Dynamo-generated magnetic fields at the surface of a massive star

Transcription

1 Mon. Not. R. Astron. Soc. 356, (2005) doi: /j x Dynamo-generated magnetic fields at the surface of a massive star D. J. Mullan 1 and James MacDonald 2 1 Bartol Research Institute, University of Delaware, Newark DE 19716, USA 2 Department of Physics and Astronomy, University of Delaware, Newark DE 19716, USA Accepted 2004 October 23. Received 2004 October 21; in original form 2004 March 29 1 INTRODUCTION 1.1 Observations of magnetic fields Positive detections of magnetic fields by optical methods have been reported for a number of hot stars. The measured quantity is the longitudinal, i.e. line-of-sight (LOS), magnetic field strength, and this is observed to vary in time. For example, in β Cephei (B1 IV), the LOS field varies between about +100 and 100 G (Donati et al. 2001), while in θ 1 Orionis C (O4-6 V), the observed LOS field varies between about +350 and +50 G (Donati et al. 2002). In three other slightly evolved B stars, the LOS field values range in value from nearly zero up to many hundreds of G (Neiner et al. 2003a,b,c). (More details on individual stars can be found in Section 6.3.) In analysing the LOS data, the investigators typically fit the data by means of an obliquely rotating dipole. These fits lead to polar fields B p of 1100 G in the main-sequence O4-6 V star θ 1 Orionis C (Donati et al. 2002). In the early B stars which are old enough to have evolved somewhat off the zero-age main sequence (ZAMS), polar field values are found to be a few hundred G, e.g. 360 G in β mullan@bartol.udel.edu (DJM); jimmacd@udel.edu (JM) ABSTRACT Spruit has shown that an astrophysical dynamo can operate in the non-convective material of a differentially rotating star as a result of a particular instability in the magnetic field (the Tayler instability). By assuming that the dynamo operates in a state of marginal instability, Spruit has obtained formulae which predict the equilibrium strengths of azimuthal and radial field components in terms of local physical quantities. Here, we apply Spruit s formulae to our previously published models of rotating massive stars in order to estimate Tayler dynamo field strengths. There are no free parameters in Spruit s formulae. In our models of 10- and 50-M stars on the zero-age main sequence, we find internal azimuthal fields of up to 1 MG, and internal radial components of a few kg. Evolved models contain weaker fields. In order to obtain estimates of the field strength at the stellar surface, we examine the conditions under which the Tayler dynamo fields are subject to magnetic buoyancy. We find that conditions for Tayler instability overlap with those for buoyancy at intermediate to high magnetic latitudes. This suggests that fields emerge at the surface of a massive star between magnetic latitudes of about 45 and the poles. We attempt to estimate the strength of the field which emerges at the surface of a massive star. Although these estimates are very rough, we find that the surface field strengths overlap with values which have been reported recently for line-of-sight fields in several O and B stars. Keywords: stars: early-type stars: magnetic fields stars: rotation. Cephei (B1 IV; Donati et al. 2001), 250 G in V2052 Oph (B2 IV V), 335 G in ζ Cas (B2 IV) and 530 G in ω Ori (B2IIIe) (Neiner et al. 2003a,b,c). In the supergiant ζ Pup [O4I(n)f], which has clearly evolved off the main sequence, the LOS field has an upper limit (1σ )ofless than 200 G (Chesneau & Moffat 2002). 1.2 Theories of magnetic field origins in hot stars The question of the origin of magnetic fields in chemically normal massive stars is a subject of great current interest, although no definitive conclusions are yet available. In some cases, the fields may be fossils; this is likely true for some chemically peculiar A and B stars (Mullan 1973; Moss 1989) where the field strengths are so large (up to 34 kg) that dynamo activity may be suppressed (Spruit 1999). In the absence of fossil fields, and because dynamos have been found useful for interpretation of magnetic properties in main-sequence stars with spectral types as early as F2 (Mullan & Mathioudakis 2000), it is natural to consider the possibility of dynamo operation in hotter stars. A principal ingredient of dynamo operation (fast rotation) is certainly available in massive stars. In fact, massive stars in this regard have a great advantage over solar-like stars. For the latter, rotational speeds are typically no more than a few km s 1, whereas in OB stars, mean rotational velocities on the main sequence are of the order of 150 km s 1 (Penny 1996; Abt 2003). C 2004 RAS

2 1140 D. J. Mullan and J. MacDonald Because dynamos in solar-like stars originate in turbulent convection, it is hardly surprising that the convective cores of massive stars have been identified as possible sites of dynamo activity (e.g. Charbonneau & MacGregor 2001). However, even if a dynamo operates in the convective core, this does not mean that the magnetic fields will necessarily be able to reach the stellar surface (thereby becoming detectable by optical methods). Quantitative studies have highlighted how difficult it is for fields from the core to rise to the surface of a massive star because of composition gradients (Moss 1989, 2001; MacDonald & Mullan 2004, hereafter MM04). In view of this, various researchers have considered the possibility that a dynamo may operate in non-convective regions of the interior of a massive star. For example, the magneto-rotational instability (MRI) may be operative (Balbus & Hawley 1994; Arlt, Hollerbach & Rudiger 2003) provided that the mean field strength lies in a certain range. For conditions that are relevant to stellar interiors, the range is typically from about 100 G to about 10 6 G (Balbus & Hawley 1994). Another candidate for dynamo operation in non-convective gas is a purely hydrodynamic instability due to shear. The onset of shear may be due, for example, to a stellar wind which carries away angular momentum from a stellar surface. In such a situation, the braking torque may create turbulent motions in a non-convective layer below the stellar surface (Vigneron et al. 1990; Lignieres, Catala & Mangeney 1996). While this layer may be subject to shear instabilities, with perhaps an accompanying dynamo, the cited papers did not provide any quantitative estimates of the field strengths that might be generated. A