c Pleiades Publishing, Ltd., 2011. ISSN 1990-3413, Astrophysical Bulletin, 2011, Vol. 66, No. 2, pp. 144–160.  c Yu.V. Glagolevskij, 2011, published in Astrofizicheskii Byulleten, 2011, Vol. 66, No. 2, pp. 158–175. Original Russian Text 

Magnetic Field Structures in Chemically Peculiar Stars Yu. V. Glagolevskij1 1

Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnii Arkhyz, 369167 Russia Received February 24, 2010; in final form, October 15, 2010

Abstract—We report the results of magnetic field modelling of around 50 CP stars, performed using the “magnetic charges” technique. The modelling shows that the sample reveals four main types of magnetic configurations: 1) a central dipole, 2) a dipole, shifted along the axis, 3) a dipole, shifted across the axis, and 4) complex structures. The vast majority of stars has the field structure of a dipole, shifted from the center of the star. This shift can have any direction, both along and across the axis. A small percentage of stars possess field structures, formed by two or more dipoles. DOI: 10.1134/S1990341311020027 Key words: Stars: magnetic fields—stars: chemically peculiar

1. INTRODUCTION So far, we have reached no agreement on the magnetic field structures on the surface of chemically peculiar (CP) stars. Nothing is known about their internal structure, what complicates the theoretical study of the stability of magnetic fields and their nature. The magnetic field is maintained throughout the stellar body by the circular currents. However, we can see only what is happening on the stellar surface in the region of formation of spectral lines. From this very region we obtain the data on the magnetic field and chemical anomalies. We are facing the problem of examining the relationship between the magnetic field and chemical anomalies, their mutual distribution on the surface in order to clarify the diffusion theory. The technique of measurements, based on the Zeeman effect as a rule allows to detect only the field component, longitudinal along the line of sight Be, averaged over the visible hemisphere. To date, there are quite a few phase dependencies Be(Φ), which can be analyzed. The data on the mean surface magnetic field, derived from the split Zeeman components are available only for a limited number stars with very narrow spectral lines (as a rule, caused by the slow rotation). In his first publications, Babcock believed that magnetic field in CP stars is described by a magnetic dipole, located at center of the star. The dipole axis can be inclined to the axis rotation at any angle β. A rotating dipole magnetic field produces a sinusoidal variation of the dependence Be(Φ). Considering the shapes of the spectral line profiles, he claimed that they always have sharp wings, indicating an almost constant magnetic field strength on the surface, and the absence of regions with a weak and very strong

field, similar to those, observed in the Sun. The presence of the so-called crossover effect in some phases of the period was also typical for the case of the dipole field, rather than for the case of the spot structure of the field. The question of the field configuration grew more complex when it became clear that not all the stars reveal harmonic phase curves of the magnetic field. To date, enough observational data was acquired, using which we can try to investigate the basic properties of global structures in magnetic stars. 2. STUDIES OF MAGNETIC FIELD STRUCTURES IN CP STARS Our views on the structure of magnetic fields of CP stars have evolved as follows. By 1950, a general view was established that magnetic CP stars are in fact the so-called rigid oblique rotators [1–3]. Within this theory, a magnetic dipole is located in the center of the star, the axis of the magnetic dipole is inclined at an angle β to the axis of rotation, and the chemical elements are concentrated in bounded regions, or the so-called “spots”. Babcock admitted the presence of multipoles of higher orders, but believed that the dipole was a dominant component. The theory of a rigid oblique rotator was developed by Stibbs and Schwarzschild [1, 2] under the assumption that a magnetic dipole is located in the center of the star. Stibbs explained why there occur periodic changes in the magnetic field, both reversible and irreversible. 1954–1958. Deutsch [3–7] has repeatedly discussed the results of observations of CP stars within the rigid oblique rotator models. He found that some 144

MAGNETIC FIELD STRUCTURES IN CHEMICALLY PECULIAR STARS

of them neither match the central dipole field, nor another axisymmetric configuration. Nevertheless, it was possible to develop a method using the observations of line intensities, radial velocities and the Zeeman effect. This technique allowed to obtain at least a rough map of the magnetic field and abundance anomalies of the surfaces of magnetic stars [8]. Here Deutsch applied a model of an oblique rotator, using the method of harmonic analysis, based on the harmonic representation of local characteristics. The magnetic field, the local equivalent widths have been decomposed into the spherical harmonics as the function of the coordinates, and the Laplacian coefficients of these expansions have been associated with the Fourier coefficients of the observed curves. He first applied this method for the star HD125248. It was found that the inclination of the stellar axis of rotation to the line of sight is i = 30◦ , and that three groups of chemical elements and the magnetic field itself are inhomogeneously distributed over the surface. The result was satisfactory qualitatively, but not quantitatively. 1967. A study of the properties of the central dipole model was made by Bohm-Vitenze [9]. She as well considered the possibilities of existence of spherically asymmetric stars and magnetic field structures, different from the central dipole to explain the observed anharmonic phase dependencies. In [10] the same author made the first attempt to solve a direct problem, to model such a distribution of chemical elements on the surface of α2 CVn, at which the observed and calculated phase variations of line intensities and radial velocities would coincide the best way. 1967. Preston [11] was the first to try to study statistically the orientation of magnetic dipoles inside the stars using the theory of Stibbs. He found that the orientation can be arbitrary, but predominantly the dipole is located in the plane of the rotation equator. Predominant orthogonality of the axis of rotation to the dipole axis was considered by some authors as an indication that the magnetic field is formed within the dynamo theory (see below). 1969. Piper [12] attempted to test the theory of oblique magnetic rotator using the technique proposed by Deutsch in 1958 on the example of α2 CVn. It was shown that the variations of brightness, color and spectral line intensities are not linked with the temperature, but are due to the inhomogeneous distribution of chemical elements on the surface of the rotating star. Magnetic configuration is presented as a combination of a dipole and a quadrupole with a common axis, inclined by 50◦ to the rotation axis (the quadrupole appears as the second term of a decomposition of the magnetic potential of a uniformly charged sphere into the harmonics). It turned out that the magnetic field of α2 CVn is symmetrical relative ASTROPHYSICAL BULLETIN

