Magnetic Spots on Hot Massive Stars Priv.Comm.)

Home , Ap and Bp stars

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OB in magnetism e-mail: Universität Bon der für Astronomie Argelander-Institut rsuesaehih a aiettesle sht brigh hot, as themselves manifest can height scale pressure ufc ants ol ersosbefrpooercvar photometric for responsible be could magnetism surface Conclusions. Results. Methods. Aims. Context. rngopeeet n srfre oa h io convecti “iron the as Th to phenomena. referred these is of and some elements for iron-group responsible C be clumping. could pr wind surface and line (DACs) turbulence, components photospheric absorption discrete are phenomena these Among hl tti tg ti di is it stage this at While ff rn eussto requests print suigdnm cinpouigmgei ed tequipar at fields magnetic producing action dynamo Assuming [email protected] efidta antcfilso su of fields magnetic that find We o uiossasso ait fpeoeai hi photo their in phenomena of variety a show stars luminous Hot ipeetmtsaemd sn h usraepoete o properties subsurface the using made are estimates Simple oaie antcfilscudb iepedi hs early those in widespread be could fields magnetic Localized .Cantiello M. : ffi antcsoso o asv stars massive hot on spots Magnetic magneticspots no. manuscript utt rdc h emtyo hs etrs eso htm that show we features, these of geometry the predict to cult ∼ ffi 5 in mltd oa to amplitude cient ff − rn rmthat from erent ff .Cnilo&J Braithwaite J. & Cantiello M. 0)o main- of 10%) c o and O for ect ,AfdmHue 1 –32 on Germany Bonn, D–53121 Hügel 71, dem Auf n, rthe or -scale spots. t nzn”(FeCZ). zone” on sure- 0kG. ABSTRACT aiiyadpa oei -a msinadwn clumping. wind and emission X-ray in role a play and iability as e und o- h- g- m niloe l 20)age htacneto oecoeto close zone convection a that argued (2009) al. et antiello fievraiiy(P) o-hra msin non-radial emission, non-thermal (LPV), variability ofile l. scnetv oei asdb eki h pct associat opacity the in peak a by caused is zone convective is d o hs antcfilsaddsuspsil bevtoa si observational possible discuss and fields magnetic these ff c h idcudeeg ttesraevamgei buoyanc magnetic via surface the at emerge could wind the ect sipratt testa ml cl antcfilso amp of fields magnetic scale small that tude stress though to previously important than fiel widespread is magnetic more (e.g., if are wonder stars 199 stars may massive al. one in et type Therefore Fullerton 2002). early 1996; al. al. et among et Prinja Kaper ubiquitous 1989; Prinja be & Howarth thes to Interestingly 2005). seem al. et nomena Henrichs 1 1997; al. al. et et Fullerton al Kaper 1996; is sur Owocki (NRP) & of pulsation Cranmer presence radial (e.g., non the possibility if with even associated fields, variabi magnetic often face profile been line have (LPV) and ity (DACs) components absorption crete rtruec scue ytepeec fteFC C9 is Przyb (C09) 2010, al. FeCZ et the (Fraser of observations presence recent observ the by by supported by obser the caused stressed that is idea recently The croturbulence been properties. surface has stellar of stars tions early-type in ers the at visible fields magnetic sta producing intermediate-mass surface. dynamo, in associated (he a present zones peak host not convective opacity could are These which an 1992). FeCZ), al. by envelop after et caused (Iglesias the iron in stars, with regions massive convective hereafte hot, (2009, of of al. properties et Cantiello the recently studied timescale this, long radia- to embarrassingly an the contrast on only through place whic upwards take 2003) to resulti Cassinelli way seems the & its MacGregor case 1989; make (Moss which envelope to tive in have core, would convective physica some field the by in sustained be or to process; have eith would operate itself to which have di 2002) by would powered dynamo envelope, radiative a the latter, in the In stars. mass di a cons to theoretical point Moreover, erations 2002). Owocki & (ud-Doula stars iini h eZ eivsiaeteocrec fsubsurf of occurrence the investigate we FeCZ, the in tition asv tr,a acltdi Dselreouinmodel evolution stellar 1D in calculated as stars, massive f pee n id hc tl akcerpyia explanati physical clear lack still which winds and spheres h elt n motneo usraecnetv lay- convective subsurface of importance and reality The ∼ 0.10Gaei rnil bet a to able principle in are G 10...100 yesasta aesbufc ovcin hstp of type This convection. subsurface have that stars type gei pt fsz oprbet h local the to comparable size of spots agnetic ff rnebtenmsieadintermediate- and massive between erence ff rnilrtto (Spruit rotation erential ff c h ido OB of wind the ect pulsations, h stellar the