quantitative study of shear instability was the subject of one of our recent papers (MacDonald & Mullan 2003, hereafter MM03). In that paper, we show how shear instability and convective instability can be treated in a unified manner in an evolution code. Shear instability is expected to lead to turbulent gas motions, with the associated diffusive processes. In fact, we have recently suggested that the gas motions associated with shear instability could serve to drive a dynamo in red giant stars (MM03) and also in massive stars (MM04). The discussions in MM03 and MM04 were, however, purely qualitative as far as field strengths were concerned; no numerical estimates of field strengths generated by shear dynamos were made. 1.3 Aim of the present paper In the present paper, we wish to be more quantitative in our estimates of dynamo-related fields. In this regard, we consider a dynamo mechanism which is not based on hydrodynamic instabilities. Instead, we consider instabilities which arise from the field itself. Spruit (1999, 2002) has demonstrated that such instabilities are relevant when we consider non-convective regions of rotating stars. Some of these instabilities may be useful in understanding certain properties of the Sun s internal rotation profile (Spruit 2002). In the present paper, we consider dynamos in massive stars using Spruit s approach; this has the distinct advantage that it leads to quantitative estimates of the field strengths created by the dynamo. Spruit s formulae require knowledge of the radial profile of various physical quantities inside a rotating star, including the angular velocity, the Brunt Väisälä frequency, and the thermal diffusivity. In Section 2, we describe how we have computed models of rotating massive stars in which the requisite radial profiles of these parameters are obtained. In Section 3, we discuss various aspects of Spruit s approach, beginning with an overview in Section 3.1. In Section 3.2, we summarize the relevant formulae from Spruit (2002) for the operation of a dynamo driven by a particular instability that is referred to by Spruit as the Tayler instability. Criteria for the operation of this instability are discussed in Section 3.3. In Section 3.4, we summarize the latitudinal behaviour of the Tayler instability. As Spruit (1999) points out, the Tayler instability is not the only magnetically driven instability that can occur in a star there is also magnetic buoyancy, which is the subject of Section 4. In Section 5, we report on the numerical results which we obtain when we apply Spruit s formulae to our massive star models, starting with a 10-M mass star on the main sequence (Section 5.1). Because Spruit s formulae yield field strengths inside the star, we need to discuss how such fields may become detectable at the surface. In this context, we argue (Section 5.2) that buoyancy plays a key role. The combination of Tayler and buoyancy instabilities leads to a prediction of the latitudinal distribution of surface fields (Section 5.3). Comparison with observations is the subject of Section 6. We note that Spruit s formulae for the Tayler dynamo have also been used by Maeder & Meynet (2003) to study how magnetic effects interact with meridional circulation inside rotating stars. Our work differs from that of Maeder & Meynet in two respects. First, we present numerical values for field strengths as a function of radius inside our models. Secondly, we include the effects of magnetic buoyancy. 2 EVOLUTIONARY MODELS FOR ROTATING STARS In two earlier papers (MM03, MM04), we reported on models of rotating stars at various phases of evolution. The results were based on a mixing length theory that treats convective and shear instability in a unified manner, including the stabilizing effects of composition gradients. In our code, we adopt the Zahn (1992) concept of shellular rotation. According to this concept, because of the anisotropy of turbulence, the angular velocity is maintained uniform on spherical shells. The composition mixing and redistribution of angular momentum due to convection and shear instabilities are modelled by diffusion equations. [A different, but related, approach to redistribution of material and angular momentum is described by Maeder (1997).] Our diffusion equations allow us to include in a consistent way the generation of shear and the change in angular velocity profile that results from shear-induced turbulence. Mass loss and angular momentum loss are included in the models. In MM04, the code was used to compute the evolution of rotating 10- and 50-M stars; these were followed from pre-main sequence on to the ZAMS and eventually to the terminal-age main sequence (TAMS). MM04 use the notation 10A and 50A to refer to ZAMS models, and 10C and 50C to refer to TAMS models. Models 10B and 50B are intermediate in age. It is these massive star models that provide the detailed numerical values of the relevant physical parameters which are needed for the present work. In the context of dynamo operation, one of the most significant parameters of a stellar model is the rotational velocity. In this regard, we note that, in the results obtained by MM04, as the 10-M star evolves off the main sequence, the equatorial rotational velocity falls from 174 to 119 km s 1 in going from model 10A to model 10C. In the 50-M models of MM04, the equatorial rotation velocity decreases by a much larger factor, from 116 km s 1 in model 50A, to 15 km s 1 in model 50B, to 2.3 km s 1 in model 50C. This large reduction occurs in part because of expansion in stellar radius (by a factor of about 4) in going between models 50A and 50C, and in part from significant mass loss. These evolutionary variations in

