SPIN AND MAGNETISM OF WHITE DWARFS

Three hypotheses have dominated previous discussions of the origin of magnetic fields in WD stars: the conservation of magnetic flux from the MS progenitor (Angel et al. 1981; Ferrario & Wickramasinghe 2005), the amplification of such a seed field by convective episodes in the core of the progenitor star (Ruderman & Sutherland 1973), and the transient amplification of the field during a merger event (Tout et al. 2008; Nordhaus et al. 2011).

We begin our investigation by reviewing some constraints on each of these hypotheses and then summarize the arguments pointing to an amplification of the magnetic field during the post-MS expansion of the WD progenitor. This process comprises three steps: (i) the advection of angular momentum inward through the convective envelope; (ii) the amplification of a magnetic field by a dynamo near the core–envelope boundary; followed by (iii) a flow of magnetic helicity into the growing core, which is sustained by a persistent angular velocity gradient and magnetic twist below the boundary.

Whether the inner envelope can support a dynamo turns out to depend on a combination of stellar properties. The core rotation rate is sensitive to the loss of angular momentum on the MS and to any interaction with a planetary or stellar companion. Clearly, solid rotation would imply $\mathrm{Co}\ll 1$ throughout the envelope of a star approaching the tip of the RGB or AGB, therefore suggesting the absence of a large-scale hydromagnetic dynamo. But when the envelope rotation profile is inwardly peaked, we find differing answers for isolated stars of initial mass ${M}_{\mathrm{ZAMS}}$ less or greater than $\sim 1.3\;{M}_{\odot }$, corresponding to the presence or absence of a strong magnetic wind on the MS, hence the greater importance of binary interaction for sustaining $\mathrm{Co}\gtrsim 1$ in solar-mass stars.

Another consideration is the proportion of the material contained in the remnant WD that is deposited during single and double shell burning. This composes about a third of the mass of the WD when ${M}_{\mathrm{ZAMS}}\sim 1$ (${M}_{\mathrm{wd}}\sim 0.55\;{M}_{\odot }$), but is strongly reduced by dredge-up (after core helium ignitition and before the onset of thermal pulses) in the intermediate-mass stars that are the progenitors of massive WDs. In this second case, the magnetized mass shell turns out to be thin enough that ohmic decay is initiated on a timescale ∼0.3 Gyr (with some dependence on the mass-loss rate on the AGB).

The source of a strong magnetic field ($1-{10}^{3}$ MG) in a WD therefore depends on both its age and mass. Indeed, expanded sampling of WD magnetism calls into question the flux-freezing hypothesis. Whereas between ∼2% and ∼10% of isolated WDs are magnetic in this sense (Liebert et al. 2003, 2005)—roughly similar to the proportion of chemically peculiar and magnetic A stars—the proportion is higher in isolated massive WDs (Kepler et al. 2013) and extends to ∼25% in cataclysmic variables (CVs; Gänsicke 2005). A large proportion of magnetic WDs in short-period CVs also appear to have very strong magnetic fields, exceeding 10 MG, which are much rarer in the isolated WD population.

It has also been noted that magnetic WDs appear to be rare in binary systems with a non-interacting MS companion (Liebert et al. 2005), suggesting that binary interaction may be involved in their genesis. Previous theoretical ideas have centered around the impulsive growth of a magnetic field during a binary merger (Nordhaus et al. 2007; Tout et al. 2008; Nordhaus et al. 2011). Here our focus is on a much longer-lived process that is sustained by the inward pumping of angular momentum.

A solar-mass star is slowly rotating at the end of the MS, so that the absorption of a giant planet significantly spins up the star and may trigger dynamo activity (Livio & Soker 2002). The critical planetary mass that has this effect near the base of the envelope depends on the angular velocity profile. For a flat distribution of specific angular momentum at $\mathrm{Co}\lesssim 1$, we find that the absorption of even a Neptune-mass planet can significantly spin up the core.

A star more massive than ${M}_{\mathrm{ZAMS}}\sim 1.3\;{M}_{\odot }$ retains enough angular momentum at the end of the MS that only a planet more massive than Jupiter will significantly supplement its rotational angular momentum. Nonetheless, we find that such an interaction can lead to stronger magnetic fields in the WD remnant.

Two properties of giants are especially relevant here. The inward flow of matter across the core–envelope boundary allows a net transport of magnetic helicity, which has been shown to be a necessary ingredient in the long-term stabilization of WD magnetic fields (Braithwaite & Spruit 2004) and has also been implicated in the establishment of solid rotation in the radiative core of the Sun (Gough & McIntyre 1998). We show that this helicity flux can be related to the magnitude of the large-scale Maxwell torque that enforces (nearly) solid rotation in the outer core.

The outer core of an AGB star is also distinguished by an intense radiation flux. This greatly facilitates the buoyancy of a toroidal magnetic field that is anchored in a tachocline layer below the convective envelope. A thickness approaching ∼10% of a scale height can maintain contact with the envelope over a dynamo period $\sim 10-{10}^{2}{\tau }_{\mathrm{con}}$. This suggests that a given level of magnetic activity (measured, e.g., by the ratio of Maxwell to Reynolds stresses) may be achieved with a lower Coriolis parameter in an AGB envelope than in a solar-type MS star. Magnetically induced mixing of core material into the envelope (Nordhaus et al. 2008) could also be significantly enhanced by this effect.

The measured spins of WDs are strongly inconsistent with solid rotation in the envelope of an AGB star, combined with a tight core–envelope coupling. But even if we allow for inward convective pumping of angular momentum, the final spin rate of a WD remnant is sensitive to the heavy loss of mass during the transition to the post-AGB phase.³

A decoupling of the rotation of core and envelope is only feasible when the large-scale poloidal magnetic field drops below the equivalent of ${10}^{2}-{10}^{3}$ G in the WD during the brief, thermally pulsating AGB phase. (Much weaker fields will couple the core and envelope of a subgiant or core helium burning star; Paper I.) This appears, at first sight, to be inconsistent with strong magnetism in the WD. In fact, a magnetic coupling between core and envelope is naturally self-limiting: once $\mathrm{Co}$ drops below a critical value, the dynamo operating in the inner envelope shuts off. A reasonable first guess (${\mathrm{Co}}_{\mathrm{crit}}\sim 0.1-0.3$) yields WD rotation periods in the range observed in most WDs.

As a result, the remnant rotation is inversely correlated with the net angular momentum in the star: the slower the rotation of the AGB envelope, the sooner the decoupling between core and envelope, when the core has a larger radius and moment of inertia. The weakly magnetized surface layer of the WD is very thin, and we show that the magnetic field below quickly diffuses to the photosphere.

Our most surprising finding is that WDs with very long (∼1 yr) rotation periods originate from stars that spin relatively rapidly near the end of the AGB phase. Then the rotation of core and envelope remains magnetically coupled during the contraction of the envelope. The endpoint is sensitive to details of how the envelope is expelled, in particular, to whether the envelope experiences a re-expansion following a late thermal pulse. This taps most of the remaining angular momentum of the core. Without such a re-expansion, the final rotation period can be shorter than a day.

It should be kept in mind that anisotropic contraction of the envelope can contribute to WD rotation (Spruit 1998). Indeed, it may be needed to generate centrifugally supported disks that provide seeds for metal enrichment of WD atmospheres.

In order to evaluate various pieces of the physics, we employ two stellar models of solar metallicity and initial mass ${M}_{\mathrm{ZAMS}}=1\;{M}_{\odot }$ and $5\;{M}_{\odot }$. Both models are constructed using the MESA code (Paxton et al. 2011, version 5527), with rotational degrees of freedom turned off. The normalization of the mass-loss rate on the AGB and the masses of the WD remants were fixed by matching the latest initial-to-final mass relation obtained from open clusters (Dobbie et al. 2009). The coefficient entering into the formula of Blöcker (1995) was taken to be η_B = 0.05. The corresponding masses of the WD remnants are $0.55\;{M}_{\odot }$ and $0.87\;{M}_{\odot }$.

The rotational evolution was followed in post-processing using the rotation models described in Section 2.1. The loss of angular momentum due to mass loss on the RGB and AGB was taken into account, along with the interaction with a companion planet of a range of masses: Earth, Neptune, and Jupiter in the $1\;{M}_{\odot }$ model, and even heavier companions in the $5\;{M}_{\odot }$ model. In the first case, the companions were started at a range of semimajor axes and orbital eccentricities. The orbit of the planet and the exchange of angular momentum with the star were computed taking into account tidal drag and the gravitational quadrupole that is generated by convection in the giant envelope. Further details are given in Paper I.

Sections 2 and 3 discuss several processes that are relevant to the growth and maintenance of a stable magnetic field in the core of a post-MS star, and which determine the spin angular momentum trapped there at the end of the AGB. Section 4 presents detailed estimates of WD dipole magnetic fields. Section 5 analyzes the magnetic decoupling of core and envelope near the tip of the AGB and provides estimates of the final WD spin. Ohmic transport through a thin, unmagnetized surface layer in a WD and the thicker magnetized layer below it is considered in Section 6.

Various outstanding issues—such as the sensitivity of our results to the assumed angular velocity profile in the progenitor envelope, the impulsive growth of a magnetic field during a merger, and effects associated with the anisotropic contraction of the residual AGB envelope—are addressed in the concluding Section 7. The Appendix gives further details of the accumulation of magnetic helicity in the stellar core.

Our understanding of the origin of WD rotation has been impeded by an incomplete knowledge of mixing processes in the radiative parts of the progenitor. Even a very weak poloidal magnetic field (corresponding to a magnetic field $\ll$ MG in the WD under the assumption of magnetic flux freezing) will enforce solid rotation in the evolving core.

A popular approach recently has been to assume that such a large-scale poloidal field is initially absent and to imagine that the radial magnetic field that is needed for angular momentum transport is generated by a current-driven ("Tayler") instability of a predominantly toroidal field (Spruit 2002; Cantiello et al. 2014, and references therein). This leads to an incomplete coupling between core and envelope. We argue that even an inefficient dynamo process operating near the core–envelope boundary will easily short-circuit this effect.

We develop, as a working alternative, a simple and deterministic model of angular momentum redistribution in evolving stars. This model applies to stars that retain (or gain) enough angular momentum to support some dynamo activity near the core–envelope boundary.

