Magnetic Fields at the Surfaces of Stars

THE The Astron AstrophysRev (1992) 4:35-77 ASTRONOMY AND ASTROPHYSICS REVIEW Springer-Verlag 1992

Magnetic fields at the surfaces of stars

J.D. Landstreet Department of Astronomy, University of Western Ontario, London, Ontario, Canada N6A 3K7 Observatoire Midi-Pyr6n6es, 14, ave. Edouard-Belin, F-31400 Toulouse, France

Summary. Magnetic fields have now been detected in stars in several parts of the Hertzsprung-Russell diagram. Roughly dipolar fields ranging in strength between 3 • 102 and 3 x 104 G are found in many chemically peculiar A and B main sequence stars. Dipolar fields are also found in some 2-3% of white dwarfs, but with strengths between 1 • 106 and 5 • 108 G. In both these types of stars, the observed fields vary as the underlying star turns, but do not change in a secular manner. In solar- type stars, structurally complex fields of a few kG are found with filling factors of the order of 0.1 to 0.8. Further indirect evidence of fields in cool main sequence stars is provided by detection of visible and ultraviolet line emission (chromospheric activity), x radiation (coronal matter), and giant starspots. In this review, we survey the observations of stellar magnetism in all these types of stars, as well as efforts to model the observed magnetic fields and associated photospheric peculiarities and activity.

Key words: Magnetic field - Stars: activity of- Stars: magnetic field - Stars: peculiar A - Stars: white dwarf

1. Introduction

Magnetic fields are now known to occur in a wide variety of types of stars. The first detection of a stellar field occurred nearly a century ago, when Hale (1908) observed the magnetic splitting of spectral lines in sunspots. The first evidence for a field in a star other than our sun was reported more than forty years ago, when Babcock (1947) discovered a large and variable magnetic field in the star 78 Vir. In 1967, pulsars (and thus neutron stars) were detected for the first time by virtue of radio emission produced in magnetic fields of the order of 1012 G (Pacini 1967; Hewish et al 1967; Gold 1968). Shortly afterwards, Kemp et al (1970) reported evidence for a large magnetic field (since estimated to have a strength of the order of 250 MG) in the white dwarf Grw +70 ~ 8247. A decade later, the circle was closed with the discovery by Robinson et al (1980) of magnetic fields in the cool main sequence stars Boo A and 70 Oph A, stars very much like the sun. Each of these discoveries has opened up a new area of research. The magnetic field of the sun has been studied extensively (see for example Stenflo's 1989 article inaugurating this journal). It is clear that the solar magnetic field is very complex. It seems to be concentrated into small flux tubes which emerge from the solar pho- 36 J.D. Landstreet tosphere in sunspots and in boundary regions between several granules. Regions of opposite polarity occur in close proximity to one another. The structure of the surface field changes on a timescale of days and perhaps even more quickly. The magnetic field is involved in almost all aspects of solar activity. It maintains the low temperature and structure of sunspots, may provide the energy source for flares, defines the structure of filaments and prominences, probably contributes to heating the chromosphere and corona, and controls the outflow of the solar wind. Our knowledge of the magnetic fields of other cool stars similar to the sun is still rather fragmentary, but it appears that all of these aspects of the solar magnetic field have analogues on other stars. We find direct evidence of kilogauss fields on a number of cool main sequence stars stars, differing from that of the sun primarily in having far greater filling factors. These magnetic stars (and others in which fields have not been detected directly) typically have active chromospheres and coronae. They sometimes exhibit giant spots, and the coolest main sequence stars (M dwarfs) frequently show flares. The magnetic fields of upper main sequence stars appear to be much simpler in structure than those of the lower main sequence. These fields, typically also of kilogauss strength, are roughly dipolar in global structure. The field distribution on a given star does not appear to change on an observable time-scale, although the measured field strength usually varies due to rotation of the underlying star. The fields are invariably found in stars whose photospheres display highly anomalous chemical abundances compared to the sun. In addition, it appears that in many of these stars the abundances of some elements are very non-uniform over the surface. As such a star rotates, we see (in some cases quite dramatic) line profile and line strength variations, and usually also small variations in brightness and colour. The observed rotation periods are generally longer than those of normal A and B stars, ranging from half a day up to many years. The magnetic white dwarfs have fields of between 1 and 500 MG. In structure, the white dwarf fields are similar to those of the upper main sequence magnetic fields, with roughly dipolar distributions. About a quarter of the known magnetic white dwarfs rotate, with periods of hours or days. A number of magnetic white dwarfs have also been found in close binary systems in which magnetically channelled mass transfer is taking place. This review presents a summary of our current observational knowledge of the fields of single (or if double, non-interacting) magnetic stars, and of efforts to understand and model the available data. It thus focusses on the directly observable surface magnetic fields detected either through the Zeeman effect or through some kind of magnetically generated activity. It does not attempt to describe in any serious way the theoretical studies of the internal structure of magnetic stars or of the generation and evolution of their magnetic fields. All the classes of stars in which magnetic fields are directly detected are discussed, with the exception of magnetic neutron stars (pulsars). The recent review of Srinivasan (1989) makes it unnecessary to include a discussion of pulsars here; in any case, the methods of observation and physical problems connected with these stars do not have a lot in common with studies of other magnetic stars. The magnetic field of the sun is recalled only rather briefly, as the excellent recent review of Stenflo (1989) makes a lengthly review here unnecessary, but the sun will of course receive a certain attention as a more-or-less typical solar-type star. The review is not intended mainly to explain to the small group of specialists in stellar magnetic fields, who know many parts of the subject better than the author, the latest developments in their own fields. Instead, it is intended to review for such Magnetic fields at the surfaces of stars 37

specialists related areas of research with which they may not be so familiar, with some emphasis on the similarities of tactics and methods in different specialities. The review is equally intended to introduce the topic in an intelligible and comprehensive way to astrophysicists whose research areas overlap this field to some extent, and to others who may be simply curious. As a result, many simple but fundamental points are discussed in detail, while other interesting but more advanced or speculative ideas are mentioned more briefly. Although many references will be included, no effort has been made to cite all the significant work in this rather large field. Instead, the intent is to provide enough references to help the interested reader to explore more deeply any topics of special interest. The paper will summarize briefly in Section 2 the methods by which magnetic fields in stars are detected and measured. In Section 3, the observations of magnetic fields and their interpretation in upper main sequence stars will be discussed; magnetic white dwarfs will be surveyed in Section 4; and finally in Section 5 we will review the fields of solar-type and other cool stars.

2. The Zeeman effect and the measurement of magnetic fields

2.1 Physics of the Zeeman effect

We start with a discussion of the effects by which the presence of a magnetic field is detected and measured, as an understanding of the underlying physics is very useful in appreciating the significance and limitations of the observations. A readable (though not modem) discussion of the interaction of an atom with a not-to-large magnetic field is found in Condon and Shortley (1951), especially in chapter XVI. An excellent recent review of both the Zeeman effect in atoms and of measurements of magnetic fields in main sequence stars has been prepared by Mathys (1989). For information on the physics of atoms in megagauss fields, the reader may consult the reviews of Garstang (1977, 1982). Consider a many-electron atom placed in a magnetic field of strength B which is small enough to alter the energy E~ of some level by considerably less than the spacing between that level and its nearest neighbors. The energy level, of total angular momentum quantum number J, splits in general into 2J 4- 1 states (sometimes called magnetic substates) of energy

E = Ei + 9MheB/47vmc, (1)

where M is the magnetic quantum number of one of the magnetic substates, (-J < M < J), h is Planck's constant, e is the charge of the electron and m is its mass, c is the speed of light, and 9, the Land6 factor, is a number of order 1 which depends on the quantum numbers of the level. If the state is described by LS coupling with total orbital and spin angular momentum quantum numbers L and S, then g is given by 9 = 1 + [J(J + 1) - L(L + 1) 4- S(S 4- 1)]/[2J(J 4- 1)]. (2)

Transitions between a level Ei and another leveler are characterized by a single frequency, u,Zfo = (Ef - Ei)/h, in the absence of a magnetic field. When a field is applied, the single spectral line splits into a number of closely spaced frequencies 38 J.D. Lands~eet

l/if = ~ifo + (gfMf - giMi)(eB/47cmc) = Uifo + [(9f - 9i)Mf + 9iAM](eB/4~mc), (3) where AM = Mf - M~. Dipole transitions between the levels E~ and Ef obey the selection rule AM = 0,-1, or +1, and the resulting spectral lines form three coiTesponding groups. The lines due to transitions with AM = 0 (~ components) are distributed symmetrically around the unsplit line formed in the absence of a field. The two groups of lines formed by transitions with AM = • (~ components) are shifted systematically to frequencies above and below uifo, with transitions of AM = 1 on one side and AM = - 1 on the other side of the unsplit line. In general, each of the and ~ groups has several lines. Examples of the splitting of particular transitions, and the rules for calculating the relative oscillator strengths of the Zeeman components in LS coupling, are found in Condon and Shortley (1951, Chapter XVI). From equation (3), the characteristic frequency spacing between 7r and ~ components is eB/47rmc, which corresponds to a wavelength spacing of A),N = eBAZ/4~rmc 2. However, considerable variation in spacing occurs from one atomic transition to another. A common way of characterizing the spacing for a particular transition is by means of the effective Lands factor ~ (or z value) of the transition; this useful quantity is the intensity-weighted mean displacement of the ~ components from the centroid of the 7r components, in units of AAN. The values of ~ found in practice are typically about 1.2, but can range from 0 (e.g., between two levels each with g = 0; Landstreet 1969), up to 3.0 in a few rare cases. The average wavelength displacement of a a component from its zero field wavelength is thus usually written AAB(A) = ~eBA2 /47rmc 2 = 4.67 • 10-2~B(kG)A(p) 2. (4)

The ~r and cr groups of lines also have quite specific polarization properties. If the magnetic field is transverse to the line of sight of the observer, the 7r components (in emission) are linearly polarized parallel to the applied field, while the ~ components are linearly polarized perpendicular to the field. If the magnetic field is parallel to the line of sight, the 7r components are not visible, and the two sets of ~ components have opposite circular polarizations. With increasing field strength, the effects of the magnetic field change somewhat. When the energy of the magnetic perturbation becomes larger than the fine-structure (spin-orbit) splitting (the energy separation between levels of a term), the magnetic splitting simplifies to normal Zeeman triplets (the Paschen-Back effect). At still larger fields, especially for large values of the principal quantum number nf of the upper level, a term in the Hamiltonian of the system that is quadratic in the field strength B, and hence negligible for small field strengths, begins to be important. This term has the effect of shifting lines of large nf systematically in frequency relative to lines of smaller nf. This effect is larger than the splitting of the normal Zeeman or Paschen-Back effect for fields of the order of ]06 G for the higher lines of the Balmer series of hydrogen. For fields much larger than 107 G, the calculation of the atomic line spectrum even for hydrogen becomes extremely complicated, as the energy of the magnetic interaction is comparable to the Coulomb energy of the central field of the nucleus. Calculations of the energies of the bound states of H, and of the transition probabilities for transitions among them, have recently been reported for fields in excess of 10 s G by several groups. An overview of the results may be found in Henry and O'Connell (1985) and recent references in West (1989). Magnetic fields at the surfaces of stars 39

At fields of the order of 106 G or more, a different kind of effect occurs that is also useful for detection (and potentially for measurement) of magnetic fields. This is the occurrence of polarization of the continuum radiation from a stellar atmosphere. The importance of this effect was first pointed out by Kemp (1970). The effect most often detected is circular polarization, but linear polarization is measured in a number of stars as well. The physical origin of this polarization lies in the fact that the magnetic field imposes preferred motions on electrons. If a field is aligned along the line of sight, the electrons all tend to spiral about magnetic field in the same sense. Perpendicular to the field, the observer sees electrons oscillating back and forth across the field lines. These motions imposed by the field have the consequence that absorption and emission of radiation are different for different polarizations. Thus, along the field lines, the absorption and emission of circularly polarized photons of the same handedness as the spiral electron motion is different than absorption and emission is photons of the opposite handedness. The result is that the continuum radiation is circularly polarized, by an amount of the order of

VII ~ UL/Z; = eB/4~mcu = e/3A/47rmc 2, (5)

where V/I is the fractional circular polarization of the emitted light, u is the observed frequency, and UL is the Larmor frequency e/3/47rmc. The expected polarization of visible light is of the order of 0.1% for a magnetic field of 1 MG, and increases with /3. References to calculations of this effect may be found in West (1989). In a stellar atmosphere permeated by a field, the effects of radiative transfer mod- ify somewhat the line profiles and particularly the polarization properties of atomic absorption and emission. This occurs especially as a result of saturation in spectral lines, but transfer effects are also substantial in the continuum polarization of white dwarfs. This is an important topic, but our brief review of it will be deferred until Section 3.5.

