Magnetic fields in stars

MAGNETIC FIELDS IN STARS

David MOSS Mathematics Department, The University, Manchester, M13 9PL, U.K.

NORTh-HOLLAND-AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 140, No. 1 (1986) 1—74. North-Holland, Amsterdam

MAGNETIC FIELDS IN STARS David MOSS Mathematics Department, The University. Manchester, M13 9PL. LI. K. Received January 1986

Contents: 1. Introduction 2. Some background magnetohydrodynamics 2.1. Basic MHD 2.2. Outline of dynamo theory 3. The magnetic chemically peculiar stars 3.1. Introduction 3.2. Observations 3.3. Origin of the fields 3.3.1. Magnetic oscillator theories 3.3.2. Battery models 3.3.3. The core-dynamo theory 3.3.4. Fossil theory 3.4. Stability of stellar magnetic fields 3.4.1. Dynamical instabilities 3.4.2. Buoyancy instabilities 3.5. Braking mechanisms: magnetic winds and accretion 3.6. Distribution of obliquities 3.7. Theoretical models of early type magnetic stars 3.7.1. General results on the structure of rotating radiative envelopes with a magnetic field 3.7.2. Polytropic models

3 4 4 8 12 12 12 16 17 17 19 20 21 21 23 24 28 30 30 32

3.7.3. Non-polytropic models 3.8. The rapidly oscillating Ap stars 3.9. Secular changes in magnetic fields 3.10. Discussion 4. Magnetic fields in lower main sequence and related stars 4.1. Introduction 4.2, Summary of observations 4.2.1. The sun 4.2.2. Late type stars other than the sun 4.3. Theoretical models 4.3.1. The solar field 4.3.2. Lower main sequence stars 5. Other topics 5.1. Degenerate objects 5.1.1. Observations 5.1.2. Theoretical interpretation 5.2. The effects of stellar fields on global oscillations 5.3. Magnetic fields and convection 5.4. Magnetic fields and star formation. 6. Summary References

33 38 40 40 42 42 42 42 43 47 47 53 58 58 58 59 63 64 66 67 69

Abstract: The observational evidence for stellar magnetic fields is reviewed. A detailed theoretical discussion of fields in the chemically peculiar starsof the middle main sequence is given, contrasting the rival claims of the fossil and contemporary dynamo theories for the origin of the field. It is shown how current theoretical ideas can lead to inferences about the interior field, even though only limited information about the surface field distribution is available. The solar field is discussed in terms of dynamo theory, and tentative extensions to similar, lower main sequence, stars are discussed. An account is also given of magnetic fields in degenerate objects (white dwarfs and neutron stars). Brief reference is made to magnetic fields in other stages of stellar evolution.

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D. Moss, Magnedc fields in stars

3

1. Introduction Rotation, magnetic fields and the presence of a close companion all perturb stars from the spherical symmetry that is usually assumed in classical studies of stellar structure. This review is directly concerned with the effects of magnetic fields on stellar structure, a discussion of the origin of the magnetic field in stars of various types, and to an explanation of howthe field structure is constrained by its immersion in a star. Some mention of rotational effects is inevitable as rotation and magnetic fields are inextricably linked, but the discussion is restricted almost entirely to single stars, and gravitational effects in close binary systems are totally excluded. A basic acquaintance with the principles of stellar structure is assumed (e.g. Schwarzschild, 1958; Tayler, 1970), and also with some of the ideas of magnetohydrodynamics, although a brief survey of the more relevant effects is given in section 2. Accounts of the effects of rotation on stellar structure can be found in the book by Tassoul (1978), and in reviews by Strittmatter (1969) and Moss and Smith (1981), amongst others. Magnetic fields often appear to be a smaller perturbation to the overall stellar structure than rotation, in that there are some stars which rotate sufficiently quickly for centrifugal force to be an appreciable fraction of gravity, whereas it appears very unlikely that the same is ever true of magnetic forces, except perhaps in very limited superficial layers. Nevertheless there are a number of interesting and even spectacular phenomena which can be attributed to stellar magnetic fields. Even weak fields present early in a star’s lifetime may significantly affect its later evolution, for example by maintaining a state of near uniform rotation (see section 3) and, even more speculatively, by modifying the supernova process. Angular momentum loss amplified by a large scale field probably is crucial in understanding star formation (section 5.4), and also rotational deceleration during stellar evolution (section 3.5). The solar field was first discovered in 1908 by Hale, and the sun’s field has been closely observed since. However, for a long time, the study of stellar fields concentrated on the peculiar A and B stars, where the presence of a strong surface field (to i04 or i05 G) is related to the occurrence of anomalous chemical abundances in the superficial layers, and also usually to relatively low rotation rates compared to many of the similar stars without strong fields. More recently there has been a great deal of interest in magnetic fields in solar type lower main sequence stars, and it seems that magnetic activity analogous to that of the sun is widespread in dwarf stars with outer convection zones, many displaying analogues of the 22 yr solar cycle. This branch of astrophysics has become known as the ‘solar-stellar connection’. Strong magnetic fields (up to about 108 G) are observed in some white dwarf stars, and the activity of the AM Her systems seems to be controlled by the field of the magnetic white dwarf primary. Pulsars are believed to be rotating neutron stars with fields of order 1012 G. Thus magnetic fields seem to be present and the source of interesting phenomena during both the early and late phases of stellar evolution at star formation and on the main sequence, and during senescence as a cooling degenerate object. Direct evidence for fields during intermediate evolutionary phases is much weaker. This review is concerned mainly with the mutual interaction of large scale magnetic fields with the pressure-density fields of the stars in which they are found. Most attention is given to stars on or near the main sequence, although mention is made of star formation, white dwarfs and other topics. The solar field is not considered in detail, but there is a discussion of its general properties in as much as they exemplify the fields present in other lower main sequence stars. Chromospheric and coronalfields, their instabilities and their relation to heating and radiative mechanisms, are not discussed. Much background and supplementary material can be found in the textbooks by Parker (1979) and Priest (1972). Zeldovich et a!. (1983) gives an often incisive account of astrophysical magnetic fields from the viewpoint of dynamo theory. Previous recent reviews on stellar magnetism include Mestel (1978, 1984a), Moss and Smith (1981) and Borra et al. (1982), and Khokhlova (1985) surveys the Ap stars. —

4

D. Moss, Magnetic fields in stars

The remainder of the article is arranged as follows. Section 2 gives background material on magnetohydrodynamics and dynamo theory. Section 3 is concerned mainly with topics relating to the magnetic A and B stars, starting with a survey of the observational evidence. One or two topics discussed here are directly relevant to later sections, for example section 3.5 on magnetically controlled stellar winds. A discussion of the observational evidence for, and theory of, magnetic fields in lower main sequence stars is given in section 4. Miscellaneous topics, including magnetic fields in white dwarfs, are grouped together in section 5, and section 6 is a brief conclusion. There has been no attempt to give exhaustive references nor does the author claim to have read or even found every relevant paper. The choice of and emphasis given to topics certainly reflects his personal interests. Apologies are offered to any reader who feels that his own contribution to the subject may have been under- or mis-represented.

2. Some background magnetohydrodynamics 2.1. Basic MHD Astrophysical magnetohydrodynamics is treated comprehensively in the monograph by Cowling (1976). This section endeavours to give an outline of a few basic concepts which are used later in this review. All but a tiny fraction of stellar material consists of a gas which is dense enough to be treated as a fluid (i.e., the particles undergo frequent collisions) and in which at least some of the atoms have been ionized, providing free electrons and ions. Even in a quite lightly ionized gas, ion-neutral collisions are usually frequent enough to prevent separation of the components. Usually the net charge per unit volume of a stellar plasma is very small (the positive and negative charges nearly cancelling one another) and in a non-relativistic plasma most of the current flow is caused by the differential motion of ions and electrons, the current from the net convection of charge being negligible. Under these conditions the analogue of Ohm’s law, J = V!R, in a simple conductor can be generalized to the Galilean invariant form (using Gaussian CGS units) [

j=ULE+

vXB

where j is the current density and E the electric field. The appropriate forms of the Maxwell equations ~=—cVxE,

(2.1)

j=~—VxB,

(2.2)

then give the magnetohydrodynamic equation in particularly simple form as ~=Vx(vxB)—Vx(ijVXB),

(2.3)

where v is the fluid velocity and i~ithe magnetic diffusivity. ij = c21(4-rru) where the conductivity a- can generally be taken as varying as (temperature)’5 in conditions of interest in stars.

D. Moss, Magnetic fields in stars

5

To order of magnitude the ratio of the convection to the diffusion term on the right-hand side of eq. (2.3) can be estimated as ULIi’ = Rm, where U and L are typical values of velocity and length scale, respectively. In regions of the flow where this ‘magnetic Reynolds number’ Rm ~ 1 it can be valid to neglect diffusion. The resulting ‘perfect MHD’ equation can then be shown to imply the ‘field freezing’ constraint. Fluid can flow freely parallel to B, but if the fluid velocity has a component perpendicular to B, then the field lines will move with the fluid as if theywere frozen to it. The fluid particles can be thought of as beads, free to slide on wires (the magnetic field lines), but unable to move perpendicular to the wires without moving the wires also. Immediate corollaries in the stellar context are the ‘isorotation’ theorem (Ferraro, 1937) and its generalization (Chandrasekhar, 1956; Meste!, 1961), which says that in a steady state, in a rotating body of infinite conductivity containing an axisymmetnc magnetic field possessing a component in meridian planes, the angularvelocity must be uniform along any poloidal field line, so that v = kB + a~I

(2.4)

where k is a scalar, a is constant on field lines, ~ is distance from the axis of rotation and i is the unit azimuthal vector. The necessity for this condition can be seen physically by considering two points on a field line which have differing angular velocities. The angular separation between the points must inevitably change with time. The points remain connected by the same field line which thus has its inclination to equatorial and meridional planes altered, in contradiction to the assumption of a steady state. In eq. (2.4) the kB part of v clearly does not distort the field; the remainder represents a uniform rotation of points along individual field lines. Equation (2.4) together with V~B = 0 and the continuity equation V~pv =0 gives pv~,= iiB~,

(2.5)

where ~ is a constant along the joint poloidal stream/field lines. Here and subsequently subscript p denotes a poloidal component; any solenoidal vector field can be expressed as a sum of poloidal and toroidal components (e.g. Chandrasekhar, 1961), and in an axisymmetric system poloidal is synonymous with meridional and toroidal with azimuthal. When a- is finite and the fluid velocities are taken to be zero then eq. (2.3) becomes =

—v x (qV X B),

(2.6)

and in Cartesian coordinates with uniform diffusivity this is =

the classical diffusion equation. From this, decay times for fields can be estimated as

(2.7)

TD~-L2I’rj’,

where L is again a typical length scale. Substituting typical stellar values, say L 10’ 1 cm, i~ i0~CGS (corresponding to T 106 K) gives the estimate TD 3 x 1010 yr for the decay time of a large scale stellar field. This estimate for the slowest decaying mode is confirmed to order of magnitude by a detailed eigenvalue calculation. In general eq. (2.3) represents a competition between convection and diffusion, -~

6

D. Moss, Magnetic fields in Stars

grossly described by the parameter Rm. Naive estimates for Rm are often extremely largein astrophysical contexts, even for small values of a- as the length scales are very large, and so flux freezing is often a very good aproximation. However care must be taken to ensure the relevant local length scale L is used, which may sometimes be very much smaller than a naive estimate of the global scale, thus permitting a local breakdown of the flux freezing condition. This point is central to the anti-dynamo theorem of Cowling (1934) see section 2.2. A number of studies have been made of solutions to eq. (2.3) which assume an ‘inexorable’ fluid velocity v (essentially assuming the kinetic energy density to be much greater than the magnetic, so that the Lorentz force is negligible and v is independent of B). Weiss (1966) considered prescribed two-dimensional cellular motions acting on an initially uniform field between two perfectly conducting boundaries, e.g. fig. 2.la. He showed that for a magnetic Reynolds number UL/i1>> 1 (L the cell size, —

U maxjv~)the field lines are wound up, almost satisfying the field freezing constraint, until the local field scale 1 is reduced to —ij I U, when reconnection can occur. The field is largely expelled from the centre of the cell and concentrated at the edges the ‘flux expulsion’ mechanism which has been much appealed to subsequently (see fig. 2.lb). This calculation shows explicitly that a global estimate Rm ~ 1 -~

—

——

—

——

~ ,.

/

If ~

——

-.-.

‘

/

\\

~\

-~ —

——

,/

- —,

—

—

I ,1

C). (a)

(b) Fig. 2.1. Field lines from a calculation with an inexorable cellular velocity in a conducting fluid, similar to Weiss (1966). (a) Initial magnetic field configuration (solid) and velocity field (dashed). (b) Steady state field configuration, with Rm —400.

D. Moss, Magnetic fields in stars

7

does not guarantee that diffusion remain unimportant everywhere. Further, although the magnetic energyis increased by this mechanism this is not a dynamo, but rather a ‘field amplifier’, as the total flux is constrained to be constant by the imposed boundary conditions. More realistic calculations which allow the fluid velocity to be self consistently determined are much harder to perform, and motions in nature may not be regular, but there is evidence that the field expulsion effect persists in modified form at least. Other consequences of finite conductivity include ohmic heating and the reconnection of field lines. The ohmic heating rate is j2/a- per unit volume. This mechanism is important in the study of solar flares and other coronal phenomena where the density is low and hi can be locally large, but appears to be relatively unimportant within stars. Dissipation and reconnection both occur where the field is strongly

curved, enabling loops of field to become disconnected and enhancing subsequent dissipation, as well as allowing changes in field topology see e.g. fig. 2.2. Under certain circumstances it is possible that ohmic —

decay can be offset by the regeneration of field by convection. It is by no means trivial to arrange for this to happen, even when a global estimate gives Rm ~ 1. This is the subject of dynamo theory (see sections

2.2 and 4.3). One further fundamental theoretical concept is that of hydromagnetic or Alfvén waves. The equation of motion together with the perfectly conducting MHD equation admit solutions corresponding to transverse waves which transmit disturbances along the magnetic field lines at a speed VA = (B2/4lrp)V2, where p is the fluid density. In the presence of finite conductivity the waves are dissipative. These waves are analogous to waves on a stretched string under tension B 2/4~. For example, if a mass of perfectly conducting fluid is in a state ofisorotation and the angular velocity of part ofit is suddenly changed, then the disturbance will be transmitted to the rest of the fluid connected by field lines to the perturbed region by Alfvén waves, in this example propagating with speed VA = B~/(4’rrp)~2. These will attempt to restore isorotation, subject to conservation of angular momentum. Alfvén waves can also transmit energy. The ratio VA/I vI is often an important parameter: from above (VA/u)

2

2

,

2

(B/8ir)I(~pu ),

i.e., the ratio of magnetic to kinetic energy densities. Crudely, when the ratio is large the field can be

(a)

(b)

(c)

(dl

Fig. 2.2. Schematic reconnection of field lines. The ‘neck’ in (a) reconnects leaving an isolatedclosed lop in (b). Two oppositely directed loops in (c) merge to give the configuration (d).

8

D. Moss, Magnetic fields in stars

expected to control the fluid motion, and when it is small the reverse will be true, as the fluid can outrun the Alfvén wave. This concept leads directly to the condition v~~ B~I(4irp) for the approximate uniform rotation of a radiative region (e.g. Mestel, 1961). 2.2. Outline of dynamo theory This addresses the question: under what circumstances can solutions of eq. (2.3) with non-zero resistivity be found such that the field B is maintained against decay? B can either be time independent (a steady dynamo), or time dependent (a periodic dynamo the periodicity need not be regular). The field of the earth is known to reverse at intervals of order i05 yr. Paleomagnetic evidence demonstrates that the earth’s surface field was of roughly the contemporary order of magnitude (—1 G) about 3 x i0~yr ago. With plausible parameters the decay time of the longest lived modes (cf. eq. (2.7)) is of order 1.5 x i04 yr. This is strong evidence for the existence of at least one cosmic dynamo in a gravitating, approximately spherical, body. The solar field is also plausibly a dynamo field (see section 4). In a solution to the full problem the velocity v appearing in eq. (2.3) should itself be a solution of the equations of fluid motion, including the Lorentz forcej X B. This is the hydromagnetic dynamo problem. It is non-linear, and only limited progress has been made towards finding relevant solutions of astrophysical interest. It is simpler initially, and more illuminating, to consider the linear kinematic dynamo problem in which the velocity field v is chosen fairly arbitrarily (but with regard to what seems plausible for the problem considered). In spite of the evidence for the existence of naturally occurring dynamos, for many years dynamo theory was beset with difficulties, even in the kinematic case. Cowling (1934) produced the first of the ‘antidynamo’ theorems. This established that a steady axisymmetric dynamo could not exist: consider such a configuration, writing the field B as the sum of poloidal and toroidal parts, B = B~+ BT. In an axisymmetric system B~lies in meridian planes and BT B~= 0. In general B~will have null lines (A in fig. 2.3) linking the axis of symmetry. In a steady state —

.

symmetry Fig. 2.3. Poloidal field lines

have a null line A (dashed). In an axisymmetric configuration the toroidal field LIT is perpendicular to B~.

D. Moss, Magnetic fields in stars

1

1/

9

vXB)\ ~ ).dr

v~XB.~.+ VTXBP ~dr =0,

using (2.1), and observing that X BT and VT X B~are both poloidal, whereas x B~= 0 on A. But fT ccV x B~ 0 on A, and so ~A j. dr cannot vanish, and the assumption of a steady state must fail. Cowling’s original proof applies to the case where fT ~ 0 on A, but if fT 0 there then B~ 0 more rapidlyand the theorem is still valid. Dynamo maintenance ofaxisymmetric fields can also be shown to be impossible under rather more general conditions; see Hide and Palmer (1982) for further details. —~

—~

The difficulties of axisymmetric dynamo maintenance can usefully be quantified thus. In an i/i for the poloidal field can be introduced, so that

axisymmetric configuration a ‘field function’ B~=V X where v =

(l/I/(r sin +

°)i~),

(r sin O)(liA, and i~is the unit azimuthal vector. The MHD equation (2.3) can be split

into poloidal and toroidal parts 9~/iI9t+V .VIp=?1D2[

~9]

,

(2.8)

~ ~..I+V.I ‘~T vl”B~vQ+ ~ D~B.,., (2.9) r sin 0 it L r sin 0 Pj r sin 0 2 = V2 1 1r2 sin2 0. The term B~VU in the toroidal equation (2.9) represents the generation of where D flux from poloidal by differential rotation, bufthere is no analogous term in eq. (2.8). Indeed if toroidal V. u = 0 it can be rigorously proved that i/i decays monotonically. ~‘

—

.

The implication is that a considerable degree of asymmetry is needed before a dynamo can operate. Producing a convincing mathematical model of such a process has proved to be quite difficult. A simple illustration can be given as follows (afterMoffatt, 1978). Consider an initial field B 0 in the x direction (fig. 2.4a) in a conducting fluid. Suppose now that there is a small scale fluid velocity consisting locally of a component parallel to z and a rotation about z. The u V X u greater and less than zero 2 a situations current j with respectively antiparallel/ parallel to B are illustrated in fig. 2.4b. After rotation by IT! 0 is produced. If motion of one helicity_dominates averaging over the small scale velocities gives a mean current j a-aB0, with a ~ 0 as u V x u ~ 0. Ohm’s law for the mean (~ large scale) quantities becomes -~

.

(2.10)

where v is the large scale (averaged) fluid velocity. Krause, Rãdler and Steinbeck and associates (see Roberts and Stix, 1971; also Moffatt, 1978; Krause and Rãdler, 1980; Zeldovich et al., 1983) formally established that turbulent motions with a suitable degree of anisotropy can maintain either steady or oscillating fields. Their mean field electrodynamics or

10

L

D. Moss, Magnetic fields in stars

(a)

B0

(C) Fig. 2.4. The initial field B,, (a) is distorted, (b) by a velocity u havingone component perpendicular to B0 and another corresponding to a rotation. In the left hand of (b) u is parallel to V x u andj~B) <0; on the right u is antiparallel to V x u andj 8) >0(c) A schematic illustration of the velocity U in an upwelling in a very compressible stratified rotating fluid. The initial motions are largely horizontal (single headed arrows), and coriolis effects cause a cyclonic rotation as shown by the double headed arrow.

‘turbulent dynamo’ approach has been heavily exploited in the last twenty years or so. In the simplest form of the theory it is necessary for small scale motions, u say, to have a degree of reflectional asymmetry (‘helicity’) such that (u ~Vx u~0, where the averaging can be either over length scales or time scales intermediate between those of the turbulence and the largest scale of the field. This is the so-called ‘a-effect’, produced by the cyclonic motion above. If v is the large scale velocity then the induction equation (2.3) after this averaging operation becomes ~-~=VX(vXB+aB)—VX(ijVXB),

(2.11)

the a-effect producing a novel source term, cf. eq. (2.10). Equation (2.11) is interpreted as applying to mean quantities. This a-effect has been demonstrated experimentally at small Rm (Steenbeck et al. 1967, translated in Roberts and Stix, 1971, p. 97). The formal steps leading to eq. (2.11) have been criticized on a number of grounds and in particular it is not clear that the usual derivations are strictly applicable to solar conditions. In particular, one of the conditions Rm 4 1, UrIL 4 1 is required to hold, where Rm is the eddy magnetic Reynolds number and U, rand L are typical eddy velocity, correlation time and length scale respectively. Under solar conditions Rm ~ 1, UrIL -~1. Arguments can be given that eq. (2.11) remains qualitatively valid see, e.g., Cowling (1981) and Priest (1982) for summaries. Equation (2.11) —

D. Moss, Magnetic fields in stars

11

remarkably robust in that a number of independent approaches to the dynamo problem result in essentially the same equation (see, e.g., Parker, 1979, Ch. 18 for a discussion), and it is generally (but not universally) believed that the parameterization outlined here captures at least some of the essential physics. The crucial ingredient of the theory is the cyclonic motions. These are naturally present in convection zones, where Coriolis effects can be expected to rotate rising and falling regions of the fluid. For example a rising motion in a rotating compressible medium will be of the form shown in fig. 2 .4c, in which case u ~VX a <0. With these assumptions, eqs. (2.8) and (2.9) can be rewritten as is

+ vp.V(,zzaBT+,lTD2[~O],

(2.12)

~ (2.13) where the diffusion coefficient is now written as ~T’ since it is usually interpreted as a turbulent diffusion coefficient. A naive estimate is ~T 0.11 all, where I is a length scale for the turbulence; usually ~. ~Now poloidal field can be generated from toroidal by the new term in eq. (2.12). If differential rotation dominates the poloidal to toroidal step then it is an ‘aw’ dynamo, if toroidal field is also generated -~

substantially by the a-effect then it is an ‘a2’ dynamo. A plethora of such kinematic models (i.e., with prescribed large scale velocity V, angular velocity 12 and coefficient a) have been computed. Typically a

=

a

0(r) cos 0 is believed to be the simplest realistic form. The crucial parameter governing the behaviour in aa dynamos in spherical geometry is the dynamo number

4, ND

=

(2.14)

a012 R

where 12’is a measure of Oh lOr, the angular velocity shear. In particular, with ND <0 it is possible to produce oscillatory models in spherical shells which reproduce the gross features of the solar butterfly diagram, see section 4. This reinforces the hope expressed above that the basic ideas may be correct,

although they can only approximately model reality. (For example, solar surface fields are extremely discontinuous in nature, whereas these dynamo models deal with smoothed out large scale fields). There are marked similarities between such an aw dynamo formulation when applied to the sun and models of the Parker—Babcock—Leighton type, as outlined in section 4.3.1. The crucial question is whether self-consistent velocity fields can be found as solutions of the full hydromagnetic equations which give rise to appropriate velocities u and v. A simple parameterization of the reaction of the field on the small scale turbulence is to reduce the magnitude of a when the field strength exceeds some initial value. Such a stratagem limits the field strength at some finite value. Large

scale meridional circulation (in a self-consistent model itself a solution of the Navier—Stokes equation) may also limit the growth of the field at a finite value. Such models have been discussed in great detail as possible explanations for solar and stellar fields, with some fairly encouraging results. It should be noted that in the sun there is some evidence for extreme inhomogeneity of magnetic fields, at the surface at least, which is perhaps a warning not to take exactly the predictions of a theory in which B, u (via a) and V are assumed to vary smoothly, over a global scale. Of course the only completely satisfactory approach would be to use the unmodified MHD equation (2.3), where the velocity v includes all scales of the

12

D. Moss, Magnetic fields in stars

motion, including the cyclonic convective motions which are averaged in the turbulent dynamo formulism to give the coefficient a. This is far beyond current computational capabilities, although a little progress in this direction is being made, see section 4.3.1.