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to the meridian, passing through the pole of rotation and magnetic poles, but it is not symmetric relative to the magnetic equator. From the data presented in this work it became clear that the distribution of chemical elements on the surface is controlled by the magnetic field, but the mechanism was unknown. This paper has hence reaffirmed the accuracy of the hypothesis that all the CP stars are oblique rotators with the dipole axes, inclined to the axis of rotation, and that there are no indications of shifts neither in the magnetic field, nor of the chemical features of the stellar surface, which means that the magnetic star behaves like a rigid rotator. The modelling revealed that the chemical elements are divided into several groups according to their distribution on the stellar surface. For example, rare earth elements are concentrated near the magnetic poles, and the iron group elements—at the magnetic equator. Thus, by this time it was finally established that the magnetic CP stars are oblique rotators with an inhomogeneous magnetic field intensity on the surface and a nonuniform distribution of chemical elements (there have been attempts to explain the observed variations by hydromagnetic oscillations). 1969. In [13] Preston clearly establishes that the structure of the magnetic field of HD215441 is not dipole either, just like in α2 CVn. 1970. While the list of stars with anharmonic phase dependencies of the magnetic field was growing, work has begun on the study of more complex structures than the central dipole. In particular, there were models proposed with a shifted dipole [14], and, later on, with the central dipole, combined with a parallel linear quadrupole [15]. Such models could describe well the structures with different magnetic field strengths on the poles. But such models corresponded only to a part of stars. 1970 is also notable for the advent of the diffusion theory [16], which explains the appearance of chemical anomalies in the atmospheres of magnetic stars under the influence of gravitational forces and radiation pressure. The diffusion theory is still supported and developed as the only valid one. 1971. Hence, the observational data show that a part of magnetic stars possesses sine phase dependencies of the magnetic field variations, and another part of them has phase dependencies different from sinusoidal. Preston [17] has developed a method for determining the parameters of stellar magnetic fields based on the assumption that the magnetic dipole is located in the center of the star (the central dipole model), which leads to sinusoidal phase dependencies. He obtained simple formulas, with the aid of which, having the observed dependencies Be(Φ), one can estimate the main parameters of the magnetic

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field—the angle between the dipole axis and the rotation axis β, the field at the poles Bp, and the mean surface field Bs. This method was widely used for analyzing the properties of magnetic stars, and typically resulted in minor differences with the parameters, obtained by more accurate methods, taking into account the deviations from sinusoidal phase dependencies. In [17] Preston comes back to the study of magnetic field orientations using additional observational data. He finds that a part of stars has a preferred orientation of the dipole axes perpendicular to the axis of rotation β ∼ 80◦ , while the other part—β ∼ 20◦ . Landstreet [14] too investigated the orientation of the shifted dipoles, and obtained principally the same result. Thus, there emerged difficulties for the dynamo theory, which requires that β = 90◦ for all the stars. In [17] Preston also for the first time investigated the distribution of magnetic stars by the field strength, and found that the peak is observed at Be∼ 2 − 3 kG, followed by a gradual decline down to tens of kilogauss. Based on the requirements of the magnetic dynamo theory, Krause and Oetken [18–20] have built equatorially symmetrical dipole-quadrupole models of some stars. They proposed two models of this construction: 1) a dipole in the equatorial plane plus a linear quadrupole, parallel to the axis of rotation, and 2) a dipole plus a linear quadrupole, parallel to each other in the equatorial plane. The latter model is similar to the model of a shifted dipole, but with the requirement β = 90◦ . It seems to be able to describe the observed phase dependencies of Be (as a simple oblique rotator) for some stars. However, such structures do not fit most of stars. Besides, most of authors now believe that magnetic fields of CP stars can not be the result of the solar dynamo, since they do not possess sufficiently strong convection zones. 1974. An attempt was made in [21, 22] to study the distribution of chemical elements in the form of round spots on the surface of the star 21Per. Later, in [23] we used the method of “magnetic charges”(see below) to build a magnetic field model of this star. It turned out that the earlier obtained coordinates of the major chemical spots coincide with the coordinates of magnetic poles. It was one of the first papers in which an attempt was made to determine the location of the regions of concentration of chemical elements on the stellar surface and find the relation between the magnetic field and chemical anomalies. 1974. Stift [24, 25] has refined Landstreet’s theory of the shifted dipole [14]. The main advantage of Landstreet’s model, compared to the central dipole by Stibbs [1] is a much larger range of applicability of this model. The generalized model includes a dipole located at any point inside the star, and

