Magnetic field breakout from white dwarf crystallization dynamos

A convective dynamo operating during the crystallization of white dwarfs is one of the promising channels to produce their observed strong magnetic fields. Although the magnitude of the fields generated by crystallization dynamos is uncertain, their timing may serve as an orthogonal test of this channel's contribution. The carbon-oxygen cores of $M\approx 0.5-1.0\,{\rm M}_\odot$ white dwarfs begin to crystallize at an age $t_{\rm cryst}\propto M^{-5/3}$, but the magnetic field is initially trapped in the convection zone - deep inside the CO core. Only once a mass of $m_{\rm cryst}$ has crystallized, the convection zone approaches the white dwarf's helium layer, such that the magnetic diffusion time through the envelope shortens sufficiently for the field to break out to the surface, where it can be observed. This breakout time is longer than $t_{\rm cryst}$ by a few Gyr, scaling as $t_{\rm break}\propto t_{\rm cryst}f^{-1/2}$, where $f\equiv 1-m_{\rm cryst}/M$ depends on the white dwarf's initial C/O profile before crystallization. The first appearance of strong magnetic fields $B\gtrsim 1\textrm{ MG}$ in volume-limited samples approximately coincides with our numerically computed $t_{\rm break}(M)$ - potentially signalling crystallization dynamos as a dominant magnetization channel. However, some observed magnetic white dwarfs are slightly younger, challenging this scenario. The dependence of the breakout process on the white dwarf's C/O profile implies that magnetism may probe the CO phase diagram, as well as uncertainties during the core helium burning phase in the white dwarf's progenitor, such as the $^{12}{\rm C}(\alpha,\gamma)^{16}{\rm O}$ nuclear reaction.


INTRODUCTION
White dwarf magnetic fields were first detected more than 50 years ago (Kemp et al. 1970;Angel & Landstreet 1971;Landstreet & Angel 1971), and strong fields of up to ∼ 10 9 G have been measured for many white dwarfs since (see Ferrario et al. 2015Ferrario et al. , 2020, for reviews), for reviews).However, the origin of white dwarf magnetism remains an open question.One idea is that white dwarfs inherited their magnetic flux from previous stages of stellar evolution (Angel et al. 1981;Braithwaite & Spruit 2004;Tout et al. 2004;Wickramasinghe & Ferrario 2005).Alternatively, white dwarfs may have been magnetized by a dynamo operating during a common envelope event (Regős & Tout 1995;Tout et al. 2008;Nordhaus et al. 2011) or a double white dwarf merger (García-Berro et al. 2012).
In a pioneering study, Isern et al. (2017) suggested that white dwarfs can drive magnetic dynamos as they evolve over time, even without interacting with a companion.When carbon-oxygen (CO) white dwarfs cool down sufficiently, their interiors begin to crystallize (solidify) from the inside out, with a crystallization front that gradually advances over several Gyr.The crystal phase is preferentially richer in oxygen, which is drawn from the liquid CO mixture above the core, rendering the liquid unstable to compositional convection (Stevenson 1980;Mochkovitch 1983;Isern et al. 1997).Such convective flows may sustain a magnetic dynamo, in analogy to the Earth's iron-nickel core, where a similar compositional instability is thought to power the geodynamo (Stevenson et al. 1983; Moffatt ★ E-mail: daniel.blatman1@mail.huji.ac.il † E-mail: sivan.ginzburg@mail.huji.ac.il & Loper 1994;Lister & Buffett 1995;Glatzmaier & Roberts 1997).
In a series of papers, Schreiber et al. (2021aSchreiber et al. ( ,b, 2022Schreiber et al. ( , 2023) ) and Belloni et al. (2021) demonstrated that crystallization dynamos may potentially solve many of the outstanding puzzles of white dwarf magnetism, specifically, why some single and binary white dwarfs are magnetic while others are not.However, the velocity of the convective flows and the scaling laws that govern magnetic dynamos are currently both under debate (Ginzburg et al. 2022;Fuentes et al. 2023;Montgomery & Dunlap 2023).Therefore, the relative role of crystallization dynamos in explaining white dwarf magnetic fields is still unclear.
Here, we circumvent the uncertainties in the magnitude of the magnetic field, and focus instead on the orthogonal question of its timing.Crystallization begins at a specific age, when the white dwarf's core has cooled down sufficiently to transition from liquid to crystal in its centre.Bagnulo & Landstreet (2022) compared this age (as a function of the white dwarf's mass) to recent volume-limited surveys, in order to identify the magnetization channels of isolated white dwarfs.Their results indicate that crystallization dynamos may contribute significantly to the magnetism of old and cool white dwarfs, but there must be other channels as well, because some magnetic white dwarfs are too hot to be crystallizing (see also Ginzburg et al. 2022).Specifically, Bagnulo & Landstreet (2022) argued that the most massive strongly magnetic white dwarfs probably resulted from double white dwarf mergers.
However, the crystallization of the white dwarf's centre might not be an accurate indicator of the age when magnetic fields can be first observed.Crystallization dynamos generate magnetic fields only in a convective mantle that surrounds the crystal core.Magnetic fields have to diffuse from the edge of this convective zone to the white dwarf's surface to be detected, and the diffusion through the star's stable (i.e.non-convecting) outer layers can take several Gyr (Isern et al. 2017;Ginzburg et al. 2022) -comparable to the evolution time.Here, we compute this diffusion time as a function of the white dwarf's mass, and evaluate the age when magnetic fields break out to the surface, where they can be observed.
Our predicted breakout times provide a more precise observational test of the crystallization dynamo theory.Moreover, the magnetic diffusion time depends on the white dwarf's carbon and oxygen distribution.Therefore, combining our work with a large enough sample of magnetic white dwarfs may help to constrain the CO phase diagram during crystallization (e.g.Blouin & Daligault 2021), as well as the 3 and 12 C(, ) 16 O nuclear reaction rates during the core helium burning stage of stellar evolution, when carbon and oxygen are synthesized.
The remainder of this paper is organized as follows.In Section 2 we lay out the theoretical principles of the crystallization dynamo and the magnetic field's breakout.Section 3 describes our computational scheme.In Section 4 we present our computed breakout times, including the sensitivity to the phase diagram and nuclear reaction rates, and compare them to the observations.We summarize and discuss our findings in Section 5. Mestel (1952) derived a basic analytical theory of white dwarf cooling, which we briefly repeat here for completeness (see also van Horn 1971).The pressure  in the white dwarf's interior is dominated by degenerate electrons  =   5/3 , where  is a constant and  is the density.In particular, the pressure at the centre (where the density is  c ∼   −3 ) balances the gravitational pressure