2.2 Measurement of the mean magnetic field modulus

The most direct and easily interpreted means of detecting a magnetic field in a star is by the observation of Zeeman splitting of spectral lines. From equation (4), if the separation of the 7r and cr components of a line of known 9i and 9f can be measured, the magnitude of the field strength < 1/31 >, averaged over the visible hemisphere of the star (often called the mean surface field or mean field modulus, B~) can be determined. However, in main sequence stars, the detection of Zeeman splitting is not easy. From equation (4) it is found that the 7r - cr separation of a triplet line of ~ = 1.5 is about 0.18 ~ at 5000 A for a field of 1 • 10 4 a. However, this broadening must compete with other sources of line broadening, the most important of which is usually rotation. For the Zeeman splitting of even such a large field to be detectable, the rotational broadening (of the order of A,~v = )w sin i/c = 0.17 A for v sini = 10km s -1 at 5000 ~) must be quite small. In the visible spectral region, the Zeeman splitting is only easily detected if/3 (in kG) is numerically larger than v sini (in km s 1). On the upper main sequence, where v sin/ values of 30 km s-1 are common even in the slowly rotating magnetic Ap and Bp stars, and fields of more than two or three kG are quite unusual, this condition is rarely fulfilled. Nevertheless, resolved Zeeman patterns are detectable in a few stars (Babcock 1960; Landstreet et al 1989; Mathys 1990; Mathys and Lanz 1992). Smaller magnetic fields 40 J.D. Landstreet

may still produce resolved Zeeman patterns if one observes in the infrared, exploiting the fact that AAB increases as A2 while AAv goes only as A (Saar and Linsky 1985; Livingston 1991). Even if the projected rotational velocity of the star is very small, a lower limit to the line broadening is set by the thermal width of the lines, which is given by A/~th ~'~ A(3kT/AmH)I/2/c, where A is the atomic weight of the ion and m/~ is the mass of the proton. The thermal velocity of iron peak ions is of the order of 2 km s 1 in the atmosphere of a typical main sequence star, corresponding to a minimal line broadening around 5000 A of about 0.035 A. Thus fields of less than about 1 kG have almost no observable effect on spectral line profiles in the visible. Above 1 kG, however, a significant magnetic broadening of line profiles may be detected in many stars in which the field is not large enough to produce resolved Zeeman patterns, and may be used to estimate the magnitude of the mean field modulus in some very sharp-lined magnetic Ap stars (Preston 1971b). It is also a powerful method for direct detection of magnetic fields in lower main sequence stars (Robinson et al 1980). Because one rarely knows a priori the non-magnetic profile of a spectral line exactly (rotational broadening and various turbulent motions may contribute), measurement of the mean field modulus B~ in stars without resolved lines is always done by comparing spectral lines with large and small values of ~. Measurement of the excess broadening of the sensitive lines gives an estimate of the field strength. If only some regions of the visible hemisphere are magnetic, as seems to be generally the case in solar-type stars, detailed modelling of the profiles of the most strongly broadened lines yields both an estimate of the mean field modulus B~ and of the fraction of the visible surface covered with magnetic flux. This is possible because the cr components in the observed line wings are produced only in the magnetic regions; an estimate of their separation from the central line core furnishes a measurement of the field strength, while the depth of the c~ components relative to that of the central (~r plus non-magnetic) component depends on the fraction of the visible surface covered with magnetic field lines. A more detailed discussion of the actual method of measurement of fields in the limit of Zeeman splitting not large compared to the field-free line width may be found in Mathys (1989, Sec. 3.2). Magnetic field searches in white dwarfs may also be carried out by looking for Zeeman splitting, but the usefulness of this method depends strongly on the absorption spectrum of the white dwarf studied. Almost all white dwarfs have optical spectra that are remarkably poor in absorption lines compared to lower-gravity stars of similar temperature. The majority (DA stars) show only (enormously Stark-broadened) spectral lines of H, reflecting atmospheric compositions of nearly pure hydrogen. This peculiar surface chemistry is probably produced by gravitational settling by diffusion of all heavier elements to levels below the observable photosphere. Another large group, the DB stars, shows only He lines; these are probably white dwarfs in which no surface H remains as a result of evolution processes that are not yet clear, and in which heavier elements have diffused downwards out of the photosphere. Many cool white dwarfs are almost without spectral lines (DC stars). These may be mainly stars with He-rich atmospheres in with the temperature is too low to populate the lower levels of the optically observable He absorption lines. A number of cooler stars are also known which show lines of C2 (DQ stars), or of one or two metals such as Ca, Mg, or Fe (DZ stars). These various spectral classes probably reflect competition among several physical processes such as gravitational settling of heavy elements, Magnetic fields at the surfaces of stars 41 convective or turbulent mixing, and accretion from or mass loss to the interstellar medium (Fontaine and Wesemael 1991). In magnetic white dwarfs, the Zeeman splitting of the visible spectral lines is usually larger than the broadening of at least the line cores, and in stars in which spectral lines are detectable the mean surface field can often be measured. The main problems connected with estimates of mean field modulus on white dwarfs have come from two directions. First, not all white dwarfs show any detectable spectral lines. This is the situation for DC stars, but it may also occur because the field has effectively "shredded" any weak lines present to the point of non-detectability. Secondly, the expected line patterns in fields of over t00 MG have only recently been calculated in detail even for H. For He and heavier elements the situation is still quite unsatisfactory. Thus one cannot always interpret the observed spectra even qualitatively in a reasonable manner.

2.3 Measurement of the mean longitudinal (and transverse)field, components

The polarization properties of a Zeeman split line furnish a second major means of measuring magnetic fields, particularly in upper main sequence stars in which the simple structure of the field leads to a non-zero component of the mean field along the line of sight. If we observe a spectral line through a polarizer that passes only one handedness of circular polarization, the observed absorption is produced primarily by one group of cr components, which are slightly displaced from the position of the line with no field present. Thus, comparing spectra taken through oppositely polarized circular polarizers, the measured wavelengths of a given spectral line are slightly different. The difference is proportional to a suitably weighted mean line-of-sight component of the magnetic field , usually known as the effective field Be (when/3z is weighted by limb darkening), or the mean longitudinal field Be (when /3z is weighted as cos 0, where 0 is the angle between the line of sight and the local surface normal). The result that the wavelength difference between the position of a line measured in right and left circular polarization is approximately proportional to the mean longitudinal field Be is demonstrated e.g. by Mathys (1989, Sec. 4.1). Four features make this a very useful method of field measurement. First, the effect may be detected even when the difference in mean wavelength of the line as seen through the two polarizers is quite a bit smaller than the width of the line, making it a rather sensitive measure of the presence of field. Secondly, the measurement requires almost no modelling of the line; in particular, one does not need to know the intrinsic non-magnetic profile of the line. Thirdly, no effect other than a magnetic field leads to this kind of wavelength difference between line positions measured in right and left circularly polarized light; and if any doubt remains, one can check to confirm that the observed separations are proportional to ~ when a number of lines are measured. Finally, many upper main sequence stars do in fact have fields that are sufficiently simple in global structure that they have non-zero mean longitudinal fields. Measuring the shift of metal lines as observed in right and left circularly polarized light has some significant limitations as a method of field measurement, however. The limiting field which can be detected with this method depends strongly on the amount by which the line is broadened by other mechanisms, particularly rotation. In practice, it is difficult to make field measurements in stars for which v sin i is much greater than about 30 km s -1, although some magnetic A and B stars have v sini values of 100 42 J.D. Landstreet

km s-1 or even more. Furthermore, the measurements are made primarily using lines of the iron peak elements, as most of the strong lines in the visible spectrum of a star are due to these elements. However, in any given magnetic A or B star, the elements producing these lines may be distributed in a highly non-uniform manner over the surface of the star. This means that the field is not necessarily sampled uniformly over the observed stellar disk, and inevitably makes interpretation of measurements as constraints on actual field structure more difficult. To avoid some of these problems, an almost equivalent method of field measurement, also based on the polarization properties of the ~ components of a spectral line, was developed by Angel and Landstreet (1970a). In this technique, the polarization due to the separation of the ~7 components in the presence of a non-zero mean line- of-sight field is observed in one of the Balmer lines of hydrogen, usually H/3. The wavelength difference of the two groups of cr components is detected by measuring the circular polarization in each of the wings of the Balmer line, which are isolated using narrow band interference filters. Circular polarization is present because the mean or central wavelength of the line is slightly different in the two senses of circular polarization, so that at any given wavelength in the line wing, the line is not quite as deep in one sense of circular polarization as in the other. The effect can be quite small in a Balmer line, because of the large width of the line produced by Stark broadening (a mean longitudinal field of 100 G produces a circular polarization of the order of 7 x 10-3%), but because the measurement is completely differential, fields of 100 G or less can be detected in bright stars. (A derivation of the result that the observed line wing polarization is approximately proportional to the mean longitudinal field may be found in Landstreet 1982 or Mathys 1989, Sec. 4.1.) Field measurements with such a Balmer-line Zeeman analyser have several interesting advantages. The most important comes from the fact that the Balmer lines are intrinsically several A wide. This means that field measurements may be made with an accuracy which is almost independent of v sin i, even for v sin i values of 2-300 km s -1. A second desirable feature of this technique is that almost no data reduction is needed, because the actual measurement is simply of the circular polarization at one point on each side of the centre of a Balmer line, together with a scan of the flux profile of the line. A third interesting feature is that the measurements are made using lines of H, which (because of its very large cosmic abundance) is the one element not expected to have significant abundance fluctuations over the stellar surface. The field measured is therefore sampled in a more-or-less uniform manner over the visible hemisphere. An important limitation of the Balmer-line Zeeman analyser is that it is not really clear exactly what is measured by this method. The difficulty is that the quantum- mechanical problem of the behaviour of the Balmer lines in the presence of both strong Stark and Zeeman effects has never been adequately solved, even numerically. In particular, the magnetic quantum number M ceases to be a good quantum number in hydrogen when the two effects are both important, and so the simple division of line components into cr and 7r types is no longer possible. Although the line still behaves qualitatively as expected from calculations made assuming the simple (LS coupling) theory of the Zeeman effect, it is not known how adequate this approximation is. The difficulty is discussed in more detail by Mathys (1989, Secs. 2.4.2 and 4.1.3). In a similar way, it is possible to measure the transverse component of the magnetic field using the Zeeman effect. This is done by measuring the linear polarization across a spectral line, using the fact that the linear polarization of the 7r component is Magnetic fields at the surfaces of stars 43 orthogonal to that of the two signa components. Because there is no net wavelength difference between the wavelength of the 7r component and that of the two o- components combined, the transverse Zeeman effect can only be detected easily in lines in which all other broadening mechanisms do not widen the line by much more than the separation between the two cr components; it is more difficult to detect in stars of weak or moderate fields than the longitudinal Zeeman effect. The transverse Zeeman effect was detected some years ago by Borra and Vaughan (1976; 1977) in the bright magnetic Ap star/3 CrB, but has not otherwise been much observed. The transverse Zeeman effect does have one very interesting (and observable) property not shared by the longitudinal Zeeman effect. Because the profile of the absorption due to the 7r components is different from the absorption profile of the cr components, the effects of saturation on the two types of components are somewhat different, and saturated lines will not in general have zero net linear polarization integrated across the line profile. (This is in constrast to the situation for the longitudinal Zeeman effect, where the symmetry of the cr components leads to null integrated circular polarization.) This saturation effect has the consequence that a spectral region with numerous saturated lines in a strongly magnetic star will show non-zero linear polarization even in broad-band light. This effect was observed in the sun many years ago, and explained by Leroy (1962). It was detected in magnetic Ap stars by Kemp and Wolstencroft (1974). Recently, Leroy et al (1991) have begun a programme to detect and study systematically this effect in magnetic A stars.

3. The rigid rotators: magnetic stars of the upper main sequence

3.1 Peculiar A and B stars

Among upper main sequence stars, fields have now been detected in well over 100 stars having spectral types that lie roughly in the range F0 to B2, corresponding to effective temperatures of about 7500 to 22,000 K. The fields of these stars have mainly been detected by using one of the methods based on the circular polarization produced by the longitudinal Zeeman effect, as described in Section 2.3. Measurements based on polarimetry of metal lines have been reported by a number of observers, particularly Babcock, Preston, and Wolff and her collaborators (see the bibliography of Didelon 1983), and Mathys (1991). Observations with the Balmer-line Zeeman analyzer technique are also catalogued by Didelon (1983); some more recent observations are given by Borra et al (1983), Thompson et al (1987), and Bohlender et al (1987). However, in a few stars, the fields have been observed mainly (or only) by means of Zeeman splitting of spectral lines in unpolarized spectra; most of the available observations are found in Babcock (1960), Preston (1971b), Mathys (1990), and Mathys and Lanz (1992). Among upper main sequence stars, some 10% have atmospheric chemical abundances that are obviously peculiar when compared to the solar pattern, and it is among these stars that magnetic fields are invariably found. The abundance peculiarities of these stars are sufficiently striking that they can usually be recognized from classification spectroscopy. Although considerable star-to-star variation occurs, these chemical peculiars have been grouped into several broad families, whose outstanding observed chemical anomalies are correlated in a rough way with effective temperature. The main families are listed in Table 1; more complete descriptions of their properties 44 J.D. Landstreet may be found in the reviews by Preston (1974) and Ryabchikova (1991), and in the monograph by Wolff (1983). The designations given in Table 1 reflect the chemical elements most obviously anomalous on classification spectrograms. However, most of these peculiar types have a number of elements for which significant abundance anomalies are found.

Table 1. Types of upper main sequence chemicallypeculiar stars Range of T~ Magnetic stars Non-magnetic stars

7-10,000 K Ap SrCrEu Am, )~ Boo 9-14,000 K Ap Si Ap HgMn 13-18,000 K He-weak Si, SrTi He-weak PGa 18-22,000 K He-strong

As indicated in Table 1, certain families of chemical peculiars are found to posess detectable fields, while other types of peculiars do not. It is generally believed that all the members of the "magnetic" families have reasonably strong fields, although there are some members of each family in which fields are not at present detectable. The presence of fields in the Ap stars of the SrCrEu and Si types was firmly established by Babcock (1947; 1958), and has been amply confirmed by subsequent measurements. The existence of fields in the He-weak stars was demonstrated by discoveries of fields in individual stars by Wolff and Morrison (1974), Wolff and Wolff (1976), and Landstreet and Borra (1976), and by the systematic survey of Borra et al (1983). The He-strong stars were established as a magnetic type from the observations of Landstreet and Borra (1978) and of Borra and Landstreet (1979); see also Bohlender et al (1987). The Am and Ap HgMn types of stars were originally reported by Babcock (1958) to have rather weak but detectable fields; later work by Conti (1969), by Borra and Landstreet (1980), and by Landstreet (1982) indicated that such fields, if actually present, must be usually less than a few hundred gauss, or complex in structure. It now appears that a weak field may finally have been found in an Am star by Mathys and Lanz (1990). The absence of detectable fields in the He-weak PGa stars was established by Borra et al (1983). The .~ Boo stars were unsuccessfully searched for fields by Bohlender and Landstreet (1990c). Finally, the absence of detectable large- scale fields (comparable to those found in the magnetic Ap stars) in normal upper main sequence stars was suggested by the work of Babcock (1958) and confirmed by Landstreet (1982). The fields observed in upper main sequence stars cover a wide range of strengths. As discussed below, most observed fields in such stars vary with time, so to indicate the typical field strength it is appropriate to consider some suitable mean field strength for each star observed. An appropriate indicative field strength is the rms field averaged over all observations, or over a period, 1/2. The distribution of 1/2 over various stellar samples is discussed by Borra and Landstreet (1980), Thompson et al (1987), and Bohlender and Landstreet (1990c). For the Ap SrCrEu, Ap Si, and the He-weak stars cooler than about T~ = 15,000 K, the median value of 1/2 is about 300 G. The observed values range from about 100 G, the smallest value that it is currently possible to detect reliably (see Bohlender and Landstreet 1990a; Donati et al 1990) up to a maximum of about 16 kG for the most strongly magnetic star known. The distribution is strongly skewed towards small field values, with a long high-field tail. Field strengths 1/2 of more than 1 kG are found Magnetic fields at the surfaces of stars 45

in only a few percent of the magnetic Ap stars. (Note, however, that because of the observational interest of such large-field stars, they occupy a disproportionate fraction of the field measurement literature.) There are still a number of fairly bright Ap SrCrEu or Si stars in which the fields are below the current threshhold of detection, which is around 200 G for stars of V = 5, decreasing to roughly 70 G at V = 2. For the hotter He-weak stars and the He-strong stars, it appears that the median value of < B 2 >1/2 may be significantly larger, perhaps around t kG (Thompson et al 1987).

