A common origin of magnetism from planets to white dwarfs

TL;DR

This paper proposes that magnetic fields in white dwarfs may originate from a dynamo process during their crystallization, similar to planetary magnetic field generation, offering a new explanation for their magnetism.

Contribution

It introduces a novel mechanism where phase separation and convection during white dwarf crystallization generate magnetic fields via a planetary-like dynamo.

Findings

White dwarf crystallization can produce a convective mantle.

Convection during solidification can generate magnetic fields up to 0.1 MG.

This dynamo mechanism could explain magnetism in single white dwarfs.

Abstract

Isolated magnetic white dwarfs have field strengths ranging from kilogauss to gigagauss. However, the origin of the magnetic field has not been hitherto elucidated. Whether these fields are fossil, hence the remnants of original weak magnetic fields amplified during the course of the evolution of their progenitor stars, or are the result of binary interactions or, finally, they are produced by other internal physical mechanisms during the cooling of the white dwarf itself, remains a mystery. At sufficiently low temperatures white dwarfs crystallize. Upon solidification, phase separation of its main constituents, 12C and 16O, and of the impurities left by previous evolution occurs. This process leads to the formation of a Rayleigh-Taylor unstable liquid mantle on top of a solid core. This convective region, as it occurs in Solar System planets like the Earth and Jupiter, can produce a dynamo able to yield magnetic fields of strengths of up to 0.1 MG, thus providing a mechanism that could explain magnetism in single white dwarfs.

Tables

Table 1. Table 1 Properties of magnetic white dwarfs within 20 pc of the Sun.

Name	Type	$B$ (MG)	Period (h)	$M\left(M_{☉}\right)$	$\log \left(L/L_{☉}\right)$	$M_s/M_{\mathrm{WD}}$	${\rho }_b$	$r_o-r_i$	$q_{\mathrm{top}}$
WD 0009+501	DA	0.2	2 – 20	$0.73\pm 0.04$	$-3.72$	0.43	$3.0\times {10}^6$	$2.3\times {10}^8$	$7.1\times {10}^{10}$
WD 0011$-$134	DA	$16.7\pm 0.6$	—	$0.72\pm 0.07$	$-3.85$	0.52	$2.3\times {10}^6$	$2.6\times {10}^8$	$5.7\times {10}^{10}$
WD 0322$-$019	DA	0.120	—	$0.63\pm 0.05$	$-4.02$	0.52	$1.6\times {10}^6$	$5.2\times {10}^8$	$3.0\times {10}^{10}$
WD 0413$-$077	DA	$0.0023\pm 0.0007$	—	$0.59\pm 0.03$	$-1.85$	0.00	—	—	—
WD 0548$-$001	non-DA	$\sim 10$	4.117	$0.69\pm 0.03$	$-3.80$	0.82	$7.0\times {10}^5$	$1.2\times {10}^8$	$6.5\times {10}^{10}$
WD 0553+053	DA	$20\pm 3$	0.97	$0.72\pm 0.03$	$-3.91$	0.35	$1.9\times {10}^6$	$2.5\times {10}^8$	$4.6\times {10}^{10}$
WD 0728+642	DA	$0.0396\pm 0.0116$	—	$0.58\pm 0.00$	$-4.00$	0.36	$1.5\times {10}^6$	$2.0\times {10}^8$	$3.7\times {10}^{10}$
WD 0912+536	non-DA	$\sim 100$	31.9224	$0.75\pm 0.02$	$-3.57$	0.71	$1.5\times {10}^6$	$1.6\times {10}^8$	$1.1\times {10}^{11}$
WD 1008+290	non-DA	$\sim 100$	—	$0.68\pm 0.01$	$-4.31$	0.93	$9.8\times {10}^4$	$3.0\times {10}^6$	$2.1\times {10}^{10}$
WD 1036$-$204	non-DA	50	$636\pm 36$	$0.60\pm 0.01$	$-4.19$	0.76	$1.9\times {10}^5$	$1.1\times {10}^7$	$2.5\times {10}^{10}$
WD 1309+853	DA	$4.9\pm 0.5$	—	$0.71\pm 0.02$	$-4.01$	0.60	$1.6\times {10}^6$	$2.3\times {10}^8$	$3.4\times {10}^{10}$
WD 1748+708	non-DA	$\gtrsim 100$	$\gtrsim 1.7\times {10}^4$	$0.79\pm 0.01$	$-4.07$	0.79	$2.0\times {10}^5$	$4.5\times {10}^6$	$4.6\times {10}^{10}$
WD 1829+547	non-DA	$170-180$	—	$0.90\pm 0.07$	$-3.94$	0.85	$2.7\times {10}^5$	$7.2\times {10}^6$	$8.3\times {10}^{10}$
WD 1900+705	non-DA	$320\pm 20$	$\gtrsim 8.5\times {10}^5$	$0.93\pm 0.02$	$-2.88$	0.18	$1.4\times {10}^7$	$2.4\times {10}^8$	$1.2\times {10}^{12}$
WD 1953$-$011	DA	$0.1-0.5$	34.60224	$0.73\pm 0.03$	$-3.38$	0.13	$5.6\times {10}^6$	$2.4\times {10}^8$	$3.3\times {10}^{11}$
WD 2105$-$820	DA	$0.010\pm 0.001$	—	$0.74\pm 0.13$	$-2.93$	0.00	—	—	—
WD 2359$-$434	DA	$0.0031\pm 0.0005$	—	$0.78\pm 0.03$	$-3.26$	0.12	$7.2\times {10}^6$	$1.8\times {10}^8$	$7.3\times {10}^{11}$