White dwarf cooling
where  and  are the white dwarf's mass and radius, and  is the gravitational constant.We omit order-unity coefficients in this section with the goal of deriving approximate scaling relations.Equation (1) yields the white dwarf's mass-radius relation  ∝  −1/3 .The efficient heat conduction of degenerate electrons implies that the white dwarf's interior is approximately isothermal, with a temperature  c .The pressure and density decrease towards the white dwarf's outer edge, until at some point the ideal gas pressure of the ions (∝ ) overcomes the electron degeneracy pressure (∝  5/3 ).We mark the density and pressure where the degeneracy is lifted by  0 and  0 , which satisfy where  is the molecular weight and  B is Boltzmann's constant.By combining equations ( 1) and (2), we find that The white dwarf's thin envelope from this point outwards can be treated as an ideal gas, and photon diffusion through this envelope regulates the cooling of the underlying isothermal core.Using equations (2), (3), and the mass-radius relation, the optical depth of the envelope is given by where  =    −2 is the surface gravity, and the opacity is given by Kramers' law  ∝  0  −7/2 c . The white dwarf's luminosity is therefore where  SB is the Stefan-Boltzmann constant.The cooling time to a given central temperature is independent of the mass with the white dwarf's heat capacity dominated by the nondegenerate ions.

Crystallization
When the white dwarf's interior cools down sufficiently, it begins to freeze into a crystal lattice.This phase transition occurs when the plasma coupling parameter (the ratio of the Coulomb to thermal energies) 200 (van Horn 1968;Potekhin & Chabrier 2000;Bauer et al. 2020;Jermyn et al. 2021);  is the atomic number,  is the electron's charge,  is the temperature, and  ∝  −1/3 is the average separation between ions.See Bauer et al. (2020) and Jermyn et al. (2021) for how to average Γ for multiple ion species .
The white dwarf's centre has a density  c ∼   −3 ∝  2 , and according to equations ( 6) and (7), it crystallizes at an age More external layers of the white dwarf's degenerate (and isothermal  =  c ) core have a lower density  <  c , and therefore reach Γ crit and crystallize at a later time.Quantitatively, we can generalize equation ( 8) and calculate the time it takes the crystal core to reach a mass of  cryst .We denote by Δ ≡  −  cryst the mass that has not crystallized yet.From hydrostatic equilibrium, in the limit Δ ≪  (which we adopt to derive a simple scaling, even though it is valid only at late times) the pressure at the crystallization front is given by The density there is given by comparing with equation ( 1) where we assume that  0 <  <  c (i.e. the crystallization front is inside the degenerate core) and define According to equation ( 7), the core has to cool down to a lower temperature  c ∝  1/3 for this shell to crystallize, which is reached at an age  >  cryst given by where we also used equations ( 6) and (10).In this way, the crystal core gradually grows in mass  cryst (i.e. decreases) as the white dwarf continues to cool.

