobe filling AGB star.This implies that the 1st white
dwarf progenitor core mass at the onset of RLOF must be already comparable to
the observed value of the DAP white dwarf in NLTT 12758. This also implies
that the 1st white dwarf progenitor ZAMS mass cannot be much larger than
${\sim 4.00}$ M⊙, since stars with masses larger than that would develop a
more massive core before the onset of RLOF.

In addition, stable mass transfer from an evolved red giant only occurs when
the ZAMS star masses are comparable, which means that the mass ratio (more
massive over the less massive) has to be $\lesssim 1.1$. Otherwise, the mass
transfer will be most likely dynamically unstable. Moreover, to reproduce the
magnetic white dwarf mass and to have stable mass transfer, the red giant has
to be on the early AGB. In case the mass transfer starts when the magnetic
white dwarf progenitor is on the thermally pulsing AGB phase, the mass
transfer will be most likely dynamically unstable, given the huge changes in
the star size during this phase.

Regarding the initial orbital period, assuming that the onset of RLOF occurs
when the star is on the early AGB phase implies that the initial orbital
period cannot be longer than ${\sim 700}$ d. On the other hand, in order to
reproduce the non-magnetic white dwarf mass in NLTT 12758, the core mass of
its progenitor needs to substantially grow before the onset of common envelope
evolution. This implies that when stable non-conservative mass transfer ends
the orbital period has to be ${\sim 3500-4000}$ d, which in turn implies that
the initial orbital period of the ZAMS binary cannot be shorter than ${\sim
500}$ d. However, we shall emphasize that this lower limit for the initial
orbital period strongly depends on the accretion efficiency during stable mass
transfer. In particular, the larger the accretion efficiency, the longer the
orbital period at the end of stable mass transfer. This correlation comes from
the fact that the increase in orbital period due to mass transfer is much
stronger than the increase caused by mass loss from the system.

To have dynamically stable mass transfer when the donor is an early AGB star,
we found that the accretion efficiency must be smaller than ${\sim 0.5}$,
which means that at least half of the mass leaving the early AGB donor is not
accreted by the donor, i.e. is lost from the binary. Such huge mass loss may
seem unrealistic at first glance, but there are several mechanisms able to
drive mass loss from double red giant binaries, such as jets, stellar winds,
circumbinary disc, accretion disc flashes/outbursts/outflows. Recently, while
modeling the system 2M17091769$+$312758, which is a semi-detached binary
composed of a red giant star transferring mass to a sub-giant star, Miller et
al. (2021) found that to reproduce the observational properties of this
system, around half of the mass leaving the donor star actually escapes the
binary, corresponding to an accretion efficiency of $\sim 0.5$ which is
consistent with our finding. However, we of course agree with one of the
greats in the field who stated concerning mass loss that "details of this
process remain obscure" (Webbink, 2008).

###  4.3 Reproducing the evolutionary history of NLTT 12758

Figure 4: HR diagram of both stars until the white dwarf formation. The
initially lower mass star (blue) is already on the first giant branch when
mass transfer starts (dashed dotted). At the onset of common envelope
evolution, the initially lower mass star evolved higher up the AGB than the
first star. This way, the higher mass white dwarf is formed second.

The best-fitting model in our approach, combining dynamically stable non-
conservative mass transfer followed by dynamically unstable mass transfer, is
illustrated in Fig. 2. The ZAMS stars have masses of 3.85 M⊙ and 3.50 M⊙ and
the orbital period is 670 days. The first mass transfer is stable but highly
non-conservative. During this mass transfer phase the lower mass white dwarf
of the system is formed, the secondary (the initially less massive) star
reaches 4.49 M⊙, and the period increases to 3814 days.

As soon as this star evolves into a TP-AGB star, the binary becomes a
symbiotic star and the white dwarf efficiently accretes mass and angular
momentum from its wind and spins up, reaching the observed values of the mass
(0.69 M⊙) and spin period ($\sim$23 minutes). Due to tidal forces, which
synchronize the rotation of the giant star with the orbital motion, the
orbital period decreases during this symbiotic phase. This finding is
consistent with observational indications that most red giants in symbiotic
stars are synchronized with the orbit (Zamanov et al., 2007).