3.2 Variations and the rigid rotator model

The observed longitudinal magnetic fields usually vary. These variations are invariably found (after enough observational effort) to be periodic. Most of the periods are in the range of 1-10 days, but the full range of observed periods covers an enormous span, from about 0.5 days up to more than 103 days. Very useful bibliographies of period determinations are found in the catalogues of Catalano and Renson (1984, 1988) and of Catalano et al (1991). In the great majority of cases, the value of Be varies more or less sinusoidally. The mean longitudinal field is observed to pass through zero (change sign) in about two-thirds of the magnetic variables. Typical data are found in Borra et al (1983) or Mathys (1991). In a few cases, such as HD 32633 (Renson 1984), HD 37776 (Thompson and Landstreet 1985) and HD 133880 (Landstreet 1990), it seems clear that the observed variation of Be is not sinusoidal. Other variations are also observed in the majority of these magnetic upper main sequence stars. Most are periodic light variables, with amplitudes sometimes as large as 0.1 mag (about 10%). The period of light variations is always the same as the period of variation of Be. The observed light variations can normally be fitted with sine curves using only the fundamental and first harmonic of the observed period (Mathys and Manffoid 1985). The curves of light variation in the various colours are often in phase, but with different amplitudes. Some stars are known, however, in which the variations in different colours do not occur in phase with one another. Light variations have also been observed in the ultraviolet for a small number of magnetic Ap stars. In general, these variations occur in antiphase with the visual light variations for wavelengths shorter than some particular value, of the order of 3000 A, and it seems likely for most stars that the total flux emitted is constant, while only the flux distribution with wavelength changes (Molnar 1973; Molnar and Wu 1978). Many magnetic Ap and Bp stars are also observed to be spectrum variables. In some cases, the variations are no more than subtle changes in detailed line profiles, but in a number of stars, quite spectacular changes are seen in the lines of a few elements, which may vary from great strength to near invisibility. When this occurs, lines of all visible ionization stages of any one element vary synchronously. In some spectrum variables, only a few elements are observed to vary strongly, while in others, the lines of most elements are obviously variable. The variations are again always periodic, and have the same period as the magnetic and light variations. To appreciate the nature of the variations, it may be interesting to examine the tracings, several hundred A long, of blue photographic spectra of CS Vir = HD 125248 (in which Cr and the rare earths are strikingly variable, with Cr strong when the rare earths are weak, and vice versa) published by Babcock (1951). One may also look at the detailed line profiles of 53 Cam = HD 65339 (in which Ca and Ti vary strongly in antiphase to one another), or Babcock's star = HD 215441 (in which all observed 46 J.D. Landstreet elements vary in unison), shown respectively by Landstreet (1988) and by Landstreet et al (1989). When we try to understand what physical mechanism underlies the observed variations, we have several very suggestive clues. First, this particular type of variation occurs only among the types of chemically peculiar upper main sequence stars that are known to be magnetic. It is therefore a phenomenon linked to the magnetic field. Secondly, the extremely large range of periods observed, even among stars of very similar mass and luminosity, and particularly the very long periods observed in some stars, makes it hard to believe that we are observing a pulsational phenomenon. The large differences between the degree of spectrum variation observed in different chemical elements in the same star is also very hard to understand as a pulsational effect. Finally, the v sin i values for different stars are inversely correlated with period of variation, as first noted by Deutsch (1956), who correctly argued that this shows that the magnetic variations occur with the rotation periods of the stars (see also Preston 1971a). We are thus led directly to what is known as the oblique or rigid rotator model for the variations of magnetic A and B stars. We imagine such a star to be an object in which the magnetic field is not usually symmetric about the rotation axis. In this case, the observer will see a varying field as the star rotates. Stibbs (1950) showed that if the magnetic field is dipolar, with an axis inclined (oblique) to the rotation axis, the sinusoidal variations of Be observed for most magnetic A and B stars may be quantitatively understood. The spectrum variations, in turn, are due to non-uniform distributions of some chemical elements over the stellar surface. This patchy structure is presumably connected with the magnetic field, as it is not observed in non-magnetic stars, even if they are chemically peculiar. A further hint of the connection between the field and the patches is the fact that in most spectrum variables, the maxima and minima of the line strength variations coincide approximately with the extrema of the observed longitudinal field, which suggests that patches may be located near the poles of the dipolar field. Finally, the light variations may be understood as a consequence of the variations of chemical abundance over the stellar surface. Different chemical compositions on different parts of the star lead to different detailed emergent flux distributions as a function of wavelength. Thus, the brightness as observed in any one limited passband may vary as the star rotates, although the overall luminosity and effective temperature are expected to be the same as measured in any direction (Peterson 1970). This is precisely what is observed for magnetic stars for which photometry in both ultraviolet and visible light is available. This simple model immediately allows us to obtain a number of interesting con- clusions concerning the magnetic field geometries present on the magnetic A and B stars. One direct conclusion from the general observability of non-zero longitudinal fields is that the field structure on these A and B stars is relatively simple, and normally posesses an important dipole component. A fundamentally more complex field, such as that of the sun, would be expected to have approximately equal amounts of emerging and re-entering magnetic flux in any large region, so that the mean longitudinal field would be quite small in comparison with the local field modulus B. In contrast, the ratio t3~/t3~ for A and B stars is commonly observed to have a maximum value of 0.3 or even 0.4 (Preston 1971b), about the ratio expected for a dipole field viewed pole-on (Deridder at al 1979). This does not show that the fields are simply dipolar, but does indicate that the field structure contains an important dipolar component. Magnetic fields at the surfaces of stars 47

The relatively simple structure of the field on most magnetic A and B stars is also shown by another interesting observation. Consider a spectral line which splits by the Zeeman effect into a simple doublet structure (the zr components separate by the same amount as the ~ components) as is the case for the 4Dr~ 2 - 4p1/a Fe ii line at A 6149.2 A. In stars in which the field is large enough (and the value of v sini small enough) that the line is completely resolved into two separate components, there is essentially no residual central absorption. That is, there is no important fraction of the stellar surface which is non-magnetic. These large-field magnetic A and B stars, and presumably the other magnetic A and B stars as well, do not have the kind of two- component magnetic structure that the sun does, in which some regions (sunspots, flux tubes) have strong fields, while other regions (interiors of granules) have undetectable fields. In fact, the relatively simple shape of the two components observed in this line when it is resolved in a really slow rotator suggests that the field on the observed hemisphere has a fairly limited range of strengths (Mathys 1990; Mathys and Lanz 1992). The dipolar component is clearly detected in the sinusoidal variation of Be observed in most magnetic Ap stars. Preston (1967) showed that the distribution over the known sample of magnetic stars of the ratio r = Bes/Bee, where Be8 is the smaller and Bee the larger (in absolute value) of the observed extrema B~ and B~- of Be, provides information about the typical values and distribution of the angle/3 that the magnetic axis makes with the rotation axis. Information about this angle is available because the fraction of stars in which the observed field Be reverses sign depends strongly on the typical/3 value. Thus, if most magnetic A and B stars have small/3, the observed field Be will be observed to change sign only in the small fraction of stars viewed nearly (rotational) equator-on. In contrast, if the typical value of/3 is around 90 ~ the sign of the field will reverse for almost all directions of the line of sight, and the ratio r will be close to -1.00. Thus, if we assume that over the sample of known magnetic stars, we observe at randomly distributed directions relative to the stellar rotation axis, we may use the distribution of the ratio r to infer the approximate distribution of angles /3. The facts that about two out of three magnetic stars show reversing fields, but that r is only rarely close to -1.00, turn out to imply that the angle /3 is usually large (of the order of 80~ but that a significant fraction of magnetic A and B stars have small/3 angles, 45 ~ or less. It is possible that the dipole axes are orientated randomly relative to the rotation axes in the observed sample of magnetic A and B stars (Preston 1967; Landstreet 1970; Hensberge et al 1979; Borra and Landstreet 1980). For the few stars for which both Be and B~ have been measured as a function of phase, more constraints on the field geometry are available. The most important observed feature of such stars is that in general the values of B~ observed at positive and negative extrema of Be are quite different, even when the two extrema of Be have similar absolute values. This is not what one would expect if the field structure were generally a simple dipole; in that case, the mean field modulus B, would be the same at the same magnetic latitude (angular distance from the magnetic equator) in both hemispheres, or equivalently at the same IBel value in the two magnetic hemispheres. The fact that one Be extremum has a strongly different mean field modulus B~ (typically by a factor of 1.5 or more) than the other extremum implies that the field structure is significantly different in the two magnetic hemispheres. A simple model that may be used to describe the observations is to assume that the field is approximately a dipole decentred along its axis (Landstreet 1970; 1980), 48 J.D. Landstreet

or equivalently (Deridder et al 1979), that the field has both dipole and quadrupole components. Since a quadrupole makes only a small contribution to the observed field Be because of field cancellation over the visible hemisphere (Schwarzschild 1950), a significant quadrupole component may be present which is not seen in the variations of Be, but does appear in the variations of/3s because the dipole and quadrupole add at one pole but subtract at the other. In fact, the polar field strength of the quadrupolar component must be about as large as that of the dipolar component before the variation of Be with rotational phase is significantly affected by the presence of the quadrupolar term (cf. Landstreet 1990), but a quadrupolar component with polar field strength as little as 30% of the dipolar polar field can be detected in the surface field variations. Only a few magnetic A and B stars have well-observed surface field variations (Landstreet 1980). In each case where enough data exists to determine whether a centred dipole is an adequate model of the field structure or not, it is found that the dipole model does not describe the data. Generally at least a quadrupolar component is required, with a ratio of polar field strengths Bq/B d of the order of at least 0.3. In a few cases (Preston 1970b; Thompson and Landstreet 1985; Landstreet 1990) the quadrupole component appears to be larger than the dipolar component. The significance of the failure of the simple dipole model, and the requirement that a significant quadrupolar field be included to describe the observations is that the local field strengths at the two magnetic poles are not equal: one pole is stronger than the other. This result resolves a troubling paradox that was found when spectrum variations were first related to magnetic field variations, and it was realized that the elements that are overabundant near one magnetic pole may be very different than those found near the other. If the two poles differ only in the sign of the field it is not obvious why the same elements should not form patches as both poles; but if the (absolute) field strength and structure are different at the two poles, one may reasonably suppose that the segregation process could lead to different results at the poles (Landstreet 1970).