3. The magnetic chemically peculiar stars 3.1. Introduction

This group of stars now encompasses a wider range of stars than the classical Ap stars, following the realization that the He weak and He strong stars also often possess strong fields. There is a variety of interesting phenomena, evidenced by an inhomogeneous body of observational data. Tantalizingly for the theorist, this seems to undergo subtle shifts from time to time, usually, it appears, just when theory and observation seem to be drawing together! Fundamental theoretical problems involve the origin, structure (internal and surface), stability, and distribution of strengths of the fields; the distribution of surface geometries; the relatively low angular velocities of the magnetic stars; evolution of the field with time; an explanation for the peculiar abundances observed; and the relationship between the magnetic field and stellar wind observed in some He strong and one He weak star. This section is organized as follows. In section 3.2 the observational material relating to field strengths and geometry is briefly outlined. Section 3.3 reviews theories of field origin, followed by discussion in sections 3.4 to 3.6 of the general properties and consequences of large scale fields of the type which might be plausible extrapolations below the stellar surface of the observed surface fields. Part of section 3.5 on braking is also relevant to the discussion of lower main sequence stars which are the subject of section 4. Section 3.7 describes calculations of field structure, in the light of both theoretical and observational constraints. Section 3.8 is devoted to the recently discovered rapid oscillations (period of order 10 mm) of some Ap stars, section 3.9 to evolutionary effects. Finally section 3.10 attempts to give an overview of these aptly named ‘peculiar’ stars. 3.2. Observations Fairly recent detailed studies of the observational evidence relating to the magnetic fields of the magnetic CP stars have been published by Hensberge et al. (1979), Borra and Landstreet (1980) and Didelon (1983, 1984). As this review is written from a theoretician’s viewpoint it is not intended to present an exhaustive treatment of the observational data. However a brief account of the main methods used to measure stellar magnetic fields will be given, followed by a summary of the salient observational data for the magnetic CP stars. An account of the observational data relating to magnetic white dwarfs will be found in the section (5.1) devoted to degenerate objects with magnetic fields. The basis of most stellar field measurements is the Zeeman effect. In its simplest form, the ‘normal linear’ Zeeman effect, an emission line in the presence of a ‘not too strong’ field is split into three components, one at the wavelength of the original line, and one on either side. If the field is parallelto the line of sight only the two displaced components are present, and they are circularly polarized in opposite senses. If the field is wholly perpendicular to the line of sight, then all three components are present, and the displaced components are linearly polarized with electric vector perpendicular to the field lines. The undisplaced component then has twice the intensity of the other two, and is linearly polarized with electric vector parallel to the magnetic field (e.g. Babcock, 1962). Roughly, the magnitude of the line

D. Moss, Magnetic fields in stars

13

splitting at wavelength A is ~A, where AA -~ 104BA2, A being measured in centimetres and B, the field strength, in gauss. The situation is rather more complicated, but similar in principle, for lines showing the anomalous Zeeman effect (Babcock, 1962; Borra, 1980). As the disks of stars other than the sun cannot be resolved using conventional methods, when stellar Zeeman patterns are recorded photographically then only the mean value of the field over the visible stellar surface (with suitable allowance for darkening effects), B~,can be determined. In practice B~can only be found in this manner for a very few favourable objects which are bright or rotate slowly, and have a strong field (Preston, 1971). Line broadening, caused by stellar rotation and other effects, soon gives sufficient line blending to make the method unusable. Nevertheless it may still be possible to find the position of the centroid of the broadened line as seen in the left- and right-hand circularly polarized light, and thus to estimate the mean longitudinal (‘effective’) field at the stellar surface, Be say (e.g. Landstreet, 1982). Note that this technique must be employed on lines of ‘metals’, that is elements other than hydrogen or helium. Effective fields have been determined photographically for a number of sharp lined Ap stars using this method (V sin i ~ 30 km s 1, V the equatorial velocity of rotation). Typically this implies a restriction to P > 5 d, depending on the inclination of the rotation axis to the line of sight. A more widely applicable technique is to measure photoelectrically the differences in polarization between the wings of a spectral line, this again yielding a measure of the effective field, Be. Now the procedure can be used on the Balmer lines of hydrogen (which are much too broad to be used by photographic techniques) as well as metallic lines, and it is also applicable to rapidly rotating stars. As hydrogen is expected to be fairly uniformly distributed over the stellar disk, whereas other elements mayform local abundance spots, the technique when applied to Balmer lines should give a better measure of the field over the whole surface. Values of B~of the order of or less than 100 G can be detected (see Borra, 1980). A comprehensive review is given by Landstreet (1980); see also Brown and Landstreet (1981) and Borra et al. (1984). Another approach to field determination has recently been developed by North (1980) and Cramer and Maeder (1980), who claim to find a correlation between the A5200 depression feature in the spectrum and effective field strength over a limited but useful range of effective temperatures and field strengths. This allows estimation of magnetic field strength from spectro-photometry, without any direct and time consuming Zeeman measurements, but the reliability and validity of the method has recently been questioned by Oetken (1985) and Thompson et al. (1985). Degenerate stars (for most purposes white dwarfs and their immediate progenitors) supply special problems, for example there may be no well determined lines in the spectra. The Balmer line technique may be applicable and for suitable objects fields of a few thousand gauss should be detectable. For fields greater than about 106 G linear Zeeman splitting is measurable from the altered line profiles, although by —‘2 x i07 G the quadratic Zeeman effect (~AxB2) must also be allowed for (Angel, 1977). At field strengths of 108 G or so the lines become almost unrecognizably smeared out (but note that Wunner et al. (1985) have identified some features); however the property of circular dichroism (differing opacities in left- and right-hand polarizations) may be used as an indicator of the presence of very strong fields, but it is difficult to use it to estimate field strengths accurately (see Angel, Borra and Landstreet, 1981). Ap stars comprise about 10% of the main sequence stars with comparable effective temperatures, and were first classified as peculiar because of their abnormal spectra, which indicate very strong enhancements of certain elements, such as silicon, some other metals and rare earths. Other elements may be deficient (e.g. Bonsack and Wolff, 1980). These stars can be classified by peculiarity type into three main classes, the HgMn, Si and SrEuCr stars. The HgMn stars do not appear to have observable magnetic fields, whereas many of the others do. The ‘superstar’ is HD215441 with B, 34000 0, which probably implies a polar field of 0.1 to 0.2 MG (Borra and Landstreet, 1978). The fields of these stars usually vary -—

14

D. Moss. Magnetic fields in stars

regularly with time (e.g. Borra and Landstreet, 1980), most being of kilogauss strength. Masses of the Ap stars are believed to be 2 — (3+)M>with radii 2 — (3+)R<, consistent with slight evolution away from the zero age main sequence. Ap stars are slow rotators compared with ‘normal’ A stars; fig. 3.1 is taken from Wolff (1975) and shows the distribution of periods among a sample of cooler (i.e. non—Si) Ap stars. This figure can be compared with a mean period of less than one day for normal, single, late A stars (e.g. Wolff, 1983). Some Ap stars have field variations with apparent periods of many years. The oblique rotator interpretation (see below) implies that these are the rotation periods (e.g. Wolff, 1975, 1983; see also Hensberge et al., 1984). The region of chemical peculiarity on and near the main sequence extends both to higher and lower effective temperatures and luminosities than those of the Ap stars, although the peculiarities are less dramatic. The Am stars are cooler and less massive than the Ap, and it seems that they do not have measurable fields (e.g. Landstreet, 1982). Recently kilogauss effective fields have been measured in both the helium-weak and helium-strong stars, which appear to be an extension of the Ap phenomenon to higher effective temperatures (maybe 22000 K) and masses (maybe 7—8 M®) the Bp stars. The rotational velocities of the latter are not as abnormally low as those of many of the Ap stars (Wolff and Wolff, 1976; Borra and Landstreet 1978, 1979; Borra, 1983). However the emission line B stars (Be), which do not have obvious abundance peculiarities, are not observably magnetic to a limiting Be 100 G, although significant toroidal fields could be undetected (Barker et al., 1985). Theoretical models (e.g. section 3.7) make predictions about the relation between surface and interior magnetic fluxes as a function of angular velocity, so any correlation between period and field strength is of considerable interest. Early impressions of a strong positive correlation have now vanished. Borra and Landstreet (1980) found marginal evidence to suggest that rapid rotators have weaker fields than slower, but without a strong correlation, whilst noting that there do exist rapid rotators with substantial fields (e.g. CU Vir, with period 0.5 d and a polar field —7000 G). Didelon (1983) is broadly in agreement with this conclusion. It may be that other parameters (e.g. age, angle between magnetic and rotational axes) are important these factors are discussed below. An interpretation of the very regular field variations is that they are due to an approximately axially symmetric surface field with axis inclined at an angle x to the rotational axis, the field configuration —

—

0.6

—

0.4

—

0.2

—

-

-~..

1.0

2.0

3.0

4.0 log P

Fig. 3.1. The distribution of periods for a sample of non-Si Ap stars, taken from Wolff (1975). f is the relative frequency distribution, P is in days.

D. Moss, Magnetic fields in stars

15

rotating with the star (the oblique rotator model (Stibbs, 1950)). A field of dipolar structure, perhaps with the nominal centre of the dipole displaced from the stellar centre (fig. 3.2) seems adequate to explain the observations of nearly all these stars. An alternative interpretation is as the sum of centred dipole and quadrupole components with the dipolar part dominant. There is just one object (HD 37776, Thompson and Landstreet, 1985) for which the oblique rotator model appears to demand a dominant quadrupolar component and so is incompatible with the displaced dipole field interpretation. Accepting the oblique rotator model for the moment (but see below), the angle x can be estimated from the Be variations, although in practice it is difficult to obtain reliable values. At first it seemed possible that the distribution was bimodal (Preston, 1970) with a primary peak near x = ir!2 and a secondary one nearx = 0, but later work (e.g. Hensberge et al., 1979; Borra and Landstreet, 1980) has been used to support the view that the distribution is more nearly random. More recently North (1985) has suggested that x is distributed randomly on the zero age main sequence, but that bimodality develops over the main sequence lifetime. An alternative interpretation to that of the oblique rotator model claims that the angle Xis always ir/2, but that the neglect of the effects of abundance inhomogeneities over the surface when reducing the spectra leads to spurious determinations of x < IT!2 (e.g. Oetken, 1977, 1979). The field is then interpreted as the sum of dipolar and quadrupolar components. (Note that this modelling is criticized by Landstreet (1978).) This has been called the perpendicular rotator model, and is closely associated with one version of dynamo theory (see sections 3.3 and 3.7). Michaud et al. (1981) and Piskunov and

Fig. 3.2. A schematic illustration of the geometry of a displaced dipole, rigid rotator model. P indicates the nearer magnetic pole.

16

D. Moss, Magnetic fields in stars

Khokhlova (1983) have demonstrated that element inhomogeneities can seriously affect the measured values of the effective field. Changes over a stellar lifetime of field strength (and other parameters) are potentially of great interest. There have been suggestions that the mean field strength declines over an evolutionary time scale (0(108) yr), both in the more massive Bp stars (Borra, 1981) and in the SrCrEu stars (North and Cramer, 1984). However Thompson et al. (1985) have reinvestigated the problem and claim there is no evidence for any significant change of field strength with age. They do suggest that there is a tendency for the hotter, more massive, CP stars to have significantly stronger fields, and that this effect can explain Borra’s observations. More work on this topic is needed to enable firm conclusions to be drawn. The situation regarding the evolution of angular velocity also has fluctuated considerably over the past ten years or so. Anomalous braking of the Ap stars compared with normal A stars clearly has occurred, but there is uncertainty as to when. Hartoog (1977) claimed there was no significant braking after arrival on the main sequence. Wolff (1981) suggested that the hotter and more massive Si stars undergo nearly all of their braking on the main sequence, whereas the SrCrEu group experience substantial pre-main sequence breaking also. Didelon (1984) says the data are consistent with some main sequence braking, whereas North (1984) claims that Si stars also lose most of their angular momentum before arrival on the main sequence. Borra et a). (1985) come to a similar conclusion from the study of stars of an assortment of peculiarity types in the upper Scorpius association. Evidence of a different nature is the presence of winds (perhaps modified by rotation via the magnetic field) in several He strong stars (Landstreet and Borra, 1978; Barker et al., preprint); and the claim that the He weak star HD21649 has a magnetically controlled wind (Brown et al., 1985). These winds could be expected to cause braking. A conclusive outcome to this debate would be very welcome to a theoretician! The most recently discovered phenomenom relating to magnetic Ap stars was announced by Kurtz (1982), who found rapid photometric oscillations (periods 6—14 mm) in six of these stars. Others have been found since (see Kurtz (1984) for a recent list). These objects are all fairly cool Ap stars, of similar position in the Hertzsprung—Russell diagram to the ~ Scuti variables, which however have periods an order of magnitude longer. In some of these Ap stars frequency triplets, separated by exactly the rotational frequency, are observed. Kurtz interpreted these observations in terms of an ‘oblique pulsator’ model, in which the star is constrained by the magnetic field to oscillate in a mode with a preferred axis along the magnetic (? displaced dipole) axis this is discussed further in section 3.8. Techniques similar to those outlined here have been used to attempt to detect fields in a variety of other non-degenerate stars. Landstreet (1982) surveyed a number of normal upper main sequence stars, and found no evidence of effective fields, down to typical upper limits of 50—100 G, although significantly smaller in some cases. Lower main sequence stars are discussed in section 4. With the possible exception of Cepheids (section 5.2) there now appears to be little evidence for substantial global fields at the surface of non-degenerate stars without significant surface convection zones, other than the chemically peculiar stars already discussed. —

3.3. Origin of the fields Four major theories have been advanced the hydromagnetic oscillator, battery, dynamo and fossil theories. For completeness each will be described in turn, although currently the major debate is between the protagonists of the latter two. —

D. Moss, Magnetic fields in stars

17

3.3.1. Magnetic oscillator theories Those do not directly deal with the origin of the magnetic field, but rather the source of the field variability. This is attributed to non-radial oscillations of the star in which, because of the large magnetic Reynolds number of the motion, the field is dragged by the stellar material causing the observed field to vary. Because the magnetic field is globally a weak perturbation to the zero-order structure, the periods of oscillation are determined essentially by the gravitational field, with small corrections due to magnetic forces (apart from some purely toroidal modes), and so, in order to obtain sufficiently long periods (of the order of days) in essentially main sequence stellar models, it is necessary to consider high-order gravity modes in which the motions are largely horizontal and are sharply restricted to the surface regions. The model then has difficulty in explaining why just one g mode, out of many with similar periods, should be excited in a particular star, and why modes with quite different periods should be excited in similar stars. Also it is difficult to explain the net polarity reversals observed during a cycle for some stars. Models based on the purely toroidal modes whose periods are totally determined by the magnetic field encounter difficulties in obtaining periods that are short enough, and a rather contrived field topology is necessary to explain field reversals (Cowling, 1965). These theories are now of historical interest only, and the displaced dipole! oblique rotator model seems a much more satisfactory explanation of the observed field variations. However Kurtz’s explanation of the rapidly oscillating Ap stars (section 3.8) could be considered as a form of oscillator theory, which slightly modifies the gross variations given by the oblique rotator model.

3.3.2. Battery models It now becomes necessary to pursue the argument of section 2.1 a little more closely. Under most of the astrophysical conditions that will concern us, matter is almost completely ionized, and a ‘two fluid’ model of the resulting plasma can be used (incomplete ionization may be important in studies of star formation, see section 5.4). It is possible to derive a non-relativistic analogue of Ohm’s law in the form a.

C

nee

(3.1)

Cnee

(j = —nev~is the current density, u~.T312theconductivity in a simple model, v the bulk velocity of the plasma, ~e’ 1~e,Ve the electron pressure, number density and drift velocity relative to the ions). Often it is adequate to approximate eq. (3.1) by E+”><~~=O,

(3.2)

which is the “MHD approximation”. This can be shown to imply the ‘flux freezing’ condition plasma cannot flow across magnetic field lines, and magnetic flux through a material circuit is conserved. Departures from flux freezing can, in the approximation leading to (3.1), arise from finite conductivity, the partial pressure gradient (cc VPe), or the Hall term. An energy equation can be derived from eq. (3.1) —

J.EZL_JPe+JXBV,

(3.3)

18

D. Moss, Magnetic fields in stars

giving the rate of working of the electromagnetic field. For net generation of magnetic energy in a region V we require J~,.j. E dV <0. The first term on the right-hand side of (3.3) represents the resistive dissipation of energy, the second can be either positive or negative, and the last is the “dynamo term” if v (j X B)Ic <0. The last two terms are possible sources of magnetic field. Broadly speaking, battery models are concerned with the first of these terms and dynamo models with the second. Note also that resistive diffusion is a crucial part of any dynamo theory which produces a net increase in flux through a material circuit. Let C be a closed circuit moving with the local electron velocity, v + v~,and let S be a surface spanning C. Then

~I B~ds=I dtis

(~—VX(v+v)XB)~dS e

Js\at

=

—Cf

V x (E

+

(v

+ Ve) X

/

B/c). dS

(using Faraday’s law) (2.1),

i\ =—p1/VP t—~——Ldr,

Jc

~

(3.4)

~

using (3.1) and Stokes’ theorem. The partial pressure gradient term can thus create flux if I Vp Jc

—p —~dr>0, and thus it is necessary that P~is not a function of ~e alone. If the star is spherically symmetric, the gravitational force density, which is experienced almost entirely by the ions, results in a slight radial separation of the ions and electrons, and the resulting electrostatic field balances the electron partial pressure gradient. Then VPe is a function of P1e~and the line integral in (3.4) vanishes. In a rotating star with angular velocity lithe equation of equilibrium is ~+V~_fl2th=0,

(3.5)

where w is the vector distance from the axis of rotation, and so IVP

P

—~

IVn. 1 2~dth, .drocP J p .drccPh J -~-

and the last integral is zero only if 12 = h(~). If, more generally, 12 = 12(th, z) then a ‘battery’ may operate and, in the absence of other effects, large toroidal fields can be shown to be built up over a time scale of i09 or iO’°yr (Biermann, 1950). However the effect is dependent on the existence of a non-conservative rotation field and, if even a weak poloidal field is present, this then makes the angular velocity almost constant along poloidal field lines (Mestel, 1961), which can be shown to effectively ‘short circuit’ the battery (Mestel and Roxburgh, 1962). It seems unlikely that poloidal fields will be completely absent, and so this version of the mechanism is thought to be generally unimportant. Dolginov (1977) has developed another version of the theory which directly uses chemical composition

D. Moss, Magnetic fields in stars

19

gradients in surface spots of anomalous helium abundance in the surfaces of Ap stars to generate the battery emf. Here, because the helium abundance and so the electron density have a horizontal variation, V x [VP~Itz~~] ~ 0, and thus a toroidal field can be generated. The poloidal field is supposed to arise from the interaction of the toroidal field with the meridional circulation. It is not clear whether this mechanism can operate for the complete range of rotational velocities of the Ap stars, and fully self-consistent models have not yet been published. Mestel and Moss (1983) constructed axisymmetric models of this type which satisfied the azimuthal component of the hydrostatic equation, (VxBT)xBp=0,

(3.6)

(i.e. the torque free condition). In contrast to Dolginov, they assumed a pre-existing poloidal field (e.g. a fossil, see section 3.3.4). They found that plausible composition gradients in the outermost stellar layers might give rise to toroidal fields, satisfying (3.6) and confined to the surface regions, of order a few hundred gauss. One motivation for this work was that surface toroidal fields confined to low temperature, low conductivity, regions will be subject to relatively rapid ohmic decay, whereas regions of horizontal surface field seem to be required by some diffusion theories for the anomalous surface abundances (e.g. Alecian and Vauclair, 1981). This work suggests a way that a helium abundance gradient might itself maintain a toroidal field against decay. Dolginov (1984) has extended the concept of battery fields being generated by chemical inhomogeneities to explain the fields of the earth, sun and neutron stars. 3.3.3. The core-dynamo theory As discussed in section 2.2 stellar dynamos appear to need a rotating convection zone in order to operate. According to current stellar structure theory the subsurface convection zone present in cooler main sequence stars has become a very thin superficial layer with little energy in its motions when the surface temperature is appropriate to all but the very coolest of the magnetic CP stars. It is implausible that these shallow residual convection zones can drive an effective dynamo. These stars do, however, possess convective cores, typically containing of order 10 % of the stellar mass. Dynamo theory for these stars proposes that their magnetic field is generated in these cores by some variant of the ‘aw’ mechanism. It follows immediately that such dynamos must be steady since otherwise the ‘skin effect’ of the deep overlying radiative envelope will prevent any field being observable at the surface (this is completely compatible with and indeed required by the oblique rotator model used to interpret the observations, see section 3.2). Problems which the theory must overcome including the following. Kilogauss effective fields are observed in some stars that are members of young groups, maybe no more than about 5 x 106 yr old. If it is assumed that the core dynamo turns on when nuclear reactions become an efficient source of energy and the central regions begin to convect, then the field has to diffuse from the core to the surface on a time scale of this order. Classical diffusion times through a radiative envelope are of order i09 yr. Detailed calculations by Schüfiler and Pàhler (1978) suggest the dynamo may ‘force’ diffusion at a somewhat faster rate, by continuously providing new field at the bottom of the envelope, but the process is still much too slow to explain the fields seen in these very young stars. An alternative mechanism might be the buoyant rise of flux tubes through a radiative envelope (see section 3.4.2), although even this might not be rapid enough. Presumably, if this is the explanation for the fields of the youngest (and most massive) magnetic stars, it must also operate in other stars. It is not apparent that the resulting surface field will possess a sufficiently regular large scale structure to be compatible with the observations. Many solar dynamo theorists now believe that, in order to avoid excessive and rapid flux loss, the seat of the solar dynamo is near the bottom of, or even in the ‘overshoot’ region just below, the convective envelope.

