Magnetic fields in non-convective regions of stars

Jonathan BraithwaiteArgelander Institut fur Astronomie

Universitat Bonn, Auf dem Hugel 71, 53121 Bonnemail: jonathan.braithwaite@gmail.com

http://www.astro.uni-bonn.de/~jonathan

Henk C. SpruitMax Planck Institut fur Astrophysik

Karl-Schwarzschild-Str. 1, 85741 Garchingemail: henk@mpa-garching.mpg.de

http://www.mpa-garching.mpg.de/~henk

(arXiv version 7 April 2017)We review the current state of knowl-edge of magnetic fields inside stars, con-centrating on recent developments con-cerning magnetic fields in stably strati-fied (zones of) stars, leaving out convec-tive dynamo theories and observations ofconvective envelopes. We include the ob-servational properties of A, B and O-typemain-sequence stars, which have radiativeenvelopes, and the fossil field model whichis normally invoked to explain the strongfields sometimes seen in these stars. Ob-servations seem to show that Ap-type sta-ble fields are excluded in stars with con-vective envelopes. Most stars contain bothradiative and convective zones, and thereare potentially important effects arisingfrom the interaction of magnetic fields atthe boundaries between them. Related tothis, we discuss whether the Sun couldharbour a magnetic field in its core. Re-cent developments regarding the variousconvective and radiative layers near thesurfaces of early-type stars and their ob-servational effects are examined. We lookat possible dynamo mechanisms that runon differential rotation rather than con-vection. Finally we turn to neutron starswith a discussion of the possible originsfor their magnetic fields.

1 Introduction

Interest in magnetic fields in the interiors ofstars, in spite of a lack of immediate observ-ability, is rapidly increasing. It is sparked byprogress in spectropolarimetric observations ofsurface magnetic fields as well as by asteroseis-mology and numerical magnetohydrodynamicsimulations.

An important incentive also comes from devel-opments in stellar evolution theory. Discrepan-cies between results and steadily improving ob-servations has led to a newly perceived need forevolution models ‘with magnetic fields’. At thesame time the demand for results from stellarevolution have increased for application outsidestellar physics itself. An example is the need forpredictable colors of stellar populations in cal-culations of galaxy evolution.

Key questions concern the rate of mixingof the products of nuclear burning, since stel-lar evolution is sensitive to the distribution ofthese products inside the star. The heavier nu-clei are normally produced later and deeper in-side the star. Outside of convective zones, thesenuclei reside stably in the gravitational poten-tial. Even weak mixing mechanisms in radia-tive zones, operating on long evolutionary timescales, can nevertheless change the distributionenough to affect critical stages in the evolu-tion of stars. Possible mechanisms include hy-

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drodynamic processes like extension of convec-tive regions into nominally stable zones (‘over-shooting’), shear instabilities due to differentialrotation, and large-scale circulations. Assump-tions about the effectiveness of such processesare made and tuned to minimise discrepanciesbetween computed evolution tracks and observa-tions. The presence of magnetic fields adds newmechanisms, some of which could compete withor suppress purely hydrodynamic processes.

As well as mixing chemical elements1, mag-netohydrodynamic processes in radiative zonesshould damp the differential rotation that is pro-duced by the evolution of a star. An indirectobservational clue has been the rotation rateof the end products of stellar evolution (Sec-tion 5.5). As the core of an evolving star con-tracts it tends to spin up. However, it is evi-dent from the slow rotation of stellar remnantsthat angular momentum is transferred outwardsto the envelope, and in many cases stellar rem-nants rotate much slower than can be explainedeven with the known hydrodynamic processes.Maxwell stresses are more effective in transport-ing angular momentum than hydrodynamic pro-cesses (they can even transport angular momen-tum across a vacuum). This leads to the study ofdynamo processes driven by differential rotationin stably stratified environments (Section 5.4).

Note that there is a difference regarding themixing of chemical elements. In purely hydrody-namic processes, the transport of angular mo-mentum and chemical elements are directly re-lated to each other, but this is not the casefor magnetohydrodynamic processes. For a givenrate of angular momentum transport, mixing bymagnetohydrodynamic processes is less effectivethan in the case of hydrodynamic processes.

Until recently, the Sun was the only star forwhich direct measurements of internal rotationwere available, made possible by helioseismol-ogy. For all other stars the only source of in-formation on angular momentum transport in-side stars was the rotation of their end products.This has changed dramatically with the astero-seismic detection of rotation-sensitive oscillationmodes in giants and subgiant stars by the Kepler

1 The term ‘chemical elements’ is used anomalouslyin astrophysics (including this review) to mean atomicspecies.

and CoRoT satellites. These data now providestringent tests for theories of angular momentumtransport in stars (Sections 5.5, 5.4).

Possible internal magnetic fields come in twodistinct kinds. One kind is time-dependent mag-netic fields created and maintained by some kindof dynamo process, running from some source offree energy. Dynamos in convective zones havebeen studied and reviewed extensively before;they are not covered in this review except fora discussion of subsurface convection in O stars(Section 6.5). Another obvious source of free en-ergy is differential rotation, and this could pro-duce a self-sustained small-scale magnetic fieldin a radiative zone. This would be the candidatefor transport of angular momentum and chemi-cal elements described above.

The other kind of internal magnetic field isfossil fields, remnants of the star formation pro-cess that have somehow survived in a stable con-figuration. The theory of such fields is discussedin Section 3.

A subset of intermediate-mass stars displaystrong magnetic fields, the chemically pecularAp and Bp stars; and some more massive starsdisplay similar fields. These are thought to besuch fossil fields. They used to be interpretedin terms of configurations resembling simpledipoles. With the improved observations of thepast decade a much larger range of configura-tions is found; this is reviewed in Section 2.Theory indicates that only a small fraction ofall imaginable magnetic equilibria in a star canbe stable as observed. Comparing these the-oretically allowed configurations with the sur-face fields actually observed in individual starsgives clues about the internal structure of thefields. Together with statistical information onobserved field strengths and configurations, thisholds the promise of telling us something aboutthe conditions under which the magnetic fieldsformed.

Though stably stratified throughout most oftheir interior, Ap stars still contain a small con-vective core. This raises the question to whatextent convection interacts with the fossil field,or whether a fossil field is compatible at all withthe presence of a convective zone somewhere inthe star. A related question is whether stars withconvective envelopes like the Sun might have fos-

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sil fields hidden in their stably stratified interi-ors. These questions are addressed in Section 6.

Also thought to be of fossil nature are themagnetic fields in neutron stars. As in upper-main-sequence stars, there is a puzzlingly enor-mous range in field strengths, spanning five or-ders of magnitude. There are two obvious waysto explain this range: either it is inherited fromthe progenitor stars, in which case one still needsto explain the range in birth magnetic proper-ties of main-sequence stars, or it is producedduring the birth of the neutron star. It is pos-sible that it is produced from the conversion ofenergy from differential rotation into magnetic,and that the same physics is at work during thebirth of main-sequence stars. These issues areaddressed in Section 7.1.

This review is organised as follows. In thenext section, we look at observations of mag-netic stars, with some focus on peculiarities thatmay hold clues on the origin of their fields.Among the main-sequence star these are theAp/Bp stars and the apparently non-magneticintermediate-mass stars, next are the massivestars, and the magnetic white dwarfs then arediscussed very briefly. Section 3 is a review ofthe theory of static ‘fossil’ magnetic fields in ra-diative zones, and in Section 4 we examine var-ious scenarios which could explain where thesefossil fields come from. In Sections 5 and 6 re-spectively we look at the interaction of mag-netic fields with differential rotation and withconvection. In Section 7 we move onto neutronstars: their observational properties as well aslikely theoretical explanations in terms of inter-nal magnetic field. Finally we summarise in Sec-tion 8.

This review goes into some depth in themagnetohydrodynamics of stars. The interestedreader may wish to look at some literatureon MHD, including the astrophysical context.The classic book by Roberts (1967) covers ba-sic MHD in general contexts but is out of print.More recent is the monograph by Spruit (2013),an introduction tailored specifically to astro-physicists and with an emphasis on physical in-tuition and visualisation rather than mathemat-ics. The books by Goedbloed & Poedts (2004)and Goedbloed et al. (2010) offer a more de-tailed look at various astrophysical contexts.

Also worth a look are the books by Choudhuri(1998) and Kulsrud (2005), which have a greateremphasis on plasma effects, i.e. not using the sin-gle fluid approximation.

Figure 1: Measured magnetic fields in a sample ofAp stars in which either no magnetic field had yetbeen detected or in which there had been an ambigu-ous or borderline detection. In this study the detec-tion limit is only a few gauss; every star in the sam-ple is found to have a magnetic field much strongerthan this. The dashed line represents the cutoff at300 G. This result confirms convincingly that allAp-type chemically peculiar stars have strong mag-netic fields. In contrast, other A stars have neverbeen found to have any magnetic field above a fewgauss; there is a clear bimodality. From Auriere etal. (2007).

2 Observed Properties of Mag-netic Stars

The observational techniques used for measur-ing magnetic fields on the surface of stars havebeen reviewed by Donati & Landstreet (2009)and Mathys (2012). In almost all types of mag-netic star the Zeeman effect is used to detect themagnetic field. (For an excellent introduction ofthe Zeeman effect and its detection in stars seeLandstreet (2011).)

The most reliable observations of magneticstars use ‘full Stokes’ spectropolarimeters thatrecord the complete information contained inlinear and circular polarization. The full setof polarization components of a single suitable

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spectral line is sufficient in principle to deter-mine the strength of the magnetic field and itsorientation with respect to the line of sight. TheV-signal, which gives information about the line-of-sight component of the magnetic field, is eas-ier to detect than Q and U because it is anti-symmetric with respect to the center of the lineand less sensitive to instrumental polarization.

To increase the signal-to-noise ratio of thesemeasurements, many metal lines can be com-bined together to give a weighted mean of theStokes I and V line profiles in a procedure knownas Least Squares Deconvolution (LSD; Donatiet al., 1997). Since this technique was intro-duced, the detection limit has dropped signif-icantly. What this gives us is a disc-averagedline-of sight component of the magnetic field, theso-called mean longitudinal field Bz; for manystars this is the only quantity that can be re-liably measured. In other stars though it hasalso been possible to get extra information fromStokes Q and U. If the star is observed at severalrotational epochs one can then construct a sim-ple model of the magnetic field on the surface,e.g., dipole + quadrupole, and work backwardsto find the best-fitting parameters of the model.

2.1 Ap stars

In this section we review the Ap stars, the spec-troscopically ’peculiar’ intermediate-mass main-sequence stars between about B8 and F0, theirspectra showing very unusual abundances of theelements.

It has gradually become clear that there isa bimodality in the population of intermediate-mass stars (1.5 to 6M), namely that all starsclassified as Ap/Bp (with exception of the so-called mercury-manganese (HgMn) stars) hostlarge-scale magnetic fields with mean longitudi-nal fields between around 200 G and 30 kG, andthat the rest of the population lack magneticfields above the detection limit of a few gauss(Auriere et al., 2007; Figure 1). Ap stars accountfor a few percent of the A star population.

Still, this leaves a factor of 100 in field strengthto be explained by models for the origin of Apstar fields. A similar problem exists in the mag-netic white dwarfs, where field strengths rangefrom < 104 to almost 109 G, and in pulsars

Figure 2: The fraction of intermediate-mass starswithin 100pc in which a magnetic field has beendetected; the sample is relatively complete. FromPower et al. (2007).

(∼ 1010−1015 G). This problem may well reflecta basic property of the formation process of fos-sil fields (see also Section 4). There are variousobservational clues to the origin of this bimodal-ity in magnetic properties amongst A and lateB stars. For instance, there is a strong corre-lation between mass and the magnetic fractionof the population. Power et al. (2007) exam-ined a volume-limited sample of intermediate-mass stars – all stars within 100 pc of the Sun– finding that the magnetic fraction of the pop-ulation increases from less than 1% at 1.5Mto & 20% at 3.5M; see Figure 2. The totalmagnetic fraction in the sample is only 1.7%.The Ap phenomenon disappears completely atmasses below 1.5M (around F0), which coin-cidences with the onset of efficient convectionin the envelope (e.g., Landstreet, 1991). Otherclues come from the rotation and binarity – seebelow.

In some strongly-magnetised, slowly-rotatingAp stars and most magnetic white dwarfs theZeeman splitting is greater than the line width,in which case it is possible to measure the Zee-man splitting directly in Stokes I (intensity),without having to use polarimetry. This gives anaverage of the field strength over the visible disc,the so-called mean field modulus Bs. Now, if astar’s magnetic field is dominated by small-scalestructure we will clearly expect that Bs |Bz|,

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since the line-of-sight component from variouspatches on the surface will cancel each other insome kind of statistical

√N manner. For the Ap

stars in which Bs is measurable, we do not findthis – an important result showing that small-scale structure, if present, is not dominant.

Ap stars display a large variety of field geome-tries. In many stars, a good fit to the data canbe achieved by assuming the simplest geometryof all, i.e., a dipole field, at the surface, whichis inclined to the rotation axis at some angle.In other stars this produces poor results and amore complicated geometry produces better re-sults, for instance dipole + quadrupole.

Improved observations have made it possibleto combine the Zeeman effect with the Dopplereffect from the rotation of the star to get, in ef-fect, some spatial resolution on the surface of thestar, without having to make prior assumptionsof this kind. Piskunov & Kochukhov (2003) havedeveloped a technique called Magnetic-Dopplerimaging and have used it to make some impres-sive maps of the magnetic field on a number ofstars, such as 53 Cam (Kochukhov et al., 2004),α2CVn (Kochukhov et al., 2002; Kochukhov &Wade, 2010) and HD 37776 (Kochukhov et al.,2011a). Two examples are shown in Figure 3. Asimilar technique called Zeeman–Doppler imag-ing, developed by Donati & Petit (see, e.g., Do-nati, 2001) has been used to make magnetic im-ages of cool stars, e.g., Petit et al. (2004), as wellas some hot stars, e.g., Donati et al. (2009) andFigure 7. Using these techniques, some rathercomplicated geometries2 have been found whichappear to indicate the presence of meanderingflux tubes just below the stellar surface.

2.1.1 Chemical peculiarities

Interesting and unique to intermediate-massstars are processes near the surface: gravita-tional settling and radiative levitation, whichcause separation of chemical elements in theatmospheres of the stars and result in a vari-ety of observed chemical abundance phenomena(Michaud, 1970). Ap/Bp stars are defined as a

2The common useage of term ‘topology’ in this con-text is sloppy. Meant is distribution on the star’s surface.Topology is by definition a global property of the entirefield configuration; nothing can be inferred about it fromobservations of the stellar surface alone.

Figure 3: The magnetic fields of α2 CVn (upperpanel) and 53 Cam (lower panel), viewed at fiverotational phases. The upper rows in each panelshow field strength and the lower rows the direction.Clearly, whilst α2 CVn has an almost perfect dipolefield, the magnetic field of 53 Cam has a much morecomplex geometry. From Kochukhov et al. (2002,2004).

class showing peculiar (hence the ‘p’ in ‘Ap star’)abundances of rare earths and some lighter ele-ments such as silicon, as well as inhomogeneitiesof these elements on the surface which show cor-relations with the magnetic field structure, al-beit not the same kind of correlation in all stars.There is a strong correlation between the Ap/Bpphenomenon and strong magnetic fields, withthe apparent exception of the subclass of theHgMn stars (Kochukhov et al., 2011b).

The origin of this phenomenon is inextricablylinked to the presence and location of surface-and subsurface convection layers resulting fromopacity bumps associated with helium and hy-drogen ionization (see also Section 6.5). Convec-tion obviously washes out the effects of any gen-tle separation processes, and a sufficiently strongmagnetic field is expected to disrupt convection.A magnetic field & 200 G is above equipartitionwith the thermal pressure at the photosphere,and should thus inhibit convection (Gough &Tayler, 1966; Moss & Tayler, 1969; Mestel, 1970,see also Section 6.5.2). Indeed there is observa-tional evidence for this in the form of a reduc-tion in microturbulence velocities in Ap stars (D.

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Shulyak, 2013, priv. comm.). In spectral type theAp/Bp phenomenon disappears around F0, cor-responding to the onset of efficient convectionat the surface, and at B8, corresponding to theappearance of stronger subsurface convection.

The chemical peculiarities apparently developafter the magnetic field is already in place,appearing at some stage during the pre-main-sequence (Folsom et al. , 2013a). Note thoughthat chemical peculiarities are not restricted tothe magnetic stars; amongst the other A stars,various other types of chemical peculiarity areseen, for instance in the slowly-rotating Am stars(of which Sirius is the best-known specimen),mercury-manganese stars and λ Bootis stars. SeeTurcotte (2003) for a review of these ‘skin dis-eases’.

2.1.2 Rotation

It has long been known that most magneticA stars rotate slowly compared to the non-magnetic stars (see e.g. Abt & Morrell, 1995).Whilst the non-magnetic A stars are generallyfast rotators, with rotation periods of a fewhours to a day, most Ap stars have periods be-tween one and ten days, and some have periodsmuch greater, with around 10% of Ap stars hav-ing periods above 100 days (Mathys, 2008). Theslowest rotation periods are of order decades andin several cases only a lower limit can be stated.Note that whilst the rotation periods of the non-magnetic stars are estimated statistically fromv sin i, those of magnetic stars can be measureddirectly from the periodicity in the Zeeman sig-nal, since the magnetic field is never perfectlysymmetrical about the rotation axis.

Some intriguing correlations between the rota-tion period and magnetic properties of Ap starshave been found. For instance, Mathys (2008)finds in a sample of slowly rotating stars thatthose with P > 100 days lack fields in excess of7.5 kG. A recent compilation is shown in Fig.4. In addition, Landstreet & Mathys (2000) findthat the slower rotators (P > 25 days) are morelikely to have closely aligned magnetic and rota-tion axes.