As soon as the giant fills its Roche-lobe dynamically unstable mass transfer
leads to common envelope evolution which then leaves behind a close double
white dwarf with a fast spinning 0.69 M⊙and a more massive 0.825 M⊙component.
For a common envelope efficiency of 0.3, considering no contributions from
recombination energy, and using $\lambda$ as calculated by bse, the orbital
period of this emerging double white dwarf binary is $1.16$ days. Within the
following $\sim 2$ Gyr its period decreases to $1.154$ days due to angular
momentum loss by the emission of gravitational radiation. The age difference
between the two white dwarfs is just 50 Myr. The two white dwarfs cool and
start to crystallize, at first the more massive DA white dwarf which does not
generate a strong magnetic field as this white dwarf is not rotating rapidly
and roughly $\sim 0.7$ Gyr later the lower mass white dwarf. When the latter
happens, the rotation and crystallization driven dynamo generates the strong
magnetic field of the DAP white dwarf we observe today.

The evolution of both stars prior to the final common envelope phase is
further illustrated in Fig. 3. We plotted the core masses, orbital period,
mass transfer rate, and the spin period as a function of the mass of the
initially less massive star.

As shown in the upper left panel, the mass of the white dwarf that later forms
the magnetic component (the white dwarf that formed first) increases between
the main two mass transfer events due to wind accretion (red dashed–dotted
line). The core mass of the initially less massive star (blue line) has
already reached more than $0.4$ M⊙ when stable mass transfer starts but
increases further after the mass transfer phase ends, until the system enters
common envelope evolution.

The upper left plot illustrates how the orbital period changes during the
evolution. During the stable mass transfer phase the period significantly
increases (red line) but it decreases before the onset of common envelope
evolution due to tidal forces when the initially lower mass star evolves on
the AGB.

The evolution of the mass transfer is shown in bottom left panel of Fig. 3.
The red line shows that the stable mass transfer reaches large values of $\sim
10^{-3}$ M⊙/yr, and that roughly three per cent of the wind mass loss rate
(blue) is accreted by the white dwarf (green) during the symbiotic phase. This
mass accretion leads to the spin-up of the white dwarf that formed first as
displayed in the bottom left panel. The final spin period depends therefore on
the mass transfer rate and on the assumed spin-up efficiency.

The full evolution of both stars prior to the white dwarf formation is also
illustrated in an HR diagram (Fig. 4). The mass ratio of both stars is so
close to one that the lower mass star is already on the FGB when the first
white dwarf is formed. During the stable mass transfer phase, as the FGB
accretor star is growing in mass, it also rejuvenates and becomes hotter.

The very good agreement in the stellar masses, orbital and spin period, and of
the ages of the white dwarfs is of course the result of adjusting the values
of the common envelope efficiency, the mass loss parameter, the initial masses
and orbital period as well as the angular momentum accretion efficiency.
However, we would like to emphasize that all other parameters for stellar
evolution and mass transfer are fixed and that the presented solution does not
represent a rare or unusual evolution. In fact, we found many solutions that
are in very good agreement with the observations and could easily produce a
broad population of double white dwarfs that formed through the channel we
propose for NLTT 12758. In addition, any model for the formation of double
white dwarfs that we are aware of contains at least the same number of free
parameters.

The agreement between theoretical predictions and observations of the DAP
white dwarf in NLTT 12758 is as good as it gets. Our new evolutionary scenario
combined with the rotation and crystallization driven dynamo scenario explains
the magnetic field generation in NLTT 12758. This dynamo mechanism is also
consistent with the observational result that most white dwarfs that are
members of close double white dwarfs are not magnetic because the vast
majority of them are not crystallizing. The only exceptions are the DA
companion to the magnetic white dwarf in NLTT 12758 and the more massive
components in NLTT 11748 and SDSS J1257+5428. In the next section, we briefly
discuss these white dwarfs.

##  5 Should the massive components in NLTT 12758, NLTT 11748 and SDSS
J1257+5482 be magnetic?

The more massive DA white dwarf in NLTT 12758 is clearly crystallizing but
according to our evolutionary scenario has never had the possibility to
accrete significant amounts of angular momentum. According to the rotation and
crystallization driven dynamo scenario it is therefore not surprising that
this white dwarf is not strongly magnetic.

In the case of NLTT 11748 the situation is more difficult to evaluate because
neither the evolutionary history of the system nor the magnetic or non-
magnetic nature of the massive white dwarf are known. Spectral observations of
NLTT 11748 by Kawka & Vennes (2009) revealed the brighter component to be a
low-mass ($<0.2$M⊙) helium core white dwarf and Steinfadt et al. (2010)
demonstrated that the companion is a relatively massive C/O white dwarf.
Kaplan et al. (2014) analyzed high-precision eclipse light curves and found
the low mass white dwarf to be hotter ($T_{\mathrm{eff}}\simeq 8700\pm 140$ K)
and with an H-rich (DA) atmosphere while the companion is cooler
($T_{\mathrm{eff}}=7600\pm 120$ K) and could therefore be crystallizing.
Unfortunately, the more massive and cooler white dwarf is not visible in the
spectra that have been taken of this object and it is therefore impossible to
tell whether it hosts a strong magnetic field or not.