Equations

$$\frac{B^2}{2{\mu }_0}=c{f}_{\Omega }\frac{1}{V}\underset{r_i}{\overset{r_o}{\int }}{\left[\frac{q_c\left(r\right)\lambda \left(r\right)}{H\left(r\right)}\right]}^{2/3}\rho {\left(r\right)}^{1/3}4\pi r^{2}dr$$

Full text

1. Introduction

White dwarfs are the most common end-point of stellar evolution (Althaus et al. 2010). Some of them have measurable magnetic fields with strengths ranging from $10^{3}$ to $10^{9}$ G (Ferrario et al. 2015). The incidence of magnetism is a matter of debate. In volume-limited surveys – those that select stars within a maximum distance from the Sun – the incidence of magnetism is $\sim 20\%$ (Kawka et al. 2007; Giammichele et al. 2012; Sion et al. 2014), while for magnitude-limited surveys – those selecting stars brighter than a given apparent magnitude – it is $\sim 8\%$ (Liebert et al. 2003; Kepler et al. 2013, 2016). Nevertheless, because a population of white dwarfs with magnetic fields weaker than $\sim 1$ kG may exist, the fraction of magnetic white dwarfs could be larger (Kawka & Vennes 2012; Koester et al. 2011; Ferrario et al. 2015). Observations also suggest (Valyavin & Fabrika 1999; Liebert et al. 2003; Sion et al. 2014; Kawka & Vennes 2014; Valyavin et al. 2014; Hollands et al. 2015) that the fraction of white dwarfs with strong magnetic fields is larger at low effective temperatures. This could indicate that magnetic fields are amplified during the evolution of white dwarfs. This interpretation, however, has been questioned (Ferrario et al. 2015) because apparently there is no clear correlation between the strength of the magnetic field and the luminosity, except for low magnetic field ($B\lesssim 0.1$ MG) white dwarfs. Interestingly enough, the observed paucity of white dwarfs with magnetic fields between 0.1 and 1 MG (Kawka & Vennes 2012; Koester et al. 2011) suggests a bimodal distribution (Ferrario et al. 2015).

The origin of magnetic white dwarfs remains unknown, and up to now three scenarios have been proposed. Within the first of them, white dwarf magnetic fields are the fossil remnants of those of their progenitors. Thus, the progenitors of magnetic white dwarfs could be main sequence Ap/Bp stars (Angel et al. 1981; Wickramasinghe & Ferrario 2005), or could be the result of field amplification during the helium-burning phase (Levy & Rose 1974). Within the second scenario, magnetic white dwarfs are the result of the evolution of binary systems. In this case the magnetic field is amplified by a dynamo that operates either during the common envelope phase (Tout et al. 2008; Nordhaus et al. 2011) or in the hot corona produced during the merger of two white dwarfs (Garcia-Berro et al. 2012). A difficulty of these two scenarios is that detailed population synthesis studies cannot reproduce the number of stars observationally found (Ferrario et al. 2015). Finally, within the third scenario the field is generated in the outer convective envelope that is formed during the evolution of single white dwarfs. However, theoretical calculations show that the strength of the resulting field is $\lesssim 0.01$ MG (Fontaine et al. 1973; Tremblay et al. 2015), well below the observed magnetic fields, so a more efficient mechanism should be invoked.

