ISSN 08845913, Kinematics and Physics of Celestial Bodies, 2010, Vol. 26, No. 4, pp. 181–191. © Allerton Press, Inc., 2010. Original Russian Text © A.F. Kholtygin, S.N. Fabrika, N.A. Drake, V.D. Bychkov, L.V. Bychkova, G.A. Chountonov, T.E. Burlakova, G.G. Valyavin, 2010, published in Kinematika i Fizika Nebesnykh Tel, 2010, Vol. 26, No. 4, pp. 52–70.

PROBLEMS OF ASTRONOMY

Magnetic Field Evolution in OBA Stars A. F. Kholtygina*, S. N. Fabrikab, N. A. Drakea, V. D. Bychkovb, L. V. Bychkovab, G. A. Chountonovb, T. E. Burlakovab, and G. G. Valyavinc a

Astronomical Institute, St. Petersburg State University, Universitetskii pr. 28, Petrodvorets, St. Petersburg, 198504 Russia bSpecial Astrophysical Observatory, Russian Academy of Sciences, Nizhnii Arkhyz, KarachaiCherkessian Republic, 369167 Russia c Observatorio Astronomico National, UNAM Ensenada, BC, Mexico *email: [email protected] Received October 1, 2009

Abstract—The applications of the spectral analysis methods discovered by Kirchhoff for the investi gation of stellar magnetic fields are considered. The statistical properties of the mean magnetic fields for OBA stars have been investigated by analyzing data from two catalogs of magnetic fields. It is shown that the mean effective magnetic field  of a star can be used as a statistically significant characteristic of its magnetic field. The magnetic field distribution functions F() have been constructed for Btype and chemically peculiar (CP) stars, which exhibit a powerlaw dependence on . A sharp decrease in F() in the range of weak magnetic fields has been found. The statistical properties of the magnetic fluxes for mainsequence stars, white dwarfs, and neutron stars are analyzed. DOI: 10.3103/S0884591310040057

INTRODUCTION The detection of stellar magnetic fields was the result of the almost centurylong development of spec tral analysis. Modern spectral analysis began 150 years ago with Kirchhoff’s works in 1859. Kirchhoff established that each chemical element is characterized by its own set of spectral lines; analysis of their intensities allows the chemical composition and physical properties of emitting bodies, including celestial objects, to be determined. A crucial step in constructing the methods of magnetic field measurements based on analysis of the spectra for celestial bodies was the discovery of sodium D1 and D2 doublet line splitting in 1897 [38]. Since then, the phenomenon of line splitting in a magnetic field called the Zeeman effect has been used to search for magnetic fields, in particular, stellar magnetic fields. A stellar magnetic field was first detected on the Sun by George Hale, who discovered sunspot magnetic fields in 1908 [16]. However, almost 40 more years elapsed before the discovery of magnetic fields in other stars. Only in 1947 did Babcock detect a magnetic field in the star 78 Vir with a dipole structure and a polar strength of about 150 mT [6]. Since then, the methods of stellar magnetic field determinations have been developed intensively and the results of stellar magnetic field measurements have been accumulated. By now, the magnetic fields have been measured for more than a thousand stars of various spectral types, from young T Tau and Herbig Ae/Be (premainsequence) stars to white dwarfs and neutron stars. However, most of the magnetic field measurements refer to mainsequence stars. The lion’s share of them is accounted for by chemically peculiar (CP) Ap and Bp stars with strong regular magnetic fields reaching several tesla. The fraction of Ap/Bp stars with magnetic fields accessible to modern measure ments reaches 10% of the total number of such stars, while it is appreciably smaller for stars of other spec tral types. Stars of spectral type F or later with masses M ≤ (1.5–2)M䉺 often possess strong and irregular magnetic fields whose generation is associated with dynamo action. The latter is eventually reduced to the conver sion of part of the mechanical energy of stellar rotation into the energy of the generated magnetic field. More massive earlytype stars with M > 2M䉺 without convective envelopes are characterized by the pres ence of regular magnetic fields [10]. Studies of the magnetic fields for stars of early spectral types O, B, and A, including the subgroup of Ap/Bp stars, are of particular interest, because the origin of the magnetic fields in these stars is not yet quite clear. A number of researchers believe that the dynamo mechanism is also efficient for hot OBA stars. In this case, it is assumed that the magnetic field is generated in the convective core and, subsequently, 181

