The Origin of the Bimodal Distribution of Magnetic Fields in Early-type Stars
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The Origin of the Bimodal Distribution of Magnetic Fields in Early-type Stars
Adam S. Jermyn and Matteo Cantiello
1, 2 Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
Submitted to ApJABSTRACTIn early-type stars a fossil magnetic field may be generated during the star formation process or bethe result of a stellar merger event. Surface magnetic fields are thought to be erased by (sub)surfaceconvection layers, which typically leave behind weak disordered fields. However, if the fossil field isstrong enough it can prevent the onset of (sub)surface convection and so be preserved onto the mainsequence. We calculate the critical field strength at which this occurs, and find that it correspondswell with the lower limit amplitude of observed fields in strongly magnetised Ap/Bp stars ( ≈
300 G).The critical field strength is predicted to increase slightly during the main sequence evolution, whichcould also explain the observed decline in the fraction of magnetic stars. This supports the conclusionthat the bimodal distribution of observed magnetic fields in early-type stars reflects two different fieldorigin stories: strongly magnetic fields are fossils fields inherited from star formation or a merger event,and weak fields are the product of on-going dynamo action.
Keywords:
Stellar magnetic fields – Stellar convection zones – Stellar interiors INTRODUCTIONMagnetic fields play many roles in stellar evolution.They are thought to modify stellar winds and enablespin-down (Weber & Davis 1967; ud-Doula et al. 2009),transport angular momentum (Spruit 2002), and influ-ence accretion (Bouvier et al. 2007). They can enhancechemical mixing (Harrington & Garaud 2019) or inhibitit (Gough & Tayler 1966). Moreover, magnetism can in-fluence heat transport, most notably by producing starspots (Berdyugina 2005; Cantiello & Braithwaite 2011,2019).Despite its importance, the origins of stellar mag-netism remains an open question. Numerical simula-tions and theoretical arguments demonstrate that dy-namo action owing to either differential rotation orconvection can amplify infinitesimal seed fields to de-tectable strength (Spruit 2002; Brown et al. 2009). Inearly-type stars with masses less than ≈ M (cid:12) enve-lope convection tends to be quite weak (Cantiello et al.2009), though, so the predicted surface field strengths Corresponding author: Adam S. [email protected] are of order 1 −
10 G (Cantiello & Braithwaite 2019).At higher mass and at solar metallicity, the presenceof the iron convection zone (FeCZ) results in strongerdynamo-generated magnetic fields, with surface ampli-tudes 10 −
300 G (Cantiello & Braithwaite 2011). Simul-taneously, the low magnetic diffusivity of stellar mat-ter means that at any strength a stable magnetic fieldconfiguration (Braithwaite & Nordlund 2006; Duez &Mathis 2010) can remain frozen in from the early for-mation of a star through to the main sequence, so longas it is not disturbed by non-diffusive processes (Cowling1945; Braithwaite & Spruit 2017). By the same token,stellar mergers are common among massive stars andmay be able to generate magnetic fields (Schneider et al.2019) which can survive through to the main sequence.Recently, observations have revealed that early-type(A/B/O) stars exhibit a bimodal distribution of surfacemagnetic field strengths (Auri`ere et al. 2007; Grunhutet al. 2017), and that there is a ‘desert’ range of fieldstrengths in which few or no stars exist (Fossati et al.2015a). This bimodal distribution raises the possibil-ity of using strength to diagnose the origin of magneticfields: perhaps strongly magnetized stars have retained a r X i v : . [ a s t r o - ph . S R ] J un Jermyn and Cantiello fossil fields while weakly magnetized ones are generatingtheir fields via contemporary dynamo processes.The interaction of a magnetic field with convectionhas been discussed in the literature (Gough & Tayler1966; Moss 1987). A strong enough magnetic field cansuppress convection (MacDonald & Petit 2019), withimportant consequences for observed stellar propertieslike macroturbulence (Sundqvist et al. 2013). On theother hand a slightly weaker large scale magnetic fieldthreading the same convective region can be twisted,losing its large scale and stability properties.We begin in Section 2 by reviewing the criterion for amagnetic field to suppress convection. We then presentcalculations of the critical magnetic field B crit at whichsubsurface convection is shut off in stars ranging from2 − M (cid:12) across the Hertzsprung-Russel (H-R) diagram.We find that this field is of order 10 -10 G. We thenshow the strength of the equipartition dynamo field forthe same stellar models (Section 3), and find that thisis generally 10 to 100 times smaller.In Section 4 we combine these results with a sim-ple physical argument for the evolution of the magneticfield in a convective region, and suggest that fossil fieldsweaker than B crit are erased by convection, while thosestronger than B crit are stable. This naturally producesa bimodal distribution of field strengths as well as theapproximate range of field strengths of the magneticdesert. We compare these and more predictions featureswith observations in Section 5 and find good agreement.We conclude with a discussion of the astrophysical im-plications in Section 6. CONVECTION CRITERIONMagnetic fields make the criterion for convective in-stability more strict (Gough & Tayler 1966). This hasbeen studied by multiple authors who gradually incor-porated additional effects such as non-ideal gas behav-ior and radiation pressure (MacDonald & Mullan 2009;MacDonald & Petit 2019). The most general stabilitycriterion of