inclined at any angle relative to the axis of rotation. A shift of the dipole can occur in any direction. Clearly, Landstreet’s model represents a special case of the generalized oblique rotator, in which the main axis of the dipole always passes through the center of the star (our cases, when the shift is + or – (Section 4)). The technique developed has allowed to explain at the first approximation the features of the stars HD126515 and HD137909, for which both phase dependencies Be(Φ) and Bs(Φ) are calculated, as well as for HD112413, HD125248 and HD153882 based on one dependency Be(Φ). The model of the central dipole is not suitable for these stars. 1977. The work of Khokhlova and her colleagues is well-known [26–28]. Based on the study of the shapes of spectral line profiles, and the polarization distribution in them, they for the first time constructed the maps of the distribution of chemical elements on the surfaces of CP stars, and determined the parameters of the magnetic field (the inverse problem or the so-called Doppler mapping). This approach is fundamentally different from what was previously done. However, such studies were possible only using the spectra with a very high signal-to-noise ratio and very high spectral resolution. Another disadvantage of this method is poor accuracy of localization of the spots and inability of mapping of the invisible side of the star, similar to Deutsch’s method. We applied this technique to study the structure of the magnetic field of α2 CVn from the Zeeman spectra with inverse ˚ dispersion of 1.3 A/mm, obtained at the 6-m BTA telescope [29], which turned out to be close to the central dipole field structure. In principle, the technique based on the analysis of line profiles can yield more accurate properties of the magnetic field than the one that uses the phase dependencies. However, it can be applied only to a small number of the brightest objects. 1980. The deviations from the sine shapes of phase dependencies for a number of stars have been eliminated by the magnetic field measurements along the hydrogen lines [30]. Thus, a significant effect of inhomogeneous distribution of chemical elements on the phase dependencies of magnetic field was discovered. 1981. An attempt to study the structure of the magnetic field of the star HD215441 was made in [31] using the analysis of relative intensities of the Zeeman split components of the lines of different chemical elements using the technique from [32]. It was found that the central dipole model gives the best results. This experiment was unique. 1982. Goossens et al. [33] applied a model of an axisymmetric magnetic field that contains poloidal and toroidal components for HD215441. The reason ASTROPHYSICAL BULLETIN

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for this construction was that only this field configuration can theoretically be stable. It turned out that the axis of the poloidal component is inclined to the axis of rotation by an angle β = 52◦ − 62◦ . This paper demonstrated that the calculated phase dependencies Be(Φ) and Bs(Φ) are in good agreement with the observations. Each model is characterized by the values Hp, ΔHp and M , which denote the field at the pole, different field strengths at the poles, and the relative strength of the toroidal and poloidal components. The application of this method to HD215441 has shown that the surface structure is primarily characterized by a dipole component and has a value of ΔHp in the range from −40 to −50 kG. The solutions with a large field difference at the poles ΔHp lead to a strong dipole contribution to the poloidal component, and have a weak toroidal component, whereas the solutions with small ΔHp values have a strong toroidal component. This technique was not applied to other stars. However, the observational data of different authors did not confirm the presence of a toroidal component in any given single star (also see the Conclusion). 2000. Currently, another magnetic field modelling technique [34] is applied with the use of collinear dipole+quadrupole+octupole. Landstreet [15], as mentioned above, has easily modelled the stars with different magnetic field strengths at the poles using the configuration of a dipole plus a parallel linear quadrupole. This allows to give a reasonable physical description of the field geometry in some cases. However, modeling of some stars [35, 36] has shown that an introduction of an octupole component was required for a more accurate representation of observations. This technique actually gives a description of the phase dependence. The parameters of the dipolequadrupole and dipole-quadrupole-octupole models, or those of a sine curve, approximately describing the phase dependencies of magnetic field and sometimes used, do not give the notions about the source of the magnetic field. They only describe the features of phase dependencies (in the case if the phase dependencies differ from sine). Any magnetic field can be decomposed into harmonics. It turns out that in many cases the first three harmonics can be used with sufficient accuracy: dipole+quadrupole+octupole. This is an implicit technique, since in fact, the magnetic field inside the star is created by elementary circular currents. When magnetic fields of elementary sources are summed up, they create a large-scale field, described by one virtual magnetic dipole with a moment M . A further development of the technique for investigating the distribution of the magnetic field on the surface of magnetic stars is done by Piskunov [37, 38]. ASTROPHYSICAL BULLETIN

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His method makes it possible to obtain the distribution of the magnetic field vector and the abundance of chemical elements on the surface. Here we do not have to set the geometry of the magnetic field. The method is based on the modern transfer theory and an effective minimization technique. The needed data are obtained from the rotational modulation of line profiles. The solution technique is based on considering the radiation transfer for 4 Stokes parameters. The program is able to reconstruct different magnetic field geometries from a simple dipole to the superposition of several multipoles in the presence or absence of chemical anomaly regions. The Doppler mapping technique has been modified by various authors [39], and the Doppler-Zeeman mapping technique was used in [40, 41]. All these methods yield good results when using the spectra of very high resolution R > 100000 and S/N over 1000, i.e. in the study of the brightest objects. In this review, we listed the key milestones in the study of magnetic field structures in chemically peculiar stars. For the further discussion let us specify some of the properties of the method we use. As far as in 1947 Babcock, analyzing the structure of spectral lines on the surface of chemically peculiar stars has assumed that the magnetic field has the structure of a magnetic dipole located in the center of a star. Its surface lacks the magnetic field “spot” structure similar to solar. The surface is uniformly magnetized in accordance with the structure of a dipole field. It is logical to assume that the field has a dipole structure at some depth, but not farther than down to the convection nucleus, inside which a large-scale poloidal magnetic field cannot exist. The magnetic field structure inside the stars is not known, which presents a large problem nowadays. In our modeling technique [42–44], in contrast to the above methods we do not set the structure of a field in advance. Instead, we successively enter the needed number of magnetic dipoles and place them in such points inside the star, that their total action would result in the calculated phase dependencies, identical to the observed ones. On the basis of such calculations we judge what type of magnetic field source is located inside a star. The calculations are carried out using the method of consecutive approximations. For the majority of stars it appears sufficient, within the limits of measurement accuracy, to assume the presence of two magnetic monopoles in the center or near the center of the star, i.e. a dipole. Some stars possess more complex structures. Using our technique we are trying to reveal the large-scale structure of a magnetic field. At the same time our modeling program allows to calculate the distribution of magnetic field intensity over the surface, to determine the value of magnetic field strength on the poles,