20

D. Moss. Magnetic fields in stars

As mass decreases along the main sequence, the convective envelope deepens. It has been proposed that when the whole star becomes convective (at mass —0.3M<), dynamo activity suddenly becomes much weaker (e.g. Spruit and Ritter, 1983), and there is some supporting observational evidence see section 4.2.2. Convective cores in CP stars also have no ~bottom’ perhaps dynamos in spheres are less efficient than extrapolation of results for spherical shells might suggest. Remember, of course that escape of flux from the core is in some way desirable see the preceding discussion and that the boundary conditions on the field at the core surface, which is overlain by a deep radiative envelope in which both buoyancy and diffusion may be important, are probably very different from those at the photosphere of lower main sequence stars. —

—

—

—

3.3.4. Fossil theory This same theory seems first to have been proposed by Cowling (1945), who observed that with a mean stellar temperature of order 106 K, eq. (2.7) gives the decay time of the longest lived (dipolar) mode of a large scale stellar field as of order iO’°yr. Thus it is possible for any field initially present when a star settles on the main sequence to survive throughout the main sequence lifetime. In the simplest form of the theory the initial field is a relic of the interstellar field which permeated the pre-stellar material. Crude estimates show that there is no difficulty in broad principle in providing an adequate amount of magnetic flux for a CP star in this manner. Flux freezing is expected to be a good approximation through a significant part of the pre-main sequence contraction. At one extreme, suppose that a region of interstellar gas with a field of 3 x 10_6 G and density 10~24g cm3 were to contract isotropically to form an Ap star of mass —5 x i0~g, radius —10” cm. With conservation of flux through the equator BR2 will remain constant, where B and R are the current field strength and cloud radius respectively. Conservation of mass implies B cc p213. On the main sequence this would lead to a mean field strength of order lOb G. The magnetic energy of such a configuration would be much greater than the gravitational, and such a contraction could not occur. Anisotropic collapse is one way that a star of normal mass can condense from such initial conditions (e.g. Mestel, 1965), but clearly any star that does form could, a priori, possess a substantial field. A more plausible picture, for the younger (population I) stars at least, is that there is a ‘molecular cloud’ phase of star formation, during which the ionization is low and flux leakage can readily occur (although the previous picture may still be valid for the stars of population II). If effective flux freezing is not reestablished until a density of order ~ g cm3, subsequent contraction then gives more modest, but still substantial, mean fields of order i~~—iü~ 0. In either case it can naively be expected that significant fields will be present in forming stars. Difficulties do arise however upon closer examination of the history of stars. There are still uncertainties in the theory of early stellar evolution but it is likely that stars of two or three solar masses undergo a largely convective phase of evolution (the ‘Hayashi’ phase) during their contraction to the main sequence (the upper mass limit for large scale convection to occur might be lower, Larson (1972)). The effect of chaotic small scale turbulence on the field is often crudely parameterized by a large value of the turbulent resistivity flT’ of order 0.1V~l,where V~and 1 are typical velocity and length scales for the turbulence. This suggests that ohmic decay might be greatly increased, or that the field might be readily expelled from the star. More sophisticated calculations suggest the possibility that the field largely will be expelled from the centre of vigorously convecting regions, but that some field may survive in the form of flux ropes, where the field is locally strong enough to resist further distortion by the turbulence (see, e.g. Weiss, 1966; Galloway et al., 1978; Galloway and Weiss, 1981). The field structure then becomes very intermittent with convection proceeding freely between the high field regions. When the turbulence eventually decays as the star approaches the main sequence the

D. Moss, Magnetic fields in stars

21

residual field could then diffuse back into a more uniform configuration pervading the bulk of the star. If the large scale field is completely destroyed/expelled, the star can be said to have undergone a magnetic ‘brainwash’ (Mestel, 1975), and the simple form of the fossil theory fails. Levine (1974) suggested that the T Tauri phenomenon (violent activity in the outer regions of stars contracting to the main sequence) might be powered from this reservoir of magnetic energy. A variant on the fossil theory proposes that a turbulent dynamo (see section 2.2) operates during the Hayashi phase, generating a large scale field. According to this ‘hybrid’ theory it is this dynamo built field which is left behind when the convection dies away, and it is the relic of this field which is now visible. The original form of the fossil theory has no trouble in explaining a random initial distribution of the angle x between rotation and magnetic axes. The initial magnetic flux is also a free parameter. Both the ‘full’ and the ‘hybrid’ fossil theories are still subject to the criticism that the fields may be subject to a variety of instabilities which may result in significant ohmic decay and flux loss in a small fraction of a main sequence lifetime. Instabilities are discussed in some detail in section 3.4 it is the kink mode dynamical instability and the slower buoyancy effects in a radiative envelope which are probably the greatest causes of concern. It is worth commenting that the toroidal field component that is necessary for stability can be generated from a poloidal field by a modest amount of differential rotation together with some reconnection of field lines. The hybrid theory automatically provides a mixed poloidal and toroidal field via the dynamo mechanism. Note also that some flux loss is by no means undesirable most near main sequence stars of several masses are not observably magnetic. This may be because their flux is concentrated in the interior (see section 3.7) and so not directly observable, but it is also consistent with the idea that substantial flux loss does occur sometimes at least. Again, there is no evidence from observations or theoretical models that the parameter A~ — Emag! Egrav B 2R41 (4 iT GM2) measuring the ratio of magnetic to gravitational energy is ever anything but a very small quantity. Even HD215441, with 3 x i04 G and a hypothetical B 106 G, has A~, i0~ and estimates are smaller still for other stars. It may be that configurations with larger values of A~,are inevitably subject to instabilities, which persist until A 1~,becomes small. —

—

—

—

-—

‘-—

3.4. Stability of stellar magnetic fields Experimental and theoretical studies of plasma confinement, especially in connection with controlled fusion experiments, have revealed the multiple instabilities to which a mixture of plasma and field is subject. Although stellar conditions are rather different, a number of instabilities have been studied. Here only a few of the seemingly more important ones are discussed. Most results are for the linear regime only, and extrapolation to the non-linear regime should always be regarded with caution. 3.4.1. Dynamical instabilities In section 3.7 the problem of finding magnetic field configurations in hydrostatic equilibrium is considered at length. Whatever the origin of the stellar field, a realistic configuration must be stable against any instabilities which have a growth rate less than, or of order of, the fields’ age on the fossil theory fields must be stable for an appreciable fraction of stellar lifetime, say 107_108 yr. The fastest instabilities are dynamical. Results for plasma confinement theory are suggestive in the stellar context. The ‘z-pinch’ is known to be unstable in two simple modes, leading to the ‘sausage’ and ‘kink’ instabilities see fig. 3.3. Consider a star with a field of simple topology for example of dipolar or —

—

—

quadrupolar angular structure. Such fields have neutral lines which are circles enclosing the magnetic axis. The local topology of the neighbouring field lines can be seen to be similar to that of the classic

22

D. Moss, Magnetic fields in stars

(a)

(b)

Fig. 3.3. (a) A simple z-pinch. Arrows indicate the magnetic field. (h) Sausage mode instability. (c) Kink mode instability. (d) z-pinch with linking longitudinal field.

z-pinch. This suggests that analogous instabilities to the sausage and kink modes might exist for these simple stellar fields. The major differences between stellar and laboratory conditions are the absence of a sharp boundary and the presence of a strong gravitational field perpendicular to the pinch in a star. If the neutral line occurs in a radiative zone in the star (which it almost certainly does in a main sequence CP star), vertical displacements are strongly stabilized. Detailed calculations confirm that displacements are indeed stabilized unless ~ g —0, where ~ is the displacement vector. This rules out the modified sausage mode, but the analogue of the kink mode is found to be unstable (Wright, 1973; Markey and Tayler, 1973, 1974; van Asche et al., 1982). These results are independent of the strength of the field and depend only on its geometry. Dynamical instabilities such as these do not in themselves destroy magnetic flux (indeed dissipative mechanisms are omitted from the analysis). The growth times are small, and growth into the non-linear regime will cause a large decrease in the length scale of the field. The perfect conductivity approximation will cease to be valid, and ohmic dissipation eventually will destroy magnetic flux and energy. Similar instabilities occur near the neutral lines of purely toroidal fields (Tayler, 1973), and appear to render untenable the suggestion of Dicke (1979) that very strong toroidal fields may exist in the central part of the sun (see also Tayler, 1980). The laboratory z-pinch can be stabilized against sausage and kink mode instabilities by the presence of a component of field parallel to the kink axis (fig. 3.3). The stellar analogue is a toroidal component of field linking the poloidal (or vice versa). The work of Wright (1973) and Tayler (1980) suggests that a toroidal field with strength of order that of the poloidal field can probably help to stabilize the configuration, and so a necessary condition for stellar fields to be stable seems to be that the fields be of linked poloidal—toroidal topology, cf. fig. 2.3. Nevertheless, for fields confined completely within a radiative core, it appears that initially stable linked poloidal—toroidal fields will diffuse into an unstable configuration in a time very much less than the main sequence lifetime of a star of a solar mass or so

D. Moss, Magnetic fields in stars

23

(Tayler, 1982; Moss, 1984a). These results strengthen the earlier prediction that strong fields are unlikely to survive in the solar interior. Uniform rotation mayalso exert a stabilizing influence, but seems unlikely to modify the general conclusions (Pitts and Tayler, 1985). These authors also suggest that instabilities may be reduced if the magnetic and rotational axes are perpendicular, rather than parallel. The general arguments which establish the existence of these instabilities are topological in nature, although the detailed calculations are for simple cases with tractable geometries. Thus the general results can be expected to hold for more realistic fields with less symmetry. 3.4.2. Buoyancy instabilities These instabilities are most easily pictured when the field is considered as composed of individual flux tubes. Consider an isolated tube with field of magnitude B, in equilibrium with its surroundings. Let the pressure, temperature and density inside and outside the tube be p1, T1, p1 and p~,Te and Pc’ respectively. The pressure equilibrium condition is 2!8ir=p~. pj+B If Te = T. then ~ < ~‘ the tube is buoyant, and starts to rise. Subsequent events depend on whether the tube is in a convectively stable region with a subadiabatic temperature gradient, or an unstable region with a superadiabatic gradient. In an unstable region, and if the rise is sufficiently rapid for changes inside the tube to be adiabatic, then subsequently T~>Te, p 1 remains less than Pc’ and the rise accelerates. Eventually the rise will be checked by a combination of aerodynamic or viscous drag (or perhaps other effects), but estimates suggest very rapid rise times as short as of the order of years through the entire solar convection zone. This instability provides a problem for simple dynamo theories of the solar cycle and a variety of stratagems have been invoked to keep fields buried for times of order the 22 (11) year solar cycle see Schüfiler (1983) and section 4.3.1. A detailed analysis by Acheson (1979a, b) suggests that rotation may exert a strongly stabilizing influence, markedly slowing the rise of flux tubes in a convection zone, even in a comparatively slowly rotating star such as the sun. If the region is stable against convection, then as the tube rises T1 < Te and p~> P~.The buoyancy decreases and the rise ceases. Any subsequent rise is limited by the rate of heat transfer into the tube. Rise times are thus long, except for very strong fields: for example ~ G or so could well remain in a radiative interior for times of order of a main sequence lifetime. See, for example, Parker (1979) for a fuller discussion. The position is not altogether clear cut, much of the theory consisting of order of magnitude estimates. It is not inconceivable that such effects could be an embarrassment to the fossil theory, allowing flux to float out of a star within a main sequence lifetime. In this case, such motions conceivably could mix the interior of the star to some extent (e.g. Hubbard and Dearborn, 1980). This could be significant in the context of the unusual isotrope ratios seen in some red giants, which seem to be characteristic of material which has been processed by nuclear reactions deep in the star. Mixing caused by a buoyancy instability could dredge up some of this material through the radiative zone into the outer convective envelope, where it would be convectively mixed to the surface. Nevertheless a general caveat should be voiced. The results outlined above are based on consideration of the behaviour of individual flux tubes in the absence of centrifugal effects. It may be inappropriate to apply these results to global fields of complex poloidal—toroidal structure. Further, molecular weight gradients inevitably will be present as the star evolves, and these seem likely to be a stabilizing influence (see, e.g. Mestel, 1975), although this has not been fully quantified. —

—

24

D. Moss. Magnetic fields in stars

3.5. Braking mechanisms: magnetic winds and accretion The topic of rotational deceleration controlled by the magnetic field is of interest both for the magnetic CP stars and for the lower main sequence stars. The general theory will be outlined here, and a discussion given of its relevance to the magnetic CP stars. The general results will be used again in section 4, applied to lower main sequence stars. If a star is surrounded by a hot corona (temperature ~1O6K), presumably somehow powered by a subsurface convection zone, then the corona cannot be confined by the star’s gravitational field, but will expand to infinity, carrying away mass and angular momentum (Parker, 1963). This flow is known as a stellar wind. If there is a large scale magnetic field present the magnetic Reynolds number of the wind will be large, and so the stream and field lines will be approximately parallel. Simple physical considerations suggest that when the energy density of the wind is much less than that of the field (v2 ~ v~j,then the field is little perturbed by the wind. Only small deviations from, for example, a force free configuration will be necessary, these exerting a torque capable of constraining the wind to flow along the almost unperturbed field lines. In the absence of the field the wind would conserve its angular momentum in an axisymmetric gravi-centrifugal potential. Thus angular momentum is transferred from the star to the wind via the magnetic field in this region. This is the ‘corotation zone’ where the wind has approximately the same angular velocity as the star. Conversely, if v2 v~,the wind will flow almost unimpeded by the field, and the field lines will be forced to be parallel to the streamlines appropriate to a flow with no field. In this region the wind will flow to infinity, approximately conserving its angular momentum, and dragging the field with it. For a large enough surface field solutions have the property that the flow is subsonic, v < a, close to the stellar surface, and v2 ~ v~in a region beyond the sonic point. However v2 = at sufficient distance. v2 = v~defines the Alfvénic surface, 5A say. A generalized Bernoulli equation can be written down for the flow along each field/streamline. Consider an axisymmetric, isothermal system. Then —

~‘

(3.7) where ~ is axial distance, ~w is the azimuthal component of wind velocity, [2is the stellar angular velocity and 8 is a constant along each streamline. The wind must flow smoothly through the sonic point (strictly the slow magnetosonic point), which determines 8. If the centrifugal terms are unimportant then a thermal wind flows. However in a sufficiently rapid rotator even a cool corona can drive a thermocentrifugal wind. In this case v = O(ulai) in the corotation zone well beyond the sonic point (Mestel, 1968). A general result of simple wind theory is that the angular momentum loss rate is the same as if the wind strictly corotated with the field out to SA and streamed freely thereafter. Thus the angular momentum loss per unit mass is enhanced by the ‘stiffening’ of the field, to be the same as if mass loss occurred from the surface SA rather than the stellar surface (Mestel, 1967; Weber and Davis, 1967; Mestel, 1968). Then -

~J~=-

~(k2R2MQ)=_

~

where H is the stellar angular momentum, kR is the radius of gyration and RA is suitably defined mean radius of the Alfvénic surface. When RA ~‘R, the stellar radius, the specific angular momentum loss can be much enhanced by the magnetic field. RA depends on field strength, but even in simple axisymmetric

D. Moss, Magnetic fields in stars

25

models this does not necessarily imply a monotonic increase of angular momentum flux with field strength for given angular velocity, and a rather more careful treatment of field topology is necessary. Conceptually the simplest models are those of Weber and Davis (1967); (see also Belcher and MacGregor, 1976). Here the basic field near the stellar surface is taken to be quasi-radial, so that mass and angular momentum loss occurs along every field line, although their analysis was only performed for equatorial stream lines. Sakurai (1985) has generalized this work. At the opposite extreme is the model of Mestel (1968) who chose a field of strictly dipolar structure near the stellar surface. In this model, those dipolar field lines on which application of Bernoulli’s equation (3.7) predicts v2 < v to their maximum distance from the star remain closed. The conclusion is that these lines must define a ‘dead zone’ in which no wind flows see fig. 3.4; the wind only flows along field lines which pass through the Alfvénic surface. Outside of SA the wind dominates and the field is severely distorted this is the wind zone. As the field increases the Alfvénic surface moves further from the stellar surface, the dead zone increases in size, and the proportion of the surface from which mass loss occurs decreases. Mestel made the simplifying assumption that the poloidal field was strictly curl free (dipolar), and corotated with the star all the way to the Alfvénic surface, and that the field followed the free streaming of the wind thereafter. Detailed numerical results show that eventually the angular momentum loss only increases very slowly, if at all, with increasing field strength i.e. the mechanism saturates, the increasing ‘lever arm’ of the receding Alfvénic surface being compensated by the decreasing number of field lines in the wind zone and the decreasing fraction of the surface from which mass loss occurs. This model almost certainly overemphasizes the importance of dead zones and an approximate intermediate model has been developed by Mestel ,~

—

—

—

I 1/

sor\\~ 1\ I

I 7\X II’J\

\_—~

\./

so

/

l~’~’

/

~‘

I

I

~ ‘—..

N

Fig. 3.4. Schematicfield lines for the wind model of Mestel (1968). S, is the reference surface in the corona, 5A is the Alfvén surface. The hatched region is the ‘dead zone’, where no wind flows,

SA

Fig. 3.5. Schematic field lines for the modified wind model of Mestel and Spruit (1986). S 0 and SA are as in fig. 3.4, but between S~and SA the field is distorted from a curl-free configuration but still corotates with the star. The reduced ‘dead zone’ is again indicated by hatching.

26

D. Moss, Magnetic fields in stars

and Spruit (1986), who limit the extent of the dead zone by requiring that at the extreme point of the limiting field line there should be balance between the magnetic and gas pressures, the gas pressure in the dead zone being enhanced by the centrifugal effects. Between this reduced dead zone and SA the field still enforces approximate corotation on the wind, but the field is distorted from its force free (dipolar) configuration see fig. 3.5. These models remove the saturation effect as the field strength increases, but still show reduced angular momentum loss rates compared with the Weber and Davis type models. The results of all these computations can be parameterized by —

dH/dt = KpOVOulR4[RAIRJm,

—

(3.8)

where K is a geometrical factor and 0 m ~ 2. Subscript zero refers to a reference level, conventionally the coronal base. RA increases with field strength. For the Weber and Davis model m = 2, and for the original Mestel (1968) model m 0. More realistic models probably have 0< m ~ 1 (e.g. Mestel, 1984b). Combine this result with the steady field freezing and mass conservation condition (2.5) ~

pVIB = PA VA! BA the definition of SA, 2

12

BAI8rrpA = ~VA, and the proportionality BA

-~

BQ[rO/RA}~

where n = 2 for a radial field and n

=

3 for a dipolar field. r

0 R; m and n are, of course, interdependent. Then for a thermal wind (i.e., one such that the centrifugal forces at approximate corotation are small), on appeal to numerical solutions which demonstrate that the wind only accelerates slowly after passing through the sonic point so that VA = 0(a), eq. (3.8) gives -—

cc

If the relation between field strength and angular velocity given, for example, by a dynamo theory, is parameterized by B0 cc u2’~then dt Similarly, for a thermo-centrifugal wind VA~RAQ, so that 2m~ )~

dt

m/(n+1)

—

cc _B

12[mI(n+1)(2p—t)+11

0

—

D. Moss, Magnetic fields in stars

27

Plausible values for p for a dynamo are p = 1 or 2. The Skumanich (1972) law, 12 cc ~~112 for main sequence stars, can be reproduced with p = 1, n = m = 2, thermal wind, or with p = 2, n = m = 2, thermo-centrifugal wind; but a wide range of other laws can also plausibly be obtained. Neither theory nor observations are yet adequate to draw definite conclusions. The theory as outlined applies to cool stars with effective subsurface convection zones, to middle main sequence (e.g. CP) stars in a convective pre-main sequence phase provided they possess a significant surface field during this period, and possibly to rapidly rotating stars with cool coronas. Recently angular momentum loss by a magnetically controlled stellar wind has been appealed to as a way of controlling the evolution of some late type contact binary systems of the WUMa type (see e.g. references in Smith, 1984). The geometry is inevitably much more complicated in such systems and little detailed work has been done — see Moss (1986) for a preliminary attempt at the problem. The application of the theory to the single lower main sequence stars is discussed further in section 4. The remainder of this section will be devoted to a discussion of the braking of early type main sequence CP stars. How relevant magnetically controlled magnetic winds are to the slow rotation of the magnetic CP stars is not clear. The absence of important sub-surface convection zones make it seem unlikely that A and B stars will have hot coronas (but note that Cash and Snow (1982) found evidence for X-ray emission, commonly thought to be associated with hot coronas, from two apparently single Ap stars). If the surface field, and so RA, is supposed to be almost independent of 12, eq. (3.8) predicts an exponential decline of angular velocity for slower rotators (thermal winds). This is rather uncomfortable, as it would be necessary to appeal to exceptionally fine tuning in order to avoid all magnetic CP stars having either their original, or essentially zero, angular velocities. This leads to the conclusion that either the excess braking of Ap stars occurs by a stellar wind in the pre-main sequence phase, or that some other mechanism is responsible. Nevertheless stellar winds and magnetic fields are observed in some He weak and He strong stars; they would appear to provide a promising area of research.

Another possibility is accretion (Havnes and Conti, 1971; Mestel, 1975; Arons and Leas, 1976a, b; Wolff, 1981). With a cool corona and so no wind, inflowing gas compresses the external magnetic field until B 2~8 IT ~ 2 Thus the flow slows at the Alfvénic surface, denser gas is supported above less dense, the configuration becomes Rayleigh—Taylor unstable, and the accreted material can enter the magnetosphere where it is spun up into approximate corotation. If [I2RA> GMIR~,the accreted material is returned to the interstellar medium by a ‘slingshot’ mechanism, braking the star. Braking continues until GM; thereafter accretion continues but without further significant braking. This mechanism has the attraction that it cuts itself off at periods of a few days, consistent with the periods of most Ap stars. Crude estimates of timescales give 107_108 yr, which is of a reasonable order of magnitude. A more detailed theoretical exploration of the problem would be welcome. Overall there is a consensus that the relatively slow rotation of the typical magnetic CP star is caused by the presence of strong surface magnetic fields, and two possible mechanisms have been identified above. If the recent observational papers, which claim that there is no significant main sequence braking, are correct then the pre-main sequence mechanism seems adequate. As comparable single non-magnetic stars are not anomalously braked, the field involved in the braking then presumably is a fossil field which must be essentially the contemporarily observed field (since there is no obvious reason why only pre-CP stars should have an envelope dynamo field and pre-normal main sequence A stars should not). On the other hand, even without magnetically controlled main sequence braking, conservation of angular —

momentum together with solid body rotation would imply measurable deceleration as the star evolves across the main sequence band — this, at least, should be observed. As noted in section 3.2, there are a few very long period magnetic stars, and these are considered by

28

D. Moss, Magnetic fields in stars

some authors to provide difficulties for the oblique rotator theory. However there is a continuous unimodal distribution of periods, and convincing evidence for the oblique rotator interpretation for periods less than about 30 d. This seems strong evidence for a universal oblique rotator model (Wolff, 1975). Have these very slowly rotating stars perhaps undergone more efficient braking than other magnetic CP stars? How do the HgMn stars, also slow rotators but not all in close binaries, fIt into the picture? A complete theory to explain all these objects still seems quite distant. 3.6. Distribution of obliquities The distribution of the angles x between the magnetic and rotation axes in the oblique rotator representation of the surface field variations is an important datum which potentially may give significant insight into or constraints upon theoretical models for the structure and evolution of the interior fields of the CP stars. The observational evidence is that x is randomly distributed, or at least the distribution is not strongly bimodal. North (1985) suggests that bimodality may develop with age. The simplest version of the fossil field theory predicts a random distribution of Xo’ the value of x with which a star is formed. A number of mechanisms have been suggested which could significantly alter x during the main sequence lifetime of a typical magnetic CP star. Consider a star in which the rotational perturbation is dominant nearly everywhere. The meridional circulation will approximate the Eddington—Sweet circulation except near the surface (and near the convective core). To some extent this circulation can be regarded as inexorable in that through most of the star it will not be much affected by ‘feedback’ effects as the magnetic field is distorted. For rapid enough rotation the magnetic Reynolds number Rm ~ 1, except near the surface where the conductivity becomes small and the velocity is finite. If xl ~ 0 then the circulation will distort the surface field distribution from its initial configuration. Idealized computations in two-dimensional geometry by Moss (1977b) suggests that if x0 is not small significant changes in the effective value of x as seen by an observer may just be able to occur within the main sequence lifetime of a typical Ap star; see fig. 3.6. (The field in the interior will take longer to adjust.) A more realistic treatment in a spherical geometry and including the reaction of the field on the motions would need, for example, a considerable

Fig. 3.6. A dipolar field, initially symmetric about an axis inclined at angle 5 i,/4 to the rotation axis, is distorted by an inexorable meridional circulation. Full curves indicate the velocity field, dashed curves the magnetic field when an approximately steady state has been reached. The surface field is then approximately symmetrical about an axis with x irI2.