• 1.  
  The outer core is assumed to rotate as a solid body, as enforced by large-scale Maxwell stresses, and to co-rotate on the average with the inner envelope.
• 2.  
  Inhomogeneous rotation is sustained mainly in convective regions of the star, especially the deep outer envelope that forms following the completion of core H and He burning.

The rotation profile that is formulated and calibrated in Paper I can be summarized as follows. Where the Coriolis force can be neglected, outside some transition radius R_c, the envelope maintains uniform specific angular momentum,

$$\begin{eqnarray}&&\bar{{\rm{\Omega }}}(r)=\bar{{\rm{\Omega }}}({R}_{c}){\left(\displaystyle \frac{r}{{R}_{c}}\right)}^{-2};\quad r\gt {R}_{c}.\end{eqnarray} \tag{ 2 }$$

Here $\bar{{\rm{\Omega }}}$ is the angular velocity averaged over a spherical shell. We identify R_c with the base of the envelope,

$$\begin{eqnarray}&&{R}_{c}={R}_{\mathrm{benv}}\quad (\bar{{\rm{\Omega }}}{\tau }_{\mathrm{con}}\lt 1\;\mathrm{at}\;{R}_{\mathrm{benv}}),\end{eqnarray} \tag{ 3 }$$

when the angular momentum of the star is small enough that $\bar{{\rm{\Omega }}}(r){\tau }_{\mathrm{con}}\lt 1$ throughout the envelope. Throughout this paper, the subscript "benv" refers to the base of the envelope.

When the star has more angular momentum, we maintain the profile (2) in the outer part of the convection zone and consider

$$\begin{eqnarray}\bar{{\rm{\Omega }}}(r) & = & \bar{{\rm{\Omega }}}({R}_{c}){\left(\displaystyle \frac{r}{{R}_{c}}\right)}^{-\alpha }\\ & = & \displaystyle \frac{{\mathrm{Co}}_{\mathrm{trans}}}{{\tau }_{\mathrm{con}}({R}_{c})}{\left(\displaystyle \frac{r}{{R}_{c}}\right)}^{-\alpha };\quad {R}_{\mathrm{benv}}\lt r\lt {R}_{c}\end{eqnarray} \tag{ 4 }$$

in the inner envelope. The Coriolis parameter at the transition radius ${\mathrm{Co}}_{\mathrm{trans}}$ is taken to be unity, and the radial index $1\lt \alpha \lt 3/2$. As the angular momentum increases, R_c moves outward and may reach the surface of the star, in which case $\mathrm{Co}$ is larger than unity everywhere in the envelope.

As we discuss in Paper I, α = 1 best matches the observed core rotation periods of sub-giant and helium burning stars, which we use to calibrate our rotational model. However, in stars of much greater size, such as those approaching the tips of the RGB and AGB, the core is more centrally concentrated and the gravitational field in the inner envelope, $g(r)\propto {r}^{-\beta }$, steepens from $\beta =1$ to β = 2. Then the value of α that results from the inward transport of angular momentum by deep convective plumes increases from $(1+\beta )/2=1$ to 3/2. It is during this more advanced phase of evolution that the helicity flux into the core is expected to peak and the rotational fate of the WD is determined. We therefore use α = 3/2 in our analysis.

Net magnetic helicity ${\mathcal{H}}=\int {\boldsymbol{A}}\cdot {\boldsymbol{B}}\;{dV}$ appears to be an essential ingredient for maintaining long-term magnetostatic stability in a static and entirely fluid star (Braithwaite & Spruit 2004). Here ${\boldsymbol{B}}$ is the magnetic field and ${\boldsymbol{A}}$ the vector potential from which it is derived. ${\mathcal{H}}$ is a topological charge in a perfectly conducting fluid without boundary. For example, when a closed poloidal magnetic loop carrying flux ${{\rm{\Phi }}}_{p}$ is also twisted through an angle ψ, the helicity is ${\mathcal{H}}={{\rm{\Phi }}}_{p}{{\rm{\Phi }}}_{\phi }=\psi {{\rm{\Phi }}}_{p}^{2}$, where ${{\rm{\Phi }}}_{\phi }$ is the toroidal flux. ${\mathcal{H}}$ evolves only on a very long ohmic timescale in an extended astrophysical fluid—unless there is an exchange of magnetic twist with the exterior of the fluid, or with a medium of low electrical conductivity.

The accumulation of helicity considered here is associated with rapid changes in the structure of the star. We expect this to be possible in the outer core and inner envelope of a giant star, since this zone is connected to the surface of the star by rapid convective motions. Core convection has a topology less favorable to the accumulation of ${\mathcal{H}}$, as does the erasure of differential rotation by the stretching of magnetic field lines in the interior of a radiative star. A simplified time-dependent model of an internal shear-driven dynamo has been developed by Wickramasinghe et al. (2014) but does not address the growth of ${\mathcal{H}}$.

Such a secular accumulation of magnetic helicity—although occuring relatively rapidly near the tips of the RGB and AGB—is nonetheless a gradual process when measured over an individual dynamo cycle. There has also been long-standing interest in the role that helicity conservation may play in limiting the exponential growth of a seed magnetic field in a turbulent magnetofluid (Gruzinov & Diamond 1994; Blackman & Field 2000a; Vishniac & Cho 2001). In mean-field dynamo models, this involves a suppression of the α effect, e.g., the toroidal electromotive force that sources a large-scale poloidal magnetic field.

This limitation on fast dynamo action could be removed by the transport of magnetic helicity on a relatively short timescale, comparable to the rotation period. Such transport may involve the removal of twist across the surface of the star (Blackman & Field 2000b), or alternatively an internal flow of helicity between rotational hemispheres (Vishniac & Shapovalov 2014). In either of these cases, rapid dynamo growth is not directly dependent on changes in the net stellar helicity, obtained by summing the contribution from both hemispheres. The longer, secular effect being considered here necessarily depends on a breakdown of reflection symmetry between magnetic hemispheres and, because it involves a change in an integral property of the star, must depend on the transport of twist across the stellar surface.

The connection between helicity growth and mass inflow can be examined with a simple model of a toroidal magnetic twist superposed on a radial magnetic field, all subjected to a radial MHD flow of speed v_r. (The sign of this flow, relative to the convective boundary, fluctuates during thermal pulses on the AGB but is negative on the average.)

Within a sphere coinciding with ${R}_{\mathrm{benv}}$, one finds

$$\begin{eqnarray}\displaystyle \frac{d{\mathcal{H}}}{{dt}} & = & \displaystyle \frac{d}{{dt}}\displaystyle \int {\boldsymbol{A}}\cdot {\boldsymbol{B}}\;{dV}\\ & = & \displaystyle \frac{\partial {\mathcal{H}}}{\partial t}+\displaystyle \frac{{{dR}}_{\mathrm{benv}}}{{dt}}\displaystyle \int {{dSA}}_{\phi }{B}_{\phi }=-\delta {v}_{r}\displaystyle \int {dS}\;{A}_{\phi }{B}_{\phi }\end{eqnarray} \tag{ 5 }$$

(Appendix). The gauge

$$\begin{eqnarray}&&{A}_{\phi }(\theta ,r)=\displaystyle \frac{1}{r\mathrm{sin}\theta }{{\rm{\Phi }}}_{r}(\theta )\end{eqnarray} \tag{ 6 }$$

is convenient, where

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{r}(\theta )={\displaystyle \int }_{0}^{\theta }\mathrm{sin}{\theta }^{\prime }{r}^{2}{B}_{r}({\theta }^{\prime })d{\theta }^{\prime }\end{eqnarray} \tag{ 7 }$$

is ${(2\pi )}^{-1}$ times the radial magnetic flux between the rotation axis and polar angle θ. Clearly, ${{\rm{\Phi }}}_{r}(\pi )=0$.

One notices that the surface integral appearing in Equation (5) has the dimensions of a torque: it is comparable in magnitude to the Maxwell stress ${B}_{r}{B}_{\phi }/4\pi$, multiplied by the lever arm $r\mathrm{sin}\theta$, and integrated over the core–envelope interface. Even if the net torque acting on the core boundary vanishes, $d{\mathcal{H}}/{dt}$ may remain finite as a consequence of the different angular dependences of ${A}_{\phi }$ and ${B}_{r}r\mathrm{sin}\theta$.

We are interested here in the case where ${B}_{r}{B}_{\phi }/4\pi$ is the dominant stress driving the core material toward solid rotation. Then we can estimate the helicity flux into the core once we know the source of the Maxwell torque. The largest effect turns out to be the latitude dependence of the Reynolds stress that is applied to the core at the base of the convective envelope.

We may estimate this stress by considering a dynamo process at work in a tachocline layer right below the convective boundary. Convection above the tachocline supports strong latitudinal variations in Ω that impose a radial mismatch ${\rm{\Delta }}{\rm{\Omega }}$ with the mean core rotation ${{\rm{\Omega }}}_{\mathrm{benv}}=\bar{{\rm{\Omega }}}({R}_{\mathrm{benv}})$. The tachocline can therefore be identified with the radial layer (of thickness Δr) in which ${\rm{\Delta }}{\rm{\Omega }}$ is concentrated,

$$\begin{eqnarray}&&{\boldsymbol{\nabla }}{\rm{\Omega }}\simeq \displaystyle \frac{\partial {\rm{\Omega }}}{\partial r}\hat{r}\sim \displaystyle \frac{{\rm{\Delta }}{\rm{\Omega }}}{{\rm{\Delta }}r}\hat{r}.\end{eqnarray} \tag{ 8 }$$

Our focus here is on the parts of the tachocline where $\partial {\rm{\Omega }}/\partial r\gt 0$ (in the Sun, it is the equatorial band at latitudes $\lesssim 45^\circ$). The magnetorotational instability (Balbus & Hawley 1994) is excited where $\partial {\rm{\Omega }}/\partial r\lt 0$ and is sustained by rapid radiative diffusion even in the presence of stable radiative convective equilibrium (Menou et al. 2004).