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to calculate the phase dependence of the mean and longitudinal surface magnetic field. The first experience of application of our method has shown that the use of the combinations dipole +quadrupole or dipole+quadrupole+octupole results in non-real field distributions over the surface [45]. Subsequent research has shown that the assumption of the central or shifted dipole results in the field distribution over the surface, which corresponds to the estimated distribution of chemical elements. Basically, we can use any number of magnetic charges to obtain the best concurrence with the observational data, but two monopoles (charges) are normally already enough to have the observed and calculated dependencies coincide well. Therefore, there are no reasons to complicate the structure additionally entering a quadrupole, an octupole, etc. The main shortcoming of our method is that it makes it possible to examine only the large-scale structures of a field, while the parameters of a field strongly depend on the measurement accuracy of the angle i to the line of sight, which is often determined with a large error. However, if both phase dependencies Be(Φ) are Bs(Φ) are known, the angle i can be determined accurately. In this case we obtain two solutions and choose the appropriate one based on indirect data. Our technique [42–44] is based on the sources of magnetic field, the circular currents of charged particles. The magnetic field is a vector variable, summing up, the elementary magnetic field sources create a common vector of the stellar magnetic field. The total vector of the magnetic field usually corresponds to one virtual dipole located in the center of the star, or at some distance from the center (a shifted dipole). This virtual magnetic dipole consists of +Q and -Q charges, spaced from each other by an average distance l. The variable l has an atomic size, but in the calculations it is adopted as about 0.1 stellar radius. With such a value of l and the same value of the magnetic moment M , the charges Q are numerically acceptable for calculations. With the present measurement accuracy values, if we change the value of l to 0.2 stellar radii, it would have virtually no effect on the shape of the phase dependence of the magnetic field. You can find papers which describe the phase dependencies of the magnetic field by a sine wave, and provide the parameters of this sinusoid. These approximations bear virtually no information. 3. STARS WITH THE CENTRAL DIPOLE This case is divided in two when: 1) the phase dependence of the longitudinal field Be(Φ) is indeed sinusoidal with a sufficient accuracy, and 2) when the

dependence Be(Φ) is constructed from the measurements with poor accuracy and the true relationship is not observed. In this case the obtained parameters of the magnetic field are approximate. In the first case, we can confidently talk about the dipole in the center of the star. In the second case, in the modelling procedure the dipole in the center of the star is set as a first approximation. Obviously, in this scenario the magnetic field parameters may differ from those that would exist if we had accurate measurements. We will further discuss what error can come out of it. Table 1 lists all the stars we studied, which revealed the central dipole field structure. The objects, belonging to the second case are marked by a question mark. In this and subsequent tables we use the following designations: β is the angle between the rotation axis and the dipole axis, i is the angle of inclination of the rotation axis to the observer’s line of sight, Bs is the mean surface magnetic field, Bp is the magnetic field on magnetic poles (in the case of the central dipole it is equal on both poles), Δa is the value of the dipole’s shift from the center of the star in the units of stellar radius. The scheme of field line geometry for the model of the central dipole, and the distribution of the field on the surface are demonstrated in Fig. 1. Only seven stars from the entire list have confirmed field structures, corresponding to the central dipole, in 11 stars the central dipole is set as a first approximation. The error of Δ, exceeding 0.05 − 0.10 is star selection criterion for the modelling using the central dipole. A comparison with results of other authors shows that the difference by angle β can exceed 10◦ , by angle i it sometimes reaches 40◦ . In fact, the accuracy of our modelling is affected the greatest by the inclination angle error. Mean surface magnetic field in our models is determined by the calculated phase dependencies as (Bs(max) + Bs(min))/2. 4. STARS WITH A DIPOLE, SHIFTED ALONG THE AXIS This is the largest group of stars the study of which led Landstreet to the assumption that the dipole is shifted in them along its axis [14]. In contrast to the stars with a central dipole, having almost sinusoidal phase dependencies of the magnetic field, in stars of this type the phase dependencies are anharmonic. In all the stars of this group studied by us, a shift of the dipole leads to a good agreement between the calculated and observed phase dependencies Be(Φ). Table. 2 lists our parameters, and the parameters obtained by other authors. The designations are the same as in Table 1. The column Δa in brackets shows the sign of the monopole, towards which it is shifted. It is outstanding that the dipole can be shifted in ASTROPHYSICAL BULLETIN

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Fig. 1. The Mercator map of the magnetic field distribution on the surface in case of the central dipole model. The abscissa is longitude, the ordinate axis is latitude. The bottom plot: the schematic representation of line geometry (the dashed line) at the central dipole in the plane of the magnetic equator. The solid line shows the surface of the star. Black half-circles mark the positions of the poles.