which we are aware is (MacDonald & Petit2019)4 − ββ ( ∇ − ∇ ad ) − v , r v , r + c (cid:18) d ln Γ d ln p (cid:19) < , (1)where ∇ ≡ d ln Td ln P (2)is the temperature gradient in the star, ∇ ad ≡ ∂ ln T∂ ln P (cid:12)(cid:12)(cid:12)(cid:12) s (3) is the adiabatic temperature gradient, β ≡ p gas p (4)is the gas pressure fraction, v , r = B r πρ (5)is the square of the radial Alfv´en speed, and Γ is thefirst adiabatic index, often just called Γ owing to itscommon use.In thermal equilibrium and in the absence of convec-tion, the temperature gradient equals the radiative tem-perature gradient ∇ = ∇ rad ≡ κL πGM σT , (6)where κ is the opacity, L is the luminosity, and M is themass beneath the point of interest. Inserting this intoequation (1) we find4 − ββ ( ∇ rad − ∇ ad ) − v , r v , r + c (cid:18) d ln Γ d ln p (cid:19) < . (7)Some algebra then yields the critical radial magneticfield which prevents convection B crit , r = (cid:115) πρc Q ( ∇ rad − ∇ ad )1 − Q ( ∇ rad − ∇ ad ) + d ln Γ /d ln p , (8)where for compactness we have let Q ≡ − ββ . (9)If the magnetic field is purely horizontal (i.e. v A , r = 0)then it does not stabilize linear motions against convec-tion. However, a mostly horizontal magnetic field onlyoccurs in a small strip at the equator of a dipole field,so we expect that if equation (8) is satisfied by the over-all magnitude of the magnetic field B then it is likelysatisfied over most of the solid angle of the star for B r .Because our arguments in Section 4 are local argumentsthey are not modified by the existence of a subcriticallatitude range, and because the observations in Section 5are of the average field over the surface our interpreta-tion of those is likewise unchanged. As such we nowdrop the subscript ‘r’ and just consider the magnitudeof the magnetic field, not its geometry.It is worth noting that equation (8) is a purely localstability criterion. There could be scenarios in whichthe global curvature of the magnetic field weakens this he Bimodal Origins of Early-Type Stars Magnetism B crit using the Modules for Experi-ments in Stellar Astrophysics (MESA Paxton et al. 2011,2013, 2015, 2018, 2019) software instrument. Details onthe microphysics inputs to this software instrument aregiven in Appendix A. The inlists used to run our modelscan be found on Zenodo (Jermyn & Cantiello 2020).For each stellar model we evaluated the maximum B crit needed to prevent the formation of any subsur-face convection zone. We did this as a function of bothmass and evolutionary history for stars ranging from2 − M (cid:12) , shown in Fig. 1. Here we present results foran initial metallicity of Z=0.02, but in Appendix C wereport results for model grids with Z = 0.014, 0.006 and0.002 as well.At the low-mass end, the critical field is of order 10 Gand is set by the Helium ionization convection zone(HeCZ). This convection zone is driven by second heliumionization, we refer to Cantiello & Braithwaite (2019) fora detailed description and classification of envelope con-vection zones. With increasing mass the HeCZ becomesweaker and the critical field falls to 5 × G. This canbe seen more clearly in Fig. 2, which shows just the fieldneeded to shut off the HeCZ for tracks ranging from2 − M (cid:12) .At higher masses, beginning around 5 − M (cid:12) depend-ing on the age of the star, the FeCZ is much strongerthan the HeCZ and so sets B crit . This results in a largejump in B crit , to 3 × G around a mass of 7 M (cid:12) , even-tually rising to 10 G for 12 M (cid:12) stars.In both regimes the critical field strength varies overthe lifetime of the star. For stars with M < M (cid:12) itvaries by of order 40 per-cent on the main sequence, be-fore rapidly declining by a similar amount when cross-ing the Hertzsprung gap. For stars with M > M (cid:12) the field strength first rises by a factor of 2 − M (cid:12) < M < M (cid:12) the star begins with just the HeCZ,then forms the FeCZ, and in some cases subsequently loses the FeCZ, returning to just having the HeCZ. Inthis last limit the variation in B crit is most dramatic,with an increase of more than 10-fold leading up to thehook, followed by a somewhat smaller decrease in thegap.Because the subsurface convection zones are very in-efficient, turning off convection in these layers producesa minimal impact on the radius and T eff of the star. A log T eff /K1.01.52.02.53.03.54.04.5 l og L / L fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl HeCZ and FeCZ l og B c r i t / G Figure 1.
The critical magnetic field B crit given by equa-tion (8) is shown on a Hertzsprung-Russel diagram in termsof log T eff and log L for stellar models ranging from 2 − M (cid:12) with Z=0.02. The sharp increase in B crit at log L/L (cid:12) ≈ . T eff andlow log L we have omitted regions where a vigorous H con-vection merges with the subsurface convection zones, and wedo not report values calculated for the H convection zone inthis diagram. Red dots show the location of 20% increase infractional age, from the zero age main sequence to hydrogenexhaustion. comparison of stellar models with subsurface convectionversus those without is included in Appendix B.Note that when the models cool below log T eff / K ≈ DYNAMO STRENGTHThe convective dynamo is capable of amplifying smallseed fields to significant amplitudes. The precise fieldstrength at which this saturates is not known, but is be-lieved to be of order the field required to quench convec-tion (Moreno-Insertis & Spruit 1989). For non-rotatingconvection zones this results in a field with small-scalestructure, coherent over distances of order the convec-tive mixing length, with an approximately equipartition
Jermyn and Cantiello log T eff /K1.01.52.02.53.03.5 l og L / L fl M fl M fl M fl M fl M fl M fl HeCZ l og B c r i t / G Figure 2.