The poloidal magnetic field threading this part of the tachocline is wound up linearly on timescales short compared with the dynamo period ${P}_{\mathrm{dyn}}$, which we normalize as ${N}_{\mathrm{dyn}}{\rm{\Delta }}{{\rm{\Omega }}}^{-1}$. We limit the magnetic field strength in the tachocline by noting that a radial magnetic field stronger than

$$\begin{eqnarray}&&{B}_{r}\sim {(4\pi \rho )}^{1/2}\displaystyle \frac{{\rm{\Delta }}r}{{P}_{\mathrm{dyn}}}\end{eqnarray} \tag{ 9 }$$

will act to redistribute angular momentum in a layer of density ρ and thickness ${\rm{\Delta }}r$, thereby shorting out the angular velocity gradient. The ratio of toroidal to poloidal field components is obtained from the linear winding term in the induction equation,

$$\begin{eqnarray}&&\displaystyle \frac{2\pi }{{P}_{\mathrm{dyn}}}{B}_{\phi }\sim \displaystyle \frac{r{\rm{\Delta }}{\rm{\Omega }}}{{\rm{\Delta }}r}{B}_{r}.\end{eqnarray} \tag{ 10 }$$

One finds that the toroidal magnetic field is in approximate equipartition with the rotational motions at the base of the convective envelope,

$$\begin{eqnarray}&&{B}_{\phi }^{2}\sim 4\pi \rho {\left(r\displaystyle \frac{{\rm{\Delta }}{\rm{\Omega }}}{2\pi }\right)}^{2}\sim 4\pi {\left(\displaystyle \frac{\mathrm{Co}}{2\pi }\displaystyle \frac{r}{{{\ell }}_{P}}\displaystyle \frac{{\rm{\Delta }}{\rm{\Omega }}}{\bar{{\rm{\Omega }}}}\right)}^{2}\rho {v}_{\mathrm{con}}^{2}.\end{eqnarray} \tag{ 11 }$$

The Maxwell stress works out to

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{r}{B}_{\phi }}{4\pi }\sim \displaystyle \frac{\rho r{\rm{\Delta }}r{({\rm{\Delta }}{\rm{\Omega }})}^{2}}{2\pi {N}_{\mathrm{dyn}}}.\end{eqnarray} \tag{ 12 }$$

We emphasize that this estimate of the stress has been obtained from dynamical considerations and side-steps the challenging question of how the α effect is actually manifested in a dynamo operating near a radiative-convective boundary. For recent approaches to this problem, in the context of lower-luminosity MS stars, see Blackman & Thomas (2015) and references therein.

We note that ${\rm{\Delta }}{\rm{\Omega }}\sim \bar{{\rm{\Omega }}}$ is seen in anelastic calculations of deep and rapidly rotating ($\mathrm{Co}\gt 1$) convective envelopes (Brun & Palacios 2009) and is suggested by the analytic treatment of angular momentum pumping in Paper I. The latitudinal gradient is milder in the Sun, where the equator rotates more rapidly than the poles, $({{\rm{\Omega }}}_{\mathrm{eq}}-{{\rm{\Omega }}}_{\mathrm{pole}})/\bar{{\rm{\Omega }}}\sim 0.1$, but the solar convective envelope also has a very different aspect ratio.

Now consider the helicity flux integrated over one rotational hemisphere. Even in a state of rotational equilibrium—where the angle-integrated Maxwell torque vanishes—the integral (5) generally remains finite. Notice also that it is the product of B_r and ${B}_{\phi }$ that is regulated according to Equation (12). The Maxwell stress does not depend on the sign of B_r that is imposed by a dynamo operating near the core–envelope boundary, nor does it depend directly on the coherence of the magnetic field.

The sum of the two hemispheric integrals for $d{\mathcal{H}}/{dt}$ would vanish if the magnetic field were reflection-symmetric about the rotational equator. But in general it is not: the distribution of radial magnetic flux across the northern hemisphere need not precisely mirror that threading the southern hemisphere. (Only the net spherical flux vanishes.) Such an incomplete hemispheric cancellation allows a net flow of magnetic twist into the core.

Other sources of helicity flow can be considered. The inflow through the core–envelope boundary itself drives $\partial {\rm{\Omega }}/\partial r$, but on a much longer timescale than the rotation period.

The WD magnetic fields that result from these two sources of ${\mathcal{H}}$ are compared in Section 3.

The magnetic field of a WD is influenced by a series of convective episodes in the progenitor. One measure of the relative importance of different convective episodes is provided by the convective Mach number ${{\mathcal{M}}}_{\mathrm{con}}={v}_{\mathrm{con}}/{c}_{s}$ (here c_s is the adiabatic speed of sound), which in a stellar core can be estimated as

$$\begin{eqnarray}&&{{\mathcal{M}}}_{\mathrm{con}}\sim {\left(\displaystyle \frac{{R}_{\mathrm{core}}{v}_{\mathrm{con}}^{2}}{{{GM}}_{\mathrm{core}}}\right)}^{1/2}.\end{eqnarray} \tag{ 13 }$$

As the core continues to contract during later stages of nuclear burning, its magnetic energy and gravitational energy scale in the same way with central density. To the extent that the generated Maxwell stress is limited by the convective stress, e.g., ${B}_{\phi }\lesssim {(4\pi \rho )}^{1/2}{v}_{\mathrm{con}}$, episodes of greater ${{\mathcal{M}}}_{\mathrm{con}}$ are capable of generating stronger magnetic fields.

One encounters much larger ${{\mathcal{M}}}_{\mathrm{con}}$ on the giant branches ($\sim {10}^{-3}-{10}^{-2}$) than during steady core H or He burning in intermediate-mass stars ($\sim {10}^{-5}-{10}^{-4}$). It is also worth noting that the intense period of convection during the AGB phase is also the shortest in duration. That implies a weaker cancellation in the magnetic helicity that accumulates in the core, due to a shifting distribution of poloidal magnetic flux (see Section 2.2).

Magnetic fields that are generated during a certain convective phase will remain trapped unless exposed to later stages of convection, or the exterior mass shells are expelled. As we examine in Section 6, ohmic diffusion in the WD remnants of intermediate-mass stars is restricted to a relatively thin magnetized layer. The relevance of dynamo activity on the RGB, horizontal branch, and AGB is determined, in good part, by the maximum inward penetration of the convective envelope: one requires $M({R}_{\mathrm{benv}})\leqslant {M}_{\mathrm{wd}}$. Only in relatively massive progenitors (${M}_{\mathrm{ZAMS}}\gtrsim 4\;{M}_{\odot }$) does MS core convection extend to the mass boundary of the WD remnant, where it can influence the visible magnetic field; we ignore this effect here.

The core helium flash occuring in stars of mass $\lt 2.3\;{M}_{\odot }$ is confined to a central subset of the eventual WD material. Because a star does not become fully convective during the flash (e.g., Bildsten et al. 2012), the magnetic helicity of the convecting core material is essentially conserved. A similar conclusion applies to the brief convective layers that are triggered by thermal pulses in the helium-burning shell near the tip of the AGB.

2.3.1. Stars with Minimal Dredge-up of Helium (${M}_{\mathrm{ZAMS}}\lesssim 2.3\;{M}_{\odot }$)

In stars of initial mass ${M}_{\mathrm{ZAMS}}\sim 1\;{M}_{\odot }$, magnetic fields generated on the RGB are buried inside the WD remnant under a layer of C/O-rich material that is produced during double shell burning (Figure 1). They are ohmically coupled on a timescale $\sim {10}^{8}-{10}^{9}$ yr to a thinner magnetized shell that appears on the AGB (Section 6).

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The ohmic timescale across the entire magnetized shell is longer than $\sim {10}^{9}$ yr, meaning that any magnetic field deposited near the center of the star during the pre-MS phase would influence the surface of the star only on a timescale of a few Gyr.

We conclude that the existence of young magnetized WDs with masses $\lesssim 0.6\;{M}_{\odot }$ points to dynamo activity in the convective envelope on the RGB and AGB.

2.3.2. Intermediate-mass Stars (${M}_{\mathrm{ZAMS}}\gtrsim 2.3\;{M}_{\odot }$)

Stars of intermediate mass are distinguished from those of solar mass in that a dynamo operating on the RGB cannot contribute directly to the WD magnetic field: the outer parts of the helium core are dispersed by dredge-up before the C/O core is assembled. In addition, only a limited mass is converted to carbon and oxygen by double shell burning (Figure 2). The fraction of the WD mass contained by this outer shell is sensitive to the rate of mass loss on the thermally pulsating AGB.

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The ohmic conduction time across this outer magnetized shell is several times 10⁸ yr. The surface magnetic field of the remnant of an isolated, intermediate-mass star is therefore expected to grow and then gradually decay on this timescale (Section 6).

Late-type MS stars generally are compact enough to sustain $\mathrm{Co}\gtrsim 1$ in their convective envelopes, as are subgiants massive enough to have retained most of their natal angular momentum at the completion of core H burning. Dynamo activity, as measured by chromospheric lines and coronal X-ray emission, begins to decline when the Rossby number $\mathrm{Ro}\equiv {P}_{\mathrm{rot}}/{\tau }_{\mathrm{con}}=2\pi /\mathrm{Co}$ is larger than about 0.1, corresponding to $\mathrm{Co}\lesssim 60$. Specifically in M dwarfs, the closest MS analogs of giants, Reiners et al. (2009) observe that surface magnetic flux scales as ${\mathrm{Co}}^{2}$ for $\mathrm{Co}\lesssim 60$. There does not appear to be a sharp cutoff of magnetic activity with Coriolis parameter, as the simplest mean-field dynamo models would suggest.

The evidence therefore points to the presence of weak, but finite, dynamo activity at $\mathrm{Co}\sim 1$ in MS stars. It also begs the question as to whether the relative strength of the dynamo-generated magnetic field (measured in terms of the convective stress at a given Coriolis parameter) differs in giants and MS stars.

First, it should be emphasized that a large-scale Maxwell torque that couples the rotation of the outer core to the envelope depends on a tiny poloidal field that is implanted from the dynamo layer. We quantify this minimal field in Section 3.3.

Some additional insight is provided by mean-field dynamo theory, which suggests that the dynamo number

$$\begin{eqnarray}&&D\equiv \displaystyle \frac{\alpha {\rm{\Omega }}{r}^{3}}{{\nu }_{t}^{2}}\sim \left(\displaystyle \frac{\alpha }{{v}_{\mathrm{con}}}\right){\left(\displaystyle \frac{r}{{{\ell }}_{P}}\right)}^{3}\mathrm{Co}\end{eqnarray} \tag{ 14 }$$

must exceed a threshold $D\gtrsim {(2\pi )}^{3}\sim {10}^{2}$ for magnetic field growth to be sustained (Charbonneau 2014). Here α is the parameter relating the toroidal magnetic field to the toroidal electromotive force, and ${\nu }_{t}$ is the turbulent diffusivity. The prospect for a dynamo is enhanced if the magnetic field can be anchored in a part of the star with relatively low (but finite) ${\nu }_{t}$.