c gnatures dwith ed S 2018 ESO s. ace on. e mi- ved y. C09) r phe- e 996; oa so illa, .It t. In . re- id- ng rs, ds li- es er 6; a- l- h - l 2 M. Cantiello & J. Braithwaite: Magnetic spots on hot massive stars priv.comm.). In the Sun stochastic convective motions excite most of the total flux (e.g. about 95% in the 60 M⊙ model of non-radial pulsations (Ulrich 1970; Leibacher & Stein 1971). Table 1). This is because the density is very low and the mean In analogy with the Sun, it has been suggested that solar- free path of photons correspondingly long. In this situation con- like oscillations are excited in hot, massive stars by the FeCZ vection is significantly superadiabatic, and the actual gradient (C09, Belkacem et al. 2010). Intriguingly, non-radial pulsations ∇≡ d ln T/d ln P has to be explicitly calculated from the mixing that are compatible with stochastic excitation from subsurface length equations (e.g. Kippenhahn & Weigert 1990). C09 found convection have recently been found by COROT in both a B velocities ranging from 1 to 10’s of kms−1 in the FeCZ of OB (Belkacem et al. 2009) and an O star (Degroote et al. 2010), but stars, where the sound speed is usually of the order 100kms−1. see also Balona et al. (2011). In addition, it is informative to estimate the values of the C09 pointed out that the FeCZ could host dynamo action diffusivities in the FeCZ. As usual in non-degenerate stars, the very close to the surface. Simulations of turbulent convection in thermal diffusivity is the largest: the presence of shear and rotation show that dynamo action is K 4acT 3 possible and can produce magnetic fields reaching equipartition χ = = , 2 (1) on scales larger than that of the convection (e.g., Käpylä et al. ρcp 3ρ cpκ 2008; Cantiello et al. 2010; Guerrero & Käpylä2011). In this paper we consider the emergence of magnetic fields produced in where a, cp and κ are the radiation constant, the specific heat a subsurface convective layer. In the following section we briefly at constant pressure and the Rosseland mean opacity. In the 20 17 19 2 −1 review the properties of the FeCZ, before examining in Sect. 3 anda60 M⊙ models χ is around 5 × 10 and 8 × 10 cm s the magnetic fields produced within them and mechanisms for respectively. The momentum diffusivity (kinematic viscosity) is those fields to reach the surface. In Sect. 4 we look at observ- (in contrast to lower mass stars) also dominated by photons, and able effects on the surface, and finally we discuss the results and is given in this case by conclude in Sect. 5 and 6. 4aT 4 ν = (2) 15cκρ2 2. Subsurface convection 9 whereinthe20anda60 M⊙ models it is approximately 6 × 10 The occurrence and properties of the iron convection zone were and 1012 cm2 s−1. Finally, the magnetic diffusivity as given by studied in detail by C09. Here we summarize their findings and Spitzer’s expression (Spitzer 1962) describe some of the fundamental properties of the outer regions of hot massive stars. The evolutionary calculations used in this 1/2 1/2 2 2 π me Ze c 11 −3/2 2 −1 paper are the same as in C09. We refer to that paper and to η = ln Λ ≈ 7 × 10 ln Λ T cm s (3) γ 8(2k T)3/2 Brott et al. (2011) for a detailed discussion of the models and E B the stellar evolution code with which these are computed. which is roughly 105 cm2 s−1 in both models (assuming a fully The peak in the opacity that causes the FeCZ occurs at ionized H plasma and a value ln Λ = 10 for the Coulomb log- 5 around 1.5×10 K. In the envelope of a massive star this region arithm). Note that photons have a negligible effect on the mag- remains relatively close to the surface during the whole main netic diffusivity. The magnetic Prandtl number (Prm ≡ ν/η) is sequence. When the star expands to become a supergiant, the consequently 105 to 107. We can compare these numbers to the surface temperature decreases and the convective zone moves −2 values usually discussed in the solar dynamo (Prm ∼ 10 in the downwards in radius, further from the surface. For photospheric tachocline) and in the galactic dynamo contexts (the ISM has temperatures . 104 K, partial ionization of H and He drive con- 12 typically Prm ∼ 10 ). vection directly at the surface. Therefore we limit the discussion of the properties of the FeCZ to spectral classes O and B (and partially A), which lack this additional convective layer1. 3. Magnetic spots The presence of subsurface convection depends also on the luminosity and metallicity of the star. At solar metallicity the The occurrence of convection zones close to the surface of hot 3.2 massive stars opens a new scenario. Magnetic fields could be FeCZ is present in models above ∼ 10 L⊙, while at the metal- 3.9 readily produced by dynamo action and reach the surface via licities of the LMC and SMC the thresholds are ∼ 10 L⊙ and 4.2 magnetic buoyancy (C09). Below we discuss this hypothesis. ∼ 10 L⊙ respectively. Atomic diffusion and radiative acceleration can in principle lower these limits (Richer et al. 2000). Due to the very low densities in the outer layers of early 3.1. Dynamo Action type stars (ρ ∼ 10−8 g cm−3), these convective regions contain very little mass despite their large radial extent. In Table 1 we In an astrophysical plasma a dynamo is a configuration of the show the properties in the outer layers of Galactic (solar metal- flow which is able to generate a magnetic field and sustain it licity) O/B stars above the luminosity threshold. Values for the against ohmic dissipation. Depending on the scale of the result- size of the convection zone and of the radiative layer above are ing magnetic field respect to the scale of kinetic energy injec- 6 shown for models of a 20 and a 60 M⊙ at ages 6.41×10 yr tion, dynamos are usually divided into small and large scale. and 2.37×106 yr respectively. These correspond to the calcula- In large scale dynamos the field has correlation length bigger tions discussed in detail in C09, where is possible to see how the than the forcing scale in the flow, while small scale dynamos re- extension and location of subsurface convection changes during sult in magnetic fields with correlation scale of order or smaller the MS evolution (See their Fig. 2 and 3). than the forcing scale. Small scale dynamos can occur in non- In the FeCZ the transport of energy by convective mo- helical turbulent flows, while anisotropic flows (e.g. shear flows) tions is relatively inefficient: radiation dominates and transports are required for a large scale dynamo. In the mean field approach, large scale dynamos are often divided into αΩ and α2, 1 The discussion in Sect. 3 is more general, and can be applied to any depending on the role of the Ω and α effect in regenerating radiative star with a convective region below the surface. the toroidal and poloidal components of the field. We refer to M. Cantiello & J. Braithwaite: Magnetic spots on hot massive stars 3