Magnetic fields
An important feature of CO crystallization is a phase separation into an oxygen-enriched solid core and a carbon-enriched liquid mantle.This liquid layer above the growing core is lighter than the surrounding CO mixture, such that it rises up due to buoyancy, driving convection (Stevenson 1980;Mochkovitch 1983;Isern et al. 1997).
We demonstrate the phase separation and convection in Fig. 1: crystallization lowers the mean molecular weight  of the liquid above the crystal core (at a radius  cryst ), which is then mixed by convection up to an outer radius  out >  cryst , where the ambient unperturbed  is low enough to stabilize the fluid.The phase separation and mixing are calculated using the mesa stellar evolution code (Paxton et al. 2011(Paxton et al. , 2013(Paxton et al. , 2015(Paxton et al. , 2018(Paxton et al. , 2019;;Jermyn et al. 2023), with its recently implemented phase separation scheme (Bauer 2023); see Section 3 for details.Fig. 1 shows how  out expands as crystallization progresses, and how this expansion depends on the white dwarf's internal structure.Convective mixing during core helium burning in the white dwarf's progenitor star leaves behind an inner homogeneous region with uniform carbon and oxygen abundances, beyond which there is a small but sharp step in  (Salaris et al. 1997;Straniero et al. 2003).At early stages of crystallization, when  cryst is small and the total mass of oxygen (carbon) that the crystal core has absorbed from (released to) the surrounding liquid is small, the liquid's  is only slightly reduced.In this case, convection stops at the small  step that corresponds to the jump in the C/O ratio (at  out / ≈ 0.5).Later on, as  cryst grows,  above the core decreases further such that this step can no longer stabilize convection and  out expands until it reaches the white dwarf's hydrogen and helium atmosphere, where the ambient  drops sharply again.Isern et al. (2017) suggested that the convective region between  cryst and  out can sustain a magnetic dynamo, analogous to the Earth's geodynamo.However, the convective velocity, and how the magnetic field scales with that velocity and with the white dwarf's rotation, are both uncertain (e.g.Christensen 2010;Augustson et al. 2019;Schreiber et al. 2021a;Brun et al. 2022;Ginzburg et al. 2022;Fuentes et al. 2023;Montgomery & Dunlap 2023).Here, we focus on an orthogonal question -when can such magnetic fields reach the white dwarf's surface, where they may be observed?In Fig. 1 we plot the magnetic diffusion time from  out to the white dwarf's surface , which is given by The magnetic diffusivity () =  2 /(4π) at each radius  is computed similarly to Cantiello et al. (2016), by interpolating between expressions that are valid in the non-degenerate, partially degenerate, and fully degenerate regimes (Spitzer 1962;Nandkumar & Pethick 1984;Wendell et al. 1987);  is the speed of light and () is the electric conductivity.
As seen in Fig. 1, as long as  out is stuck at the small  step that is deep inside the CO core, the diffusion time is prohibitively long  diff ∼ 10 Gyr.Only once  out expands towards the white dwarf's hydrogen and helium atmosphere (when  cryst grows sufficiently),  diff shortens and enables the magnetic field to reach the surface.Specifically, the magnetic field's breakout time (age)  =  break is found by solving where  cryst marks the onset of crystallization in the white dwarf's centre (i.e.breakout is defined by when diffusion becomes faster than crystallization; see Section 3 for details).We note that other processes, such as advection (Charbonneau & MacGregor 2001) or magnetic buoyancy (MacGregor & Cassinelli 2003;MacDonald & Mullan 2004), may in principle transport magnetic fields from  out to the surface faster than diffusion.These mechanisms have been invoked -with limited success -for massive stars, where magnetic fields generated in the convective core have to penetrate through the radiative envelope to be observed.Their efficiency in the white dwarf context has not been studied yet, and we assume here that diffusion is the sole transport mechanism.