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individual magnetic flux tubes buoyantly rise in the radiative stellar envelope [23]. The authors of this paper suggest that a stellar surface magnetic field with a strength of several tens of millitesla can be gener ated in this way. At the same time, however, it remains unclear how a regular field structure can be formed through the random process of the buoyant rise of magnetic flux tubes. The hypothesis about a fossil nature of the magnetic fields in OBA stars is presently more popular. In this hypothesis, the stellar magnetic field observed at the present epoch is assumed to be the remnant (fos sil) of the magnetic field of the molecular cloud in which the star was formed [10]. Wolf–Rayet (WR) stars represent a special case. Until now, despite numerous attempts, no magnetic field has been detected in these stars. At the same time, WR stars are a late evolutionary phase of O stars whose magnetic fields have been detected. Therefore, the presence of a magnetic field in WR stars also seems quite likely. Statistical studies of the correlation between basic characteristics of stars and their magnetic fields can shed light on the mechanisms of magnetic field generation and evolution in OBA stars. The recently pub lished catalog [11] provides information about the magnetic field measurements for more than 1000 stars of spectral types from O to M. We used data from this catalog, along with those from the catalog [2] and results of recent measurements that were not included in these catalogs, for our investigation. STATISTICAL CHARACTERISTICS OF STELLAR MAGNETIC FIELDS The stellar magnetic field strength is determined from the Zeeman shift between the left and right hand circularly polarized components of the line profile [6, 25]. The shift is proportional to the longitu dinal magnetic field component Bl averaged over the stellar disk, which is called an effective magnetic field. To measure Bl, a statistically significant signal is often sought for in the line profiles directly in the Stokes V parameter. To increase the measurement accuracy, the Stokes V parameter is averaged over a large number of lines using the LSD (leastsquares deconvolution) technique [13]. The effective magnetic field Bl depends on the stellar rotation phase and changes from a minimum value of Bmin to a maximum value of Bmax, with Bmin and Bmax often having opposite signs. Thus, Bl is unsuitable for statistical studies of the magnetic fields for a large ensemble of stars. Therefore, a global field characteristic that can be obtained from observations and that depends little on when and at which stellar rotation phases ϕ the measurements were performed should be used. Since the largest variations in Bl with rotation phase are observed for a dipole stellar magnetic field, to choose the soughtfor magnetic field characteristic, it will suffice to analyze the case of a dipole field. In the model of a rotating magnetic dipole, Bl is described by the expression [30, 36] 15 + u  [ cos β cos i + sin β sin i cos 2π ( ϕ – ϕ ) ], B l = B l ( ϕ ) = B p  0 20 ( 3 – u )

(1)

where Bp is the polar field strength, β is the angle between the magnetic dipole axis and the rotation axis, i is the inclination of the rotation axis to the line of sight, ϕ0 is the rotation phase at which Bl(ϕ) is at a maximum, and u is the limbdarkening coefficient. For Btype stars, u = 0.35 [33]. Since the dependence of Bl on u is weak, the same value can also be used for stars of other spectral types. As the soughtfor characteristic of the mean stellar magnetic field that depends weakly on the random values of i and β and is barely sensitive to the rotation phases, we will consider the rootmeansquare (rms) field (see [8])

 =

1 n

n

∑ (B ) . i 2 l

(2)

i=1

i

The summation is extended over all of the performed field measurements B l , where i = 1, …, n. The choice of other statistically significant characteristics of the stellar magnetic field described in [4] is also possible. The following quantity is commonly used to estimate the mean accuracy of a field measurement: σ =

1 n

n

∑ (σ

2

i

Bl

).

(3)

i=1

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Here, σ Bi is the rms error of the ith field measurement. The quantity σ is not the standard deviation of l

the rms field , because the  itself is not a normally distributed random variable. Nevertheless, σ can serve as a measure of the accuracy of the entire series of field measurements. It is generally believed that if

 > 2σ  ,

(4)

then the measured field strengths are real. The standard χ2 statistics is also used to estimate the reliability of the performed field measurements: n