This effect provides part of the motivation for considering a thin, stably stratified layer below the convective envelope as the locus of strong differential rotation and a reservoir for a toroidal magnetic field (Parker 1993). However, the sharpness of the onset of stable stratification below the envelope boundary has presented a difficulty for this hypothesis. The solar convective motions have a relatively low Mach number (${{\mathcal{M}}}_{\mathrm{con}}\sim {10}^{-4}$), and so the overshoot layer below the envelope is very thin. A toroidal magnetic field reaching equipartition with the convective stresses overcomes the stable stratification in a layer that is not much thicker.

In Section 3.2, we show that the intense flux of radiation emanating from the core of an RGB or AGB star will induce an upward drift of magnetized material, one that more than counterbalances the downward flow to the burning shell(s). A tiny seed field that is generated by the convective motions will sustain this feedback (Section 3.3). The interchange between toroidal and poloidal fields will persist in a layer of considerable thickness $\sim (0.01-0.1){{\ell }}_{P}$ over a dynamo cycle $\sim (10-{10}^{2}){\tau }_{\mathrm{con}}$. Convectively driven turbulence is strongly suppressed within this layer.

We now examine the competition between the downward drift of hydrogen-rich material across the tachocline and the upward drift of magnetized material (Figure 3). Consider a horizontal (toroidal) magnetic flux tube, of a radial thickness comparable to the distance ${\rm{\Delta }}r={R}_{\mathrm{benv}}-r$ to the base of the convective envelope. The flux tube starts in buoyancy equilibrium, with a density equal to that of its unmagnetized surroundings. The temperature deficit in the flux tube (labeled "B") is

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{B}-T}{T}=-\displaystyle \frac{{B}_{\phi }^{2}}{8\pi P},\end{eqnarray} \tag{ 15 }$$

which maintains a differential energy flux

$$\begin{eqnarray}&&\delta {F}_{\mathrm{rad}}\sim -\displaystyle \frac{4{ac}}{3\kappa \rho {\rm{\Delta }}r}({T}_{B}-T){T}^{3}\end{eqnarray} \tag{ 16 }$$

into the flux tube. This drives a buoyant rise of the flux tube at a speed v_r^B, which is determined by

$$\begin{eqnarray}&&\rho {k}_{{\rm{B}}}{{Tv}}_{r}^{B}\displaystyle \frac{{dS}}{{dr}}=-\displaystyle \frac{{{dF}}_{\mathrm{rad}}}{{dr}}\sim \displaystyle \frac{\delta {F}_{\mathrm{rad}}}{{\rm{\Delta }}r}.\end{eqnarray} \tag{ 17 }$$

The background entropy gradient is usefully expressed in terms of the Brunt–Väisälä frequency

$$\begin{eqnarray}&&N=\sqrt{g\displaystyle \frac{\gamma -1}{\gamma }\displaystyle \frac{{dS}}{{dr}}},\end{eqnarray} \tag{ 18 }$$

so that

$$\begin{eqnarray}&&{v}_{r}^{B}\sim \left(\displaystyle \frac{{B}_{\phi }^{2}}{8\pi P}\right)\displaystyle \frac{{F}_{\mathrm{rad}}}{\rho {(\Delta rN)}^{2}}.\end{eqnarray} \tag{ 19 }$$

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The background flow of mass into the radiative core is

$$\begin{eqnarray}&&\displaystyle \frac{{{dM}}_{\mathrm{core}}}{{dt}}=4\pi {R}_{\mathrm{benv}}^{2}\rho ({R}_{\mathrm{benv}})\left(\displaystyle \frac{{{dR}}_{\mathrm{core}}}{{dt}}-{v}_{r}^{\mathrm{nuc}}\right)\sim \displaystyle \frac{L}{{\varepsilon }_{i}}.\end{eqnarray} \tag{ 20 }$$

After the star passes onto the thermally pulsating AGB, the relation between core growth rate and luminosity L only holds after averaging over pulsations. The nuclear energy released per unit mass of new core material is ${\varepsilon }_{1}=6.3X$ MeV/m_u during the first step of converting material with hydrogen mass fraction X to helium; and ${\varepsilon }_{2}={\varepsilon }_{1}+0.8$ MeV/m_u during double shell burning. The corresponding velocity is

$$\begin{eqnarray}&&{v}_{r}^{\mathrm{nuc}}\sim -\displaystyle \frac{{F}_{\mathrm{rad}}}{\rho {\varepsilon }_{i}}.\end{eqnarray} \tag{ 21 }$$

The ratio ${v}_{r}^{B}/| {v}_{r}^{\mathrm{nuc}}|$ is plotted in Figure 4 at various intervals during the RGB and AGB evolution of our $1\;{M}_{\odot }$ model. One observes that the most strongly magnetized parts of the tachocline will not be subducted if the magnetic pressure approaches the kinetic pressure at the base of the convective envelope. It should be kept in mind that this effect depends on a difference in magnetization: subduction of the less weakly magnetized parts of the tachocline must continue even where ${v}_{r}^{B}\gt | {v}_{r}^{\mathrm{nuc}}|$.

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Magnetic buoyancy has the effect of expanding the active dynamo region below ${R}_{\mathrm{benv}}$, by allowing a layer with strong $d{\rm{\Omega }}/{dr}$—the tachocline—to remain in contact with the convection zone.⁴ We can estimate the size of this region by comparing the buoyant rise time ${\rm{\Delta }}r/{v}_{r}^{B}(r)$ with the dynamo timescale ${P}_{\mathrm{dyn}}$ (set equal to $100{\tau }_{\mathrm{con}}$)

$$\begin{eqnarray}&&{\rm{\Delta }}r\equiv {R}_{\mathrm{benv}}-r={v}_{r}^{B}(r)\cdot {P}_{\mathrm{dyn}}.\end{eqnarray} \tag{ 22 }$$

To invert this expression, we fit ${v}_{r}^{B}({\rm{\Delta }}r)$ to a power law in ${\rm{\Delta }}r$, measuring it in two different radial zones.

Figure 5 shows the resulting buoyant shell thickness, in comparison with the local scale height, during the AGB phase of the $5\;{M}_{\odot }$ model. The buoyant shell thickness reaches $\sim 0.1{l}_{P}({R}_{\mathrm{benv}})$ before and during the first thermal pulse and then relaxes to a value about 10 times smaller.

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We now consider the minimal seed field that will trigger dynamo feedback in the tachocline, thereby facilitating the magnetic coupling of core and envelope. We suppose that this seed field is generated within the convective envelope and is deposited in the outer radiative core by the mass flow toward the burning shell(s).

This seed field must pass a certain threshold if linear winding in the tachocline is to generate a buoyantly unstable toroidal field. The time available for linear winding depends on the depth ${\rm{\Delta }}r$ below the radiative-convective boundary. Consider an unmagnetized core in which matter flowing across the convective boundary sustains an angular velocity profile ${\rm{\Omega }}={{\rm{\Omega }}}_{\mathrm{benv}}{(r/{R}_{\mathrm{benv}})}^{-2}$. Near the boundary, the toroidal magnetic field that is sourced by a seed radial field ${B}_{\mathrm{seed}}$ is ${B}_{\phi }=-2{B}_{\mathrm{seed}}{{\rm{\Omega }}}_{\mathrm{benv}}{\rm{\Delta }}r/{v}_{r}^{\mathrm{nuc}}$. Substituting this into Equation (19) and setting ${v}_{r}^{B}\gt | {v}_{r}^{\mathrm{nuc}}|$ gives

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{\mathrm{seed}}^{2}}{4\pi \rho ({R}_{\mathrm{benv}}){v}_{\mathrm{con}}^{2}}\gt \displaystyle \frac{{N}^{2}}{2{{\rm{\Omega }}}^{2}}\displaystyle \frac{P{| {v}_{r}^{\mathrm{nuc}}| }^{3}}{{v}_{\mathrm{con}}^{2}{F}_{\mathrm{rad}}}.\end{eqnarray} \tag{ 23 }$$

This minimal seed field for dynamo feedback turns out to be quite weak. Figure 6 shows ${B}_{\mathrm{seed}}/\sqrt{4\pi \rho ({R}_{\mathrm{benv}})}{v}_{\mathrm{con}}$ during the final AGB evolution of a $5\;{M}_{\odot }$ star, measured at a distance ${\rm{\Delta }}r={l}_{P}({R}_{\mathrm{benv}})/10$ below the core–envelope boundary. Much of the C/O mass deposition from double shell burning is concentrated around $t\sim 1.0716\times {10}^{8}$ yr, when this quantity is as small as $\sim {10}^{-9}$; it rises to $\sim {10}^{-4}$ during the phase of rapid thermal pulses.

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Hydrodynamic turbulence is strongly suppressed in the tachocline. The mean-field approach to dynamo action then suggests (see the review in Section 3.1) that dynamo activity should be easily sustained in the tachocline as long as the toroidal magnetic field can be rotated back into the poloidal direction. This effect is present in numerical simulations of a shear layer (Cline et al. 2003) and is conjectured to operate in a more global dynamo model where toroidal magnetic field formed in the tachocline diffuses into the convective envelope (Parker 1993). Estimates of the equipartition toroidal magnetic field and the large-scale Maxwell stress are given by Equations (11) and (12).

The zone of interest here is the tenuous mantle of radiative material that is sandwiched between the hydrogen-depleted core and the base of the convective envelope. Here only a small mass $\lesssim {10}^{-3}\;{M}_{\odot }$ lies within a scale height, and so we model the mass inflow through the mantle as quasi-steady,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{core}}=4\pi {r}^{2}| {v}_{r}| \rho (r).\end{eqnarray} \tag{ 26 }$$

The poloidal magnetic field is approximated as radial with flux function (7), and the toroidal magnetic field is expressed in terms of a radial magnetic twist as ${B}_{\phi }(r,\theta )=(\partial {\phi }_{B}/\partial r)r\mathrm{sin}\theta {B}_{r}$. Then helicity accumulates in the core at the rate

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathcal{H}}}{{dt}}=-2\pi \delta {v}_{r}\int d\theta {{\rm{\Phi }}}_{r}\displaystyle \frac{\partial {{\rm{\Phi }}}_{r}}{\partial \theta }\displaystyle \frac{\partial {\phi }_{B}}{\partial r},\end{eqnarray} \tag{ 27 }$$

where $\delta {v}_{r}$ is the flow velocity relative to the core–envelope boundary.