3.3 Origin and evolution of fields in Ap and Bp stars

In addition to studying the field structure of magnetic A and B stars, the simple observations of Be may be used to provide some information about the evolution of the magnetic structure with time, provided a reasonably large sample of stars of approximately known ages is available. One could then proceed by comparing mean values or distributions of the rms field strength in samples of different ages to find out whether the field strengths change with (main sequence) age on average. The main problem with carrying through this programme is the rather small number of magnetic stars known in clusters and associations, as it is not yet clear how accurately ages may be determined in field magnetic stars from spectroscopic gravity determinations. Nevertheless, a preliminary attempt to detect large-scale, systematic fields strength changes with main sequence age was made by Thompson et al (1987), who compared rms fields in magnetic stars which are plausibly members of the very young Sco-Cen and Orion associations to rms fields of a sample of (presumably rather older) field stars. The conclusion from this comparison is that the typical fields of hot magnetic stars do not decrease by more than about a factor of two between the zero-age main sequence and the the middle of the main sequence stage. A second, and somewhat surprising, result to emerge from this study is some evidence that typical magnetic Magnetic fields at the surfaces of stars 49 field strengths do depend on stellar mass; the typical fields found for stars of spectral types earlier than about B4 (more massive than about 4-5 MG) seem to be about three times larger than the fields of cooler magnetic stars. We may also possibly obtain some information about the fundamental source of the observed fields from the available observations. Two basic field origins seem plausible. One is that the observed fields are maintained by some kind of a currently operating dynamo, perhaps somewhat like the one thought to underly the solar field (e.g. Brandenburg and Tuominen 1991). In this case, we would expect that the field should depend in some obvious way on the stellar rotation rate. This is certainly the case in lower main sequence stars, where the total observed magnetic flux at the stellar surface is correlated fairly closely with angular velocity (e.g. Saar 1991). This behaviour is also expected theoretically (Moss 1989, 1990a). However, no significant relationship between field strength or structure and rotation has been found in magnetic A and B stars (e.g. Borra and Landstreet 1980). The observed field variations suggest very similar global field geometries in stars of periods of 1 day and of 102 to 10 4 days, and large B~ or Be values are found both in stars of very short and very long rotation periods. It thus seems unlikely that the fields observed in upper main sequence stars are generated by currently acting dynamos. The other possibility is that the observed fields were produced at an earlier stage in the lives of the magnetic stars (for example by compression of the weak galactic magnetic field as a gas cloud collapses to become a star), and have simply not yet died away. That is, the observed fields may be essentially "fossil" fields. This is not as unreasonable an idea as it may appear at first glance. The fundamental physical principle underlying this possibility, which is that the natural decay time of a simple global field in a star is so long that the field may be essentially preserved through the whole main sequence lifetime of the star, was discussed by Cowling (1945) even before the discovery of magnetic fields in Ap stars. He showed that in a main sequence star, the time scale for ohmic decay is of the order of 10 l~ yr (Cowling 1976). Thus, if a star starts its main sequence life with a simple dipole field, it will take far longer than the main sequence lifetime of the star (3 x 107 - 1 x l09 yr) for the field to decrease by an important factor. It is therefore quite plausible to consider the possibility that the observed fields are basically fossil fields. The lack of secular change in the the observed fields, apart from the effects of rotation, is at least consistent with the fossil field hypothesis. So is the observation that the observed magnetic fields of a sample of presumably young magnetic Ap stars form a distribution very similar to that observed in a sample of field stars of average ages (Thompson et al 1987), as discussed above. On this hypothesis, there is no difficulty in understanding the occurrence of similar fields in both extremely slowly rotating stars and in stars rotating hundreds of times faster. Thus the fossil field hypothesis, while still far from proven correct, is the most commonly accepted explanation for the occurrence of fields in some A and B stars. It should be emphasized that even if these fields are indeed fossils, some slow secular change is to be expected. This will occur as a result of the advection of magnetic field lines by meridional circulation currents induced by stellar rotation, which may substantially distort a field on an evolutionary time scale. Secular change will also occur due to direct effects of stellar evolution, in which the stellar core shrinks and the envelope expands, producing significant changes (especially dilution) of field lines. Finally, decay of structures small compared to the size of the star as a whole, especially if they are confined to the relatively less conductive stellar envelope, 50 J.D. Landstreet can proceed on time-scales shorter than the main-sequence lifetime. These effects have been investigated numerically in a series of papers by Moss (see especially Moss 1989, 1990b). Other evolutionary effects may occur as a result, for example, of magnetic distortion of the stellar figure leading to a convergence or divergence of the axes of rotation and of the field (Mestel et al 1981). A more comprehensive discussion of some of these effects is given by Borra et al (1982).

3.4 Other phenomena observed in magnetic A and B stars

Not all the variability observed in magnetic A stars is due to rotation. Pulsational light variations were discovered by Kurtz and Wegner (1979) in one of the very coolest magnetic Ap stars; similar variations have since been found in about a dozen cool magnetic stars (Kurtz 1990). These pulsating stars are observed to vary in brightness as measured in white light, generally with an amplitude of a few millimagnitudes. The variations are periodic, sometimes with a single stable period, but more often with several related periods. The observed periods range from about 4 to 15 minutes, strongly reminiscent of the (much lower amplitude) five-minute solar oscillation. The shape of the light curve is not usually constant with time, but may be modulated in two ways. First, it is usually observed that the amplitude of the pulsations varies as the star rotates. The amplitude is largest when the line of sight approaches either of the magnetic poles, and decreases to zero as the line of sight crosses the magnetic equator, and the observed field Be goes through zero. This type of modulation indicates that the pulsations are aligned with the magnetic axis of the star. Secondly, in some of the rapidly oscillating Ap stars the individual modes of pulsation are observed to grow and then die away over time intervals not related to the star's rotation, sometimes in as little as a few days. This occurs particularly in stars which have a large number of frequencies present simultaneously. In other stars, the pulsations are stable over time intervals of a year or more. The observed pulsations have been identified as non-radial p-mode oscillations of high order (many radial nodes, and hence rather short periods) but low degree (simple surface structure, so that the pulsations are detectable in the light integrated over the disk of the star). The possible periods of oscillation depend on the internal structure of the star, as well as on the rotation and magnetic field if these are sufficiently large to perturb the envelope structure. Thus, once the nature of individual modes has been identified, the observed periods provide sensitive probes of the physical conditions inside the star. Many aspects of the pulsations of the cool magnetic Ap stars are not yet well understood. Identification of individual modes of pulsation is not at all complete. Typically, some modes in a given star can be identified in a reasonably unambiguous manner while others cannot. Furthermore, it is not yet clear what drives the pulsations. The pulsating Ap stars are within the ~ Scuti instability strip, and several variations of the n mechanism ("valve" action by an ionization zone, which alternately traps and releases outflowing energy as the star pulsates, in such a way as to use the star's luminosity to drive the pulsations) have been suggested. The oscillations may be driven by the He n ionization zone (note that, depending on the results of gravitational settling of He, this zone may be present over the whole star, over part of it, or absent altogether), or the mechanism might involve overabundant Si. Driving by magnetic overstability has also been suggested. Finally, it is not yet at all clear how the star Magnetic fields at the surfaces of stars 51

selects out of the very large range of possible pulsation modes the few which are excited to a detectable level (Kurtz 1990). A quite different and very interesting phenomenon is found in a number of the hottest magnetic B stars. Some of these stars show clear evidence of the presence of significant circumstellar matter, and perhaps for mass loss. The phenomenon is sometimes found in He-weak stars. It is common in He-strong objects, which are the hottest of the known magnetic B stars, and which (unlike any of the cooler magnetic stars) have higher He abundances in their atmospheres than normal stars. Ultraviolet spectra of these stars often show strong C IV resonance lines in absorption or emission, sometimes with a strong asymmetry towards shorter wavelengths that strongly suggests mass outflow. In at least two or three of these objects, the resonance lines are modulated by the rotation of the star. This appears to be due to an asymmetry in the circumstellar matter or wind introduced by the effect of the stellar magnetic field (Barker et al 1982; Shore at al 1987, 1990; Shore and Brown 1990). In the prototypical He-strong star cr Ori E, the circumstellar matter appears to be largely trapped in a magnetosphere which is dense enough to scatter several percent of the outflowing light when one of the magnetospheric clouds passes in front of the star. This leads to weak periodic eclipses of the star which for some years greatly confused efforts to understand the nature of ~r Ori E (Groote and Hunger 1976; Landstreet and Borra 1978; Nakajima 1985; Hunger 1986). It is also from these hottest magnetic stars that non-thermal radio emission has been detected (Drake et al 1987). The winds of the He-strong stars constitute a very interesting subject which has not been studied as intensively as it deserves.

3.5 Mapping the detailed field structure

One means of making further empirical progress towards understanding the fields of upper main sequence stars is to try to obtain detailed maps of the magnetic structure, and of the distributions over the stellar surface of various chemical elements in the same coordinate reference frame. It seems clear that we may expect to be able to do this even for the most suitable stars (i.e., ones which we observe from near the equatorial plane, so that we actually can see most of the stellar surface at some time during the rotation, and in which Be reverses, so that we see both magnetic hemispheres) only by extracting all the available information about the field from flux and polarization measurements of some well-chosen spectral lines, obtained at a number of times through the rotation cycle of the star. Such data sets may then be compared with synthetic spectra calculated for various assumed magnetic field geometries and chemical element abundance distributions to determine from modelling the field and abundance maps that are consistent with the observations. This kind of modelling has recently beome a major source of interesting new information about the magnetic Ap and Bp stars. It should be clear that the resolution of the resulting maps will depend very strongly on the quality of the available data set. As a simple example, consider the problem of mapping the distribution of one element over the stellar surface, using the information about this distribution contained in the variation of a single spectral line of the element as the star turns. If we have, say, ten spectral resolution elements across the line, and we observe the line profile at ten rotational phases, we have in effect some 100 (approximately) independent numbers with which to constrain 52 J.D. Landstreet the abundance of the element on the stellar surface. Roughly speaking, we cannot reasonably expect to determine the abundance for more than about 100 representative points on the stellar surface with such data (with an accuracy that will depend on the signal-to-noise ratio of the data, of course). This means that we will be able to divide the star into units of surface area of a characteristic diameter of 20 ~ This is not a very fine mesh; on earth, such low surface resolution would reveal little more that the general sizes and locations of the continents. This resolution limit cannot easily be evaded by observing more spectral lines, as roughly the same information is contained in each line. To improve the surface resolution of the map, observations of higher spectral and temporal resolution are needed. To determine the magnetic field structure, which means determination of three components of a vector at each representative surface point, we need still more information, most readily provided by circular and linear polarization measurements of one or several lines, again with equally good resolution and phase coverage and a signal-to-noise level adequate for the desired accuracy. The point is that detailed maps require data sets which are really extraordinary from the point of view of resolution, phase coverage and signal-to-noise ratio. Such data sets are only now becoming available for even a few stars. To obtain reliable maps, observations must be compared with synthetic spectra of adequate accuracy. If the magnetic field is weak, the local spectral line profiles will be almost the same as in the non-magnetic case. In this situation, one may assume that the local line profiles are those predicted by the solution of the usual equation of transfer, in which the local chemical abundance, and perhaps the local turbulent velocity, are the important free parameters. For strong fields, however, the local line profiles are greatly altered by the field from their non-magnetic shapes. The calculation of spectral lines in the light emerging from a stellar atmosphere permeated by a magnetic field has been studied for some years by solar physicists. The theory, assuming local thermodynamic equilibrium (LTE), was first derived in a heuristic manner by Unno (1956). It has been extended and generalized by a number of others; references are given by Mathys (1989, Section 2.1). Useful reviews of the subject are those of Stenflo (1971; a clear and complete account of the relationship of various forms of the tranfer theory for polarized light, particularly as applied to the solar situation), Rees (1987; a brief introduction in which the essentials of the theory are made accessible), and most recently Mathys (1989; a careful and comprehensive article with many recent references, in which the assumptions behind various forms of the equations are explicitly discussed). During the past few years there has been considerable interest in the theoretical problems connected with the generalization to non-LTE situations. This aspect of line profile calculations is also surveyed by Rees (1987); further discussions may be found in the following three chapters in the same book. The most straight-forward mapping problem is to infer the distributions of one or more chemical elements over the surface of a star in which the local line profiles are not much affected by the magnetic field, which requires that the stellar field be everywhere small (~ 1- 2 kG). In this case, the locations of individual patches of overabundance of a particular element are revealed fairly clearly by the increase and decrease of equivalent width of a suitable spectral line as the star turns, and by the change in shape of the profile due to the Doppler effect as each patch rotates into or out of the visible hemisphere. It is found from numerical experimentation with synthetic spectra (e.g. Piskunov 1991) that the best maps may be obtained for stars with appreciable rotational velocity (v sin i ~> 20 km s-l), so that the observed profile Magnetic fields at the surfaces of stars 53

is several times wider than the intrinsic local profile, but not so wide that most lines are badly blended or vanish into a slightly wavy continuum (v sin i ~ 50 km s-l). It is also useful for the stellar inclination to satisfy 30 ~ ~< i ~ 70 ~ so that at least part of both hemispheres is observed, and so that some discrimination between the two (rotational) hemispheres is possible. In this situation, detailed maps of abundance distributions may be deduced from observations of sufficient quality. Mapping based on this idea has developed very much in the past decade. A comprehensive and clear review of the subject was prepared a few years ago by Khokhlova (1985), one of the pioneers in the field. Maps have recently been published for a number of weak-field Ap stars (Khokhlova et al 1986; Hatzes 1988; Hatzes et al 1989; Rice and Wehlau 1991). In general, such maps reveal surprisingly complex surface distributions of the elements studied (usually the iron peak elements Ti, Cr, and Fe because of the large number of strong lines of these elements), with large abundance contrasts, as much as two dex (!), from one region to another. Some maps show rough axisymmetry about an axis inclined to the rotation axis, but this is not always found. Note that because the stars mapped have been selected to have local line profiles not much affected by the magnetic field, they are usually stars for which relatively little information (usually only the variation of Be) constraining the field geometry is available. The problem of deducing the abundance distribution of an element from observations of profile changes through the rotation cycle is quite a lot more complicated if the local line profile is strongly widened by the magnetic field, as then the contributions of regions at different Doppler shifts within the observed line profile are mixed together. However, in this situation, considerably more information about the field structure is available, and so one may hope to produce useful maps of the field geometry. Because of the interest in determining the magnetic field structure and especially its relationship to the abundance patches, efforts are now being made to obtain observational data sets adequate to define both field and abundance distributions, and to develop methods of deducing efficiently the field geometry as well as abundance distributions from the observations. To obtain both kinds of maps, clearly stars must be observed in which the line profiles are strongly perturbed by the field, so that more than just longitudinal field measurements are available to constrain the field structure. The first efforts to deduce both abundance distributions and field geometries were made by Deutsch (1958) and Preston (1972). More recently, modelling based on detailed comparisons of calculated and observed magnetically distorted line profiles has been reported for the cool Ap star 53 Cam (Landstreet 1988) and the hotter Bp star HD 215441 = Babcock's star, the star with the strongest known surface field (Land- street et al 1989). Both of these stars have line profiles that are strongly distorted by the magnetic fields present at the surface, from which considerable information about the field structure may be deduced; they also show strong spectrum variations from which maps of several elements, such as Si, Ca, Ti, Cr and Fe, may be deduced. Models of these stars were obtained by parameterizing both the field and the abundance distributions of the elements of interest in a simple way, and attempting to determine the set of field and distribution parameters that produce the best fit to the rotational variations of a number of observed line profiles. The problem of uniqueness of the resulting maps was dealt with by keeping to such a small number of free parameters for each quantity studied (typically four magnetic parameters, and between three and six parameters for the distribution of each element, plus the inclination i of the rotation axis) that only one acceptable solution could be found. The 54 J.D. Landstreet

adequacy of the simple models adopted for field geometry and abundance distribution was judged by the quality of the fits to the line profiles (note, however, that a good fit does not guarantee that the correct generic model has been chosen!). In these models, the field distribution was chosen to be an axisymmetric superposition of a dipole, quadrupole, and linear octupole. The abundance distribution of each element was supposed to be axisymmetric about the magnetic axis, so that only the variation of the abundance from one magnetic pole to the other had to be specified. The free parameters were then varied until the best possible fit was obtained. The resulting maps yield reasonable, although certainly not exact, fits to the observed variations of the spectral lines studied for both stars; presumably in each case the best fitting model is only a rough approximation to the true magnetic field and element distribution geometries. However, in the absence of better maps, even the information contained in such rough models is valuable. In both stars, the field structure is found to depart significantly from a pure dipole, in agreement with previous modelling of these stars (Borra and Landstreet 1977, 1978). Both stars appear to have a more intense magnetic field at one pole than at the other. In 53 Cam, deduced abundances of several iron peak elements are typically one dex higher than in the sun, and surface variations of one to two dex with colatitude are found for the elements Ca and Ti, while Cr, Fe, Sc and Sr appear more uniformly distributed. On HD 215441, the abundance of Fe is enhanced by at least one dex over the solar value, Si and Cr are two dex overabundant, and Ti is enhanced by three dex. All four elements show surface variations of at least one dex over the half of the star that is visible. Maps based on the same technique have been reported for three He strong stars by Bohlender (1989) and Bohlender and Landstreet (1990b). Other maps are currently being obtained by Bohlender, Landstreet, and Lanz. In addition, mapping programmes similar to the ones in use for abundance mapping in the absence of strong magnetic fields, but suitable for situations in which the field does greatly perturb the line profiles, are now under development by Donati, Rees and Semel and by Piskunov. We may expect rather rapid progress in mapping in the next few years.