The slow rotation of Ap stars holds only in avery broad sense; rapidly rotating examples likeCU Virginis (0.5 day) exist as well. Any given

Figure 4: Observed average of the mean magneticfield modulus against rotation period. Dots: starswith known Prot; triangles: only a lower limit of Protis known. Open symbols : stars for which existingmeasurements do not cover the whole rotation cycle.From Mathys (2016)

explanation for the slow average rotation maywell miss the most important clue: the astonish-ingly large range in rotation periods, of at leastfour orders of magnitude.

2.1.3 Binarity

The binary fraction amongst the magnetic starsis lower than in the non-magnetic stars (Abt &Snowden, 1973; Gerbaldi et al., 1985; Carrier etal., 2002; Folsom et al., 2013b). There is appar-ently a complete lack of Ap stars in binaries withperiods of less than about 3 days, except for oneknown example (HD 200405) with a period of1.6 days. This speaks in favour of more than oneof the formation scenarios (see Section 4).

2.1.4 Ages

Pre-MS stars are known as T Tauri stars if theyare later than spectral type F5 (log T ≈ 3.8)and Herbig Ae-Be (HAeBe) if they are earlier.We now know of over 100 HAeBe stars (e.g.,Herbig & Bell, 1988; Vieira et al., 2003). Starsbetween about 1.5 and 4M leave the birthline as fully-convective T Tauri stars. Eventu-

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ally they develop a radiative core, at which timethey stop moving downwards on the HR dia-gram and move instead to the left on what isknown as the Henyey track; the convective en-velope shrinks and they become HAeBe stars.More massive stars leave the birthline as HAeBestars. Depending on the local and/or accretionconditions as well as on the mass, these starsmay or may not become visible before they reachthe MS; the most massive HAeBe stars observedare around 20M.

It is clear now that some fraction – compara-ble to the fraction amongst main-sequence A andB stars – of HAeBe stars are magnetic. Wade etal. (2005) presented the first detections of mag-netic fields in HAeBe stars, and Alecian et al.(2013b,c) present the results from a survey of 70Herbig Ae-Be stars: see Figure 5.

Figure 5:Magnetic (red squares) and non-magnetic(black circles) HAeBe stars on an H-R diagram. Thethick blue dashed line is the birth line, the thin bluedashed lines are isochrones, the dot-dashed line isthe ZAMS and the solid lines are theoretical evo-lutionary tracks from the birth line to the ZAMSfor the masses indicated. The orange dot-dot-dot-dashed line represents a radiative/convective tran-sition: to the left of this line, the convective enve-lope accounts for less than 1% of the star’s mass. Aconvective core appears towards the end of the PMSwhen the star moves downwards on the HR diagram.The open circles correspond to HD 98922 (abovethe birthline) and IL Cep (below the ZAMS), whosepositions cannot be reproduced with the theoreti-cal evolutionary tracks considered. From Figure 4 inAlecian et al. (2013b).

There was a claim (Hubrig et al., 2000) thatall Ap stars have passed through at least 30%of their main-sequence lifetime. This result de-pended on determining the ages of stars by plac-ing them on the HR diagram, which is very chal-lenging since stars move very slowly across theHR diagram during the first part of the main-sequence, and because of the distortion of ap-parent surface temperatures by the atmosphericabundance anomalies. In light of this, and ofthe recent results on pre-MS stars, it looks un-likely that this result is correct. Landstreet etal. (2009a) looked instead at Ap stars in clus-ters – where the ages can be determined muchmore accurately – and found the opposite: thatthere is a negative correlation between the fieldstrength in Ap stars and their age, greater thanone would expect from flux conservation as thestar expands along the main sequence. Possibleexplanations for this field decay are discussed inSection 3.8.

Figure 6: Variation in the rotation period ofCU Virginis. The left panel shows the variationin the rotation period, and the right panel is thephase residue, assuming a constant period. FromMikulasek et al. (2011a).

2.1.5 Variability in Ap stars

Though Ap stars are characterised by stability oftheir magnetic and spectral signatures on all ob-servable time scales (apart from rotational mod-ulation), there exist a few curious cases wherechanges have been observed. The rapidly ro-tating star CU Virginis (P = 0.52 days, withobservations stretching back to 1949 (Deutsch,

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1952)), has increased its rotation period byabout 50 ppm (Figure 6). The O-C data (thephase drift relative to a fixed period) can be fitby an increase in rotation period within a fewyears around 1984 (Pyper et al., 1998, 20133).This would be reminiscent of the ‘glitches’ seenin pulsars. However, a more gradual change inperiod also seems compatible with the observa-tions. Stępień (1998) suggested that the changesin CU Vir may not actually be monotonic, butcould reflect some form of magnetic oscillationin the star, with typical time scales of severaldecades. This would imply that, at a given pointin time, one would find period decreases aboutas frequently as increases. Period increases havebeen seen in other stars: spindown rates of or-der P/P ∼ 105−6 yr−1 have also been reportedin V901 Ori, σ Ori e, HR 7355, SX Ari, and EEDra (Mikulasek et al., 2011b). No clear case of aperiod decrease case has so far been seen in thesample of stars with changing periods, but thereare hints that both CU Virginis and V901 Ori-onis may now be spinning up (Mikulasek et al.,2011a). The current limited sample looks there-fore still to be statistically compatible with theidea; finding alternative explanations would bechallenging. Speaking strongly in favour of theoscillation idea is that the spindown timescalesmeasured are of order 105−6 yr, which wouldotherwise be hard to reconcile with the ages in-ferred for these stars, of the order of 108 yr.

Mikulasek et al. (2013) offer two alterna-tive interpretations of the data on CU Virginis,namely that the rotation period is undergoingeither some kind of Gaussian variation and willreturn to its original value, or a periodic vari-ation with a minimum rotation period in 1968and a maximum in 2005. In the oscillation sce-nario, the amplitude (at the surface) of the tor-sional oscillation phase residue in CU Vir wouldbe approximately 2.3π and in V901 Ori at least0.5π (Mikulasek et al., 2013). A challenge forthe oscillation idea might be that the Alfventimescale in CU Vir, with a surface field strengthof around 3 kG, would be only about 3 years,assuming that field strength also in the star’sinterior. One expects the fundamental magneticoscillation mode to have this period, and higherharmonics even shorter periods.3 The title of the paper incorrectly says ‘decrease’.

A remarkably rapid change in the field config-uration, on a time scale of years, was reportedin the pre-MS star HD190073 (Alecian et al.,2013a). It now appears that this was a spuriousresult (Hubrig et al., 2013).

2.2 Other stably stratified stars

Having looked in some detail at the stronglymagnetic subset of intermediate-mass stars,which have a long history in the literature, wenow turn to other stars which are stably strat-ified, at least on the outside. First of all, therest of the intermediate-mass population and themore massive stars. We look then briefly at whitedwarfs. Neutron stars are discussed separately inSect. 7.

2.2.1 Vega & Sirius

As far as the ‘non-magnetic’ part of the popula-tion of intermediate-mass stars is concerned, anexciting discovery has been the detection of mag-netic fields in Vega and Sirius, the two brightestA stars in the sky. Zeeman polarimetric obser-vations of these two stars have revealed weakmagnetic fields: in Vega a field of 0.6 ± 0.3 G(Lignieres et al., 2009; Petit et al., 2010) and inSirius 0.2 ± 0.1 G (Petit et al., 2011). The fieldgeometry is poorly constrained, except that thefield should be structured on reasonably largelength scales, as cancellation effects would pre-vent detection of a very small-scale field. Vegaseems to have a strong (∼ 3 G) magnetic fea-ture at its rotational pole. The existence or oth-erwise of time variability is unknown – Petit etal. simply note that in Vega ‘no significant vari-ability in the field structure is observed over atime span of one year’. These two stars have sim-ilar mass but have significant differences: Vegais a rapidly rotating single star, and Sirius is aslowly rotating Am star which may well haveaccreted material from its companion (de Val-Borro et al., 2009). Given that we so far havedetections in both stars observed, it seems verylikely that the rest of the ‘non-magnetic’ popula-tion also have magnetic fields of this kind. Thereare also theoretical grounds to expect this (seeSection 3.7).

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2.2.2 Massive main sequence stars

Direct detection via the Zeeman effect ofmagnetic fields in stars above around 6 or8M is significantly more challenging than inintermediate-mass stars. Firstly, because thereare fewer lines in the spectrum, and since the sig-nals from each line are normally added togetherwith the LSD technique (see above) this leadsto a smaller signal. Secondly, because there arevarious line-broadening mechanisms. Whilst theatmospheres of A and late B stars are very quiet,O and early B stars display a number of obser-vational phenomena such as discrete absorptioncomponents, line profile variability, wind clump-ing, solar-like oscillations, red noise, photomet-ric variability and X-ray emission. Much of thishas to do with winds and wind variability, andmuch is not yet understood (see e.g. Michaud etal., 2013, and refs therein). This complicates thelife of the Zeeman observer, with the result thatthe detection limit on magnetic fields is perhaps30 or 100 gauss, much higher than in A stars.See for instance Henrichs (2012) for a review ofmagnetism in massive stars.

The result of this is that magnetic fields werenot detected in hot stars until relatively recently(Henrichs et al., 2000), but with the completionof recent surveys (e.g., the MiMeS survey, Wadeet al., 2013) we have a much better picture of theincidence of large-scale magnetic fields in mas-sive stars. It seems that around 7% of the popu-lation host large-scale fields (Petit et al., 2013).The magnetic stars have fields of 300 G – 10 kG,and a variety of geometries, as is found in theA stars. Whilst some of the magnetic stars havean approximately dipolar field, others are foundto have a more complicated geometry – see Fig-ure 7 for an example. In several other stars sim-ilarly complex magnetic fields have been found,dubbed the ‘τ Sco clones’. Recently, it has beenfound that the magnetic flux in magnetic OBstars decays during the main-sequence ((Fossatiet al. , 2016)), the same as is found in Ap stars(Section 2.1.4). Possible explanations are dis-cussed in Section 3.8.

It is tempting to conclude, therefore, thatthe phenomenon is simply a continuation ofthat seen in intermediate-mass stars, and in-deed there are no particular theoretical rea-

Figure 7: The observed field geometry on the main-sequence B0 star τ Sco, at two rotational phases, us-ing Zeeman-Doppler imaging. The paths of archedmagnetic field lines are clearly visible; it seems likelythey are associated with twisted flux tubes buriedjust below the surface – see Section 3.5. From Do-nati et al. (2009). Observations of this star takenseveral years apart show the same topology, confirm-ing a lack of variability also in stars with complexmagnetic topologies (Donati & Landstreet, 2009).

sons to think otherwise. The historical divisionbetween magnetism in intermediate- and high-mass stars is probably due only to the obser-vational difficulty of observing the Zeeman ef-fect in the hotter stars, and perhaps more im-portantly the fact that magnetic fields in hotterstars do not give rise to chemical peculiarities asseen in intermediate-mass stars, making themharder to identify. The reason for the lack ofsuch peculiar abundances is presumably that thewind removes the outer layers before chemical-separating mechanisms have time to work.

The ‘non-magnetic’ majority of the popula-tion may well have weaker and/or smaller-scalemagnetic fields of some kind, and the various ob-servational phenomena mentioned above couldplausibly be at least partly the result of mag-netic activity, not unlike that in the Sun – seeSection 6.5.

2.2.3 White dwarfs

The stable fields observed in the subclass of mag-netic white dwarfs (mWD) show parallels withthose of Ap stars, though it is not clear if this ismore than a coincidence. The subclass is sim-ilarly small, the range in field strength againlarge, ranging from a few times 103 to 5×108 G.As in the case of Ap stars, the large range in ro-

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tation periods of mWD is puzzling. Most rotateat periods from a fraction of a day to severalweeks, but some of them have inferred periodsas long as decades. For more on their proper-ties we refer to reviews by Putney (1999) andSchmidt (2001).

White dwarfs have been important for thetheory of fossil fields because, unlike early-typemain-sequence stars, they do not contain anysignificant convective zones. Consequently, as faras the nature of their magnetic fields is con-cerned, there is essentially no alternative to amagnetic equilibrium. They are the same in thisrespect to the neutron stars, see Sect. 7; indeedmuch of the physical processes discussed in thatsection also applies to white dwarfs. In Sect. 3.1we discuss magnetic white dwarfs in the contextof the origin of fossil fields in general. They alsooffer clues regarding the rotation of the coresof giant stars and angular momentum transportwithin stars in general (see Sect. 5.5).

3 Theory of Fossil Fields

In radiative stars, the main challenges are toexplain not just how these stars come to hostmagnetic fields at all, but to explain the largerange in the magnetic and rotational propertiesof otherwise similar stars. There are similaritiesin these respects between Ap stars, magneticmassive stars, magnetic white dwarfs (mWD),and to some extent also the neutron stars.

An early idea triggered by these similarities,still widely referred to, is that of ‘flux freezing’.It does not attempt to explain the fields of Apstars themselves, which is just taken as an ob-servational given. Instead, it interprets the mag-netic flux of white dwarfs and neutron stars as‘inherited’ from magnetic main sequence progen-itors, the Ap, Bp and magnetic O stars. Ideasfor the origin of these progenitors themselves areeven less well developed. An obvious connectionthat must play a role is that with the magneticfields observed in protostellar clouds, but this initself does not tell us what to make of the puz-zling range of field strengths in Ap stars, theirlow flux-to-mass ratio, and overall low frequency.This is discussed some more in Sect. 4.

3.1 Flux freezing

The large field strengths of mWD can be under-stood as a consequence of the greater compact-ness of WD compared to main sequence stars.An mWD may have a radius of ∼ 5 × 108 cm,a factor ∼ 100 smaller than the core of a mid-A main sequence star. A popular hypothesis isthat mWD inherited their fields from Ap pro-genitors. Under the assumption of ‘flux freezing’,the magnetic field lines would stay anchored inthe star during its evolution through the giantbranch, the planetary nebula (PN) phase andthe evolution of the remaining core to a WD.This would predict the field strength of an mWDto be ∼ 1002 higher than the field of the MSstars it started with, which would put it in theobserved range.

The flux inheritance hypothesis is neverthe-less somewhat dubious, as it implies that the lossof more than half of the star’s mass in the PNphase does not significantly affect its magneticflux. Figures 9 and 11 do suggest that some frac-tion of the surface flux of the Ap star may pos-sibly pass through the core that ends up to formthe WD. This flux consists, however, of poloidalfield lines. The subsurface torus surrounding thispoloidal flux is what maintains the stability ofthe configuration. It would be lost during theejection of the envelope in the PN phase. The re-maining poloidal configuration would be highlyunstable via the Flowers–Ruderman mechanism(Sect. 3.3). Numerical results of this process sug-gest that not much of the poloidal field wouldsurvive from such a configuration.

Another way of looking at the field strengths isby comparing magnetic energy density (B2/8π)with other energy densities in the star. For amagnetic star to be bound, its magnetic energymust be less than the gravitational binding en-ergy. Per unit mass this energy is of the order ofthe central pressure pc. The central pressure instars of mass M and radius R scales as M2/R4

(e.g., Kippenhahn et al., 2013). Comparing thetwo predicts a scaling of the maximum possi-ble field strength, Bmax ∼ M/R2. In terms ofthe magnetic flux: Φmax ∼ BmaxR

2 ∼ M . Asthe mass of a WD is only a few times smallerthan an Ap star, the similarity of the maximummagnetic flux in the two cases is effectively the

10

same as predicted by ‘flux freezing’. Magneticfield strengths from a flux freezing argument cantherefore not be distinguished from argumentsrelating field strength to energy density (whichimply very different physics).

The maximum values of the surface fieldstrength observed are of order 2 × 104, 5 × 108

and 1015 G for Ap stars, mWD and neutronstars, respectively. These numbers are a factor104–103 smaller, respectively than the maximumfield strengths allowed by equipartition with thegravitational binding energy of the stars. It isnot clear if this factor, or its similarity betweenthe different types of star, has a special mean-ing. In any case these maxima do not constraintheories for the origin of the field very much.

3.2 The nature of magnetic fieldsin radiative stars

Since the discovery of magnetic fields in Apstars, there have been two theories to explaintheir presence: the core-dynamo theory and thefossil-field theory. According to the former, theconvective core of the star hosts a dynamo ofsome kind, which sheds magnetic field into theoverlying radiative layer. This magnetic fieldthen rises, under the action of buoyancy, to-wards the surface. The reason for this buoyantrise can be understood as follows. A magneticfeature (e.g., a flux tube) must be in total pres-sure equilibrium with its non-magnetised sur-roundings: the sum of gas, radiation and mag-netic pressures inside the feature must be equalto the sum of gas and radiation pressures inthe surroundings. To prevent buoyant rise, thedensity of the feature must be the same asthat of the surroundings, which is only pos-sible if the temperature is lower. This causesheat to diffuse into the magnetic feature, caus-ing it gradually to rise. However, it turns outthat the timescale for this buoyancy process islonger than the main-sequence lifetimes of thesestars unless very small flux tubes can be gener-ated (Parker, 1979c; Moss, 1989; MacGregor &Cassinelli, 2003). This does not agree with theobservations, which suggest mainly large-scalestructure at the surface. In addition, the retreat-ing convective core (in mass coordinates) leavesbehind a strong composition gradient in the ra-

diative layers, enormously slowing down the es-cape process (MacDonald & Mullan, 2004). Alsopuzzling in this theory is the enormous range offield strengths between different stars which arepredicted to have similar convective cores.