Now consider the torque imparted to a mass shell by the Maxwell stress ${B}_{r}{B}_{\phi }/4\pi$,

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}\left(\delta m\displaystyle \frac{2}{3}{r}^{2}\bar{{\rm{\Omega }}}\right)=\delta \left[\int {dS}\ r\mathrm{sin}\theta \displaystyle \frac{{B}_{r}{B}_{\phi }}{4\pi }\right].\end{eqnarray} \tag{ 28 }$$

Here the Lagrangian time derivative follows the radial flow at speed v_r. Since $d(\delta m)/{dt}=0$, one has, following Equation (7),

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}\left(\displaystyle \frac{2}{3}{r}^{2}\bar{{\rm{\Omega }}}\right)=\displaystyle \frac{1}{8\pi \rho {r}^{2}}\int d\theta \mathrm{sin}\theta {\left(\displaystyle \frac{\partial {{\rm{\Phi }}}_{r}}{\partial \theta }\right)}^{2}\displaystyle \frac{{\partial }^{2}{\phi }_{B}}{\partial {r}^{2}}.\end{eqnarray} \tag{ 29 }$$

Both hemispheres generally contribute in the same sense to the torque.

4.1.1. Helicity Sourced by Radial Inflow

We suppose that the magnetic field is strong enough that nearly solid rotation is maintained in the radiative layers, $| \partial {\rm{\Omega }}/\partial r| \ll \bar{{\rm{\Omega }}}/r$. Each mass shell starts with a mean angular frequency ${{\rm{\Omega }}}_{\mathrm{benv}}$ that is imposed by the lower part of the convective envelope. The twist that is required to maintain a weak angular velocity gradient is found from Equation (29),

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}{\phi }_{B}}{\partial {r}^{2}}\sim -\displaystyle \frac{2{\dot{M}}_{\mathrm{core}}{{\rm{\Omega }}}_{\mathrm{benv}}r}{{({B}_{r}{r}^{2})}^{2}},\end{eqnarray} \tag{ 30 }$$

where the coefficient on the right-hand side corresponds to B_r independent of θ. From Equation (69) one has

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\phi }_{B}}{\partial r}\sim \displaystyle \frac{r}{| {v}_{r}| }\displaystyle \frac{\partial {\rm{\Omega }}}{\partial r}.\end{eqnarray} \tag{ 31 }$$

Nearly solid rotation is maintained if

$$\begin{eqnarray}&&{v}_{A,r}^{2}\equiv \displaystyle \frac{{B}_{r}^{2}}{4\pi \rho }\gg {v}_{r}^{2}.\end{eqnarray} \tag{ 32 }$$

Now consider the magnetic helicity that is advected into the growing radiative core. We evaluate the magnetic twist in Equation (27) from (30), obtaining

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathcal{H}}}{{dt}}=f({\rm{\Delta }}{M}_{\mathrm{core}})\cdot \pi {\dot{M}}_{\mathrm{core}}{{\rm{\Omega }}}_{\mathrm{benv}}{R}_{\mathrm{benv}}^{2}\delta {v}_{r}({R}_{\mathrm{benv}})\end{eqnarray} \tag{ 33 }$$

at $r={R}_{\mathrm{benv}}$.

The factor f, with indeterminate sign but magnitude $| f| \lt 1$, represents the cancellation between hemispheres. We discuss its origin in Section 4.1.3.

4.1.2. Magnetic Twist Compensating a Latitude-dependent Convective Torque

A greater Maxwell stress results from the application of convective stresses to the outer radiative core. A convection zone generally sustains latitudinal gradients in rotation. Part of the differential Reynolds stress is transmitted through the tachocline to the core by a dynamo-generated magnetic field (Equation (12)). As a working estimate, we use

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{r}{B}_{\phi }}{4\pi }={\varepsilon }_{B}\rho ({R}_{\mathrm{benv}}){({{\rm{\Omega }}}_{\mathrm{benv}}{R}_{\mathrm{benv}})}^{2},\end{eqnarray} \tag{ 34 }$$

with ${\varepsilon }_{B}\sim {10}^{-3}$, corresponding to a dynamo period ${P}_{\mathrm{dyn}}\sim 10{[{\rm{\Delta }}{\rm{\Omega }}({R}_{\mathrm{benv}})]}^{-1}$ and a pole-equator offset ${\rm{\Delta }}{\rm{\Omega }}({R}_{\mathrm{benv}})\sim {{\rm{\Omega }}}_{\mathrm{benv}}$.

Given that the deeper parts of the radiative core are coupled magnetically to the tachocline, uniform rotation can be maintained in the outer core only if this stress is divergence-free below the tachocline. The corresponding magnetic helicity flow into the core through the advection of this twisted field is

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathcal{H}}}{{dt}}=f({\rm{\Delta }}{M}_{\mathrm{core}})\cdot \pi {\varepsilon }_{B}{\dot{M}}_{\mathrm{core}}{R}_{\mathrm{benv}}{({{\rm{\Omega }}}_{\mathrm{benv}}{R}_{\mathrm{benv}})}^{2}.\end{eqnarray} \tag{ 35 }$$

The contributions to ${\mathcal{H}}$ from Equations (33) and (35) are labeled in the figures as "inflow" and "${\rm{\Omega }}(\theta )$," respectively.

4.1.3. Cancellation between Hemispheres

One feature of Equations (33) and (35) stands out: the growth of ${\mathcal{H}}$ is independent of the seed radial magnetic field, as long as it is strong enough to satisfy the inequality (32). This means that the contribution to ${\boldsymbol{A}}\cdot {\boldsymbol{B}}$ in one hemisphere is not strongly affected by fluctuations in the magnitude and sign of the poloidal magnetic field that is supplied to the radiative layer by a dynamo operating near the convective boundary.

A net mass ${\rm{\Delta }}{M}_{\mathrm{core}}$ flows through the convective-radiative boundary over the duration of dynamo activity. In the case of an AGB star, this is approximately the increase in the mass of the C/O core due to double shell burning. Note that ${\rm{\Delta }}{M}_{\mathrm{core}}$ is generally much larger than the mass of the rarefied core mantle ($\sim {10}^{-3}\;{M}_{\odot }$). The low mass of the mantle imposes a short-timescale cutoff to fluctuations in the summed ${\mathcal{H}}$ from the two hemispheres. We express this cutoff in terms of the mass within a scale height below the core–envelope boundary,

$$\begin{eqnarray}&&{\rm{\Delta }}{M}_{\mathrm{min}}\sim {\left.\displaystyle \frac{4\pi {{Pr}}^{4}}{{{GM}}_{\mathrm{core}}}\right|}_{{R}_{\mathrm{benv}}}.\end{eqnarray} \tag{ 36 }$$

Then we take

$$\begin{eqnarray}&&f({\rm{\Delta }}{M}_{\mathrm{core}})\sim {\left(\displaystyle \frac{{\rm{\Delta }}{M}_{\mathrm{min}}}{{\rm{\Delta }}{M}_{\mathrm{core}}}\right)}^{1/2}.\end{eqnarray} \tag{ 37 }$$

First consider the ${M}_{\mathrm{ZAMS}}=1\;{M}_{\odot }$ model. The angular momentum profile of the star is evolved in post-processing using the method of Section 2.1 and Paper I. Companion planets with a range of masses (Earth, Neptune, and Jupiter) are placed at an initial semimajor axis ${a}_{i}=[0.5,0.75,1,1.5,2]$ AU and with a range of orbital eccentricities.

Magnetic fields generated during the RGB and AGB phases both contribute to the remnant WD field, but at different depths in the star (see Figure 1). A net mass $\sim 0.135\;{M}_{\odot }$ passes through the core–envelope boundary while conditions are favorable for magnetic field growth.⁵

Figure 7 shows the helicity that accumulates in the brief period ${t}_{\mathrm{drift}}={{\ell }}_{P}/| {v}_{r}^{\mathrm{nuc}}|$ when stellar material settles through the outer scale height of the radiative core. The dominant contribution comes from the "${\rm{\Omega }}(\theta )$" channel.

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In the $5\;{M}_{\odot }$ star, only helicity accumulated on the AGB contributes directly to the final WD magnetic field (Figure 2). During this phase, $\sim 0.044\;{M}_{\odot }$ passes through the active dynamo region (given our normalization ${\eta }_{B}=0.05$ of the Blöcker mass-loss formula). We see in Figure 8 that the initial phase of the dynamo, which occurs during the first thermal pulse, contributes the majority of ${\mathcal{H}}$. In the 5 ${M}_{\odot }$ model, we simplified the treatment of the thermal pulses by linearly interpolating all of the relevant quantities between the local minima in ${R}_{\mathrm{benv}}$ after each He shell flash, so as to focus on the mean growth of core mass and ${\mathcal{H}}$. The stellar model considered in Figure 8 has no planetary companion.

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The resulting WD dipolar field is shown in Figure 9 for both the $1\;{M}_{\odot }$ and $5\;{M}_{\odot }$ models, using the mass–radius relation of Zapolsky & Salpeter (1969). A strong magnetic field in the $0.55\;{M}_{\odot }$ WD remnant of the $1\;{M}_{\odot }$ progenitor depends on the injection of angular momentum from a planetary companion (as considered here), or a tidal interaction with a stellar companion. We find that ${B}_{\mathrm{wd}}$ is an increasing function of the angular momentum absorbed; the various points correspond to different realizations of the orbital evolution.

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Dipole fields approaching 10⁸ G are achievable through the $"{\rm{\Omega }}(\theta )^{\prime\prime}$ channel, following the absorption of a Jupiter-mass planet. Note that an Earth-mass planet does not signicantly augment the angular momentum left over at the end of the MS spindown, and so this case corresponds closely to an isolated star that does not interact with planets.

Figure 9 focuses on those runs that end in the engulfment of the planet. However, there are cases in which the planet is pushed out into a large orbit owing to the convective quadrupole. The planet therefore gains angular momentum from the star, which in turn causes the star to spin in the opposite direction. In some cases this leads to the development of a dynamo as well. We ignore these particular cases.

An external source of angular momentum injection is not required to sustain a dynamo in the $5\;{M}_{\odot }$ progenitor. Nonetheless, ${B}_{\mathrm{wd}}$ depends on the C/O mass that accumulates during double shell burning and therefore on the mass-loss rate during the expulsion of the envelope.