any direction, towards (+) and (–) equally probably. Obviously, this fact is important for determining the nature and evolution of magnetic fields, just like the fact that this is a prevailing type of magnetic CP stars. The scheme of field line geometry for the cases Δa = 0 is shown in Fig. 2. It is clear from the data in Table 2 that in 8 stars the dipole shift occurs towards the positive monopole, and in 7 stars—in the direction of the negative monopole, i.e. no sign is preferential, the shift is equiprobable. The maximum shift amounts to Δa ∼ 0.6. Here the ratio of the magnetic field strength at the poles reaches Bp(max)/ Bp(min)= 35 (HD45583). However, this star is unique and has yet to be additionally examined. Another star HD147010 has Δa = 0.45, and the ratio of the field strengths at the poles is equal to Bp(max)/ Bp(min)= 15. This shows that the deformation of the magnetic field configuration can reach a considerable magnitude. In such structures (marked by colons), the angle β does not posses ASTROPHYSICAL BULLETIN

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such a great value as in the case of a central dipole. The angles β in the studied stars range from 5◦ to 90◦ . It stands out that there is a disproportionately great number of large angles β, as already noted by Preston and Landstreet [17, 47]. In other words, among the stars from Tables 1 and 2, there is a marked tendency for the dipoles to be located in the equatorial plane. This is one of the most important properties of magnetic fields of CP stars. 5. MODELS OF MAGNETIC FIELD WITH THE DIPOLE, SHIFTED ACROSS THE AXIS The list of stars with magnetic field structures of the dipole, shifted across the axis is given in Table. 3 The star HD21699 reveals a structure, where both monopoles are shifted from the center by the value Δa = 0.4 across its axis. There are very few of such stars detected as yet. The latitude of the dipole is near the equatorial plane. As a result, closely spaced

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GLAGOLEVSKIJ Table 1. Models of magnetic stars with a central dipole №

HD

Type

β, deg i, deg Bs, G Bp, G

Δa

Source

1

3360

He-r

87

18

294

±517

0

[46]

79

18

–

±335

0?

[47]

2

2453

SrCrEu 80

14

3750 ±6560

0

[48]

3

4778

SrCrEu 65

70

–

–

0?

[49]

80

40

6000 –

0?

[50]

81

56

2600 ±4030

0?

[46]

88.5

20

3190 ±5600

0?

[46]

89.5

5000: ±8100 :

0?

[51]

68

47

–

±3000

0?

[30]

75

50

1000 ±1550

0?

[52]

4

5737

He-w

5

9996

SrCrEu 12:

6

11503 Si+

7

12098 Si+

46

55

2450 ±2720

0

[52]

8

12288 SrCrEu 66

24

8080 ±13400

0

[48]

9

18296 SrCrEu 88

14

890

±1580

0?

[23]

10

19832 Si

80

74

–

±1200

0?

[30]

90

34

495

±840

0?

[23]

11

22470 Si

90

25

2350 ±4100

0

[23]

12

24712 SrCrEu 33

49

–

0?

[49]

38

44

1250 ±1800

0?

[23]

55

26–90 –

0?

[53]

69

42

1000 ±1620

0?

[52]

14

112185 SrCrEu 82

51

330

0

[68]

15

115708 SrCrEu 87

55

3850 ±6100

0?

[55]

75

50

–

–

0?

[49]

77

60

–

–

0?

[56]

13

34452 Si

–

±520

16

178892 SrCrEu 74.5

9

17440 ±30180 −

0

[52]

17

192678 SrCrEu 70

8

4700 ±7300

0?

[54]

60

10

–

–

0?

[49]

60

27

–

±6500

0?

[57]

34

3800 ±6210

18

201601 SrCrEu 85.5

magnetic poles of positive and negative signs appear at the equator on the stellar surface, and the distance between the magnetic poles is hence about 55◦ ,

0? – [58]

instead of the usual 180◦ in the case of the central dipole. In HD40312 both monopoles are also equally spaced from the center of the star, Δa = 0.2, but are ASTROPHYSICAL BULLETIN

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Fig. 2. Same as in Fig. 1, for a model of a dipole, shifted along the dipole axis.

located at different latitudes, at ±5◦ relative to the equator, which also leads to a considerable l value. In longitude the position of poles differs by 110◦ . In contrast to the discussed stars, the dipole axis in HD119419 is oriented almost parallel to the axis of rotation, the distance between the monopoles is also large. The star HD126515 is a rather complicated case. Its monopoles are shifted across the axis by Δa = 0.4(–) and 0.3(+), but above the plane of the rotation equator: the width of both monopoles amounts to δ = 30◦ . Therefore, the distance between the magnetic poles is equal to 60◦ , and the distance between the monopoles is about 0.4 stellar radius. At the discussed configurations, the angle β ceases to be as relevant as in the case of the central dipole. In addition, an important point is that there is a considerable distance between the monopoles, and hence the field structure that occurs is not dipolar, but rather a rod field structure. From these examples we see that the dipole can be shifted from the center of the ASTROPHYSICAL BULLETIN

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star not only along but also across the axis. We should emphasize that the shift of the dipole along the axis affects the phase dependencies much stronger than a shift across the axis. Probably this is the reason why such a configuration is less frequent. The effect on the phase curve is not as prominent. Within the magnetic field measurement accuracy, the variation in the shape of the phase dependencies is noticeable only at very large deviations from the center of the star. The obtained results hence have a low accuracy, and have to be treated mainly qualitatively. A small number of stars with a transverse shift of the dipole seems unnatural from the point view of physics. The scheme of the orientation of magnetic field lines for the case is shown in Fig. 3. Figure 4 demonstrates the case where Δa for the monopoles is different. The main problem lies in the fact that the dipoles obtained are not point-like, as in previous cases, but have large l values. This case will be investigated specifically.