3.6 Physics of element separation

In parallel with the recent developments in modelling has gone a considerable development of theoretical work aimed at understanding the physical processes that lead to strong surface abundance anomalies and to very inhomogeneous surface distributions of some chemical elements. This work is of rather general interest because the surface abundance patterns observed in the magnetic stars reveal processes going on in the sub-photospheric envelopes, and perhaps also above the atmosphere, that may occur to a greater or lesser extent in all upper main sequence stars. In fact, the abundance anomalies seen in magnetic stars are only the most extreme of the anomalies observed in upper main sequence stars; many kinds of A and B stars (see Sec 3.1) show more or less marked deviations from the solar abundance pattern that also characterizes most lower main sequence stars. It is clear that powerful chemical segregation processes act in upper main sequence stars, processes that we wish to understand so that it is possible to know under which circumstances observed stellar abundances represent the composition of the medium from which the star formed and when they represent essentially the results of further sorting. Magnetic fields at the surfaces of stars 55

It is now believed that one of the main processes acting to produce anomalous surface abundance in upper main sequence stars is diffusion of trace elements in the predominant hydrogen of the stellar atmosphere and envelope. If diffusion acted only under the effect of gravity, we would find that all elements heavier than H would gradually sink below the visible photosphere, as actually appears to happen in many white dwarfs (Fontaine and Wesemael 1991). This would lead to stellar atmospheres greatly deficient in everything except the lightest element, not at all like the observed magnetic A and B stars. An important discovery was made by Michaud (1970), who realized that in main sequence stars gravity must compete with the upward radiative acceleration acting in the spectral lines. For some elements this radiative acceleration can exceed gravity, so that the elements tend to diffuse upward rather than downward, at least in certain regions of the stellar envelope and atmosphere. The levitating effect of radiation is most marked for elements which are relatively low in cosmic abundance and have rich line spectra, so that momentum upward may be extracted from the radiation field at many frequencies and is not strongly reduced by saturation of spectral lines. Thus from a competition between gravity and radiation pressure, we may expect abundant He, with its simple spectrum, to sink under almost any circumstances (and in fact He is usually underabundant in magnetic A and B stars), while elements such as the rare earths should often be observed to rise (such elements are frequently overabundant in cooler Ap stars). The situation is complicated by the abundance gradients produced by the diffusion, which tend to oppose further concentration, and by the possibility for some elements of being driven completely out of the star into the interestellar medium. It is also clear that diffusion must compete with a number of other processes that may act in the outer layers of stars. If convection occurs, the abundances within the convection zone will be homogenized by the rapid mixing motions, although elements may still diffuse into or out of a convective layer at its boundaries, and thus be transported through the layer. Furthermore, the large-scale circulation motions induced by stellar rotation (Mestel 1965; Tassoul and Tassoul 1982) compete with diffusion, both by the direct mixing they produce in rapidly rotating stars (e.g. Charbonneau and Michaud 1988) and also perhaps by the turbulence of the flow, which may indirectly lead to enhanced diffusion (Charbonnel et al 1992). If a stellar wind exists, the diffusive motions must compete with the slow levitation of the whole atmosphere (Vanclair 1975; Michaud and Charland 1986). The fact that diffusion, turbulence, meridional circulation, and mass loss may all affect the observed surface abundances in a star means in turn that study of surface abundances should provide precious direct information about these processes. The very patchy distribution of some elements over the surfaces of many magnetic A and B stars, while no such surface inhomogeneities are recognized in any non- magnetic stars, makes it clear that diffusion is also strongly affected by the presence of a sizeable magnetic field. This feature of the separation process in magnetic stars is one of the least well understood aspects of the problem. During the past two decades, much work has been done to understand the observational effects of diffusion, and how it may compete with other envelope hydrody- namical processes, in a variety of types of upper main sequence stars. Good general reviews of the basic physics of diffusion in stars are given by Vauclair and Vauclair (1982) and by Vanclair (1983). A more recent review has been given by Michaud (1991). Some recent efforts to account for specific classes of abundance peculiarity in 56 J.D. Landstreet non-magnetic stars are reported by Michaud and Charland (1986) and Charbonneau and Michaud (1988). The first steps towards understanding the development of chemical peculiarities specifically in the presence of magnetic fields were taken by Vauclair et al (1979) and Alecian and Vauclair (1981), who considered whether a horizontal magnetic field could support a significant amount of Si in the atmospheres of magnetic B stars. In many such stars, more Si is observed in the atmosphere than can be readily explained by the effects of radiative levitation. Vauclair et al pointed out that in a B star atmosphere, neutral Si tends to rise under the effect of radiation while ionized Si tends to sink, and they proposed that a roughly horizontal field could impede the sinking of ionized Si, at least high in the atmosphere where collision rates are low, while not affecting the neutrals, thus enhancing the accumulation of Si in the photosphere. Since this effect would not occur where the magnetic field is nearly vertical, the mechanism also would lead to surface inhomogeneities. Another interesting mechanism producing inhomogeneities in magnetic stars was proposed by Michaud et al (1981) and more extensively explored by Megessier (1984). They considered possible horizontal transport of certain ions by the guiding effects of an oblique magnetic field. The basic idea is again based on the different behaviour of neutrals and ions in the low collision frequency region high in the stellar atmosphere. Consider an atomic species that is levitated as a neutral by the radiation field, but that sinks in the once-ionized state. A particular atom will tend to rise vertically during intervals when it is neutral, but when ionized it will tend to sink along field lines. If the field lines are oblique, it will not fall vertically, but will drift towards the nearest magnetic pole, since in a dipole-like field, this is the direction imposed by the overall field structure. A stellar wind too weak to be directly observable may have interesting consequences for both the vertical and horizontal stratification of He near the stellar surface. Vauclair (1975) has sugggested that a weak stellar wind is required to explain the observed occurrence of He-rich atmospheres in the He-strong magnetic stars. Helium has such a simple spectrum and such large cosmic abundance that in any atmosphere stable enough for gravitational diffusion downward to occur, it will sink out of sight. The radiative acceleration on this element is not sufficient to support a solar abundance of He in or near the atmosphere. To counter this effect in the hottest magnetic stars, Vauclair supposed that a stellar wind occurs with about the right mass loss rate so that in the region where He is ionized, it is levitated more rapidly by the wind than it can diffuse downward. Within the atmosphere, however, where He is largely neutral and hence interacts much less. strongly with the outflow of plasma, the He atoms are not lifted by the wind. Helium thus is expected to accumulate in the atmosphere until a large enough concentration is present to prevent further accumulation. This idea may also account for the inhomogeneous distribution of He over the stellar surface, if the wind is controlled or modulated by the magnetic field. Levitation of He into the visible atmosphere may occur near the magnetic poles, where the wind is allowed by the field to flow, but around the magnetic equator the wind is suppressed and He presumably sinks out of sight (Shore 1987; Vauclair et al 1991). If this same picture is extended to lower mass stars, the region of He accumulation drops below the visible photosphere, but may lie in just the region where it could excite pulsations in rapidly oscillating Ap stars (see Sec. 3.4) by the ~ mechanism, with the excitation occuring preferentially near the magnetic poles because considerably more He is present at the Magnetic fields at the surfaces of stars 57 level of the ionization zones near the poles than around the equator (Vanclair et al 1991). Each of the papers discussed above above focussed on the effects of a single concentrating mechanism. A more comprehensive effort to consider all the significant mechanisms for producing abundance anomalies and inhomogeneities in magnetic stars has recently been undertaken by Babel and Michaud (1991a, 1991b, 1991c) and Babel (1992). They have tried to theoretically explain the abundance distributions observed by Landstreet (1988) for the cool Ap star 53 Cam (see Sec. 3.5), considering the effects of diffusion both vertically and horizontally and examining also the role of possible competing effects, such as convection and mass loss. They find that the theory as developed so far can account qualitatively for the general level of overabundances found for this star, but still has great difficulty in explaining the strong abundance contrasts found between one magnetic pole and the other for Ca and Ti, and the absence of important concentrations around the magnetic equator. Babel (1992) has shown that in the presence of a weak stellar wind (too weak to observe directly, and thus in effect a free parameter in the problem), it is possible to explain the abundance distribution of Ca, whose surface abundance is sensitive to the competition between vertical diffusion and wind levitation, by assuming that the wind is suitably modulated in strength by the magnetic field over the surface. His model also accounts for the rather peculiar Ca II K line profiles observed in this star, which he argues are produced by pronounced vertical stratification within the observed photosphere. However, the model is still unable to explain, even qualitatively, the distribution of Ti.

4. White Dwarfs with Megagauss Fields

4.1 Observations

The earliest searches for magnetic fields in white dwarfs were stimulated by Blackett's (1947) proposal that the angular momentum of a white dwarf might be associated with a field of several megagauss. However, observations by Thackeray (1947) of the DA white dwarf Wolf 1346, and by Babcock (1948) of 40 Eri B, revealed no sign of the predicted fields. During the 1950's, the fossil field theory gradually gained popularity as an explanation of the fields observed by Babcock in the Ap stars. The idea that a field initially present in a main sequence star could remain largely unaltered throughout the main sequence lifetime led to the consideration of possible consequences of flux constancy through the full evolution of a star. If the total flux ~o~ ~ 7rR213 passsing through the magnetic equator of star can be roughly conserved, then the expected typical field strength in and at the surface of the star should vary roughly as

(B/Bins) ~ (Rms/R) 2, (6)

where the subscript ms refers to the main sequence. This result suggests that a magnetic field of 103 G on a main sequence star (R ,-~ 1011 cm) could lead to a field of 10 7 G in a white dwarf (/~ ~ 109 cm) or 1013 G in neutron star (/~ ~ 106 cm). The idea remained speculative until the discovery of pulsars (Hewish et al 1967) and their identification as rotating magnetized neutron stars having fields of 1012 G (Gold 1968), about the value predicted by equation (6). 58 J.D. Landstreet

The discovery of neutron star fields led to considerable renewed interest in possible magnetic fields in white dwarfs. At the suggestion of Prof. L. Woltjer, Angel and Landstreet (1970a) carried out a systematic survey of several bright DA white dwarfs using the first Balmer-line Zeeman analyser. No longitudinal fields were detected, although standard errors ranged from 10 4 up to 10 5 G, presumably an adequate sensi- tivity to detect the expected megagauss fields if they were roughly dipolar. At about the same time, Preston (1970b) showed that the internal consistency of radial velocity measurements of several Balmer lines for a large sample of DA white dwarfs implied the absence of the quadratic Zeeman effect at a level which placed upper limits of about 106 G on the stars of this sample. It was also at this time that Kemp (1970) showed that a field of 106 - 107 G could produce detectable circular polarization of the continuum radiation of a white dwarf. He then adapted a laboratory polarimeter for astronomical observations and mounted it on a 60-cm telescope. After a few unsuccessful measurements of bright DA stars, at the author's suggestion Kemp observed several DC white dwarfs, and discovered continuum circular polarization of well over 1% in the visible region of the spectrum of Grw +70 ~ 8247 (Kemp et al 1970). Continuum linear polarization was also soon detected in the same star (Angel and Landstreet 1970b). The discovery of the magnetic field of Grw +70 ~ 8247 was followed by the detection of continuum circular (and sometimes linear) polarization in several other white dwarfs. Most of the magnetic white dwarfs found in these early searches showed peculiar spectra, either unusual combinations of elements, or spectral features for which no ready explanation was available. Within a few years, it became clear that the fields detected in white dwarfs are usually large enough to greatly perturb the spectrum of even a simple DA star, and that classification spectroscopy also offers a powerful tool for finding new magnetic white dwarfs. Starting with GD 90, a magnetic DA discovered by Angel et al (1974), most of the recent discoveries of white dwarf fields have been made by spectroscopy. The most sensitive searches have been made with spectral line (mostly Balmer line) Zeeman analysers, with limiting detection thresholds (for longitudinal fields Be) of the order of 105 G (Angel et al 1981), corresponding roughly to limiting surface field values (for dipolar field structures) of about 3 • 105 G. Surveys of DA and DB stars based on classification spectroscopy have a detection threshhold (for surface fields/3~) of at best about 1-2 • 106 G (Liebert et al 1983). The weakest field detected so far through continuum polarization is of order 8 • 106 G, and this is probably near the detection threshhold for this method at present. Fields have now been detected in more than two dozen white dwarfs. A list of the known single magnetic white dwarfs is presented in Table 2. Successive columns list a common name for each star, an approximate effective temperature, the element(s) detected in the atmospheres, period of variation (in hours) if any is observed (NV means no variations have been observed even though they have been searched for), typical values for the observed circular and linear polarization in the visible spectral region, inferred typical field strength /3~, and a recent reference. The field strength t38 assigned to each star is not very accurate; if a dipolar model of the observed spectrum or polarization is available, the tabulated value of/3s is taken to be about 0.75 of the polar field strength t3d of the dipole. If only the longitudinal field is observed, the value of/3s given is two or three times larger than the maximum observed or deduced value of Be. Fields for some of the most polarized objects are estimated by Magnetic fields at the surfaces of stars 59