3 rotational velocity are expected to translate into evolutionary variations in magnetic field strength. 3 SPRUIT S DYNAMO THEORY Spruit (1999) has discussed three different classes of magnetic instabilities which can arise in non-convective material: Tayler instability, buoyancy instability and MRI. Tayler instability derives its energy from nearly horizontal interchanges of magnetic flux tubes. This class of unstable modes was first analysed in detail by Tayler (1957, 1973) in the context of cylindrical plasmas. Buoyancy instability occurs when vertical interchanges of flux tubes lead to energy release. MRI, which depends on an interaction between the magnetic field and differential rotation, is best known in the context of two-dimensional flows (accretion discs), although it has also been applied to stars (e.g. Balbus & Hawley 1994; Arlt et al. 2003). In a review of these instabilities, Spruit (1999) has shown that, in the conditions which typically hold true in the interior of a rotating star, the Tayler instability is likely to be the first instability to set in. In view of this result, Spruit (2002) has derived formulae for the saturation state of a dynamo which is driven solely by Tayler instability. It is these formulae that are of primary interest to us in the present paper. In order to set the stage for the numerical results we will present below, we first give a summary of Spruit s reasoning. 3.1 Spruit s approach Let the magnetic field strength B at radius r inside a star be characterized by an Alfvén frequency ω A defined by ω 2 A = V 2 A /r 2 B 2 /4πρr 2. The region of the star in which we are interested is convectively stable, with a local (real) Brunt Väisälä frequency that is defined by N = (g/h) 1/2 ( ad ) 1/2. Consider a displacement of amplitude ξ in this medium. Because of the energy in the magnetic field, the displacement results in the release of kinetic energy (ω 2 A ξ 2 )/2 per unit of mass. In a highly stabilized medium, where N is large, the amount of work that has to be done in order to move material vertically against gravity is so large that vertical motions are unlikely. Instead, if we wish to ensure that the released kinetic energy can overcome the strong ambient stabilization (thereby leading to instability), the displacements must occur in an almost horizontal direction. [See fig. 1 of Spruit (1999), illustrating the slipped disc nature of the Tayler instability.] In order that the magnetic energy be able to overcome the stabilizing effects of gravity, the ratio of radial displacement L r to horizontal displacement L h cannot exceed the value ω A /N.Inview of the upper limit on L h (<r), this leads to an upper limit on L r.onthe other hand, alower limit can be set on L r by requiring that, in the time-scale on which Tayler instability grows, the field should not diffuse away significantly. Combining the two limits on L r, Spruit shows that for Tayler instability to occur, the field strength ( ω A ) must exceed a certain critical value [ ω A (crit)]. The value of the Tayler critical field strength depends on N and on the diffusivity coefficient η (see equation 8 in Spruit 2002). Now, inside a differentially rotating star, the effects of field-line stretching are such that the azimuthal field may be continuously amplified at a certain rate. This winding-up can be treated as a kinematic process as long as the field is not too strong. In the kinematic regime, the time-scale τ a,onwhich the field grows, depends linearly on the (inverse of the) angular velocity gradient parameter q = d ln / dlnr. The essence of Spruit s model enters at this point. He asserts that the magnetic field strength inside a differentially rotating star will grow until it reaches the critical value, i.e. the field strength that is Dynamo-generated magnetic fields 1141 just large enough to drive the Tayler instability. Spruit bases this marginal instability assertion on an analogy with convection: when astrophysical convection sets in, the eddies take on diffusive properties which ensure that the effective Rayleigh number of the flow is just equal to the critical Rayleigh number required for convective instability. By analogy, Spruit asserts that, inside a magnetically unstable star, the effective diffusivity of the medium takes on the value η e, which is just marginally consistent with the onset of Tayler instability. Given a value for η e, the time-scale τ d for diffusive decay of the field from eddies with dimensions L r and L h can be evaluated. As well as τ a and τ d,athird time-scale in the problem is the growth time of the Tayler instability: τ g = /ω 2 A.Byequating the various timescales, Spruit (2002) derives expressions for the azimuthal and radial field strengths in marginally unstable conditions. The equilibrium strengths to which the fields grow depend on (among other factors) the value of q.for the sake of consistency, in order to ensure that ω A exceeds a certain value (thereby driving the Tayler instability), it is necessary that the gradient q exceed a certain limit. We will make this condition more specific below (Section 3.3). As well as deriving the effective diffusivity η e in marginal instability, Spruit also obtains other transport coefficients. These coefficients, while not of direct interest to us in the present work, were of special interest to Maeder & Meynet (2003) in their study of the transport of angular momentum and chemical elements; they found that the effects of magnetic diffusion may overwhelm the effects of meridional circulation. We report below on estimates of Tayler dynamo field strengths in the interiors of our own models; Maeder &Meynet do not report on the numerical values of the field strength in their paper. 3.2 Dynamo formulae In a medium where thermal diffusivity dominates over composition gradients, Spruit finds that the equilibrium azimuthal field in chemically homogeneous material is given by B ϕ = r(4πρ) 1/2 q 1/2 ( /N) 1/8 (κ/r 2 N) 1/8 G (1) and the equilibrium radial field is given by B r = B ϕ ( /N) 1/4 (κ/r 2 N) 1/4 G. (2) In these formulae (which are equations 22 and 23 in Spruit 2002), the quantity N is the Brunt Väisälä (or buoyancy) frequency, g is the local gravity, H is the local pressure scaleheight, and κ = 16σ T 3 / (3κ R ρ 2 C p )isthe thermal diffusivity. In regions of the star where thermal diffusion can be ignored, and composition gradients are responsible for stabilization of the gas, the formulae for the field components are somewhat different (see Spruit 2002, equation 21). Throughout most of our models, equations (1) and (2) are the relevant formulae for the fields. In Spruit s marginal instability theory, we note that, apart from some coefficients of order unity, there are no free parameters. Given a snapshot of the internal structure of a star at any particular stage of evolution, the above formulae yield values for the local field components at each point in that star. 3.3 Criteria for Tayler dynamo activity Although there are no free parameters in the predicted field strengths once the Tayler dynamo reaches a state of marginal instability, nevertheless certain criteria must be satisfied before the dynamo can operate at all.