The critical magnetic field B crit given by equa-tion (8) is shown on a Hertzsprung-Russel diagram in termsof log T eff and log L for stellar models ranging from 2 − M (cid:12) ,which is the range in which the FeCZ is absent and the HeCZis the most important convection zone. field (Cantiello et al. 2011) B π ≈ ρv , (10)or B ≈ (cid:112) πρv , (11)where v c is the root-mean square of the convective ve-locity.It is instructive to compare saturation field strength(equation 11) to the critical magnetic field (equation 8).In the simple limit of an ideal gas with v A (cid:28) c s , theratio of these fields is B dynamo B crit = v c c s (cid:112) ( ∇ − ∇ ad ) . (12)Convection is inefficient in these stars, so √∇ − ∇ ad isof order unity and v c (cid:28) c s , so the dynamo typicallysaturates at very sub-critical field strengths . This maybe seen in Fig. 3, which shows the dynamo saturationfield strength (equation 11) and the critical field strength B crit as functions of temperature in the subsurface con-vection zones of a main-sequence 2 . M (cid:12) stellar model. This can change in the subsurface convection zones of very mas-sive stars, where the large luminosity can drive turbulent veloci-ties close to c s (Grassitelli et al. 2015; Jiang et al. 2015, 2018) l og B / G HeII HeI H2.4 M fl B Crit B Eq B Eq , Ω log τ log T / K2.55.0 l og v c [ c m / s ] v c Rossby ≡ P rot /P con l og τ R o ss by Figure 3. (Upper) The critical magnetic field strength re-quired to prevent convection (equation 8, green), the non-rotating dynamo saturation stregth (equation 11, blue), thesame with a rotation rate of 150 km s − (equation 15, pink),and the optical depth τ are shown as functions of log T /
Kfor a 2 . M (cid:12) stellar model. (Lower) The convection speed v c and Rossby number ( P rot | N | / π ) are shown as functionsof log T /
K on the same horizontal scale. The model wasextracted at a fractional main sequence age of 0.79.
In all three subsurface convection zones the critical mag-netic field is much stronger than the saturated dynamofield. The same is true for a 5 M (cid:12) model (Fig. 4) and a9 M (cid:12) model (Fig. 5), though in the latter case note thatthe FeCZ has a moderate Mach number ( ≈ .
07) and sothe two scales are only separated by a factor of a few.This more than compensates for the fact that the FeCZhas a lower superadiabaticity ∇ − ∇ ad .In rotating stars the situation is more complex. In nu-merical simulations the magnetic field has been shown tobecome super-equiparition once the Rossby number be-comes small (Christensen & Aubert 2006; Aubert et al.2017). Simulations by Augustson et al. (2016) suggestthat in this limit B ρv ≈ Ω hv c , (13)where Ω is the angular velocity of the convection zone, h ≡ Pρg (14)is the pressure scale height and g is the accelerationowing to gravity. Combining this scaling with the non- he Bimodal Origins of Early-Type Stars Magnetism l og B / G HeII5.0 M fl B Crit B Eq B Eq , Ω log τ log T / K2.55.0 l og v c [ c m / s ] v c Rossby ≡ P rot /P con l og τ R o ss by Figure 4.
Same as Fig. 3 but for a 5 M (cid:12) stellar model at afractional main sequence age of 0.77. l og B / G FeCZ HeII9.0 M fl B Crit B Eq B Eq , Ω log τ log T / K2.55.0 l og v c [ c m / s ] v c Rossby ≡ P rot /P con l og τ R o ss by Figure 5.
Same as Fig. 3 but for a 9 M (cid:12) stellar model at afractional main sequence age of 0.95. rotating limit in equation (11), we find B ≈ (cid:115) πρv (cid:18) hv c (cid:19) . (15)In this rapidly-rotating limit the convection speed is alsoreduced by rotation (Stevenson 1982; Currie et al. 2020),so if we evaluate equation (15) with the non-rotating convection speed we should obtain an upper limit onthe saturation strength of the convective dynamo. Anexample of this upper limit is also shown in Figs. 3, 4,and 5 (dashed) for models rotating at 150 km s − . Thisis a typical value for the equatorial rotational velocityof OBA stars (Huang & Gies 2008; Zorec & Royer 2012;Sim´on-D´ıaz & Herrero 2014). In the 2 . M (cid:12) HeII and9 M (cid:12) Fe convection zones rotation is slow relative toconvection, so the effect of rotation is minimal. By con-trast, the other four subsurface convection zones in thesemodels are slow, so the effect of rotation is to increase B dynamo by a factor of up to 3 − B surface ≈ B dynamo (cid:18) ρ surface ρ CZ (cid:19) / , (16)where ρ CZ is the density in the convection zone and ρ surface is the density at the photosphere.We adopt this correction in Fig. 6, which shows thedynamo field adjusted using equation (16) across theH-R diagram. For M > M (cid:12) the surface field is domi-nated by the FeCZ, producing fields of 10 −
100 G. Forlower masses the magnetic field is generated by the HeIIconvection zone (Fig. 7) and is considerably weaker, oforder 0 . − v c ∝ α , where α is the mixing length parame-ter (Cantiello & Braithwaite 2019). The uncertainty inthis parameter is of order a factor of 2, which translatesinto an order of magnitude uncertainty in the dynamosaturation field strength. MAGNETIC EVOLUTION MODELWe model the evolution of fossil fields through a sim-ple physical argument. When the fossil field strength B fossil > B crit , the fossil field is stable and convectionis shut off. When B fossil < B crit convection is able toproceed.Convection twists the fossil field on a turnover time-scale τ ≈ | N | − , where N is the Brunt-V¨ais¨al¨a fre- Jermyn and Cantiello log T eff /K1.01.52.02.53.03.54.04.5 l og L / L fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl HeCZ and FeCZ fl fl fl l og B Su r f a ce / G Figure 6.