COMPUTATIONAL SCHEME
We construct a set of CO white dwarfs with masses  ≈ 0.5−1.0M ⊙ similarly to Bauer (2023), by using the mesa test suite make_co_wd, which evolves the progenitor ≈ 2.6 − 6.7 M ⊙ stars through the various stages of stellar evolution, starting from the pre-main sequence.We then calculate the cooling and crystallization of these white dwarfs using the wd_cool_0.6Mtest suite.We use the mesa version r23.05.1, which implements the 'Skye' equation of state (Jermyn et al. 2021) and the new CO phase separation scheme of Bauer (2023).The carbon and oxygen profiles before and after phase separation can be seen in fig.8 of Bauer (2023).
Skye determines the phase (crystal or liquid) by minimizing the Helmholtz free energy, such that the critical Γ crit for crystallization varies between Γ crit ≈ 200 − 230, as a function of the C/O ratio (Jermyn et al. 2021).The crystallization time  cryst is defined by the phase transition of the innermost cell.In the following time steps, the phase_separation scheme propagates the crystallization front  cryst outwards as the white dwarf cools and more cells crystallize.For each crystallizing cell, the scheme increases the oxygen mass fraction  O and reduces the carbon mass fraction  C ≈ 1 −  O by Δ, which is a function of  O , given by the Blouin & Daligault (2021) phase diagram (which has a similar Γ crit to Skye).The total amounts of carbon and oxygen in the star are conserved by adjusting the composition of the cells above  cryst .This liquid becomes carbon-enriched and oxygen-depleted (i.e.opposite to the crystal core), triggering convection.At each time step, the scheme propagates the outer radius of the convection zone  out iteratively, by mixing the cells at radii  cryst <  <  out until the  profile satisfies the Ledoux criterion for stability (see Bauer 2023, for details).We note that, in principle, thermohaline mixing may operate in regions that are stable according to the Ledoux criterion, but where ∇ > 0 (e.g.Paxton et al. 2013;Fuentes et al. 2023).In practice, however, the Ledoux criterion for stability is reduced in our case to ∇ < 0 (the Ledoux term ∝ ∇ dominates the criterion) -the role of the entropy gradient in stabilizing the fluid is negligible.
In a previous paper, Ginzburg et al. (2022) used a simplified postprocessing scheme to evaluate  out .As described above, here we employ more accurate variable Δ ( O ) and Γ crit ( O ) instead of their constant Δ = 0.2 and Γ crit = 230.More crucially, the white dwarfs in Ginzburg et al. (2022) do not experience phase separation as they crystallize.Instead, at each time ,  out () is found by assuming that the unperturbed liquid profile at  >  cryst () is enriched by a carbon mass of Δ cryst (), and depleted by a similar amount of oxygen.As demonstrated in our Fig. 1  Breakout of the crystallization dynamo's magnetic field to the surface.Top left: helium, carbon, and oxygen abundances inside a 0.6 M ⊙ white dwarf before the onset of crystallization at  cryst (dotted lines), and later (solid lines), when the crystal core has reached a radius  cryst .Bottom left: the corresponding mean molecular weight .Crystallization lowers  of the liquid above the crystal core, rendering it unstable to convection up to an outer radius  out , where the ambient unperturbed  is sufficiently low.Convection mixes the region  cryst <  <  out , and also sustains a magnetic dynamo there.Top right: as crystallization progresses over time, a lower  is required in order to stabilize the liquid, such that  out expands.Most of the time, convection stops at a steep  gradient.At early times  out ≈ 0.5, corresponding to the jump in the C/O ratio of the original profile.Only at a later time convection reaches the white dwarf's atmosphere.Bottom right: the magnetic diffusion time  diff from  out to the white dwarf's surface.The magnetic field breaks out to the surface at separation gradually changes the liquid carbon and oxygen profiles as crystallization progresses, leading to a difference between the two methods -our current calculation with built-in phase separation is more precise.
The magnetic diffusion time from  out to the surface is calculated using equation (13), and the breakout time  =  break is found in accordance with equation ( 14), when the condition  diff () <  −  cryst is first met.These equations provide a good estimate for  break at high white dwarf masses, for which  diff () drops very sharply.At lower white dwarf masses,  diff () declines more gradually (Fig. 1), necessitating the incorporation of a time-dependent magnetic diffusion solver in mesa, which is beyond the scope of this work.This difference with  stems from the structure of the initial (i.e.before crystallization) C and O profiles beyond the original C/O jump (Fig. 1).As seen in fig.8 of Bauer (2023), these profiles are flatter at high , leading to a sharper transition of  out () from the small  step (that corresponds to the C/O jump) to the larger  drop at the helium layer -where the diffusion time is much shorter (Fig. 1).

RESULTS
Fig. 2 shows our computed crystallization and breakout times as a function of the white dwarf's mass .The crystallization times follow the analytical model of Mestel (1952) - cryst ∝  −5/3 .The breakout times  break are significantly longer: roughly half of the white dwarf's mass has to crystallize before the magnetic field emerges to the surface (Fig. 3), with an even higher fraction  cryst / ≡ 1 −  at low masses  (because of the difference in the initial carbon and oxygen profiles; see fig. 8 in Bauer 2023).This variation in  () explains the deviation from a  break ∝  −5/3 scaling, as implied by equation ( 12), but with an order-unity fitting coefficient (though formally our  ≪ 1 approximation breaks down).
Bagnulo & Landstreet (2022) compared their volume-limited white dwarf sample to  cryst ().They concluded that massive white dwarfs ( ≳ 1.0 M ⊙ ) were likely magnetized by mergers, with strong magnetic fields often appearing before crystallization (see also Caiazzo et al. 2021;Fleury et al. 2022).Such massive white dwarfs are beyond the scope of our CO crystallization dynamo model because they may harbour oxygen-neon cores (e.g.Camisassa et al. 2022).Similarly,  ≲ 0.5 M ⊙ white dwarfs potentially harbour helium cores and are thus also beyond the scope of this work (e.g.Prada Moroni & Straniero 2009).In the relevant ≈ 0.5 − 1.0 M ⊙ range, Bagnulo & Landstreet (2022) found that the frequency and strength of magnetic fields increase after the crystallization line is crossed, signifying the potential contribution of the crystallization dynamo mechanism (see also Amorim et al. 2023;Caron et al. 2023;Hardy et al. 2023).
However, the breakout time  break () >  cryst () provides a more accurate test of the crystallization dynamo channel, because it accounts for the magnetic diffusion to the surface.As seen in Fig. 2, the difference between crystallization and breakout is of order a few Gyr, potentially explaining the observed delay between crystallization and the appearance of strong magnetic fields - break fits better the age at which magnetic fields  ≳ 1 MG first emerge.Bagnulo & Landstreet (2022) made the distinction between white dwarfs with thick hydrogen envelopes (blue symbols in Fig. 2) and hydrogen-deficient white dwarfs, which exhibit helium spectra (red symbols).The difference in  cryst between the two spectral types (due to their different cooling rates) can be seen in fig. 2 of Bagnulo & Landstreet (2022).All our simulated white dwarfs possess thick hydrogen envelopes (10 −6 − 10 −4 M ⊙ ; see Bédard et al. 2020, for comparison).We do not calculate  break for white dwarfs with helium atmospheres, and focus instead on the sensitivity to other parameters that influence the breakout process more directly.