4 1142 D. J. Mullan and J. MacDonald One criterion, which can be tested a priori, states that the gradient q in angular velocity inside the star must exceed a minimum value in order that the stabilizing (diffusive) effects of the ambient medium can be overcome. The minimum value required of the angular velocity gradient q min can be written as the sum of two terms, q 0 + q 1, each term being determined by a different source of diffusivity (Spruit 2002). The separate terms q 0 and q 1 depend on the stabilizing effects of gradients in molecular weight and temperature, respectively. These two stabilizing effects are quite distinct from each other in the sense that for fluctuations with small enough length-scales, the temperature gradient is no longer an effective stabilizer. (The fact that stabilizing effects of subadiabatic gradients disappear on short length-scales was used by MM03 in their discussion of mass loss from cool giants.) In contrast, negative molecular weight gradients have a stabilizing influence on all length-scales. A second criterion is based on the requirement that a Tayler dynamo (which depends for its existence on magnetic energy) cannot operate unless the seed field exceeds a lower limit. Spruit (1999) gives formulae (his equations 47 and 49) for the lower limit on the seed field for Tayler instability in terms of a lower limit on ω A in the limits of strong or weak thermal diffusion. Because of the approach Spruit takes to dynamo operation, this criterion turns out in practice to be equivalent to the a priori condition on q. One of Spruit s criteria cannot be tested beforehand, but is subject only to a posteriori test. This concerns the local Alfvén frequency ω A. Spruit s treatment requires that ω A should be small compared to the local angular velocity. Actually this is not strictly necessary; it merely allows us to take advantage of the convenient (and simplified) formulae given by Spruit (2002). In the models we have computed, we have verified that this a posteriori criterion is in fact satisfied. Also to be tested a posteriori are criteria that we will consider below for the onset of buoyancy instability. Finally, as we have already noted, the requirements of kinematic winding-up set an upper limit on the initial field. Spruit (1999) refers to this limit in the context of magnetic white dwarfs. It appears that there is a window of opportunity of initial field strengths within which Spruit s treatment of the Tayler instability is applicable. We note that in the context of another magnetic instability (MRI), a similar window exists, with well-defined upper and lower limits on field strength (Balbus & Hawley 1994). 3.4 Tayler instability in spherical geometry The Tayler instability was originally derived in the context of a magnetic cylinder. The motions which are involved in the instability are mainly in the ϖ -direction, which increases radially outward from the axis of the cylinder. Tayler applied this class of instability in the context of the stably stratified gas in a star: applying the analysis of the instability to polar caps close to the axis, the ϖ -direction is effectively perpendicular to the direction of gravity. As a result, even if the gas is strongly stratified (i.e. highly subadiabatic), the stabilizing effects of gravity are largely circumvented. This is an important aspect of the process whereby dynamo action can occur in convectively stable gas. In his 1999 paper, Spruit points out that latitudinal effects play a role when Tayler instability is applied to a sphere. The instability operates most readily at latitudes close to the poles, and less effectively as one moves towards the equator. However, in the dynamo paper (Spruit 2002), the latitudinal properties of the dynamo are mentioned only in passing in a footnote. Neither are latitudinal effects discussed by Maeder & Meynet (2003) in their application of Spruit s dynamo model to transport processes in stellar models. In order to understand how variations in latitude might affect a Tayler dynamo, it is important to consider how the Tayler instability is to be treated in the context of a spherical object such as a star. The most general discussion of instabilities of toroidal magnetic fields and rotation is that provided in a lengthy paper by Acheson (1978). Although Acheson does not refer to the Tayler instability by that name, he discusses the instability in a subsection entitled almost horizontal motions (his p. 490). [Acheson s discussion in that subsection refers to non-rotating stars, whereas Spruit (1999) includes the effects of rotation.] Acheson finds that, in the weak field limit (V 2 A gr), Tayler instability in a sphere occurs in its most easily excited mode (m = 1, i.e. non-axisymmetric) provided that the quantity B 2 sin 2θ increases outward along a spherical surface. (Here, θ is the magnetic co-latitude, and B is the field strength.) Now, in the stellar models which we use for our work (i.e. those taken from MM04), we use the approximation of shellular convection, i.e. quantities are constant along spherical shells. As a result, the field values B which we derive from Spruit s formulae are inevitably constant on spheres. Therefore, as we move outward from the magnetic pole along a spherical surface, the condition for Tayler instability reduces to an increasing value of the quantity sin 2θ; this condition is satisfied for magnetic co-latitudes up to π/4. Thus, in the weak field limit, Tayler instability occurs as non-axisymmetric modes in polar caps which extend from the North and South magnetic poles to magnetic latitudes of 45 in both hemispheres (see also Goossens, Biront & Tayler 1981; Spruit 1999). Therefore, even though the Tayler instability is strongest near the magnetic poles, the mechanism is not confined only to small regions in the immediate vicinity of the poles. Instead, the instability operates over some 30 per cent of the surface area of the sphere. In fact, the spreading may be even more extensive than 30 per cent of the area. For example, in the strong field limit (V 2 A gr), Acheson finds that Tayler-unstable modes spread over 50 per cent of the surface, extending down to latitudes as low as 30 in each hemisphere. Moreover, in the same limit, axisymmetric Tayler-unstable modes appear even in the equatorial belts, extending from the equator to latitudes of ±30. Thus, in the strong field limit, Tayler instability occurs over the entire sphere (in the context of shellular convection). Below, in our post-processing analysis of our results, we shall test the fields created by Spruit s dynamo in our models to determine if they are in the weak-field limit or the strong-field limit. 4 MAGNETIC BUOYANCY INSTABILITY Tayler instability is not the only process that is important when we consider the properties of magnetic fields in massive stars. A second type of magnetic instability, i.e. buoyancy, also merits consideration. 4.1 Onset of buoyancy instability At first sight, in view of the results presented by Spruit (1999), it may seem that buoyancy ought to be unimportant compared to Tayler instability in the stellar context. After all, according to fig. 2ofSpruit (1999), the onset of buoyancy instability requires field strengths which are three or four orders of magnitude stronger than those required for onset of Tayler instability. However, there is no contradiction: Spruit s plot refers only to the onset of the instabilities. We shall find that, when the Tayler instability has become non-linear and reached saturation (in the sense discussed by Spruit 2002), the local field strengths have risen to values that are orders of magnitude larger than the strengths required for onset. Such strong fields may indeed then become subject also to buoyancy instability.