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Fig. 3. Same as in Fig. 1, for a model of a dipole, shifted across the dipole axis.

6. COMPLEX MAGNETIC FIELD STRUCTURES Unlike the above examples, where the magnetic field is formed by a shifted dipole, there are objects with more complex configurations. By now four such objects are known, they are listed in Table. 4. The phase dependence Be(Φ) of the star HD32633 is unique in that the descending side, after reaching a “positive” peak reveals another maximum. The calculations show that no single-dipole orientation can yield this shape of dependence. Indeed, if we insert another dipole in the model, we obtain the calculated phase dependence, which is in good agreement with the observed one. Both dipoles are located near the equatorial rotation plane on both sides of the rotation axis, and at the same distance from it, Δa = 0.6. This configuration can not be considered a quadrupole due to the large distance between the dipoles and different values of magnetic moment.

The distance between monopoles is also large, and hence the field structure corresponds to the structure of the two rods with opposite signs, rather than to the point dipoles. The longitudes of magnetic charges of the first dipole are equal to 27◦ and 47◦ , and of the second dipole: 215◦ and 217◦ , and their latitudes are −5◦ and +12◦ , −5◦ and +23◦ , respectively. Consequently, they are located near the plane of the equator. In this configuration the angle β has not got the same sense, as in the case of a central dipole. There appear two pairs of closely spaced magnetic “spots” on the stellar surface. The star HD35502 has not yet been studied well enough, but there is convincing evidence that its magnetic field has a structure of two dipoles, as in the case of HD32633. The monopoles are also located near the equatorial plane at the latitudes ±10◦ . In longitude, they are spaced from each other by about 90◦ . The star HD37776 has a more complicated phase ASTROPHYSICAL BULLETIN

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Fig. 4. Same as in Fig. 1 for a model with monopoles, shifted by different values.

dependence than HD32633 and HD35502, three maxima and three minima are clearly visible on it. Consequently, the method of successive approximations led to the configuration of three dipoles. The monopoles are located near the equatorial plane, the positive monopoles being located in one hemisphere, and negative in the other. In longitude the dipole centers are spaced by approximately the same distance equal to 120◦ , while the distances of the monopoles from the axis of rotation are equal. Together they lead to the configuration of the poloidal field with an axis almost parallel to the axis of rotation. The star HD137909 has a characteristic field structure too. It is formed by two dipoles, shifted ASTROPHYSICAL BULLETIN

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across the axis by Δa = 0.3 in opposite directions from the axis. The monopoles are located near the equator of rotation. They form four magnetic poles with the same strength Bp = ±14500G. The observed configuration can not be considered a quadrupole since the monopoles form an elongated rectangle. Thus, the three stars considered (except for HD37776) share a common property—their fields are formed by two dipole sources shifted from the center, but all four stars have dipoles, located near the equatorial plane of rotation (see diagram in Fig. 5). We have to note that is not yet clear what causes

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GLAGOLEVSKIJ

Fig. 5. Same as in Fig. 1 for a model with two shifted dipoles.

Fig. 6. Possible internal structures of the magnetic field with a central (a) and shifted (b) dipole in the presence of a convective core.

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the large l values. Is this a methodical, or a physical cause? More research is needed here.

ples, when the data is obtained under the assumption of central dipole and shifted dipole models.

7. ON THE POSSIBILITY TO USE THE CENTRAL DIPOLE MODEL IN UNCLEAR CASES

The table shows that as a rule the angles of inclination of the dipole axis to the axis of rotation β differ little, except in the case of a complex field in HD32633. The star HD12288 also reveals a large difference. The values of the surface field Bs are also of the same order. The field at the poles Bp can be different, but the mean value of (Bp(max) + Bp(min))/2 is approximately equal to Bp for the central dipole. Thus, in the case of statistical studies, we can use the parameters obtained under the assumption of a central dipole.

The question naturally arises how great the error may be in the calculated parameters of the magnetic field if we apply the central dipole model for the stars with different configurations. When there are few measurements available, or their accuracy is low, and the phase dependence has a large scatter of points, we always compute the parameters, assuming the model of the central dipole. Table 5 lists the exam-

Table 2. Magnetic field models with a dipole, shifted along its axis №

HD

Type

1 12288 SrCrEu

β, deg i, deg Bs, G 12

78.5

8100

Bp, G

Δa

Source

–15800

0.08(–)

[48]

+9700

2 12447

21

61

8100

–

0.01(+)

[59]

90

38

782

+2260

0.2(+)

[52]

–670 3 14437 SrCrEu 1 var.

14:

65

7665

–

0.23(+)

[59]

5:

89+

7665

+8900

0.25(+)

[52]

0.15(+)

[52]

0.6:(–)

[60]

–6100 2 var.

88

6

7665

+22100 –8900

4 45583

Si

90:

39

16000: +9400: –314000:

5 62140 SrCrEu

90

58

+2019

0.045

[61]

–2639 6 65339 SrCrEu

85

58

13250 +12600

0.14(–)

[45]

–29600 80

50

–

–28400

0.145(–)

[62]

80:

60

–

–

0.67(–)

[63]

1 var.