D. Moss, Magnetic fields in stars

29

extension of the calculations of Moss (1984b) — see the discussion in the section 3.7.3. The simple kinematic computations discussed above do not reveal a strong dependence on the degree of concentration of the field to the interior; but it can be expected that the effect will be less marked as field strength increases. Braking by wind and accretion mechanisms was outlined in section 3.5 for the simplest case, x = 0. When x ~ 0 there will be a torque perpendicular to the rotation axis, as well as the component parallel to Ill which will continue to brake the star. The component of torque perpendicular to 12 will cause the instantaneous axis of rotation to precess through the star, and the magnetic axis to rotate in space. The analysis is complicated, but in the only (rather artificial) case worked out in detail the rotation axis sought the region of the surface with the strongest magnetic field (Mestel and Selley, 1970). Arguments were given to suggest that this maybe a general result. For a dipolar field this would imply x —~ 0. The timescale is uncertain but conceivably short enough to be of interest. However, if braking by a wind on the main sequence is unimportant (section 3.5) then this mechanism will not be effective. There is another dynamical mechanism which may be more important than the effect of wind torques. An oblique rotator can be considered as having first-order distortions symmetrical about each of the magnetic and the rotation axes, and so in general will possess three unequal moments of inertia. Ignoring the rotational distortion, this is just the classical problem of a top precessing under no external forces. In the frame rotating with the star the motion consists of a slow spin w about the magnetic axis, where w A~12.In a first-order perturbation theory the effects of rotation and the magnetic field on the stellar structure can be superimposed. If the rotational distortion is now included, the spin about the magnetic axis alters the pressure-density distribution. Dynamically driven motions, also of frequency w, are induced in order to restore hydrostatic equilibrium. The consequent distortions of the internal magnetic field give rise to a dissipation of energy, which comes ultimately from the rotation field. Angular momentum is conserved, so that the star will tend to a minimum energystate for that angular momentum. Thus x —* i~!2if the configuration is dynamically prolate about its magnetic axis and x —*0 if it is dynamically oblate (Spitzer, 1958; Mestel and Takhar, 1972). Note that it is the dynamical, not surface, prolateness or oblateness that is relevant, and that toroidal fields tend to give prolate configurations and poloidal fields oblate. Realistic fields are likely to contain comparable poloidal and toroidal components (see section 3.4). Mestel et al. (1981) investigated a model with a purely poloidal field, assumed to be excluded from the convective core, and found that the timescale for x to change significantly decreased rapidly with increasing ratio of internal to surface field. If the maximum ratio of internal to surface radial field is about 13 then x changes on a timescale of 108 to 1012 yr for rotation periods between 1 and 100 d, the timescale decreasing with decreasing period. For field concentrations only about 3 times as large the corresponding timescales are i05 to i09 yr, and field ratios of around 350 give i04 to 108 yr. There are, of course, still considerable uncertainties associated with the theory as outlined above. Preliminary work on purely toroidal fields by Nittman and Wood (1981) suggests that the timescales in this case may be considerably longer. No calculations are available for the more realistic case of linked comparable poloidal and toroidal field components, nor for decentred dipole configurations. Molecular weight gradients in the interior also could conceivably alter the quoted figures. Theoretical ideas about the internal structure of magnetic fields are discussed in detail in section 3.7. Clearly if the above estimates are valid to order of magnitude, and a wide range of x values really are observed in stars with age of order 108 yr or so, then this potentially is a powerful constraint on allowable field configurations. Finally, these dynamically driven motions could cause a limited degree of mixing of the stellar interior, enabling extra nuclear fuel to be burnt, and thus somewhat extending main sequence lifetimes (Nittman and Wood, 1981).

3t)

Lb. Moss. Magnetic fields in stars

3. 7. Theoretical models of early type magnetic stars A considerable effort has been exerted in the last twenty or so years in solving the equations of stellar structure incorporating magnetic fields and the associated Lorentz force. One of the basic difficulties is that the Lorentz force is in general non-conservative, and has a non-radial component, so the problem immediately becomes two-dimensional. Stellar rotation by itself presents similar, although perhaps slightly less severe, problems and as most magnetic stars have a significant angular velocity, rotational effects also have to be included. Notwithstanding such difficulties a considerable amount of progress on this problem has been made recently, and a plausible if not universally agreed picture of the magnetic CP star phenomenon appears to be within sight. Firstly some general ideas and results applicable to radiative envelopes with a large scale magnetic field are discussed. These are relevant whether the field is of fossil or dynamo origin, provided that the dynamo is not located in the region concerned. A discussion of some detailed models, both fossil and dynamo, follows. 3.7.1. General results on the structure of rotating radiative envelopes with a magnetic field This section will be concerned entirely with configurations that are in hydrostatic equilibrium, satisfying _~_V~+X~ p

+Q2~=O,

(3.9)

4irp

(i.e., neglecting inertial and Coriolis terms). Stars must also satisfy Poisson’s equation =

47rGp,

(3.1(J)

the equation of radiative transfer in radiative regions. F=—KV(aT4),

(3.11)

where K = c/3Kp, and the energy equation pe —V.F = C~pT

(In T/p~’)= C~pT(~+ v .V)(ln T!p~~),

(3.12)

where v is any laminar velocity field present, K is the opacity, e the energy generation rate and d/dt is the Lagrangian derivative. The other symbols have their conventional meanings. The equation of state is usually adequately approximated by the perfect gas law, Pz~RpT//2,

(3.13)

where ~ is the molecular weight. Von Zeipel (1924) first pointed out that in a uniformly rotating star in which the energy generation rate e is a prescribed function of pressure and density, strict radiative equilibrium (i.e., with v = 0 in eq. (3.12)) is impossible. The necessary circulation in meridional planes was first accurately calculated by Sweet (1950), who showed that to order of magnitude throughout the star the radial component of the Eddington—Sweet circulation is

D. Moss, Magnetic fields in stars

31

LR2

v~I ~

A

0, Using representative figures of L = 40L 2R3IGM. where A0 = I2 2.5M®,andR =descends 2R®, Pat = 3the d, 1. The circulation rises 0,at M the= poles gives A0 —-5 X i0~ and v0~ i0’~cms equator, as defined by the rotation axis. A similar meridional circulation, VH, is in general driven by large scale magnetic fields in a stellar radiative zone (although special field configurations can be found which have no associated circulation, see section 3.7.3). (The discussion of this section is restricted to fields of global scale. In contrast, Parker (1984c, d) considers this type of circulation driven by local inhomogeneities in the field within a radiative region and discusses the effectiveness of such a circulation in mixing stellar interiors. Applied to the sun in conjunction with the observed 7Li abundances, his analysis yields upper limits to any field in the outer part of the radiative interior in the range 105_106 G). The Eddington—Sweet circulation in an initially uniformly rotating nonmagnetic star will transport angular momentum and so cause departures from uniform rotation. The subsequent history of such configurations is still disputed, but it seems likely that there are no realistic steady solutions. Happily if there is even a weak large scale magnetic field present this problem does not arise — see below. The presence, in the general case, of a large scale circulation together with a magnetic field means that a form of the MHD equation (2.3) must also be satisfied, where the circulation velocity v is itself determined by the magnetic field and stellar rotation as part of the solution to the set of equations (3.9) to (3.13). The problem is thus intrinsically extremely non-linear. Under stellar conditions with a relatively short mean free path, a simple form of Ohm’s law is valid (see section 2.1). In the perfectly conducting approximation with a steady circulation, poloidal streamlines and field lines coincide, and the most general form of the fluid velocity is a flow parallel to the total field, together with an angular velocity which is constant on individual poloidal field lines (Mestel 1961; section 2.1). The toroidal component of eq. (3.9) implies a strictly torque-free configuration in the absence of Coriolis terms. If the convection of angular momentum by the circulation is included, a differential rotation is generated in general (cf. section 2.1) and consequently a toroidal component of field, from eq. (2.3). The Lorentz force now has a toroidal component, ((V x B 1) x B~)!4iT,whose torque opposes the toroidal acceleration caused by the transport of angular momentum along the field lines by the circulation, and so maintains a state of approximate uniform rotation on each poloidal field line. This process will be efficient if the circulation speed v~ ~ VA = B(4irp)-1/2 the Alfvén speed (Mestel 1961; section 2.1), and even a very weak poloidal field satisfies this criterion. This is an important result as stable solutions with an approximately uniform rotation can now be sought, in contrast to the -—

situation discussed for rotating non-magnetic radiative zones where the evolution of the angular momentum distribution is not fully understood (e.g. Moss and Smith, 1981). This is true even when the Lorentz force provides only a small perturbation to the poloidal equations of motion. In addition, a large scale magnetic field may be important as a star evolves, in keeping a condensed core and extended envelope approximately corotating*. 7T3’2 CGS (Cowling, 1945), and putting is given approximately a- cm) = 10we see that Rm> 1 even at T = i05 K if L =In2 aX star 1010 the cm conductivity (compared with a stellar radius R by 1011 U ~ iO~cm s~.However, velocities may well be much less than this, and length scales may be locally —

*

In a recently circulated preprint Tassoul and Tassoul claim that a state of approximately uniform rotation will not be established. The author

believes their conclusion to be mistaken, but this is not the place to give a detailed refutation.

32

Lb. Moss, Magnetic fields in stars

small, so that the field-freezing approximation (3.2) is not universally valid. This, indeed, is essential if we are to observe magnetic fields in stars possessing a meridional circulation, since otherwise if the streamlines are contained within the star, then so are the field lines. The comparatively low temperatures (.-.10~K) and small velocities at the surface do enable some field lines to ‘leak out’. Substantial flow across the field also occurs in a star containing a magnetic field of dipolar-like structure (i.e., of odd parity) and a circulation of even parity (such as the quadrupolar Eddington—Sweet circulation). In this case the flow must distort the field near the equator until the length scales are reducd sufficiently that Rms51. This discussion suggests that the introduction of a magnetic field into the stellar structure equations will be a considerable complication, both conceptually and computationally, leading, in particular, to departures from radial symmetry. Several authors have attempted to estimate the gross effects of magnetic fields on stellar structure whilst keeping strict radial symmetry by introducing a ‘magnetic pressure’, which is a function of radius only. This is considered to be an order of magnitude representation of the Lorentz force VxBxB 1B21 B~VB 4 =_VL~—j+ ~ neglecting the curvature term and assuming that the field is of sufficiently small spatial scale (‘tangled’) that it can be averaged over a spherical shell to give a meaningful magnetic pressure P 2/8ir (e.g. 8 = B Trasco, 1970; Stothers, 1980, 1982; Hubbard and Dearborn, 1982). The results of any such computation must be treated with great care. Note that even a strictly force-free field (e.g. curl free dipolar) will have P 8 0 and so exert a radial force according to this formulation. In these models the ratio of P8 to P8. the gas pressure, has to be less than unity, but is otherwise fairly arbitrary, and is often taken to be quite large. Predictably if is significantly greater than zero, stars of a given mass tend to have rather larger radii and lower effective temperatures than standard models, and if is non-negligible in the innermost regions the luminosity is also reduced. Models of this type are rather ad hoc and there is no direct evidence for strong tangled fields with P8 a significant fraction of P8. There are also worries about the stability of some configurations. Comparison with more exact models suggest that these ‘tangled field’ models overestimate the effects of magnetic fields. (The interior field strengths are, in some cases, very large.) Landstreet (1986) gives a discussion and criticism of Hubbard and Dearborn’s models.

Polytropic models Some insight into the general problem can be gained by considering polytropic model stars. These satisfy hydrostatic equilibrium (eq. (3.9)) and Poisson’s equation (3.10) but explicit solution of the energy transfer and generation equations (3.11) and (3.12) can be avoided by making an assumption of the form p =p(p), and often 3.7.2.

p=Kp

1+-i/n

where n is the polytropic index, is adequate. (The gross structure of upper main sequence stars is reasonably well approximated by 3 ~ n ~ 4, and that of fully convective stars by n = 1.5.) Note that in this approximation a thermally driven circulation does not occur nor does the MHD equation (2.3) have to be satisfied explicitly. It turns out that the structure of static non-rotating magnetic models of the type described in the next section is reasonably well approximated by polytropic models with appropriate

D. Moss, Magnetic fields in stars

33

Table 3.1

n

b(0)/b(R)

1.5 2.0 3.0 4.0

7.4 11.7 37.9 270 1930

4.5

values of n. If 122~ is derivable from a potential, as is the case for uniform rotation, then (3.6) severely restricts the form of the magnetic field, since from eq. (3.5) (VxB)XBpVØ for some scalar 0. Writing p = p0 (the unperturbed density) as a first approximation eq. (3.7) can be solved directly for an arbitrary choice of 0. Note that in this case the field B is independent of U. Table 3.1 lists the ratio of central to surface (r = R) radial field on the magnetic axis (0 = 0) when 0 is chosen so as to make B have dipolar angular dependence, i.e., B,. = b(r) cos 0, and with no toroidal field component (Moss, 1983b). The internal concentration of field increases with the central condensation as measured by n. Inasmuch as stellar evolution results in an increased central condensation, these results hint that

evolved stellar models may have larger ratios of internal to surface field than zero-age models. It should be remembered however that, amongst other shortcomings, there is no constraint exerted by the MHD equation. This point is returned to later. 3.7.3. Non-polytropic models As soon as the polytropic assumption is abandoned, Poisson’s equation and the equations of radiative transfer and energy conservation must be solved simultaneously with eqs. (3.11) and (3.12). The velocity must also satisfy the MHD equation. This complicates the problem considerably. It ismagnetic convenient to (To use 2R4/4GM2, where H is now a measure of the surface field. the parameters A0 and = Hbe the ratio of the magnetic to gravitational energy if the field were to be order of magnitude AH AH would uniform throughout the star. Thus Emag/Egrav f2A 11, where f is4 the ratiomore of mean surface G, and ofteninternal about a to kilogauss. field.) In the starssequence the observed are less than about 3_~10~1 x i0 and so interior fields might be much Thus near theCP main AH S fields i0~,and more typically larger than those at the surface, without the total magnetic energy being more than a small fraction of the

gravitational energy. In a rotating magnetic star we can consider the circulation driven by both the rotation and magnetic fields as given separately by first-order perturbation theories, and then the corresponding velocities, v 0 and VH say, can be added linearly. The rotationally driven meridional circulation then satisfies the perturbation theory energy equation c~p0T0v0.V(lnT0/p~~)rz(pe_V.F)01,

(3.14)

where F is the radiative flux, subscripts 0 and 1 denote the zeroth-order and first-order quantities respectively and the Eulerian time derivative has been neglected. The circulation due to the distortion of the thermal field by the Lorentz force similarly satisfies

34

Lb. Moss. Magnetic fields in stars

cVpOTOvHV(ln

T01p0

)=

(~E—VF)111.

(3.15)

The simplest problem is to impose a uniform rotation 12 and then to seek a solution satisfying (3.9) to (3.11), (3.14) and (3.15) (the appropriate forms of Poisson’s equation and the energy balance and transfer equations), subject to the condition that the total circulation velocity, v~= v~+ VH, is zero everywhere. That is, to seek a magnetic field yielding from (3.15) and the continuity equation a velocity VH = v0. This is a substantial simplification as the steady MHD equation is trivially satisfied with infinte conductivity, and there are no problems concerning flow across the field. Finite conductivity can he considered to modify the picture by giving a slow decay of B~,to which the star can steadily adjust. Such models are certainly not wholly realistic, but they do provide us with a certain amount of insight into the more general problem. A number of such investigations have been made, with various assumptions about the field topology, zero-order stellar model and so on. For reasons of mathematical tractability most of the calculations have been with rotational and magnetic axes assumed parallel, even though observations demonstrate that in many cases the angle x between the axes is non-zero. Some general conclusions can be drawn and these are conveniently illustrated by reference to the model of Wright (1969), which has a strictly poloidal magnetic field of dipolar angular structure, with axis parallel to the axis of (uniform) rotation, and no toroidal field component. The solutions are a function of the single parameter AQ/AJf, and table 3.2 shows how the ratio of central to surface field on the axis of symmetry increases with A11!A11, and also gives the ratio,2. F1!F5, internal to surface magnetic For larger large values of surface AQIAH, field, Br(0~ 0)/Br(R, 0).~(A12/ Even of though the internal magnetic fieldflux. is much than the the magnetic energy A11)’ is asymptotically only about 10% of the centrifugal energy for large values of A 11 /AH. An alternative interpretation of these results is to say that as F1 is reduced at constant 12, then the surface flux tends to zero at a finite value of F1. These results can be understood in terms of the magnetic field distributing itself to oppose the rotationally driven currents which are in a sense ‘inexorable’ given the assumed angular velocity field. As F1 decreases, more of the field must be concentrated to the interior, and eventually there is ‘not enough’ field left to produce a v such that v0 + V11 = 0. More generally, solutions can be sought with ~ 0. Now it is essential to have a non-zero resistivity so that fluid can flow across the field. This allows field lines to escape from the stellar surface even though streamlines close within the star, and also permits the coexistence of magnetic and velocity fields of differing parities (cf. fig. 3.7). For example when the magnetic field is of dipolar angular structure, since the circulation is quadrupolar flow across the field must occur at the equator. in addition to that at the —

I ‘~

‘

/\ ~(~~J\

(a)

\~\

,I

“

I /

\\f~\,i\\ (b)

Fig. 3.7. (a) The schematic relation between a modified Eddington— Sweet circulation and magnetic field in the aligned rotator. Field lines are solid, stream lines dashed. (h) As (a). but for the )+‘averaged circulation in the perpendicular rotator.

Table 3.2 Ali/Afl

B,(O,0)/B,(R,O)

~

2 x III

119(1

6(1

D. Moss, Magnetic fields in stars

35

poles. A purely quadrupolar field could, of course, have v~,and B~parallel through much of the interior. From the anti-dynamo theorems steady axisymmetric solutions do not exist. However, as global decay times are longer than main sequence lifetimes of Ap stars, it is reasonable to look for approximately steady solutions which satisfy explicitly the steady form of the MHD equation (2.3). Such solutions will be referred to as ‘quasi-static’. Calculations have been performed for both parities of strictly poloidal field with generally comparable conclusions (Mestel and Moss, 1977). One group of solutions are essentially finite resistivity modifications of the previous zero-circulation solutions with v~ v01 in general (magnetically dominated), whilst another, rotationally dominated, group has v~, v0 through much of the interior, although v~ —*0 near the surface. Qualitatively these finite circulation models are similar to those with v~, 0 in that the ratio of surface to interior field decreases as A0 IAH increases. Quite typically the ratio of interior to surface field strength is of the order of several hundred or a thousand. (For reasons of stability a toroidal field component is probably essential (see section 3.4). Models can be calculated which include such a field, and qualitatively results are not altered very dramatically (e.g. Moss, 1975, 1977a). It is possible in some cases for this field to be of similar strength to the poloidal field.) At first these models appeared to be a fairly satisfactory representation of the observational data. The apparently quite strong anti-correlation of angular velocity with field strength was fairly naturally explained. Moreover a preliminary version of the ‘f-motion’ theory (Mestel and Takhar, 1972) suggested that x = 0 and x = ir/2 were indeed asymptotic states for an ‘oblique rotator’, but the time to approach these values was relatively long. Consistent with this, the observed x distribution could be interpreted as being approximately bimodal, with peaks near x = 0 and ri-!2. This picture was spoilt by both observational and theoretical developments. Improved observational techniques revealed the existence of kilogauss effective fields in quite rapidly rotating stars (P ~ 1 d even). Although there was still a general tendency for field strength to increase with period the trend obviously was much weaker than had been thought previously. Also, Mestel et al. (1981) published an improved ‘4-motion’ analysis (see section 3.6). This predicted that the times for x to reach its asymptotic values was very much less than a

main sequence lifetime if the ratio of internal to surface field was anywhere near as large as that predicted by the existing ‘quasi-static’ models. Finally the view that the observed x distribution was consistent with a random distribution of x received more support. More sophisticated model calculations included a two-dimensional numerical treatment of the time-dependent MHD equation (Moss, 1984b). These revealed a much greater richness of structure of the solutions available. In particular solutions could be started at and followed from a wide and fairly arbitrary range of initial fluxes; the previous ‘quasi-static’ solutions were recovered as special cases. An interesting set of calculations was that begun with a fairly uniform distribution of magnetic field through the interior of the star. A critical period, P,~’can be defined such that, when P < P,~,a mean value of the magnetic Reynolds number of the Eddington—Sweet circulation is rather greater than unity, and when P> P, Rm s 1. Unless the field is strong, corresponding to mean strengths of more than about 105_106 G, the total magneto-centrifugal circulation is approximately the rotationally driven Eddington— Sweet circulation through most of the star, except in the outermost regions (I v is finite and a- —*0 at the stellar surface, so Rm—* 0 always there). P~ 4 days for typical CP star parameters, increasing with increasing stellar mass. If P> P~then an initially fairly uniform flux distribution is found to persist more or less indefinitely,9 changing only slowly approximately an below ohmicthe decay time forconcentrated the envelope yr). In contrast, if P < on P,~the field is dragged surface and to (typically a few i0 the interior, the xprocess occurring roughly on a circulation time scale. For relevant parameters this time is of order a few x i0~to a few x 108 yr, varying roughly as the square of the period and having a weak dependence on field strength, in that stronger fields are better able to resist burying. Main sequence

36

Lb. Moss, Magnetic fields in stars

lifetimes for Ap stars are typically a few x i05 yr, so it is quite possible to observe even rapid rotators when their fields are not strongly concentrated to the interior. Calculations with initial fields corresponding to displaced dipole surface fields show very similar trends. All the calculations discussed so far have been for the case of parallel rotation and magnetic axes. This restriction was made largely for conceptual and computational expediency. As mentioned in section 3.2 most observed magnetic CF stars have their axes inclined at an angle x > 0, with many large values of x Early work with x ~ 0 by Monaghan (1973), and subsequently Moss (1977c), was rather inconclusive, although Galea and Wood (1985) have thrown some light onto these difficulties. However the recent time dependent calculations seem to have clarified the situation somewhat. Consider a star with an initial magnetic field with axis of symmetry at angle x to the rotation axis. Suppose the field is sufficiently weak that through the bulk of the star the circulation is approximately the Eddington—Sweet circulation (this means that B ~ i04—i05 G). Work in a corotating spherical coordinate polar system (r, 0, A) with axis along the initial field axis. In this coordinate system the modified Eddington—Sweet circulation takes the form Vr(t~~ 0, A) = V(r) P~(cosA) P 2x P~(cos0) cos 2A + ~V 2,y P~(cos0) 2(cos x) ~V(r) sin 2(r)sin (3.16) —

where the circulation in a coordinate system (r, 0, A) with axis the rotation axis is given by Vr =

V(r) P 2(cos 0).

V(r)

V0(r) except near the stellar surface and convective core. 2. Then First take x = IT! Vr=~~~V(r)P 2(cos0)~ ~V(r)P~(cos0) cos2A. —

Averaged over A this becomes =

—

~V(r) P2(cos 0).