5 We consider two limiting approaches to investigating instability due to magnetic buoyancy. In one limit, the filling factor of magnetic flux tubes is low; then we may consider the motion of a discrete magnetic flux tube rising up through essentially field-free gas. This is the approach that was used by MM04. If a dynamo generates magnetic flux in the form of discrete flux ropes, then it is possible that any particular flux rope may be surrounded by field-free material, i.e. the filling factor of the magnetic field may be small. In this context, MM04 determined the limiting field strengths that are necessary at a prescribed initial radial location inside a given stellar model such that a flux tube with that initial field strength will be just able to rise to the surface by means of buoyancy. In calculating the buoyancy effects, it is essential to include radial gradients of chemical composition in the star; such gradients can be strongly stabilizing. MM04 parametrized the limiting field strength in terms of β 0, the initial value of the plasma β parameter. (The plasma β is defined to be the ratio of gas plus radiation pressure to magnetic pressure, i.e. stronger magnetic fields correspond to smaller values of β.) Values of β 0 as a function of radial location inside models of 10- and 50-M stars at various stages of evolution were obtained by MM04 (their fig. 11). Not surprisingly, a flux tube that starts its upward rise deep inside the star requires a stronger initial field (i.e. its β 0 value is smaller) to reach the surface than a tube which starts its rise far out in the envelope. A second limiting approach to buoyancy instability assumes that the filling factor of the magnetic field is high. In this limit, a particular radial profile of magnetic field strength can be subjected to linear magnetohydrodynamics instability analysis. Acheson (1978, p. 489) discusses the buoyancy instability in the presence of doubly diffusive conditions. In the presence of thermal and magnetic diffusivities, buoyancy instability occurs if any one of three conditions is satisfied (see his equations ). In the context of our stellar models, where spherical symmetry is assumed to hold, Acheson s analysis is not immediately applicable, because his formulae are expressed in terms of cylindrical coordinates. However, by considering Acheson s general dispersion relation (his equation 3.20) in the limit of vertical motions, his equation (8.12) the planar analogue of equation (8.9) can be rederived in spherical coordinates to obtain for the instability condition ) ln BR m 2 R 2 sin 2 θ + ηg ln pρ γ κva 2 R < ( 2 R g a 2 where a is the isothermal sound speed. A brief derivation of this formula is given in Appendix A. Although the similarity between equation (3) and Acheson s equation (8.12) is apparent, we stress that R in the above formula represents the radial coordinate of spherical geometry; it is to be distinguished from the (cylindrical) r coordinate (sometimes written as ϖ ) in Acheson s equation (8.9). The usefulness of equation (3) is that, by setting sin θ to a particular value, we can apply it a posteriori to any of our shellular models once we have derived the radial profile of magnetic field strength. This will allow us to test if our field profiles are subject to buoyancy in the equatorial plane (say) if we set sin θ = 1, or at latitude 45 (say), if we set sin θ = 1/ 2, or at any other latitude we choose. 4.2 Comparison and contrast between Tayler and buoyancy instabilities The first difference to be noted between the two instabilities has already been noted. In terms of the magnitude of the magnetic field, Tayler instability sets in more readily than buoyancy; the difference R (3) Dynamo-generated magnetic fields 1143 in threshold field strength is several orders of magnitude (Spruit 1999). Secondly, because the sin 2 θ term appears in the denominator in equation (3), buoyancy instability (in the limit of large filling factor) tends to set in more readily near the magnetic equator than at high magnetic latitudes. (This is the case if the sin 2 θ term is large enough to play a significant role in equation 3 in some models, the radial gradient of field strength is so dominant that latitudinal effects are small.) This favouring of equatorial zones for the buoyancy instability is in contrast to Tayler instability, which sets in more readily near the magnetic poles. Thirdly, in discussing the Tayler instability, with its almost horizontal motions, gravity is largely irrelevant: the instability is driven largely by magnetic tension effects (including curvature). In contrast to this, buoyancy instability relies in an essential manner on gravity, involving as it does motions with significant vertical components. Despite the significant physical distinctions between the two instabilities, it is important for our purposes to note that the instabilities are not mutually exclusive. In fact, the combined operation of the Tayler instability and the buoyancy instability is an essential aspect of the present paper. This combined operation leads to surface field manifestations occurring at mid-latitudes. This is reminiscent of the solar surface, where the strongest fields (those occurring in sunspots) are observed to be confined almost entirely to a band of latitudes between about 40 and 10 in both hemispheres. 