The evolution of white dwarfs can be described as a simple gravothermal process (Althaus et al. 2010). The core of the vast majority of white dwarfs is made of a completely ionized mixture of 12C and 16O, and some minor chemical species like 22Ne and 56Fe. During their evolution, two physical processes modify the internal chemical profiles of white dwarfs. The first one is gravitational settling of neutron-rich species in the liquid phase (Bravo et al. 1992; Bildsten & Hall 2001; Garcia-Berro et al. 2010; Camisassa et al. 2016), which occurs at moderately high luminosities. The second one is phase separation upon crystallization (Isern et al. 1997, 2000; Garcia-Berro et al. 2010), and takes place at lower luminosities. In both cases, the energy involved is large, $\sim 2\times 10^{46}$ erg.

For the sake of simplicity, here we will only consider white dwarfs made of 12C and 16O. The phase diagram of the carbon-oxygen mixture is of the spindle form (Segretain & Chabrier 1993; Horowitz et al. 2010). Since the solid phase is oxygen-rich, hence denser than the liquid phase, when white dwarfs crystallize it settles down, while the carbon-rich liquid left behind is redistributed by Rayleigh-Taylor instabilities (Isern et al. 1997, 2000). This configuration, a solid core surrounded by a convective mantle driven by compositional buoyancy is similar to that found in the core of the Earth, where the light element release associated with the inner core growth is a primary driver of the dynamo (Lister & Buffet 1995). Given this analogy, in this Letter we analyze if the convective mantle of crystallizing white dwarfs can produce magnetic fields of the observed strengths.

2. The stellar dynamo

For a typical white dwarf of mass $\simeq 0.61\,M_{\sun}$ solidification starts at $\log(L/L_{\sun})\simeq-3.5$, whereas for a massive one of $1.0\,M_{\sun}$ crystallization sets in at $\simeq-2.6$. At this point of the evolution the temperatures of their nearly isothermal cores are, respectively, $\log T_{\rm c}\,({\rm K})=6.3$ and $6.6$, whereas their central densities are $\log\rho_{\rm c}\,({\rm g\,cm^{-3}})=6.6$ and $7.5$ (Salaris et al. 2010). For the sake of conciseness, and unless specified otherwise, here we will only discuss the case of the heavier white dwarf, since it is observationally found that magnetic white dwarfs are more massive than usual (Ferrario et al. 2015).

As can be seen in Fig. 1, for a $1.0\,M_{\sun}$ white dwarf, when solidification starts the Rayleigh-Taylor unstable region encompasses a large fraction of the radius of the star. This also holds for the $0.6\,M_{\sun}$ model star. As the $1.0\,M_{\sun}$ white dwarf cools, the outer edge of the convective mantle barely moves, while the inner edge moves progressively outwards. At $\log(L/L_{\sun})\simeq-2.9$, the outer edge of the mantle is located at $r/R\simeq 0.65$, and the inner edge at $\simeq 0.4$. At this moment, the outer edge moves abruptly outwards and when $\log(L/L_{\sun})\simeq-3.1$ it almost reaches the surface, while the inner edge continues its progression at a slower pace. The main difference with the $0.6\,M_{\sun}$ white dwarf is that in this case the outer radius stabilizes at $r/R\simeq 0.9$. Finally, in both cases, the convective region disappears at about $\log(L/L_{\sun})\simeq-4.5$.

The density contrast between the carbon enriched material and the ambient liquid mixture is $\delta\rho/\rho\sim 10^{-3}$, leading to effective accelerations $a_{\rm eff}=g(\delta\rho/\rho)\simeq 2\times 10^{6}$ cm s*-2*. However, since the viscosity of Coulomb plasmas is very small, drag cannot be neglected and results in a limiting velocity of the turbulent eddies, $v_{\rm s}=\sqrt{(3/8)C_{\rm b}a_{\rm eff}D_{\rm b}}$, where $D_{\rm b}$ is the radius of the bubble, and $C_{\rm b}\approx 0.2$ in the spherical case (García-Senz & Woosley 1995). The value of $D_{\rm b}$ is not yet known, but it cannot be larger than the curvature radius at the edge of the crystallized core. When the same value found for the Earth is adopted $D_{\rm b}\sim 0.1R_{\rm core}$ (Moffatt & Loper 1994), we obtain velocities of $\sim 35$ km s*-1*.