80

65

–

–28000

0.15(–)

[64]

2 var.

65

80

–

–28000

0.15(–)

[64]

68:

9

27430 –72268

0.17(–)

[61]

0.1(+)

[46]

7 75049 SrCrEu

+25592 8 112413 SrCrEu

82

55

2600

+5300 –2890

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GLAGOLEVSKIJ Table 2. (Contd.) №

HD

Type

9 116458 He-w?

β, deg i, deg Bs, G

Bp, G

Δa

Source

+5100

0.2(+)

[13]

+4900

0.05(+)

[29]

65

65

76

50

65

71

+4400

0

[64]

65

65

+10000

0.2(+)

[64]

60:

63

+26700

0.4(+)

[64]

12

75

+9510

0.07(+)

[65]

2900

4650

–6220 10 118022 SrCrEu

65

22

–

+8000

0.2(+)

[66]

11 124224

87:

60

2200

–7900

0.30(–)

[67]

0.24:(–)

[65]

0.36(–)

[13]

0.45(+)

[68]

0.10(–)

[69]

–

0.055(–)

[70]

–7400

0.07(–)

[51]

0.08(+)

[48]

–

0.09

[71]

–

0.2–0.3(+)

[72]

+55100

0.03(+)

[46]

Si

+1200 12 126515 SrCrEu

86:

22

13500 –45800: +11100:

13 147010

Si+

95

17

25:

76

-19400 12000: +103000 –7000

14 187474

Si+

24

86

5500

–11600 +6300

15 188041 SrCrEu

37

89

7

83

3600

+4850 16 200311

Si+

86

30

8500

+18520 –11420

90

28

17 215441 SrCrEu 30–35 30–35 34000 10

67.5

–45900

8. CONCLUSION The data, discussed in Sections 3–6 (and Fig. 1) confirm that magnetic stars possess large-scale poloidal field structures with an arbitrary orientation of the dipoles. The relative number of stars with symmetric magnetic fields (central dipole) is small (see Table 6), only 17 %. A large number of stars possess structures, different from the central dipole. The biggest ratio of stars reveal the structures a the dipole, shifted along the axis. Obviously, there is a physical reason behind this, but to be able to make

any confident conclusions we have to increase the statistics. It is likely that a small number of stars with the dipole, shifted across the axis is due to their weak effect on the phase dependence Be(Φ). These objects are just difficult to detect. A small number of complex configurations may indicate a low probability of occurrence of physical mechanisms, strongly distorting the magnetic field structure. As already mentioned, the structure of the magnetic field of CP stars, listed in the third and fourth rows of Table 6 most likely do not correspond to a ASTROPHYSICAL BULLETIN

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Table 3. Magnetic field models with a dipole, shifted across the axis №

HD

Type

1 21699

i,deg Bs,G Bp, G

He-w

32

Δa

6150 +21800 0.4 (both monopoles)

Source [73]

–21800 2 40312

Si

52

650

+1420 0.2 (both monopoles)

[52]

–1420 3 119419

Si

89 23000 +37000 0.05 (both monopoles)

[55]

–37000 4 126515 SrCrEu 22 13000 –60000

0.4 and 0.3

[65]

+20000 Table 4. Complex structures of magnetic fields in CP stars №

HD

1

32633

Type Si+

Bs, G 12000

Bp, G ±42447

Δa

Source

0.6 (2 dipoles)

[74]

0.1 (2 dipoles)

[75]

±19302 2

35502

He-w

3

37776

He-r

–

–9000:

+70000 ∼ 0.18 (3 dipoles)

[76]

–67000 4

137909 SrCrEu

5500

dipole in the mathematical sense, i.e. having the value of the axis of l, comparable to the size of the eddy currents, but rather to a bar magnet with a large l, comparable to the radius of the star (this source of magnetic field can be called a “long dipole”). But even in those cases when the field of the star is described by the central dipole, we can not assert that l has atomic dimensions, because at the calculations the variations of Δa from 0 to 0.2 has little effect on the calculated phase dependence Be(Φ). Such an effect is beyond the accuracy of measurements. It is possible that the source of the field in the stars with the central dipole also has a configuration of the rod. This assumption requires a special study. A refinement of the field structure inside the star can be important for the theory of formation and evolution of magnetic fields. Besides, the fact that the source of the field is similar to a rod magnet can indicate the presence of structural field features in internal and external regions of the star. For example, it is unknown what kind of effect the existence of a convective core has on the field structure in the surface layers. In the body of the star, electric currents are flowing from a multitude of elementary vortices, supporting the magnetic field. In simple structures ASTROPHYSICAL BULLETIN

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±14500

0.3 (2 dipoles)

[77]

the elementary magnetic vectors sum up into a common vector, described by one virtual magnetic dipole. It is obvious that the features of the internal structure affect the total magnetic field vector. If a large number of sources are summed inside the star, the resulting dipole can be oriented in any direction, and shifted from the center of the star. However, if there are heterogeneities in the depth of the atmosphere, where the spectral lines are efficiently formed, then such a sum, in view of the entire hemisphere, is not converted into a single vector from one virtual dipole, but appears distorted. An effect of stratification of chemical elements is possible. Figure 6 demonstrates the schemes of probable positions of magnetic field lines in the presence of a convective core in the case of the central and shifted dipoles. A magnetic field can not exist in the convective core, and the field lines have to bypass it. However, the direction of field lines on the surface is such that it creates an illusion of a central or shifted dipole. It is quite possible that it is this effect that leads to the anomalous magnetic field gradient, studied by the authors of [78, 79]. Such stars as HD32633 and HD137909 have phase dependencies Be(Φ), not exactly described by