It also remains unclear which of the two white dwarfs formed first. White
dwarfs with masses below $0.2$ M⊙, such as the brighter component in NLTT
11748, can experience stable H burning for several Gyr (Driebe et al., 1999;
Serenelli et al., 2002; Panei et al., 2007). We can therefore not fully
exclude that the cooler and more massive white dwarf formed after the low-mass
white dwarf. This scenario requires the first mass transfer to be stable,
similar to the scenario we developed in this paper which was inspired by
earlier work (e.g Woods et al., 2012). In this case, the more massive white
dwarf might not have accreted sufficient mass and angular momentum and the
dynamo scenario would not apply. If, in contrast, the more massive white dwarf
formed first, the accretion of some mass and angular momentum during the
second mass transfer phase could be expected and the conditions for the dynamo
to work would be met.

The third double white dwarf with one component most likely crystallizing is
SDSS J1257+5482. In this system the lower mass white dwarf seems to be
significantly older than the more massive component. It is currently not clear
how this system formed (Bours et al., 2015) and we therefore restrain
ourselves from a discussion of the potential accretion of mass and angular
momentum of the massive component of the system.

Interestingly, both Kulkarni & van Kerkwijk (2010) and Marsh et al. (2011)
found that the Balmer lines of the massive component are broadened
significantly relative to what is expected from just pressure broadening.
Mechanisms that have been suggested for the additional broadening are fast
rotation and/or a magnetic field. It might thus be that the massive component
in SDSS J1257+5428 is the second fast spinning and crystallizing magnetic
white dwarf in a double degenerate binary. Testing this hypothesis would
provide important constraints on the evolutionary history of the binary.

Given the above discussion, we conclude that the predictions of the
crystallization and rotation driven dynamo scenario are consistent with the
observed incidence of magnetism among close double white dwarfs. This
separates the dynamo from the common envelope dynamo scenario previously
suggested. According to the common envelope idea for the origin of the
magnetic fields in white dwarfs (e.g. Tout et al., 2008), a large number of
hotter white dwarfs that are members of double white dwarfs should be magnetic
as shown by Belloni & Schreiber (2020). This does not seem to be the case:
among the relatively well studied 57 double white dwarfs listed in Table 1
only for one white dwarf, the fast spinning component in NLTT 12758, the
detection of a magnetic field has been reported.

##  6 Could all (strong) magnetic fields in white dwarfs be generated by the
dynamo?

While representing a promising scenario for a relatively large number of
magnetic fields in white dwarfs in different settings that are otherwise
inexplicable, the rotation and crystallization driven dynamo is certainly not
the only mechanisms producing strongly magnetic white dwarfs. As illustrated
in figure 16 of Ferrario et al. (2015), white dwarfs with field strength
exceeding 100 MG are found to be slow rotators which is inconsistent with one
of the conditions proposed for the dynamo. In addition, carefully inspecting
the full sample of known magnetic white dwarfs in close binaries reveals that
not even all magnetic fields of white dwarfs in these systems can be explained
by the dynamo.

The magnetic CV HY Eridiani contains a white dwarf that is most likely
consisting of a He-core as its mass is below $0.47\mbox{M${}_{\odot}$}$ at a
90 per cent confidence level. The dynamo proposed by Schreiber et al. (2021a),
however, relies on the crystallization of a C/O core which appears unlikely
(although not impossible) for HY Eridiani.

One of the biggest strengths of the crystallization and rotation driven dynamo
scenario is that is explains the large number of magnetic CVs and the absence
of strongly magnetic white dwarfs among young detached post common envelope
binaries. However, in two cases, weakly magnetic white dwarfs have been
detected in young close detached white dwarf binaries with main sequence star
companions.

The first example is the V471 Tau, with a estimated field strength of a few
hundred kG (Sion et al., 2012), and a white dwarf temperature of $34500$ K
which clearly excludes crystallization and thus the dynamo to be operating. An
alternative scenario that can be excluded to be the dominant formation
mechanism of magnetic white dwarfs might be at work here: fossil fields. V471
Tau is most likely the descendant of a compact triple star system as the white
dwarf is the youngest and most massive white dwarf in the Hyades (O’Brien et
al., 2001). This potential history offers an explanation for the magnetic
nature of the white dwarf as magnetic main sequence Ap/Bp stars are likely the
result of main sequence star mergers (Ferrario et al., 2009; Schneider et al.,
2019). It might therefore be that the origin of the magnetic field of the
white dwarf in V471 Tau is best explained by the fossil field scenario.