The surface magnetic field B surface from a dynamoin the dominant subsurface convection zones is shown on aHertzsprung-Russel diagram in terms of log T eff and log L for stellar models ranging from 2 − M (cid:12) . We only considerthe HeCZ and FeCZ, and we plot values calculated usinga radial average of the convective velocities. The surfacevalues are calculated assuming a scaling B ∝ ρ / . Theridge at log L/L (cid:12) ≈ . T eff and low log L corresponds to the vigorous onset of H convection, rapidlymoving deeper in the model and merging with the subsurfaceconvection zones. We do not report values calculated for theH convection zone in this diagram. quency, which is imaginary in a convection zone. Thisdestroys the large-scale structure of the field, pushingfield energy towards ever-smaller scales until dissipa-tion reduces the field strength to the dynamo satura-tion strength. The net effect is to expel magnetic fluxfrom the convection zone (Zeldovich 1957; Avdeev et al.1989).The field in the surface radiative zone is unlikely tobe left in a stable configuration and so reconnects on anAlfv´en time. Even if it was in a stable configuration, inthe shallow surface radiative zone of low-mass stars thefield diffuses on a timescale t D ≈ H /η ≈ yr, whilefor higher-mass stars it is expelled by winds on a time-scale of order ∆ M/ ˙ M ≈
50 yr (Cantiello & Braithwaite2011).This process is shown schematically in Fig. 8. Notethat this need not alter the field in deeper regions signif-icantly: it may simply be that the field within the con-vection zone is reduced, decoupling the field observed log T eff /K1.01.52.02.53.03.5 l og L / L fl M fl M fl M fl M fl M fl M fl HeCZ l og B Su r f a ce / G Figure 7.
The average surface magnetic field B surface from a dynamo in the HeII convection zone is shown on aHertzsprung-Russel diagram in terms of log T eff and log L forstellar models ranging from 2 − M (cid:12) . We plot values calcu-lated using a radial average of the convective velocities. Thesurface values are calculated assuming a scaling B ∝ ρ / . at the surface from that deeper in. Moreover, for shal-low convection zones this process likely only expels afraction of the poloidal component of the field, sincereconnecting the toroidal component requires bunchingfield lines close together on scales of order h (cid:28) r tore-route them through the convection zone. Since weassume that the magnetic field was initially stable, theoutcome of this process is also a stable magnetic fieldconfiguration. This is because reconnection events onlyoccur in or above the convection zone, so that magnetichelicity is conserved in deeper regions. In addition, thepredominant loss of poloidal magnetic flux increases theratio of toroidal magnetic energy to poloidal magneticenergy, increasing the configuration’ stability (Braith-waite 2009). The toroidal component is not directly ob-servable though, so in stars which have undergone thisprocess it is possible that a toroidal component remainsin the convection zone.Because τ is much less than the main-sequence lifetimeof the star, the result is that a sub-critical fossil field israpidly erased and replaced with a less structured fieldon the order of the dynamo saturation stregth (equa-tion 15). Thus we predict a bimodal distribution of fieldstrengths, with each mode pointing to a distinct mag-netic field origin. This is shown schematically in Fig. 9. he Bimodal Origins of Early-Type Stars Magnetism BB B
Figure 8.
A fossil field (upper, black) is shown twisted(middle) by convective motion, resulting in reconnection andejection from the near-surface region (lower).
Our predictions based on this analysis are that:1. Early-type stars should either have a weak mag-netic field owing to a dynamo, or a strong fossilfield above the largest B crit encountered at anypoint in its evolutionary history.2. Because B crit generally decreases with increasingmass for M < M (cid:12) , in this mass range we expectto see a higher fraction of stars strongly magne-tized at higher masses.3. Because B crit initially increases with stellar age,we expect the fraction of stars with strong mag-netic fields to decline with age. This is in addi- Fossil Field StrengthIncidence B crit PreserveErase B dynamo Amplify Current Field StrengthIncidence B crit PreserveErase B dynamo Amplify NGC 1624-2Vega
Figure 9.
The evolution of the distribution of stellar mag-netic field strength is shown schematically. Initially thereis a broad distribution. Fields stronger than the criticalfield strength are preserved. Stars with weaker magneticfields see theirs either amplified ( B fossil < B dynamo ) or erased( B fossil > B dynamo ), resulting in a pileup at B dynamo . Vegaand NGC 1624-2 are provided on the lower panel as examplesof stars with dynamo-driven and fossil magnetic fields. Theamplification or erasure takes place on a time-scale of orderthe convective turnover time. However, an unstable mag-netic field may remain above a convection zone for an ohmicdiffusion time (of order 10 yr) or for more massive stars thetime it takes for stellar winds to expel the near-surface ma-terial (of order 50 yr), and so the observable surface fieldshould only decay on the shorter of these time-scales. tion to the effect where flux conservation causes afrozen-in magnetic field to weaken as the star agesand expands.4. Following MacDonald & Petit (2019), if macrotur-bulence is caused by subsurface convection thenstars with B > B crit ought to have little or nomacroturbulence.5. As shown in Appendix C, B crit increases with in-creasing metallicity, so we expect to see fewer starswith strong fossil fields at higher metallicity. Sim- Jermyn and Cantiello ilarly, B dynamo increases with increasing metallic-ity, so we expect to see stronger dynamo-drivenfields at higher metallicity.Note that while this model is physically distinct fromthe magnetic instability model of Auri`ere et al. (2007),the two are similar in that both invoke a critical mag-netic field above which fossil fields are stable against theinstability of interest. In our case, however, the instabil-ity is convective and so we expect B crit to be principallya function of the thermal structure of the star, whereasin their model this depends on differential rotation andso the scale potentially varies with rotation rate and ro-tational history even among stars of the same mass andage. We thus expect the two models to predict differenttrends of magnetization with mass, age, and rotation,though we leave a detailed analysis of the differences forthe future.Finally Braithwaite & Cantiello (2013) discussed thepossibility that some of the observed magnetic fields inrotating early-type stars might be failed fossils. Thesefields are not in a stable configuration (like in Ap andBp stars), but they still evolve relatively slowly thanksto the balance of Coriolis and Lorentz force in themomentum equation. Failed fossils are also expectedto be erased by subsurface convection for amplitudes B < B crit , so our conclusions about a dichotomy inthe origin of magnetic fields and the resulting magneticdesert are independent on the specific fossil field scenarioadopted. Below subsurface convection zones, subcriticalfailed fossil fields can still exist and evolve slowly, andhave an impact on stellar interior properties. OBSERVATIONSWe now compare our predictions to observations.5.1.