6 Table 1. Properties of the outer layers in a 20M⊙ and 60M⊙ model main-sequence stars of solar metallicity at ages 6.41×10 yr and 2.37×106 yr respectively, corresponding to about 90% and 80% of the main-sequence lifetime.

a b c d e f g h i Mini R⋆ RFeCZ ∆RFeCZ HP 3c ρ ∆MFeCZ ∆Mtop τturn τconv M˙ −1 −3 −1 M⊙ R⊙ R⊙ R⊙ R⊙ km s g cm M⊙ M⊙ days days M⊙yr 20 10.46 10.20 0.28 0.08 - 0.24 10.74 7.4 × 10−8 3.6 × 10−6 5.8 × 10−7 0.53 18250 7.3 × 10−8 60 22.04 21.34 2.84 0.23 - 1.93 69.26 6.2 × 10−9 1.6 × 10−5 9.8 × 10−7 0.61 1570 3.7 × 10−6

a Radial coordinate of the top of the FeCZ. b Radial extension of the FeCZ. c Pressure scale height at top/bottom of the FeCZ. d Maximum of the convective velocity inside the FeCZ. e Density at 3c. f Mass contained in the convective region. g Mass in the radiative layer between the stellar surface and the upper boundary of the convective zone. h Convective turnover time, τturn := HP/3c. i Time that a piece of stellar material spends inside a convective region, τconv := ∆MFeCZ/M˙ .

Brandenburg & Subramanian (2005) for a very nice review of B (G) astrophysical dynamos. The FeCZ is a turbulent layer close to the surface of a mas- 0 500 1000 1500 2000 2500 sive star and as such could be the site of a small scale dynamo 1500 0 120 MSun 1000 500 (e.g., Moss 1994). Moreover main sequence massive stars usu- 6.0 2000 ally rotate rapidly. A typical equatorial rotational velocity of 1

−1 00 150kms (typical for Galactic B-type stars, e.g. Dufton et al. 0 5.5 1500 MSun 2006) corresponds to a rotational period of the order of the 35 500 convective turnover timescale (Rossby number is in the range 5.0 2 1...10), therefore an efficient α or αΩ-dynamo could be possi- 1000 20 MSun ble. Assuming magnetic fields at equipartition with the kinetic LogL 4.5 energy of the convective motion, gives magnetic fields up to ∼ 2kG. This is supported by simulations of turbulent convection 4.0 in the presenceof rotation and shear,which show dynamo excita- 500 10 MSun tion with magnetic fields reaching equipartition on scales larger 3.5 0 than the scale of convection (Käpyläet al. 2008; Cantiello et al. GAL 7 MSun 2010). 3.0 Hence dynamo action in the FeCZ could depend on parame- 4.8 4.6 4.4 4.2 4.0 logT ters like the stellar rotation (as measured by the Rossby num- eff ber) and the shear profile in the region of interest. The scale Fig. 1. Values of maximum magnetic fields (in gauss) in the of the magnetic field depending on the type of dynamo occur- FeCZ, as a function of the location in the HR diagram. This ring in the relevant layers. The dynamo may also be affected by plot is based on evolutionary models between 5M and 120M a (presumably fossil) large-scale magnetic field penetrating up- ⊙ ⊙ for solar metallicity (some evolutionary tracks shown as dotted wards from the radiative zone below (see Sect. 5). However it lines). The amplitude of magnetic fields is calculated assuming is interesting to note that many of the photospheric and wind equipartition with the kinetic energy of convective motion. The phenomena observed in hot massive stars are ubiquitous (e.g., full drawn black line roughly corresponds to the zero age main Howarth & Prinja 1989; Kaper et al. 1996; Fullerton et al. 1996; sequence. The two star symbols correspond to the location of the Prinja et al. 2002), in contrast to fossil fields which have been 20 and 60M models discussed in the text and in C09. found at the surfaces of some (as yet badly measured) fraction ⊙ of massive stars. As it is difficult to predict the exact rotational properties of the plasma inside the FeCZ, here we will not focus on a detailed study of subsurface dynamo action. As a prelimi- of 3c. Values of magnetic fields calculated using the average con- nary study of subsurface magnetism we will just follow the re- vective velocity and density typically differ by less than 30%. sults of Cantiello et al. (2009, 2010), and assume that magnetic fields at equipartition are produced in the FeCZ. In Fig. 1 we 3.2. Magnetic buoyancy show the expected maximum magnetic field inside the FeCZ, assuming equipartition of magnetic and kinetic energy: There are various possible mechanisms which can bring a magnetic field generated in the FeCZ through the overlying radiative 2 Beq 1 layer to the surface, and it is a useful exercise to compare their = ρ 32, (4) 8π 2 c timescales. As the title of this section suggests, it turns out that among the mechanisms studied here, the dominant one is mag- 32 where we adoptedthe maximumvalue of the ρ c inside the FeCZ netic buoyancy. (as computed in the non-rotating models of C09). Due to the The magnitude of the mass loss from massive stars is such higher power dependency on the velocity, the location of the that material resides in the outer, radiative layer for just a short 32 −7 −1 maximum of ρ c always roughly corresponds to the maximum time. Assuming a rate of ∼ 10 M⊙ yr , this time is ∼ 1 yr. 4 M. Cantiello & J. Braithwaite: Magnetic spots on hot massive stars