Phase diagram
The degree of oxygen enrichment Δ during crystallization is a key factor in determining the delay between crystallization and breakout.A higher Δ requires more oxygen to be drawn from the surrounding liquid to form a crystal core with a given  cryst , reducing the liquid's  faster.As can be understood from Fig. 1, this faster reduction in  leads to a more rapid expansion of  out , and thereby to an earlier breakout of the magnetic field.Our nominal Δ ( O ), given by the phase diagram of Blouin & Daligault (2021), is plotted in Fig. 4.
In Fig. 5 we test the dependence of  break on the uncertainty in the phase diagram by varying Δ by a factor of 2 in each direction (more precisely, we change the enrichment in the number fraction Δ, which is almost identical to changing Δ; see Fig. 4).This variation roughly represents the difference between Blouin & Daligault (2021) and other phase diagrams (Horowitz et al. 2010;Medin & Cumming 2010).In our  ≈ 0.5−1.0M ⊙ white dwarfs, the initial central liquid  O ≈ 0.6 − 0.7 (Bauer 2023), for which the nominal Δ ≈ 0.1.The interplay between this Δ and the  O step (of comparable height) at the edge of the inner homogeneous region in the initial profile sets  break (Fig. 1).The height and location of this step change with  (fig.8   Figure 2. onset of crystallization  cryst , and the magnetic field's breakout to the surface  break , as a function of the white dwarf's mass . cryst is approximated well by equation ( 8), whereas the breakout time is fit better by equation ( 12):  break ∝  cryst  −1/2 , with the mass fraction  that remains uncrystallized at breakout extracted from mesa (plotted in Fig. 3).The observed white dwarfs (blue for hydrogen atmospheres and red for helium) are from the volume-limited sample of Bagnulo & Landstreet (2022).Empty dots represent non-magnetic white dwarfs and filled dots mark magnetic ones, with the strength of the measured magnetic field indicated by the number of surrounding circles.In this paper, however, we choose to focus on Δ because it directly changes the delay time Δ ≡  break −  cryst .

Reaction rates
The second key factor in determining  break , as can be appreciated from Fig. 1, is the initial C/O profile before crystallization.Specifically, the breakout time is sensitive to the details of the white dwarf's inner homogeneous zone: its radial extent, the central C/O ratio, and the C/O jump beyond this region (followed by a more gradual rise in the C/O ratio).All these features are set during the core helium burning phase of the white dwarf's progenitor star (Salaris et al. 1997(Salaris et al. , 2010;;Straniero et al. 2003).The magnetic field's breakout to the surface may therefore probe the various uncertainties of this stellar evolution phase, such as convective overshooting and the 3 and 12 C(, ) 16 O nuclear reactions, which synthesize carbon and oxygen (e.g.Chidester et al. 2023).
As a specific example, we focus here on the poorly constrained  capture reaction 12 C(, ) 16 O, which converts carbon into oxygen (deBoer et al. 2017).This reaction is of great astrophysical importance, and it has recently gained attention in the context of LIGO's detection of gravitational waves from binary black hole mergers.Pair-instability supernova theory predicts a gap in the black hole mass distribution at ∼ 50 − 130 M ⊙ (e.g.Belczynski et al. 2016;Woosley 2017;Woosley & Heger 2021).However, the location of this gap is sensitive to the 12 C(, ) 16 O reaction (Farmer et al. 2019(Farmer et al. , 2020;;Shen et al. -leading to uncertainties in interpreting the results from LIGO/Virgo/KAGRA. In Fig. 6 we show how our results depend on the 12 C(, ) 16 O reaction rate during core helium burning.We followed Chidester et al. (2022) and used the deBoer et al. ( 2017) reaction rates that were expanded by Mehta et al. (2022) to include ±3 uncertainties.The nominal deBoer et al. ( 2017) rates produce  cryst and  break that are very similar to the ones obtained in previous sections, using the older Kunz et al. (2002) rates (for consistency with Bauer 2023).The breakout process depends on the degree of oxygen enrichment Δ during crystallization, which is plotted in Fig. 4. The onset of crystallization  cryst is independent of Δ, and we do not vary the crystallization curve Γ crit ( O ).
When using the ±3 rates, however, make_co_wd produces white dwarfs with different carbon and oxygen distributions -changing both the crystallization and breakout times.