2

χ =

∑ ( B ) /σ i 2 l

2 i. Bl

(5)

i=1

The reduced ratio χ2/n is also commonly used instead of χ2 [11]. When the star has no magnetic field, i the expectation values of B l are zero. The smallness of χ2/n compared to unity suggests that the hypothesis i

of B l = 0, i.e., the absence of a magnetic field in the star, is valid, while χ2/n Ⰷ 1 suggests that the per formed measurements are real. To ascertain whether  can be used as a characteristic of the mean stellar magnetic field, let us calcu ∞) distributed uniformly in stellar late its values in the limit of an infinite number of measurements (n rotation phase. Using Eq. (1) and replacing the summation with integration over the phases ϕ, we will obtain



n→∞

2 2 2 1 – u/5 1 2 B p ⎛  cos β cos i +  sin β sin i⎞ . ⎝ 4 ( 1 – u/3 ) ⎠ 2

(6)

The rotationphaseaveraged ratios /Bp vary (depending on i and β) within the range 0.12–0.30 with a mean value close to 0.20. In real observations, the number of magnetic field measurements is usually small. To understand how the values of the quantities we consider depend in this case on the number of field measurements as a func tion of i and β, let us model the field measurement process in the following way. Suppose that the field measure ments for each of the possible angles i and β were performed Nmf times, where Nmf can be equal to one. We will assume that the stellar rotation phases ϕ at which the field was measured are random variables distributed uniformly in the interval [0, 1]. Suppose also that i are random variables determined by arbi trary orientations of the stellar rotation axis within the full solid angle of 4π. At the same time, the angle β probably varies over a narrower range (±15°) with a mean value of about 45° [5, 18]. For this reason, we performed our calculations for two ranges of β: 30–60° and 0–90°. The calculations were performed as follows: we chose about 5000 random values of the angles i and β that varied within the above ranges. The number Nmf of random rotation phases ϕ was determined for each pair of i and β. Bl was determined for all these phases ϕ from Eq. (1). The values of Bl obtained were used to calculate the ratio /Bp from Eq. (2). Since this ratio does not depend on Bp, the latter was taken to be equal to 1. We estimated the means  /B p and the standard deviation σ for the calculated ratios  /B p in a stan dard way. The results of our numerical experiment show that the ratio  /B p varies within a narrow range from 0.17 to 0.20 with a median value of 0.19. The values of σ are relatively low even for Nmf = 2 and  is statistically significant for any angles i and β [4]. MAGNETIC FIELD MEASUREMENTS The catalog [11] contains the published data accessible to the authors along with the results of unpub lished observations, both their own observations and those of other researchers. The catalog provides information about the magnetic field measurements for 1222 mainsequence stars, Herbig Ae/Be stars, white dwarfs, and giants, 610 of which are CP stars. The correlation between atmospheric chemical anom alies and stellar magnetization has long been known. The number of CP stars with respect to the normal ones has been variously estimated to be no more than 10% [7, 26, 31]. However, despite their relatively small number, they have been studied better in terms of magnetic field measurements. Consider a sample of OBA stars from the catalog [11]. It provides information about the magnetic field measurements per KINEMATICS AND PHYSICS OF CELESTIAL BODIES

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0.1

0

0409 B1 B2 B3 B4B5 B6 B7 B8 B9 Spectral type

Fig. 1. Distribution function of the spectralsubtypeaveraged magnetic field for OB stars.

formed for 11 Otype stars, 466 Btype stars, and 472 Atype stars. Since not all of the performed mea surements satisfy criterion (4), the number of real detections is considerably smaller than that in the catalog. New magnetic field measurements for OBA stars have appeared since the catalog [11] was submitted for publication [9, 18, 20, 24, 29, 32, 33]. Below, we use the new magnetic field measurements for OB stars that are presented in the cited papers and that were not included in the catalog [11] to analyze the statis tical properties of the magnetic fields for OB stars. Figure 1 presents the spectralsubtypeaveraged magnetic fields of OB stars based on data from the cat alog [11]. Since the number of magnetic field measurements for Otype stars that satisfy criterion (4) is small (eight O stars), the figure shows the mean value for all O stars. Although the spread in  for stars of the same subtype is very large and its standard deviations for spec tral subtypes B1–B3 and B5–B9 are comparable to the values of  themselves, we can reach a tentative conclusion about a jump in the mean values when passing from Otype stars to Btype ones. Since for B4subtype stars the catalog provides the magnetic field measurements only for one object (HD 175362), the values of  were averaged for all stars of spectral subtypes B4–B5. The cause of such a jump in  still remains unclear. If the magnetic flux for OB stars is assumed to be approximately constant [29], then this jump could be partly explained by larger radii of O stars than those of B ones. This effect may also be related to a significant rate of mass loss by O stars and the removal of magnetic flux by a stellar wind. MAGNETIC FIELD DISTRIBUTIONS FOR B AND ATYPE STARS To ascertain how the mean stellar magnetic fields depend on stellar parameters, we used  from the catalogs [2, 11] and the new values of  presented in papers cited in the previous section. There are values of insufficient accuracy that are statistically insignificant among the magnetic field measurements in the catalog [11]. To select significant values of , we applied criterion (4). Let us construct the differential magnetic field distribution function f() (magnetic field function) introduced in [3] based on the above catalogs. This function is defined as N ( ,  + Δ  ) ≈ Nf (  )Δ  ,