Magnetic field could also be broughtto the surface by Ohmic to magnetic pressure, β: diffusion. The timescale over which it produces effects over the l 1/2 c l 1/2 relevant radial length scale, i.e. the pressure scale height HP, is 3 ∼ 3 ∼ s . drag A 1/2 (10) HP ! β HP ! 2 2 3/2 HP HP T 7 τOhm ∼ ∼ 10 yr, (5) Putting the numbers into (7) and (10) gives η 1010 cm 105 K −2 ff l −1 10 13 −1 where η is the magnetic di usivity. We shall now see that this 3therm ≈ β 10 − 10 cm s , (11) timescale and that of advective transport are much longer than HP ! the buoyancy timescale. 1/2 l −1/2 7 7 −1 Buoyancy causes magnetic features to rise through a radia- 3drag ≈ β 10 − 3 × 10 cm s , (12) tive zone because magnetic field provides pressure without con- HP ! tributing to density. Since the sound and Alfvén timescales are where ranges in the numbers come from the difference between shorter than the thermal/buoyancy timescale (the buoyant rise is 20and60 M⊙ models and from differences between top and bot- subsonic and sub-Alfvénic), we can assume that a magnetic fea- tom of the surface radiative layer. It is clear that unless l ≫ HP, ture is in pressure equilibrium with its surroundings. Calling the for which the approach above breaks down anyway, or the field sum of gas and radiation pressure inside and outside the feature is implausibly weak (and therefore undetectable and unimpor- Pi and Pe we have tant), 3therm is larger and the rise is therefore limited by drag. Pe = Pi + Pmag, (6) For a magnetic feature with l ∼ HP and β ∼ 100 (a conserva- tive estimate), the time taken to rise one scale height is of order where the value of the magnetic pressure Pmag depends on the geometry of the feature. For instance, in a self-contained struc- 2 hours in both models, much shorter than the advective and 2 2 Ohmic timescales. ture it is B /24π, and in a flux tube of fixed length it is Bax/8π 2 This is in contrast to the thermal-diffusion-limited regime where Bax is the component along the axis of the tube . Note the implicit assumption that the size of the structure l is much which was studied by MacGregor & Cassinelli (2003) to check smaller than the scale height H ; it is quite likely (see below) the possibility that magnetic fields generated in a convective core P could rise to the surface. In that case the timescale appears to ex- that in fact l ≈ HP but this will make a difference just of factors of order unity. Now, in the absence of thermal diffusion ceed the main sequence lifetime unless very small-scale features are used. Moreover Mullan & MacDonald (2005) argue that the the feature reaches an equilibrium where ρi = ρe, made possible by the internal temperature being lower than the external inclusion of compositional gradients may suppress the buoyancy, making this scenario unlikely4. Ti < Te. With the addition of a small thermal diffusion, the feature absorbs heat from its surroundings and rises quasistatically The situation in massive star surface layers is in some sense upwards through surroundings of increasing entropy. The speed more similar to that of flux tubes in the solar convection zone of this rise is given by: in that the rise is limited by drag forces, although the drag force in the solar CZ does not scale simply as v2 but depends also on 2χH 1 the convective motion. A stationary flux tube in the solar CZ 3 ∼ P · therm 2 (7) l β (∇ad − ∇) 4 − 3α experiences a net downwards aerodynamic force once the average of upwards- and downwards-moving convective cells has where β ≡ Pe/Pmag, l is the size of the feature, ∇ and ∇ad have been taken, with the result that there is a field-strength thresh- their usual definitions and take values ∇ = ∇rad ≈ 0.24 and old below which the tube moves downwards despite its intrinsic ∇ad ≈ 0.25 in the radiative zones between the FeCZ and the buoyancy. photosphere, and χ is the thermal diffusivity given in (1). Once a magnetic field element reaches the surface of the star If on the other hand thermal diffusivity is large, the feature it would be useful to estimate its lifetime tspot. This is not trivial, rises so fast that the speed is limited by aerodynamicdrag. In this as the photosphere of OB stars is complicated by the presence of regime the speed is independent of thermal conductivity, and we strong winds and, likely, turbulence and shear. However we can can make the approximation that Ti = Te; note that unlike above, put some limits on the lifetime. in this regime we have ∆ρ ≡ ρe − ρi , 0. The buoyant force as As we discussed at the beginning of this section, if the mate- given by Archimedes’ principle is balanced by the drag force, rial in the radiative layer above the FeCZ has a mass ∆M, it will be removed by mass loss in a time ∆M/M˙ . This sets an upper 1 32 CdAρe drag = Vg∆ρ (8) limit for the time a magnetic element, which is not anchored to 2 the convective zone, can exist at the surface: this time is about where A and V are the cross-sectional area (projected onto a hor- 8 years for the 20M⊙ and 3 months for the 60M⊙ model. Of izontal plane) and volume of the magnetic feature, and Cd is the course this holds true only as long as the field is not annihilated drag coefficient whose value depends on geometry. This can be by dissipative processes like magnetic reconnection. If the mag- rearranged to netic element is anchored to the convective zone, it can exist at P the surface as long as the stellar wind does not remove all the 32 2V mag drag ≈ . (9) mass contained in both the radiative layer and the FeCZ. It turns CdHPA! ρe out that this limits tspot to be less than 50 years (20M⊙ model) ≈ Since the length scale of the feature l V/A, the term in brackets and 4 years (60M⊙ model). We refer to Table 1 in C09 for the 3 is approximately equal to l/HP. The remaining part is approx- data used for these simple estimates. These are solid upper lim- imately the square of the Alfvén speed, which one can alterna- its, and it is likely that such features would survive for a much tively express in terms of the sound speed cs and the ratio of total 4 Mullan & MacDonald (2005) also discuss the possibility of buoy- 2 See Braithwaite (2010) for a discussion of this point and a review ant rise of magnetic fields generated in the radiative zone by the Tayler- of the stability of flux tubes. Spruit dynamo. However it is not clear how this dynamo process actu- 3 If the feature is spherical, Cd ≈ 0.75 (Churazov et al. 2001). ally works (Braithwaite 2006; Zahn et al. 2007). M. Cantiello & J. Braithwaite: Magnetic spots on hot massive stars 5