The second magnetic white dwarf in a young close detached binary, CC Ceti, was
discovered only recently with a field strength of $600-700$kG (Wilson et al.,
2021). The white dwarf in this system is most likely a He-core white dwarf and
even if it was not, it is clearly too hot ($25203\pm 42$,K) to be
crystallizing.

While the crystallization and rotation driven dynamo offers a consistent
explanation for the origin of the magnetic fields in many white dwarfs, the
examples listed above clearly demonstrate that it cannot be the only mechanism
producing magnetic white dwarfs. For massive and young magnetic single white
dwarfs stellar or common envelope merger events may offer an explanation. The
magnetic field in V471 Tau might be of the fossil field origin as the
progenitor star of the white dwarf was likely formed by a stellar merger. The
origin of the magnetic fields in the close binaries CC Ceti and HY Eri remains
unclear.

##  7 Concluding discussion

The magnetic white dwarf in NLTT 12758 is rapidly rotating and its temperature
and mass are consistent with the core having started to crystallize.
Therefore, the crystallization and rotation driven dynamo can explain the
occurrence of the detected magnetic field. Furthermore, the model also
explains that in all other known double white dwarfs no magnetic component has
been found.

To understand the fast rotation of the magnetic white dwarf in NLTT 12758, we
investigated the evolutionary history of the system and found that a
previously suggested sequence consisting of two common envelope events appears
to be unlikely as additional energy sources are required and given that the
cooling ages of both white dwarfs are rather similar.

We propose that instead a phase of stable mass transfer, a symbiotic phase
during which the first formed white dwarf accretes mass and angular momentum,
and common envelope evolution as the second main mass transfer phase led to
the formation of NLTT 12758. We performed MESA simulations that illustrate how
well this revised evolutionary scenario can reproduce the observations of the
system. Our simulations show how the combination of stable mass transfer and a
subsequent common envelope event can lead to a double white dwarf binary
consisting of two relatively massive C/O white dwarfs. In particular, and most
importantly in the context of this paper, they offer an explanation for the
increased rotation rate of the magnetic white dwarf, which represents a key
ingredient for magnetic field generation according to the rotation and
crystallization driven dynamo.

Our finding that the dynamo scenario can consistently explain the magnetic
nature of NLTT 12758 (and the absence of clear signs of magnetism among close
double white dwarfs otherwise) adds another piece of evidence to the already
available support for the dynamo scenario for the generation of strong
magnetic fields in white dwarfs: the dynamo is the only scenario that can
explain the absence of strongly magnetic white dwarfs among young post common
envelope binaries, it offers an explanation for the existence of the radio
pulsing white dwarf binary AR Sco, it is consistent with the high occurrence
rate of magnetic white dwarfs in cataclysmic variables (Schreiber et al.,
2021a), it naturally explains the absence of high accretion rate intermediate
polars in globular clusters (Belloni et al., 2021), and seems to be the reason
behind the observed relation between magnetism and metal pollution (Schreiber
et al., 2021b). Considering that also single white dwarfs that are currently
not metal polluted, might have accreted planetary material in the past and
therefore gained angular momentum (Schreiber et al., 2021b), the model might
also explain the magnetism of old white dwarfs that currently do not show
signs of accretion (Bagnulo & Landstreet, 2021).

All this recent evidence shows that the idea initially put forward by Isern et
al. (2017) and further developed by Schreiber et al. (2021a) has the potential
to significantly help to finally solve the long standing mystery of the origin
of strong magnetic fields in white dwarfs.

However, despite this success, the case cannot be closed yet. Magnetic white
dwarfs exist that the dynamo model fails to explain. This concerns slowly
rotating magnetic single white dwarfs as well as three magnetic white dwarfs
in close binary stars (V471 Tau, CC Ceti, HY Eri) which either contain He-core
white dwarfs or a non-crystallizing C/O white dwarf. At least for these
objects, alternative scenarios for the generation of magnetic fields in white
dwarfs need to be considered.

In addition, on the modeling side, the scenario remains rather
phenomenological as we currently do not have theories that allow us to
properly calculate how long it takes the magnetic field to emerge as soon as
the dynamo kicks in. We also have no clue if there are upper and lower limits
for the crystallized mass fractions for the dynamo to properly work (and if
so, what are their values?). Finally, despite the rather detailed discussion
presented in Schreiber et al. (2021a), proper population synthesis that
include magnetic field generation through the dynamo have still to be
performed.