The fossil field theory, on the other hand, ap-pears better able to explain the observations,in particular the large-scale geometry, and largefield strengths. The basic idea is that instead ofbeing continually regenerated in some ongoingdynamo process feeding off the star’s luminos-ity, the field is in a stable equilibrium in a staticradiative zone. In MHD, the magnetic field Bevolves according to the induction equation

∂B∂t

= ∇× (u×B− η∇×B) , (1)

where u is the fluid velocity and η is the mag-netic diffusivity, the reciprocal of the electricalconductivity. In turn, the velocity field is relatedto the forces acting on the gas, i.e., the pressuregradient, gravity and Lorentz forces via the mo-mentum equation:

dudt

= −1ρ∇P + g +

14πρ

(∇×B)×B , (2)

where P , ρ and g are pressure, density andgravity. In an equilibrium unmagnetised star,the pressure gradient and gravity balance eachother. Upon the addition of an arbitrary mag-netic field, the Lorentz force will cause the gasto move at approximately the Alfven speed vA =B/√

4πρ, and the system evolves on an Alfventimescale τA = R/vA (where R is the radius ofthe star) which in a star with a 1 kG field isaround ten years. Eventually one might hope toreach an equilibrium – a so-called fossil field –where the three forces balance each other andu = 0, so that the first term on the r.h.s. ofthe induction equation disappears and the fieldevolves only on a diffusive timescale R2/η. Cowl-ing (1945) first realised that this timescale is oforder 1010 years in the radiative core of the Sunand that a field in equilibrium there could there-fore persist for the entire lifetime of the star. Themagnetic equilibrium must also be stable, how-ever, since instability time scales are of the or-der of the Alfven timescale, which as mentionedabove can be as short as a few years.

11

Much effort has been put into finding such sta-ble equilibria, which was historically also moti-vated by the need for magnetic plasma confine-ment in nuclear fusion devices. Unfortunately, itturns out to be a difficult problem to solve, andwith analytic techniques the existence of thesestable equilibria was never convincingly demon-strated. This was historically a major weaknessof the fossil field theory but, in light of the weak-nesses of the core dynamo theory (including thediscovery of magnetic fields with similar proper-ties in white dwarfs, which contain no convectivecore), there was a widespread feeling that stableequilibria must exist.

Using analytic techniques, attempts to findsuch equilibria are split into two parts: first find-ing an equilibrium, and then checking its stabil-ity. The first step should not, prima facie, repre-sent any major problem. To see this (without anactual proof) consider the following. The mag-netic field has two degrees of freedom (reducedfrom three by the ∇ ·B = 0 constraint) and sothe Lorentz force should also have two degreesof freedom. Ignoring thermal diffusion, the ther-modynamic state of the gas also has two degreesof freedom. Where the magnetic field is weak –in the sense that the plasma-β = 8πP/B2 1– equilibrium can be obtained by suitable ad-justments of the thermodynamic variables, saypressure and entropy4. Where the field strengthis not small in this sense, for instance close to orabove the surface of a star, an equilibrium mustbe close to a force-free configuration. We cantherefore divide the star conceptually into twodomains: the interior of the star where β 1and the allowed range of field geometries is notstrongly constrained (as long as equilibrium isthe only concern and stability is ignored), andthe exterior where β 1 and the field is close toforce-free. Near the photosphere of the star thegas pressure scale height is much smaller thanthe length scales on which the magnetic fieldchanges, and so for any Ap star the β = 1 surfacewill be fairly close to the photosphere. In fact ifthe field strength at the photosphere is 300 G,as in the weakest-field Ap stars, then β = 1 co-

4 This is not possible in a hypothetical convectivelyneutral star where entropy is constant. In this case thefield must be of the much more restricted class for whichthe Lorentz force has a potential. See also Sect. 3.6.

incides with the photosphere; in Ap stars withstronger fields this surface is a little lower down.

In addition to the MHD processes above, fieldgeneration by microscopic processes have beenconsidered: the ‘Biermann battery’, Biermann(1950), cf. Kulsrud (2005), and the thermo-electric effect Dolginov & Urpin (1980), Urpin& Yakovlev (1980). In the stellar environment,both operate very slowly, as they depend on dif-fusion. For them to be effective, a very stableenvironment is required. In and A star, such anenvironment can be supplied by a stable fos-sil field, but appealing to this would obviouslycause circular reasoning. The situation is bet-ter for neutron star crusts, where its solid statecan provides a stable environment there. It hasbeen concluded that the process is unlikely tobe effective in this case, however, since the fieldproduced in this way early in the life of the neu-tron star would decay when it cools (Blandfordet al. 1983).

3.3 Stability of fossil equilibria

To check the stability of an equilibrium usinganalytic techniques is rather trickier. Most stud-ies use the energy method of Bernstein et al.(1958) where the energy change in a configu-ration is calculated as a displacement pertur-bation is applied. If the energy change is pos-itive for all possible perturbation fields, the con-figuration is stable. In the stellar context, thismethod was more successful in uncovering in-stabilities than in demonstrating stability. Forreasons of tractability, almost all effort has beenconcentrated on axisymmetric equilibria. Tayler(1973, see also Spruit 1999) looked at purelytoroidal fields, that is, fields that have only anazimuthal component Bφ in some spherical co-ordinate frame (r, θ, φ) with the origin at thecentre of the star. He derived necessary and suf-ficient stability conditions for adiabatic condi-tions (no viscosity, thermal diffusion or mag-netic diffusion). The main conclusion was thatsuch purely toroidal fields are always unstableto adiabatic perturbations at some place in thestar, in particular to perturbations of the m = 1form (m being the azimuthal wavenumber). Nu-merical simulations have also been used to lookat the stability of purely toroidal fields (Braith-

12

waite, 2006b), reproducing many of the analyticresults.

The opposite case is a field in which all fieldlines are in meridional planes (Bφ = 0, see Fig-ure 8). In subsequent papers Markey & Tayler(1973, 1974) and independently Wright (1973)studied the stability of axisymmetric poloidalfields in which (at least some) field lines areclosed within the star (right-hand side of Fig-ure 8). These fields were again found to be un-stable.

A case not covered by these analyses was thatof a poloidal field in which none of the field linesclose within the star. An example of such a fieldis that of a uniform field inside, matched by adipole field in the vacuum outside the star (left-hand side of Figure 8). This case has been con-sidered earlier by Flowers & Ruderman (1977)who found it to be unstable, by the followingargument. Consider what would happen to sucha dipolar field if one were to cut the star in half(along a plane parallel to the magnetic axis),rotate one half by 180, and put the two halvesback together again. The magnetic energy insidethe star would be unchanged, but in the atmo-sphere, where the field can be approximated bya potential field, i.e., with no current, the mag-netic energy will be lower than before. This pro-cess can be repeated ad infinitum – the mag-netic energy outside the star approaches zeroand the sign of the field in the interior changesbetween thinner and thinner slices. Marchant etal. (2011) present a more rigorous analysis of thisinstability. The reduction of the external fieldenergy is responsible for driving the instability.Since the initial external field energy is of thesame order as the field energy inside the star,the initial growth time of the instability is ofthe order of the Alfven travel time through thestar, as in the cases studied by Markey & Taylerand Wright.

Prendergast (1956) showed that an equilib-rium can be constructed from a linked poloidal-toroidal field, but stopped short of demonstrat-ing that this field could be stable. Since bothpurely toroidal fields and purely poloidal fieldsare unstable, an axisymmetric stable field config-uration, if one exists, should presumably be sucha linked poloidal-toroidal shape. Wright (1973)showed that a poloidal field could be stabilised

by adding to it a toroidal field of comparablestrength. However, the result was again some-what short of a proof.

These results were all valid only in the absenceof dissipative effects and rotation. The only casein which dissipative effects have been investi-gated in detail is that of a purely toroidal fieldAcheson (1978), (see also Hughes & Weiss 1995,where it is found, for instance, that some com-binations of diffusivities can have a destabilis-ing effect on configurations which are stable tothe non-diffusive Parker instability). These ef-fects have still to be investigated in a more gen-eral geometry such as that of the mixed poloidal-toroidal equilibria.

The effect of rotation was investigated byPitts & Tayler (1985) for the adiabatic case (i.e.,without the effects of viscosity, magnetic andthermal diffusion). These authors reached theconclusion that although some instabilities couldbe inhibited by sufficiently rapid rotation, otherinstabilities were likely to remain, whose growthcould only be slowed by rotation – the growthtimescale would still be very short comparedto a star’s lifetime. Also, diffusion can reducethe stabilising effect of rotation (Ibanez-Mejıa& Braithwaite, 2015). Importantly though, ro-tation does not introduce any new instabilities:a configuration which is stable in a non-rotatingstar should also be stable in a rotating star.

3.4 Numerical results

More recently, it has become possible to findstable equilibria using numerical methods. Var-ious equilibria are found. The simplest equiib-rium consists of a single flux tube around themagnetic equator of the star, surrounded by aregion of poloidal field. More complex equilib-ria (Section 3.5) can have more than one tubein various arrangements. From observations ofmagnetic A, B and O stars, we see that boththe simple and the more complex equilibria dooccur in nature.

Essentially, the numerical method consists inevolving the MHD equations in a star contain-ing initially some arbitrary field. Braithwaite& Spruit (2004) and Braithwaite & Nordlund(2006) modelled a simplified radiative star: aself-gravitating ball of gas with an ideal gas

13

Figure 8: Poloidal field configurations. Left: all field lines close outside the star, this field is unstable byan argument due to Flowers & Ruderman (1977). For the case where some field lines are closed inside thestar, instability was demonstrated by Markey & Tayler (1973, 1974) and Wright (1973).

equation of state, ratio of specific heats γ = 5/3and a stratification of pressure and density asin a polytrope of index n = 3, embedded inan atmosphere with low electrical conductivity.Over few Alfven timescales, the field organisesitself into a roughly axisymmetric equilibriumwith both toroidal and poloidal components ina twisted-torus configuration, illustrated in Fig-ure 9. This corresponds qualitatively to equilib-ria suggested by Prendergast (1956) and Wright(1973).

The stability of these axisymmetric fields,and in particular the range of possible ratios oftoroidal to poloidal field strength, was examinedfurther by Braithwaite (2009) with a mixture ofanalytic and numerical methods. It was foundthat the fraction of energy in the poloidal com-ponent must satisfy a(E/Egrav) < Ep/E . 0.8where E and Egrav are the total magnetic energyand the gravitational energy, Ep is the energy ofthe poloidal field and a is some dimensionlessfactor of order unity. Akgun et al. (2013) getthe same results with more analytic methods.To give some numbers, for an A star the dimen-sionless factor a ∼ 15 and the ratio E/Egrav isonly about 10−6 even in the most strongly mag-netic Ap star (B ≈ 30 kG), so that in this starwe require for stability 10−5 . Ep/E . 0.8. In aneutron star a ∼ 400 (Akgun et al., 2013), and

assuming a magnetar field strength of 1015 G,the condition is 4×10−4 . Ep/E . 0.8. In starswith weaker fields of course the lower limit tothe ratio Ep/E is even lower.

Physically, the upper limit comes from theneed for a comparable-strength toroidal field tostabilise the instability of a purely poloidal field,and is in rough agreement with the result ofWright (1973). The lower limit is different be-cause the instability of a purely toroidal field,unlike that of a poloidal field, involves radial mo-tion and so the stable entropy stratification hasa stabilising effect, preferentially on the longerwavelengths which involve greater radial motion.The poloidal field stabilises preferentially theshorter wavelengths, and at sufficient poloidalfield strength the two effects meet in the mid-dle and all wavelengths are stabilised. The sta-ble stratification is more effective if the field isweaker, hence the presence of the total field en-ergy in the threshold. Since E/Egrav is alwaysa very small number, only a relatively smallpoloidal field is required. See Braithwaite (2009)for a more thorough explanation.

This result is of particular interest in the con-text of neutron stars, where the deformation ofthe star from a spherical shape depends cruciallyon this ratio Ep/E since poloidal field makes thestar oblate and toroidal field prolate. In the pres-

14

ence of a suitable mechanism to damp torque-free precession, a prolate star should ‘flip over’until the magnetic and rotation axes are perpen-dicular, which is the state in which the rotationalkinetic energy is at a minimum, given a constantangular momentum. In this state the star emitsgravitational waves (see e.g. Stella et al., 2005).

Figure 9: The shape of the stable twisted-torus fieldin a star, viewed along and normal to the axis ofsymmetry. The transparent surface represents thesurface of the star; strong magnetic field is shownwith yellow field lines, weak with black. Figure fromBraithwaite & Nordlund (2006).

3.5 Non-axisymmetric equilibria

From further simulations (Braithwaite, 2008) itbecame clear that depending on the initial condi-tions, a wide range of equilibria can form, includ-ing non-axisymmetric equilibria – see Figure 10.

Crucial is the distribution of magnetic energyand the amount of flux passing through the stel-lar surface to the low-conductivity medium out-side. It is important to note that during relax-ation to equilibrium there is essentially no radialtransport of gas and magnetic flux, fluid mo-tion being confined to spherical shells, so thetotal unsigned flux through any spherical shell∮|B · dS| can only fall. Therefore an initial field

which is buried in the interior of a radiative staror zone evolves into a similarly buried equilib-rium. It turns out that in this case, an approx-imately axisymmetric field forms. At the oppo-site end of the spectrum, an initial field witha flat radial field-strength profile with a finiteamount of magnetic flux at the surface evolvesinto a non-axisymmetric equilibrium – see Fig-ure 12. It seems that both axisymmetric andnon-axisymmetric equilibria do form in nature,and both types can be found amongst A, B andO main-sequence stars as well as amongst whitedwarfs: see Figures 3 and 7.

Figure 10: A typical non-axisymmetric equilibriumas found in simulations, viewed from both sides ofthe star. This corresponds qualitatively to those ob-served on stars such as τ Sco (see Figure 7). Figurefrom Braithwaite (2008).

The geometries of these various equilibriahave one feature in particular in common,namely that they can be thought of in termsof twisted flux tubes surrounded by regions ofpurely poloidal field. The simple axisymmetricequilibrium can be thought of as a single twistedtube wrapped in a circle (a ‘twisted torus’); pass-ing through this circle are poloidal field lineswhich pass through the stellar surface. The morecomplex equilibria contain one or more twistedflux tubes meandering around the star in ap-parently random patterns; these do not touch

15

each other but are surrounded by regions wherethe field is perpendicular to the flux-tube axis.In the equilibria found thus far, the meanderingis done at roughly constant radius a little be-low the surface. Equilibria where the flux tubesdo not lie at constant radius seem plausible butit also seems plausible that they are difficult toreach from realistic initial conditions, especiallyin view of the restriction of motion to spheri-cal shells. Figure 11 shows cross-sections of boththe axisymmetric and more complex equilibria.The toroidal field is confined to the region wherethe poloidal field lines are closed within the star– this region resembles a twisted flux tube –toroidal field outside this region would ‘unwind’through the atmosphere and disappear.

Qualitatively the difference in equilibria re-sulting from differing radial energy distributionsin the initial conditions can be understood in thefollowing way. If the magnetic energy distribu-tion is flatter, the axis of the circular flux tubein an axisymmetric equilibrium will be closerto the surface of the star and the bulk of thepoloidal flux goes through the surface, leavinga smaller volume in which the toroidal compo-nent can reside. Beyond some threshold, thismeans that the toroidal field is not able to ful-fill its role in stabilising the instability seen inpurely poloidal fields, and the field buckles, theflux tube first attaining a shape reminiscent ofthe seam on a tennis ball and then somethingmore complex. This lengthening of the flux tube(at constant volume) increases the toroidal (ax-ial) field strength and decreases the poloidalfield strength, until stability is regained. Thesame process can also be thought of in termsof the tension in a flux tube, which is equal toT = (2B2

ax − B2h)a2/8 where a is the radius of

the tube, and Bax and Bh are the r.m.s. ax-ial (toroidal) and hoop (poloidal) components ofthe field 5. In the simple axisymmetric equilib-rium, the tension in the tube must be positive.If the tube is too close to the surface, there isnot enough space for the toroidal field and thetension can become negative, causing the tubeto buckle into a more complex shape until thelengthening of the tube causes the tension goesto zero. A fuller discussion is given in Braith-

5In the literature one often finds that the factor of 2inside the brackets is missing.

waite (2008). Figure 13 shows a cross-section of

Figure 11: Cross-sections of magnetic equilibria,both of which contain a twisted flux tube surroundedby a volume containing just poloidal field. The stel-lar surface is shown in green and poloidal field lines(in black) are marked with arrows. The toroidal field(direction into/out of the paper, red shaded area) isconfined to the poloidal lines which are closed withinthe star. (Toroidal field outside this area would un-wind rather like a twisted elastic band that is notheld at the ends.) Above: the axisymmetric casewhere the flux tube lies in a circle around the mag-netic equator (corresponding to Fig. 9). Below: anarrower flux tube (corresponding to Fig. 10). Inthis case, the flux tube must be longer than in theaxisymmetric case in order to occupy the whole stel-lar volume; it meanders around the star in an appar-ently random fashion, and there may also be two ormore such tubes. Figure from Braithwaite (2008).

a non-axisymmetric equilibrium found in a sim-ulation. Note how in this figure, as well as inthe cross-sections of axisymmetric equilibria inFigure 12, we can see that the poloidal field isparallel to contours of the toroidal field multi-plied by cylindrical radius. This condition wasderived analytically in the case of axisymmetricequilibria from the need for the toroidal part ofthe Lorentz force to vanish (Mestel, 1961; Rox-

16

burgh, 1966), and the same condition applies innon-axisymmetric equilibria.

Figure 12: A sequence of equilibria resultingfrom initial conditions with different degrees ofcentral concentration of the magnetic energy –above left, centrally concentrated initial conditions;above right, medium concentration (still leading toa roughly axisymmetric equilibrium); and below,flatter distributions; below right the radial energydistribution is completely flat. For the three ax-isymmetric equilibria, the (relatively small) non-axisymmetric component is ignored: plotted are con-tours of the flux function of the poloidal field, whichare also poloidal field lines, and the shading repre-sents the toroidal field multiplied by the cylindri-cal radius. For the two non-axisymmetric equilibriashown, field lines are plotted in red and the surfaceof the star is shaded blue. Note that these two equi-libria have flux tube(s) of different widths (corre-sponding to the angle α in Figure 11). Figures fromBraithwaite (2008, 2009).