(7)

where N(,  + Δ) is the number of stars in the interval of mean magnetic fields (,  + Δ), N is the total number of stars with measured . To construct the magnetic field function from the data of [11], we use a sample that includes only Btype stars. The constructed distribution function f() is shown in Fig. 2a. We chose the bins of mean magnetic fields in such a way that at least eight stars would fall within each bin. Only in the regions  < 6 mT and  > 500 mT did this number turn out to be smaller than eight, because the number of stars with very small and very large magnetic field strengths was small. The derived function f() for  ≥ 40 mT can be fitted by a power law



–γ

f (  ) = A 0 ⎛ ⎞ . ⎝  0⎠ KINEMATICS AND PHYSICS OF CELESTIAL BODIES

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(a)

185

(b)

1

0.1

0.01 0.01

0.1

1

0.01

0.1

B, T

Fig. 2. Distribution functions of the stellar magnetic field B based on data from the catalog [11]. The arrows indicate the values of f() at  < 40 mT. The right panel is the same as the left one but for CP stars based on data from the catalog [2].

It turned out that in a wide range of  (0.04–1.2 mT), the distribution function f() for Btype stars could be described by a single dependence (8) with parameters A0 = 0.33 ± 0.04 and γ = 1.82 ± 0.07. A catalog of magnetic fields for CP stars, including 355 B and Atype objects, is presented in [2]. The magnetic fields of 282 objects from the catalog [2] satisfy criterion (4). The magnetic field function con structed from the data of the catalog [2] also has the form (8). The parameters of the fit are A0 = 0.37 ± 0.06 and γ = 1.80 ± 0.19. Note that the parameters of the fits to the magnetic field functions constructed from the data of the two catalogs we consider coincide, within the limits of the fitting errors, although these contain different objects. Thus, we can reach a tentative conclusion about a single magnetic field function, at least for earlytype magnetic stars. Monin et al. [1] constructed the surface magnetic field function from a sample of 57 bright (V < 4.0m) magnetic mainsequence B3–F9 stars. The magnetic field function was fitted by a power law. For  > 0.4 T, the authors obtained γ = 2.2, which is close to our value. In the range of magnetic fields 0.1–0.6 T, Monin et al. [1] obtained γ ≈ 1 and concluded that there was a break in the magnetic field function in the range Bs = 0.3–0.5 T, where Bs is the stellar surface field, which is a factor of 3–4 larger than  [1]. Our data are consistent with the conclusion about the existence of such a break (see Fig. 2). However, since the number of stars with measured magnetic fields in the above range is relatively small, the value of γ cannot be firmly established. The behavior of the function f() at relatively low values of  ≤ 40 mT is of particular interest. The corresponding values of f() calculated using Eq. (7) are indicated in Fig. 2 by the arrows. At such values of , the behavior of f() does not follow the dependence (8). At small mean magnetic field strengths  ≤ 10 mT, the values of the function f() are lower than those obtained from the fit (8) by more than an order of magnitude. Thus, we can assume that the empirical magnetic field distribution function decreases sharply at  below a threshold value of th ≈ 40 mT. The deviations of f() from the law (8) may be due to a low probability of detecting relatively weak magnetic fields. Consider the following model. Let a large number of magnetic field measurements for various stars be performed with a spectropolarimeter that measures the field with an accuracy σ. Let us describe the technique for calculating the probability of detecting the weak magnetic field of a star with a mean magnetic field  for n random field measurements. Suppose that the model star for which the field is measured possesses a dipole magnetic field with a polar field strength p and Ntot Ⰷ 1 random values of the angles i and β and that the observations are performed at n random rotation phases ϕ. In many papers, it is assumed that if B l Ⰷ 3σ, where σ is the rms measurement error, at least for one measurement, then the field is considered to have been detected. For greater generality, suppose that if the absolute value of the measured longitudinal field component B l exceeds 3σ for k ≤ n random rotation phases at the time of field measurement, then the field will be considered to have been detected. The field detection probability is then N det P ( n, σ , , k ) =  , N tot KINEMATICS AND PHYSICS OF CELESTIAL BODIES