5 MAGNETIC FIELD ESTIMATES IN MODELS OF MASSIVE STARS We have applied Spruit s formulae to the detailed quantitative models of rotating stars which we reported in MM04. Model 10A (50A) refers to a 10-M (50-M ) model on the ZAMS. Model 10C (50C) refers to a 10-M (50-M ) model at the TAMS. Models 10B and 50B are of intermediate age. 5.1 Model 10A Applying the test q > q min,wefind that this criterion is satisfied throughout the non-convective envelope of model 10A. Therefore, it is consistent to assume that the Tayler dynamo, as modelled by Spruit, can operate in a star that is described by this model. In Fig. 1 we show the radial profile of the internal azimuthal and radial magnetic fields which are predicted by Spruit s formulae when applied to our 10A model. The gap inside radial coordinate 0.22 is due to the presence of a convective core; there, Spruit s formulae do not apply. Also, although it is more difficult to notice, there are two narrow convection zones near the surface, at r/r = and r/r = Spruit s formulae are also not applicable in those narrow zones. Moreover, in the layers closest to the surface, where differential rotation falls essentially to zero in our models, the field strength generated by the Spruit dynamo falls formally to zero. Referring to Fig. 1, we see that, in the non-convective regions, the equilibrium azimuthal field strengths range from MG values just outside the core to kg values far out in the envelope. As regards the radial field component, the strength varies from a few kg near the core, to a few hundred G close to (but underneath) the surface. Applying the a posteriori test mentioned in Section 3.2, we find that, as regards the upper-limit criterion on ω A, the numerical values of the ratio ω A / in the non-convective regions lie in the range Thus, it is permissible to use the simplified formulae as given by Spruit (2002). Moreover, we have tested the results in Fig. 1 and found them to be in all cases in Acheson s weak-field limit.

6 1144 D. J. Mullan and J. MacDonald Figure 1. Radial profiles of azimuthal (solid curve) and radial (dotted curve) components of the magnetic fields generated by a Tayler dynamo in a rotating 10-M star on the ZAMS. We use model 10A from MM Buoyancy onset in massive stars As regards the matter of observational detection of magnetic fields on the surface of massive stars, Maeder & Meynet (2003) write that, because the field strength predicted by the Spruit Tayler dynamo formally tends to zero near the surface, this may explain why there is in general no strong magnetic field observed at the surface of OB stars. However, we consider the following question to be key: are there any regions in the deep interior of the star from which flux tubes can rise to the surface? If the answer is yes, then the surface field strength will be related to the field strength in those deep regions. Magnetic buoyancy (if it can occur) is a mechanism which can bring magnetic field from deep inside a star up to the surface. In this context, let us examine our 10A model in the context of the two criteria mentioned above for buoyancy instability. First, consider buoyancy in the limit of a flux tube model. Results for the limiting β 0 from fig. 11 of MM04 are plotted in Fig. 2 of the present paper as a dotted line labelled β (limit). In the present paper, Spruit s theory gives us a local value of magnetic field strength at each radial location. Comparing this with the local gas and radiation pressures, we can obtain the radial profile of the plasma β parameter predicted by Spruit s model. When the value we have computed locally for β in model 10A falls below the β (limit) curve, then the locally generated dynamo field ( 1/ β)issostrong that the buoyancy force is large enough to ensure the flux tube will reach the surface (in the context of a flux-tube model). Fig. 2 indicates that, in the 10A model, this condition is satisfied at all radial locations with radial location in excess of r = r(buoy) = 0.47R. Secondly, consider the criterion for buoyancy in the limit of a large filling factor; the criterion for buoyancy instability in the equatorial plane is given by equation (3) with sin θ = 1. Applying this to the results of our 10A model, we find that buoyancy instability occurs within the ranges of radial distance shown in Fig. 2 by the dash-dotted line labelled equation (3). According to this criterion, the fields in the equatorial plane are unstable to buoyancy at radial locations between about 0.4R and 0.7R, and again between 0.77R and 0.975R. Thus, according to both limiting cases of buoyancy instability (flux tube and large filling factor), the fields that we have derived from Spruit s dynamo theory (based on Tayler instability) are contin- Figure 2. The solid curve is the radial profile of the plasma β parameter (= total pressure/magnetic pressure) in model 10A. The dotted curve is the limiting value of β such that buoyancy forces can raise a flux tube launched at radius r all the way to the surface. The dash-dotted curve shows the ranges of radial distance within which the buoyancy instability occurs. At radial distances in excess of r(buoy), the dynamo-generated fields are strong enough to reach the surface. uously subject to buoyancy instability in an annulus in the equatorial plane that extends from just beneath the surface (0.975R )toadepth that is at least as far in as 0.77R,ormay be even deeper (0.47R ). 5.3 Latitudinal distribution of surface fields The Tayler instability operates at magnetic latitudes of 45 and higher (in the weak-field limit which applies to our models). On the other hand, buoyancy favours lower latitudes. As a result, the fields which emerge at the surface as a result of the combination of Tayler instability and buoyancy are expected to occupy a band of intermediate latitudes. In order to quantify the extent of this band, we have considered the case of buoyancy instability at non-zero latitudes, specifically in regions where the Tayler instability operates (θ = 0 45 ). To test for buoyancy instability at 45,weinsert sin θ = 1/ 2inequation (3); we find no detectable variation in the dash-dotted line in Fig. 2. The reason for this behaviour is that the term involving the radial gradient of the field dominates in equation (3). We must go well into the polar cap to see any significant change in this result. For example, at sin θ = 0.1 (i.e. within a few degrees of the pole), we find that the inner border of the buoyant unstable annulus moves out by about 