3.6 Recent analytic work

There has recently been renewed interest in find-ing stable equilibria with analytic and semi-analytic methods. In stably-stratified stars, therange of possible equilibria is large or per-haps even essentially unlimited within the zero-divergence constraint (as described above in Sec-tion 3.2), although of course only some subsetwill be stable. To find analytic equilibria, various

Figure 13: Cross-section of a non-axisymmetricequilibrium. The curved grey line towards the rightis the surface of the star and the centre of the staris on the left; the coordinate system used for theplot is cylindrical. The blue/red shading representsthe toroidal field component (out of/into the page)multiplied by the cylindrical radius. The poloidalcomponent (in the plane of the page) is representedby the arrows and by contours of its scalar potential,calculated by ignoring spatial derivatives perpendic-ular to the plane. In fact it can be seen that thearrows are very nearly parallel to the contours ofthe scalar potential, showing that the length scaleof variation in the direction perpendicular to thepage is much greater than the length scale in theplane of the page, i.e., that the flux tubes meanderaround the star over scales much greater than theirwidth. Note also that neighbouring flux tubes canhave toroidal field in either the same or opposite di-rections: since the toroidal field is absent from thespace between the tubes, one tube is not ‘aware’ ofthe direction of toroidal field in its neighbours, so theequilibrium and stability properties are independentof its direction. Figure from Braithwaite (2008).

assumptions and choices are made to constrainthe solutions. For instance, all analytic works sofar have assumed axisymmetry.

Furthermore, except for a few recent papers(see below), all works have assumed a barotropicequation of state (e.g., Yoshida et al., 2006;Ciolfi et al., 2010; Lyutikov, 2010; Fujisawa etal., 2012), which represents a significant restric-tion on the range of equilibria, as explained inSection 3.2. The barotropic assumption is alsoquite artificial: regions in stars are either sta-bly stratified or convective. In a convective re-gion in a non-rotating star the stratification isnearly barotropic, but convective flows are in-compatible with static field configurations. Ina rotating star, convective zones are not evennearly barotropic and thermal winds arise. The

17

parameter space in between convection and sta-ble stratification is a ‘set of measure zero’,of rather academic interest. Pursuing the lineof thought nevertheless, taking the momentumequation (2), setting the left-hand side to zero(as in equilibrium) and taking the curl, gives

0 = ∇×(− 1ρ∇P

)+ ∇× g + (3)

∇×(

14πρ (∇×B)×B

). (4)

Gravity being a conservative force, the curlof g vanishes. With a barotropic equation ofstate ρ = ρ(P ), the pressure gradient force−(1/ρ)∇P = −∇h where h = h(P ) is a newvariable; the curl of this force is then obviouslyzero. We are left with a condition of curl-freeLorentz acceleration:

∇× [(∇×B)×B/ρ] = 0. (5)

In other words, the Lorentz acceleration musthave vanishing curl because it is balanced bytwo other forces with vanishing curl. Thus thebarotropic equation of state imposes a restric-tion on the equilibrium which does not existwith an equation of state ρ = ρ(P, T ). Assum-ing axisymmetry and a barotropic EOS, findingan equilibrium is a matter of solving the Grad-Shafranov equation, derived from the equilib-rium condition that comes from setting the LHSof (2) to zero.

Most works have constructed simple twisted-torus equilibria of the form in the upper part ofFig. 11, with some more complex axisymmetricequilibria with two or more tori. Some examplesare shown in Fig. 14.

A common feature of the simple equilibria isthat the volume containing closed poloidal linesand toroidal field is rather small, the neutral line(where the poloidal field vanishes) being closeto the stellar surface (e.g. Lyutikov, 2010, seefig. 14). This is possibly something to do withthe requirement that the equilibrium can be ex-pressed mathematically as the sum of low-orderspherical harmonics – indeed the interior fieldis often matched to a pure dipole field outsidethe star. Physically, an equilibrium can formout of an initial field which has most of its fluxburied away from the stellar surface and so moredeeply buried equilibria must be possible, and

such buried equilibria are found in simulations(see Fig. 12). Alternatively this may have to dowith the use of a barotropic EOS.

Other authors (e.g., Haskell et al., 2008; Duezet al., 2010) have found equilibria where thepoloidal field does not penetrate through thestellar surface at all, so that toroidal field occu-pies the entire volume of the star. Strictly speak-ing, these are of course of academic interest ifthe application is to objects with an observedfield at the surface, but equilibria in nature mayhave only a modest fraction of their flux pass-ing through the surface, if the magnetic energyis relatively concentrated in the middle of thestar.

Several models (Broderick & Narayan, 2008;Ioka & Sasaki, 2004; Colaiuda et al., 2008, in-ter alia) include a current sheet at the stel-lar surface. This means that the Lorentz forceis infinite, which is impossible in nature, espe-cially at a location where the fluid density goesto zero. In for instance Broderick & Narayan(2008) the current sheet is an unavoidable con-sequence of the assumption made that the fieldin the interior is force-free (i.e., j×B = 0). Tosee this, recall a classical result: the ‘vanishingforce free field theorem’. It says that a magneticfield which is force free everywhere in space van-ishes identically. Force-free fields can exist onlyby virtue of a surface where the Maxwell stressin the field is taken up. (For the 3-line proof seeRoberts, 1967, also reproduced in Spruit, 2013).The Lorentz force density (the divergence of thisstress) may vanish; the stress itself however van-ishes only when the field itself vanishes.

Whilst most studies use the approximationsthat (a) the field is too weak to have a signif-icant effect on the shape of the star, (b) thestar is not rotationally flattened and (c) gen-eral relativistic effects can be ignored, some au-thors drop these assumptions. For instance Fu-jisawa et al. (2012) have a rotating model witha strong magnetic field and Ciolfi et al. (2009,2010) include general relativity (relevant in thecontext of neutron stars). Perhaps reassuringly,including these effects does not seem to resultin any qualitative difference to the geometry ofthe equilibria found. Also interesting in the con-text of neutron-star magnetic equilibria is theHall effect, which is essentially an extra term

18

in the induction equation (1) to account for thevelocity difference between the electron fluid, towhich the magnetic field is frozen, and the bulkflow: see, e.g., Gourgouliatos et al. (2013).

3.6.1 Stability

Having found an equilibrium, one needs to checkits stability. Some authors have tried to ensurethe stability of their equilibria by finding an en-ergy minimum with respect to some invariants.A popular invariant to use is magnetic helicityH =

∫A · B dV where A is the vector poten-

tial defined by ∇ × A = B, which is approxi-mately conserved in a highly-conducting fluid.For instance, Ciolfi et al. (2009) use a maxi-mum helicity argument to find the ratio betweenpoloidal and toroidal field strengths. This ap-proach was also used by, for instance, Broderick& Narayan (2008) and Duez & Mathis (2009),who also introduce further higher-order invari-ants. The efficacy of these higher-order invari-ants is not completely established; indeed it isnot even certain to what degree helicity shouldbe conserved when a significant fraction of theflux passes through the stellar surface, abovewhich helicity conservation breaks down.

As mentioned above, almost all of the analyticwork has assumed a barotropic equation of state,such as would apply to a convective star. This isdone mainly for mathematical convenience; how-ever it may be of physical relevance in neutronstars, where beta processes eliminate the stablestratification over some timescale (see Reiseneg-ger, 2009). In a barotropic star, one can imaginethat an equilibrium magnetic field might be ableto ‘hold itself down’ against magnetic buoyancysomehow by means of magnetic tension. Thiswould presumably only work if the magneticfield has organised itself globally, with buoyancyacting in opposite directions on opposite sides ofthe star. Physically relevant of course is not justthe question of whether equilibria are possiblein principle, but whether they can actually formfrom realistic initial conditions. It was suggestedby Braithwaite (2012) that buoyancy, acting ona disorganised initial magnetic field, pushes themagnetic field to the surface faster than it isable to organise itself into an equilibrium. Us-ing star-in-box numerical methods Mitchell et al.

(2014) have also investigated this issue. They usean ideal gas EOS with a heating/cooling termwhich maintains a uniform entropy in the star’sinterior. This removes the stabilising effect of thestratification and so is equivalent to a barotropicequation of state, but is numerically easier to im-plement. It is found that the small-scale randommagnetic field which does evolve into a stableequilibrium in a stably stratified star does notreach an equilibrium in an isentropic star.

It seems possible in principle however that anequilibrium could form while the neutron staris non-barotropic, and could somehow adjust tothe changing structure of the star, remainingin a quasi-statically evolving stable equilibriumas the star becomes barotropic. Mitchell et al.(2015) construct various mixed toroidal-poloidalaxisymmetric torus fields in a barotropic star,and use numerical methods to test their stabil-ity in the linear regime. All of the equilibria con-structed prove to be unstable, and the authorstentatively suggest that stable equilibria mightnot exist in barotropic stars. The decay involvesglobal-scale modes and happens on an Alfventimescale. The condition for instability is simplythat the buoyancy frequency is less than the in-verse Alfven timescale. In light of these results, itis probably safe to assume that barotropic starscannot host MHD equilibria. Since neutron starsdo host magnetic fields, it seems they cannot beperfectly barotropic, or that the crust plays animportant role.

3.7 The ‘failed fossil’ hypothesis

Above it was described how an arbitrary mag-netic field evolves towards an equilibrium on anAlfven timescale τA, but the discussion ignoredany (solid-body) rotation of the star, which addsa Coriolis acceleration −2Ω × u to the mo-mentum equation (2). As a general principlein MHD, it can be shown (Frieman & Roten-berg, 1960) that if the rotation is slow, suchthat ΩτA 1, the rotation has little effect onthe evolution and stability of magnetic fields;this can also be seen from a comparison of therelative sizes of terms in the momentum equa-tion. If however ΩτA 1 then one expects theLorentz force to be balanced not by inertia but

19

0 0.5 1 1.5 20

0.5

1

1.5

2

I=F=0

I=0F=0

I=0F=0

Figure 14: Magnetic equilibria found analytically. On the left, an equiibrium from Lyutikov (2010) wherethe toroidal field occupies a volume around the neutral line, and on the right, an equiibrium from Duez etal. (2010) where the magnetic field is confined entirely to the star. Both studies use a barotropic equationof state and Newtonian gravity.

by the Coriolis force, and a comparison of thesizes of these two terms shows that the evolu-tion timescale is no longer equal to the Alfventimescale τevol ∼ τA but instead τevol ∼ τ2

AΩ.This general principle is seen in various contextsin MHD; an early reference is Chandrasekhar(1961, Sections 84 and after, and Figure 101)and it is seen for instance in the growth ratesof various instabilities such as the Tayler in-stability (Pitts & Tayler, 1985; Ibanez-Mejıa& Braithwaite, 2015). The idea of the Corio-lis force, rather than inertia, balancing what-ever is driving fluid motion, is of course alsowell known from atmospheric physics (quasi-geostrophic balance, see, e.g., Pedlosky, 1982).

As described by Braithwaite & Cantiello(2013), this principle should also apply to the re-laxation of an arbitrary initial magnetic field ina star towards an equilibrium. In a non-rotatingstar the field evolves on a timescale τA, its energyfalling as it does so. If its energy falls by a largefactor as it does so (as seems likely; see discus-sion in Section 4) then τA will increase by a largefactor as the relaxation progresses, and the timetaken to reach equilibrium can be approximatedsimply to the Alfven timescale at equilibrium. Ina rotating star the evolution timescale becomesτ2AΩ, and increases in the same way as relaxation

progresses. Putting in some numbers, it takesone year to form an equilibrium of strength 10

kG in a non-rotating star, but 5 × 1011 yr toform an equilibrium of 1 G in a star rotatingwith a period of 12 hours. It may be that in starslacking strong fields, the magnetic fields are stillevolving ‘dynamically’ towards a weak equilib-rium. During this time, the evolution timescaleτ2AΩ will be roughly equal to the age of the star,

and so given the age and rotation period of astar, it should be possible to calculate the fieldstrength. With Vega, assuming a rotation pe-riod of 12 hours and an age of 4×108 years, thisargument predicts a field strength of 20 gauss;for Sirius it predicts about 7 gauss. It may wellbe then that Vega and Sirius (see Section 3.7)contain a non-equilibrium fields undergoing dy-namic evolution. It is easy to reconcile the pre-dictions with observed field strengths of 0.6 and0.2 G: the observations will underestimate thestrength of a smaller-scale field, and one wouldnaturally expect the field strength on the surfaceto be lower than the volume-average predictedby the theory.

Finally, if it can be assumed in all stars whichhosted a pre-main-sequence convective dynamo,that the dynamo leaves behind a magnetic field,then a magnetic field of this order of magni-tude should be visible at the surface of all suchstars during the main sequence. However this as-sumption is not certain – the slow retreat of thepre- main-sequence convective envelope contain-

20

ing a time-dependent dynamo will leave a fieldof small radial length scale, causing the field todecay more quickly via magnetic diffusion. Itssurvival will also be affected by processes likemeridional circulation and differential rotation.see Section 3.8 for a more detailed discussion ofthis point.

3.8 The evolution of fossil fields

As mentioned in Sections 2.1.4 and 2.2.2, thestrength of the fields observed in magnetic early-type stars falls during the main-sequence, and itfalls faster than expected from simple flux con-servation while the star’s radius increases. Var-ious explanations spring to mind, the most ob-vious of which is Ohmic diffusion. The globaltimescale for Ohmic diffusion is of order 1010

years, somewhat longer than the main-sequencelifetime of the least massive stars in question.It may however be possible to massage thistimescale downwards, perhaps by making use ofthe lower conductivity near the surface of thestar – conductivity goes as T 3/2. In the absenceof other effects, one would expect the electriccurrent associated with the magnetic field to dieaway reasonably quickly in the outer part of thestar, so that after some time the field at the sur-face is simply a potential-field extrapolation ofthe field further inside. Depending on the initialgeometry and radial distribution of the magneticfield, this could cause the surface field either torise or fall during the main sequence. This willdepend on the origin of the magnetic field. Themain weakness of finite conductivity in explain-ing this decay though is that it is also observed inmassive stars with much shorter main-sequencelifetimes and somewhat longer Ohmic timescalesthan intermediate-mass stars.

Another possibility is the combination ofbuoyancy and thermal diffusion. In short, a mag-netic feature is in pressure balance with its less-strongly magnetised surroundings, and so its gaspressure must be lower; to avoid moving on adynamic timescale its temperature must there-fore be lower than its surroundings. Heat con-sequently diffuses into it, resulting in a buoyantrise to the surface (Parker 1979c; Acheson 1979;Hughes & Weiss 1995; MacGregor & Cassinelli

2003; Braithwaite 2008). Note that this mecha-nism is distinct from the so-called buoyancy in-stability (or Parker instability) where diffusionis not required. In the low-density environmentnear the surface of a star where heat diffusion isvery efficient, the rise is limited by aerodynamicdrag and takes place at the Alfven speed (seeSection 6.5). Deeper down, the process is lim-ited by heat diffusion, and for a global magneticfield structure its timescale can be expressed interms of the Kelvin-Helmholtz timescale and theplasma-β as τdecay ∼ β τKH (Braithwaite, 2008).On the one hand, this would immediately ex-plain why a similar flux decay is seen in A,B and O stars, since the thermal timescale isroughly the same fraction of the main-sequencelifetime and β falls in roughly the same range atall spectral types. On the other hand, it mightbe tricky to get this process to work fast enough:even assuming that the interior field is ten timesstronger than the surface field, the timescalefor the most strongly magnetised stars (e.g. anAp star with a 30 kG field) would be compa-rable to the main-sequence lifetime. Since thetimescale goes as B−2, the effect in stars withmuch weaker fields would be negligible. How-ever, as with Ohmic diffusion, it may be possibleto massage the numbers in light of the fact thatthe thermal diffusion timescale is much shorternear the surface of the star.

Another possibility is meridional circulation.Its characteristic time scale is the Eddington-Sweet time, τdecay ∼ τKHEgrav/Erot, making itoccupy roughly the same range, as a fraction ofmain-sequence lifetime, at all masses. The geom-etry of this flow has been clarified in the sem-inal work of Zahn (1992). Note the similaritywith the timescale of diffusive buoyancy in theprevious paragraph – the only difference is thatrotational energy has replaced magnetic energy.Since the ratio of energies in this case can bemuch closer to unity than in the previous case,the timescale can also be much smaller, indeedthe fast-rotating magnetic stars should experi-ence relatively fast decay, unless the magneticfield finds some way to coexist alongside themeridional flow. It may be though that this flowis simply inhibited by the magnetic stress in thefossil field. Essentially nothing is present in theliterature regarding how this inhibition might

21

work. A first guess might be that the magneticenergy would have to be greater than the cir-culation kinetic energy, leading to the stabilitycondition Emag > MR2τ−2

KHE−2gravE

2rot, or

β <τ2KH

τ2rot

E3grav

E3rot

. (6)

Alternatively an interaction with the convec-tive core could be the crucial process – the con-vective motion and waves that it sends into theradiative envelope could somehow result in anenhanced diffusion. In this case, one would cer-tainly expect a correlation with mass, since thecore is very small in late A stars but reaches toaround a third of the stellar radius in O stars.This topic is explored in Section 6.3. In any case,important clues will come if/when correlationsare observed between flux decay and mass, rota-tion and field strength.

4 The Origin of Fossil Fields

The discussion in the previous section begs aquestion: where does the variation in magneticfields in otherwise similar stars come from? Animportant clue must be the observed extremerange of field strengths in A stars, with roughly‘equal numbers per decade’ between 200 and2×104 G, as well as the bimodality, with no starshaving fields between a few gauss and 200 gauss(Auriere et al., 2007). The existence of very weakfields such as in Vega and Sirius (Section 2.2.1)highlights the suspicion that the field in the pro-tostellar cloud from which a star forms is notparticularly relevant in determining the fieldsobserved in stars (contrary to the classical viewof the origin of fossil fields). There are variousways in which one might explain the observedrange in field strength; the relevant processesare discussed below approximately in order ofdecreasing scale and/or increasing time.