B (G)

0 5 10 20 40 80 160 > 320

40 Optical depth = 2/3 320 120 MSun 10 20 5 Radiative Layer Emerging B ﬁeld 6 160

35 MSun 0 5 4 20

20 MSun 10

LogL Convective zone 4 5 Log(L x/L bol ) > -7 10 MSun -7 > Log(L x/L bol ) > -8

Log(L x/L bol ) < -8 GAL MSun 3 7

4.8 4.6 4.4 4.2 4.0 logT Fig. 2. Schematic representation of the effect of an emerging eff magnetic element at the surface of a hot massive star. A magnetic Fig. 3. Minimum values of expected surface magnetic fields (in field (dashed line) rising from the subsurface convection zone gauss) as a function of location in the HR diagram. This plot threads the radiative layer and reaches the stellar photosphere. is based on evolutionary models between 5M⊙ and 120M⊙ The solid, grey line shows the location of the stellar surface, as for Galactic metallicity (some shown as dotted lines). Surface defined by the value 2/3 for the optical depth. Notice that inside magnetic fields are calculated scaling the equipartition fields the magnetic field this line reaches deeper (and hotter) layers of in the FeCZ (see Fig. 1) with ρ2/3 (see discussion in Sect. 4). the star, as compared to regions not affected by the field. The full drawn black line roughly corresponds to the zero age main sequence. Circles correspond to the observed stars in the Cohen et al. (1997) sample, with symbol size coding different shorter time, owing to turbulence, shear and magnetic reconnec- range of Lx/Lbol, as explained in the legend. The two star sym- tion. As a lower limit for tspot we could assume the time a feature bols correspond to the location of the 20 and 60M⊙ models dis- of size l takes to cross the photosphere while rising with a ve- cussed in the text and in C09. locity vdrag. For l ∼ HP this gives a timescale of the order of hours. Finally, note that it is not inconceivable that the magnetic dynamo does not saturate properlybecause of the finite time the gas tube cases. The photospheric field could be significantly greater spends in the FeCZ. However, we consider this unlikely, since if a tube arches upwards and fluid is allowed to flow along it – the convective turnover times (∼ hours- a day) appear to be very the limit should then be of the order of equipartition with the much shorter than the aforementioned mass-flux timescales of photospheric pressure (about 300 G) or somewhat more, since 50 and 4 years. the photosphere (i.e. the τ = 2/3 level) in the tube is lower. In Fig. 3 we show the expected amplitude of surface magnetic fields emerging from the FeCZ, assuming a scaling B ∝ 4. Observable effects ρ2/3. We argue that this represents a lower limit for the mag- Let us first estimate the likely field strength at the photosphere. netic field that should be found at the surface of hot massive Whilst one might expect it to be proportionalto the field strength stars, if equipartition of kinetic and magnetic energy is reached produced in the convective layer, the difference in field strength in their FeCZ. It might seem plausible that magnetic features ris- of a magnetic feature between the top and bottom of the radiative ing through the radiative layer partially annihilate each other on layer depends on its geometry. In a self-contained magnetic fea- the way up. However, if these magnetic bubbles have a size at ture (a ‘blob’ or ‘plasmoid’) the field strength scales as B ∝ ρ2/3, the bottom of the radiative layer comparable to the local scale while a horizontal flux tube of fixed length has Bax ∝ ρ and height, they only have to rise through a distance about three 1/2 Bh ∝ ρ where Bax and Bh are the axial and ‘hoop’ compo- times their own size to reach the photosphere, expanding as they nents. Knowing that P ∼ ρ4/3 (approximately, since ∇≈ 1/4) in do so. They rise at approximately the Alfvén speed, and since the radiative layer, we see that these two scenarios give β = const any reconnection would presumably take place at some fraction and β ∝ P−1/2 respectively. In constrast, if the central section of of the Alfvén speed, there is limited time available to destroy a flux tube rises to the surface while its ends are still in the con- much of the original flux. It seems more likely that significant vective layer, plasma can flow downwards along the tube and reconnection does take place above the photosphere. the field strength at the surface can in principle reach equipar- While these fields appear to have very small amplitude, their tition with the surrounding (gas plus radiation) pressure, as we role at the stellar surface should not be underestimated. In Fig. 3, 2 2 ˙ 32 see happening in the solar CZ. Assuming equipartition magnetic the ratio of the magnetic to wind pressure (η ≈ BeqR∗/M ∞, field in the convective layer gives up to ≈ 2 kG at the base of ud-Doula& Owocki 2002) is of the order of unity over the the radiative layer, and β ≈ 100. The density contrast across the whole region where magnetic fields are present. This means that radiativelayer is around50 and 100 in the 20M⊙ and 60M⊙ mod- already such low magnetic fields can alter the dynamic of the els respectively, which gives field strengths at the photosphere of stellar wind, for example contributing to seed instabilities in the 40−150Gand20−100 G respectively in the blob and horizontal flow. 6 M. Cantiello & J. Braithwaite: Magnetic spots on hot massive stars