Metallicity
The carbon and oxygen profiles may also depend on the progenitor star's initial metallicity .However, as seen in Fig. 7, while the white dwarf's final mass  is sensitive to its progenitor's metallicity, the  cryst () and  break () curves are largely unchanged for metal-poor ( = 0.001) and metal-rich ( = 0.04) stars.
We emphasize that our models do not include the effect of gravitational energy release by 22 Ne sedimentation on the white dwarf's cooling (Bildsten & Hall 2001).For  = 0.6 M ⊙ and our nominal  = 0.02, Bauer (2023) finds a modest cooling delay due to sedimentation of ≈ 0.3 Gyr, which is below the noise level of our  break () curves.However, recently observed cooling delays for some white dwarfs suggest that 22 Ne settling might be more significant (Althaus et al. 2010;Cheng et al. 2019;Bauer et al. 2020;Camisassa et al. 2021).Specifically, in Section 5.1 we discuss the distillation process, which may transport 22 Ne towards the centre more efficiently (e.g.Blouin et al. 2021).The dependence of such mechanisms on the 22 Ne abundance implies a much larger -yet currently uncertainsensitivity to  than Fig. 7 suggests.We defer the investigation of the influence of 22 Ne to future work.2017) rates (see also fig. 1 in Chidester et al. 2022).The nominal rates produce similar curves to Fig. 2, where we used the older Kunz et al. (2002) reaction rates.