4.1 Variations within the ISM

According to the traditional model, variationsin magnetic field strength in the interstellarmedium (ISM) are simply carried forwards intothe star. In light of recent results though, includ-ing the very weak fields in Vega and Sirius, this

scenario, at least in its simplest form, now looksvery unlikely – the range in field strengths instars is far greater than that in the ISM. In addi-tion, this model requires an additional ingredientto produce the observed bimodality between Apand other A stars. It is perhaps a clue that thelower threshold of 200 G in Ap stars is the sameas the equipartition field strength at the photo-sphere – in the merger hypothesis (Section 4.6)this might have to be a coincidence. Also, thismodel is compatible with the lack of magneticstars observed in binaries, since collapsing cloudcores with a strong magnetic field will spin downefficiently, whereas cores lacking a strong fieldwill retain too much angular momentum to forma single star and so form a binary. This effect hasbeen seen in simulations (Machida et al., 2008).To summarise, the simple ISM-variation modelignores much of the star formation process andso alone, it will not explain what we see in stars.It may play some role however; in any case it isworthwhile to take a more detailed look at starformation from the perspective of magnetic fieldevolution.

4.2 Core collapse

In star formation, the relative strength of thegravitational and magnetic fields is often ex-pressed as a dimensionless mass-to-flux ratio,defined as λ ≡ 2πG1/2M/Φ, where M and Φare the mass and magnetic flux, or locally ina disc context as 2πG1/2Σ/Bz where Bz andΣ are the field normal to the disc, and surfacedensity; this ratio is conserved if flux freezing isvalid. It is related to the gravitational and mag-netic energies by (ignoring factors of order unity)λ2 = |Egrav|/Emag. A cloud with λ & 1 is saidto be ‘magnetically supercritical’ and will col-lapse, in the absence of significant thermal or ro-tational energy. Conversely a cloud with λ . 1 is‘magnetically subcritical’ and the magnetic fieldsupports the cloud against gravity. Once a cloudhas become supercritical, it can collapse dynami-cally. Magnetic braking becomes ineffective oncethe collapse is super-Alfvenic, so the rotationalenergy becomes larger in relation to the otherenergies. Normally this leads to formation of adisc of radius 100 – 1000 astronomical units.

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4.3 The role of the accretion disk

There is strong evidence that discs containstrong, ordered, net poloidal flux – there are di-rect measurements of the magnetic field in pro-tostellar discs at various radii from 1000 AUright down to 0.05 AU (Vlemmings et al., 2010;Levy & Sonett, 1978; Donati et al., 2005) andthe mass-to-flux ratio is always found to be oforder unity.6

There is additional evidence for the presenceof ordered magnetic fields in discs. Systems rang-ing from protostars to active galactic nuclei usu-ally (not always, and not all of them) show evi-dence of a fast outflow in the form of a collimatedjet. The default model for its origin is the rota-tion of an ordered magnetic field in the inner re-gions of the disk. That is, a field crossing the diskwith a uniform polarity over a significant regionaround the central object. Models assuming theexistence of such an ordered field (as opposed tothe small scale field of mixed polarities gener-ated in MRI turbulence) have been particularlysuccessful in producing fast magnetically drivenoutflows. Accepting this as evidence of the exis-tence of such ordered fields, they might also bethe fields that are accreted to form magnetic A,B and O stars.

The origin of the ordered field in disks is lesscertain. An important constraint on the possibil-ities is the fact that the net magnetic flux cross-ing an accretion disk is a conserved quantity (adirect consequence of div B = 0). It can changeonly by field lines entering or leaving the diskthrough its outer boundary, i.e., any net flux ina disc must be accreted from the ISM.

Given the strong ordered fields in discs, it issomething of a puzzle that in stars we observemass-to-flux ratios λ ∼ 103 in the most stronglymagnetised Ap stars and ratios up to at least 108

in other stars. In other words, we have an ex-tra phenomenon to explain: why even the moststrongly magnetised stars have such weak fields.If in the steady state the star is accreting massand flux in the ratio λ∗, then mass and flux mustbe passing through each surface of constant ra-dius in the disc in the same ratio (ignoring out-

6 Note that β ∼ λ2(h/r)M∗/Mdisc where β =8πP/B2; in a disc therefore it is possible to have bothβ > 1 and λ < 1 at the same time, in constrast to starswhere β ∼ λ2.

flow), even though the local ratio will in generalbe much lower λ(r) λ∗, requiring almost per-fect slippage at all radii. Either there is a fun-damental problem accreting flux through a disc,or there is a bottleneck further in through whichmagnetic flux cannot be accreted, located eitherin the star’s magnetosphere or in the star itself.

This may be related somehow to the fact thataccretion disks are turbulent. Simple estimatesshow that accretion of an external field is veryinefficient if the disk has a magnetic diffusiv-ity similar to the turbulent viscosity that en-ables the accretion (van Ballegooijen, 1989). Nu-merical simulations (Fromang & Stone, 2009)show that this is in fact a good approximationfor magnetorotational turbulence. Though intu-itively appealing, accretion of the field of a pro-tostellar cloud as the source of Ap star fields istherefore not an obvious possibility.

The flux bundles that drive jets from the innerregions of the disk, inferred indirectly from ob-servations, must somehow be due to a more sub-tle process. Numerical simulations for the caseof accretion onto black holes (cf. Tchekhovskoyet al., 2011, and references therein) have shownthat ‘a flux bundle’ of uniform polarity in theinner disk can persist against outward diffusionin a turbulent disk. These results are still some-what artificial since the flux of the bundle insuch simulations depends on the magnetic fieldassumed in the initial conditions. The specula-tion is (cf. Igumenshchev et al., 2003) that thisflux itself starts as a random magnetic fluctua-tion further out in the disk, which somehow isadvected inward with the accretion flow (e.g.,Spruit & Uzdensky, 2005).

It may be then that magnetic stars have ac-creted random magnetic features from the accre-tion disc. Whether such a scenario is realistic isan open question (for recent suggestions in thisdirection see Sorathia et al., 2012). If somethinglike this happens, it might explain the appar-ently unsystematic presence of jets from accre-tion disks. In the context of A stars, the assumedrandomness of the sources in the disk on a rangeof length and time scales, might perhaps explainthe range in field strength of magnetic Ap stars.

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4.4 Destroying flux in or near thestar

Given the evidence for strong ordered fields inthe inner part of the accretion disc, it might benecessary to prevent the bulk of it from enter-ing the star. Star-disc interaction seems to beintrinsically complex and is poorly understood;and it may be difficult to accrete flux from theinner edge of the disc. What flux does reach thestar, however, is certainly not obliged to remainthere in its entirety – excess flux can easily bedestroyed after it reaches the star (Braithwaite,2012). If the protostar is convective, magneticfield is prevented by its own buoyancy from pen-etrating into the star, and if the star is alreadyradiative then the quantity of magnetic helicityis crucial.

Magnetic helicity is a global scalar quantitydefined as H =

∫A ·B dV where A is the vec-

tor potential defined by ∇ × A = B. It canbe shown (Woltjer, 1958) that this quantity isconserved in the case of infinite conductivity. Inplasmas of finite but high conductivity, it hasbeen demonstrated that it is approximately con-served, for instance in the laboratory (Chui &Moffatt, 1995; Hsu & Bellan, 2002) and the so-lar corona (Zhang & Low, 2003).

If the magnetic field in a radiative star isallowed to relax, it will evolve into an equi-librium (Section 3). Once an equilibrium hasbeen reached, energy and helicity are related byH = EL where L is some length scale which iscomparable to the size of the system or, in thiscontext, to the size of the star. Therefore, onecan predict the magnetic energy of the equilib-rium from the helicity present initially – helicityis a more relevant quantity than initial magneticenergy. The observations then imply that thereis an enormous range in the magnetic helicitywhich stars contain at birth, as well as a possi-ble bimodality.

In the symmetrical ‘hourglass model’ of starformation, the helicity is zero. Any field accu-mulated from such an hourglass should thereforedecay to zero energy (cf. Flowers & Rudermanninstability, Section 3.3 and Figure 8). In realityof course, one expects some asymmetry. How-ever, an unrealistically high degree of symmetrywould be required to be left with a field of only

1 G from an initial mass-to-flux ratio of orderunity.

4.5 Pre-main-sequence convection

As they decend from the birth line down theHayashi track, intermediate-mass stars are fullyconvective, before turning onto the Henyey track(leftwards on the HR diagram), during whichtime the convective zone retreats outwards anddisappears (see e.g. Stahler & Palla 2005). Thesituation with stars above very approximately 6M is less well understood; they may have a ra-diative core throughout the pre-main-sequence.In any case, as suggested by Braithwaite (2012)and shown more thoroughly by Mitchell et al.(2015) a star with a constant entropy – as ina convective star – cannot hold onto any pre-vious magnetic field: it rises buoyantly towardsand through the surface on a Alfven timescale.This process does not actually require the con-vective motion itself, just the flat entropy profileit creates. Instead of a pre-existing field, the starwould have a convective dynamo field.

As the convective zone retreats towards thesurface, it should leave something behind in theradiative core. Assuming that the dynamo fluc-tuates, as in the case of the Earth and the Sun,layers of alternating polarity are deposited intothe growing radiative core, rather like we see inthe rocks of the mid-Atlantic ridge, which keepa memory of the polarity of the Earth’s mag-netic field at the time when it solidified frommagma. However, given that the star is fluid,and evolves on a thermal timescale which isvery much greater than the dynamo fluctuationtimescale, the layers will be very thin and shouldannihilate each other by finite resistivity almostimmediately. It is therefore not clear whetheranything could be left behind at all; the best onemight hope for is some net north-south asymme-try from

√N statistics. One way out of this may

be if the convective region retreats more rapidly.This could happen in the inner part of the star:the Schwarzschild criterion is breached first atsome intermediate radius and a thin radiativelayer forms, cutting off the supply to the interiorof deuterium from the ongoing accretion, and re-sulting in a relatively fast transition to a radia-tive core (Palla & Stahler, 1993). This would not

24

work though in the more massive stars, in whichdeuterium burning is less important to the stel-lar structure.

The most obvious serious weakness of pre-main-sequence convection as the origin of Ap-star magnetic fields is that one would expect itto produce a range of magnetic fields strengthsaccording to the rotation rate of the star; it isdifficult to produce any bimodality. Bimodalitycould though perhaps be produced afterwards,if some mechanism existed which destroys mag-netic fields below some threshold. This high-lights the problem that it is not enough to havea theory, whatever it is, that explains a certaintypical field strength. The enormous scatter inthe observed field values is the actual challengeto theory here.

4.6 Mergers

Mergers are a strong candidate to create strongfossil fields in a subset of stars (e.g., Bidel-man, 2002; Zinnecker & Yorke, 2007; Maitzenet al., 2008). Bogomazov & Tutukov (2009) sug-gest that Ap stars may be the result of mergersbetween binary stars with convective envelopeswhose orbits shrink as a result of magnetic brak-ing. This is seems more likely on the pre-main-sequence. Ferrario et al. (2009) point out thatthe observed correlation between mass and mag-netic fraction (Power et al., 2007) could be ex-plained by the need for the merger product tobe radiative, i.e., on the Henyey track. Exactlyhow a merger should produce a fossil field is notunderstood, but we can at least expect plentyof free energy in the form of differential rota-tion. This model would also explain the lack ofclose binaries containing an Ap star, althoughthere are one or two peculiar counter exam-ples to this observational result, for instance thebinary HD 200405 with a period of 1.6 days.Even periods of 3 days would be tricky to ex-plain as the result of a merger in a triple sys-tem; it might be necessary to reduce the orbitalperiod of the resulting binary after the merger.This might be related to a similar issue for themerger hypothesis, namely that a merger prod-uct will initially be rotating close to break-upbut that magnetic stars are observed to rotateslowly. In both cases, angular momentum must

be extracted. One could imagine perhaps thatthe material ejected in a merger, which is esti-mated to be around 10% of the total mass of thetwo merging stars, could absorb angular momen-tum as it flows outwards. Some form of magneticcoupling between the stars and the circumstellarmaterial might be important, just as is thoughtto work to slow down the rotation of single mag-netic stars, as discussed in the next section.

4.7 The rotation periods

Apart from the large range in field strengths inradiative stars, we also want to explain the rangein rotation periods. On the one hand we need anexplanation of how sufficient angular momentumcan be removed to form a star at all, but on theother hand we need to form some stars rotat-ing at close to break-up and others with peri-ods of several decades or more. The correlationbetween the presence of a significant magneticfield and rotation period is strong, but there isstill a large range in rotation period amongststars with similar field strengths. How is it pos-sible at all to spin a star down to a rotationperiod of 50 years, or in other words up to aKeplerian co-rotation radius of ≈ 17 AU (as-suming M = 2M)? Assuming a stellar radiusR = 2R, a surface dipole of 3 kG and the stan-dard r−3 radial dependence, we arrive at just0.5 µG at the co-rotation radius – it is obvi-ously a major challenge for any kind of disc-locking model, where the co-rotation and mag-netospheric (Alfven) radii are comparable, for amagnetic field weaker than the galactic averageto be in equipartition with gas of much greaterdensity than the galactic average.

5 Magnetic Fields and Differen-tial Rotation

Since strong stable fields are already foundamong pre-MS (Herbig Ae-Be-) stars, their ori-gin must lie in earlier phases of star formation,when the protostar was in a state of rapid, possi-bly differential, rotation. The sequence of eventsthat led to the formation of a stable magneticfield is not known, but may have involved pro-cesses of interaction between magnetic field and

25

differential rotation. This interaction takes dif-ferent paths depending on the relative strengthof initial field and initial rotation. Since we donot know the star formation process well enough,we have to consider diffferent possibilities.

Take as a measure of the strength of the mag-netic field the Alfven frequency ωA = vA/R,with vA = B/(4πρ)1/2, where B is a representa-tive field strength, ρ the star’s average densityand R its radius. If ωA is large compared with (arepresentative value of) the differential rotationrate ∆Ω, the evolution of the field configurationunder the Lorentz forces is fast and the field willrelax to a stable equilibrium, if it exists. On topof this, there will be oscillations with frequen-cies of order ωA, reflecting the aftermath of therelaxation process and the differential rotationthat was present initially.

5.1 Phase mixing

These oscillations are then damped by a pro-cess of phase mixing. As a model for this damp-ing consider an idealised case, where the field isin a stable equilibrium to which an azimuthalflow in the form of differential rotation has beenadded as an initial condition. The deformationof the field lines in this flow reacts back on theflow; the result is an Alfvenic oscillation. Theoscillation period is given by an Alfven traveltime. Since the energy of Alfven modes travelsalong field lines, neighbouring magnetic surfacesoscillate independently of each other. Their fre-quencies are in general different, with the resultthat neighboring surfaces get out of phase. Thelength scale in the flow in the direction perpen-dicular to the surfaces decreases linearly withtime. The result is damping of the oscillationon a short time scale. For more detailed discus-sions of this process, see e.g. Heyvaerts & Priest(1983) and the references in Spruit (1999).

5.2 Rotational expulsion

In the opposite case ωA ∆Ω, the differen-tial rotation flow is initially unaffected by thefield. The non-axisymmetric component of thefield (with respect to the axis of rotation) gets‘wrapped up’, such that lines of opposite direc-tion get increasingly close together, increasing

the rate of magnetic diffusion. As a result mag-netic diffusion cancels opposite directions in a fi-nite time. The nonaxisymmetric component de-cays, it is effectively expelled from the regionof differential rotation. The process (‘rotationalexpulsion’, Radler, 1980) is similar to the evolu-tion of a weak field in a steady convective cell(‘convective expulsion’, see Section 6.1.1 below).

This process, if allowed to proceed to com-pletion, will therefore ‘axisymmetrise’ the ini-tial field configuration. In this idealised form, itis probably somewhat academic, however, sincethe wrapping process increases the field strengthlinearly with time (the non-axisymmetric as wellas the axisymmetric component). It may wellhappen that magnetic forces become importantbefore magnetic diffusion has become effective(for discussion see Spruit, 1999). Magnetic in-stabilities, Tayler instability being the first toset in, then take over and determine the furtherevolution (Section 5.4). As discussed below, thiscan have important consequences for the rota-tional properties of the star.

5.3 Angular momentum transportin radiative zones

In A stars, the detection limit for large-scalemagnetic fields is of order few gauss, and some-what lower in very bright stars like Vega and Sir-ius where subgauss fields have been found (Sec-tion 2.2.1 above). It is reasonable to assume thatthe internal field strengths of these stars is ratherhigher than the measured value at the surface,since stronger smaller-scale fields at the surfacewould escape detection and since it would not besurprising if the magnetic field were weaker atthe surface than deeper inside the star. The in-ternal field strength even in these ‘non-magnetic’stars is likely to be of order 10 G or perhaps more(see also Section 3.7).

Even fields below current detection limits forthese stars can have dramatic effects though onthe internal rotation of stars. If r is the distancefrom the center of the star, the torque exertedby Maxwell stresses in a field with toroidal (az-imuthal) component Bφ and radial componentBr is of the order r3BφBr. The torque in a ‘ge-ometric mean’ field B = 〈BφBr〉1/2 of the order1 gauss is sufficient to redistribute angular mo-

26

mentum on a time scale of the star’s main se-quence life time, and to keep the core corotatingwith its envelope as the star spins down by astellar wind torque (a classical argument datingfrom the 1950s).

This idea may however be a little over-simplistic. As mentioned above, the MHD insta-bilities are expected to set in. This could resultin a magnetic dynamo.

5.4 Dynamos in radiative zones

Fields generated by some form of dynamo activ-ity have traditionally been associated with con-vective envelopes, to the extent that dynamo ac-tion in stars was considered equivalent with aprocess of interaction between magnetic fields,convective flows, and differential rotation. Thisis only one of the possibilities, however. In fact,differential rotation alone is sufficient to producemagnetic fields. The most well known exampleis that of magnetorotational fields generated inaccretion disks (Balbus & Hawley, 1991). Theidealization of an accretion disk in this case isa laminar shear in a rotating flow, with a rota-tion rate Ω decreasing with distance r. Dynamoaction triggered by magnetorotational instabil-ity (MRI) quickly (10 – 20 rotation periods). Itgenerates a fluctuating field with a small scaleradial length scale (l ∼ cs/Ω, where cs is thesound speed, comparable to the thickness of theaccretion disk).