4.1. Surface bright spots and photometric variability bright spot. In a rotating star the luminosity would also decrease (increase) when the spot moves out of (into) the visible part of Once a magnetic field element reaches the stellar photosphere, the stellar disc. It is difficult at this stage to give a firm estimate ff one would like to know what e ect this can have on observable of the total expected change in luminosity, since this will depend properties of the star. In the Sun, magnetic fields of amplitude not only on the intensity of the magnetic field, but also on the ∼ ff up to 3 kG emerge at the surface as sunspots. The main e ect geometry and number of magnetic spots. Also, it is not clear ex- of such magnetic fields is to locally inhibit convection. As the actly where the extra luminosity in the spots should come from, energy flux below the surface of the Sun is mainly transported in terms of underluminous areas. Nevertheless, if we assume a by convection, this produces a decrease of the flux at the sur- circular spot of radius r, with filling factor f = (r/R)2, where R face, resulting in a visible dark spot. In contrast, the hot massive is the stellar radius, then the effect on the stellar luminosity can ff stars considered here have radiative surfaces and the e ect of an be estimated if r is known. A simple approach, based on prelim- ff emerging magnetic field is very di erent. inary models of subsurface convection (Cantiello et al. 2010), is Under the simple assumption of magnetohydrostatic equi- to equate r to the pressure scale height in the FeCZ. These simu- librium we can make use of Eq. 6. Assuming again that also lations show dynamo action producing magnetic fields on scales = the gas temperature inside and outside is the same (Te Ti), comparableto or larger than the scale of convection,i.e. the local we obtain once more that the density inside the magnetic ele- pressure scale height H . We obtain: ment is lower than outside. Since we are at the stellar surface, P this makes the element more “transparent”, i.e. photons can es- 2 ∆L 4 ∇rad HP cape from deeper down the star compared to outside the mag- = , (15) netic flux. This is somehow similar to what occurs in the solar L β R faculae, which are regions of emerging magnetic fields with a which, for typical values of subsurface convection, gives ∆L/L ∼ scale much smaller than the scale of convection. In that case 10−5. This is the relative change in luminosity caused by the ap- Keller et al. (2004) found, using MHD calculations, that “The pearance/disappearance of one magnetic spot of about 100 G opacity in the magnetic flux concentration is strongly reduced with size 0.2 R⊙ at the surface of a hot massive star. Recall that owing to its low density and temperature and thus provides a these stars have radii around 10R⊙. clear sight straight through the magnetic field onto the adjacent Of course spots with stronger magnetic fields and bigger fill- nonmagnetic granule”. Their calculations basically verified the ing factors are certainly possible. Such spots would lead, most explanation for solar faculae proposed by Spruit (1976, 1977). likely, to larger changes in the integrated light from the star. An Photons emerging from deeper down the star are emitted estimate of the amplitude of the variability as function of filling from regions where the temperature is higher. The tempera- factor and surface strength is much more difficult in this case, ture stratification, assuming that the magnetic pressure is much as one can no longer consider the magnetic element thin (in a smaller than the gas + radiation pressure (i.e. β ≫ 1), is given thermal sense). MHD calculations of magnetic flux tubes in a by the radiative gradient ∇rad radiative environment are required to study the details of this d ln T problem. −3 ∇rad ≡ . Photometric variability at the micro level (10 magni- d ln P ! rad tudes) and with timescales of the order of days seems to be Therefore it is easy to estimate the temperature difference be- widespread in O stars (Balona 1992). Using COROT to perform tween the region of the star visible through the magnetic element high-precision photometry, Degroote et al. (2010) found solar- and outside of it: like oscillations in a young O-type star (HD46149). These are modes with a finite lifetime that are believed to be excited by ∆T T∆(ln T) T = ≈ ∇rad the presence of subsurface convection (Belkacem et al. 2010). ∆P P∆(ln P) P Intriguingly, this system also shows a low-frequency variability with time scale of days and amplitude of order 10−3 magni- ∆T ∇rad ≈ . (13) tudes. The authors could not relate this variability to any type T β of pulsations. Instead they argue that this could be the signa- This means that regions with an emerging magnetic field will ture of spots, stellar winds or chemical inhomogeneities modu- look hotter. In the case of the FeCZ, assuming the minimum lated by the stellar rotation (Degroote et al. 2010). Pápics et al. value of magnetic fields shown in Fig. 3, preliminary estimates (2011) obtained high-precision photometric observations of the give temperature differences up to a few hundred K. In the case B0.5IV star HD51756 with COROT. While no solar-like oscil- of a surface magnetic field in equipartition with the photospheric lations were identified in this case, they found cusp-like fea- pressure (gas plus radiation) equation 13 is not strictly correct, as tures in the light-curve. These features have amplitudes of or- the magnetic field can affect the radiative gradient. Nevertheless, der 10−3 magnitudes, and recur on a time scale of days. Similar assuming for a moment that the correction to this relation is to HD46149, Pápics et al. (2011) propose as a possible source small, one expects temperature differences up to a few kK. for this variability photospheric features like spots or chemical Since the stellar luminosity is related to the effective temper- inhomogeneities, modulated by stellar rotation. ature through the relation L = 4πR2σT 4, magnetic fields in stars having radiative surfaces will produce bright spots. We can estimate the local luminosity contrast between the magnetic bright 4.2. X rays spot and the non magnetic regions of the stellar surface OB stars have been found to emit X rays (Harndenetal. ∼ −7 ∆Lloc ∆T 4 ∇rad 1979). O stars follow the relation Lx/Lbol 10 (see ≈ 4 ≈ . (14) e.g. Pallavicini et al. 1981) while B star X-ray luminosities L T β loc show a lot of scatter (Meursetal. 1992; Berghoeferetal. Temporal changes in the observed stellar luminosity might be 1997). Among possible physical mechanisms for the produc- expected, as a result of the appearance/disappearance of such a tion of X rays are magnetic coronae (Cassinelli & Olson 1979; M. Cantiello & J. Braithwaite: Magnetic spots on hot massive stars 7