Table 2. Properties of single magnetic white dwarfs

Star Chem T~ Period Circ Lin Bs Reference Comp poln poln (kK) (hr) (%) (%) (MS)

PG 0136+251 H 45 0 1.3? Liebert et al 1983 G 141-2 H 5.6 <0.1 2? Greenstein 1986 PG 1658+441 H 30 NV <0.1 2.3 Liebert et al 1983 GD 90 H 15 <0.1 7 Wickramasinghe 1987 G 99-37 H,C 6.3 NV 1 <0.2 8 Angel 1977 KUV03292+0035 H 13 8 Wegner et al 1987 PG 1312+098 H 5.4 1.7v 8 Schmidt and Norsworthy 1991 GD 356 H 7.5 NV <0.1 11 Greenstein and McCarthy 1985 HS 1254+3430 H 15 13 Jordan 1989 KPD 0253+5052 H 18 4.1 0.4v 13 Schmidt and Norsworthy 1991 BPM25114 H 20 67.2 14 Wickramasinghe and Martin 1979 LBQS 1136-014 H 18 Foltz et al 1989 G 99-47 H 5.7 NV 0.5 <0.2 18 Wickramasinghe and Martin 1979 KUV2316+123 H 11 428.6 0.8v 21 Schmidt and Norsworthy 1991 PG 1533-057 H 20 24 Achilleos and Wickramasinghe 1989 Feige 7 H,He 20 2.2 0.3v 27 Martin and Wickramasinghe 1986 GD 116 H 16 50 Saffer et al 1989 PG 1015+015 H 14 1.6 lv 90 Schmidt and Norsworthy 1991 ESO 439-162 C 6.3 100: Ruiz and Maza 1989 G 195-19 ? 7: 31.9 lv <0.2 100: Angel et al 1981 G 227-35 ? 7 NV 2 <0.5 100: West 1989 LP 790-29 C 8 NV 8 1 200: West 1989 G 240-72 ? 6 NV 0.5 1.5 200: West 1989 Grw +70 ~ 8247 H 14 NV 4 3 240 West 1989 GD 229 v 16: NV 0.5 5 250: Schmidt et al 1990 PG 1031+234 H 15: 3.4 12v 5v 350: Latter et al 1987 rough interpolation of polarization levels between polarized stars whose fields have been modelled. The presence of a magnetic field in the first two objects listed, PG 1036+251 and G 141-2, is based only on unusual broadening of the Balmer lines, and should be considered still uncertain. The observed fields range from about 1 x 106 up to 3.5 • 108 G. The distribution of field strengths is roughly uniform between 3 • 106 and 3 x 108 G when plotted as a function of log B, with a weak peak around 2 x 107 G (Schmidt 1987). However, this distribution is probably affected by selection effects at both extremes. At the low field 60 J.D. Landstreet end, the perturbation of a DA spectrum by a field of 1 2 • 10 6 G is small enough to be missed in classification. At very high fields, the spectrum is so perturbed that few features of any strength remain. Overall, the probability of finding a magnetic field in a white dwarf is roughly uniform at about 0.5-1% per decade of field strength between 3 • 106 and 3 • 108 G; megagauss magnetic fields are found overall in about 2-3% of white dwarfs. The rest of the white dwarfs, if they have fields at all, probably mostly have fields of below 106 G, and the majority are probably below 2 - 3 • 105 G (Angel et al 1981; Schmidt 1987). Spectra of known magnetic white dwarfs provide information about the surface chemistry as well as about field strength. Most magnetic white dwarfs with fields below 108 G show only H lines, although two have unsual composition. No magnetic fields have been found in any DB stars, or in any star whose atmosphere is known to contain elements heavier than C. At fields of about 108 G or more, identification of atomic species becomes much more difficult. Two stars show profoundly altered spectra of H; two others have still recognizable lines of C2. The rest either show no lines, or lines which are apparently not produced by H. Lack of atomic calculations of energy levels and transition probabilities for elements heavier than H in fields of more than about 5 x 107 G have so far frustrated attempts to identify even the main chemical constituents in these stars. Effective temperatures of magnetic white dwarfs may be estimated from energy distributions or colours. Fields are found in stars of temperatures ranging from nearly 50,000 K down to below 6,000 K. As a white dwarf is simply an object cooling by radiative energy loss, the observed effective temperature is effectively an age indicator. One may try to explore the time evolution of white dwarf fields by looking for a temperature-field strength correlation. Apart from the rather peculiar fact that the two hottest magnetic white dwarfs are among the three known (or suspected) magnetic white dwarfs with the weakest fields, no correlation is apparent. Certainly among the cooler stars, with temperatures ranging from 20,000 K to about 5,600 K (which represents a range of cooling times since formation between about 1 x 10' and 4 x 109 yr; cf Winget et al 1987), no evolution of field strength is obvious. About one-fourth of the magnetic white dwarfs are observed to be variables; this number represents about half of the white dwarfs that have been seriously studied for variability. Generally, the variations are observed in continuum polarization, but for BPM 25114 only spectrum variations have been detected. The variations observed are simply due to magnetic field strength changes; no cases are know of spectrum variability that suggest important surface abundance inhomogeneities such as those found among the magnetic A and B stars. The variations are invariably strictly periodic, with periods ranging from 1.5 hrs up to almost 18 days. In about half the white dwarfs with variable circular polarization, the observed polarization changes sign during the cycle of variation. For examples of polarization variations see Schmidt and Norsworthy (1991); spectrum variations are discussed by Latter et al (1987) and Wickramasinghe and Cropper (1988). It should be mentioned that a number of magnetic white dwarfs are also found in close (generally interacting) binary systems of the AM Her and DQ Her types. The magnetic white dwarfs in AM Her systems generally have surface fields in the range of 20-40 MG. Those in DQ Her systems are thought to have somewhat weaker fields. These objects lie outside the range of this review; for further information, see the recent review of AM Her stars by Wickramasinghe (1989) or articles in the Magnetic fields at the surfaces of stars 61 proceedings of the Vatican conference on polarized radiation of circumstellar origin, particularly those by Berriman (1988) and Wickramasinghe (1988).

4.2 More rigid rotators: models of magnetic white dwarfs

The observed periods of magnetic variation of the magnetic white dwarfs are extremely long compared to the dynamical time scales of these stars (a few seconds) and to the known periods of pulsation of a number of (non-magnetic) white dwarfs (half an hour or less; cf Kawaler and Hansen 1989). Unlike the great majority of pulsating white dwarfs, the variations of the magnetic white dwarfs are found to occur at only a single frequency. Furthermore, the range of observed periods is large; the most slowly varying magnetic white dwarf has a period more than 250 times larger than the most rapidly varying, even though most of these stars probably have rather similar radii and masses. These characteristics, together with the fact that the sign of Be often reverses, all strongly suggest that we are seeing variations produced simply by the rotation of magnetic structures that are not axi-symmetric about the rotation axis. The variable magnetic white dwarfs are thus aparently rigid rotators like the magnetic A and B stars. The fact that only about a fourth of magnetic white dwarfs are known to vary is in some contrast with the situation for the well-studied magnetic A and B stars, almost all of which are variable. It is certainly possible that a few more stars from the present magnetic white dwarf sample will be found to vary after more thorough observation (in fact, Bues and Pragal 1989 claim to have observed variations in G 99-37 and G 99-47, although most of their data have not yet been published), but it is clear already that a substantial fraction of the magnetic white dwarfs do not vary. A particularly sensitive test of variation is available for the stars which show continuum linear polarization, as one may look for rotation of the position angle of linear polarization. The position angle 0 of polarization in the blue for Grw +70 ~ 8247, for example, has not rotated by more than 5 ~ in 15 years, and a number of other high field white dwarfs show similar lack of variation (West 1989). On the oblique rotator model, the lack of variability observed for many magnetic white dwarfs could be due to extremely long rotation periods. For Grw+70 ~ 8247, for example, the lack of change of 0 could be clue to a rotation period of ~ 103 years; in other cases, lower limits of the order of 102 years are derived. The lack of variability could also be due to alignment of the axis of symmetry of the magnetic field geometry with the rotation axis. It is not easy to find a way to choose between these two possibilities. No direct evidence of rotation, in the form of Doppler broadening of the cores of spectral lines, has been detected in any of the non-variable magnetic stars. However, this would only be detectable even in principle in stars with fields sufficiently weak and/or homogeneous that the spectral lines could have sharp cores, and then (at least with presently practical spectral resolution and signal-to-noise ratios) only if v sin i is at least of the order of 30 km s -1, corresponding to a maximum detectable rotation period of ~ 15 min, less than the observed periods of any of the known variables. This possibility thus provides no useful constraint on possible rotation. On the other hand, one may argue that the lower limits to rotation periods derived from lack of observed variability are so long that it is unlikely that white dwarfs could rotate so slowly, especially considering the spin-up in angular velocity expected 62 J.D. Landstreet when a star shrinks to become a white dwarf. This argument is strengthened by the apparent lack of rotation periods in the range between 20 d and 102 yr. However, the demonstrated existence of very long rotation periods (hundreds of days or more; cf Catalano and Renson 1984) for a very small number of magnetic Ap stars, as well as the possibility of transferring magnetically almost all of the stellar angular momentum to a stellar wind during the giant phase, suggest caution witk this line of argument. At present, it does not seem possible to decide whether the non-varying magnetic white dwarfs are extremely slow rotators, or have magnetic structures that are symmetric about the rotation axes. Whichever of these two situations applies, it seems on the basis of the current statistics to be somewhat more common for magnetic white dwarfs than for upper main sequence magnetic stars. Efforts to model magnetic geometries of individual white dwarfs have a somewhat different set of problems to confront than comparable efforts for magnetic A and B stars. On the one hand, no spectrum variations due to non-homogeneous surface element distribution have yet been recognized, which eliminates an important complication found in Ap and Bp stars. In addition, the broadening of spectral lines is often dominated by Zeeman splitting, and even when this is not the case, the splitting is generally detectable in the line cores. This provides an important constraint on field structure not usually available for magnetic main sequence stars. Both of these points act to simplify the modelling. On the other hand, a number of new problems appear. First, spectra of magnetic white dwarfs usually have low signal-to-noise ratios (G 50) and low resolution (< 5000). Even in the most rapidly rotating stars, no information whatever is available about spatial location on the stellar surface from Doppler shifts, because for stars of such small radii the value of the equatorial velocity (about 10 km s -1 for PG 1015+015, the most rapidly varying magnetic white dwarf) is far below the detectable level. Secondly, the available polarimetry often consists of observations in very broad bands, and is difficult to interpret or model quantitatively. Even when higher resolution is available, individual spectral features are often not resolved, so that the line-of-sight field component is usually much less securely con- strained than in main sequence stars. Finally, at the level of the fundamental spectral physics, a further problem is that calculation of wavelengths of absorption features and of oscillator strengths are available only for H (up to about 1 x 109 G) and for some lines of He I (up to only about 2 x 107 G; see Wickramasinghe 1987). No calculations of wavelength shifts or 9f values are available for any lines of any heavier elements beyond the range of validity of the usual low-field theory. Line broadening parameters have not been calculated for any chemical elements. Furthermore, the behaviour of the atomic continuous opacity (and especially the dichroism of the continuous opacity, that is, its ability to absorb various states of polarization) has only been estimated, even in the low-field limit. No calculations are available at field strengths appropriate to the fields present in the white dwarfs showing continuum polarization (West 1989), except for some very interesting recent work by Whitney (1991a,b) on the absorption and dichroism of a sea of free electrons. For this reason, it is not yet possible to model with confidence the wavelength dependence of continuum polarization in any of the magnetic white dwarfs. A final possible complication, especially serious for white dwarfs with H rich atmospheres, is the likely presence of significant Lorentz forces in the atmospheres of stars with surface fields in excess of about 108 G. These forces could potentially alter considerably the atmospheric pressure stratification from that calculated from ordinary hydrostatic pressure equilibrium (Landstreet 1987). None of these problems have yet received adequate study. Magnetic fields at the surfaces of stars 63