0.04R. That is, buoyancy instability can occur at radial distances between 0.81 and 0.975R even near the pole. Because the Tayler instability occurs at θ values between 0 and 45, and buoyancy instability occurs at θ values which extend from 90 to nearly zero degrees, these results provide quantitative support for the statement made above that Tayler instability and buoyancy instability are not mutually exclusive. Our analysis of the two instabilities leads to the following important conclusion: material which occupies a spherical shell extending over the outermost per cent of the stellar radius generates Tayler dynamo fields (according to Spruit s formulae) which are so strong that buoyancy forces will bring the fields in those regions up to the surface. Thus, the emergent fields will occur in caps extending from about 45 latitude towards the magnetic poles in both hemispheres, covering at least 30 per cent of the surface area. In fact, some

7 Dynamo-generated magnetic fields 1145 Figure 3. Dynamo-generated magnetic fields in a 50-M star on the ZAMS. For details, see caption of Fig. 1. overflow of field to latitudes that lie even lower than 45 is expected due to the expansion of individual flux tubes as they are buoyed up to the surface from deep within. For example, if a tube were to reach the surface with an angular diameter of the order of 10 (corresponding in fractional surface area to some of the largest active regions ever observed on the Sun), such a tube could extend to latitudes on the surface of the star that are as low as 40. Moreover, it is well known (from studies of fields in planets and stars) that astrophysical dynamos do not automatically produce a magnetic axis that is aligned with the rotation axis: the two axes may be tilted, or even offset spatially. The discussion in the above two paragraphs was couched in terms of magnetic latitude only. If, in fact, the magnetic axis is tilted and/or offset relative to the rotation axis, then, depending on our viewing angle, the caps of emergent flux may occupy at times an even larger fraction of the visible disc than the 30 per cent quoted above. Clearly, the caps in which the field emerges under the combined effects of Tayler and buoyancy instabilities are by no means small features on the surface of the star. On the contrary, they are macroscopically large, in both Northern and Southern hemispheres. Therefore, they can be expected to exhibit macroscopic signatures in observations which involve integration over the disc of the star. 5.4 Model 50A In the 50A model, we find that, as in the 10A model, the differential rotation parameter q exceeds the minimum value for Tayler instability throughout the non-convective zone of the star. As a result, Spruit s dynamo formulation is relevant, and it yields the radial profile of azimuthal and radial field components as shown in Fig. 3. We find that in the deep interior of the 50A model, azimuthal field strengths reach values of several hundred kg, while radial fields reach values of a few kg. As regards the effects of buoyancy, we show in Fig. 4 the radial profile of β values obtained by Spruit s formulae, and also how these compare with the limiting β values (taken from fig. 11 of MM04) in the low filling-factor limit. In the limit of a large filling factor, the results of equation (3) are plotted in Fig. 4 as a dashdotted line; there are more gaps in this line than in the case of model 10A, because of differences in the detailed radial gradient of field Figure 4. Same as Fig. 2, but for model 50A. strength predicted by the Spruit formulae. In both limits, the criteria indicate that buoyant instability exists in a continuous range of radial distances from about 0.77R to the surface convection zone which extends down to 0.97R.Testing equation (3) at non-zero latitudes, we again find that the buoyancy instability extends to latitudes of 45 and larger. Thus, as for the earlier 10A model, we find the important result that the radial range of buoyant uplift of magnetic fields overlaps with the region of dynamo operation in the 50A model. At all radial positions external to about 0.77R, the local fields are so strong that they will be continually buoyed up to the surface. 5.5 Other models We have applied Spruit s approach to the remaining four MM04 models: 10B, 10C, 50B and 50C. In all cases, we find that the a priori condition q > q min is satisfied throughout the non-convective zones. As the 10-M star evolves off the main sequence, v rot falls by about one-third (from 174 to 119 km s 1 )ingoing from the ZAMS (model 10A) to the TAMS (model 10C); see MM04. The decrease in rotation results in some reduction in the field strengths generated by the Tayler dynamo in models 10B and 10C compared to model 10A. However, the reductions are relatively small: values of B ϕ deep inside model 10C still lie in the range MG, while values of B r remain at the level of a few kg throughout most of the interior. As a result, the fields which are buoyed up to the surface of a TAMS 10-M star are expected to be only slightly smaller than those on a ZAMS star with the same mass. In contrast to the 10-M models, the 50-M models of MM04 show a very large reduction in v rot in the course of evolution from ZAMS to TAMS. The value of v rot decreases from 116 km s 1 in model 50A, to 15 km s 1 in model 50B, to 2.3 km s 1 in model 50C (see MM04). This large reduction occurs in part because of expansion in stellar radius (by a factor of about 4) in going between models 50A and 50C, and in part from significant mass loss. Because the stellar radius is also increasing as we go from model 50A to 50C, the decreases in (=v rot /R) are even more pronounced than those in v rot.asaresult, dynamo fields are much weaker in the evolved 50-M models. For example, in the buoyancy region of model 50B [i.e. at radial locations outside r = r(buoy) = 0.71R ], the radial field strength lies in the range G, i.e.