Given a sufficiently high magnetic Reynoldsnumber, the energy source of differential rota-tion is sufficient for field generation, in the pres-ence of a dynamical instability of the magneticfield itself. In disks the magnetic instabilities in-volved in ‘closing the dynamo cycle’ are magne-torotational instability (Balbus & Hawley, 1991)and magnetic buoyancy instability (Newcomb,1961; Parker, 1966). In the case of the solar cy-cle, the phenomenology strongly indicates mag-netic buoyancy as the main ingredient in closingthe dynamo cycle (see review by Fan 2009). Thiscontrasts with conventional turbulent mean-fieldviews of the solar cycle (e.g. Charbonneau 2010);for a critical discussion see Spruit 2011, 2012.

In the radiative interior of a star, the highstability of the stratification allows buoyant in-stability only at very high field strengths (cf.

the review in Spruit, 1999). Instead, in such astable stratification a pinch-type instability islikely to be the first to set in. A dynamo cy-cle operating on differential rotation combinedwith this ‘Tayler instability’ has been describedin Spruit (2002). Its application to the solar in-terior predicts field generation at a level justenough to exert the torques needed to keep thecore in nearly uniformly rotation and to trans-port the angular momentum extracted by the so-lar wind (Spruit, 2002; Eggenberger et al., 2005)The magnetic field generated in this model isextremely anisotropic: in the radial (r) direc-tion, the length scale for changes of sign of fieldline direction is very small. It should thus beregarded as a ‘small scale dynamo’ instead ofa global one7. This reflects the dominant roleof the stable stratification, but also the natureof Tayler instability, which is a local one in ther and θ directions. In the azimuthal direction,however, its length scale is large, dominatedby the fastest growing nonaxisymmetric Taylermode, m = 1.

Tests of this dynamo cycle through some formof numerical simulation would be desirable. Ina proof-of-principle simulation by Braithwaite(2006a), a dynamo cycle in a differentially ro-tating stable stratification including thermal dif-fusion was observed with properties as pre-dicted. Simulations for realistic stellar condi-tions present serious obstacles, however. Thepredicted fields are very weak compared withthe stability of the stratification, while the cy-cle time is very large and its length scale smallcompared with the other physical scales of theproblem. A widely cited simulation by Zahn etal. (2007), claimed to be valid for the physicalconditions in the Sun, did not yield dynamo ac-tion. Inspection of the parameter values actuallyused in this simulation shows that the negativeresult is caused by damping of magnetic pertur-bations by the high magnetic diffusivity assumed(orders of magnitude off). With the thermal andmagnetic diffusivities used, the differential rota-tion in this simulation is actually three ordersof magnitude below the threshhold for dynamoaction (eq. 27 in Spruit 2002). The case studied

7The two uses of the term ‘dynamo’ in the literaturemay lead to confusion here. The process is not a globaldynamo as envisaged in ‘mean field’ models.

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in Zahn et al. (2007) is therefore not relevant,neither for questions of existence or otherwiseof differential rotation driven dynamos in stablystratified zones of stars, nor as a test of a givendynamo model.

The large range in length and time scalesinvolved makes simulations for realistic condi-tions in stellar interiors impossible to achieve atpresent. The dynamo mechanism should be di-rectly testable, however, by appropriate simula-tions which lie in a parameter space that satisfiesthe minimum criteria (such as those that werereported already in Braithwaite 2006a).

Recently Jouve et al. (2015) performed sim-ulations of a differentially-rotating star withan incompressible constant-density equation ofstate, finding that the MRI is the dominantdynamo process. In a more realistic stably-stratified star, it is not immediately obviouswhether the MRI or the Tayler instability shoulddominate. Whilst under adiabatic conditions theTayler instability is the first to set in, the situ-ation may be different once thermal diffusion isincluded.

5.5 Observational clues

Only very indirect observational evidence isavailable for or against the existence of mag-netic fields in radiative interiors. The nearly uni-form rotation of the solar interior, as well as itscorotation with the convective envelope, havelong posed the strongest constraints on possi-ble angular momentum transport mechanisms8.In addition, the rotation rates of the end prod-ucts of stellar evolution, the white dwarfs andneutron stars, may provide clues. To the ex-tent that the rotation of these stars descendsdirectly from their progenitors (AGB stars andpre-supernovae), they also contain informationregarding the degree to which the cores of theprogenitors were coupled to their envelopes.The very high effectiveness of Maxwell stress attransporting momentum makes magnetic fieldsthe natural candidate. Transport of angular mo-mentum by internal gravity waves (e.g. Char-bonnel & Talon 2005) may also be an important

8The standard recipes used in stellar evolution calcu-lations “with rotation” in fact fail this constraint ratherspectacularly when applied to the Sun.

mechanism, however (for recent theoretical de-velopments see Alvan et al. 2013).

Figure 15: Specific angular momentum of whitedwarf and neutron stars as a function of initialmass of the progenitor star, as computed by Suijset al. (2008). Green circles: initial core angular mo-mentum, blue triangles: including known hydrody-namic angular momentum transport mechanisms,red squares: including the Tayler-Spruit magnetictorque prescription. The dashed horizontal line in-dicates the spectroscopic upper limit on the whitedwarf spins obtained by Berger et al. (2005). Starsymbols represent astroseismic measurements fromZZ Ceti stars and the green hatched area is pop-ulated by magnetic white dwarfs. The three blackopen pentagons correspond to the youngest Galacticneutron stars, while the green pentagon is thoughtto roughly correspond to magnetars, where thevertical-dotted green line indicates the possibilitythat magnetars are born rotating faster. See Suijset al. (2008) for more details.

It is somewhat uncertain, however, whetherthere is a direct connection between the endproducts and the internal rotation of their pro-genitors. The observed asymmetries in planetarynebulae, for example, indicate highly asymmet-ric mass loss in the final evolution stages of AGBstars. Such asymmetric ‘kicks’ may have resetthe angular momentum of the cores, such thatthe rotation rates of WD actually reflect the‘kicks’ imparted by the last few mass loss eventsrather than the initial core rotation (Spruit,1998). Supernova kicks (Wongwathanarat et al.,2013) may also be the main process determin-

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ing of the rotation of neutron stars (Spruit &Phinney, 1998).

Leaving these complications aside, stellar evo-lution calculations are used to make predic-tions of the rotation rates of end products byincluding (parametrisations of) known mecha-nisms of angular momentum transport such asmeridional circulation (e.g., Zahn, 1992) and hy-drodynamic instabilities. Figure 15 shows re-sults from evolution calculations of massive andintermediate-mass stars. Including the hydrody-namic processes predicts rotation rates that aretoo high by ∼ 2 orders of magnitude, clearly in-dicating that much more effective processes ofangular momentum transport must be presentin stars. When magnetic torques according toSpruit (2002, see also Section 5.4), are included,agreement with observations is better, but dis-crepancy of 1 order of magnitude nevertheless isstill present.

5.5.1 Asteroseismic results

The new asteroseismic results of giants and sub-giants from the Kepler mission have greatly ex-panded the evidence on the internal rotation ofstars other than the Sun (Mosser et al., 2012).The cores of these stars rotate faster than theirenvelopes, with typical periods of 10 to 200 days(see Figure 16). This shows a degree of decou-pling between envelope and interior. The torquesrequired to explain these rotation rates are stillmuch stronger than can be explained by theknown non-magnetic processes, however (withthe possible exception of angular momentumtransport by internal gravity waves, cf. Ogilvie &Lin, 2007; Mathis et al., 2008; Barker & Ogilvie,2011; Rogers et al., 2013). Results from stel-lar evolution calculations using the estimate inSpruit (2002) can be compared with the Keplerrate of Figure 16. As with the rotation rates ofthe end products, the predicted rotation ratesare up to a factor of 10 too high (Cantiello etal., 2014)). (The fact that the disagreement isby a similar factor in both cases may be a coin-cidence.)

Figure 16: Mean period of core rotation as a func-tion of the asteroseismic stellar radius, in log-logscale. Crosses correspond to RGB stars, triangles toclump stars, and squares to secondary clump stars.The color code gives the mass estimated from theasteroseismic global parameters. The dotted line in-dicates a rotation period varying as R2. The dashed(dot-dashed, triple-dot-dashed) line indicates the fitof RGB (clump, secondary clump) core rotation pe-riod. The rectangles in the right side indicate thetypical error boxes, as a function of the rotation pe-riod. From Mosser et al. (2012).

6 Interaction between Convectiveand Radiative Zones

In this section we explore the interaction ofsteady magnetic fields with convection. In par-ticular, we look at the possibility of steady fieldsin the radiative interiors of solar-type stars,shielded from becoming observable at the sur-face (sect. 6.1) and how these might interactwith the convective envelope above (sect. 6.2).In 6.3 we look at at how fossil fields in the radia-tive envelopes of early-type stars might interactwith their convective core; and finally we take abrief look at fully convective stars and the possi-bility of producing magnetic fields in subsurfaceconvective layers in early-type stars.

In contrast with A, B and O stars, amongwhich a significant minority has a detectablefield and the rest of the population has at mostvery weak fields, the cooler stars (those with con-vective envelopes) seem all to display magneticfields, but never of the stable kind seen in A,B and O stars. In addition, there is apparentlysome difference among the cooler stars, between

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magnetic fields in stars such as the Sun whichhave a radiative core and a convective envelope,and in fully convective stars (this is looked atbelow in Section 6.4).

For observation results, the reader is referredto recent spectropolarimetric surveys (Marsdenet al., 2014; Vidotto et al., 2014) which have pro-vided excellent results on magnetic propertiesof late-type stars. Also Pizzolato et al. (2003);Wright et al. (2011) obtained interesting resultswhich link dynamo activity to rotation. In solar-type stars the magnetic fields vary much like theSun’s field, without a consistent steady compo-nent; the small-scale field is much stronger thanthe large-scale field. Upper limits on a steadydipole component on the Sun (which would beobservable as a North-South asymmetry duringthe solar cycle), for example, are probably of theorder of only a few gauss.

In these stars a field anchored in the radia-tive core could in principle be present, but if itis, it seems not to manifest itself at the surface.Two kinds of explanation suggest themselves. Aconvective envelope might somehow be incom-patible with the presence of a fossil field, eitherby preventing it from forming during star forma-tion, or by destroying it soon after (Section 6.2below). Alternatively, it might be that a fossilfield is actually still present in (some) solar typestars, but somehow confined or ‘shielded’ by theconvection zone (Section 6.1).

6.1 Confinement of steady fields inthe interior of sunlike stars

To the extent that a steady internal field con-nects to the convective envelope it might becomedetectable at the surface as a time independentcomponent, superposed on the cyclic dynamo-generated field characteristic of stars with con-vective envelopes. The Sun provides some limitson this possibility. Its cyclic dipole componenthas an amplitude of about 20 G at the magneticpoles (Petrie, 2012). In most cycles some asym-metries are seen between the north and southhemispheres, but no signal of a long-term av-erage polarity has been reported. The implieddetection limit is probably on the order of a fewgauss. This indicates either that a steady field inthe solar interior has a dipolar component below

a few gauss, or that a (possibly stronger) fieldis somehow actively shielded by the convectionzone.

A steady, shielded field (called ‘inevitable’ byits authors) was discussed by Gough & McIn-tyre (1998). Their model assumes a meridionalcirculation near the base of the convection zone,downward at the poles and the equator and up-ward at mid-latitudes. A magnetic field in theradiative zone, such that the field lines are par-allel to its interface with the convective zone, isin contact with this circulation. The authors finda steady solution to a set of reduced equations,with the property that the interior field does notspread into the convection zone.

This is a somewhat surprising result. At itsupper boundary, field lines from the interiorspread upward into the convection zone by mag-netic diffusion. At the poles and equator of themodel, the downward advection of field lines bythe circulation opposes this spreading. At itsmid-latitude, however, advection in the model isupward, into the convection zone, carrying thefield lines in the same direction as the spreadingby diffusion. This does not depend on the par-ticular configuration of the circulation: there isalways at least one region where diffusion andadvection both act to spread the field into theflow. A circulation therefore cannot prevent afield in the radiative interior from diffusing intothe convective envelope. The conflicting resultby Gough and McIntyre is a consequence of thereductions made to the induction equation9.

The conclusion from the above is that a fieldin the radiative interior is inevitably connectedto some degree with the convection zone by fieldlines crossing the interface, even if it does not ex-tend far enough into the convective envelope tobecome visible at the surface. An internal fieldcan be shielded from the visible surface in thisway, while at the same time remaining mechani-cally coupled to flows in the envelope, the differ-ential rotation for example. The construction inGough & McIntyre makes the assumption thatthe interior field is both shielded and decoupled.

9 The model does not include the radial advection anddiffusion of the poloidal component of the magnetic field(Bp). It is instead replaced by an assumed fixed valueB0. The model includes an equation for diffusion of theazimuthal field component (Bφ), but leaves out its radialadvection by the circulation.

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Observations of the interior rotation of theSun through helioseismology show that the ra-diative interior has approximately the same spe-cific angular momentum as the convective enve-lope. Since the Sun spins down through angu-lar momentum loss in the solar wind, this ob-servation indicates the existence of an efficientcoupling between interior and envelope, on anevolutionary time scale. The coupling resultingfrom magnetic interaction between interior andenvelope naturally fits this observation.

Decoupling is a far stronger assumption thanshielding. The contrasting conclusions reachedin numerical simulations (Strugarek et al., 2011;Acevedo-Arreguin et al., 2013) can be traced toconfusion of these two concepts.

6.1.1 Shielding: convective expulsion

Shielding of a sufficiently weak internal fieldcould be achieved by a process of convectiveexpulsion (Zeldovich, 1956; Parker, 1963). Thefirst numerical study of expulsion, idealised as asteady circulating flow interacting with a mag-netic field was by Weiss (1966). It shows howa convective cell can create a field-free regionby pushing the field lines passing through it tothe sides of the cell. This happens in two stages.First the field lines in the cell are ‘wrappedaround’ by the flow, changing the field into acomplex configuration with changes of directionon small scales. This happens on a short timescale, a few turnover times of the cell. In thesecond stage magnetic diffusion reconnects fieldlines of different direction until the wrapped fieldhas disappeared from the cell. In the field whichis now concentrated at the boundaries of the cellthe flow is suppressed. The process is effectiveup to a certain maximum field strength roughlygiven by equipartition between the magnetic andconvective energy densities. In more realistic,time-dependent convective flows, the separationbetween flows and magnetic fields is observed tobe stable once established. An example is thehighly fragmented magnetic field seen at the so-lar surface (Carlsson et al. 2004; Rempel 2014and references therein).

In the case of shielding of a field in the radia-tive interior of stars with convective envelopes,both convective flows and meridional circulation

would contribute to shielding. But as argued inthe above and in Strugarek et al. (2011) bothwould also lead to mechanical coupling of theenvelope to the interior by the poloidal field com-ponent.

6.1.2 Coupling

The above (Section 6.1.1) shows that by a con-vective expulsion process the envelope may beable to effectively shield a field in the radiativeinterior from manifesting itself at the surface.This does not imply, however, that the enve-lope and interior are also mechanically decou-pled. There is always some connection betweenthe two through field lines diffusing from the in-terior into the envelope (see 6.1).

The simulations by Acevedo-Arreguin et al.(2013) focus on the shielding aspect. As in Stru-garek et al. (2011), however, their simulationsalso show a poloidal field connecting the interiorto the envelope. The consequence of this con-nection is that flows in the envelope, whether inthe form of convection or differential rotation,keep the field in the interior in a time-dependentstate. It excludes the steady internal field as-sumed by Gough & McIntyre.

The time dependences that can be covered innumerical simulations are on the order of tens ofrotation periods of the star. This is to be com-pared with the age of the fossil field (nine ordersof magnitude longer).A poloidal field componentthat is inconsequential on numerically accessibletime scales can wreak havoc on the internal fieldalready on time scales that are very short com-pared with the age of the star.

The differential rotation of the envelope, forexample, stretches the connecting poloidal fieldlines into an azimuthal component. Mestel’s wellknown estimate (Mestel, 1953, remark on p735)shows that even a field as weak as a few mi-crogauss would, on an evolution time scale, beamplified to a strength sufficient to affect therotation of the interior. Long before this hap-pens, however, the wound-up azimuthal field willdevelop magnetically driven instabilities. As ar-gued in Spruit (2002), this sequence of eventsis likely to lead to a self-sustained, time depen-dent magnetic field independent of the initialfield configuration.

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6.2 Strong magnetic fields belowconvective envelopes

Discussions of magnetic field evolution are eas-iest when a ‘kinematic’ view is (implicitly) as-sumed: when the fields considered are suffi-ciently weak that their Lorentz forces can be ig-nored to first order. One may wonder what starswith convective envelopes would look like if theycontained magnetic fields as strong as those ofAp stars in their radiative interiors and Lorentzforces cannot be ignored.

The type of behavior of a magnetic field ina flow (convective or otherwise) depends on itsstrength relative to the kinetic energy density inthe flow. Field lines connecting the interior witha convective envelope are subject to advection(i.e., being moved around) by the differential ro-tation ∆Ω between pole and equator that takesplace in the envelope. Take the Sun as a rep-resentative example, where the differential rota-tion rate is ∆Ω is ∼ 10−6 s−1, and the density ρat the base of the convection zone is 0.2 g/cm3.Equipartition of magnetic energy density B2/8πwith the energy density 1

2ρ(r∆Ω)2 in this differ-ential rotation corresponds to a field strength ofabout 105 G. Above this strength the field wouldbe able to affect differential rotation in the con-vection zone. Equipartition with the typical en-ergy density in convective flows at the base of thesolar convection zone on the other hand wouldbe somewhat less, a few kG.