In spite of the difficulties discussed above, it is quite feasible to try to model the magnetic field geometries of the magnetic white dwarfs, at least for those with H- and He-rich atmospheres for which the necessary atomic data are available. Such a modelling effort is practical because the positions and intensities of the observed spectral lines are primarily determined by the field strength and geometry. Uncertainties in the line broadening, continuous opacity and dichroism, and atmospheric structure do not lead to major ambiguities in the models. Partly because of the limitations of the available data (especially the lack of Doppler shift information and the low signal-to-noise ratio of most white dwarf spectra), and partly for conceptual simplicity, modelling of magnetic white dwarfs has so far been limited to simple field geometries specified by only a small number of parameters (centred or decentred dipoles, quadrupoles, or combinations of such fields with simple local spots), which are fit to the available data as well as possible by judicious choice of parameters. Models have by now been constructed for a number of stars. Considerable modelling has been carried out primarily by fitting the wavelengths of observed spectral features to the calculated Zeeman shifts of spectral lines of H; essentially, these are models created without any discussion of effects of radiative transfer. More elaborate models which include transfer effects have been produced by Martin and Wickramasinghe, by O'Donoghue, and by Jordan. Many of the available models have been reviewed by Wickramasinghe (1987). Some more recent modelling of individual stars has been reported by Latter et al (1987), Achilleos and Wickramasinghe (1989), Foltz et al (1989), Jordan (1989), Saffer et al (1989), Bues (1991), and Whitney (1991b) In general, approximate agreement with observed flux spectra (and for the low- field stars, with circular polarization spectra when these are available) is found by assuming centred or slightly decentred dipole fields, although for PG 1031+234 and perhaps for Feige 7 there is evidence for a localized region of intense field (a magnetic spot?). For high-field stars, the polarizations spectra are not generally reproduced by models even when the flux spectrum can be explained, reflecting either shortcomings in the assumed field geometries, or deficiencies in the atomic data used, or both. It appears likely that most magnetic white dwarfs have field geometries which are at least grossly dipolar in structure. However, remaining discrepancies indicate that this is probably only a rough description of actual field distributions. Further high quality observations and modelling are clearly worth the effort involved. As in the case of the magnetic A and B main sequence stars, the lack of any observed field variation except that due to rotation, and the absence of any obvious driving machinery for a current dynamo in most white dwarfs, suggests the possibility that the fields of the magnetic white dwarfs are fossil fields, left from an earlier stage of evolution. This is, of course, the hypothesis that originally led to searches for megaganss fields in white dwarfs (cf. Sec 4.1). Calculations of the ohmic decay times for simple field structures (Wendell et al 1987) show that ohmic decay is so slow in these stars that no important decay should be observed. This is entirely consistent with the observed lack of any systematic variation of field strength with effective temperature, and makes the fossil field hypothesis very attractive. Of course, the question then arises as to what previous state of the star gave rise to the white dwarf fields. One reasonable possibility is that the fields of white dwarfs are the evolutionary descendants of the fields of magnetic main sequence stars, as most of these stars have low enough mass to evolve to the white dwarf State rather than to a more compact final configuration. Angel et al (1981) have shown that the empirical probability of 64 J,D, Landstreet

finding a strong field in a white dwarf is approximately consistent with the fraction of white dwarfs which should be strongly magnetic if this is indeed the origin of the strong fields. No more direct proof of this plausible connection has yet been found; as it is not obvious how a fossil field would be able to survive the intense and deep convection of the giant stage of evolution which separates the main sequence from the white dwarfs, the question of the origin of the white dwarf fields should perhaps be left open until this puzzling point is clarified.

5. The complex fields of solar-type stars

5.1 Direct measurements of magnetic fields in cool stars

The magnetic field of the sun is observed directly, principally by means of the circular polarization of the longitudinal Zeeman effect, but also through the linear polarization of the transverse Zeeman effect, and by the splitting of magnetically sensitive lines in large field regions. In fact, Hale (1908) observed both circular polarization and line splitting at the time when he discovered the kiloganss fields of sunspots. On the sun, only the sunpots contain easily detectable fields. The general field of the sun was finally detected only after the development by Babcock and Babcock (1952) of the solar magnetograph, a device for measuring the longitudinal field component from high precision circular polarization observations in the wings of a magnetically sensitive spectral line, which allows detection of fields of the order of a few gauss. The observed field of the sun is found to be strongly inhomogenous in structure. Regions of particularly strong field (typically 1-4 kilogauss) are found in sunspots. The sunspots often occur in pairs, with opposite polarities in the two spots; it appears that the magnetic measurements map out the footprints of rather small-scale loops of magnetic flux that emerge from the photosphere in one spot and return to the solar interior through the other spot, connecting the two by a horseshoe-shaped magnetic flux tube. Around pairs of sunspots, one generally finds an active region in which the measured field strength is considerably smaller than in the spots, but still substantial. Active regions usually also show both field polarities. Outside of the active regions, the measured field is still weaker, often of the order of a few gauss; the distribution of polarities is complex. Measurements of field strength in an active region, outside of the sunspots, give measured fields of a few tens of gauss. If these fields are really weak, the circular polarization measured with spectral lines of different ~ values should be proportional to ~ for lines of a given strength (Mathys 1989, Sec. 2.5.2). Instead, lines of equal strength but different values of ~ show similar polarization. This result reveals that the fields are not intrinsically weak, but in fact are large (some kilogauss). Polarization corresponding to only a few tens of gauss is measured because the flux tubes are present in only a small fraction of the sampled area. Thus, the solar photospheric magnetic field is present mainly in the form of isolated flux tubes embedded in a largely non-magnetic atmosphere; the filling factor of the field is mostly small, of the order of 10 -1 to 10 .2 in active regions and closer to 10 -3 elsewhere. The field strength within the flux tubes is typically more than one kilogauss, comparable to the value found in sunspots (Stenflo 1973; 1989). The magnetic field of the sun is intimately connected with virtually every kind of solar activity. Sunspots, visible in integrated (white) light images of the sun, represent Magnetic fields at the surfaces of stars 65 the most important concentrations of magnetic flux. Around the sunspots, solar active regions constitute regions of enhanced non-radiative heating of the upper photosphere and the chromosphere, a fact revealed by emission observed in the cores of the H and K lines of Ca ~ and in numerous ultraviolet spectral lines. The solar corona is hottest, densest and brightest in regions where it is confined by loops of magnetic field; conversely, in regions where the magnetic field lines do not close near the solar surface, the so-called coronal holes, the corona is not confined, but streams away from the surface as the solar wind. This structure is clearly evident in x-ray photographs of the sun. (For a more extensive description of both observations and models, see Priest 1982.) The solar field changes on a time-scale of weeks or less, sometimes (as during a flare) perhaps even in minutes. The entire polarity structure of the field reverses roughly every 11 years. This is clearly not a fossil field. Instead, it is generally believed that the field of the sun is generated by a dynamo, operating perhaps near the base of the deep solar convection zone (e.g. Brandenburg and Tuominen 1991). This dynamo is probably driven by a combination of the convective motions and the differential rotation of the sun. These motions are not at all unique to the sun, and it thus seems reasonable to expect to find magnetic fields in many other solar-type stars which share the properties of a deep convection zone and reasonably rapid rotation with the sun. This plausible idea is supported by the observations of analogues of solar activity in other solar-type stars, which often show the emission lines characterstic of chromospheric heating, are frequently found to be emitters of x-rays at about the same level of luminosity as the sun, and sometimes (especially in close binary systems) even reveal the presence of giant starspots. The structurally complex solar field posesses a non-zero longitudinal field component only because it is so close that we can observe the surface with sufficient spatial resolution to measure the field in regions of more-or-less unmixed polarity, even though we cannot resolve individual fluxtubes. The visible hemisphere of the sun has generally no detectable net longitudinal field component when integrated over the surface. We thus do not expect that it will be easy to detect longitudinal fields in stars other than the sun, and observational searches have confirmed this suspicion (Brown and Landstreet 1981; Borra et al 1984). However, the problem is even worse than this; the sun posesses a strong field in only a tiny fraction of the visible surface, of the order of 10 .2 of the total. If this is typical of other solar-type stars, field detection by any method whatever may be expected to be difficult. Searches for fields on solar-type stars thus proceed from the hope that if such stars also have flux tubes with kilogauss fields, the filling factor of the field will be much larger than on the sun, at least 10 -~, in some stars. It is natural to look first at the stars which show the strongest symptoms of solar-like activity. The critical development that made the direct study of magnetic fields in cool stars other than the sun accessible to direct study was the suggestion of Robinson (1980) to use Preston's (1971b) technique for measuring the mean field modulus by comparing the broadening of spectral lines of widely differing ~ values, as discussed in Sec 2.2. Robinson et al (1980) reported the discovery of kiloganss fields in two lower main sequence stars. Searches by others soon resulted in further detections (Marcy 1984; Gray 1984; Saar and Linsky 1985). Fields have now been detected in more than 30 main sequence stars, ranging in spectral type from G0V to M4.5V. No fields have been detected in any F stars. (No dipolar fields of the sort found in upper main sequence stars are found in F 66 J.D. Landstreet

stars either; the existence of this non-magnetic gap on the main sequence is rather puzzling; see Landstreet 1991). Fields are also found in two subgiants, members of close binary systems. No fields are observed directly in any giants or supergiants. A critical selection of particularly accurately measurements is given by Saar (1990). The fields detected range in strength from 1 to 5 kG, very similar to the values found in the magnetic regions on the surface of the sun. This represents a much more restricted range than is found among the upper main sequence magnetic stars, and is consistent with the idea that fields in the cooler stars are dynamo-generated, while those of upper main sequence stars are fossil fields. The fields are essentially always found to have filling factors f less than 1.0; a few (especially some M dwarfs) have filling factors of 0.8 or so, but for most of the detectably magnetic stars, the filling factors range downward from about 0.5 to a low of about 0.1, the present lower limit for detection. It is found (Saar 1990) that the total flux f/3 correlates fairly well with angular velocity, as would be expected for dynamo-generated fields. There is however considerable dispersion around the mean relationship. The actual field strength/3 observed does not correlate with angular velocity; most of the observed correlation of fB with angular velocity is due to an increase in f with decreasing rotation period, which may be taken to mean that in a more rapidly rotating star, the dynamo manages to fill a larger fraction of the surface with magnetic flux. On the other hand, Saar (1990) has argued that the mean field strength/3 is correlated with the photospheric gas pressure. This may mean that the actual field strength observed in the atmosphere is set by the ability of the gas pressure to confine the flux in magnetic regions. As a result, the observed fields tend to be larger in M dwarfs that elsewhere, because of the low continuum opacity and consequently high photospheric pressure; in subgiants, with relatively low gas pressure, the observed fields are at the limit of detectability. Measurements of field strength in cool main sequence stars by Preston's technique have some very important limitations. The extra magnetic broadening is difficult to detect even in measurements of high resolution (50,000 or more) and high signal-to- noise ratio (250 or larger), and cannot be detected at all if v sin i is much above 7 or 8 km s -1 . Furthermore, any measurement using intensity profiles has the disadvantage that the measurement is diluted by the light from the non-magnetic regions. Recently, new methods of field measurement have begun to be used that circumvent these limitations. One method uses the fact that in a saturated spectral line, the Zeeman effect widens the local absorption profile in a manner similar to that of thermal and microturbulent velocity fields, and hence increases the equivalent width of the line over the value that would be produced in the absence of a field. This "magnetic intensification" increases with increased 7r - ~7 separation in the magnetically split line, and also with increasing separation of the 7r components (or cr components) from one another. It is possible to look for this magnetically induced excess line strength in the curve of growth of a star with v sin i too large to make line profile studies rewarding (Leroy 1962). This method has recently been applied by Basri and Marcy (1991), who report the first probable field detection (f/3 ~ 1000 G) in a T Tauri star. Another very interesting development is a first successful exploration of the use of spectrally resolved polarimetry (sometimes called "Zeeman Doppler imaging") to detect the polarimetric signature of small magnetic regions on fairly rapidly rotating (15 km s t ~ vsini ~ 40 km s -1) stars by using the Doppler effect to separate regions of opposite magnetic polarity in wavelength position within a spectral line. This method, first proposed by Semel (1989), has recently been used by Donati et Magnetic fields at the surfaces of stars 67

al (1990) to detect a large magnetic region on the surface of the primary component of the RS CVn binary HR 1099. Not only does this method break the v sini barrier, but the polarimetric information in the observations potentially provides considerable information about the spatial distribution of polarity in the magnetic structure(s) observed. By analogy with the rigid rotators of the upper main sequence, it should be possible to use stellar rotation to obtain maps of the distribution of magnetic flux over a stellar surface. Many active stars rotate with periods of days or weeks (Baliunas et al 1983), a time interval short enough for one to hope that magnetic structures would not change too much in one rotation. In fact, not much progress has yet been made in this direction. Searches for variations in fB with rotational phase, and on longer time scales, have so far failed to reveal much change in the quantity f/3 in most of the stars studied (Basri and Marcy 1988; Saar et al 1990). The one fairly clear example of variation so far found has been in the active G8V star ~ Boo A. This star has had a history of detections and non-detections of its field; a campaign by Saar et al (1988) to study both field strength and other activity indicators through the rotation of this star showed variations in ft? and in activity at one phase which were used to obtain a rough map of magnetically active longitudes on the star.