From an observational point of view, we need to expand the work of Bagnulo &
Landstreet (2021) who presented the first volume limited (but still rather
small) sample of white dwarfs and investigated how the incidence of magnetism
depends on key parameters. As in SDSS-V white dwarfs are for the first time
prime targets of an SDSS survey, we may indeed be able to analyze a large
volume limited sample of white dwarfs in the near future. Combining this
sample with population models of single white dwarfs and those that are part
of binary systems would provide key constraints on the dynamo scenario and
should bring us closer to finally understand the origin of strongly magnetic
white dwarfs.

## Acknowledgements

We thank the researchers that provided the microphysics necessary to develop
mesa. The mesa equation of state is a blend of the OPAL (Rogers & Nayfonov,
2002), SCVH (Saumon et al., 1995), PTEH (Pols et al., 1995), HELM (Timmes &
Swesty, 2000), and PC (Potekhin & Chabrier, 2010) equation of states.
Radiative opacities are primarily from OPAL (Iglesias & Rogers, 1993, 1996),
with low-temperature data from Ferguson et al. (2005) and the high-
temperature, Compton-scattering dominated regime by Buchler & Yueh (1976).
Electron conduction opacities are from Cassisi et al. (2007). Nuclear reaction
rates are a combination of rates from NACRE (Angulo et al., 1999), JINA
REACLIB (Cyburt et al., 2010), plus additional tabulated weak reaction rates
(Fuller et al., 1985; Oda et al., 1994; Langanke & Martínez-Pinedo, 2000).
Screening is included via the prescription of Chugunov et al. (2007). Thermal
neutrino loss rates are from Itoh et al. (1996).

MRS and MZ acknowledge support from Fondecyt (grant 1221059). MRS is
furthermore spported by ANID, – Millennium Science Initiative Program –
NCN19_171. DB was supported by ESO/Gobierno de Chile. BTG was supported by the
UK STFC grant ST/T000406/1. SGP acknowledges the support of a STFC Ernest
Rutherford Fellowship. This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Grant agreement No. 101020057).

For the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

## Data Availability

The MESA files required to reproduce our simulations will be provided upon
request.

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##  Appendix A Close double white dwarf binaries

We compiled a comprehensive list of close (periods below 35 days) double white
dwarfs. The resulting Table 1 should be complete with respect to double white
dwarfs that contain a C/O white dwarf component with measured effective
temperature. Such systems are highlighted in bold and represent those that are
plotted in Figure 1 of this paper.

We did not include all systems from the recently published large sample of
almost hundred extremely low-mass (ELM) white dwarfs (Brown et al., 2020) as
in all but one of these systems the stellar components with a measured
temperature are helium core low-mass white dwarfs which are irrelevant for the
rotation and crystallization driven dynamo. The only exception, SDSS
J1638+3500, is included in the table.

Table 1: Parameters of the known close double white dwarfs with measured
orbital periods. We ignored many ELM systems where the only component with
measured temperature is certainly a helium core white dwarf. For a
comprehensive list of such systems see Brown et al. (2020). Components
highlighted in bold are (potential) C/O white dwarfs with measured
$T_{\mathrm{eff}}$ which are the white dwarfs of relevance for this paper and
which are plotted in Fig.1.