The typical Ap star field (several kG) wouldtherefore significantly interfere with convectionthroughout a convective envelope. Even in sucha strong field, however, some form of reducedenergy transport is likely to exist, since the con-vective flow and the magnetic field can disengagefrom each other by the convective expulsion pro-cess discussed above. The field gets concentratedin narrow bundles in which flows are suppressed.In the gaps between the bundles the fluid is in anearly field-free convective state10.

An example of how this might work is seenin sunspots, which have surface field strengthswell above equipartition with the surrounding

10This separation is reminiscent of the formation offlux bundles in a superconductor in an imposed mag-netic field, cf. http://hyperphysics.phy-astr.gsu.edu/hbase/solids/scbc.html

convective flows. Below the visible surface of asunspot a ‘splitting’ process is present in thefield configuration, extending to just the visiblesurface. It produces gaps through which convec-tive flows can transport heat. This explains boththe inhomogeneities observed in sunspots, andthe relatively large heat flux in sunspots (Parker,1979a; Spruit & Scharmer, 2006). It has convinc-ingly been seen in operation in realistic radia-tive MHD simulations of sunspots (Schussler &Vogler, 2006; Heinemann et al., 2007; Rempel,2011).

Stars with strong, stable fields and with suchsunspot-like phenomenology are not known. Theprocess of accumulation of magnetic flux froma protostellar disk might be different in starswith a final mass like the Sun compared withmore massive stars, since the star’s magneticfield could affect the accretion process. However,it seems more likely that survival of a fossil fieldsomehow is not compatible with a convective en-velope. If this turns out to be the case, it alsoraises the possibility that the convective core ofan Ap star may affect the evolution of its fossilfield. This is discussed further in Section 6.3.

Convective motions in the envelope would im-pose random displacements of the field lines ex-tending through the interior. Since convectiveflows have length scales smaller than the stellarradius, the displacements of field lines by thesemotions are incoherent between their points ofentry and exit from the interior. This ‘tangles’the field lines in the interior: neighboring fieldlines get wrapped around each other. This raisesthe issue of reconnection : if in the course of tan-gling neighboring field lines can exchange pathsby reconnection, the cumulative effect of manysuch events on small scales would act like an ef-fective diffusion process, allowing field lines todrift at a rate much higher than resulting frommicroscopic resistivity.

6.2.1 Reconnection

The consequences of this wrapping process havebeen studied extensively in the context of thesolar corona by Parker (1972, 1979a, 2012, andrefs. therein), who finds that it leads to rapidformation of current sheets (on the length andtime scale of the displacements), through which

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reconnection takes place. Numerical simulationsof this process have been done for the case ofcoronal heating of the Sun driven by convectivedisplacements of the footpoints of the coronalfield in the photosphere (Galsgaard & Nordlund,1996). Using simple MHD simulations, Braith-waite (2015) finds this phenomenon not only ina low-β plasma (as in the corona) but also inorder-unity (e.g., the ISM) and high-β (e.g., stel-lar interiors) plasmas. Particularly relevant forthe present case of reconnection by small scaleflows in a strong background field is the exten-sive study by Zhdankin et al. (2013).

The tangling process transports a certainamount of energy from the convection zone intothe interior, but more importantly the continuedreconnection of field lines in the interior drivenby flows in the envelope effectively also acts asan enhanced diffusion of those field lines that areaffected by the tangling process.

If the effect is large enough, the field could al-ready disappear from the star when accretionof magnetic flux ends, toward the end of thestar’s formation. There is some observational ev-idence relevant for this, since large scale fieldshave been observed also in Herbig Ae-Be stars(cf. Section 2.1.4 above). Though only a few havebeen found so far, they occupy the same range ofsurface temperatures as the main sequence mag-netic Ap stars. The onset of efficient convectionaround F0 coincides with the disappearance ofthe Ap phenomenon (e.g., Landstreet, 1991). Inthe above interpretation it would also mark theonset of enhanced magnetic diffusion. The ob-servations would then imply that the decay ofthe Ap-type field from a star with a convectiveenvelope is in fact effective on a time scale no-ticeably less than the pre-main sequence life ofan F0 star, i.e., less than about 107 yr, or about afactor 100 shorter than the decay time expectedfrom purely Ohmic diffusion.

6.3 Convective cores

The above line of thought about enhanced diffu-sion by ‘tangling’ is also relevant to convectionin the cores of the magnetic early-type stars, es-pecially the more massive ones. Going from es-sentially fully radiative at around 1.5M, theconvective core extends to around r/R ≈ 0.16

in late B stars, around 5M, and up to aroundr/R ≈ 0.3 at even higher masses. In this casesome of the field lines emerging at the surfacecould pass through the core. In the simple dipo-lar configuration of Figures 9, 11 and 12 thesefield lines would populate the magnetic poles.The azimuthal field torus that stabilises the con-figuration as a whole (Section 3.3) needs to belocated in the stably stratified radiative zoneoutside the core. Shuffling of field lines by con-vection could keep the polar field region in asomewhat time-dependent state. This tanglingby convective motion may have a similar effectto the tangling of field lines in the solar corona,where it keeps the field above the convectivezone close to a potential field. Unlike the case ofa convective envelope, however, the stabilizingpart of the field is not connected to the convec-tive region; it is unaffected by reconnection pro-cesses taking place on the polar field lines. Wehypothesise that this explains how Ap star mag-netic fields can coexist with a convective core.

Convection in the core would have the sec-ondary effect of exciting some level of internalgravity waves in the surrounding radiative en-velope (see e.g. Rogers et al. 2013). Waves canalso increase the rate of magnetic diffusion, but,being periodic, their effect (at the same velocityamplitude) is not comparable with the reconnec-tion processes resulting from the wholesale tan-gling of field lines discussed above. The magneticdiffusion time scale by Ohmic diffusion alone isof the order 1010 yr in an Ap star, their main se-quence life time of the order 108 yr. A possiblewave-induced increase of the diffusion rate by afactor 10–100 would still be compatible with thefields seen in Ap stars.

Another possibly important difference be-tween convective core and convective envelopeis the direction of gravity at the boundary be-tween radiative and convective zones. In solar-type stars, cold, fast downflows should penetratesome distance into the radiative zone, but incontrast rising bubbles in convective cores arenot expected to overshoot significantly. In ad-dition to this, magnetic fields have an inherentbuoyancy since they provide pressure withoutcontributing to density, and have a tendency torise, either on some thermal timescale (which inmost contexts is very long) or on a dynamical

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timescale if conditions for buoyancy instability(Newcomb, 1961; Parker, 1966) are met. Bothof these effects would tend to make it easier toexpel a large-scale field from a convective core,and keep it from re-entering, than to prevent thefield in a radiative core from interacting with aconvective envelope. It may be that fossil fieldsdo not enter the convective core. The structureand stability of the non-axisymmetric equilibriain particular (Figure 11, lower panel) would belittle affected by expulsion from the core.

Convective cores are expected to host a dy-namo; this has been seen in the simulations ofBrowning et al. (2004) and Brun et al. (2005).Interestingly, a fossil field in the radiative zonemight have an important effect on the nature ofthe core dynamo. Featherstone et al. (2009) per-formed simulations of a core dynamo in the casewhere the surrounding radiative zone contains afossil field. Without a fossil field, an equiparti-tion field is generated in the core; the additionof a fossil field (significantly weaker than thisequipartition strength) switches the dynamo toa different regime in which the field generated ismuch stronger, which not surprisingly changesthe properties of the convection. This would beanalogous to the situation in accretion discs,where the presence or absence of even a weak netflux through the disc appears to have a funda-mental effect on the nature of the dynamo. Thisresult could be relevant for any compact stellarremnant born out of the core, as it would enableneutron stars and perhaps also white dwarfs toinherit in some way the magnetic properties oftheir progenitors.

6.4 Fully convective stars

In the light of interesting recent observationsof pre-main-sequence stars and low-mass stars,in this section we deviate from the main fo-cus of this article – non-convective zones – todiscuss briefly magnetic fields in fully convec-tive stars. In constrast to solar-type stars, bothmain-sequence stars below about 0.4M andT Tauri stars often display dipole fields of or-der 1 kG (see, e.g., Morin et al., 2010; Yang &Johns-Krull, 2011; Hussain, 2012, and referencestherein). Some recent results are summarized inFigure 17. Gregory et al. (2013) find that fully

convective pre-main-sequence stars tend to havestrong dipolar fields roughly aligned with the ro-tation axis. Those which have a small radiativecore tend to have both strong dipolar and oc-tupolar components, and those with larger ra-diative cores have more complex fields and onlya weak dipole component. Intuitively, this is per-haps not surprising since it is difficult in a thinconvective envelope to get different parts of theenvelope to ‘communicate’ with each other andform a global magnetic field. The length scaleof sunspot systems is comparable to the depthof the convective layer; extrapolation to a fullyconvective star might explain the large scale oftheir field.

Browning (2008) performed simulations of afully convective star, finding that a dipole-likefield can indeed be generated. It is thought thatthe rotation is a key ingredient in producing acoherent large-scale field from smaller-scale mo-tion, as in standard mean field dynamo models.

One can make an analogy here with plan-etary magnetic fields. The Earth, Jupiter andSaturn all contain convective, conducting fluids:the Earth (the outer core) between about 0.19and 0.44RE, Jupiter from 0 or 0.1 out to 0.78RJ,and Saturn between about 0.15 and 0.5RS. Im-portant for the nature of the dynamo is the ratiobetween these inner and outer radii, and in thatsense these planets are analogous to stars withsmall radiative cores. Looking at the results ofGregory et al. (2013) we should expect Jupiterto have a predominantly dipolar field alignedwith the rotation axis, which it does. We ex-pect Earth and Saturn to have probably also asignificant octupole component, but since thisdrops off faster with radius in the overlying non-conducting fluid than the dipole component, wewould still expect the field to be predominantlydipolar at the surface and approximately alignedwith the rotation axis, which is also consistentwith observations.

By analogy though with the Earth’s dynamo,where the polarity of the field changes, we mayspeculate that the same happens in low-massstars, in which case even the strong dipole fieldsobserved would not be steady like those in ra-diative stars.

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Figure 17: Properties of the surface distributionof the magnetic field (derived from Zeeman-DopplerImaging) of the M dwarfs observed with the spec-tropolarimeters ESPaDOnS (Canada-France-HawaiiTelescope) and NARVAL (Telescope Bernard Lyot)as a function of rotation period and mass. Largersymbols indicate stronger fields, symbol shapes de-pict the degree of axisymmetry of the reconstructedmagnetic field (from decagons for purely axisymmet-ric to sharp stars for purely non axisymmetric), andcolours the field configuration (from blue for purelytoroidal to red for purely poloidal). Solid curves rep-resent contours of constant Rossby number Ro = 0.1(saturation threshold) and 0.01. The theoretical full-convection limit (0.35M) is plotted as a horizontaldashed line, and the approximate limits of the threestellar groups discussed in the text are representedas horizontal solid lines. Compiled from the studiesby Morin et al. (2008a,b, 2010); Donati et al. (2008)and Phan-Bao et al. (2009).

6.5 Subsurface convection in earlytype stars

In early-type stars, whilst it seems very unlikelythat a magnetic field generated by a dynamoin the convective core could rise all the way tothe surface on a sensible timescale, these starsalso have convective layers close to the surfacewhich could in principle generate fields whichthen rise to the surface. The convection is drivenby bumps in the opacity at certain temperatures,caused by the ionisation of iron, helium and hy-drogen. Massive stars (above about 8M) havetwo or three such layers, the deepest and ener-

getically most interesting of which is driven byionization of iron (see Cantiello et al., 2009). Ifthis layer hosts a dynamo, there is no difficultyfor the resulting magnetic field to reach the sur-face very quickly via buoyancy, since the thermaltimescale is so short at the very low density inthe overlying radiative layer (Cantiello & Braith-waite, 2011); see Figure 18.

At the surface of the star, magnetic pres-sure supports magnetic features against the sur-rounding gas pressure, meaning that at a givenheight, the gas pressure inside the magnetic fea-tures is lower than in the surroundings. Sincethe photosphere is approximately located wherethe column density of gas above it has a cer-tain value, the photosphere in magnetic featuresis lower than in the surroundings. In a radia-tive star, this means that magnetic spots ap-pear bright. This contrasts to convective starswhere the magnetic field has the additional ef-fect of suppressing convection and therefore heattransfer, and magnetic spots are dark.

For solar metallicity field strengths of ap-proximately 5 to 300 G are predicted – as de-picted in Figure 19. The field strength dependson the mass and age of the star: higher fieldsin more massive stars and towards the end ofthe main sequence. These fields are expected todissipate energy above the stellar surface andcould give rise to, or at least play some rolein, various observational effects such as line pro-file variability, discrete absorption components,wind clumping, solar-like oscillations, red noise,photometric variability and X-ray emission (see,e.g., Oskinova et al., 2012, for a review of thesephenomena). Indeed, if the X-ray luminositiesof various main-sequence stars are plotted onthe HR diagram (Figure 19) a connection withsubsurface convection does seem apparent. Ofcourse, we cannot be certain that it is a magneticfield which is mediating transfer of energy andvariability from the convective layer to the sur-face; one could also imagine that internal gravitywaves are involved. Simulations of the genera-tion and propagation of such gravity waves werepresented by Cantiello et al. (2011).

35

Radiative Layer

Convective zone

Optical depth = 2/3

Emerging B !eld

Figure 18: Schematic of the magnetic field gener-ated by dynamo action in a subsurface convectionzone. Note that magnetic features should appear asbright spots on the surface, rather than as dark spotsas in stars with convective envelopes. From Cantiello& Braithwaite (2011).

6.5.1 Intermediate-mass stars

Stars below about 8M have no such iron-ionisation-driven convective layer, but do havesimilar layers caused by helium and hydrogenionisation. In intermediate-mass stars such asVega and Sirius (spectral types A0 and A1 re-spectively) a dynamo-generated field could floatto the surface from a helium-ionization-drivenconvective layer beneath the surface Cantiello& Braithwaite (2011). Field strengths of a fewgauss are predicted. With the observations doneso far though, it might be difficult to distin-guish between this and the failed fossil hypoth-esis (Braithwaite & Cantiello, 2013) describedabove in Section 3.7. The main difference wouldbe the length scales: the convective layer is verythin and it would be difficult to generate mag-netic features large enough to be detectableas a disc-averaged line-of-sight component (cf.Kochukhov & Sudnik, 2013). In comparison tomore massive stars, late-B and early-A stars arevery quiet in X rays, and so far only upper lim-its on X-ray emission have been established. Forinstance, Pease et al. (2006) obtained an upperlimit of 3 × 1025 erg s−1 in X rays from Vega.Pre-main-sequence stars in this mass range areknown to emit X rays, but Drake et al. (2014)obtain a limit of 1.3×1027 erg s−1 from the 8 Myr

4.8 4.6 4.4 4.2 4.0logTeff

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Figure 19: The field strength predicted at the sur-face of massive stars. The prediction assumes anequipartition dynamo in the convective layer anda B ∝ ρ2/3 dependence in the overlying radiativelayer. Also shown are the X-ray luminosities mea-sured in a number of stars: there does seem tobe some connection between subsurface convectionand X-ray emission. The dotted lines are evolution-ary tracks of stars of various initial masses. FromCantiello & Braithwaite (2011).

old A0 star HR 4796A, demonstrating that thisX-ray activity shuts down roughly when the starreaches the ZAMS. This is consistent with weakdynamo activity; Drake et al. (2014) predict anX-ray luminosity of very approximately 1025 ergs−1 from magnetic activity in Vega, and lower lu-minosites in slower rotators. Cooler than aroundtype A5 (Teff ∼ 10 000 K), X rays become de-tectable again, presumably owing to convectionat the surface; this convection is also detectedmore directly as microturbulence (Landstreet etal., 2009b).

6.5.2 Interaction with fossil fields

The various observational phenomena in mas-sive stars, listed above, seem to be ubiquitous.Notably, they are present also in stars in whichstrong large-scale fields have been detected. Thismeans that if these phenomena are caused bysubsurface convection, that this convection isnot disturbed significantly by the fossil field.However, a fossil field of order 1 kG is in equipar-tition with the predicted convective kinetic en-ergy and should at least have some effect on it

36

(see, e.g., Cantiello & Braithwaite, 2011). Themagnetic field may simply force the entropygradient to become steeper until convection re-sumes. This is often assumed in parametrisa-tions of the effect of magnetic fields in stellarevolution calculations (e.g. Feiden & Chaboyer2013). As discussed above (6.2), however, inter-action of convection with fields at strengths oforder equipartition is very inhomogeneous, andeven small gaps between strands of strong fieldcan allow a nearly unimpeded convective heatflux. The effects of such inhomogeneous fieldson stellar structure are much smaller than inconventional parametrisations based on averagefield strengths (Spruit & Weiss, 1986; Spruit,1991).

In the case of the early type stars (6.5),subsurface convection only transports a modestfraction of the total energy flux and one couldimagine it is easier to suppress. There is re-cent observational evidence that a very strongmagnetic field can indeed suppress the subsur-face convection: Sundqvist et al. (2013) measuremacroturbulent velocities in a sample of mag-netic OB stars, finding that one star in the sam-ple (NGC 1624-2), which has a field of around20 kG, lacks significant macroturbulence, whilstthe rest, which have fields up to around 3 kG,have vigorous macroturbulence of over 20 kms−1. The thermal energy density in the convec-tive layer corresponds to an equipartition field ofaround 15 kG, so this results appears to confirmthat convection can be suppressed only by a fieldcomparable in energy density to the thermal– rather than convective kinetic – energy den-sity, even when convection is weak. Obviouslyit would be useful to improve the observationalstatistics, and to look at stars with fields be-tween 3 and 20 kG. Finally, whilst it seems likelythat macroturbulence (the part thereof which isnot the result of stellar rotation) is essentiallygravity waves produced by subsurface convec-tion, the origin of the various other observationalphenomena in massive stars is less certain; itwould therefore be a very useful to determinewhether strongly magnetic stars lacking macro-turbulence display these other phenomena.