The effect of adding a relatively massive companion to the $5\;{M}_{\odot }$ model is considered in Figure 10. The absorption of a $3\;{M}_{J}$ planet significantly augments the remnant dipole field in the $0.87\;{M}_{\odot }$ WD. We see a steady increase in the remnant field as the companion mass increases to the brown dwarf range, or even to $\sim 0.1\;{M}_{\odot }$, for the angular velocity profile considered here ($\alpha =3/2$). For comparison we show the magnetic fields obtained by a shallower inner rotation profile ($\alpha =1$), which may be applicable depending on the strength of the Coriolis back-reaction. In this case an increase in angular momentum results in a decrease of remnant magnetic field, caused by a decrease in ${\mathrm{Co}}_{\mathrm{benv}}$ due to the shallow ${\tau }_{\mathrm{conv}}(r)$ profile.

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The WD spin period that results from core–envelope decoupling at ${\mathrm{Co}}_{\mathrm{benv},\mathrm{crit}}=0.1$ and 0.3 in the $1\;{M}_{\odot }$ model is shown in Figure 13. For the more conservative estimate ${\mathrm{Co}}_{\mathrm{benv},\mathrm{crit}}=0.1$, we find that ${P}_{\mathrm{wd}}$ ranges over 0.5–1.5 days.

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In this situation, where the timing of the transition to weak magnetization in the envelope depends on ${J}_{\star }$, ${P}_{\mathrm{wd}}$ depends on the initial orbital angular momentum of the planetary companion. A lower total angular momentum results in faster decoupling and therefore a larger trapped core angular momentum: the core rotation frequency at decoupling is tied to the convective timescale in the inner envelope and to the structure of the star.

This behavior is explained in Figure 14, which shows the minimum angular momentum required to maintain ${\mathrm{Co}}_{\mathrm{benv}}=0.1$ in the $1\;{M}_{\odot }$ model near the tip of the AGB. This is compared with the actual angular momentum evolution that results from the absorption of a Jupiter, Neptune, or Earth, initially orbiting at a_i = 1 AU with eccentricity e_i = 0.

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We observe a significant difference in the remnant angular momentum in two models that maintain a tight core–envelope coupling through to the post-AGB phase: the $5\;{M}_{\odot }$ model, and the $1\;{M}_{\odot }$ model that absorbs a Jupiter-mass planet. In the first case, downward angular momentum pumping in the convective envelope allows ${J}_{\mathrm{core}}$ to remain relatively constant during the ejection of the envelope, and only modest spindown is caused by a wind from the surface of the post-AGB star (Figure 12). The spin of the WD remnant is directly proportional to the initial spin period of the star, except in the case of the absorption of a relatively massive ($\gtrsim 3{M}_{J}$) companion. An equatorial rotation speed 50 km s⁻¹ on the zero-age MS results in ${P}_{\mathrm{wd}}\simeq 0.6$ days.

The initial contraction of the envelope also does not cause much of a reduction in ${J}_{\mathrm{core}}$ in the $1\;{M}_{\odot }$ model; however, we observe a much stronger decrease following the re-expansion and re-contraction of the envelope that is triggered by a late helium shell instability (Figure 11). This means that the WD remnant of the $5\;{M}_{\odot }$ star has a spin period ${P}_{\mathrm{wd}}\sim 15$ hr, as compared with ∼2 yr for the remnant of the $1\;{M}_{\odot }$ star.

To understand how such a dramatic difference in final WD spin could arise, we consider a simplified analytic model of mass and angular momentum loss from an AGB star. The envelope of the model star has uniform specific angular momentum, the core has a fixed radius ${R}_{\mathrm{benv}}$, and the core rotates as a solid body with angular velocity ${{\rm{\Omega }}}_{\mathrm{core}}={{\rm{\Omega }}}_{\mathrm{benv}}$. Mass is lost from the outer boundary of the star and is also exchanged between core and envelope,

$$\begin{eqnarray}&&\displaystyle \frac{{{dM}}_{\mathrm{env}}}{{dt}}=\displaystyle \frac{{{dM}}_{\star }}{{dt}}-\displaystyle \frac{{{dM}}_{\mathrm{benv}}}{{dt}}.\end{eqnarray} \tag{ 38 }$$

Here ${M}_{\mathrm{env}}$ is the mass of the convective H-rich envelope, and ${M}_{\mathrm{benv}}$ is the mass of (mainly radiative) material inside the base of the envelope. The moment of inertia of the core is initially a small fraction of the total effective moment of inertia

$$\begin{eqnarray}&&{I}_{\mathrm{eff}}=\displaystyle \frac{{J}_{\star }}{{{\rm{\Omega }}}_{\mathrm{benv}}}=\displaystyle \frac{2}{3}{M}_{\mathrm{env}}{R}_{\mathrm{benv}}^{2}+{I}_{\mathrm{core}},\end{eqnarray} \tag{ 39 }$$

where we parameterize ${I}_{\mathrm{core}}=(2{\epsilon }_{\mathrm{core}}/3){M}_{\mathrm{benv}}{R}_{\mathrm{benv}}^{2}$.

The coefficient ${\epsilon }_{\mathrm{core}}$ is very small, but a key consideration is its relative size compared with the fraction ${M}_{\mathrm{env}}/{M}_{\star }$ of the stellar mass that is contained in the envelope. We find that ${\epsilon }_{\mathrm{core}}$ ranges over $\sim 3\times {10}^{-5}-{10}^{-3}$ during the superwind and post-AGB phases. The envelope of the $1\;{M}_{\odot }$ model completes its first contraction while it retains $\sim {10}^{-3}\;{M}_{\odot }$, but following its re-expansion and re-contraction, this mass has dropped to ${M}_{\mathrm{env}}\sim {10}^{-4}\;{M}_{\odot }$. In the $5\;{M}_{\odot }$ model, the envelope experiences a single contraction when its mass drops below $\sim {10}^{-4}\;{M}_{\odot }$. See Figure 15.

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Consider first the stage(s) when the envelope is still inflated and the core radius ${R}_{\mathrm{benv}}$ changes only slowly. Then

$$\begin{eqnarray}\displaystyle \frac{{{dJ}}_{\star }}{{dt}} & = & \displaystyle \frac{2}{3}{R}_{\star }^{2}{\rm{\Omega }}({R}_{\star })\displaystyle \frac{{{dM}}_{\star }}{{dt}}\\ & = & \displaystyle \frac{2}{3}{R}_{\mathrm{benv}}^{2}{{\rm{\Omega }}}_{\mathrm{benv}}\displaystyle \frac{{{dM}}_{\mathrm{env}}}{{dt}}+{I}_{\mathrm{eff}}\displaystyle \frac{d{{\rm{\Omega }}}_{\mathrm{benv}}}{{dt}}\end{eqnarray} \tag{ 40 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{1}{{{\rm{\Omega }}}_{\mathrm{benv}}}\displaystyle \frac{d{{\rm{\Omega }}}_{\mathrm{benv}}}{{dt}}=\displaystyle \frac{{{dM}}_{\mathrm{benv}}/{dt}}{{M}_{\mathrm{env}}+{\epsilon }_{\mathrm{core}}{M}_{\mathrm{benv}}}.\end{eqnarray} \tag{ 41 }$$

Changes in the radius of the star do not enter into these expressions as a result of the uniform specific angular momentum profile in the envelope.

The core mass increases relatively slowly compared with the rapid drop in ${M}_{\mathrm{env}}$ during the peak of the superwind. So ${{\rm{\Omega }}}_{\mathrm{benv}}$ increases only slightly up to the contraction of the envelope, where it takes the value ${{\rm{\Omega }}}_{\mathrm{benv},\mathrm{col}}$. Beyond that point, the entire star rotates nearly as a solid body. A wind from its surface, which carries away a mass ${\rm{\Delta }}{M}_{\star }$, causes a net spindown

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{benv},\mathrm{col}}^{\prime }}{{{\rm{\Omega }}}_{\mathrm{benv},\mathrm{col}}}\simeq \mathrm{exp}\left[-\displaystyle \frac{1}{{\epsilon }_{\mathrm{core}}}\displaystyle \frac{{\rm{\Delta }}{M}_{\star }}{{M}_{\star }}\right].\end{eqnarray} \tag{ 42 }$$

This works out to a factor of ∼0.2 decrease in ${{\rm{\Omega }}}_{\mathrm{benv}}$ in the $1\;{M}_{\odot }$ model at the onset of the late helium shell flash.

The key step in the dramatic spindown of the post-AGB star involves the re-expansion of the envelope. This occurs by the transfer of $\sim {10}^{-3}\;{M}_{\odot }$ of hydrogen-rich material from the radiative part of the star into a rejuvenated convective layer, all occuring at nearly constant total angular momentum. Now the core rotation rate (still equal to the rotation rate at the base of the envelope owing to a tight magnetic coupling) decreases to

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{benv}}}{{{\rm{\Omega }}}_{\mathrm{benv},\mathrm{col}}^{\prime }}=\displaystyle \frac{{\epsilon }_{\mathrm{core}}{M}_{\mathrm{benv}}}{{M}_{\mathrm{env}}+{\epsilon }_{\mathrm{core}}{M}_{\mathrm{benv}}}\sim \displaystyle \frac{{\epsilon }_{\mathrm{core}}{M}_{\mathrm{benv}}}{{M}_{\mathrm{env}}}\sim \displaystyle \frac{1}{30}.\end{eqnarray} \tag{ 43 }$$

(Although the star contracts by a factor ∼10 in between the initial envelope contraction and the late helium shell flash, ${\epsilon }_{\mathrm{core}}$ returns to its pre-contraction value after the re-expansion of the envelope.) The rotation rate of the core remains nearly constant at the value (43), as most of the rejuvenated envelope is expelled.

The net spindown from the initial contraction of the envelope works out to ${{\rm{\Omega }}}_{\mathrm{benv}}/{{\rm{\Omega }}}_{\mathrm{benv},\mathrm{col}}\sim 1/150$. This behavior is neatly encapsulated by the trend of ${J}_{\star }$ and ${J}_{\mathrm{core}}$ with the mass ${M}_{\star }-{M}_{\mathrm{wd}}$ remaining to be expelled (Figure 15).