Name | $M_{1}[\mathrm{M_{\odot}}]$ | $M_{2}[\mathrm{M_{\odot}}]$ | $T_{\mathrm{eff},1}[K]$ | $T_{\mathrm{eff},2}[K]$ | Period | Reference
---|---|---|---|---|---|---
NLTT11748 (WD 0342+176) | 0.136-0162 | 0.707-0.740 |  8706$\pm$136 | 7597$\pm$119† | 0.24 day | 1
CSS 41177 |  0.38$\pm$0.02 |  0.32$\pm$0.01 | 24500 | 11500 | 2.78 hr | 2,3
GALEXJ171708.5+675712 |  0.185$\pm$0.01 |  $\geq$0.86 |  14 900$\pm$200 | – | 5.90 hr | 4
SDSS J065133.33+284423.3 | 0.25 | 0.55 |  16 400$\pm$300 | – | 12.75 min. | 5
SDSS J075141.18-014120.9 |  0.19$\pm$0.02 |  0.97${}^{+0.06}_{-0.01}$ |  15 750$\pm$240 | – | 1.92 hr | 6
SDSS J115219.99+024814.4 | 0.47$\pm$0.11 | 0.44$\pm$0.09 | 25 500$\pm$1000 | 14 350$\pm$500 | 2.40 hr | 7, 8
ZTF J153932.16+502738.8 | 0.61${}^{+0.017}_{-0.022}$ |  0.210${}^{+0.014}_{-0.015}$ | 48 900$\pm$900 |  $\leq$10 000 | 6.91 min. | 9
ZTF J190125.42+530929.5 | – | – |  28 000$\pm$500 |  17 600$\pm$400 | 40.60 min. | 10
ZTF J2243+5242 |  0.349${}^{+0.093}_{-0.074}$ |  0.384${}^{+0.114}_{-0.074}$ |  22 200${}^{+1800}_{-1600}$ |  16 200${}^{+1200}_{-1000}$ | 8.8 min | 11
PG 1632+177 |  0.392${}^{+0.069}_{-0.059}$ | 0.526${}^{+0.095}_{-0.082}$ |  8 800$\pm$500 | 11 200$\pm$500 | 2.05 day | 12
WD 1534+503 |  0.392${}^{+0.069}_{-0.059}$ | 0.617${}^{+0.110}_{-0.096}$ |  8 900$\pm$500 | 8 500$\pm$500 | 0.71 day | 12
SDSS J033816.16-813929.9 |  0.23$\pm$0.015 | 0.38+0.05-0.03 |  18 100$\pm$300 |  10 000$\pm$1000 | 30.6 min. | 13
SDSS J063449.92+380352.2 | 0.452${}^{+0.070}_{-0.062}$ |  0.209${}^{+0.034}_{-0.021}$ | 27 300${}^{+4000}_{-2900}$ |  10 500${}^{+300}_{-200}$ | 26.5 min | 13
WD 1434+503 (SDSS J143633.29+501026.8) | 0.23(01) | – | 17 120(200) | – | 1.15 hr | 14
WD 1050+522 (SDSS J105353.89+520031.0) | 0.22(01) | – | 16 150(200) | – | 0.96 hr | 14
SDSSJ1337+3952 | 0.46$\pm$0.02 |  0.26$\pm$0.01 | 9 450$\pm$80 |  7 520$\pm$170 | 99 min. | 15
WD0028-474 | 0.60$\pm$0.06 |  0.45$\pm$0.04 | 18 500$\pm$500 |  17 000$\pm$500 | 9.35 hr | 16
SDSSJ0318–0107 |  . 0.4$\pm$0.05 | 0.49$\pm$0.05 |  14 500$\pm$500 | 13 500$\pm$500 | 45.9 hr | 16
HE0410-1137 | 0.51$\pm$0.04 |  0.39$\pm$0.03 | 16 000$\pm$500 |  19 000$\pm$500 | 12.2 hr | 16
NLTT12758 | 0.83$\pm$0.03 | 0.69$\pm$0.05 | 7950$\pm$50 | 7220$\pm$180 | 1.154 day | 17
WD1202+608 (Feige 55) | 0.3-0.487 |  $\geq$0.25 | 56 300$\pm$1000 | – | 1.493 day | 18
SDSSJ125733.63+542850.5 | 0.24 | 1.06$\pm$0.05 |  6400$\pm$50 | 13 030$\pm$150 | 4.6 hr | 19
WD 1704+481 |  0.39$\pm$0.05 | 0.56$\pm$0.07 | 9000 | 10 000 | 0.145 day | 20, 21
WD0136+768 | 0.47 | 0.37 | 18 500 | 10 500 | 1.41 day | 21, 22
WD1704+481 |  0.39$\pm$ 0.05 | 0.54 | 9000 | 10 000 | 0.145 day | 21
WD0957-666 |  0.37$\pm$0.02 |  0.32$\pm$0.03 | 30000 | 11 000 | 1.46 hr | 21, 23, 24
WD1204+450 | 0.46 | 0.52 | 31 000 | 16 000 | 1.603 day | 21
WD0135-052 (L870-2) | 0.47$\pm$0.05 | 0.52$\pm$0.05 | 7470$\pm$500 | 6920$\pm$500 | 1.556 day | 25, 26
WD1101+364 (=PG1101+364) | 0.29 | 0.33 | 15 500 | 12 000 | 0.145 day | 27
PG1115+166 | 0.70 | 0.70 | – | – | 722.2 hr | 28
SDSS J174140.49+652638.7 |  0.17$\pm$0.02 | $\geq 1.11$ |  10540$\pm$170 | – | 1.47 hr | 6
WD0225-192 (HE0225-1912) | 0.55 | 0.23 | 20488 | – | 0.22 day | 29
WD0315-013 (HE0315-0118) | 0.50 | 0.49 | 12720 | – | 1.91 day | 29
WD0320-192 (HE0320-1917) | 0.31 | 0.45 | 13 248 | – | 0.86 day | 29, 30, 31
WD0326-273 | 0.364 |  $\geq$0.96 | 9158 | – | 1.88 day | 29, 30, 31
WD0453-295 | 0.40 | 0.44 | 16 360 | 1330 | 0.36 day | 29, 32
WD1013-010 | 0.32 |  $\geq$ 0.62 | 8080 | – | 0.44 day | 29, 30, 31
WD1022+050 | 0.37 |  $\geq$0.28 | 14 693 | – | 1.16 day | 29, 31, 33, 34
HS1102+0934 | 0.38 |  $\geq$0.45 | 16 961 | – | 0.55 day | 29, 30, 31, 35
WD1210+140 | 0.33 |  $\geq$0.44 | 32 127 | – | 0.64 day | 29, 30, 31
HS1334+0701 | 0.35 | – | 16 891 | – | 0.23 day (uncertain) | 29
WD1349+144 | 0.55 | 0.33 | 19917 | – | 2.21 day | 29, 30, 31
HE1414-0848 | 0.52 | 0.74 | 11133 | – | 0.52 day | 29, 31, 34
HE1511-0448 | 0.50 |  $\geq$0.67 | 50 899 | – | 3.22 day | 29, 30, 31
WD1824+040 | 0.4 |  $\geq$0.73 | 14 787 | – | 6.27 day | 25, 29, 31, 34
WD2020-425 | 0.81 | 0.54 | 34 004 | – | 0.3 day | 29, 30
HE2209-1444 | 0.43 | 0.72 | 8471 | – | 0.28 day | 29, 31, 34
WD1428+373 | 0.35 |  $\geq$0.23 | – | – | 1.14 day | 34
WD2032+188 | 0.41 |  $\geq$0.47 | – | – | 5.08 day | 34
SDSSJ0755+4800 | 0.42 |  $\geq$0.90 |  19 890$\pm$350 | – | 0.55 day | 35
SDSSJ1104+0918 | 0.46 |  $\geq$0.55 |  16 710$\pm$250 | – | 0.55 day | 35
SDSSJ1557+2823 | 0.49 |  $\geq$0.43 | 12 550$\pm$200 | – | 0.41 day | 35
WD2331+290 | 0.39 |  $\geq$0.32 | – | – | 0.17 day | 36
WD1713+332 | 0.35 |  $\geq$0.18 | – | – | 1.12 day | 36
WD1241-010 | 0.31 |  $\geq$0.37 | – | – | 3.35 day | 36
WD1317+453 | 0.33 |  $\geq$0.42 | – | – | 4.87 day | 36
SDSSJ1638+3500 | 0.698$\pm$0.030 | – | 37250$\pm$570 | – | 0.91 day | 37