8 1146 D. J. Mullan and J. MacDonald about an order of magnitude weaker than the kg fields we found in model 50A. Also, in model 50C, at the TAMS, our predictions of surface field strength are no more than a few G. Our work suggests that evolutionary effects are expected to lead to more severe reduction of field strengths in 50-M objects than in 10-M objects. 6 SURFACE FIELD STRENGTHS Given that fields rise from inside a star of 10 or 50 M, possibly from as deep as 0.47R,weexpect the surface of such stars at any instant of time to contain a number of discrete magnetic structures. Each structure represents a flux tube which has emerged in the recent past due to buoyancy instability. Such a structure would be analogous to an active region in the Sun. The structure will survive for a longer or shorter interval of time, depending on the processes which can erode it. Can we estimate the field strength in the surface structures on M stars, now that we know some details about the internal field strengths? A quantitative answer to this question would require us to model in detail how a flux tube from a Tayler dynamo (with its slipped-disc topology ) rises up through the outer layers of the star. Such a model calculation has not yet been performed. However, even without such a calculation, we may be able to make some preliminary estimates based on a solar analogy. 6.1 Guidance from solar fields In an application of his Tayler dynamo formalism to the Sun, Spruit (2002) has found that in the outermost region of the solar core, the field created by a Tayler dynamo has an azimuthal component of the order of 10 kg, and a radial component of the order of 1 G. The location of the Tayler dynamo in Spruit s calculation is at radial distances of ( ) R, i.e. at depths of about 0.3 R below the surface. These depths are comparable to the depths from which magnetic fields in hot stars may be buoyed upward. The question is, how do the field strengths in the Tayler dynamo region compare with the strengths of the magnetic fields that are observed on the surface of the Sun? The answer to this question depends on angular resolution. When there is enough angular resolution to allow observers to examine individual active regions in detail, and the field can be measured with high resolution (in pixels as small as 2 2 arcsec 2 ), the photospheric field strength in non-spotted regions is observed to be about 200 G (e.g. Gopasyuk et al. 2000). [Although fields in localized patches (sunspots) are stronger than this, most of the flux in active regions occurs in non-spotted gas.] However, if the resolution is poorer, there is increased contribution from field-poor regions, as well as significant cancellation among different active regions. As a result, the mean field strengths over large areas of the active Sun typically have values that are no larger than G (e.g. Foukal 2004). Such field strengths are not greatly in excess of the global (dipole) field of the Sun, which has values in the range 5 10 G (Snodgrass, Kress &Wilson 2000). If we adopt the hypothesis that the fields in solar active regions owe their origin ultimately to the azimuthal Tayler dynamo fields in the tachocline, then the solar surface fields in small pixels are some two orders of magnitude weaker than the fields in the generating region. Also, when we step back and view the Sun globally, the cancellation effects are such that the large-scale fields are weaker than the fields in the dynamo region by some three orders of magnitude. 6.2 Applying the solar analogy to hot stars Before we apply the solar estimates to hot stars, we note a significant difference between the Sun and the stars we study in the present paper. In the Sun, as the fields rise up from the generating region (in the tachocline) to the surface, the flux tubes must pass through the entire convection zone of the Sun. Flux tubes passing through the convection zone are subject to significant stresses due to the turbulent convective flows. These are expected to lead to significant distortion of the flux tubes, with shredding and reconnecting processes at work. Some (at least) of the two to three orders of magnitude reduction in field strength between the Tayler dynamo site and the solar surface must be due to these distortions. In contrast, when we consider the rising of flux tubes through the envelope of a massive star, the flux tubes do not have to pass through any significant intervening convection zone (apart from very shallow regions near the surface). As a result, it seems plausible that the weakening factors between source and surface in a massive star would not be as severe as those in the Sun. In other words, the reduction factor of three orders of magnitude between dynamo source and the global surface field is expected to be an upper limit in the case of massive stars. Now, as we have seen, in the interior regions of model 10A where buoyancy is expected to occur (at a radial distance of R ), the azimuthal field is in the range kg. Applying a reduction of no more than 10 3,weexpect global surface fields of >(60 500) Ginmodel 10A. In model 50A, with buoyant flux tubes rising up from radial locations of 0.77 R, the local field strengths are of the order of kg. Applying a reduction factor of no more than 10 3,weexpect global surface fields of >( ) G. We recognize the limitations of the approach outlined above. Only a full model calculation will allow us to derive more reliable estimates of surface field strengths. However, it is a matter of some interest to compare our rough estimates with observations. 6.3 Comparison with observations We note that a 10-M star corresponds to a star with a spectral type of early B. Therefore, our models 10A, 10B and 10C represent roughly the evolution of an early B star from ZAMS (luminosity class V) to subgiant (luminosity class IV). Our results suggest that dynamo action in such stars should lead to global surface fields of at least G. We also note that a 50-M star on the ZAMS corresponds to a spectral class of mid-o, so we expect to see surface fields of at least G. As regards B stars, we note that there now exists a sample of four slightly evolved B2 stars for which field measurements exist (see Section 1). In order to see whether it is permissible to make comparisons between any of these stars and our models, we note that the rotational velocities of the four stars (in km s 1 ) are reported to lie in the ranges (ω Ori), (V 2052 Oph), 55 ± 28 (ζ Cas) and (β Cep). (We obtain these numbers by combining ranges of v sin i and ranges of i in the papers cited above.) The fastest of these four stars has a rotational velocity that is comparable to (or faster than) the fastest rotator among our models (10A). It is also the star which shows the largest observed LOS field strengths. According to Neiner et al. (2003c), individual measures of the LOS field