Bimodal Field Strengths
First, we predict a bimodal distribution of magenticfield strengths. This is generally what is seen (Ligni`ereset al. 2014).Auri`ere et al. (2007) detected magnetic fields in 28 Apstars with log T eff / K ranging from 3 . − . L/L (cid:12) from 1 . − .
7. For these stars we predict a criticalfield strength of order 700 G. Given that the Ap phe-nomenon is believed to be a result of strong magneticfields altering the chemical mixing of the star (Babel1993), we expect most of this sample to exhibit fossilfields with strengths of order B crit or larger. Indeed allbut one of the other stars in their sample have best-fitfield strengths above 200 G, all but two are consistentwith B >
700 G, and half of the sample have best-fitfield strengths greater than 700 G. At the other end of the spectrum, Auri`ere et al. (2010)placed upper limits of order 5 G on the typical magneticfields of A stars, while spot mesaurements (Balona 2017,2019; Trust et al. 2020) indicate that most of these starsdo have some weak magnetic field. Likewise Petit et al.(2010) detected a magnetic field strength of order 1 G inVega and Petit et al. (2011) detected a field strength of0 . −
100 G and the emergent sur-face flux is of order 1 −
10 G (Cantiello & Braithwaite2019). Moreover, except for rapid rotators we expectthe convective dynamo to primarily generate a small-scale magnetic field with almost no dipole component,so it is consistent that of the positive detections themore rapidly rotating star (Vega) exhibits a strongerlarge-scale field.At the high-mass end, Fossati et al. (2015a) report thedetection of a 60 −
230 G field in β CMa, and a lower limitof 13 G for (cid:15)
CMa. Despite their claim that this is in-consistent with a magnetic desert model, these observa-tions are consistent with what we expect if the fields aredynamo-generated, though it is perhaps somewhat sur-prising that so much power lies in the dipole mode giventheir moderate rotation ( v sin i ≈
20 km s − ). β CMaand (cid:15)
CMa are extremely luminous, massive stars with M ≈ M (cid:12) , log L/L (cid:12) ≈ .
4, and log T eff / K ≈ .
4. Iftheir magnetic fields are generated by subsurface con-vective dynamos, the expected surface field strength isof order 30 G, which is consistent with observations of (cid:15)
CMa and not far from what is observed in β CMa, es-pecially given the uncertainties in deriving equipartitionmagnetic fields from mixing length theory (Cantiello &Braithwaite 2019).We feel compelled to mention that despite this gen-eral agreement with our model, it is possible that themagnetic desert results simply from observational in-completeness (Kholtygin et al. 2010). This makes ourfurther predictions especially salient, as they are lesssusceptible to this difficulty.5.2.
Mass Distribution
Because B crit generally decreases with increasing massfor M < M (cid:12) , also we expect to see a higher fraction ofstars strongly magnetized at higher masses. Sikora et al.(2019) report a volume-limited sample of 52 chemicallypeculiar A and B stars, and find that the fraction of de-tectable magnetic fields rises from 10 − at M = 2 M (cid:12) toover 10 per-cent at 3 − M (cid:12) . A majority of the increasein magnetic fraction lies between 2 . M (cid:12) and 3 M (cid:12) . Ifour model for the origins of these magnetic fields is cor-rect, and if the typical fossil field strength of chemically he Bimodal Origins of Early-Type Stars Magnetism B crit (3 M (cid:12) ) ≈
700 G and B crit (2 . M (cid:12) ) ≈
900 G.At higher masses, Kholtygin et al. (2010) report thatthe fraction of O stars with measured magnetic fields is1 / B crit risingby 10 −
30 fold in the mass range 5 M (cid:12) < M < M (cid:12) .If the initial fossil field strength in O stars is similarto that in B stars, this increase means that convectionzones erase the fossil fileds in a much greater fraction ofO stars than B stars, resulting in weaker fields and moredifficult detections.5.3. Age Distribution
Because B crit initially increases with stellar age, weexpect the fraction of stars with strong magnetic fieldsto decline with age. Moreover the radius of the starincreases with age, so if magnetic flux is conserved thefossil magnetic field should decrease in strength as thestar ages, pushing ever-more stars below the critical fieldstrength and erasing their fossil fields. The combinedeffect of B crit increasing and R increasing with time re-sults in B/B crit falling by a factor of 3 −
10 over themain-sequence evolution of these stars (Fig. 10).This prediction is consistent with observations of Oand B stars (Fossati et al. 2016), which suggest thatthe magnetic fraction declines starting around 0 . e − t/τ , where τ is the main-sequencestellar lifetime. Some of this decline can be attributedto flux conservation as the stars expand, which couldproduce a factor of a few, though it is likely that thisdoes not explain the full decline.That the decline in field strength happens on the evo-lutionary time-scale of the star despite the wide massrange considered suggests that it is a matter of stellarstructure as our model predicts, rather than being set bythe magnetic diffusion time. Medvedev et al. (2018) finda similar decrease in the mean magnetic field strengthwith age. Their figure 9 shows that while there is a slightdecrease in the strongest O/B stellar magnetic fieldswith age, the bulk of the reduction in mean magneticfield arises due to the appearance of O stars with weak(30 −
300 G) magnetic fields around t/τ ≈ . −
1. This isconsistent with the dynamo fields of less massive starsbeing too weak to detect, and with a sub-populationof O stars developing subsurface convection zones andreplacing their fossil magnetic fields with much weakerequipartition ones. log T eff /K1.01.52.02.53.03.54.04.5 l og L / L fl M fl M fl M fl M fl M fl M fl M fl M fl M fl l og B c r i t ( R ∗ / R fl ) Figure 10.