For the thermodynamical conditions found in white dwarf interiors, viscosities are very small and conductivities rather high (Nandkumar & Pethick 1984). In particular, for the heavier white dwarf, the electrical conductivity is $\sigma=1.3\times 10^{21}$ s*-1*, the magnetic diffusivity is $\eta=5.6\times 10^{-2}$ cm2 s*-1* and the kinematic viscosity is $\nu=3.13\times 10^{-2}$ cm2 s*-1*. Adopting $\sim 2\times 10^{8}$ cm for the characteristic size of the convective mantle (see Fig. 1, upper panel), the Reynolds and magnetic Reynolds numbers111The magnetic Reynolds number, Rm$=ul/\eta$, where $u$ and $l$ are typical velocities and lengths and $\eta$ is the magnetic diffusivity, measures the relative importance of induction versus dissipation, while the magnetic Prandtl number is the ratio between the magnetic and the ordinary Reynolds numbers, Pm=Rm/Re. range from $\sim 10^{14}$ to $10^{15}$, and the magnetic Prandtl number is $\approx 0.58$. Furthermore, the characteristic Ohmic decay time $\tau_{\Omega}\sim R^{2}_{\rm WD}/\eta\simeq 10^{18}$ s is much longer than the age of the white dwarf. Consequently, once the magnetic field is generated, it will remain almost constant for long times (Cumming 2002).

The aim of dynamo scaling theories is to explain the intensity of magnetic fields in terms of the properties of the region hosting the dynamo. The most comprehensive of these theories takes into account the balance between the ohmic dissipation and the energy flux available to the dynamo (Christensen 2010). The magnetic fields of the Earth and Jupiter are generated by convective dynamos powered by the cooling and chemical segregation of their interiors and it has been shown that the scaling law based on the energy fluxes can be extended to fully convective stars like T Tauri and rapidly rotating M dwarfs (Christensen et al. 2009). Therefore, it is natural to use these theories to compute the magnetic field produced by the convective mantle of crystallizing white dwarfs.

If the dynamo is saturated, the magnetic Reynolds number is large enough, and convection is described by the mixing length formalism, this scaling law can be written as (Christensen 2010):

[TABLE]

where the integral is the energy of the convective mantle, $E$, $c$ is an adjustable parameter, $\mu_{0}$ is the vacuum permeability, $f_{\Omega}\leq 1$ is the ratio of the Ohmic dissipation to the total dissipation, $q_{\rm c}$ is the convected energy flux, $H$ is the scale height (temperature and compositional), $V$ is the volume of the convective region, $r_{o}$ and $r_{i}$ are its outer and inner radii, and $\lambda$ is the mixing length (the minimum between the density scale height and the size of the convective zone, $r_{o}-r_{i}$). Here we adopt $f_{\Omega}=1$ and $\lambda=H$. Also, we use the BasTI cooling sequences (Salaris et al. 2010) which, in addition to the release of latent heat, consider the energy released by chemical differentiation (Mochkovitch 1983; Segretain et al. 1994).

3. Results and discussion

Figure 2 displays the density of convective energy available to the dynamo, namely the r.h.s. of Eq. (1) excluding the factor $cf_{\Omega}$, for several white dwarf masses. The dynamo starts when solidification sets in, then reaches a maximum and, finally, slowly declines as the bottom of the convective mantle moves outwards at fainter luminosities. This is the result of two effects. First, the mass of the star, that determines when crystallization starts. White dwarfs with larger masses crystallize at larger luminosities because their central densities are also larger. The luminosity of the star also plays a role, since it determines the rate at which crystallization takes place. This luminosity, in turn, depends on the transparency of the atmosphere and the temperature of the core. The maximum energy available to the dynamo ranges from $\log E=8.2$ to 9.6, and from 8.3 to 9.7 (c.g.s. units) for DA and non-DA white dwarfs, respectively, when masses ranging from 0.54 to $1.00\ M_{\sun}$ are considered. The scaling law relating the magnetic fields of the Earth, Jupiter, T Tauri and M dwarf stars can be approximated by $\log B=0.5\log E-5.42$ – see Fig. 3, solid line – and predicts that the maximum field that can be generated at the top of the dynamo range from $\sim 0.05$ to 0.25 MG.

The X-ray emission of stars with convective regions correlates with the rotation period, and with magnetic activity. The existence of a plateau in the relationship linking magnetic activity and X-ray luminosity suggests that dynamos saturate when the Rossby number Ro$=P_{\rm rot}/\tau<0.1$, being $\tau$ the convective turnover time (Wright et al. 2011; Wright & Drake 2011). For a star of mass $1.0\,M_{\sun}$, and adopting the typical values found in Sect. 2, the turnover time is $\sim 90$ s. Since the critical stability rotation period is $\sim 6$ s, we conclude that rapidly rotating white dwarfs can have saturated dynamos. Because white dwarf spectral lines are broad, measuring the rotation periods is a difficult task, and have only been measured in pulsating and magnetic white dwarfs. Thus, their distribution of angular velocities is not well determined. Observations indicate that white dwarfs are slow rotators, but strictly speaking, the existence of rapidly rotating white dwarfs cannot be discarded (Kawaler 2015). Within our scenario white dwarfs with high magnetic fields would rotate rapidly and have saturated dynamos, while stars with weak fields would rotate slowly and have non-saturated dynamos. Therefore, our mechanism could easily explain white dwarfs with magnetic fields $B\lesssim 0.1$ MG.