Waldron & Cassinelli 2007, 2009) and the intrinsic instability to the presence of subsurface convection (e.g. microturbulence, of line-driven winds (Lucy & White 1980; Owocki et al. 1988; C09). One can make an analogy here with the intermediate-mass Feldmeier 1995). stars (A and late B), where fossil fields do appear to suppress or 38 −1 The Lx/Lbol relation only applies for Lbol > 10 erg s , i.e. at least significantly reduce the weak helium-ionisation convec- earlier than B0-B1 type stars. It breaks down at lower luminosi- tion at the photospherewhich manifests itself as microturbulence ties, with the ratio Lx/Lbol two orders of magnitude smaller at (D. Shulyak, priv.comm.) B3 than at B1 (Cassinelli et al. 1994; Cohen et al. 1997). This is shown in Fig. 3, where we plot the data from Cohen et al. (1997) C09 proposed the FeCZ as a possible unifying mechanism ff together with the theoretical expectation for the minimum val- for the formation (or seed) of di erent observational phenom- ues of surface magnetic fields (assuming dynamo action in the ena at the surface of hot, massive stars. The presence of turbu- FeCZ). While there could be different explanations for such a lent subsurface convection appears to be able to trigger at least ff correlation, it is interesting to note that in the sample stars with three di erent classes of physical phenomena:running waves (g- modes and p-modes, e.g. Goldreich & Kumar 1990, C09), solar- Lx/Lbol > −7 correspond to regions of the HRD with larger values of predicted surface magnetic field. Recently Gagnéet al. like oscillations (e.g., C09; Belkacem et al. 2010) and magnetic (2011) have studied with Chandra the X-ray emission from OB fields (C09; this work). Each of these perturbations can, in prin- ff / stars in the Carina Complex. They also found that most of their ciple, results in observable e ects at the surface and or in the stellar wind. Estimating the amplitude, length-scale and time- O stars follow the Lx/Lbol relation, while this breaks down for B stars. Interestingly they identify a group of B stars with high scale of the surface perturbations produced by each mechanism X-ray emission that cannot be explained by a distribution of or- is of primary importance to establish the connection between dinary coronal, pre main-sequence companions. A possibility is FeCZ and observable phenomena. For example wind clumping / that in O stars wind embedded shocks dominate and are respon- could be produced by small-scale velocity and or density fluc- tuations at the stellar surface. These perturbations could excite sible for the Lx/Lbol relation, while the observed X-ray luminosity in B stars is mainly due to the presence of subsurface gener- the line-driven instability (Runacres & Owocki 2002) closer to ated magnetic fields. the stellar surface than originally predicted from the theory. This would reconcile theory with observations of the radial stratification of wind clumping (Puls et al. 2006). However both run- 5. Discussion ning waves and localized magnetic fields could be responsible for such a perturbation at the base of the wind. To understand ff We showed that, among advection, magnetic di usion and mag- which one dominates, hydrodynamic simulations of the line- netic buoyancy, the most likely process that can bring to sur- driven instability are required. These calculations need to in- face magnetic fields generated by dynamo action in the FeCZ clude, as boundary conditions at the base of the stellar wind, is magnetic buoyancy. However there could be other ways for the different velocity and density perturbations predicted in the a magnetic field to escape the subsurface convection region and case of running waves and surface magnetic fields. While simple reach the stellar surface. For example, Warnecke & Brandenburg estimates for these perturbations exist (C09 and this paper), real- (2010); Warnecke et