Here we evaluate the minimal magnetic flux that must thread the core–envelope boundary in order for (i) the outer radiative core to remain in solid rotation and (ii) the rotation of the inner core to remain coupled to the outer core and inner envelope. A related (although less quantitative) estimate has been made by Spruit & Phinney (1998) for the central cores of massive stars during the later stages of nuclear burning.

First, consider the outer core. Although it contains a small fraction of the core mass, it can contribute significantly to the moment of inertia. We define a characteristic magnetic flux threading a single hemisphere by setting ${B}_{r}/\sqrt{4\pi \rho ({R}_{\mathrm{benv}})}=| {v}_{r}-{{dR}}_{\mathrm{benv}}/{dt}| :$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{solid}}=\pi {R}_{\mathrm{benv}}^{2}| {v}_{r}-{{dR}}_{\mathrm{benv}}/{dt}| \sqrt{4\pi \rho ({R}_{\mathrm{benv}})}.\end{eqnarray} \tag{ 44 }$$

Next, consider changes in the angular momentum of the entire core, which are sensitive to the evolving rotation profile of the envelope. The core angular momentum responds to the applied Maxwell stress ${B}_{r}{B}_{\phi }/4\pi$ according to

$$\begin{eqnarray}&&{I}_{\mathrm{core}}\displaystyle \frac{d{{\rm{\Omega }}}_{\mathrm{core}}}{{dt}}\sim \displaystyle \int \displaystyle \frac{{B}_{r}{B}_{\phi }}{4\pi }{R}_{\mathrm{benv}}\mathrm{sin}\theta \cdot 2\pi {R}_{\mathrm{benv}}^{2}\mathrm{sin}\theta d\theta .\end{eqnarray} \tag{ 45 }$$

Taking a second time derivative, making use of the induction equation, we get

$$\begin{eqnarray}&&{I}_{\mathrm{core}}\displaystyle \frac{{d}^{2}{{\rm{\Omega }}}_{\mathrm{core}}}{{{dt}}^{2}}\sim \displaystyle \frac{1}{2}\int {B}_{r}^{2}\displaystyle \frac{\partial {\rm{\Omega }}}{\partial r}{R}_{\mathrm{benv}}^{4}{\mathrm{sin}}^{3}\theta d\theta .\end{eqnarray} \tag{ 46 }$$

Here the poloidal magnetic field is assumed to be purely radial, so that ${\partial }_{r}({r}^{2}{B}_{r})=0$, and the integral is evaluated at radius ${R}_{\mathrm{benv}}$. Integrating over θ then gives

$$\begin{eqnarray}&&{I}_{\mathrm{core}}\displaystyle \frac{{d}^{2}{{\rm{\Omega }}}_{\mathrm{core}}}{{{dt}}^{2}}\sim \displaystyle \frac{2}{3}{B}_{r}^{2}{R}_{\mathrm{benv}}^{4}\displaystyle \frac{\partial {\rm{\Omega }}}{\partial r}.\end{eqnarray} \tag{ 47 }$$

We take the rotation rate to vary according to

$$\begin{eqnarray}&&{\rm{\Omega }}(r,t)\sim {{\rm{\Omega }}}_{\mathrm{benv}}{\left(\displaystyle \frac{{R}_{\mathrm{benv}}}{r}\right)}^{2}\mathrm{exp}\left[-\displaystyle \frac{t}{{\tau }_{{\rm{\Omega }}}}\right].\end{eqnarray} \tag{ 48 }$$

Then to avoid spindown on a timescale ${\tau }_{{\rm{\Omega }}}$, the magnetic flux must exceed

$$\begin{eqnarray}{{\rm{\Phi }}}_{\mathrm{couple}} & = & \pi {R}_{\mathrm{benv}}^{2}{B}_{r}\sim \sqrt{\displaystyle \frac{3}{4}}\pi \displaystyle \frac{{I}_{\mathrm{core}}^{1/2}{R}_{\mathrm{benv}}^{1/2}}{{\tau }_{{\rm{\Omega }}}}\\ & \simeq & 7.1\times {10}^{20}\;\mathrm{Mx}{\left(\displaystyle \frac{{I}_{\mathrm{core}}}{{10}^{-4}\;{M}_{\odot }\;{R}_{\odot }^{2}}\right)}^{1/2}\\ & & {\left(\displaystyle \frac{{R}_{\mathrm{benv}}}{{R}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{\tau }_{{\rm{\Omega }}}}{{10}^{3}\;\mathrm{years}}\right)}^{-1}.\end{eqnarray} \tag{ 49 }$$

We note that ${\tau }_{{\rm{\Omega }}}$ is controlled by the rate at which material is ejected from the envelope during the final superwind phase, and as a result ${{\rm{\Phi }}}_{\mathrm{couple}}$ can in principle be larger or smaller than ${{\rm{\Phi }}}_{\mathrm{solid}}$ (which is determined by the requirement that the core itself remain in solid rotation).

These estimates of the magnetic flux that will enforce a tight rotational coupling between core and envelope are compared in Figures 16 and 17 with the radial flux ${({\rm{\Delta }}{\mathcal{H}})}^{1/2}$ that accumulates at the core–envelope boundary. This increment ${\rm{\Delta }}{\mathcal{H}}$ is calculated using Equation (35), taking the timescale for helicity accumulation to be the minimum of the radial drift time ${{\ell }}_{P}/| {v}_{r}^{\mathrm{nuc}}|$ and ${\tau }_{{\rm{\Omega }}}$.

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The larger value of the dynamo-generated flux supports our choice of ${\mathrm{Co}}_{\mathrm{benv}}\lt 1$ for core–envelope decoupling. Recall also from Section 3.3 that a tiny poloidal magnetic field—corresponding to ${10}^{-4}\sqrt{4\pi \rho ({R}_{\mathrm{benv}})}{v}_{\mathrm{con}}$ during the later stages of the AGB—will seed a toroidal field in the tachocline that is strong enough to interact buoyantly with the layers above (see Figure 6).

Sensitivity to strength of angular-momentum pumping in giant envelope. The Kepler asteroseismological data give a calibration of the envelope rotation profile in subgiants—as investigated in detail in Paper I—but not in stars larger than $\sim 10\;{R}_{\odot }$. Here we have considered the consequences of uniform ${dJ}/{dM}$ in the slowly rotating parts of the envelope. This zone with $\mathrm{Co}\lt 1$ extends deep into the star near the tips of the RGB and AGB, unless the star interacts with a relatively massive binary companion. A rotation profile ${\rm{\Omega }}\propto {r}^{-2}$ in the outer envelope is supported by the numerical simulations of Brun & Palacios (2009).

The orbital angular momentum of a Jupiter can be compared with the minimum rotational angular momentum that will sustain a dynamo near the tip of the AGB. Starting the planet at a_i = 1 AU, the excess is about a factor of $\sim 30{({\mathrm{Co}}_{\mathrm{benv},\mathrm{crit}}/0.3)}^{-1}$ in our $1\;{M}_{\odot }$ model. The shallowest rotation profile that could maintain a minimal Coriolis parameter ${\mathrm{Co}}_{\mathrm{benv}}\sim 0.3$ in an envelope of depth ${R}_{\mathrm{benv}}/{R}_{\star }\sim 0.01$ is ${\rm{\Omega }}(r)\propto {r}^{-4/3}$. In this situation, the amplitude of the helicity flux into the core, which is proportional to ${{\rm{\Omega }}}_{\mathrm{benv}}^{2}$ (Equation (35)), would be reduced by a factor of $\sim {10}^{-3}$. The corresponding reduction in ${B}_{\mathrm{dipole}}$, by a factor of ∼0.03, would still allow for fields of order MG (see Figure 9), but not magnetic fields $\gtrsim 10$ MG.

A related consideration is the steepness of the angular velocity profile in the inner envelope, where $\mathrm{Co}\gtrsim 1$, and its effect on the upper range that is obtained for the WD magnetic field. In this paper we did not simply take the profile that was determined in Paper I from asteroseismic models of sub-giant and helium burning stars. During these relatively compact evolutionary phases, an inner rotation profile ${\rm{\Omega }}(r)\propto {r}^{-1}$ corresponds to ${\mathrm{Co}}_{\mathrm{benv}}\sim 10$–30. This profile is consistent with the inward pumping of angular momentum by deep convective plumes in an adiabatic envelope with gravity $g(r)\propto {r}^{-1}$. A lower inner Coriolis parameter would be maintained near the tips of the RGB and AGB if the same rotation profile were sustained there.

In fact, the rotation profile is expected to steepen to ${\rm{\Omega }}(r)\sim {r}^{-3/2}$ in the presence of a similar convective structure, owing to the contraction of the core and the steepening of the gravity profile. For this reason, we allow the inner index in Equation (4) to increase to $\alpha =3/2$ during the expanded phase where magnetic helicity growth is concentrated and the remnant WD spin angular momentum is determined. The net effect is to raise the upper envelope of the magnetic field distribution in massive WDs from ∼3 MG ($\alpha =1$) to 10–30 MG ($\alpha =3/2$). The limiting magnetization and spins of lower-mass WDs hardly change over this range of α.

What happens to any core magnetic field left over from the MS phase, when exposed to the turbulent motions in a slowly rotating convective envelope (${\mathrm{Co}}_{\mathrm{benv}}\;\ll \;1$)? Consider taking the magnetic flux threading a WD and spreading it across the core–envelope boundary of the progenitor during its AGB expansion. Even a MG magnetic field in the WD becomes dynamically insignificant, ${B}_{r}({R}_{\mathrm{benv}})\sim {10}^{-4}{(4\pi \rho {v}_{\mathrm{con}}^{2})}^{1/2}$. Such a weak poloidal magnetic field would rapidly diffuse over the boundary sphere, causing fluid elements that are threaded by radial fields of opposing signs to be mixed together. In this situation it is difficult to see how the material flow from the envelope into the core could maintain a large-scale poloidal magnetic field unless it were self-consistently maintained by a dynamo process. A numerical experiment studying the diffusion and reconnection of such a seed field across the spherical boundary between radiative and convective layers is feasible and would provide interesting results.

Relaxation of helical magnetic field to the "maximum-dipole" configuration. The surface dipole magnetic of the WD remnant has been estimated assuming that ${{\rm{\Phi }}}_{r}\sim {{\rm{\Phi }}}_{\phi }\sim {{\mathcal{H}}}^{1/2}$. Other magnetic configurations are possible, especially those with relatively stronger toroidal fields, corresponding to ${{\rm{\Phi }}}_{\phi }\gg {{\rm{\Phi }}}_{r}={({\mathcal{H}}/{{\rm{\Phi }}}_{\phi })}^{1/2}$ (Braithwaite 2009). Ohmic diffusion in WD stars with strong toroidal fields may cause some transient growth of the dipole field, as the magnetic field relaxes to a more isotropic configuration.