SUMMARY AND DISCUSSION
Convective dynamos operating during crystallization present a promising channel to white dwarf magnetism (Isern et al. 2017).Even though the magnitude of the generated magnetic fields is uncertain (Schreiber et al. 2021a;Ginzburg et al. 2022;Fuentes et al. 2023;Montgomery & Dunlap 2023), their timing is an orthogonal question that may be used to constrain the relative contribution of this channel.The dynamo operates in a convective mantle that surrounds the crystallizing core.In order to be observed, the generated magnetic field has to diffuse out from this mantle to the white dwarf's surface through convectively-stable layers.The delay time -due to diffusion -between the onset of crystallization  cryst and the breakout of the magnetic field to the surface  break is the focus of our paper.
We computed the crystallization and breakout times in CO white dwarfs with masses  ≈ 0.5 − 1.0 M ⊙ , using the phase separation scheme that has been recently implemented in mesa (Bauer 2023).This scheme utilizes the Skye equation of state (Jermyn et al. 2021) and the Blouin & Daligault (2021) phase diagram to separate the CO mixture into an oxygen-rich crystal and a carbon-rich liquid which is unstable to convection due to its reduced molecular weight .The scheme updates the () profile iteratively to determine the outer radius of the convection  out , from which we calculate the magnetic diffusion time to the surface  diff .
As crystallization progresses, and a larger fraction  cryst / of the white dwarf crystallizes,  out expands in two distinct stages (Fig. 1).At early times, convection is stabilized by a small  step inside the CO core -where the C/O ratio jumps -a relic of core helium burning and convective mixing in the white dwarf's progenitor star.At later times, In previous plots we used the nominal  = 0.02.Sedimentation and possible distillation of 22 Ne, which are sensitive to , are not considered in the plot and are discussed in the text.For this plot, we used the Kunz et al. (2002) nuclear reaction rates.
as  cryst grows and  of the surrounding liquid is reduced further, convection extends all the way to the white dwarf's helium layer.The magnetic diffusion time changes dramatically between these two stages.As long as  out is deep inside the CO core,  diff ≳ 10 Gyr -trapping the magnetic field.Once  out reaches the helium layer, on the other hand,  diff ≲ 1 Gyr -enabling the field to reach the surface.At high masses , the transition between these regimes is sharp, with a well defined breakout time  break .At lower , the transition is more gradual, and we approximately estimate  break ≈  cryst +  diff ( break ), i.e. when the diffusion time becomes shorter than the time elapsed since the onset of crystallization.
Our computed  cryst and  break are presented in Fig. 2, where they are compared to the Bagnulo & Landstreet (2022) volumelimited white dwarf sample.The crystallization times are fit well by  cryst ∝  −5/3 , which is derived from the analytical cooling theory of Mestel (1952).The breakout times deviate from this scaling and are fit better by the generalized  break ∝  cryst  −1/2 , where  ≡ 1 −  cryst /.In other words, a somewhat different fraction  cryst / has to crystallize in white dwarfs with different masses  before the magnetic field breaks out to the surface (a result of the difference in initial C/O profiles before crystallization).
As seen in Fig. 2, strong magnetic fields  ≳ 1 MG first appear in proximity to  break () for  ≈ 0.5 − 1.0 M ⊙ , except for a single 0.9 M ⊙ outlier -ostensibly signifying the dominant contribution of the crystallization dynamo channel to strongly magnetic CO white dwarfs.Lower-mass white dwarfs may contain helium cores, whereas higher-mass white dwarfs may contain ONe cores, or might have been magnetized by mergers -both of these white dwarf classes are beyond the scope of this work.Given the potential challenges of the crystallization dynamo theory (Fuentes et al. 2023;Montgomery & Dunlap 2023), it is noteworthy that even if crystallization cannot sustain a strong magnetic dynamo on its own, the triggered convection may help to convey existing ("fossil") fields to the white dwarf's surface.In order to explain the observed late appearance of strong magnetism, such fossil fields must initially be buried deep inside the CO core, because the magnetic diffusion time from the edge of the CO core to the surface is shorter than a Gyr (Fig. 1).If the fossil field is confined to deeper layers of the CO core (where the diffusion time is much longer), then it must wait for crystallization-driven convection to transport it closer to the surface -where it will emerge at a similar time to our computed  break .For example, a convective dynamo operating during core helium burning might leave behind a fossil magnetic field that is confined to about half of the CO core's radius (the inner initially homogeneous region of the profiles in Fig. 1, which is a relic of convection during this stage).
Nevertheless, several of the observed magnetic white dwarfs with  ≈ 0.7 M ⊙ are in fact slightly younger than  break , challenging the crystallization dynamo (or a related) channel.This result seems to hold even when considering the uncertainties in the phase diagram (Fig. 5) and in the nuclear reaction rates (Fig. 6).Besides the possibility that these white dwarfs were actually magnetized by a different process, we identify two potential reasons for their offset from  break ().One source of uncertainty is convective overshootingpenetration and mixing of material beyond a convection zone (see Anders et al. 2022;Jermyn et al. 2022;Blouin et al. 2023Blouin et al. , 2024, for recent progress).Chidester et al. (2023) demonstrated that overshooting during core helium burning in the white dwarf's progenitor star may significantly alter the initial C/O profiles before crystallization, which may in turn shift  break once crystallization unfolds.Calculating the effects of overshooting is beyond the scope of our work, because the non-monotonic initial C and O profiles in that case (fig. 1 of Chidester et al. 2023) complicate our analysis.Another source of inaccuracy is our definition of  break .For  ≈ 0.8 M ⊙ ,  diff () drops quickly, such that the breakout time is well defined.For  ≈ 0.6 M ⊙ ,  diff () declines more gradually (Fig. 1), leading to inconsistencies in our analysis.A consistent solution requires the incorporation of a magnetic diffusion solver in mesa -instead of equation ( 13) -which is also beyond the scope of the current paper.
While our technique may be used to constrain the relative role of different magnetization channels, we can also turn the argument on its head and use white dwarf magnetism as a tool.If we assume that crystallization dynamos are indeed the main channel to produce ≳ 1 MG magnetic fields in  ≈ 0.5 − 1 M ⊙ white dwarfs, then the observed  break () -i.e. when strong fields first appear -may be used to study several physical processes that govern the formation and evolution of CO white dwarfs.Specifically, the breakout time  break and the mass fraction that crystallizes at this time  cryst / ≡ 1 −  are sensitive to the initial C/O profile before crystallization, as well as to the CO phase diagram which determines how this profile changes during crystallization.
The dependence of our results on the phase diagram is estimated in Fig. 5.In Fig. 6 we show the dependence on the poorly constrained 12 C(, ) 16 O nuclear reaction rate, which together with the 3 reaction and convective overshooting sets the white dwarf's C/O profile during core helium burning (e.g.Salaris et al. 2010;Chidester et al. 2022Chidester et al. , 2023;;Tognini et al. 2023).While the variations that we find in  break are relatively small, a large enough white dwarf survey may in principle utilize  break () to distinguish between different theoretical models.Thus, white dwarf magnetism may complement asteroseismology as an independent tool to probe the internal structure of white dwarfs (Metcalfe et al. 2001(Metcalfe et al. , 2002;;Metcalfe 2003).
Interestingly, the seismic signature is sensitive to similar features of the C/O profile as the breakout process, namely the region between the jump in the C/O ratio and the helium layer (Chidester et al. 2022(Chidester et al. , 2023)).Specifically, several asteroseismic studies indicate that the oxygen-rich central homogeneous zone is larger than predicted by standard stellar evolution models (Giammichele et al. 2018(Giammichele et al. , 2022)), possibly due to extra convective boundary mixing during the core helium burning phase (see also Constantino et al. 2015;Paxton et al. 2018).The timing of white dwarf magnetism could provide additional constraints on such mixing mechanisms.
As demonstrated by Farmer et al. (2020), the 12 C(, ) 16 O reaction rate also determines the location of the pair-instability supernova mass gap, which is probed by LIGO/Virgo/KAGRA observations of merging black holes (Abbott et al. 2019(Abbott et al. , 2023)).A more accurate determination of  break (again, assuming that the crystallization dynamo is the dominant magnetization channel of CO white dwarfs) might therefore shed some light on these gravitational wave observations.