The critical magnetic flux B crit ( R/R (cid:12) ) isshown on a Hertzsprung-Russel diagram in terms of log T eff and log L for stellar models ranging from 2 − M (cid:12) withZ=0.02. We do not report values calculated for the H con-vection zone in this diagram. Red dots show the locationof 20% increase in fractional age, from the zero age mainsequence to hydrogen exhaustion. Other lines of evidence point in the same direction.Briquet et al. (2007) report that Bp stars are on aver-age much younger and more strongly magnetized thanSPB stars, suggesting either that magnetic fields inter-fere with the pulsations of SPB stars or that many Bpstars lose their magnetic fields en route to becoming SPBstars. The former possibility is inconsistent with obser-vations of the SPB star o Lup with B ≈ Macroturbulence
If macroturbulence is a result of subsurface convec-tive motions, we expect stars with
B > B crit to showlittle or no evidence of macroturbulence . In a similaranalysis for more massive stars, first Sundqvist et al.(2013) and later MacDonald & Petit (2019) suggest thisas the reason that NGC 1624–2 lacks macroturbulence,and with a measured field strength of 16 −
20 kG it is agood candidate for this effect. Similarly, since microturbulence is believed to be caused by sub-surface convection (Cantiello et al. 2009), it should be absent ornegligible in stars with
B > B crit Jermyn and Cantiello
HD 215441 (Babcock’s star) provides a similar exam-ple, with T eff ≈ ,
500 K and B ≈
67 kG. Landstreetet al. (1989) report that the spectrum of HD 215441 re-quires macroturbulence ξ > − and is consistentwith zero macroturbulence.The other candidate of which we are aware isHD 54879. This star has T eff ≈ ,
000 K and is stronglymagnetized, with mean longitudinal field B = − ± ξ = 4 ± − from HARPS and5 ± − from FORS 2 (Castro et al. 2015), comparedwith more typical values of 20 −
60 km s − (Sundqvistet al. 2013; Sim´on-D´ıaz et al. 2017). DISCUSSIONWe predict that strong magnetic fields in early-typestars are fossil fields, and that weak magnetic fields inthese stars emerge from dynamo action in subsurfaceconvection zones. In our model these convection zonesserve to erase any near-surface evidence of fossil fieldsby twisting them down to small length-scales where theymay be dissipated. An important task for future workis to test this basic physical picture in numerical simu-lations.If correct, this prediction means that the observed bi-modal distribution of magnetic fields is really an indica-tion of two populations: one in which the magnetic fieldwas strong enough to prevent near-surface convection,and one in which it was not. These populations oughtto exhibit very different surface magnetic field evolution,and may appear qualitatively different in terms of re-lated near-surface phenomena like macroturbulence andmicroturbulence.Our scenario may be distinguished from thatof Auri`ere et al. (2007) by noting that their critical mag-netic field strength is dependent on the stellar rotationrate while ours is not. As a result their model predictsthat the population of strongly-magnetized stars shiftsto weaker field strengths as the rotation period increaseswhile our model predicts no dependence on stellar rota-tion rate. This comparison is potentially complicated inpractice because magnetic breaking should reduce stel-lar rotation rates, but we are hopeful that these effectsmay be disentangled by comparing population synthesismodels produced with our scenario and that of Auri`ereet al. (2007).Recently low-frequency variability has been detectedin massive stars (Bowman et al. 2019). There are com-peting explanations for this phenomenon, including in-ternal gravity waves emitted by core convection (Bow-man et al. 2019) and motions excited by subsurface con-vection (Lecoanet et al. 2019). If the same phenomenon is detected in any star with a magnetic field
B > B crit that would be strong evidence against the hypothesisthat the variability originates in a subsurface convectionzone. Conversely, an absence of low-frequency variabil-ity in strongly magnetized massive stars could point toan origin in subsurface convection zones.Our model also has consequences for angular momen-tum transport. While we posit that subsurface convec-tion erases any subcritical fossil field at the surface, itlikely leaves any magnetic field in the deeper interiorlargely unaltered. Given the large fraction of early-B/late-O type stars with magnetic fields (Power et al.2007; Fossati et al. 2015b; Grunhut et al. 2017), it seemsplausible that a large fraction of early-type stars with-out significant surface magnetism could still be stronglymagnetized below the subsurface convection layers.A strong fossil magnetic field hidden in the interior of astar could be subcritical yet still play an important rolein angular momentum transport, potentially enforcingnearly rigid rotation via magnetic tension. For instancea field strength of 10 G in a medium of typical stellardensity ρ ≈ . − suffices to generate a specifictorque of order 10 erg g − , which yields an angular ac-celeration over r ∼ × cm of 10 − s − ≈ Ω (cid:12) / Myr,where Ω (cid:12) is the mean angular velocity of the Sun. Soover the main-sequence lifetime of a massive star evensuch a weak and highly-subcritical field is enough toredistribute the entire angular momentum of the starmany times over.Conversely, differential rotation can amplify magneticfields via the Spruit-Tayler dynamo (Spruit 2002; Fulleret al. 2019). With significant (order unity) differentialrotation this can generate magnetic fields comparable to B crit and so could provide another source of supercriti-cal fossil fields, shutting off convection if the dynamo isactive before subsurface convection layers form.Finally, in stars with multiple subsurface convectionzones, it is possible that the magnetic field from theFeCZ is strong enough to shut off the weaker