In intermediate-mass stars, there is also someevidence that fossil fields suppress convection.Although late-A stars normally display micro-

turbulence, it seems to be lacking in the mag-netic subset of the population (Shukyak, priv.comm.) However, this is perhaps not directlycomparable to massive stars because the sur-face convection is very weak, and all fossil fieldsare at least an order of magnitude stronger thanequipartition with the convective kinetic energy.

7 Neutron Stars

Field strengths of neutron stars observed as pul-sars and magnetars span the range of 1010 –1015 G (see Fig. 20). (In addition there arethe ‘recycled millisecond pulsars’, with fieldstrengths around 108 G. Their fields are believednot to be representative of their formation, in-stead representing a process related to the recy-cling. See Harding, 2013 for a review of differentclasses of neutron star.) The width of this range:several decades, is similar to that seen in mWDand Ap stars. The presence of a solid crust makesa difference compared with the other classes,however, since it can anchor fields that otherwisewould be unstable. Before this anchoring takesplace, the magnetic field of the proto-neutronstar is subject to the same instabilities and de-cay processes as in Ap stars. This is discussedbelow in Section 7.2.

7.1 Mechanisms of field genera-tion

Three mechanisms have been proposed. Thefield could be inherited from the pre-collapsecore of the progenitor star (the ‘fossil’ theory11),it could be generated by a convective dynamoduring (or shortly after) core collapse, or it couldbe generated by differential rotation alone, viaa magnetorotational mechanism like that oper-ating in accretion disks. This mechanism looksespecially promising for field generation during

11 There is some inconsistency in the literature con-cerning the meaning of ‘fossil field’. In the Ap-star con-text it is normally taken to mean a field left over from anearlier epoch, for instance the pre-main-sequence or theparent ISM cloud, rather than being the result of somecontemporary dynamo process. In the neutron star con-text it means not just this but also that the earlier epochis not the proto-neutron star phase but the pre-collapseprogenitor.

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Figure 20: The well-known P − P diagram on which two readily measurable quantities are plotted,period derivative against period, together with inferred magnetic dipole field strengths and spindown ages(diagonal lines). The main clump is the radio pulsars (red dots), of which around 2000 are known. Atthe top right are the magnetars (green circles), and on the lower left we have old neutron stars with veryweak dipole fields which have been spun up by accretion. Blue circles represent binary systems and starssupernova remnants. The so-called magnificent seven are the pink triangles on the right. Figure providedby T. M. Tauris.

or shortly after collapse of a rotating core (cf.Spruit 2008).

7.1.1 Inheritance

In this scenario the field is simply a compres-sion of what was present in the progenitor, whichmay be either an evolved high-mass star or anaccreting white dwarf. We discuss first the pre-collapse evolution of massive stars.

During the main sequence the core of the staris convective, and afterwards various convectivezones appear and disappear at different loca-tions as burning moves in steps to heavier el-ements, as is illustrated in Figure 21 (Heger2013, personal communication). These convec-tive zones may become relevant to the mag-netic field of the star, especially in the core, outof which the neutron star will eventually form.During the main sequence, we expect an activedynamo in the core, perhaps similar to thosein low-mass stars. For instance, Browning et al.(2004); Brun et al. (2005); Featherstone et al.(2009) performed simulations of an A-star con-

vective core, finding a sustained dynamo. Afterthe main sequence the star moves onto the red-giant branch. Magnetic fields have been observedin several red giants, such as EK Boo, an M5 gi-ant, where a field of B ∼ 1 − 8 G was detected(Konstantinova-Antova et al., 2010). Anotherexample is Arcturus, a K1.5 III giant, where asubGauss field has been found (Sennhauser &Berdyugina, 2012). The nature of these fields ispoorly constrained at present but they are pre-sumably generated by a dynamo in the convec-tive envelope, and may or may not be directlyrelevant for the neutron star’s magnetic field.

Under flux freezing (see also Section 3.1), thefield in a pre-collapse core of radius R0 ∼ 3×108

cm collapsing to a neutron star of radius R =106, will be amplified by factor 105. To explainmagnetar strength fields of 1015 G (more real-istically, internal fields of 1016, see Section 7.3)requires an initial field B0 of 1010 G. This ismore than a factor 10 larger than the largest fieldstrengths seen magnetic white dwarfs (whichhave about the same size and mass as suchcores). The question what could lead to such a

38

Figure 21: A Kippenhahn diagram of a star with an initial mass 22M. Along the horizontal axis islog time before the supernova, so the main sequence takes up only a short space on the left. Note theappearance and disappearance of convective layers. Figure provided by A. Heger.

strong field in the progenitor core is still open.In the accretion-induced collapse (AIC) sce-

nario, a white dwarf passes over the Chan-drasekhar mass limit and collapses into a neu-tron star, the composition of the star being suchthat there is insufficient nuclear energy to pro-duce a supernova. Accreting white dwarfs have avariety of magnetic properties, with large-scalefields observed with strengths up to at least108 G. This does not seem to be quite sufficientto explain fields in magnetars.

7.1.2 Convection

Thompson & Duncan (1993) proposed that mag-netic fields in neutron stars are generated in theyoung neutron star (NS) by a dynamo deriv-ing its energy from the convection with the helpof the differential rotation, estimating an upperlimit to the field strength of 3× 1015 G, a littleabove the highest dipole component measured insoft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXPs). A challenge for the theory

is how to transfer this energy from the lengthscale of convection (around 1 km) to the dipolecomponent on ten times that scale. If magnetichelicity is conserved (see Section 4) then most ofthe energy should be lost once convection diesaway; in fact even if the magnetic field in everyconvective element is twisted in the same direc-tion in some sense (corresponding to maximumhelicity) then one still expects to lose 90% of theenergy (a reduction in field strength of a factorof 3) if the dominant length scale is to rise by afactor of ten. If symmetry breaking doesn’t workand convective elements are twisted in randomdirections then one loses a further factor of

√N

of the energy. It is a general challenge for dy-namo theory to produce large-scale structures,but the problem does not seem insurmountable:rotating fully-convective stars are observed tohave dipolar fields (Morin et al., 2010) and thesame has been reproduced in simulations (e.g.Browning 2008); see also Sect. 6.4. It may bethat magnetars were born very quickly rotating;energetically one is in a much more favourable

39

position if the star is rotating with a period of3 ms or 1 ms at birth. However, one expects sucha fast-rotating, highly-magnetised NS to spindown within a day or so, or even faster if it has astrong wind, injecting 2×1051 to 2×1052 erg intothe supernova and creating a ‘hypernova’. Thereis evidence from supernovae remnants that thisis not the case: Vink & Kuiper (2006) comparedenergies of SN remnants around magnetars andother neutron stars and found no significant dif-ference, concluding that the spin period of mag-netars at birth must be at least 5 ms.

7.1.3 Differential rotation

Energy in the differential rotation could alsobe tapped by the magneto-rotational instabil-ity (MRI, Chandrasekhar, 1960; Acheson, 1978;Balbus & Hawley, 1991). Its importance in thecontext of neutron star magnetic fields comesfrom the exponential growth of MRI on adifferential-rotation timescale. It may be mostrelevant in the proto-neutron star, after the con-vection has finished but when the star is stilldifferentially rotating.

In main-sequence stars, the dynamo mecha-nism proposed by Spruit (2002; see also Braith-waite, 2006a) should dominate over the MRIsince it is inhibited less by the strong entropystratification and works better than the MRIwhen differential rotation is weak. In proto neu-tron stars, which probably have quite strong dif-ferential rotation, it could also convert some ofthe energy in the differential rotation into mag-netic form, but it works more slowly because itinvolves an initial amplification stage where thefield strength increases only linearly with time.The time available in the collapse and immediatepost-collapse phase is likely to be insufficient forthis process. In proto neutron stars, field gener-ation from the magnetorotational instability is amore likely process.

In any case, the fact that older neutron starsare known to have magnetic fields, despite hav-ing used up their natal energy reservoir of dif-ferential rotation, brings us to the next section.

7.1.4 Magnetic helicity

Important for generation of a magnetar field willbe the magnetic helicity present, as when the

driving processes switch off, the star will try torelax into a minimum energy state for its level ofhelicity. The process that determines the helicityproduced by the combination of a field amplifica-tion process, as well as the buoyant instabilitiesthat bring the field to the surface of the star, areunclear. This is just as in the case of the Ap andmWD stars.

Helicity need not be conserved if the field isbrought to the surface since we only expect con-servation in a highly-conducting medium, so itmay still be possible to build or destroy helicityduring the star’s early evolution. To build he-licity, the symmetry (between positive and neg-ative helicity, or in other words between right-and left-handed twist), has to be broken.

In terms of explaining the apparent diversityin neutron star properties, beyond the two de-grees of freedom which place the star on theP − P diagram, it helps to think in terms of therange in available magnetic field configurations(Section 7.3).

7.2 Field evolution before crustformation

When neutron stars form they are differentiallyrotating and convective in most of the volume.As the star cools by neutrino emission the strat-ification becomes stable. Some time later – esti-mates vary from 30 seconds to 1 day after for-mation – a solid crust forms (see, e.g., Suwa,2013, and references therein). At some stage (ei-ther before or after crust formation) any field-generating dynamo present will die away, af-ter which time the field will relax towards anMHD equilibrium; this happens on a dynamicaltimescale, the Alfven timescale τA. This timescale is short, ranging from 3 hours down to0.1 s for field strengths ranging from 1010 to1015 G respectively, so in a non-rotating starone would expect that in most cases an MHDequilibrium must be established before the crustforms. Rotation however should slow down theformation of equilibrium to a timescale given byτevol ∼ τ2

AΩ where Ω is the angular rotation ve-locity of the star (see Braithwaite & Cantiello,2013, and Section 3.7 for a discussion of this ef-fect in A stars). Assuming an initial rotation pe-riod of 30 ms, an equilibrium would then take

40

600 years to form for a field of 1010 G or 2 sec-onds for a field of 1015 G. So it may be thatonly the magnetars really find an MHD equi-librium before the crust forms. Making an esti-mate of the strength of a crust (shear modulusand breaking strain, see e.g. Horowitz & Kadau2009; Hoffman & Heyl 2012), it seems that anon-equilibrium magnetar-strength field wouldnot be stopped from evolving anyway, in con-trast to radio-pulsar-strength fields which couldbe held in position ‘against their will’, so tospeak. Of course it might not be a coincidencethat most neutron stars have fields at about thecrust-breaking threshold; the magnetic field de-cays until it reaches this threshold and is pre-vented from decaying further. In this picture themagnetars would, for some reason, be born witha greater magnetic helicity than the other neu-tron stars; however the so-called central compactobjects (CCOs) with fields of 1010 G would in-dicate intrinsically less efficient field generation.In any case, the location of the crust-breakingthreshold is subject to large uncertainty, becauseof our poor understanding of the properties ofthe crust and perhaps more importantly of thegeometry of the magnetic field and crustal frac-turing.

7.3 The energy budget of magne-tars

There is a consensus that magnetars are poweredby the decay of their magnetic field, whereby it isnecessary for energy to be dissipated in or abovethe crust rather than the interior, to avoid los-ing the energy to neutrinos (e.g., Kaminker etal., 2006). A magnetar with an r.m.s. strength1015 gauss in its interior contains around 3×1047

erg in magnetic energy, enough to maintain amean luminosity of 3 × 1035 erg s−1 for a life-time of 3 × 104 yr (see, e.g., Mereghetti, 2008,for a review of the observations). However, wehave now seen giant flares in three objects, themost energetic of which is thought to have re-leased (very approximately) 3× 1046 erg in justa few seconds; if it is not to be a coincidence tohave observed this many giant flares then the en-ergy source needs to be larger. In addition, mostSGRs and AXPs have somewhat weaker mea-sured dipole fields than 1015 gauss, some are be-

low 1014 gauss. It looks as if the field strength inthe interior of the star needs to be greater thanwe infer from the spindown rate, which gives usjust the dipole component at the surface.

Fortunately this seems very possible. It maybe that the magnetars have a strong toroidalfield in relation to the poloidal component whichemerges through the surface; Braithwaite (2009)and Akgun et al. (2013) found that for a givenpoloidal field strength a much stronger toroidalfield is permitted, since the stratification hindersradial motion and therefore also the instabilityof a toroidal field. Also the magnetar fields couldbe more complex than a simple dipole; Braith-waite (2008) found a range of non-axisymmetricequilibrium where a measurement of the dipolecomponent gives an underestimate of the actualfield strength. Alternatively the field could belargely buried in the stellar interior with only asmall fraction of the total flux actually emerg-ing through the surface; whether this is possi-ble depends both on where the field is originallygenerated and on diffusive processes which bringthe field towards the surface over long timescales(see Braithwaite, 2008; Reisenegger, 2009). Inany case, there are various degrees of freedomavailable; see Section 3.

8 Summary and open questions

The subject of magnetic fields in the interior ofstars inevitably relies to a large extent on theo-retical developments. The increasing quality andquantity of observational constraints, however,is providing more clues and constraints on the-ory than ever before. In parallel, the increasingrealism of numerical MHD simulations makesthem an effective and indispensable means oftesting theoretical speculation. An example isthe subject of ‘fossil fields’ that have becomethe preferred interpretation of the steady mag-netic fields seen on Ap, Bp and O stars. Herenumerical MHD has not only convincingly repro-duced the range of observed surface distributionsof these stars, but also provided physical under-standing of their stability and internal structure.Axisymmetric purely toroidal or purely poloidalfields are unstable, and so they must exist to-gether, in a certain range of strength ratios; the

41

two components can either be comparable toeach other, or the toroidal field can be stronger.The average interior field of a neutron star couldbe much higher than the surface dipole compo-nent inferred from spindown, which would ex-plain how magnetars appear to have a more gen-erous energy budget than one would estimatefrom their dipole components alone.

A key to understanding the nature of suchstable equilibria is magnetic helicity (Sections3.6.1, 4.4 and 7.1.4). To the extent that it is con-served during relaxation of a field configurationit guarantees the existence of stable equilibriaof finite strength: a vanishing field has vanish-ing helicity. An open issue, however, concernshelicity generation: the question of which mech-anism(s) has/have given the field the magnetichelicity that is essential for its long-term sur-vival. Perhaps stochastical coincidences duringthe dynamical phases of star formation and evo-lution (Section 4.3) may be all that is needed.

The recent finding that the fields in Ap starsappear to have a minimum of about 200 G (Sec-tion 2.1) may be a clue, still to be deciphered, forthe formation mechanism of the fields. The muchlower fields of the order of a gauss discovered inthe two brightest A stars, on the other hand,may just be the result of an initially strongerfield that is presently still in the process of de-caying (Section 2.2.1).

Stars in the A-B-O range have convectivecores, which are likely to interact to some ex-tent with the stable fossil field in (the remainderof) the star. This is still a very open question.Some inconclusive speculations on the physicsthat may be involved in such interaction weregiven in Section 6.3. The question may have anobservational connection, however. The distribu-tion of field strengths of ABO stars across theHRD has recently been shown to indicate decayon a somewhat shorter time scale than can be at-tributed to finite (‘Ohmic’) resistivity, perhapsan indication of enhanced diffusion related to in-teraction with the convective core, or with a con-vective envelope developing as the star evolvesoff the main sequence (Section 3.8, 6.3). In starsgoing through a fully convective phase, any fos-sil field (such as that inherited from a molecu-lar cloud) should be lost by magnetic buoyancy.Common to ABO stars, white dwarfs, and neu-

tron stars is the large range in fields strengthsin the population, and the ratio of magnetic togravitational energy, which ranges from about10−16 to 10−6 in all three classes of star. Thesize of this range is a puzzle for any theory ofthe origin of these magnetic fields (Sections 2.1,4.1).

Solar type stars do not show magnetic fieldsremotely resembling the stable Ap configura-tions and strengths. This has led to the spec-ulation that the radiative interiors of these starsmight still have a fossil field, but shielded fromthe surface by the convective envelope. Such afield can then be invoked to explain the near-uniform rotation of the Sun’s radiative interior.A clear distinction has to be made here be-tween shielding by the convective envelope anddecoupling from it (Section 6.1). Convective pro-cesses are known that could shield an internalfield from becoming observable at the surface,but in the presence of magnetic diffusion it isimpossible to avoid mechanical coupling acrossthe boundary between interior and envelope. Itcauses the internal field to evolve on time scalesgoverned by interaction with the differential ro-tation of the envelope; the result will probablylook more like the differential rotation-drivendynamo process discussed in Section 5.4.

Further clues on magnetic fields in the inte-riors of stars come from asteroseismic resultson the internal rotation of giants and sub-giants (Section 5.5.1). The coupling betweencore and envelope deduced from these resultsis far stronger than can be explained with ex-isting hydrodynamic coupling processes, such asshear instabilities. In stably stratified zones ofstars, time dependent self-sustained magneticfields powered only by differential rotation andmagnetic instabilities (i.e. ones governed by theMaxwell stress) are likely to operate, except invery slowly rotating stars. The favored scenariofor such a dynamo process (Section 5.4) faresmuch better in matching the asteroseismic ob-servations, but still misses the target by a sig-nificant factor. Magnetic fields, while probablyinvolved in stably stratified zones of stars, mayhave modes of behavior not yet recognised.

Future progress on the questions raised bythe observations is likely to benefit increasinglyfrom numerical simulations. The main obstacle

42

is the fact that in almost all cases simulationsfor the actual physical conditions in stellar inte-riors will remain out of reach in the foreseeablefuture, irrespective of expected increases in com-puting power. Experience shows that extrapo-lation over the missing orders of magnitude inphysical parameter space cannot be done simplyfrom the simulations themselves. Extrapolationneeds physical understanding formalised in mod-els that cover the asymptotic conditions encoun-tered in stars. The validity of such models canoften be tested well with targeted simulations ofreduced scope (i.e. not with ‘3D stars’). The for-mulation of the models themselves requires moreclassical style theoretical effort, however (com-pare the discussion on p. 11 in Schwarzschild(1970).

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