This is of the same form with respect to the field as the Eddington—Sweet circulation in the aligned (x = 0) case, but of opposite sense (see fig. 3.7). Simple kinematic considerations suggest that if Rm ~ 1 throughout the bulk of the stellar interior, then the field will not be buried, but merely concentrated towards the poles. Inclusion of the A-dependent terms may increase the angular structure present in the surface field but will not change the overall effect. Thus it seems there is indeed a gross difference between the properties of perpendicular and aligned rotators when the rotation is sufficiently rapid that (Rm) ~ I: when x = 0 the field eventually is buried whereas when x = rrI2 it is never buried. These predictions are supported by preliminary trial calculations, for initial configurations of both centred and displaced dipole type (Moss2 1984b, 1985). are shown in Initial fig. 3.8.field lines, and field lines after some adjustment has occurred, for x = 0 and IT! Examination of eq. (3.16) shows that there is a critical angle x~such that P 2(cos x~)= 0 (x~‘=55°). When x x~it is in the same sense as that in the perpendicular rotator. Thus for x
37

D. Moss, Magnetic fields in Stars

(a)

(b)

(c)

Fig. 3.8. (a) Schematic field lines for initial ‘quasi-uniform’ field configuration. The small quadrant indicates the boundary of the convective core from which the field is assumed to be excluded. (b) Aligned axes, field lines in quasi-steady configuration (t — few x iO~yr), P ~ P,. (c) Perpendicular axes, field lines in quasi-steady configuration, P ~ P~.

the circulation is reduced by a factor P2(cos x), so that the critical angularvelocity is increased by a factor 112. When x > x~the field is never buried. Galea and Wood (1985) also deduce that [P2(cos x)1 x = x~—55°is critical in separating quasi-aligned from quasi-perpendicular behaviour, from study of the Lorentz force term in the hydrostatic equation alone. This effect undoubtedly plays some role, however it seems likely that in most circumstances the magnetohydrodynamic process discussed above will dominate. What does seem clear is that distinctly different behaviour can be expected when x is small compared with x near IT!2. The analysis given for the field structure in the radiative envelope in general applies to either fossil or dynamo fields. Because the dynamo operates in the core, as far as the envelope is concerned the main difference between calculations assuming a fossil or a dynamo origin for the field concerns the boundary conditions applied to the field at the base of the radiative envelope. In the fossil model the convective core is essentially passive — any magnetic field passing into the core may interact in some not very well understood way with the convection, but it makes little difference to the model if, for example, the field is assumed to be completely excluded from the core or, at the other extreme, to penetrate it smoothly. In a dynamo model the core plays an active role, generating the field, which then passes smoothly into the envelope. An appropriate (steady) dynamo solution (satisfying the Navier—Stokes equations including the Lorentz force, as well as a dynamo induction equation, such as (2.11)) must be solved in the core region, this solution essentially providinga boundary condition for the envelope calculation. There is now a potential competition between the expected tendency of dynamo field strength to increase with increasing angular velocity, and the known tendency for quasi-static and time-dependent envelope calculations with rapid rotation to bury the field for small values of x. The only detailed calculations for this sort of composite model so far published appear to be the quasi-static, aligned axes, calculations of Moss (1982, 1983b). In these calculations the growth of the core dynamo field appeared to be limited in a steady state by a combination of the effects of meridional circulational and suppression of differential rotation by the Lorentz force. The results show a tendency for field burying to win the competition, in the sense that for sufficiently rapid rotation the ratio IB,.(0, 0)!B~(R,0)l.~(AOIAH)~3, to be compared with (A t / 2 for the purely fossil calculations. These results should not be taken too seriously for a number 0 !A11)

38

D. Moss. Magnetic fields in star.s

of reasons. As discussed above the quasi-static models are not fully representative of the fossil solutions available and they may not be the only dynamo solutions either. In particular these solutions assume a steady state to have been attained throughout the envelope. Further the aw dynamo formulism adopted is just one from a variety of possible dynamo models. It has been demonstrated that fundamentally different results can be obtained from the aligned and perpendicular configurations in a fossil calculation; it is clearly unreasonable to expect aligned dynamo results necesarily to be representative of those for perpendicular configurations. Unfortunately these are the only results so far available for fully consistent global core-dynamo models. Naive application of dynamo theory might seem to predict that two convecting spheres of similar masses, structure and angular velocity would produce fields of similar structure and strength. However magnetic fields in apparently similar CF stars are not strongly correlated. Leaving aside age effects, dynamo theory would seem to lack the extra degree of freedom possessed by the fossil theory with its arbitrary initial flux. Nevertheless it is only possible to observe the surface angular velocity, and a version of dynamo theory which may be called the ‘fossil rotator’ theory proposes that the observed fields may be correlated with the initial internal angular velocity, or equivalently with a measure of the differential rotation. Rãdler (preprint) points out that differential rotation and a field component with symmetry axis perpendicular to the rotation axis cannot persist. If the magnetic Reynolds number of the differential rotation is large enough field lines of opposite sense will inevitably be brought close together, and ohmic dissipation of the perpendicular component will be much enhanced. If the field strength is large enough Alfvén waves may be able to transfer angular momentum along the field lines rapidly enough to reduce the envelope to a state of uniform rotation. Otherwise the surface field will be much diminished. No such markedly enhanced dissipation will occur for a field component with symmetry axis parallel to the rotation axis. For a field consisting of a mix of parallel and perpendicular components this sort of process may alter the angle x between the effective field axis and the rotation axis. Note, however, that according to the ideas associated with Krause and his co-workers (e.g. Krause, 1983) the core dynamo theory of CF stars usually requires a purely perpendicular field, again implying approximately uniform rotation in the observably magnetic stars. A further complication might be the interaction of ‘f-motions’ with a continuously regenerated perpendicular field. The strength of a dynamo built field cannot be expected to increase indefinitely with angular velocity, for reasons outlined in section 4.3. Other suggestions to account for the observed anti-correlation of field strength with angular velocity in the context of dynamo theory include that of Moss (1980b) who pointed out that dynamo models have the general property that they tend to be oscillatory for larger angular velocities. If such a transition occurred at an angular velocity appropriate to that of the most rapidly rotating magnetic CF stars, then this mode change, together with the skin effect of the radiative envelope, might preventthe field of rapidly rotating cores from penetrating to the stellar surface. It might be further argued that the relatively few strong field rapidly rotating stars have larger cores or some other structural diference which changes the cut off angular velocity. In addition, any dynamo field has to be consistent with constraints imposed by the overlying radiative zone and, as discussed above, it is possible that the ratio of surface to internal field could decrease sufficiently as angular velocity increases so that the net surface field decreases as the period decreases. 3.8. The rapidly oscillating Ap stars Observations of low amplitude short period oscillations (periods of order of, or less than, hours) in the sun have excited great interest in recentyears. Apart from the intrinsic interest for stellar pulsation theory

D. Moss, Magnetic fields in Stars

39

such observations have the potential of revealing details of the structure of the solar interior — the topic can generally be called ‘solar seismology’. Whilst the short period photometric variations in Ap stars clearly cannot be studied in anything like the detail of the solar oscillations, the apparent intimate connection between the variations (generally interpreted as some sort of oscillation) and the magnetic field is novel, and an understanding of the phenomenon potentially could lead to a deeper understanding of both the stellar and field structure in these objects also. Salient properties of the rapidly oscillating Ap stars were summarized in section 3.2. Immediate problems for the theorist include explaining the excitation mechanism, why only one or two modes are excited, and the cause of the frequency splitting observed for some modes, where the components appear to be separated by an exact multiple of the rotational frequency 12. It should be noted immediately that the ratio of magnetic to gravitational potential energy is very small, and that although magnetic fields can perturb slightly the frequencies of oscillation of non-magnetic configurations, lifting partially the degeneracy with respect to the azimuthal wave number, in general this purely magnetic splitting can be shown to be much smaller than that caused by rotation (cf. section 5.2). Moreover, there is certainly no reason to expect any magnetic splitting to cause frequency shifts that are near exact multiples of the rotational frequency. Kurtz (1982) introduced the ‘oblique pulsator’ model. The pulsations are assumed to be axisymmetric non-radial pulsations, with the magnetic axis as axis of symmetry, inclined at angle x to the rotation axis, ‘md so consistent with the oblique model adopted for modelling the field variations of the Ap stars. With respect to the rotation axis the oscillations are, of course, non-axisymmetric. It is well known that such modes will precess, with angular velocity estimated at about 99% of the stellar rotation frequency (e.g. Dolez and Gough, 1982). However Kurtz’s observations suggest that this precession does not occur. Dolez and Gough also point out that only one or two modes, from a densely packed spectrum, appear to be excited. The precession problem could be avoided if some mechanism maintains the alignment of the field and pulsation axes — a possible candidate is discussed below. An alternative suggestion by Dolez and Gough is that precession is present, but that excitation only occurs when oscillation and rotation axes are approximately parallel. Preliminary calculations do, however, pose further problems for a simple K-effect (opacity driven) instability, of the kind thought to drive the oscillations of the classical variable stars — Cepheids, RR Lyrae, ô Scuti, etc. Naive applications of diffusion theory suggest that the gravitational settling of helium might be most marked near the magnetic poles, where the field lines are nearly vertical and the stability of the atmosphere is maximal. Removal of helium from the outermost layers would effectively remove the K-mechanism destabilization in the polar regions. Axisymmetric modes have their greatest amplitude along the symmetry axis, and thus they should be strongly affected by stabilization in this region. These considerations, although the supporting calculations are somewhat preliminary, prompted investigations of alternative oscillation mechanisms, although later work by Dziembowski and Goode (1985) seems to overcome some of these difficulties and supports a modified version of Kurtz’s original model. Shibahashi (1983) proposed that a magnetically driven overstability occurred in the superadiabatic part of the convection zone where the field lines are approximately vertical — i.e., near the magnetic poles. The estimated periods are of the correct order of magnitude. Such a mechanism would explain both the continuing alignment of the oscillation and magnetic axes, and also why the oscillations only appear in the cooler Ap stars (which have the most effective sub-surface convection zones). Detailed pulsational calculations by Shibahashi and Saio (1985) give general support to the oblique pulsator interpretation. An alternative mechanism of a radically different nature is proposed by Mathys (1985). He suggests that the observed photometric variations are caused by the star oscillating in a mode symmetric with respect to the rotation axis, but possessing surface inhomogeneities, presumably caused by the presence

40

D. Moss. Magnetic fields in stars

of the magnetic field. The inhomogeneities might, for example, be symmetric about the field axis, inclined to the rotation axis. He shows that such a model can, in principle, produce the type of variation observed by Kurtz. Furthermore, if a frequency is split, the splitting will be an exact multiple of the rotational frequency 12. 3.9. Secular changes in magnetic fields In section 3.7 calculations have been described which follow the evolution of the surface field in time, as described by the MHD equation (2.3), the velocities being self consistently calculated, but with the assumption that the stellar model itself is unchanging (Moss, 1984b) (see also Roberts and Wood (1985) for calculations with ad hoc velocity fields). However over the timescales involved the stars will evolve across the ‘main sequence band’ in the Hertzsprung—Russell diagram, with corresponding changes to their internal structure. The degree of central condensation (of the stellar material) increases whilst the mean density decreases. If the pressure—density distribution of the initial main sequence star can be approximated by a polytrope of index n =3, then this evolution corresponds to an increase in mean polytropic index. The figures in table 3.1 suggest that this might be accompanied by an increasing concentration of the field (remembering that these figures are for equilibrium configurations with aligned axes, and without any thermal effects, and so may not be directly applicable). Moss (1983b) followed the evolution of fields of axisymmetric ‘quasi-static’ initial structure (section 3.7) as the star evolved away from the main sequence and found a decline of surface field on an evolutionary timescale (O(10~)yr). These results (which are independent of those effects described in Moss (1984b) which can be attributed to the relaxation of an initial configuration in a relatively rapid rotator) seemed to suggest that a decrease in surface field strength with an e-folding time of order of a few x i08 yr might be expected in some stars at least. This appeared to be in reasonable general agreement with the preliminary findings of Borra (1981) and Cramer and North (1984), but at variance with Thompson et al. (1985). who criticize the earlier work. The latter correctly point out that the field burying effect of the meridional circulation decreases with increasing angle x between the rotation and magnetic axes, and is reversed when x ~55° (section 3.7.3), and so, over a set of stars with a broad distribution of x values the average field strength might not change significantly with time. Moss (in preparation) has reinvestigated the problem, using the improved formulation described in Moss (1984b) to follow the evolution of fairly arbitrary initial field structures, as stars of various masses evolve from the zero age main sequence. In the absence of any information about the initial distribution of magnetic flux through a star it is impossible to decide conclusively between the varying interpretations of the observations. With a plausible initial field configuration and distribution of angle x it seems likely that any observed decline in surface field strength averaged over a group of stars might be quite small. Substantial braking on the main sequence, if it occurs, could also result in readjustment of the field structure, and could well reduce the amount of field burying that occurs. Note that in these calculations ohmic decay contributes a very small effect over timescales of the order of the main sequence lifetime of stars of several solar masses. An additional mechanism might operate if the field is of dynamo origin, as the field diffuses outwards from the core to the surface after the dynamo is turned on (e.g. SchüBler and Pähler, 1978). This presumably would result in an increasing surface field; but the effect would be competing with the other changes described above. 3.10. Discussion The preceding sections will have demonstrated the large gaps in both the theoretical understanding of

D. Moss, Magnetic fields in Stars

41

magnetic fields in the early type CP stars, and also in the observational evidence about what form the fields really take, and how their properties correlate with other stellar parameters. Of crucial importance here is the role played by element inhomogeneities at the stellar surface when determining the surface field configuration. The perpendicular rotator / core dynamo model would be much advanced by a detailed demonstration of how a particular surface element distribution when properly interpreted gives 2, but when inhomogeneities are neglected gives x < ir!2. Note however that this would not be a xconclusive = IT! proof of the validity of the dynamo theory — such a demonstration would be just as compatible with a fossil field in the asymptotic state x = ir!2 (see discussion in section 3.6). Some progress has been made recently towards a theoretical understanding of fossil fields, and perhaps the crucial point to emerge is the importance of the initial conditions, and also of time-dependent effects. In other words, both the initial field distribution and the age are important parameters for the fossil model. Some consequences and predictions of the latest calculations are presented in section 3.7, and in particular it does appear possible to advance a coherent (although still tentative) explanation of the gross features of the magnetic field distribution of the magnetic CP stars that is consistent with the fossil hypothesis. Hopefully this can be tested against a growing body of data in the next few years. In this context the recent observation of HD37776 (Thompson and Landstreet, 1985) seems to be of particular interest, as the authors interpret its field variations as the effects of a sum of dipole and quadrupole surface components with the quadrupole term dominant. This is not consistent with a displaced dipole model interpretation according to existing calculations (unlike the usual situation where the dipole component dominates), but naively suggests that the field within the stars is predominantly quadrupolar. A simple version of the fossil theory, where the large scale stellar field is the direct descendant of a very large scale background field pervading the material from which the star forms, would suggest that fields should be of odd parity with respect to the magnetic equator. (Of course there is no such restriction of a ‘hybrid’ fossil theory.) So far HD37776 is a unique object. Are there other stars with similar field structure? Maybe it should be telling us something significant — if so, what? The other major problem posed by the CP stars, that has only been mentioned in passing, is the source of the spectral anomalies. Currently the most favoured theory is based on mechanisms of selective diffusion induced by radiation pressure (e.g. Michaud et al., 1981, 1983; Megessier, 1984; and references therein). Such processes need a relatively quiescent atmosphere, and it is probably significant that the Ap and Am stars are generally slow rotators, since rapid rotation would be expected to induce subsurface turbulence which would destroy any stratification (e.g. Vauclair, 1976). Magnetic fields do not appear to be essential, as they have not been detected in stars of the Am and HgMn subgroups. The Am stars probably owe their slow rotation to their membership of close binary systems, with spin and orbital periods synchronized, whereas a magnetically controlled braking mechanism plausibly can decelerate the magnetic stars. However, not all HgMn stars are in close binary systems, and the origin of the slow rotation of these objects is not apparent. The presence of a magnetic field may help to stabilize residual subsurface convection zones, and possibly to produce abundance spots on the subsurface, depending on local field topology and the properties of the element concerned. Conversely it may be possible to determine the spatial distribution of the elements over the stellar surface, see e.g., Megessier et al. (1979), Goncharski et al. (1983), Piskunov and Khoklova (1983), Khoklova and Pavlova (1984), and references therein. As mentioned earlier, ideally element and surface field distributions should be determined simultaneously and self-consistently from the observations. An alternative theory to explain the abundance anomalies is the selective accretion of interstellar matter in the magnetosphere (Havnes and Conti, 1971; Havnes, 1974, 1975, 1979; Havnes and Goertz, 1984). The idea of contamination of the outer stellar layers by r-process material produced either in the magnetosphere or by a nearby supernova (Fowler et al., 1965; Guthrie, 1967; Kuchowicz, 1973) is less favoured now.

42

Lb. Moss, Magnetic fields in stars

4. Magnetic fields in lower.main sequence and related stars 4.1. Introduction Magnetic fields were first directly detected in the sun by Hale in 1908 who (nearly 300 years after the demonstration of the structure of the Earth’s field by W. Gilbert) observed Zeeman splitting from the vicinity of sunspots. Since then a mass of data on the spatial structure and time variation of the solar surface field has been accumulated. Although the sun is a late type (G2) dwarf star, until quite recently virtually all other studies of stellar magnetism have concentrated on the earlier type, chemically peculiar, stars discussed in section 3. As outlined in section 1, the period since the late 1970s has seen a remarkable surge of interest and increase in knowledge of the activity and rotation periods of the lower main sequence stars loosely defined as those possessing significant outer convection zones. Included in this group are cool subgiants such as the FK Comae and (some) RS CVn stars. This major increase in knowledge (to use the word ‘understanding’ would perhaps be a little premature) results in part from the availability of space based UV and X-ray observations, but largely from ground based optical work of increasing precision. These observations give information of a qualitatively different nature from that of the solar field, telling us very little directly about the spatial structure of stellar fields but giving information about how gross features vary with stellar properties. Thus it is convenient to split the discussion and to consider solar and other late type stellar fields separately. Salient features of solar fields are described in section 4.2.1, but there is no attempt to give a comprehensive review (for recent reviews see, e.g., Bumba and Kleczek, 1976; Stenflo, 1982; Guyenne and Hunt, 1984). An attempt is made in section 4.2.2 to pick out the more important facts which seem to emerge from the observational data for stars other than the sun, although again there is certainly no attempt to give an exhaustive review. Attempts at modelling the solar field are outlined in section 4.3. 1 these may give some hints for or constraints on models for other late type stars which are discussed in section 4.3.2. —

4.2.

Summary of observations

4.2.1. The sun The somewhat irregular, approximately eleven year, solar sunspot cycle is a symptom of an underlying ‘twenty two year’ magnetic cycle. The cycle is rather irregular and there was an interval about 70 yr in the latter part of the 17th century when the sunspot cycle appeared to die out completely the ‘Maunder minimum’. C’4 records suggest other periods of extended inactivity (e.g. Eddy, 1977). Sunspots, with associated magnetic fields of -.-3000 to 4000 G, erupt in mid-latitudes in the early stages of a cycle, and the activity zones migrate towards the equator as the cycle proceeds (the ‘butterfly diagram’). There may be some asymmetry between northern and southern hemispheres. Sunspot magnetic fields are thought to he ‘upwellings’ of a submerged toroidal field. The total flux present can be of order 1023 Mx, suggesting a mean toroidal field in the convection zone of order 102 G. A remarkable feature of the solar field in regions away from sunspots is its concentration into isolated flux tubes of strength of the order of i0~G or so, even where the mean field is only a few gauss. There is a weak, approximately axisymmetric, poloidal field present, which is only readily detectable consistently near the poles. Its polarity is observed to reverse roughly at the time of sunspot maximum, but the two poles do not necessarily reverse polarity simultaneously. At times a major component of net poloidal field maybe in quasi-radial sector structures, —

D. Moss, Magnetic fields in stars

43

which extend into interplanetary space. In the equatorial plane there are two or (more usually) four sectors of alternating sign. Regions of solar activity as indicated by a variety of markers in the chromosphere and corona are intimately connected with regions of strong magnetic field — this correlation when assumed to extend to other late type stars provides a powerful probe of their magnetism — see section 4.2.2. Other relevant observations include the surface differential rotation, which can be approximated by 12(R, 0) = 12~(1+ b sin2 0 + c sin4 0), where 0 is the colatitude. (e.g. Howard and Harvey, 1970). Grossly, the equatorial regions rotate more rapidly than the poles. The pole—equator temperature difference at the surface is very small — this is a strong constraint on theories of differential rotation and energy transport in the convection zone. Finally there is no observable modulation of the overall solar luminosity during the solar cycle. 4.2.2. Late type stars other than the sun In contrast to the situation for the magnetic CP stars, there are very few direct measurements of magnetic fields in lower main sequence stars, and the existing determinations are subject to considerable uncertainty. However there is a substantial body of inferential evidence which will be discussed first — the direct measurements can then be used as some sort of test of the inferences. Mangeney and Praderie (1983), Rosner (1983) and Baliunas and Vaughan (1985) give comprehensive overviews of the topic. The story really starts with Wilson’s (1978) publication of his results of the long term monitoring of the Call H and K line core emission fluxes in a sample of 91 relatively cool lower main sequence stars. This work demonstrated that emission flux variations are often large, and that in a substantial fraction of the observed stars the emission varies cyclically, in a manner reminiscent of the solar cycle. Subsequent work by Vaughan and Preston (1980), Vaughan (1980) and Simon et al. (1985) showed that chromospheric activity in single stars declined with stellar age. The angular velocity of single stars appears to decline monotonically with age, and it is the rotation period that is the significant parameter, rather than the age per Se. This is supported by the observations of RS CVn stars (close binary systems with synchronized rotation and orbital periods and which so do not slow down with age in the manner of single stars) and other synchronized binary systems which exhibit the activity appropriate to single stars of the same period, and not of the same age (e.g. Middelkoop and Zwaan, 1981). Smoothlyvarying solar type activity cycles, with periods of the order of 10 yr (corresponding to a half solar period, since overall activity will be independent of the sign of the field) are only found in the older stars. When the H and K flux is plotted against B—V colour the points appear to populate two distinct regions (see fig. 4.1), the upper population consisting almost entirely of the younger stars without distinct quasi-regular cycles (although hints of periodicity underlying the chaotic behaviour may be detected). The lower part of the diagram contains older stars which display analogues of the solar cycle. The reality of the ‘Vaughan—Preston’ gap separating these populations has been questioned and, for example, Baliunas et al. (1983) find no evidence of a discontinuity when stars in a very limited range of effective temperature are considered. If real, the gap is potentially of great value in testing dynamo theories (see e.g. section 4.3.2) or other mechanisms involved in generating the stellar activity. In the sun Call H and K emission enhanced above ‘normal’ values is particularly associated with strong magnetic fields in the plage and supergranular network, the magnitude of the enhancement being approximately proportional to the field strength. Observations such as discussed above were perhaps some of the first evidence for the existence of magnetic fields in stars similar to the sun, and marked the

44

D. Moss, Magnetic fields in stars

A42 0

. 044

—02

.

0

.

46 -04

.

.

4.~

0

o

-06 o

—08’

0

:~~ 5

0

—5’O

00

0

000

0

I

—10 ________________________________________ 06 08 10 12 14

I

0’4

I

—02

I

0

-

log0~2 (P,~

04 5/T~)

B-V

Fig. 4.1. Mean values (S) of the Mount Wilson H—K flux index determined from the data of Wilson (1978). Closed circles represent

Fig. 4.2. (Rl’,K) plotted against P,50/r, for mixing length of 2 pressure scale heights. (From Noyes et al.. 1984.)