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0.8

f 100

(a)

p κ=1

2

(b)

10 3

1

6

0.1

0.4

2

0.01 0

0.01

0.1

1

0.001 0.001

4

8

16

32 mT

0.01

0.1

B, T

Fig. 3. (a) Probability of detecting the magnetic field of a star with a mean magnetic field  (σ = 5 mT). (b) Magnetic field distribution function calculated by assuming that the power law (10) is valid in the entire range of magnetic field mea surements (heavy solid line). The dotted lines indicate the magnetic field distribution functions for OB stars corrected for the possible nondetection of weak magnetic fields at fixed measurement accuracy σ. The values of σ are indicated near the corresponding curves. The thick dots and vertical arrows have the same meaning as those in Fig. 2.

where Ndet is the number of measurements in which the field will be detected according to the above cri terion. Figure 3a presents the probabilities P(n, σ, , k) for a measurement error σ = 5 mT, n = 6 field measurements, and k = 1, 2, 3, 6. We see that the criterion k ≈ n is too stringent, because the field detection probability for this criterion is 1/2 even at  ≈ 50 mT. In contrast, k = 1 seems insufficient, because the probability that the condition Bl > 3σ will be met as a result of the random field measurement error is very high. Therefore, we will use k = 2 adopted in many observational works and denote P(n, σ, , 2) = P(n, σ, ). Note that the field detection probabilities P(n, σ, ) are almost the same for n > 6. Let us ascertain how the magnetic field distribution function would appear at a given measurement error σ under the assumption that the probability of detecting weak fields is described by Eq. (9). Since  > 40 mT and the function P(B, σ, n) ≈ 1 at σ = 10 mT, which generally exceeds the errors of present day magnetic field measurements, we will assume that a magnetic field in the region  > c = 40 mT was detected in all of the stars for which the corresponding measurements were performed. Let min be the minimum strength of the mean magnetic field that can still be determined at the presentday accuracy of measurements. Based on the data of [5, 24, 32], we may conclude that min ≈ 2.5 mT. Suppose that the magnetic field distribution function in the entire region  > min is described by the power law (8). The normalized distribution function will be f

real

–γ

() = A  , *

(10) γ–1

where we assume that 0 = 100 mT, i.e.,  is measured in kilogauss, and the factor A∗ = (γ – 1)  min is defined by the normalization condition

∫

∞ 

f min ()d = 1. Let the field detection probability be equal to the real

function P(, σ, n) described above. The number of stars that will be detected in the interval (,  + Δ) at a given σ is then N ( ,  + Δ  )Δ  = N σB f σB (  )Δ  = N P ( , σ , n ) f *

real

(  )Δ  .

(11)

Here, N σB is the number of stars in which a magnetic field will be detected at a given σ, N∗ is the total number of stars in the ensemble of magnetic stars with mean field strengths distributed according to the law (10). It follows from Eq. (11) that f σ B (  ) = P (  , σ , n ) f

real

real N 1 (  )  * = P ( , σ , n ) f (  )  . N σB Q σ

(12)

In the correction factor Q σ , we took into account the fact that N σB < N∗ at nonzero σ. KINEMATICS AND PHYSICS OF CELESTIAL BODIES

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Using Eq. (11), it is easy to find that Q σ

Nσ = B = N *

∞

∫ P ( , σ

,

n) f

real

(  ) d .