158

GLAGOLEVSKIJ Table 5. Comparison of results of use of different models Star, HD

β, deg Bs, G

Bp, G

Δa

Source

12447 (central dipole)

90

715

±1190

0

[52]

12447 (shifted dipole)

90

782

+2260 –670

0.2 (+)

[52]

12288 (central dipole)

66

–

13400

0

[39]

12288 (shifted dipole)

12

–

–15800+9700

0.08(–)

[39]

32633 (central dipole)

40

–

±12970

0

[48]

32633 (shifted dipole)

48

–

±42447

0.6(across)

[48]

±19302

0.6(across)

5 40312 (central dipole)

75

365

550

0

[52]

40312 (shifted dipole)

85

650

+1420–

0.2 (across)

[52]

0.2 (across) 116458 (central dipole)

10

–

±7290

0

[66]

116458 (shifted dipole)

12

–

+9510–6220

0.07(+)

[66]

–

±22600

0

[66]

126515 (central dipole) 88.5 126515 (shifted dipole)

86

–

–45800+11100

0.24(–)

[66]

187474 (central dipole)

14

–

±8630

0

[69]

187474 (shifted dipole)

24

–

–11620+6320

0.1(–)

[69]

200311 (central dipole)

86

–

14560

0

[39]

200311 (shifted dipole)

86

–

+18520-11420

0.08(+)

[39]

Table 6. Rate of magnetic stars of different types №

Field structure

Ratio of stars

1

Central dipole

17%

2 Dipole, shifted along the axis

38%

3 Dipole, shifted across the axis

9%

4

Complex configurations

9%

5

Unknown structure

26%

a shifted dipole. They demonstrate a relative shift of the extrema of variations of the longitudinal field Be and the mean surface field Bs. The field structure in these stars is adequately described by two dipoles. It is obvious that the deformation of the magnetic field could occur in the early stages of the origin and evolution of stars by an as yet unknown mechanism. One of the basic properties of magnetic configurations is that the dipole axes are inclined at any angle β relative to the axis of rotation, but the number of stars with large β appears to be disproportionately large. At the

same time, our distribution of magnetic stars by the angle β lacks the maximum at β ∼ 20◦ –30◦ , which was obtained by Preston. It is very important for the theory of magnetic field formation. Another important feature is that the dipole shift from the center of the star can occur in any direction, and sometimes by a large value, up to a half of the radius of the star. The results, presented here can be useful for the primary discussion of the problem of magnetic fields in CP stars. Attempts to explain the nature of the observed magnetic configurations have not yet yielded any significant results. The most promising scenario was developed in [80], where a numerical simulation of the formation of a stable magnetic equilibrium in a star with a complex initial magnetic field was made. In the stellar conditions such a field “self-organizes” with time into a poloidal-toroidal structure. Moreover, it was found that the initial field, concentrated in the center of the star evolves into an approximately axisymmetric configuration, close to poloidal plus a toroidal component. A more extensive initial field evolves into a more complex non-axisymmetric ASTROPHYSICAL BULLETIN

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structure, consisting of twisted tubes, coiling under the surface of the star. The author believes that the latter result can help explain the existence of CP stars with non-dipolar fields. Obviously, the processes taking place inside the magnetic stars, are not so simple and require more sophisticated approaches. The main difficulty of the proposed mechanism is that it leads in general to a more complex, intricate system of power tubes than the one observed. In addition, the observed configurations are oriented absolutely arbitrary relative to the axis of rotation (any angle β is possible), but this can not explain the proposed mechanism. Long-term measurements of magnetic fields of stars do not reveal the presence of a toroidal component, which is predicted by the calculations. We have to note that in practically all models the dipole axes lie near the equatorial plane. In the future we have to pay particular attention to this property. If the conclusion about the preferential shift of magnetic structures in the equatorial plane is confirmed, it would be in favor of the hypothesis on the suspected effect of the accreting mass fall-out or even a merger of close binary components [81] on the formation of different magnetic field structures. Any asymmetry is theoretically unstable, but the complete alignment of the structure during the evolution up to the main sequence does not occur due to the slow relaxation process. It is comparable to the lifetime of stars on the main sequence. This is confirmed by the data on the secular stability of magnetic field structures and the associated chemical anomalies in CP stars [82]. This issue is important not only to investigate the nature of magnetic stars, but also for the understanding of the formation and evolution of stars in general. Price et al. [83] show that during the star formation from a magnetized cloud, the magnetic field prevents merging of dense areas, due to what in the future the given magnetic star can reveal magnetic field inhomogeneities. The resulting models show no apparent relationship between the period of stellar rotation and the angle β, which is one of indications that the assumption about the loss of angular momentum by Ap stars through the “magnetic braking” is incorrect. This paper presents a primary discussion of results of magnetic field modeling in CP stars, and will be continued. REFERENCES 1. D. W. N. Stibbs, Monthly Notices Roy. Astronom. Soc. 110, 395 (1950). 2. M. Schwarzschild, Astrophys. J. 112, 222 (1950). 3. A. J. Deutsch, Astrophys. J. 105, 283 (1947). 4. A. J. Deutsch, Trans. IAU. 8, 801 (1954). ASTROPHYSICAL BULLETIN

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