Distillation of minor species
This paper sets the stage for studying the related process of distillation.Despite their low abundance, neutron-rich trace elements such as 22 Ne and 56 Fe may significantly alter the crystallization process (Isern et al. 1991;Segretain et al. 1994;Segretain 1996;Blouin et al. 2021;Caplan et al. 2023).If the solid crystals are sufficiently depleted in such elements compared to the liquid, then they float upwardsaway from the crystallization front -until they melt and mix in the lower density environment.This process gradually enriches the liquid at the bottom, until it crystallizes later at a higher Γ crit (due to the different composition) -forming a crystal layer enriched in these trace elements (i.e.crystallization "distills" the trace elements).Recently, Blouin et al. (2021) demonstrated that the gravitational energy release by 22 Ne distillation may explain the cooling delays observed for a fraction of massive white dwarfs (see also Shen et al. 2023b).
In principle, the convective motions during the distillation process may sustain a magnetic dynamo.The large amount of released gravitational energy suggests that such distillation-driven dynamos might even generate stronger magnetic fields than classical crystallizationdriven dynamos.In order to compute the breakout time of these magnetic fields to the surface,  out of the convection has to be recalculated using a three-component phase diagram and phase separation scheme that include 22 Ne.Such a calculation is beyond the scope of this paper.
According to Blouin et al. (2021), the distillation process is sensitive to the metallicity, and specifically to the 22 Ne abundance.Interestingly, for a standard 22 Ne abundance and for  O = 0.6 (similar to our models), Blouin et al. (2021) find that distillation does not start at the onset of crystallization, but rather only when  cryst / ≈ 0.6 of the core has already crystallized.Coincidentally, this is similar to the mass fraction that has to crystallize before the magnetic field from a crystallization dynamo breaks out to the surface (Fig. 3).We therefore conclude that -if generated -magnetic fields from distillation dynamos first appear close to our computed  break , potentially providing an alternative explanation for the observations.This preliminary conclusion motivates future work on distillation dynamos.thank Evan Bauer, Simon Blouin, Jay Farihi, Jim Fuller, Na'ama Hallakoun, Selma de Mink, Ken Shen, and the anonymous reviewer for helpful discussions and comments which improved the paper.We acknowledge support from the Israel Ministry of Innovation, Science, and Technology (grant No. 1001572596), and from the United States -Israel Binational Science Foundation (BSF; grant No. 2022175).MesaScript (Wolf et al. 2023) was used to automate some of the steps in this work.
Figure1.Breakout of the crystallization dynamo's magnetic field to the surface.Top left: helium, carbon, and oxygen abundances inside a 0.6 M ⊙ white dwarf before the onset of crystallization at  cryst (dotted lines), and later (solid lines), when the crystal core has reached a radius  cryst .Bottom left: the corresponding mean molecular weight .Crystallization lowers  of the liquid above the crystal core, rendering it unstable to convection up to an outer radius  out , where the ambient unperturbed  is sufficiently low.Convection mixes the region  cryst <  <  out , and also sustains a magnetic dynamo there.Top right: as crystallization progresses over time, a lower  is required in order to stabilize the liquid, such that  out expands.Most of the time, convection stops at a steep  gradient.At early times  out ≈ 0.5, corresponding to the jump in the C/O ratio of the original profile.Only at a later time convection reaches the white dwarf's atmosphere.Bottom right: the magnetic diffusion time  diff from  out to the white dwarf's surface.The magnetic field breaks out to the surface at  break =  cryst +  diff ( break ).
of Bauer 2023), possibly explaining the converging  break at low  in Fig.5.In principle, an additional uncertainty in Γ crit ( O ) of the phase diagram would shift both  cryst and  break ; see e.g.fig.7inJermyn et al. (2021) for the (small) difference between Skye (our Γ crit ) andBlouin & Daligault (2021), which is the same asBlouin et al. (2020).

Figure 3 .Figure 4 .
Figure3.The white dwarf's uncrystallized mass fraction  at the moment of the magnetic field's breakout to the surface  break .In lower mass white dwarfs, a larger fraction  cryst / = 1 −  has to crystallize before the field breaks out, leading to a deviation from a  break ∝  −5/3 scaling (Fig.2).

Figure 5 .
Figure 5.The sensitivity of  break to the uncertainty in the CO phase diagram.The breakout process depends on the degree of oxygen enrichment Δ during crystallization, which is plotted in Fig.4.The onset of crystallization  cryst is independent of Δ, and we do not vary the crystallization curve Γ crit ( O ).

Figure 6 .
Figure 6.The sensitivity of  cryst and  break to the uncertainty in the 12 C( , ) 16 O nuclear reaction rate during core helium burning in the white dwarf's progenitor star.Nominal and ±3 values are from fig. 9 of Mehta et al. (2022), who expanded the deBoer et al. (2017) rates (see also fig.1inChidester et al. 2022).The nominal rates produce similar curves to Fig.2, where we used the olderKunz et al. (2002) reaction rates.

Figure
Figure sensitivity of  cryst and  break to the progenitor star's initial metallicity .In previous plots we used the nominal  = 0.02.Sedimentation and possible distillation of 22 Ne, which are sensitive to , are not considered in the plot and are discussed in the text.For this plot, we used theKunz et al. (2002) nuclear reaction rates.