‘young’ stars, open circles ‘old’ stars. 0 represents the sun. (From Noyes et al., 1984.)

commencement of the studies of what is now known as the ‘solar-stellar connection’, although on theoretical grounds such fields had long been expected. By studying the short-term modulations of the Call H and K emission it is possible to deduce accurate rotation periods for a number of these stars (Vaughan et al., 1981; see also Baliunas et al., 1983). The technique may give results for equatorial velocities as small as 1 km s~,a far smaller figure than attainable spectroscopically from Doppler effects. This confirms that older stars are slower rotators (expected on a number of observational and theoretical grounds), and that smoothly varying quasi-regular solar type cycles are generally found in stars with rotational periods greater than 20 d. Emission fluxes in the H and K lines seem to be the most useful, or at least most widelystudied, measure of stellar activity. It soon became apparent that main sequence stars of roughly similar spectral type (equivalently mass or B—V colour) fell on quite a well-defined locus in the activity/period diagram although there was also a dependence on spectral type. This observation was refined by Noyes et al. (1984a), in perhaps one of the most significant papers on the topic. They defined a mean chromospheric emission ratio (R~K) (a mean because the emission varies not only on timescales interpreted as cycles (years) and rotation (days), but also over shorter periods — perhaps signalling occurrences of events similar to solar flares (?)), and showed that the programme stars define a very tight relation if log (R~K)is plotted against a quantity log (ProtITc) — see fig. 4.2. r, is a theoretically derived convective turnover time determined near the bottom of the convection zone of models of stellar envelopes of appropriate effective temperature and surface gravity. The envelope models are calculated with a conventional mixing length theory, with ratio of mixing length to pressure scale height of approximately 2. This remarkable result, that a single parameter governs overall activity levels, is highlighted if it is noted that this choice of parameter gives significantly less scatter than if other quantities (e.g. trot’ Prot!Pcycie) are used for the abscissae; and also that the scatter is very much larger if, for example, r~is calculated with a mixing length of one pressure scale height rather than two. The parameter ProtIr.~,is significant in the context of aw dynamo theory as it can be interpreted as measuring the dynamo

D. Moss, Magnetic fields in stars

45

number. Note also that the Rossby number R0 = (Prot!Tc)’. (Interestingly this choice of mixing length is consistent with other empirical determinations, although given the admitted inadequacies of mixing length theory it would be rash to use this result to argue that this is the ‘correct’ choice of mixing length. Also the mechanism connecting fields generated deep in the envelope with the various emissions in the chromosphere is extremely uncertain). The importance of the parameter R~’ has been further emphasized by Vilhu (1984a), who extended somewhat the range of spectral type considered, and also investigated the variations of other activity parameters thought to monitor behaviour at different heights above the stellar surface as functions of R~’.See also Simon et al. (1985). Noyes et al. (1984b) deduce a further empirical relation, PcycieZ(Prot/Tc)~, where n —1.25. Such relations potentially can constrain theories of field generation and activity mechanisms — this aspect is discussed in section 4.3.2. A final piece of observational evidence is of an apparent decline of magnetic activity on the main sequence for very cool stars spectral class M5 corresponding to a mass of about 0.3 M0 — cooler than those considered by Noyes et al.). This evidence exists for Ha emission (Giampapa, 1983), Ca H and K line emission (Soderblom, 1983) and X-rays (Golub, 1983). Close binary systems are of particular interest since, as mentioned above, they maycontain lower main sequence stars which are both rapidly rotating and old. W UMa systems are the ultimate close binary systems, having components in physical contact with a common convective envelope and, with rotation and orbital periods synchronized, possessing very short periods (of a few hours). Whereas the detached RS CVn systems appear to follow the single star relations quite closely, Vilhu (1984a) claims that some activity indicators eventually saturate as R~”decreases, and the X-ray emission relation may even be different for these stars. It is interesting to speculate whether differences might be due to the differing envelope structure in the contact configuration. Light variations from some stars have been studied in rather more detail. The rapidly rotating BY Dra stars (young and single) exhibit photometric variations typically of order a tenth of a magnitude which can be traced for as long as several months (see e.g., the review by Rodono (1983)). Similar phenomena are exhibited by the binary RS CVn stars which have periods from about 0.5 d to 15 d (Catalano, 1983). The accepted interpretation is that the variations are caused by the rotation of dark areas — ‘star spots’ — which retain their coherence for a number of rotation periods. Dark spots on the primary have also been appealed to as an explanation of the slightly irregular light curves and cooler primaries of W type W UMa systems (Mullan, 1975; Eaton et al., 1980). Modelling is crude and solutions are not unique, but there seems to be evidence for spots drifting with respect to the local frame of rotation of the stellar surface. There is a temptation to draw a close analogy with sunspots, but these starspots are very much larger in extent, covering perhaps as much as 20% of the visible disk, with a temperature contrast, z~ T, of order 1000—2000 K, and they are obviously of much greater global importance than sunspots. Perhaps a closer analogy to the solar case is given by Dorren and Guinan (1982) who observed five of the more active stars studied by Wilson (1978) and found evidence for starspots, with i~T —200 K, covering less than 5% of the disk. If it is assumed that activity mechanisms in late-type main sequence stars are broadly related to those in the sun, a strong circumstantial case can be made for the association of magnetic fields with the various types of atmospheric activity. In any case, on general grounds there is no reason to expect the sun to be unique, or even unusual, in its possession of a complex field structure. However there are only a few accurate direct determinations of field strengths. The main problem basically is that if the field topology is similar to that of the sun with the radial component changing sign many times across the disk, then methods which rely on the net linear polarization will suffer from the large amount of cancellation present. This is true of most of the methods used to determine fields in the CP stars (section 3.2). If there (‘—.-

46

U. Moss, Magnetic fields in stars

is a single isolated field region (e.g. a sunspot) any associated Zeeman polarization will be diluted. Mullan (1979) showed that such a field of circa i04 G could well give a measurable effective field of circa 102 G, or less, and so be only marginally detectable. Thus it is not surprising that early surveys found no detectable net longitudinal field on late-type main sequence stars. A more appropriate technique measures the magnetic line broadening, which depends only on field strength, and not on sign. In essence the method is to compare the profiles of two lines, one of which is known to be sensitive to a magnetic field and one which is not, with their profiles in the absence of a field (see e.g. Robinson, Worden and Harvey, 1980). This technique yields both an estimate of field strength and of an area filling factor. Marcy (1984) detected fields in 19 out of 29 G and K main sequence stars. Field strengths of up to about 2000 G and filling factors for the visible disk up to about 50% were found. There was no evidence for any marked dependence of field strength on spectral type, hut the area filling factor appears to be larger for the cooler stars of spectral type K. Marcy fitted his results for the total magnetic flux, 4, by the formula =

KT~V~ 0i,

(4.1)

where VRO, is the equatorial rotation speed, and a = —2.8 ± 1.1, b = 0.55 ±0.2, with the comment that b 1 may eventually prove a better estimate. This result is very preliminary, being based on a small sample and, indeed, Gray (1985) presents evidence suggesting that for the cool, observably magnetic, stars ~ — constant. However, such results do provide a first yardstick to judge dynamo (or other) theories by when they are applied to main sequence stars in general, rather than just the sun — see section 4.3.2. Further refinements are awaited with interest. Borra et al. (1984) have made a very sensitive search for mean longitudinal fields on late type stars (main sequence dwarfs, RS CVn systems and Cepheids). They detected mean longitudinal fields of 5—25 G in ~ Boo A (spectral type G8), which varied substantially over 24 h, and probably detected a non-zero mean longitudinal field on the RS CVn star UX An. Apart from marginal detections at about the 2u level in a number of Cepheids these were the only non-null measurements. ~ Boo A has a field strength estimated at —~1000G with a filling factor of about 50% (Marcy, 1984). These apparently contradictory results can be reconciled if the surface field geometry is complex (e.g. with many bipolar regions) so that there is heavy cancellation when determining the mean longitudinal field. Giampapa et al. (1983) reported a 1290 G field covering half the visible surface of the RS CVn star A And, and Doiron and Mustel (1984) find radio observations of the RS CVn AR Lac to show circular polarization consistent with the presence of a field of between 5 and 80 G. Working in the infrared part of the spectrum, Gondoin et al. (1985) found no indication of any field in ~Boo A, and evidence for a field significantly less than that found by Giampapa et al. in A And. These and other results provide evidence for the variation of field strengths over periods from days to years. Note that this discussion has covered only a selection of positive field detections. There is strong observational evidence for the rotational braking lower main sequence as they 7 yr) of have a mean equatorialstars velocity of age. For example, early G dwarfs in the Pleiades (age 8 x i0 12—15 km/s, compared with the solar value of about 2 km/s. Skumanich (1972) deduced the relation 12 ~ I /2, with a decay time of order ~ yr. Rotational periods of lower main sequence stars have been determined with increasing accuracy during the last few years, but later results do not, in general, seem to depart very markedly from the Skumanich relation (e.g. Rengarajan, 1984) although there may be exceptions. Baliunas (1984) gives a review and many references. Overall there appears to be no reason to doubt the widespread if not ubiquitous presence of fields of —

D.

Moss, Magnetic fields in stars

47

complex structure at the surface of cool dwarf stars. Filling factors and total unsigned flux may be much greater than that of the sun. Interestingly, however, there is no evidence for field strengths substantially greater than those seen in the sun (i.e., of order several kilogauss). While there is more detailed information available than outlined here, and more is being acquired, enough has been said to define the salient phenomena that must be explained by a successful theoretical interpretation. Important points include the dependence of field strength and geometry, and filling factor, on rotational period and depth (and other structure) of the convective zone; the relation of chromospheric and coronal fluxes to the photospheric field strength and structure; an explanation for the Vaughan—Preston gap, if it is real; a theory of stellar braking; special effects in close binary systems; the energetics of starspots. At present there is no completely satisfactory theory for any of these phenomena, but current thinking will be outlined in the following sections. 4.3. Theoretical models The current consensus of opinion is that the solar field is a dynamo field, although there are those who dissent strongly from this view. Thus most of section 4.3.1. will be devoted to a brief description of dynamo theory as applied to the sun, followed by an outline of alternative theories for the solar field. Section 4.3.2 is still more speculative, consisting of descriptions of attempts to extrapolate ideas about stellar dynamos to late type stars other than the sun. Recent reviews on dynamo theory as applicable to solar and stellar fields include Stix (1976, 1981, 1984), Cowling (1981), SchüBler (1983). 4.3.1. The solar field

The ideas leading to the first plausible models of the stellar field are found in Parker (1955), Babcock (1961) and Leighton (1969). A brief outline of the general principles is given below which, however, does not follow exactly any of these papers. Consider a flux tube in an initially purely toroidal field, fig. 4.3a. Suppose that a buoyancy instability forms a loop in this tube. The convection zone is very compressible and strongly stratified vertically, so the fluid motions will be of the type indicated in fig. 4.3b. In the northern hemisphere the Coriolis effect will rotate the bulge in the flux tube as shown by the double headed arrow in fig. 4.3b, and thus poloidal field is created; the associated helicity, u V x u, is negative. Consider a set of such toroidal flux tubes each creating its own elements of poloidal field. The poloidal elements can be averaged, appealing to a suitable dissipation mechanism to allow reconnection, and if the toroidal field is antisymmetric with respect to the equator a loop of poloidal field is formed as in fig. 4.3c. If there is also a radial gradient of angular velocity (latitudinal gradients seem less effective), then the poloidal field can be drawn out to create new toroidal field, which restarts the cycle. A crucial feature of this mechanism is the existence of a large effective resistivity, which must be able to effect reconnection and dissipation on a timescale of 0(10) yr. The usual molecular resistivity is far too small to assist directly, and all dynamo theories of the solar cycle rely on an enhanced turbulent resistivity to accelerate these processes, although there is still no detailed theory of such processes for vector fields. This point is taken up again later. The brief account given above does not attempt to describe how the Parker—Babcock—Leighton models explain various solar phenomena, such as the butterfly diagram (but see below). The similarity of the basic mechanism to the aw dynamos described in section 2.2 is apparent, and there is a close relationship between models of these types. It is instructive to consider in detail what happens in a simple aw dynamo the behaviour of the model described above will be generally similar. Assume that helicity is negative in the northern hemisphere and positive in the southern, as above, so —

48

Lb. Moss. Magnetic fields in stars

a is positive in the north and negative in the south. Further assume that 312/or < 0, SO that ND <0 (eq. (2.14)), and that there is initially a toroidal field with antisymmetry with respect to the equator — fig. 4.4a. The poloidal dynamo equation (2.12) generates poloidal field from the a-effect in the sense shown in fig. 4.4b. The effect of differential rotation on this field is to generate new toroidal field. With 312/Or < 0 this additional toroidal field is as shown in fig. 4.4c. When this is added to the original field, the new toroidal field is relatively reinforced at lower latitudes, giving a net field as shown in fig. 4.4d, the belts of toroidal field moving equatonwards. The cycle continues, producing new poloidal field in the sense shown in fig. 4.4d; then new toroidal field, and so on. Eventually the old toroidal field disappears in the equatorial region, leaving toroidal field of the opposite sign in high latitudes, ready to begin the second half of the solar cycle (fig. 4.4e). (This argument is taken essentially from Stix (1976)). A little thought shows that if ND a(OIuI9r) >0, then the toroidal field migrates in the opposite sense, from equator to pole. Sunspots are believed to be associated with strong subsurface toroidal fields, and so the butterfly diagram can he explained. With suitable fine tuning models of this type can explain the observed solar phenomenon in considerable detail. -‘-

(a)

Ic)

Fig. 4.3. A schematic dynamo of the Babcock—Parker—Leighton type. (a) A toroidal flux tube. (b) A cyclonic upwelling. which twists the loops of toroidal field by approximately rrI2, to produce loops of poloidal field. (c) After suitable merging, the loops of poloidal field produced from toroidal field in both hemispheres give a poloidal field as shown by the arrows. The toroidal field into the page is indicated by ®. out of the page by (I).

49

D. Moss, Magnetic fields in stars

(ac~

~

(c~

~

(e)

Fig. 4.4. aw dynamos and the butterfly diagram. (a) The initial toroidal field: ® into the page, Gout of the page. (b) The a-effect generates poloidal field. (c) The additional toroidal field produced by differential rotation from the poloidal field in (b). (d) The sum of the toroidal fields in (a) and (c) gives the toroidal field illustrated here. The a-effect creates new poloidal field as shown. (e) The cycle (a) to (d) repeats until the opposing flux tubes migrate to the equatorial plane where they disappear. The tubes of opposite sign appearing at high latitude in (d) restart the cycle, with opposite polarity.

Stix (1984) argues that it is necessary to have 012/Or <0 in order to explain the observed phase relation between poloidal and toroidal fields; theories of the detailed structure and dynamics of the convection zone do not necessarily agree with this statement — see below. However, 012/Or <0 is, superficially at least, consistent with the observed more rapid rotation of sunspots and active regions with respect to the ambient plasma (e.g. Golub et al., 1981), with the braking of the outer regions of the sun by the solar wind (cf. section 3.5), and with suggestions from observations of short period low amplitude solar oscillations that the interior rotates more rapidly than the surface (e.g. Duvall and Harvey, 1984; Duvall et al., 1984). Parameterized a -effect models (and Boussinesq calculations, see below) ignore the explicit effects of magnetic buoyancy (section 3.4.2). Estimates (e.g. Parker, 1975) suggest that even a 102 G flux tube will rise through the solar convection zone on a timescale of 2 or 3 yr, which would not allow time for effective amplification. Stronger tubes rise faster, but rise times are longest near the bottom of the convection zone. Mechanisms have been identified which might oppose buoyant flux rise. In particular, Acheson (1979a and b) performs a careful stability analysis and finds that rotation is strongly stabilizing, greatly slowing the rise, especially in the deeper parts of the solar convection zone. The situation is complex: Van Ballegooijen (1982) and Schüfiler (1983) review a number of such mechanisms. These and other considerations (e.g. Golub et al., 1981; Van Ballegooijen, 1982) lead to the prediction that the dynamo will be localized near the bottom of the zone or even in the ‘overshoot’ zone just below the region of formal instability. Gilman and Miller (1981) and Gilman (1983a, b) have taken the first steps towards a three-dimensional time-dependent dynamically consistent dynamo calculation for the sun. They do not prescribe an ad hoc coefficient a, leading to the parameterization of the poloidal MHD equations in the form (2.12), but instead try to follow the motions in the convection zone with sufficient resolution that the ‘a-effect’ appears explicitly. Parameterized ‘turbulent’ viscosity and resistivity transport coefficients are, however, still used. They also follow the transfer of heat by convection and radiation. In principle this should automatically include any non-linear limitation of the dynamo by suppression of the a-effect or differential rotation as the field strength increases. These calculations were for Boussinesq (i.e. basically incompressible) fluids. The first models did not show periodic behaviour, but were rather chaotic. On

50

U.

Moss. Magnetic fields in stars

close examination it was found that the kinetic energy in the convective motions dominated that of the differential rotation, so that the dynamo was operating essentially in an a2 mode, rather than an aw mode (a2 dynamos are known to be less likely to give periodic behaviour). Periodic behaviour can be produced by reducing the (turbulent) viscosity and thermal conductivity; however even then the cycle period is only 2 or 3 yr and the active regions migrate from equator to pole. As the helicity is negative in the northern hemisphere and the angular velocity decreases inwards in these models, the latter effect is consistent with predictions of aw dynamo theory, but in contradiction to the observed solar cycle. It is suggested that the importance of the a-effect may be decreased if the field is concentrated into flux tubes, with most of the convection occurring in the field-free space between the tubes, so that the field only ‘feels’ a small part of the helicity. In an idealized calculation Childress (1979) suggests that the a-effect might be reduced by as much as a factor Rm’ 2 If this is true then very much finer resolution in a compressible calculation will he needed. Calculations for a compressible fluid are more difficult and time consuming, but some results are now available (Glatzmaier, 1985a, b) in the anelastic approximation. In the calculations described in Glatzmaier (1985a), the dynamo is assumed to act throughout the convection zone, as in the Boussinesq calculations just mentioned. Glatzmaier points out that, as the variation of density with depth is much smaller deep within the sun, the dominant contribution to helicity generation there does not come from the expansion and contraction of rising and falling elements, as discussed in section 4.3.1, but rather from the convergence of ascending flows and divergence of descending flows in cellular motions near the bottom of the convection zone (fig. 4.5). This generates helicity of the opposite sense to that in the outer part of the zone. Thus in the northern hemisphere the helicity is negative over the outer part of the convection zone and positive in the inner part, giving however a net negative value; and similarly an overall positive value in the southern hemisphere. Differential rotation is approximately constant on cylinders through a substantial part of the convection zone (the Proudman—Taylor constraint); this coupled with the fit to the observed surface equatorial acceleration gives an angular velocity that decreases inward, and so the toroidal field belts still migrate polewards. This cycle period is about 10 yr. These contradictions with observations seems difficult to avoid with Glatzmaier’s model, and clearly are rather embarrassing for, at least, this form of dynamo theory. Glatzmaier comments that the dependence of angular velocity on depth is consistent with recent

radiative interior

convective envelope

Fig. 4.5. A schematic representation of how ascending flows converge and descending flows diverge at the base of an envelope convection zone, thus giving a contribution to the net helicity of opposite sign to that produced by the process illustrated in fig. 4.3b.

D. Moss, Magnetic fields in stars

51

observations of frequencysplitting of solar oscillations (Duvall and Harvey, 1984; Duval et al., 1984). He reiterates that his calculations deal with continuously distributed fields, and that if fields really are concentrated into small flux tubes between the cells through most of the convection zone, thus decoupling to some extent the velocity and magnetic fields, and if the dynamo does operate at the base of the convection zone, then different results may be obtained. See also, e.g., Galloway and Weiss (1981). Glatzmaier (1985b) applies his program to a simulation of an ‘overshoot zone dynamo’ operating beneath the base of the convection zone. There are severe numerical difficulties which prevent definite conclusions from being drawn. The calculations do suggest that, consistent with the helicity distribution described above, the toroidal field deep within the sun may indeed migrate equatorward, even though 012/Or >0 through most of the convection zone. Calculations were not continued for a complete cycle but Glatzmaier says that an estimate of the period was roughly compatible with the solar cycle period. Unless the numerical technique can be considerably improved it seems that investigations of this nature will be costly and rare. ‘Flux tube’ dynamos have generated quite a lot of speculation, but they are difficult to model. Essentially a two phase medium must be considered, one phase magnetically dominated, the other almost field free. Two extreme cases are where the field is in many ‘fibrils’ with diameters of several hundred kilometers and fluxes of 10t7_1018 Mx, similar to the observed photopheric fields; or with the field concentrated intobottom relatively fewconvection flux ‘ropes’zone) with containing fields of approximately equipartition strength (maybe 4 G near the of the fluxes of maybe o~1022Mx each. In the first —i0 case the fibrils will be dynamically dominated by the convective motions through much of the convection zone, and the small scale of the tubes makes it possible meaningfully to consider average fields, with the provision that the mean field will be much less distorted by the fluid motions than the equivalent diffuse field. Also, as mentioned above, the net a-effect may be much reduced. In the flux rope case the large diameter (—~10~ cm) of the ropes means that averaging to obtain mean field equations is not really possible. Such strong fields will resist distortion by convection throughout the convection zone, and even modulate the convective flux of energy (see e.g. SchüBler, 1980). This leads to the argument (e.g. Golub et al., 1981; Schüfiler, 1983) such that a ‘flux rope’ dynamo will probably be situated just below the convection zone, in the overshoot region where most energy is carried by radiation; otherwise significant luminosity variations would be observed during the solar cycle, see also Spiegel and Weiss (1985). These remarks can be compared with the preceding comments about non-linear dynamo simulations. SchüBler gives a review of the expected properties of such an overshoot zone dynamo. This discussion, and much of the theoretical work, has concentrated on mechanisms which produce regular (or fairly regular) cycles. Ruzmaikin (1981, 1983, 1984) has stressed that the solar field may be exhibiting the properties of a ‘strange attractor’, which might naturally explain phenomena like the Maunder minimum — see also Cattaneo et al. (1983), Jones (1984), Jones et al. (1985) and section 4.3.2. Although some form of dynamo is believed by most theorists to operate in or just below the solar convection zone, there are dissenters from this view. The most persistent is Piddington, whohas attacked the dynamo theories in detail in a long series of publications (see e.g. Piddington, 1983, for references). Although he makes a number of criticisims, the most fundamental is his attack on the concept of a greatly enhanced turbulent resistivity (of order 0.1 V~l,V~and 1 being turbulent velocity and length scale respectively) which is central to all forms of solar dynamo theory. Piddington correctly points out that there is no convincing detailed theory for the turbulent diffusion of a magnetic field, especially when Rm ~ 1. The usual argument is that the turbulent motions will continue to reduce the length scale of the magnetic field until eventually ohmic diffusion is able to be effective. This is supplemented by appealing to ‘fast’ reconnection mechanisms such as the Petschek (1964) mechanism, in which fields of opposite

52

Lb. Moss, Magnetic fields in stars

sense when pushed together by dynamically driven motions only need to invoke ohmic diffusion effects in the small region of large field gradient between the opposed fields in order for rapid reconnection to occur. Priest (1985) gives an extensive discussion of reconnection processes. Piddington claims that these processes will not be rapid enough to give an effective resistivity of the magnitude required. He also argues that the usual analogy between turbulent viscosity and turbulent resistivity is dangerous: for example, as the eddy scale decreases energy cascades down the velocity spectrum from the large to the small eddies, where eventually it is dissipated. In contrast, in magnetoturbulence, additional magnetic energy is created at small wavelengths (at the expense of the kinetic energy) rather than degrading that at larger wavelengths. In place of dynamo theory, Piddington proposes a ‘magnetic oscillator’ theory. Piddington (1971) appears to be the only reasonably accessible reference which gives details of the actual field generation process; further discussion is given in Piddington (1981). In this model there is an underlying permanent poloidal field, B~,anchored ‘deep within the sun’, with strength of the order of gauss, which supposed to have survived from the time the sun was formed. In its time averaged state the field lies along isorotational surfaces. A meridional oscillation of period 22 yr (velocities —100 cms ‘) is presumed to occur, distorting the field so that B() ~VQ0, and so generating a toroidal field component BT, whose direction changes during the succeeding half cycles of the meridional oscillation. BT~increases to hundreds of gauss before some mechanism such as magnetic buoyancy brings it to the surface, causing sunspots and associated phenomena — Piddington likens this to a hydromagnetic wave with component BT travelling along B~.It is claimed that the details of the observed solar field can be reproduced. However no detailed analysis of the underlying mechanism is given. Some speculations have also been made about the effect of a residual primordial field on the dynamo process. Levy and Boyer (1982) and Boyer and Levy (1984) in the context of standard aw dynamo theory show that such a field would introduce polarity asymmetries in the solar cycle, and they use the observational evidence to put limits of one or two gauss on any large scale poloidal field in the solar core, or at least on the component that penetrates into the convective envelope. Pudovkin and Benevolenskaya (1984) perform analogous calculations with a version of the Leighton (1969) model, and claim that a primordial poloidal field of order 0.5 G gives an improved representation of solar activity as recorded by asymmetries in the Zurich sunspot numbers. In this context it is interesting that Rosner and Weiss (1985) suggest that the variation in angular velocity with radius, deduced from studies of solar oscillations, requires a weak field in the radiative zone to explain the approximately constant angular velocity inferred between fractional radii of 0.2 and 0.7. They further suggest that the inner 20% by radius of the sun may be magnetically isolated from the outer regions, and possess its own weak primordial field. Layzer et al. (1979) also criticized dynamo theories, and in particular the concept of turbulent resistivity. They propose a somewhat modified oscillator theory. A dynamo-like mechanism is invoked in the Hayashi phase to generate (without turbulent resistivity) a very disordered field in the young sun. As time proceeds the smaller scale components decay, but the present day sun is supposed to retain a ‘large scale irregular’ field with a linked poloidal—toroidal structure within the radiative interior. The field is assumed to be largely excluded from the differentially rotating envelope by the turbulent motions, but there is a fuzzy transition region near the base of the convection zone. Here the overshooting convective motions penetrate into the field region and buoyancy raises some field into the envelope proper. In this region differential rotation drags out poloidal field to form toroidal, very much as in an aw dynamo, the toroidal field generation being assumed to occur at different rates at different latitudes. When a critical level is reached the differential rotation is reversed, and the toroidal field unwinds and rewinds in the opposite sense. This toroidal oscillation is the basis of the solar cycle. Losses, from buoyancyfor example.