Table 1 shows the properties of white dwarfs with carbon-oxygen cores within 20 pc of the Sun (Kawka et al. 2007; Giammichele et al. 2012). We list the name of the star, the spectral type, the magnetic field, the rotation period in hours, the mass and luminosity in solar units, the relative mass of the crystallized core and the properties of the convective mantle (the density at the inner edge, the size, and the flux at the top, all in c.g.s. units). It is important to realize that all of them, except WD 0413$-$077 (40 Eri B) and WD 2105$-$820 (L24-52) have crystallized cores. For each of these stars we computed the energy density available to the dynamo according to Eq. (1) employing the value of $c$ derived using the scaling law for the Earth and Jupiter, T Tauri stars and M dwarfs. In the top panel of Fig. 3 we show the set of calculations in which the energy density of the dynamo has been computed adopting the actual size of the convective mantle corresponding to the observed luminosity of the star, whereas in the bottom panel we show the magnetic field strength when the maximum energy density of the dynamo shown in Fig. 2 is adopted. This is a reasonable assumption given the very long ohmic decay timescale. In both cases, to relate the outer and inner magnetic fields we used the prescription of Christensen et al. (2009). The distribution of magnetic field strengths is clearly bimodal. Most DA white dwarfs cluster around the scaling law (the solid line in Fig. 3), while non-DA white dwarfs have magnetic fields larger than DA stars with similar energy density of the dynamos. This points towards to a different origin of the dynamo, and/or to other scaling laws like the Coriolis-Inertial-Archimedian balance or the Magnetic-Archimedian-Coriolis balance (Christensen 2010), because the mixing length scaling law considered here demands $\sim 10^{15}$ erg s*-1* to produce fields of $\sim 100$ MG.

It is important to realize that the time necessary for the magnetic field to diffuse through the radiative outer layers is very long, and only the convective mantle of massive white dwarfs is close to the surface, as clearly shown in Fig 1. The final magnetic field strength will depend on how this mantle interacts with the convective envelope of cool white dwarfs. Furthermore, only white dwarfs with progenitors more massive than $2.3\,M_{\sun}$ can have rotation periods of minutes (Kawaler 2015). This, together with the fact that the energy of the dynamo induced by crystallization is larger for massive white dwarfs could naturally explain why magnetic white dwarfs are more massive than average. This scenario could also explain why magnetic white dwarfs are preferentially cool. Moreover, it is likely that low-mass white dwarfs would have weaker magnetic fields confined in their interiors. Therefore, this scenario puts forward a tantalizing possibility, namely, that the magnetic fields observed in planets, non-evolved stars and white dwarfs share a common physical origin. In summary, our calculations indicate that the magnetic field observed in a sizable fraction of all white dwarfs could be the result of a dynamo generated by phase separation upon crystallization.

Furthermore, our scenario does not preclude other possibilities, but instead alleviates one of the main drawbacks of the current hypotheses to explain magnetic white dwarfs, since both the fossil field model and the binary scenario predict an insuficient number of magnetic white dwarfs. For instance, it has been long suspected that at least a fraction of high-field magnetic white dwarfs could originate from the merger of two white dwarfs (Wickramasinghe & Ferrario 2000). During the merger, a hot, differentially-rotating, convective corona forms. The temperatures reached during the coalescence are so high that hydrogen is burned during the early phases of the merger. This corona is prone to magnetorotational instability, and it has been shown that can produce magnetic fields with the energy required to explain the magnetic fields of non-DA stars shown in Fig. 3 (Garcia-Berro et al. 2012). This could explain why many white dwarfs with very high magnetic fields are H-deficient. In this case a dynamo of a completely different nature would be operating.

This work has been supported by MINECO grants ESP2013- 47637-P, ESP2015-66134-R (J.I.), and AYA2014-59084-P (E.G.-B.), by the European Union FEDER funds, by grants 2014SGR1458 (J.I.), 2014SGR0038 (E.G.-B.) of the AGAUR, and by the CERCS program of the Generalitat de Catalunya.