overlyingHeCZ. Because these dynamo-generated magnetic fieldshave significant power at small scales, this likely doesnot happen everywhere in the HeCZ at the same timeand so manifests with patches of active Helium-drivenconvection and patches of quiescence. he Bimodal Origins of Early-Type Stars Magnetism A. MESA MICROPHYSICSAll calculations were done with MESA version 11701. The MESA EOS is a blend of the OPAL Rogers & Nayfonov(2002), SCVH Saumon et al. (1995), PTEH Pols et al. (1995), HELM Timmes & Swesty (2000), and PC Potekhin &Chabrier (2010) EOSes.Radiative opacities are primarily from OPAL (Iglesias & Rogers 1993, 1996), with low-temperature data fromFerguson et al. (2005) and the high-temperature, Compton-scattering dominated regime by Buchler & Yueh (1976).Electron conduction opacities are from Cassisi et al. (2007).Nuclear reaction rates are a combination of rates from NACRE (Angulo et al. 1999), JINA REACLIB (Cyburt et al.2010), plus additional tabulated weak reaction rates Fuller et al. (1985); Oda et al. (1994); Langanke & Mart´ınez-Pinedo (2000). (For MESA versions before 11701): Screening is included via the prescriptions of Salpeter (1954);Dewitt et al. (1973); Alastuey & Jancovici (1978); Itoh et al. (1979). (For MESA versions 11701 or later): Screeningis included via the prescription of Chugunov et al. (2007). Thermal neutrino loss rates are from Itoh et al. (1996). B. EFFECTS OF SUPPRESSING CONVECTIONHere we look at the possible effects of suppressing subsurface convection zones on the effective temperature andluminosity of stars. The suppression of subsurface convection is achieved by forcing the flux to be purely radiativebelow a temperature of 500,000 K. This is to mimic the effect of a magnetic field with amplitude
B > B crit . Fig. 11shows that the effect of shutting off the subsurface convective regions in a 10 M (cid:12) model is pretty much negligible. Thisis expected, since the flux carried by convection in these regions is very small (Usually 1% or less, see e.g. Fig.6 inCantiello & Braithwaite 2019). C. GRIDS AT DIFFERENT METALLICITIESHere we present results for the critical and surface magnetic field at metallicities of Z=0.014, Z=0.006, and Z=0.002,representing early-type stars in the Galaxy (MW), Large Magellanic Cloud (LMC), and Small Magellanic Cloud (SMC)respectively (e.g. Yusof et al. 2013). The initial Helium content of the grids is Y=0.2659 (MW), Y=0.2559 (LMC), andY=0.2508 (SMC). The metallicity is initialized scaling the standard solar composition of Grevesse & Sauval (1998).REFERENCES
Alastuey, A., & Jancovici, B. 1978, ApJ, 226, 1034,doi: 10.1086/156681Angulo, C., Arnould, M., Rayet, M., et al. 1999, NuclearPhysics A, 656, 3, doi: 10.1016/S0375-9474(99)00030-5 Aubert, J., Gastine, T., & Fournier, A. 2017, Journal ofFluid Mechanics, 813, 558, doi: 10.1017/jfm.2016.789Augustson, K. C., Brun, A. S., & Toomre, J. 2016, ApJ,829, 92, doi: 10.3847/0004-637X/829/2/92 Jermyn and Cantiello log T eff /K3.83.94.04.14.2 l og L / L fl M fl M fl B > B crit
Figure 11.
Impact of suppressing subsurface convection on the evolution of a 10 M (cid:12) model. To mimic the presence of amagnetic field with B > B crit , for
T < ,
000 K we force the flux to be exclusively radiative. The effect is negligible duringthe main sequence.Auri`ere, M., Wade, G. A., Silvester, J., et al. 2007, A&A,475, 1053, doi: 10.1051/0004-6361:20078189Auri`ere, M., Wade, G. A., Ligni`eres, F., et al. 2010, A&A,523, A40, doi: 10.1051/0004-6361/201014848Avdeev, E. I., Dogel’, V. A., & Dolgov, O. V. 1989, ZhurnalEksperimentalnoi i Teoreticheskoi Fiziki, 96, 885Babel, J. 1993, Astronomical Society of the PacificConference Series, Vol. 44, Diffusion Models for MagneticAp-Stars, ed. M. M. Dworetsky, F. Castelli, &R. Faraggiana, 458Balona, L. A. 2017, MNRAS, 467, 1830,doi: 10.1093/mnras/stx265—. 2019, MNRAS, 490, 2112, doi: 10.1093/mnras/stz2808Berdyugina, S. V. 2005, Living Reviews in Solar Physics, 2,doi: 10.12942/lrsp-2005-8Bouvier, J., Alencar, S. H. P., Harries, T. J., Johns-Krull,C. M., & Romanova, M. M. 2007, in Protostars andPlanets V, ed. B. Reipurth, D. Jewitt, & K. Keil, 479.https://arxiv.org/abs/astro-ph/0603498Bowman, D. M., Burssens, S., Pedersen, M. G., et al. 2019,Nature Astronomy, 3, 760,doi: 10.1038/s41550-019-0768-1Braithwaite, J. 2009, MNRAS, 397, 763,doi: 10.1111/j.1365-2966.2008.14034.xBraithwaite, J., & Cantiello, M. 2013, MNRAS, 428, 2789,doi: 10.1093/mnras/sts109 Braithwaite, J., & Nordlund, ˚A. 2006, A&A, 450, 1077,doi: 10.1051/0004-6361:20041980Braithwaite, J., & Spruit, H. C. 2017, Royal Society OpenScience, 4, 160271, doi: 10.1098/rsos.160271Briquet, M., Hubrig, S., De Cat, P., et al. 2007, A&A, 466,269, doi: 10.1051/0004-6361:20066940Brown, B. P., Browning, M. K., Brun, A. S., Miesch, M. S.,& Toomre, J. 2009, Astronomical Society of the PacificConference Series, Vol. 416, Dynamo Action and Wreathsof Magnetism in a Younger Sun, ed. M. Dikpati,T. Arentoft, I. Gonz´alez Hern´andez, C. Lindsey, &F. Hill, 369Buchler, J. R., & Yueh, W. R. 1976, ApJ, 210, 440,doi: 10.1086/154847Buysschaert, B., Neiner, C., Martin, A. J., et al. 2018,arXiv e-prints, arXiv:1808.05503.https://arxiv.org/abs/1808.05503Cantiello, M., & Braithwaite, J. 2011, A&A, 534, A140,doi: 10.1051/0004-6361/201117512—. 2019, ApJ, 883, 106, doi: 10.3847/1538-4357/ab3924Cantiello, M., Braithwaite, J., Brandenburg, A., et al. 2011,in IAU Symposium, Vol. 272, Active OB Stars:Structure, Evolution, Mass Loss, and Critical Limits, ed.C. Neiner, G. Wade, G. Meynet, & G. Peters, 32–37,doi: 10.1017/S174392131100994X he Bimodal Origins of Early-Type Stars Magnetism log T eff /K1.01.52.02.53.03.54.04.5 l og L / L fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl HeCZ and FeCZMW l og B c r i t / G log T eff /K1.01.52.02.53.03.54.04.5 l og L / L fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl HeCZ and FeCZMW l og B Su r f a ce / G Figure 12.