$\dagger$ values correspond to assuming a thin envelope. References: (1)
Kaplan et al. (2014), (2) Parsons et al. (2011), (3) Bours et al. (2014), (4)
Vennes et al. (2011), (5) Brown et al. (2011), (6) Kilic et al. (2014), (7)
Hallakoun et al. (2016), (8) Parsons et al. (2020), (9) Burdge et al. (2019),
(10) Coughlin et al. (2020), (11) Burdge et al. (2020), (12) Kilic et al.
(2021a), (13) Kilic et al. (2021b), (14) Mullally et al. (2009), (15) Chandra
et al. (2021), (16) Rebassa-Mansergas et al. (2017), (17) Kawka et al. (2017),
(18) Holberg et al. (1995), (19) Bours et al. (2015), (20) Maxted et al.
(2000) (21) Maxted et al. (2002a) (22) Bergeron et al. (1992), (23) Moran et
al. (1997), (24) Bragaglia et al. (1990) (25) Saffer et al. (1988), (26)
Bergeron et al. (1989), (27) Marsh (1995), (28) Maxted et al. (2002b), (29)
Napiwotzki et al. (2020), (30) Koester et al. (2009), (31) Nelemans et al.
(2005), (32) Wesemael et al. (1994), (33) Maxted & Marsh (1999), (34) Morales-
Rueda et al. (2005), (35) Brown et al. (2013), (36) Marsh et al. (1995), (37)
Brown et al. (2020)

##  Appendix B Understanding the BSE input parameters

Figure 5: White dwarf mass versus orbital separation after first (left) and
second (right) common envelope phase for different fractions of the
ionization/recombination energy. The gray area is for the measured white dwarf
masses (with their error), the black dashed lines correspond to the orbital
separations obtained by (Kawka et al., 2017) after each of the two common-
envelope episodes, while the red dotted line in the right panel is the current
orbital separation (Kawka et al., 2017). Apparently (and potentially without
being aware of it) between 70 and 80 per cent of the available recombination
energy was assumed to contribute to expelling the envelope which can be
considered as unrealistic.