Dispersion in magnetic field strength in massive WDs. Although a greater fraction of massive WDs exhibit strong magnetic fields than do their $\sim 0.55-0.6\;{M}_{\odot }$ cousins, a majority do not. Intermediate-mass stars retain more angular momentum at the end of the MS, thereby facilitating a dynamo. This angular momentum can, nonetheless, be enhanced by the absorption of a massive planet, brown dwarf, or low-mass star, or a tidal interaction with a stellar companion. Other effects causing dispersion in the visible surface magnetic field include exchange between toroidal and poloidal magnetic fluxes and the time spent by the progenitor star during the thermally pulsating AGB phase.

Emergence of "frozen" magnetic field generated during the pre-MS phase (or during MS core convection)? The outer $0.3\;{M}_{\odot }$ of the $0.55\;{M}_{\odot }$ WD is processed through a deep convective envelope during the RGB and AGB phases. The proportion of this material that experiences rapidly rotating convection and develops a strong magnetic field depends on the initial placement of planets around the star. For example, if the closest Neptune or Jupiter orbits at a distance of ∼1 AU, then only the outer half of the processed layer is strongly magnetized.

The ohmic diffusion time from the base of this processed layer approaches a Hubble time, independent of the architecture of the planetary system. From this we conclude that the magnetization of all but the very oldest ∼0.55–$0.6\;{M}_{\odot }$ WDs is not significantly influenced by a "frozen" core magnetic field.

The assembly history is very different for the massive WD remnants of intermediate-mass stars. Now the bulk of the WD mass is processed during core H and He burning, which leaves unaltered the magnetic helicity stored in the stellar core. In the case of an isolated $\sim 0.9\;{M}_{\odot }$ WD, a core magnetic field could emerge ohmically through the outer $\sim 0.05-0.1\;{M}_{\odot }$ that is processed on the thermally pulsating AGB, but only with a delay of $\sim (2-5)\times {10}^{8}$ yr (depending on the normalization of mass loss during this phase). A growth in the incidence of magnetism in older WDs could therefore point to a contribution of such a "frozen" field to the visible surface field (but not exclusively so, given the competing possiblity of dipole field growth mediated by the decay of a toroidal field that is buried in a thin outer mass shell).

Implications of binary magnetic WDs. The incidence of strong magnetism (exceeding 10⁷ G and implying classification as a "polar") exceeds $\sim 30\%$ in short-period CVs with ages exceeding ∼1 Gyr (Gänsicke 2005). Our calculations suggest that these fields might decay significantly in more massive isolated WDs. Long-lived magnetism in old binary WDs with periods $\lt 2$ hr (below the "period gap") could be maintained by the accretion of $\gtrsim 0.1\;{M}_{\odot }$ from the companion, which would have the effect of pushing down the magnetized layers to a greater depth and a longer ohmic timescale.

The final contraction of an AGB envelope, beginning at a photospheric radius $\sim {10}^{2}\;{R}_{\odot }$, leaves behind a hydrogen layer of mass ${M}_{\mathrm{env},f}\sim {10}^{-4}\;{M}_{\odot }$. Mass loss slows down significantly as the star contracts inside a radius ${R}_{\mathrm{pAGB}}\sim {R}_{\odot }$ (Figure 21).

It is possible that the process driving mass loss is so anisotropic that the collapsing envelope carries with it a significant angular momentum, equal and opposite to that lost during the final stages of the superwind (Spruit 1998). An upper bound to the angular momentum thus added to the post-AGB star can be obtained as follows. The collapsed envelope will experience a bar instability and spread outward into a disk if its angular velocity exceeds $\sim 0.3{({{GM}}_{\mathrm{wd}}/{R}_{\mathrm{pAGB}}^{3})}^{1/2}$ at radius ${R}_{\mathrm{pAGB}}$. Any angular momentum in excess of ${J}_{\mathrm{max}}\sim 0.3{M}_{\mathrm{env},f}{({{GM}}_{\mathrm{wd}}{R}_{\mathrm{pAGB}})}^{1/2}$ is transported outward by viscous torques in the disk and is presumably lost in a wind.

Setting the envelope angular momentum to this limiting value, and assuming that it is shared with the rest of the star at radius ${R}_{\mathrm{pAGB}}$, we obtain a minimum spin period for the cold WD remnant,

$$\begin{eqnarray}&&{P}_{\mathrm{wd}}\geqslant \displaystyle \frac{2\pi {I}_{\mathrm{wd}}}{{J}_{\mathrm{max}}}=0.4{\left(\displaystyle \frac{{M}_{\mathrm{env},f}}{{10}^{-4}\;{M}_{\odot }}\right)}^{-1}{\left(\displaystyle \frac{{R}_{\mathrm{pAGB}}}{{R}_{\odot }}\right)}^{-1/2}\;\mathrm{hr}.\end{eqnarray} \tag{ 57 }$$

A comparison with Figure 13 shows that somewhat faster WD spins can be imparted by rotating envelope contraction than would result from a cutoff of the envelope dynamo before the final intense phase of mass loss. However, the fastest WD spins may depend on a merger between two WDs.

The formation of magnetic WDs in stellar binaries is considered by Tout et al. (2008). It is noted that the injection of angular momentum from a binary companion (either a low-mass MS star or a substellar companion) into a giant star will trigger strong differential rotation in its core and envelope. This differential rotation may spark the impulsive growth of the magnetic field in the core, further details of which have been examined by Wickramasinghe et al. (2014).

Nordhaus et al. (2011) consider the tidal shredding of a planetary companion close to the giant core. They argue that this material initially forms a disk in nearly Keplerian rotation about the core, in which a magnetic field is generated by the magnetorotational instability. The most promising application of this process is to isolated magnetic WDs, since it may be difficult for a planet and a stellar companion (later the mass donor in a CV system) to coexist on similar orbits.

The contribution made by such an impulsive dynamo to the net magnetic helicity stored in the stellar core can be compared with the longer-term effect examined in this paper. When the companion is tidally disrupted (either within the convective envelope or within the rarefied mantle surrounding the burning shells), the mass M₂ deposited would be re-distributed back through the convective envelope on the thermal timescale

$$\begin{eqnarray}&&{t}_{\mathrm{th}}\sim \displaystyle \frac{{{GM}}_{\mathrm{core}}{M}_{2}}{{R}_{\mathrm{tide}}{L}_{\mathrm{AGB}}}\sim 600\left(\displaystyle \frac{{M}_{2}}{0.1\;{M}_{\odot }}\right)\left(\displaystyle \frac{{R}_{\mathrm{tide}}}{{R}_{\odot }}\right)\;\mathrm{years}.\end{eqnarray} \tag{ 58 }$$

As envisaged by Nordhaus et al. (2011), a centrifugally supported disk may spread on a shorter timescale than (58), in which case the internal heat generated in the disk is advected around without being radiated back into the surrounding stellar envelope. Studies of advective accretion flows in other contexts (e.g., Blandford & Begelman 1999) suggest that a fraction of the disk material would be accreted deeper into the gravitational potential of the degenerate core, which may be enough to transiently increase its rotation.

Since the energy released by H burning easily suffices to unbind the ashes from the degenerate core, any such rapidly accreted material must remain H rich. On the longer thermal timescale, this nuclear energy can be radiated away, accompanied by a relaxation of the outer core to its equilibrium structure. The mass $\delta {M}_{\mathrm{nuc}}$ processed by H and He burning during that relaxation is small compared with the total deposited mass M₂. Given that a mass ${M}_{\mathrm{acc}}$ is accreted close to the burning radius ${R}_{\mathrm{burn}}\sim 0.1\;{R}_{\odot }$, one has

$$\begin{eqnarray}\delta {M}_{\mathrm{nuc}} & \sim & \displaystyle \frac{{{GM}}_{\mathrm{core}}}{{\varepsilon }_{\mathrm{nuc}}{R}_{\mathrm{burn}}}{M}_{\mathrm{acc}}\\ & \sim & 2\times {10}^{-3}{\left(\displaystyle \frac{{R}_{\mathrm{burn}}}{0.1\;{R}_{\odot }}\right)}^{-1}\left(\displaystyle \frac{{M}_{\mathrm{acc}}}{{M}_{2}}\right){M}_{2}.\end{eqnarray} \tag{ 59 }$$

The normalization is ∼10 times larger if H burning is extinguished, so that only the He burning shell remains active.

We see that a relatively small mass is incorporated into the C/O core before the star adjusts to a new equilibrium structure. Given that the magnetic coupling between core and envelope is sustained, most of the added angular momentum in this new equilibrium state will be stored in the outer envelope.

The large-scale Maxwell stress ${B}_{r}{B}_{\phi }/4\pi$ in the outer core may easily increase by a factor of $\sim {10}^{4}$ in the immediate aftermath of the tidal disruption, when the rotation frequency increases by $\gtrsim {10}^{2}$. However, as the envelope dredges up the injected angular momentum, we expect the Maxwell stress to adiabatically adjust downward, since little mass flows into the core. Such a downward adjustment could be avoided if the magnetic field lines became decoupled from the envelope, but that does not seem consistent with the rapid buoyant motion of such a strong magnetic field (see Equation (19)).

From this perspective, the impulsive growth of the magnetic field following the tidal disruption of a low-mass star or substellar companion may leave little permanent mark on the magnetization of the C/O core. A merger during the early post-MS expansion could induce some direct hydrodynamic mixing with the hydrogen-depleted core, but only when the accumulated helium mass was still well below the final WD mass.

We argue that the dominant effect of such a merger is, instead, to (i) push the envelope above the threshold for a gentler and more persistent dynamo process in low-mass giant stars, ${M}_{\mathrm{ZAMS}}\lesssim 1.3\;{M}_{\odot }$; and (ii) augment the inward flux of magnetic helicity in stars that remain rapidly rotating at the end of the MS.

The merger of two WDs (or of a WD with the core of an evolved companion) provides the main counterexample to these conclusions. Now hydrogen-depleted material in the more massive WD experiences direct hydrodynamic mixing with the tidally shredded remnants of the companion. The rate of such events appears to be relatively small and may therefore only accomodate the most strongly magnetized WDs.