5.2 Modelling problems: a description of the stellar surface and field

The solar-type stars with detectable magnetic fields present quite a different set of modelling problems than the magnetic Ap and Bp stars. As discussed above, not much progress has yet been made towards mapping directly the macroscopic field structure of solar-type stars. Nor is there any evidence or expectation of surface chemical inhomogeneities in cool main sequence stars with deep convection zones. Instead, the most important problems currently receiving attention concern the small- scale structure of the stellar photospheres. Much work is presently being done to understand the nature of the emitting regions that contain magnetic flux, and how these differ from the field-free regions whose existence is implied by the observed filling factors. To interpret the measurements of photospheric field strength, one must have a model of the emitting region. In the earliest measurements, the model used was a simple superposition of a Zeeman split line (from the magnetic regions) and an unsplit line (from the non-magnetic regions). Both lines were assumed to be weak (so that various contributions added linearly), and to be constant in profile over the visible hemisphere, so that disk integration and velocity broadening could be treated as convolutions. In addition, because the relative strength of the 7r and cr components depends on the orientation of the lines of force relative to the line of sight, a mean angle of tip was generally adopted for the whole visible hemisphere (Robinson 1980; Marcy 1982; Gray 1984). This simple model made it possible to interpret the measurements of magnetically induced excess line broadening in terms of field strengths and filling factors. However, the assumption of optically thin lines ignores the effects of saturation in the line cores, which are important in many of the lines measured; to obtain the relatively strong cr components deduced from the shapes of line wings, unrealistically large filling factors have to be adopted. This kind of limitation in the interpretation of the measurements has been gradually addressed through increasingly sophisticated models of the observed lines with 68 J.D. Landstreet

spectral synthesis programmes that treat effects of radiative transfer, at first in the Milne-Eddington approximation and then with more realistic techniques (Saar and Linsky 1985; Saar et al 1986; Saar 1988; Basri and Marcy t988; Marcy and Basri 1989). Disk integrations now replace convolutions. The magnetic field is now radial over the star instead of having one "average" angle of tip. Other atmospheric parameters (v sini, macroturbulent velocity, abundance of Fe) are determined with some accuracy. Most importantly, the effects of saturation in the line profiles are dealt with, resulting in more accurate determinations of the mean field/3 and the filling factor f. As a result of the rather large efforts involved in constructing the necessary spectrum synthesis codes, the tools are now available to probe further into the nature of the observed photospheres. A first effort to obtain some information about the difference between the magnetic and non-magnetic regions was made by Mathys and Solanki (1989), still making the simple assumptions of optically thin spectral lines and a mean angle of field tip, but analysing statistically a large sample of Fe i lines from each star, using a technique devised by Stenflo and Lindegren (1977) for the study of spatially um'esolved magnetic flux tubes in the sun. For the star e Eri, their data were good enough to examine the effects of magnetic broadening in lines of low and high excitation potential separately. They concluded that the magnetic regions in c Eri responsible for the measured field are somewhat hotter than the non-magnetic regions, and thus resemble solar plage regions more nearly than sunspots. A more explicit attempt to determine the ways in which the observed spectral lines are affected by plausible multi-component atmosphere models, and to assess the errors in field determination that may be incurred in assuming that the magnetic and non-magnetic regions are physically similar in atmosphere structure (i.e., they produce similar spectral lines except for magnetic splitting, and emit similar continuum levels), has recently been reported by Basri et al (1990). They try to construct reasonable two-component models for magnetically active cool stars, starting from the observations of flux tubes and non-magnetic regions on the sun. In fact, the magnetic flux tubes may differ considerably from the non-magnetic regions. The magnetic field contributes to the pressure inside an isolated flux tube, and almost certainly suppresses convection in the interior of the tube. Wave dissipation inside the tube is probably different from that in non-magnetic regions, changing the local non-thermal heating. Energy exchange between thin flux tubes and their non-magnetic surroundings may have further effects in altering the temperature structure of both kinds of regions. In addition, the emission from a small flux tube depends on whether one looks nearly vertically into it, or obliquely through it. After considerable comparison of various models with one another and with observations, Basri et al conclude (1) that (not surprisingly) errors arise when observations of a star that in fact thas a two-component atmosphere are analysed assuming an atmosphere which is the same for both magnetic and non-magnetic regions, with larger errors in/3 and f separately than in the product f/3, (2) that at least some models of solar active regions are inconsistent with observed line profiles, and (3) that the actual surfaces of cool magnetic stars may have structures that cannot even be represented with a two-component model. This work is quite important, both in indicating the complexity of the problem of deducing the surface structure of magnetic stars from observations, and also in making a real effort to establish what observations and modelling will be required to make further progress. Similar modelling, to consider the effects of vertical variations in the magnetic field Magnetic fields at the surfaces of stars 69 strength on measured /3 and f values, has been reported by Grossman-Doerth and Solanki (1990). A summary of the present situation is offered by Saar (1991).

5.3 Information about fields provided by measures of stellar activity

Considering the close connection observed on the sun between the presence of a magnetic field and such symptoms of solar activity as the occurrence of sunspots, local heating of the chromosphere and attendant emission in the cores of the Ca II H and K lines, and the x-ray bright regions of the corona, it is not surprising that correlations between magnetic fields and indicators of activity have been looked for and found (Marcy 1984; Saar and Schrijver 1987_). Both the strength of emission in the cores of the Ca n H and K lines and the observed x-ray fluxes correlate with fB, although at flux values fB above about 300 G, the correlation with Ca n emission saturates. Jordan et al (1987) and St~piefi (1988) have predicted relationships between magnetic flux and activity with some success. It thus seems very likely that further interesting information about stellar magnetic fields is to be found in observations and modelling of stellar chromospheric and coronal activity indicators. A nice example of the possibilities is furnished by the long-term observations of the equivalent widths of the emission cores in the H and K lines of a sample of about 50 lower main sequence stars (ranging from F0 to M0) and half a dozen cool giants, started by Wilson (1978) and continued by a large group (e.g. Vanghan et al 1978; Baliunas et al 1983, 1985; Baliunas and Vaughan 1985). These emission cores are presumably produced, as in the sun, in regions of local chromospheric heating channeled or controlled by the stellar magnetic field. It is found that, although some stars exhibit no significant variability, a number are variable on a variety of interesting time-scales. The first of these scales is apparently a rotational time-scale; more than half of the observed stars show periodic variations that seem to originate in rotation. The derived rotation periods range from about 2.6 days up to nearly two months, and are consistent with the v sin i values. Variation is not observed in stars with a generally low level of emission; those with strong emission, or spectral types later than K0, almost always show rotational modulation. From these data, one can infer that the (presumably magnetic) structures underlying the H and K line emission are not distributed uniformly over the surface of the star (Baliunas et al 1983). In principle, continued monitoring of the H and K emission in the more rapidly rotating stars at very high resolution (e.g. Beckman et al 1991) may allow actual mapping of chromospherically active regions, but observations of sufficient phase coverage are not yet available. In about a quarter of the stars of the Wilson sample, another kind of variation is observed (Baliunas et al 1985). Multiple periods (beat phenomena) or changes in period from one season to another are detected. These are the sorts of variation that would be produced by differential rotation of active regions at different latitudes, or by the growth and decay of large active regions at different latitudes. In a few cases, it appears that differential rotation is indeed the explanation of the observed period structure. A longer interesting time-scale is found when observations covering some years are examined (Wilson 1978; Baliunas and Vaughan 1985). Changes in overall emission level, analogous to the solar activity cycle, may be observed. Some stars show quite strong and regular oscillations of activity level; others vary irregularly or show an 70 J.D. Landstreet

almost constant activity level. Among the regular variables, cycle lengths that range from about 2.5 years up to nearly two decades (the longest period detectable in the present data) have been found. Clearly, detection of stellar activity cycles represents an important advance in our ability to study stellar magnetism in cool stars, especially in view of the lack of observed variations in most of the direct field measurements of lower main sequence objects. Another interesting category of supplementary information is available from x-ray observations of cool stars (e.g. Pallavicini 1989). In the sun, most of the brightness observed in direct x-ray photographs comes from magnetically confined loop-like structures. The gas in these loops is slightly hotter and considerably denser than in other parts of the corona, from which the lower temperature gas escapes as the solar wind. Clearly the magnetic field loops on the sun both confine the plasma and somehow control its heating, although the physics of the heating is not yet well understood. Thus we expect that observations of x-rays from cool stars may provide interesting indirect information about coronal magnetic fields on these stars. All lower main sequence stars from spectral class F to M emit x-rays, with luminosities Lx ranging from about 1027 erg s -1 to almost 1030 erg s -1. Little dependence of Lx on spectral type is observed; instead, there is a strong correlation between Lx and rotation rate. Various authors derive different relationships (e.g. Pallavicini et al 1982; Fleming et al 1989), but generally it is found that Lx increases with increasing angular or linear rotational velocity. This is reminiscent of the correlation of magnetic flux Bf with rotation, and is consistent with the idea that Lx and the magnetic field are related. At any one rotation rate a considerable spread of x-ray luminosities is observed; it seems that additional parameters beyond the rotation rate also affect the emission. Among cool giants, x-ray emission at luminosity levels comparable to those of cool main sequence stars (and therefore at x-ray fluxes per unit area considerably less than ,in main sequence stars) is observed for many stars of spectral types earlier than about K2 III but not very commonly for cooler stars (Maggio et al 1990). The observed x-ray luminosities are very loosely correlated with surface rotational velocity. The decline in x-ray emission near K2 III approximately parallels the onset of cool massive winds among the cooler giants at about the same spectral type. This is superficially reminiscent of the dichotomy on the surface of the sun between closed loop, x-ray bright regions and open field line, x-ray dark areas, but the correspondence is not exact; winds of the cool giants that emit little or no x-rays are cool and slow while that of the sun is hot and fast. However, it seems likely that magnetic fields must be present on the surfaces of the x-ray emitting giants; as pointed out by Rosner et al (1991), a gas as hot as the solar corona is not gravitationally bound to a G giant in the absence of non-gravitational attachment, such as that provided by closed magnetic loops, and would simply expand freely at the sound speed. The necessity to invoke a magnetic field to retain the observed x-ray emitting gas in the corona of a giant star, as well as the clear relationship on the sun between the coronal x-ray emission and closed magnetic loops, have led theorists to try to explain the decline in frequency of x-ray emission cooler than K2 III, and the corresponding onset of massive cool winds for the cooler giants, by changes in the structure of the magnetic field at about this spectral type, or by changes in the thermodynamic properties of magnetic loops at the temperature of the "dividing line". Antiochos et al (1986) have suggested that the cut-off of x-rays in cool giants is due to a change in the thermal structure of stable magnetic loops, from a structure filled with hot gas Magnetic fields at the surfaces of stars 71

(T ~ 106 K) for earlier type giants, to a cooler (T < 105 K) gas in magnetic loops in stars cooler than the dividing line. This model, however, does not account for the coincidence of the decline of x-ray emission with the onset of massive winds. Rosner et al (1991) propose that both the decline of x-radiation and the onset of the massive wind are due to a change in magnetic structure, from predominantly closed loops earlier that K2 to mainly open loops on cooler stars, which leads to a drop in coronal temperatures and a considerable increase in the reflection efficiency of Alfvdn waves, in turn driving a massive wind. A third indirect means of studying the magnetic fields of cool stars is provided by the occurrence of starspots, presumably stellar analogues of sunspots. Such structures are detected in many lower main sequence and giant stars through the observation of light variations that may range in amplitude from the detection limit of about 0~.01 (1%) up to about 0t~3 in main sequence stars, and up to 0~75 in giants. The light variations are generally accompanied by colour changes, as well as by spectral line profile variations. Such light variations are commonly observed in close binary systems such as RS CVn stars and BY Dra stars, but may also be found in single stars (Strassmeier and Hall 1988). The light and spectrum variations are found to be more or less periodic, and the observed variations apparently represent in individual stars various combinations of stellar rotation with growth, decay, and sometimes differential migration of spots (Hall 1991). Because the spots often persist for times considerably longer than one stellar rotation, as indicated by light curves that are fairly stable for several oscillations, it should be possible to obtain maps of the spots by methods similar to those used for the surface chemical inhomogeneities of upper main sequence stars, and in fact the art of surface mapping has undergone considerable development in this direction. The earliest efforts were mainly concerned to model photometric observations alone, and it now seems clear that except in eclipsing binaries, where fewer geometrical degrees of freedom exist than in most systems, there is a problem in deriving unique maps even with rather simple representations of the spots. Hence much of the recent mapping activity has concentrated either on eclipsing systems (e.g. Zeilik 1991), or has been based on modelling variations of spectral lines. Recent progress has been reviewed by Vogt (1988), Vogt and Hatzes (1991), and Piskunov (1991). A very interesting experiment was recently organized by Strassmeier, who had three different groups, using different programmes, analyze the same spectroscopic data set to obtain maps (Strassmeier et al 1991). Encouragingly, the three groups obtained fairly similar maps from the data, although there were definite differences in detail. Recent mapping suggests that spots are common both at high and low latitudes, in constrast to the solar situation, where most spots are confined to within about 30 degrees of the equator. Like sunspots, the starspots are 500 to 1500 K cooler than the surrounding photosphere. The spots are enormous by solar standards, however. A starspot cannot even be detected reliably if it has a radius of less than about 5 ~ and the inferred sizes of observed spots range up to 20 or 25 ~ in radius, roughly comparable in size to solar active regions. In stars with spots at a variety of latitudes, it is found that differential rotation occurs (e. g. Hall 1991), sometimes in the opposite sense to that found in the sun (i.e., the poles may rotate faster than the equator; cf Vogt and Hatzes 1991). In photometric and spectroscopic studies of spots on cool stars, no direct information is obtained about the magnetic structures that presumably underly the spots. This deficiency has recently been addressed through spectropolarimetric observations. 72 J.D. Landstreet

As mentioned in Sec. 5.1, Donati et al (1990; 1992a) have recently succeeded in detecting the direct Zeeman circular polarization due to the magnetic field in the line profiles of several RS CVn binary systems, thus providing the first clear information about fields in the spots of any cool stars. These data have been used by Donati et al (1992b) to produce maps not only of the surface brightness but also of the magnetic structure for one particularly well-observed system, HR 1099, for which data from two different epochs, two years apart, are available. Remarkably, in the image recon- struction from the first data set, the magnetic field lines seem to emerge more or less radially from an equatorial warm region; in the second data set, the field seems to be directed toroidally, in one sense around a cool polar spot, and in the other close to an equatorial cool spot. Like other manifestations of stellar magnetic activity, the occurrence of starspots seems to be linked to rotation. Hall (1991) has shown that stars with and without (photometric) spots are rather cleanly separated according to whether the Rossby number Ro = (rotation period)/(convective turnover time) is smaller or larger than 2/3, regardless whether the stars are single or in binaries. The predominance of spots in close binaries probably results from the tidally forced synchronization of the rotation and orbital periods to small values of rotation period. For giants, again a critical value of Ro separates non-variables from variables. Further parameters are presumably involved in the occurence of spots; once into the spotted regime, spot size (amplitude of variation) has little relation to the value of Ro. In none of the manifestations of stellar magnetism discussed above - chromospheric activity, coronal x-ray emission, or starspots - is it yet clear how to relate the phenomenon observed to the underlying magnetic field. However, these phenomena are certainly quite direct tracers of magnetism, and we may expect considerable progress in coming years in the exploration of the nature of stellar activity and its relationship to the underlying magnetic field structure.

6. Conclusion

Over the past few decades the observational study of magnetic fields in stars has progressed enormously, from a subject with the sun as its only known example to one encompassing objects as different as dynamos in solar-type stars and megaganss fossil fields in white dwarfs. The subject is still unified to a large extent, however, by methods of measurement and analysis; and its relevance to other fields of astronomy is greater than ever.

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Received 9 March 1992; accepted 8 April 1992

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