D. Moss, Magnetic fields in stars

53

bring toroidal field to the surface where it manifests as sunspots and associated phenomena. To obtain a period of about 20 yr the typical poloidal field at the base of the convection zone is estimated about 100 G. If the energy is primarily extracted from the cycle where the toroidal field grows most rapidly (at latitude -3S’), it is suggested that the oscillation will propagate to high and low latitudes - the basis of an explanation of the butterfly diagram. Interestingly, these mechanisms operate at the same place as it is now widely believed a dynamo must operate - that is near the bottom of the convection zone. Such criticisms, especially of the concept of turbulent resistivity, deserve more study and refutation than the proponents of dynamo theory have so far provided. However it is not clear that the alternative theories are more convincing. Piddington does not give a consistent analysis of his oscillations, preferring to go into great detail concerning surface manifestations of the resulting field. Layzer et al. provide a little more detail, but make a number of rather ad hoc assumptions about the properties of their model. It is not clear how either of these oscillator theories would explain the variation of magnetic activity markers and inferred field that are now observed among lower main sequence stars. In particular the suggestion that an estimate of a dynamo number seems to be the crucial parameter (section 4.2) seems a very striking piece of evidence (of course this was not known when the oscillator theories were first proposed). Magnetic fields are subtle entities, and we should not be deceived by the apparent simplicity of the governing equations into believing that we understood even qualitatively all facets of their behaviour. Parker (1979) gives perhaps the most complete exposition of the properties of cosmic magnetic fields. (See also Parker (1982, 1984a) on ‘flux expulsion’ dynamos, ignored in most kinematic solar dynamo models, and Parker (1984b) for the suggestion that concentration of the surface solar field into fibrils may represent an attempt to relax to a state of minimum total (thermal + gravitational + magnetic) energy.) Cowling (1981) gives an incisive review of both dynamo and oscillator models, and his overall conclusion seems very pertinent - “the dynamo theory of the sun’s magnetic field is subject to a number of unresolved objections, but alternative theories proposed so far are open to much greater objections”. 4.3.2. Lower main sequence stars This brief survey of theories of the solar magnetic field demonstrates both the potential richness of the behaviour of stellar magnetic fields and our lack of detailed understanding of relevant processes in even our nearest star. Given this, it might seem somewhat premature if not presumptuous to attempt to model the fields of other lower main sequence stars, about which we have very much less information. Nevertheless, because these stars do exhibit a wide range of the relevant parameters (of which probably the most important are rotation period and depth of convection zone, or equivalently effective temperature), their study does offer the prospect of testing the gross predictions of any proposed general theory for parameters other than the solar values. In some ways it might be argued that one of the problems with studying the solar field is that there is too much detailed information, so that the wood is hidden by the trees. This certainly is not the problem for other stars! Detailed explanations of the variations of properties with stellar parameters cannot be expected yet. The best that can be hoped for at present perhaps is to predict the gross variation of field strength, total flux and cycle behaviour as rotation period and effective temperature vary. Belvedere, Paterno and Stix (1980a) produce differential rotation and meridional circulation fields by a theory which attempts to explain the observed surface solar differential rotation as the consequence of an anisotropic convective heat flux in the solar envelope. These calculations are calibrated with the solar differential rotation, and along the main sequence the predicted differential rotation is minimal around spectral type G5 to KO and increases markedly towards earlier and later spectral types. These

54

D. Moss. Magnetic fields in stars

predetermined macroscopic velocity fields are input into a kinematic (and linear) aw dynamo calculation (Belvedere et al., 1980b). Because of the linear nature of the theory (e.g. no Lorentz force included) the only parameter directly determined by the model is the cycle period. This period is found to increase significantly towards later spectral types. Belvedere et al. (1981) give order of magnitude arguments to estimate mean field strengths and X-ray luminosity increasing towards earlier and later spectral types. with a minimum around G5. The underlying differential rotation distribution is not consistent with null observations of differential rotation in F-type stars (Gray, 1982), and observations do not give a significant variation of cycle period with spectral type although the observed X-ray luminosities do vary approximately as predicted. (Note that Rodono (1986) finds that differential rotation decreases towards shorter periods and later spectral type for solar type single stars.) The shortcomings of the models seem to be of detail in that the model used to calculate the underlying velocity fields is inadequate (see detailed criticisms in Moss and Vilhu (1983)) and of principle in that non-linear effects probably must be included in order to make accurate estimates of cycle periods. This is not to say that a related mechanism does not operate, only that these calculations do not seem to describe it adequately. Some general remarks can be made. Linear dynamo theory may be useful for predicting when a dynamo is excited, but it cannot be relied upon to estimate limiting field strengths and cycle periods. Cattaneo et al. (1984) study an idealized system of dynamo-like differential equations of 6th order, including a simulation of the effects of the Lorentz force on the differential rotation, and demonstrate how dramatically different the behaviour of non-linear systems can be from that of related linear systems. They show that as the relevant dynamo number, N, increases, for suitable parameters there is a transition from strictly periodic behaviour to chaotic behaviour for N ~ 4N~,where N~is the critical value for dynamo excitation. Episodes of relative quiescence (— Maunder minima?!) occur. Figures 4.6a and b are B

______t

B~

Fig. 4.6. (a) Doubly periodic toroidal field as function of time for N Cattaneo et al. (1984).

3N~.(b) Chaotic toroidal field when N

16N~.These figures are taken from

D. Moss, Magnetic fields in stars

55

taken from this paper and show how periodic behaviour of the field strength at smaller values of N becomes irregular for larger values. The maximum field strength in a stellar dynamo obviously is determined by some form of non-linear limitation. One possible mechanism in the context of aw dynamo theory is the suppression by the magnetic field of the small scale motions which generate the a -effect, e.g. Stix (1972), Jepps (1975) among others. This may be linked to ideas about spatial intermittency of field structure, with the turbulence largely occurring in field free regions, effectively diluting the a-effect. Other suggestions include the effects of large scale meridional circulation, e.g. Proctor (1977), Schüfller (1979); suppression of differential rotation by Lorentz forces; suppression of the a -effect by Coriolis forces; and the removal of magnetic flux from the dynamo region by the buoyant rise of flux tubes (e.g. Leighton, 1969; SchüBler, 1980; Durney and Robinson 1982). Noyes et al. (1984b) argue from their empirically determined relation, PcycteZ(Prot/Tc)1~25, that the suppression of the a-effect, or of differential rotation, by the growing magnetic field cannot be the relevant mechanism. They point out that if the dynamo number N (eq. (2.14)) is supercritical then N will be a monotonically decreasing function of lB I; as the field grows the effective dynamo number, Nett~decreases until Neff —-. N~.The cycle will then stabilize at a period which is approximately that at critical excitation, and so ~cycte will be independent of trot (i.e. 12), contrary to observations. Jones (1984) discusses several field limitation mechanisms; he concludes that his simulations do not support suppression of differential rotation as the dominant mechanism, but that he cannot distinguish between others. Knobloch, Rosner and Weiss (1981) attempted a semi-quantitative explanation of the dichotomy between the stars with period greater than about 20 d which exhibit solar type cycles with period of order 10 yr, and the more rapidly rotating stars with predominantly aperiodic behaviour. They suggested that a change in the pattern of convection may occur when the inverse Rossby number, R~1= 212L1U =

41T•T/Proi,

becomes large. (U a typical convective velocity, L a characteristic length, T a convection turnover time). The Proudman—Taylor constraint (that for rapid rotators in approximately steady motion 11 •Vu c~eO) then becomes more effective, favouring convection in rolls with axis parallel to the rotation axis a pattern resembling a cartridge belt wrapped around the stellar equator. (There is only marginal and occasional evidence for such patterns on the sun, where R~1~ 1, and probably the solar rotation is too slow for the constraint to be effective.) Knobloch et al. propose that stellar dynamos are limited when buoyancy removes the toroidal field as rapidly as it is created. The buoyant rise timescale is estimated at (IL 2IV~,and the timescale for generation of toroidal field from poloidal by differential rotation as BTI(R12’BP), from eq. (4.2). Thus —

B~.IB~ ce

1212’L2R4ITp — 122[4rrpRL2h~R],

(4.2)

where I~sR— depth of convection zone--. scale of variation of 12. When R~t> R~rit convection is predominantly in rolls, at least near the equator. V x u is approximately parallel to the rotation axis and 11 is small. Thus the coefficient a is much reduced in magnitude. The authors argue that poloidal field reversals, which depend on a, are then much less likely to occur, although the toroidal field generation mechanism is essentially unchanged. They also comment that a reduction in dynamo number ND = afl’L 41q2 might be accompanied by an increase in the field BT if increasing 11’ means decreasing all’. They suggest that as the angular velocity of a star falls as it ages, there is a transition from a regime with convection largely in rolls to a pattern not dominated by rotation. (Observations determine the critical

56

U. Moss, Magnetic field.s in stars

period as about 20 d and R~~1130). The field strength changes discontinuously on passing through this critical value, and the slower rotators with enhanced a show quasi-regular cycles. The convective rise time increases with decreasing stellar mass for given (land BT, so the theory predicts larger (interior) fields for smaller values of the stellar mass. The authors argue that R~rjtwill be an increasing function of mass on the main sequence, so that non-oscillatory behaviour is more likely—constant for coolerthen stars. Finally, if 213,from eq. (4.2); if BT/BP BTZ.ul. B~Order —constant for R~’ > R~rit then BT~ul of magnitude estimates of the consequences of dynamo limitation by buoyant flux loss are made by Durney and Robinson (1982), who also take the field amplification as being limited by the rise of flux tubes out of the lower part of the convection zone where new flux is assumed to be generated. As Knobloch et al. they note that rise times decrease with increasing field strength in the flux tube (Parker, 1979) and so a crude estimate of the limiting field can be obtained by equating the growth time for the dynamo field with the rise time out of the convection zone. Rise and amplification times can be estimated for suitable models of the convection zone and the aw dynamo equations, making (superficially at least) plausible assumptions about scalings. In spite of the superficial similarity with the procedure adopted by Knobloch et al. (above) it should be noted that the estimates for the buoyant rise times differ. The most significant assumptions probably are that both the differential rotation and the a-effect coefficient are proportional to (l(cf. Knobloch et al. where a decreases with increasing 11 if R~’> R~rjt).The model can be calibrated to reproduce the 22 yr solar cycle and observed surface differential rotation. With these assumptions, internal field strengths, B, and cycle periods, P,~,can be estimated as a function of spectral type and rotation period. For angular velocity (la 2Q~,Bz 12, and B also increases with decreasing effective temperature, T~.Simple estimates suggest that surface area filling factors also increase as decreases. The amplification time, and so P,, decreases with increasing (1. A crude model linking internal and surface fluxes predicts that total surface flux should vary approximately as (I2R )2. 3T -1.5 where R is the stellar radius. Robinson and Durney (1982) pursued these ideas further, and their analysis is given in a little more detail below. Start with the basic dynamo equations in the form —

~aBT+~V2A,

(4.3)

[vQXV(ArsinO)—~VX Vx

OBT

BTI~l—

(44)

where ~ is the magnetic diffusivity, assumed to be constant, and B = B~+ BT,BP = V x (Al) with respect to a spherical polar coordinates system with the rotation axis as its axis. The last term in eq. (4.4) is an ad hoc simulated buoyant loss term. Assume the dynamo operates in the lower part of the convection zone, with lower boundary at radius r~,and consider only solutions varying locally in latitude. Thus write A sin 0 = A’ exp(ikr~O),

B = B’ exp(ikr~0),

and assume that a and Vu can be represented by appropriate fixed values, assumed characteristic of radius r~.These approximations transform eqs. (4.3) and (4.4) into ordinary differential equations for A and B. The effects of increasing field strength on a and the differential rotation are modelled by writing a

IlL2

a 0

r~

012

e

,

LQ

-~—-l yr

=

—i- e r~

D. Moss, Magnetic fields in stars

57

where a0, /3~and ‘y are disposable parameters, and e = [B2/81T]![~pv2],where p and v are typical values of density and convective velocity near radius r~.L is a scale height at the bottom of the convection zone. Note again the explicit assumption that both a and 012/Or are proportional to 12. Growing solutions to the equations from (4.3) and (4.4) exist when 1/2

ND

[alV12Irc/(~2k3)]

>

NDC

=

0(1).

Choosing kr~= 2 corresponds to one toroidal flux tube in each hemisphere (consistent with our ideas about the solar field). a 0 and /3~can be chosen so that a solar model has the 22 yr cycle and values of V121 comparable with the observed solar surface value. The equations can be integrated with plausible values of y, and some results are given in table 4.1. Decreasing y shortens P (by a factor 0.5 at MS with y = 0), and increases the maximum field strength. Salient results are that P,~decreases with decreasing Te and also with increasing 12 for a given spectral type. Remember that the calculated fields are toroidal fields deep within the convective zone, and there is no really adequate theory linking these fields to the observable surface fields, nor to relate surface field strength and envelope structure to activity indicators. (Paterno and Zuccarello (1983) discuss the generation of X-ray flux in a Durney and Robinson dynamo model, and deduce L~ce (I2 Vilhu (1984b) considers the generation of transition level radiation, also in the context of 25, the Durney and Robinsonwith model, estimates the ratio of transition level to total flux to vary as R4~ in reasonable agreement the and observed values.) The trend to shorter cycles at later spectral types is in general agreement with observations, and the change is of the correct magnitude. There is, however, no hint of any transition to a more chaotic behaviour for the faster rotators this is perhaps rather much to expect from such a simple model (but, again, see Cattaneo et al., 1984). The result 4, —-. (12R)2~3Te~’~5 can be compared with Marcy’s (1984) preliminary parameterizations (eq. (4.1)) and Gray’s (1985) estimate that 4, -~ constant. Theory and observation thus seem quite disparate. These models are very crude (and for example, predicted area filling factors are unity for cooler stars for a wide range of angular velocities). The assumptions that a scales as 12 by both Knobloch et al. (1981) and Robinson and Durney (1982) is not very convincing as, when rotation is rapid enough so that the Rossby number R 0 ~ 1, the cyclonic convective rotation may be too large for an effective a -effect to operate. Similarly Rüdiger (1983) finds that I V12 ~ 2 for R0 small, and speculates that a dynamo of type different from the aw dynamo must operate in the more rapidly rotating stars. It should be stressed again that, even if we have a correct internal dynamo model, the relation between internal and observed fields is very uncertain. It is certainly premature to argue from this discrepancy that buoyancy is not an important mechanism. However other mechanisms should also be —

Table 4.1 Magnetic cycle periods in days and toroidal field amplitudes, taken from Robinson and Dumey (1982); 11 = 120, y = 0 and 5 Spectral type

P~(y0) P~(y5)

logB,(y—5)

GO

35 21 17 15 11 9

2.6 2.8 3.2 3.4 3.5 3.7

G5 KO K5 MO M5

35 22 20 21 23 19

58

U. Moss, Magnetic fields in stars

considered, but unfortunately no models with the degree of detail of those discussed above appear to be available at present for alternative mechanisms. Observational determination of braking laws on the lower main sequence (section 3.5) potentially provides a further constraint on dynamo mechanisms. However the discussion in section 3.5 shows that a range of wind models and dynamo parameterizations are consistent with the observational evidence. At least, it can be said that no glaring contradictions are so far apparent. 5. Other topics 5.1. Degenerate objects 5.1.1. Observations Magnetic fields have been detected in a number of single white dwarfs. Angel, Borra and Landstreet (1981) give a comprehensive review of the situation to 1980, which is still substantially valid, although there have been a number of additional detections since then. Field strengths are from circa 106 G in one or two objects (e.g. Liebert et al., 1983) to about 3 x i05 G (Angel et al., 1985; Wunner et al., 1985). Apart from the large magnitude of these fields the most remarkable feature is the apparent absence of fields s106 0, even though there are upper limits of a few kilogauss from some bright white dwarfs, and fields of a few hundred kilogauss should be measurable in many objects. Angel, Borra and Landstreet deduce that the distribution of field strengths is approximately flat in the range —‘3 x 106 to —3 x 108 G and that the deficiency of fields much less than about 106 G is real. The distribution is thus bimodal, with peaks above 106 G and near zero. They do not find any clear correlation between occurrence of a measurable field and other parameters such as effective temperature or spectral classification, in contrast to some earlier comments. However Liebert (personal communication to 0. Chanmugam) believes that surveys are incomplete and the observations are consistent with a uniform distribution of field strengths above i04 G. The observed fields seem to be representable by a dipole model, possibly displaced (Wickramasinghe and Martin, 1979; Martin and Wickramasinghe, 1984). Four objects have rotation periods that can be deduced from their field variations (periods from 1.6 h to 2.8 d). The others are either exceedingly slow rotators, or have rotation and magnetic axes aligned. Non-magnetic white dwarfs appear to be slow rotators, although perhaps more rapid than the measurably magnetic objects (Pilachowski and Milkey, 1984; see also below). The white dwarf primaries of the AM Her systems appear to be strongly magnetic, giving rise to polarized optical cyclotron and X-ray emission (e.g. Angel, 1978), explicable by the accretion of material onto the white dwarf from the Roche lobe-filling low mass main sequence secondary. Fields of order (2—3) x i0~G have been detected, for example by measurement of the cyclotron emission lines (e.g. references in Chanmugam and Ray, 1984). An apparent preference for this figure may be a selection effect although the absence of much larger fields may be real (Bond and Chanmugam, 1982). The DO Her systems appear to be closely related, but generally of somewhat longer period. Direct detection of fields has proved difficult, although fields of order 105_106 G have been inferred and still stronger fields suggested on theoretical grounds — see below. Recently Penning et al. (1985) have discovered weak circular polarization in a DO Her object, suggestive of the presence of a field considerably in excess of 106 G. There are theoretical grounds for expecting the polarization to be much diluted by radiation from the accretion disc which, together with the large polar caps in these systems, would make even strong fields difficult to detect (e.g. Barrett and Chanmugam, 1984). The AM Her and

D.

t~

Moss, Magnetic fields in stars

59

10

—1 -

I—Period

Gap

~

~AM

~ 5-

Phr Fig. 5.1. The period distribution of magnetic cataclysmic variables (from King et al., 1985). AM denote AM Her systems, IP intermediate polars.

DO Her stars together have an orbital period distribution between 80 mm and about 10 h, with a marked gap between 2 and 3 h (fig. 5.1). This is briefly discussed below. Detailed accounts of the observational evidence relating to these systems can be found in Livio and Shaviv (1983) and Lamb and Patterson (1985). Neutron stars occurring in pulsars are the other class of degenerate objects believed to possess strong magnetic fields, although again mostly they have not been measured directly. (Trümper et al. (1978) detected a field of 5 X 1012 G in Her X-1, from an X-ray cyclotron line.) A brief discussion is given at the end of the next section. 5.1.2. Theoretical interpretation Perhaps the most striking property of the single magnetic white dwarfs, apart from the strengths of their fields, is the distribution of field strengths, with all observed fields greater than one or two megagauss, even though much smaller fields could be detected in some favourable instances. If real the obvious theoretical possibilities to explain this distribution are that some white dwarfs are formed with large amounts of flux, others with essentially none; or that the structure of white dwarfs with large scale fields is such as to only reveal the field when the surface field is large. An associated problem is why observably magnetic single white dwarfs are all slow rotators, Veq S 10 km s 1, with the upper limit very much less than this in most cases. Angel (1978) and Angel, Borra and Landstreet (1981) suggest that the single magnetic white dwarfs are the descendants of the magnetic Ap stars, which in turn suggests that if other main sequence stars do have substantial hidden magnetic fields (e.g. because they rotate rapidly, see section 3.7) then this flux remains hidden at later stages of evolution. For illustration, a 1000 G field throughout a star of radius 2 x 1011 cm would be amplified to _~-.108G if contraction to a typical white dwarf radius of 7 x 10~cm were to occur without flux loss. Thus even allowing for the possibility of flux and mass loss occurring in association, for example during evolution through giant configurations, this mechanism seems capable in principle of providing fields of the observed magnitudes. Ruderman and Sutherland (1973) and Levy and Rose (1974) propose that neutron star fields, and possibly those of white dwarfs also, are built by dynamo action in the convective carbon burning core of the progenitor (once again providing a clash between ‘fossil’ and ‘dynamo’ theories!). As only massive stars are believed to develop carbon burning cores, this would explain the relative scarcity of magnetic white dwarfs. A further proposal is that the loss of the outer layers of the pre-white dwarf star could expose the underlying field. Presumably this mechanism would produce magnetic white dwarfs from the

60

Lb. Moss. Magnetic fields in stars

magnetic CP stars as well as from non-observably magnetic stars. It is difficult to see how it explains the distribution of white dwarf fields any better than the simple ‘fossil’ theory mentioned above. Further, planetary nebulae nuclei which are believed to have suffered substantial mass loss do not appear to be strongly magnetic (Angel, 1978). A radically different mechanism for generating magnetic fields in close binary systems containing a white dwarf is due to Dolginov and Urpin (1979), who discuss a dynamo mechanism similar in principle to the two-sphere dynamo of Herzenberg (1958). There have been few detailed theoretical studies of the structure of white dwarfs with large scale magnetic fields. In some ways this is because in the lowest approximation the problem is straightforward whilst in the next approximation it is difficult and subject to considerable uncertainties. Before discussing theoretical models it should be emphasized that even a field of 108 6 is a weak field dynamically, in that the parameter A,1 = H2R4/4 GM2 -~1 for typical white dwarf parameters, unless the mean field is very much greater than the surface field. The bulk of a white dwarf can he well approximated by a polytrope, with index n = 1.5 for non-relativistic degeneracy of the electrons, and n = 3 for relativistic degeneracy. If the magnetic field is assumed to be the equilibrium field in a polytrope it can be computed readily. With the unrealistic boundary condition BT = B~= 0 at the surface, models with very large internal fields can be constructed, leading to pronounced surface distortion and a significant increase in the Chandrasekhar limiting mass (Ostriker and Hartwick, 1968). More realistic polytropic models (axisymmetric) have poloidal and toroidal field components which do not vanish at the surface, and which are torque free inside and outside of the star (Raadu, 1971). These have modest ratios of internal to surface fields, consistent with the simple polytropic models discussed in section 3.7.2, and there is no significant change of the global stellar properties for plausible field strengths (see also Shul’man, 1976). The consequences of allowing for the effects of radiative transport through the outermost layers (thin, but crucial in that they determine the luminosity and cooling rate) were investigated by Moss (1979) who performed calculations analogous to those for main sequence stars described in section 3.7.3, for models appropriate to hotter white dwarfs without significant convection zones. The equations solved were those for the special case of exactly zero meridional circulation and so possibly do not provide a complete description of the possible field configurations, giving perhaps only an approximation to a subset of solutions (cf. section 3.7.3). Moss found that for a given total magnetic flux there is a threshold value of the angular velocity, Ll~say, above which the ratio of surface to interior field is very small. (l~increases with increasing internal flux. Thus significant surface fields would only be visible if 11 < flu. For the minimum observed field to be of megagauss strength it is necessary for ul~to correspond to an equatorial velocity of about 10 kms ‘. Clearly any object with an initially small flux will never appear as a magnetic white dwarf. but magnetic white dwarfs with internal fluxes large enough to give megagauss surface fields in slow rotators will appear magnetic unless V~ 01~ l0kms’, in which case the surface field becomes very small. Stars which have somewhat smaller total fluxes would have surface fields less than about 106 6 (the lower limit to the observed fields) even in the limit of very slow rotation; but again, if 11> Il~(corresponding to rotational velocities of 10 kms’ or somewhat less) the surface field will be unobservably small. Thus megagauss fields should not be observed in objects with V~~ 10 kms -1, and if non-magnetic white dwarfs have 11< 12~,they must have very small fluxes. The theory a maximum 2, corresponding to internal a mean main sequence field also in thepredicts progenitor of a few internal flux of order 1026 Gcm thousand gauss (assuming perfect flux conservation). At the time only an upper limit of circa 50 kms was available for the rotational velocity of non-magnetic white dwarfs but, as mentioned above, it was known that most magnetic white dwarfs either were very slow rotators, or had aligned magnetic and rotational axes. The (now four) magnetic white dwarfs with measurable angular velocities all have Vr,,t < 10 kms~.Recently Pilachowski and Milkey (1984) have made a preliminary determination of