al. (2011) studied the magnetic flux pro- istic MHD simulations of the FeCZ and the radiative layer above duced by a turbulent dynamo in Cartesian and spherical geom- it are necessary. Understanding the exact role and properties of etry, respectively. They found that magnetic flux can rise above each perturbation is also fundamental to explain the puzzling co- the turbulent region without the need of magnetic buoyancy. This existence of small- and large- scale phenomena observed at the appears to be related to the release of magnetic tension, which surface and in the wind of hot, massive stars (e.g. microturbu- leads to the relaxation and emergence of the field. Convective lence/macroturbulence and wind clumping/DACs). An effort is overshoot might also contribute to the transport of magnetic flux clearly required to improve existing MHD calculations of sub- out of the FeCZ. However, given the fact that the FeCZ usually surface convection (Cantiello et al. 2010). sits a few pressure scale heights from the stellar surface, this effect is likely marginal. The presence of surface magnetic fields might have impli- Some fraction of massive stars display large-scale, appar- cations for the evolution of OB stars. This is because, for wind ently fossil fields of strengths (1−3 kG) comparable to the FeCZ confinement parameters η ∼> 1, the magnetic field can in prin- equipartition strengths derived above. A sufficiently strong field ciple affect the stellar wind mass-loss and angular momentum- will suppress convection, forcing an increase in temperature gra- loss rates. For large scale fields at the surface, the impact on dient so that the entire energy flux can be transported radiatively. angular-momentumloss has been studied (Weber & Davis 1967; However, it is not obvious where the field strength threshold ud-Doula et al. 2009). In the case of the magnetic star σ Ori E should be; one might na¨ıvely expect convection to be suppressed the theoretical spin-down is in agreement with observations by a field of greater energy density than the convective motion, (Townsend et al. 2010). The effect of large scale fields on the but the work of Gough & Tayler (1966); Moss & Taylor (1969); angular momentum evolution of main sequence massive stars Mestel (1970) suggests that in that case the temperature gradient has been investigatedby Meynet et al. (2011). Their results seem would simply steepen further above the adiabatic gradient until to show that only fields above a few hundred gauss can have a convection resumes, and that to suppress convection completely, substantial impact. For a given surface field amplitude, higher a field above (approximate) equipartition with the thermal en- order multipoles (i.e. less coherent fields, as expected in the ergy would be required. However, this work considers a situation case of dynamo action in the FeCZ) result in a reduced angu- where the energyflux is almost entirely convective,such as in the lar momentum-loss. Moreover in the case of small-scale fields bulk of the solar convective zone. In the FeCZ context the situ- a lot of the flux may reconnect and disappear fairly close to the ation is different, in that only a small fraction of the stellar heat surface, below where most of the wind acceleration takes place. flux is carried by convection, and that the temperature gradient Therefore we expect magnetic fields produced in the FeCZ to is already significantly above adiabatic. It is plausible therefore have only a modest effect on the angular momentum evolution that a fossil field of a few kG could indeed suppress convec- of massive stars. Nevertheless further investigation of their ge- tion, which would have consequences on observables connected ometry and amplitude is required to confirm this statement. 8 M. Cantiello & J. Braithwaite: Magnetic spots on hot massive stars

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