Left: The critical magnetic field B crit given by equation (8) is shown on a Hertzsprung-Russel diagram in terms oflog T eff and log L for stellar models ranging from 2 − M (cid:12) with Z = 0.014. The ridge at log L/L (cid:12) ≈ . T eff and low log L corresponds to the vigorous onset of H convection, rapidlymoving deeper in the model and merging with the subsurface convection zones. We do not report values calculated for the Hconvection zone in this diagram. Right: The surface magnetic field B surface from a dynamo in the dominant subsurface convectionzones is shown on a Hertzsprung-Russel diagram in terms of log T eff and log L for stellar models ranging from 2 − M (cid:12) with Z= 0.014. We only consider the HeCZ and FeCZ, and we plot values calculated using a radial average of the convective velocities.The surface values are calculated assuming a scaling B ∝ ρ / . The ridge at log L/L (cid:12) ≈ . T eff and low log L corresponds to the vigorous onset of H convection, rapidly movingdeeper in the model and merging with the subsurface convection zones. We do not report values calculated for the H convectionzone in this diagram.Cantiello, M., Langer, N., Brott, I., et al. 2009, A&A, 499,279, doi: 10.1051/0004-6361/200911643Cassisi, S., Potekhin, A. Y., Pietrinferni, A., Catelan, M., &Salaris, M. 2007, ApJ, 661, 1094, doi: 10.1086/516819Castro, N., Fossati, L., Hubrig, S., et al. 2015, A&A, 581,A81, doi: 10.1051/0004-6361/201425354Christensen, U. R., & Aubert, J. 2006, Geophysical JournalInternational, 166, 97,doi: 10.1111/j.1365-246X.2006.03009.xChugunov, A. I., Dewitt, H. E., & Yakovlev, D. G. 2007,PhRvD, 76, 025028, doi: 10.1103/PhysRevD.76.025028Cowling, T. G. 1945, MNRAS, 105, 166,doi: 10.1093/mnras/105.3.166Currie, L. K., Barker, A. J., Lithwick, Y., & Browning,M. K. 2020, MNRAS, 493, 5233,doi: 10.1093/mnras/staa372Cyburt, R. H., Amthor, A. M., Ferguson, R., et al. 2010,ApJS, 189, 240, doi: 10.1088/0067-0049/189/1/240Dewitt, H. E., Graboske, H. C., & Cooper, M. S. 1973,ApJ, 181, 439, doi: 10.1086/152061 Duez, V., & Mathis, S. 2010, A&A, 517, A58,doi: 10.1051/0004-6361/200913496Ferguson, J. W., Alexander, D. R., Allard, F., et al. 2005,ApJ, 623, 585, doi: 10.1086/428642Fossati, L., Castro, N., Morel, T., et al. 2015a, A&A, 574,A20, doi: 10.1051/0004-6361/201424986Fossati, L., Castro, N., Sch¨oller, M., et al. 2015b, A&A,582, A45, doi: 10.1051/0004-6361/201526725Fossati, L., Schneider, F. R. N., Castro, N., et al. 2016,A&A, 592, A84, doi: 10.1051/0004-6361/201628259Fuller, G. M., Fowler, W. A., & Newman, M. J. 1985, ApJ,293, 1, doi: 10.1086/163208Fuller, J., Piro, A. L., & Jermyn, A. S. 2019, MNRAS, 485,3661, doi: 10.1093/mnras/stz514Gough, D. O., & Tayler, R. J. 1966, MNRAS, 133, 85,doi: 10.1093/mnras/133.1.85Grassitelli, L., Fossati, L., Sim´on-Di´az, S., et al. 2015,ApJL, 808, L31, doi: 10.1088/2041-8205/808/1/L31Grevesse, N., & Sauval, A. J. 1998, SSRv, 85, 161,doi: 10.1023/A:1005161325181 Jermyn and Cantiello log T eff /K12345 l og L / L fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl HeCZ and FeCZLMC l og B c r i t / G log T eff /K12345 l og L / L fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl HeCZ and FeCZLMC l og B Su r f a ce / G Figure 13.
Same as Fig. 12 but for stellar models ranging from 2 − M (cid:12) with Z = 0.006. log T eff /K123456 l og L / L fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl HeCZ and FeCZSMC l og B c r i t / G log T eff /K123456 l og L / L fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl M fl HeCZ and FeCZSMC l og B Su r f a ce / G Figure 14.