According to the Kawka et al. (2017), agreement with the observed
characteristics of NLTT 12758 was achieved with bse assuming a common-envelope
efficiency of $\alpha_{\mathrm{CE}}=0.15$, a Reimer’s mass-loss parameter
$\eta=1.0$, and solar metallicity. However, the value that was adopted for the
binding energy parameter $\lambda$ is not given in their paper. This parameter
is of crucial importance in the bsecode as its value determines to what degree
recombination energy is included in the energy budget of common envelope
evolution.

We therefore evolved the initial binary they proposed as a progenitor for NLTT
12758 using bseassuming a large range of values for $\lambda$. We used the
same parameters as Kawka et al. (2017) but different values for the fraction
of recombination energy by varying $\alpha_{\mathrm{rec}}$ from 0 to 1 in
steps of 0.1. Figure 5 shows the results we obtained from this exercise. The
two panels show the orbital separation of both stars after the first (left
panel) and the second (right panel) common envelope phase as a function of the
white dwarf mass. The assumed contributions from recombination energy are
color coded and the dashed lines indicate the separation given for the
corresponding stage in Kawka et al. (2017).

Apparently, a high efficiency $\alpha_{\mathrm{rec}}$ between 0.7 and 0.8 was
used by Kawka et al. (2017), i.e. they assumed that more than 70 per cent of
the available recombination energy contributes to the envelope ejection
process. Under these assumptions, during the first common envelope the binding
energy is reduced by a factor of $\sim 64$ and $\sim 73$ respectively, i.e.
$\lambda$ is 64 and 73 times larger for $\alpha_{\mathrm{rec}}=0.7$ and $0.8$
with respect to $\lambda$ for $\alpha_{\mathrm{rec}}=0$. This implies that
while the orbital separation is reduced to $\sim 1$ per cent during common
envelope evolution for the model without recombination energy
($a_{f}/a_{i}\sim 0.01$), the simulations with $\alpha_{\mathrm{rec}}=0.7$ and
$0.8$ predict the separation after common envelope to be $\sim 41$ and $\sim
44$ per cent of the separation at the onset of common envelope evolution
($a_{f}/a_{i}\sim 0.41$ and $\sim 0.44$).

We consider this assumption highly unlikely given that even simulations in
favor of contributions from recombination energy conclude that at most 60 per
cent of the initially available hydrogen recombination energy contribute to
expelling the envelope (Ivanova et al., 2015). Other works provide much
stricter limits such as Soker et al. (2018) who found that at maximum 10 per
cent of the available recombination energy can help expelling the envelope.

The fact that we could not reproduce the evolution suggested by Kawka et al.
(2017) unless an unrealistic fraction of the recombination energy of the
envelope contributes to the ejection (in both common envelope phases) might
indicate that the authors perhaps misinterpreted the input parameter $\lambda$
of the bse code. As noted by Zorotovic et al. (2014), according to the
comments written in the code by its developers one can read in the main
routine (bse.f) that the input parameter called lambda is "the binding energy
factor for common envelope evolution (0.5)". This leads many users to believe
that $\lambda$ is fixed with the value they write in the input file. However,
the code has been updated after publication. In the README_BSE file it says
that the updates made after the publication of the bse paper are documented in
the header of the evolv2.f routine. One of the updates documented there (since
March 2001) says "The value of lambda used in calculations of the envelope
binding energy for giants in common-envelope is now variable (see function in
zfuncs)". And if we take a look at the file called zfuncs.f it is explained
that the input parameter lambda (that in this routine is called "fac") is the
fraction of the recombination/ionization energy used in the energy balance.
Digging into the equations, which were fitted by Onno Pols and published in
Robert Izzard’s thesis (Izzard 2004) and Claeys et al. (2014, their Appendix
A), one can confirm that the input parameter called lambda, if positive,
corresponds to the fraction of recombination energy that is assumed to
contribute to the envelope ejection (called $\lambda_{\mathrm{ion}}$ in Claeys
et al. 2014 or $\alpha_{\mathrm{rec}}$ in Zorotovic et al. 2014). A fixed
value for $\lambda$ can still be assumed in bse by selecting a negative value.
For example, if the input parameter is set to $-0.5$, then the common-envelope
equations will use $\lambda=0.5$. If, on the other hand, the input parameter
is set to $+0.75$, 75 per cent of the recombination energy is going to be used
in the calculation of $\lambda$, which leads to an unrealistically small
reduction in orbital separation during common envelope evolution.