61

D. Moss, Magnetic fields in stars

v sin i values for some non-magnetic white dwarfs, and find lOkms’ ~ v sin i ~ 60 kms’. The correspondence between the theoretically determination of value of fi~and the limiting observed values of Vrot for magnetic and non-magnetic objects is striking, but probably fortuitous in view of the crudeness of the theoretical models! Note also that the frequency splitting in the (non-magnetic) oscillating ZZ Ceti white dwarfs has been interpreted as being caused by rotation of period order 1 d that is they are slow rotators (e.g. O’Donoghue and Warner, 1982). However, it may be significant that non-magnetic white dwarfs are found with a range of V~01,extending to values significantly greater than the critical —

value, whereas those with measurable fields appear to have V~0less than the critical value. This discussion has been in terms of isolated white dwarfs, but it should be equally applicable to those in close binary systems, such as the AM Her and DO Her objects (see below). The rapidly rotating primaries of some DO Her systems such as AE Aqu (Prot —-. 33 s) may provide objections to these ideas. Possibly such systems are in the process of burying their surface fields (King, 1985) — or maybe field burying just is not important in degenerate objects. A more sophisticated treatment of the problem, including meridional circulation, time dependence, and possibly lattice effects, would be welcome. The theories mentioned above assume that a single white dwarf inherits its magnetic field. By contrast, Dolginov and Urpin (1980) propose that a thermomagnetic instability mayspontaneously generate a field in the degenerate interior, which subsequently rises to the surface. As their analysis is linear no estimates of field strength are given. It may possibly be difficult to produce a field with large scale order from the ‘bubbling’ process they discuss; however they point out that the widespread presence of such fields could have important effects on white dwarfs, even if it does not explain the observed fields. Cooler white dwarfs have deep subsurface convection zones (but the energy density of the turbulence seems likely to be less than that of a megagauss field, which may make an envelope dynamo mechanism unlikely (Fontaine et al., 1973)). A number of speculations have appeared concerning the interaction between the convection and magnetic fields. These include magnetic fields inhibiting convection and reducing cooling rates (D’Antona and Mazzitelli, 1975), magnetic fields being brought to the surface as the star cools and convection develops (Greenstein et al., 1971), and fields less than about a megagauss being expelled from and trapped beneath the convection zone (Moss, 1979). Theories multiply in the absence of evidence! Recently attention has focussed on the AM Her systems (polars). These are believed to be close binary systems containing a magnetic white dwarf primary with a quasi-dipolar field of a few x i0~G and a Roche lobe filling lower main sequence secondary. The DO Her systems (which are among the group of objects known as intermediate polars) are felt to be somewhat similar to the AM Hers, although there has been only one direct determination of magnetic fields via measurements of circular polarization or Zeeman spectroscopy in these systems. Their orbital periods are systematically longer than those of the AM Her systems, all but one of the DO Hers having periods greater than the 2 h upper limit valid for nearly all the AM Hers. There is a broad consensus that the field of the AM Her primaries is strong enough to disrupt (or prevent from forming) an accretion disk around the white dwarf, and that the field controls the accretion, channelling it down the field lines towards the magnetic poles. This is consistent with the suggestion that the primary rotates in synchrony with the orbital motion. The synchronization is probably brought about by the presence of the field, and both dissipation of energy via the penetration of the primary’s field into the envelope of the secondary (e.g. Campbell, 1984), and the direct action of magnetohydrodynamic torques (Lamb, Aly, Cook and Lamb, 1985; Campbell, 1985) have been proposed as mechanisms. The DO Her stars are believed to possess accretion disks around the white dwarfs, which rotate asynchronously. Estimates based on the spin up properties for disk accretion have led to approximate

62

Lb. Moss. Magnetic fields in star.,

field estimates in the range few x i05 to few x 10~G, an order of magnitude or so smaller than those in the AM Her systems (e.g. Lamb and Patterson, 1983); the only positive measurement (Penning et al.. 1985) is at the upper end of this range. Taken together with the measured AM Her fields, this would imply a very different distribution of fields in binary white dwarfs than in single ones, if the apparently bimodal distribution for the fields of the latter is true. There appears to be no theoretical explanation for this. Chanmugam and Ray (1984) attempt a unified theoretical explanation of both these types of system (see also Lamb, 1985). They suggest that if the white dwarf’s field is too weak, then it can never control the accretion, and the accretion disk is not disrupted, even when the separation between the binary components decreases subsequent to angular momentum loss by. for example, a magnetically controlled wind. They estimate that an initially asynchronous long period system will evolve into an AM Her system (in that the white dwarf’s rotation becomes synchronous) if its field strength is greater than about 3 x 10~G. Their mechanism explains approximately the period distribution of the AM Her and DO Her systems. This would suggest that some DO Her systems have fields comparable with those of the AM Her systems. There remain some outstanding difficulties, discussed by the authors, and it would certainly be comforting to detect positively more large scale megagauss fields in DO Her systems. This theory seems to be consistent with the existence of low field, short period systems, which are not seen. King et al. (1985) give a broadly similar discussion. They estimate a magnetopheric radius r~,out to which the field of the white dwarf can control the accretion flow, r,~ 1~ p,~7( — M) 2 where M) is the mass accretion rate from the secondary and ~i R~DBis the magnetic moment of the white dwarf. 3, A minimum moment for a white dwarf primary of an AM Her system is estimated as of order i0~6 cm by demanding that r 8, be as large as the Roche lobe radius. Orbital evolution is driven by angular momentum loss via by a stellar wind controlled by the dynamo maintained magnetic field of the lower main sequence secondary, which fills its Roche lobe. As the orbital period decreases the separation of the components and the mass of the secondary decrease. An upper limitHer on systems the whiteall dwarf 3 is both obtained by noting that AM have magnetic~5moment of the the field ordercannot of i0~G cm periods h, and so be strong enough for r,~to exceed the Roche radius at longer periods and larger separations. As the mass of the secondary decreases it eventually becomes fully convective, at a period of about 3 h. Both King et al. and Chanmugam and Ray use the idea of Spruit and Ritter (1983) and Rappaport et al. (1983) that the field of the secondary then becomes much weaker as the dynamo is less effective in the absence of a radiative core (cf. section 4.3), reducing the braking by the wind to a very small value. The decline in orbital separation becomes much slower, mass loss by the secondary halts, and the secondary relaxes to the radius appropriate to a star of constant mass, which is smaller than in the presence of mass loss. Thus the secondary shrinks within its Roche lobe and, in the absence of accretion onto the white dwarf, the system ‘turns off’. Orbital angular momentum loss is now only by gravitational radiation, until the separation has reduced sufficiently that the secondary again contacts its Roche lobe. This is estimated to occur at a period of about 2 h, thus neatly explaining the period ‘gap’ in fig. 5.1. The system then ‘turns on’ again, as an AM Her or intermediate polar (DO Her), according to whether the white dwarf field is yet strong enough to completely control the accretion. See also Ritter (1985). King et al. further point out that the observed absence of DO Her systems with P ~ 2 h, which this evolutionary scheme predicts should exist, suggests that white dwarfs in binary systems with i03°~ ~ i0~are very rare, and so white dwarfs in binaries either have ~

or

~t~103~Gcm3,

corresponding to fields

(—

D.

B~fewx106G or

Moss, Magnetic fields in stars

63

Bs104G.

They point out that this is similar to the proposed bimodal distribution for single white dwarfs. King (1985) suggests that a “field burying” mechanism, similar to that discussed above for single white dwarfs, also may be relevant to the distribution of the fields of the DO Her and AM Her primaries. Further discussion of those objects can be found in Livio and Shaviv (1983) and Lamb and Patterson (1985), in particular the reviews by Warner, and Liebert and Stockman, respectively. The rotating neutron stars which power pulsars are estimated to have field strengths of order 1012 G, from theories which explain their braking as the result of magnetic dipole radiation (e.g. Ruderman, 1972). Again, dynamically this is a small field. Interestingly this corresponds to a total magnetic flux of the same order of magnitude as that found in single magnetic white dwarfs. It is tempting to believe that there is a common origin for the field in a progenitor star, the exact details of the foregoing stellar evolution determining whether a white dwarf or neutron star is the end product. Alternative mechanisms proposed to produce pulsar fields include thermomagnetic instability in the neutron star crust (Blandford et al., 1983) and a form of battery mechanism (Dolginov, 1984) these would seem to require most if not all neutron stars to have significant fields. —

5.2. The effects of stellar fields on global oscillations Magnetic fields which slightly perturb the stellar structure share with rotation (and any other non-spherically symmetrical perturbation) the property of shifting slightly the oscillation frequencies with respect to those of the non-magnetic configuration, partially lifting the degeneracy of non-radial modes of oscillation with respect to the azimuthal wave number m (writing the Lagrangian radial component of the displacement vector in spherical polar coordinates as ~(r)P~(cos 0) &m4~).There are also some purely toroidal modes which have no non-magnetic analogues, which will not be considered here. The frequency shifting is relatively larger for high-order pressure and gravity modes of non-radial oscillation. Detailed calculations show that for magnetic stellar models of the types discussed in section 3, the purely magnetic splittings are too small to be of any interest (e.g. Goossens et al., 1976), except possibly marginally in rapidly rotating upper main sequence stars if they possess large ratios of internal to surface field (Moss, 1980a). There is no observational evidence that any such effect occurs. The only investigation that seems to have been published on the possibilityof magnetic effects contributing to the instability of such models is that by Shibahashi (1983). In the outer photospheric region of a star even a weak field will effectively constrain the motions, and it is possible that the shapes of the radial velocity curves in, for example, Cepheid variables, could be altered (Biront et al. 1982; Roberts and Soward, 1983). An extreme example of this is the oblique oscillator theory for the rapid oscillations of Ap stars (Kurtz, 1982; see also section 3.8) where the oscillation is assumed to be constrained to be symmetric about the magnetic axis, inclined to the rotational axis. Until the recent interest in rapid oscillations in Ap stars, classical Cepheids have attracted the most attention as candidates for pulsating stars with large scale fields. Fields have been reported from a variety of Cepheids, starting with the observations of Babcock (1958). Photographic methods indicated longitudinal field strengths up to hundreds of gauss in several Cepheids (see, e.g., references in Wood et al. (1977)). Photoelectric measurements by Borra et al. (1981) detected fields of order tens of gauss at a few standard deviations in two Cepheids, with null measures (effectively upper limits of a similar order of magnitude) in other, less accurately observed, stars. At least one star with a photographically detected field gave a null photoelectric result, and generally the photoelectrically detected fields are much smaller than those detected photographically. Borra et al. conclude that

64

Lb. Moss, Magnetic fields in stars

longitudinal fields of tens of gauss plausibly could be quite widespread in this class of stars. They also note that there is sufficient microturbulent broadening to hide (or even to be partly caused by) a small scale surface field of order i0~G. Stothers (1979, 1982) has investigated the possibility of magnetic fields significantly affecting the oscillations of Cepheid variables. He posited a highly tangled magnetic field through the pulsating layers, the significance of the tangling being that the effects of the field on the stellar structure and pulsations can be represented simply by an extra, radially varying, ‘magnetic pressure’ term, and the problem then retains its spherical symmetry. The assumption is consistent with the small integrated longitudinal field found by Borra et al. (1981). For ratios of magnetic to gas pressure throughout the pulsation region (less than 1% of the stellar mass in these giant stars) of order 0.5 Stothers found that the period of the fundamental radial mode of oscillation increases sufficiently to remove the long-standing discrepancy between Cepheid masses as calculated from comparison between evolutionary tracks and the observed positions of Cepheids in the HR diagram, and those estimated from their pulsational properties. However, the more detailed, non-linear models are less successful in explaining some features of the Cepheid population, such as ratios between the periods of the fundamental and overtone modes in the double mode Cepheids. It is difficult to assess the validity of the detailed calculations. The strong tangled field is an ad hoc assumption. Such a disordered field allows a high magnetic energy density and low integrated longitudinal fields (consistent with observations) to coexist. Presumably the field cannot be very old, otherwise the small scale components would have decayed. Prior to the Cepheid stage, stars have a deep convective envelope, where dynamo action might be appealed to in order to create the field. However giant stars tend to be slow rotators, so it seems likely that any field created in this manner would have large scale order. If the field is globally ordered then permitted field strengths are much less, and magnetic effects are unimportant (Moss, 1980c). Finally it appears that the masses derived from evolutionary theory and those from the periods of the fundamental mode now have been almost reconciled by the ‘standard’ theory (Cox, 1980), whereas the outstanding problems associated with the double mode variables are not significantly lessened by the tangled field hypothesis. Stothers (1980) also suggests that a strong tangled field may be responsible for some period changes observed in RR Lyrae variables. These stars do have significant convective envelopes, but it seems that the most significant contribution must come from the deeper, convectively stable, region, where some of the same comments as made above apply. 5.3. Magnetic fields and convection The preceding section and section 3.4 were concerned with the role of magnetic fields on instabilities with a global scale. Convection is a small scale dynamical instability of considerable importance for stars. The interaction of convective motions and magnetic fields poses a number of interesting problems. Classical work in the Boussinesq (‘almost incompressible’) approximation is described in Chandrasekhar (1961). The introduction of compressibility widens the range of effects, even in the perfectly conducting approximation — for example a rising fluid element will expand horizontally, and this will be resisted by a magnetic field which has a vertical component. The problem becomes richer still when finite conductivity is included. Idealized studies in which a large scale magnetic field threads a convectively unstable layer (plane or spherical shell) sandwiched between convectively stable regions have been made (with the assumption of perfect conductivity) by Gough and Tayler (1966), Kovetz (1967), Moss and Tayler (1969, 1970); and from a different viewpoint by Moss (1978). For illustration consider a uniform field permeating a star with a relatively thin convective shell, fig. 5.2. It seems that a field with energy density a

D.

Moss, Magnetic fields in stars

65

A, ~

/

~

F\

/ 7’s~-~1~\

/ /‘7 l’-.I fc4

II \~N \I \I\

S S

I I

/ F I~

\

I

lid I id! I Ill Ill I J(.Y \cC’~-_J~...~’~A /I

P1

II

\d.\ \ \

B A

Fig. 5.2. Convection occurs in the shaded spherical shell. On a field line such as AN a modest field strength gives local stability. On lines such as BB’ a satisfactory criterion for stability cannot be found.

small fraction of the thermal energy density gives stabilization on field lines near the axis (e.g. AA’, fig. 5.2). However on field lines that are nearly horizontal for a significant distance in the potentially unstable region, e.g. BB’, criteria for stabilization cannot be found. It appears unlikely that a magnetic field can completely stabilize a convective shell of finite thickness. Locally the convective transport of energy can probably be significantly altered (sunspots would seem to be an obvious example). The strong fields present in the outer layers of the magnetic CP stars may reduce the instability of the small subsurface convection zones and so contribute to the stabilization of the surface regions that is needed for diffusion theories of the anomalous abundances. A small convective core, such as found in a main sequence star a little more massive than the sun, plausibly could be stabilized by a field whose energy density is a fraction of the local thermal energy density. Note however that an equipartition field would be of order 2 x i09 G, so this still would imply very substantial fields, whose continuation probably would dominate the thermal pressure in the outer layers of the star. In any case such minor core inhibition appears unlikely to have any significant effects on stellar properties, at least near the main sequence. This discussion has been about fields which are strong enough to affect the convective motions. The other extreme that is often relevant is when the magnetic field is so weak that the motions are inexorable, reducing the MHD equation (2.3) to a linear kinematic equation for B with prescribed v. This problem was discussed in section 2.1. Sunspots are often referred to as an example of the interaction of strong fields and convection. This is undoubtedly true, but the exact manner of the interaction is still debated. The most obvious idea is that the field inhibits the convective transfer of heat (Biermann, 1941; and many later authors), the surplus heat being redistributed through the deep part of the convection zone so there is no appreciable bright ‘ring’ around the spot. Spruit (1983) gives a review. An alternative theory (e.g. Parker, 1974a, b; Roberts, 1976) is that convective overstability converts the heat flux into Alfvén waves which predominantly transmit energy downwards. Alfvén waves, together with fast and slow mode waves, are also appealed to explain the heating of the chromosphere and corona, see, e.g., Ulmscheider and Stein (1982). To summarize, the interaction of magnetic fields and convection appears unlikely to have any large scale, global, consequences for the structure and evolution of the great majority of stars, but may well give rise to important and interesting local phenomena.

66

Lb. Moss, Magnetic fields in stars

5.4. Magnetic fields and star formation The topic of this review is magnetic fields in stars. The discussion of section 3 will have made it obvious that the pre-stellar history of the material may be very relevant to the magnetic properties of the resulting star. Moreover magnetic fields almost certainly have an important role to play in star formation processes, even if they do not survive to the main sequence stage. A full discussion of star formation in the presence of magnetic fields (and necessarily including the effects of rotation), would need as much space as has been used already. Reviews of the topic have been given by Mestel (1965, 1977), Mouschovias (1983), and in Mestel and Paris (1984) where references to the recent literature can be found. A few remarks will be made here, in an atttempt to illustrate the salient points of the problem. The estimates given in section 3.3.4 show that a naive assumption of flux freezing together with isotropic contraction from an interstellar density p1 10 24 gem to a mean stellar density ~ — I gem “-

would result in a very large magnetic energy, far in excess of the gravitational binding energy, and so could not occur. Clearly magnetic fields are strong enough to modify the star formation process. More careful analysis gives a condition for isotropic contraction in three dimensions to occur (i.e., perpendicular to, as well as along, the magnetic field), 2> M

-~-~-

3irG

F~=M~,

(5.1)

where F is the total flux, and K a numerical factor of order unity which depends on the cloud and field structure. Note that once contraction starts it will continue since it is the invariant flux that appears in (5.1). If a mass M forms isotropically from a standard galactic background with p 3, 0 —by1024 gcm B 8M~,.The situation can be alleviated contraction 1, —3 x 10~6, then relation M ~enhancing 10 occurring preferentially parallel(5.1) to therequires field, thus the mean density (and so mass within a given radius) whilst leaving the field strength essentially unchanged. As M approaches or exceeds M~, contraction across the field becomes more important. Detailed numerical calculations of this sort of process have been produced by Mouschovias (1976a, b) and Scott and Black (1980), amongst others. The relation B ~ p213 is replaced by Bxp~,

K~.

It is possible to argue that, with a suitable degree of anisotropy, stars with comparatively modest fields can form without any flux loss. However the accumulation length parallel to the field is then long. For example, consider a cylinder of length L 11, radius r11, density p11 and field B11 contracting to form a spherical star of radius R and mean density Flux and mass conservation combine to give ~.

(5.2) 5 G (i.e., of the order of the interior fields that might be With -— 1024, B0 = Ap 3 x 106 3 xcm, i0 then L found ~/p0 in the magnetic stars),G,RB= =lOll 5 pc, an unrealistically large figure (e.g. 11 —3 x i0together with the flux freezing condition Mestel and Paris, 1984). Note that conditions (5.1) and (5.2), for contraction perpendicular to the field, BR2 = constant, can be used to estimate the minimum accumulation length for contraction as L~= ~(B0Ip11)R,

Lmin

-—

B

2). 0!(p0G’

D. Moss, Magnetic fields in stars

67

This estimate does not contain a scale perpendicular to B0, and so Lmjn is independent of mass. For 8M® for isotropic contraction. Putting p example, putting R = Lmjn gives the previous estimate of 10 0= 3, B 3 pc, which is just plausible, if still rather uncomfortably 1024 g cm 0 = 3 X 1O’~ G, gives Lmjn -— i0 large. But in this case the resulting mean magnetic field in a resulting stellar object of typical main sequence dimensions would be of order i07 or 108 G. Automatically, such objects are strongly magnetic, with magnetic energy comparable to the gravitational energy. The problem is now to explain why no stars are observed to be strongly magnetic in this sense. In fact, current ideas abut field strengths in the interiors of even the apparently strongly magnetic stars (section 3) make it unlikely that B is significantly larger than the figure of order i05 or 106 G used above. For the majority of stars, which are not observably magnetic, there is no evidence that B is anywhere near this figure. Clearly the approximation of flux freezing must break down at some time. For population I stars this might plausibly be by ambipolar diffusion (drifting of ions and electrons with respect to neutrals) during early stages of the star formation process (Mestel and Spitzer, 1956; and rough estimates made in section 3.3.4.). However for the generally older population II stars it may be necessary to appeal to flux loss in the relatively dense, opaque, proto-star phase, and some of the instabilities discussed in section 3.4 may then be important. This mechanism may still be relevant to population I stars. Recent work is by Norman and Heyvaerts (1985). Note that Levine (1974), Lynden-Bell (1977) and Gershberg (1982) have suggested that the T Tauri phenomena may be connected with the destruction of accumulated magnetic energy (see also Appenzeller and Dearborn, 1984). All the discussion so far has ignored the effects of rotation. If M M~then collapse will occur at approximately the free fall rate until angular momentum conservation increases the angular velocity so much that contraction can be halted. Subsequent evolution will be controlled by the rate at which angular momentum can be lost, by the agency of the magnetic field. If M M~effective magnetic braking will occur, and the evolution will be controlled by the rate of flux loss. If M — M~,then detailed analysis is necessary. The exact geometry (e.g. rotation axis aligned with or inclined to the magnetic axis, e.g. Mestel, 1965) will influence the subsequent evolution. In particular field lines can be expected to experience severe distortion locally, and in these regions flux freezing will break down. If, for example, field lines in the contracting cloud largely can disconnect substantially from the background field at large distances, then angular momentum loss may be severely inhibited. It should be clear by now that magnetic fields play an important role in star formation, which still has to be fully elucidated. However there is nothing in the above discussion (or referenced papers) which gives obvious reasons why consideration of the pre-stellar phases should explain why some middle main sequence stars should be slow rotators with strong fields, and others should be rapid rotators without detectable fields. The possibility that the amount of flux loss is substantial, but not uniformly distributed, has been mentioned earlier, but this could occur in the opaque pre-main sequence stage of evolution when the object is recognizably a ‘star’. ~‘

‘~

6. Summary This review has gone into some detail about the two major groups of stars which display evidence for the widespread possession of magnetic fields. In the chemically peculiar stars of the middle main sequence the origin of the field and correspondingly, the interpretation of the observations, is a matter of active debate between the protagonists of the fossil and contemporary dynamo theories. The problem is intimately connected with details of the process of star formation, together with unsolved problems

68

U. ‘etoss, Magnetic fields in stars

concerning the interaction of large scale fields and strong turbulence. In the last few years it has become increasingly clear that the inclusion of time-dependent effects in theoretical studies is crucial. Such work is in its infancy. An understanding of the fields in the CP stars may also contribute to ideas about magnetism in white dwarfs and neutron stars. Conversely, an understanding of the fields in these objects conceivably could throw light on the fossil/dynamo controversy. For example, if most white dwarfs do not possess significant magnetic flux, this could be interpreted as evidence for the fossil theory, in a form which only stars retaining significant flux appear as magnetic middle/upper main sequence stars. By contrast, there is almost universal agreement that the fields seen (or more often inferred) in the cooler stars of the lower main sequence are of dynamo origin. However a complete working model of even the solar dynamo still has not been demonstrated, although it is widely felt that the principles are understood. A full theory will, of course, not only explain the solar cycle, but also the variation in magnetic activity with spectral type and rotation speed. Again, a comprehensive theory seems to be rather remote. Apart from the effects of magnetic fields in main sequence stars, most attention has been paid to degenerate objects which represent the final stages of stellar evolution. The intermediate evolutionary phases, perhaps sweeping out spectacular loops in the HR diagram, which have attracted much of the attention of workers in stellar evolution over the last twenty years, have hardly been mentioned. Apart from the instance of Cepheid variables, there is little direct evidence for the presence of magnetic fields in objects other than the stars in the main sequence band or degenerate stars. Nevertheless, this is no reason why quite strong fields should not be present during the intermediate stages. If a star does retain a coherent interior field, one important consequence could be that uniform rotation is enforced as the star evolves. This hypothesis may be testable against the proposition that angular momentum is conserved in shells, but the subject cannot be fully discussed without going into the current ideas about angular momentum transport and redistribution, see for example the discussion in Moss and Smith (1981). Hubbard and Dearborn (1981) speculate on the possibility of strong internal fields suppressing convection in the central regions as a star evolves. They point out that the evolutionary track could be distorted, but such strong hidden fields may find it difficult to survive the kinds of instabilities discussed in section 3.4. The implied presence of very strong magnetic fields in those supernova remnants which appear as pulsars has led to a number of attempts to incorporate strong magnetic fields into models of the supernova explosion, especially of type II. Even relatively weak rotation and magnetic fields may be considerably amplified by angular momentum and flux conservation, especially in the final stages of dynamical core collapse. An attraction is that the presence of a dynamically important field may help to eject substantial amounts of matter into the interstellar medium. The physics is still very uncertain and detailed calculations encounter numerical difficulties, so it is difficult to know whether this is really a relevant mechanism. One of the more recent studies, by Symbalisty (1984), draws a negative conclusion. If strong magnetic fields are an integral part of the supernova process it might be expected naively that all neutron stars should be born with very strong fields it is not clear that this is true. Of course it could just be that when a strong field is present the course of the event is different to when it is not. Other references include Le Blanc and Wilson (1970), Levy and Rose (1976), Kundt (1976), Ardelyan et al. (1979). This review has sought to demonstrate the wide range of effects that are caused by stellar magnetlc fields, even though to a first approximation they might be dismissed as unimportant because they are, almost without exception, dynamically weak. (Recall that even a 1012 G field in a pulsar has a ratio of magnetic to gravitational energies A~— 10~). A considerable amount of progress has been made towards elucidating the interaction between magnetic fields and their “host/parent” stars, but theorists in the field need have no fear of lack of problems for the foreseeable future! —

D. Moss, Magnetic fields in stars

69

Acknowledgements The author thanks G. Chanmugam, J. Landstreet, L. Mestel and 0. Vilhu for their comments on a preliminary version of this paper.

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