(13)

B min

The values of f () calculated from Eq. (12) are presented in Fig. 3b. The quantity f σB () is the field distribution function that would be constructed from the observations of an ensemble of magnetic stars with mean fields distributed according to the law (10) using an ideal spectropolarimeter that measures the longitudinal field component with an error σ for all of the observed stars. Analysis of Fig. 3 leads us to conclude that, although the function f() for  > 40 mT derived from magnetic field observations can be described by the reconstructed function f σB () at σ = 2 mT, the mag netic field function in the range of weak magnetic fields cannot be reproduced at any σ. The explanation of the cutoff in the magnetic field function for  < 40 mT only by observational selection effects may be incomplete, because the field measurement accuracy σ is currently fairly high, 2–10 mT [5, 24, 32]. This allows magnetic fields to be also detected in the range 4–12 mT. We see from Fig. 3 that a considerably larger number of stars with magnetic fields must be detected in this range at such values of σ. There exists an alternative explanation for the rapid decrease in f() in the range  < 20–40 mT. Glagolevskij and Chuntonov (2000) suggested that if the mean stellar magnetic field is below some thresh old value of th, then the field strength in the stellar atmosphere decreases almost to zero in a short time due to the processes of meridional circulation. Auriere et al. [5] described the magnetic field measurements for 28 Ap/Bp stars; for 24 program stars, they fitted the phase dependence of the measured longitudinal field Bl in the model of an oblique rotating dipole and obtained the polar field strengths Bp. A histogram of the number of stars in bins ΔlnB = 0.2 dex was constructed from the values of Bp obtained. The number of stars with Bp ≈ 30 mT was found to be very small. Based on this fact, the authors suggested that there are stable configurations of global stellar mag th netic field at Bp > B p , while the global magnetic field is destroyed on the Alfven time scale through mag real

th

netic field instabilities at Bp < B p . According to [35, 36], the most important type of instability that destroys the magnetic field is the pinch one. th

th

The threshold value of B p ≈ 30 mT obtained in [5] corresponds to Bth ≈ B p /5 = 6 mT, which is a factor of 6–7 lower than our threshold value of Bth ≈ 40 mT. Such a significant discrepancy may stem from the fact that the number of objects with measured Bp in [5] is small. In addition, analysis of Fig. 6 from [5] shows that there may be another sharp decrease in the number of stars at ln(Bp) ≈ 1.2, which corresponds to a mean magnetic field  = 30 mT close to our threshold magnetic field. STATISTICS OF STELLAR MAGNETIC FLUXES The total magnetic flux  from a star is an important characteristic of its magnetic field. In spherical coordinates (θ, ϕ), it is π 2π

 =

∫ ∫ ( B ⋅ n ) dS = R* ∫ ∫ B sin θ dθ dϕ. 2

(14)

r

S

–π 0

Here, B is the stellar surface magnetic field vector, Br is its radial component, n is the normal vector, and R∗ is the stellar radius. Integration in Eq. (14) for a dipole field gives d = (4/3)πBp R * . Below, we will estimate the magnetic fluxes from stars with known mean magnetic fields  using the formula 2

 = 4π  R 2* .

(15)

This formula gives good magnetic flux estimates for stars, because  ≈ 5/3d even in the extreme case of a dipole field. For a more complex field structure, the differences between the estimate (15) and the exact value calculated from Eq. (14) are considerably smaller. In many theoretical studies, the magnetic flux from a star is assumed [14] to be constant during star’s evolution. Let us first consider the dependence of the magnetic fields of stars on their radii in the ideal case KINEMATICS AND PHYSICS OF CELESTIAL BODIES

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(a) NS

6 WD

2 A

−2

B O RGB

−6 10

(b) WD

6 NS

1E24

2

A 1E20

−2 −6

−4

−2

0

B

O

RGB

2 log(R夹/R䉺)

Fig. 4. (a) Dependence of the magnetic field of a star on its radius in the ideal case of a constant magnetic flux,  = 7 × 1023 T cm2, for all stars and (b) real dependence of the magnetic fields of stars on their radii. The positions of normal stars from the catalog [11] (dots), white dwarfs (squares), and neutron stars (triangles) are shown. The dotted lines mark the lines of constant magnetic fluxes: 1020, 1021, 1022, 1023, and 1024 T cm2 (from the bottom upward).

of a constant magnetic flux for all stars. Let us assume that the magnetic flux for all stars is equal to 0 = 7 × 1023 T cm2, the magnetic flux from the O6type supergiant θ1 Ori [29]. According to Eq. (15), the mean 2 stellar magnetic field strength is then  = 0/(4π R * ). The dependence (R∗) calculated in this way is presented in Fig. 4a. The ranges of radii for O, B, and A stars, white dwarfs (WD), neutron stars (NS), and red giants (RGB) are marked in the figure. We see that the approximation of a constant magnetic flux can qualitatively explain the wide variety of stellar magnetic fields. For the real dependence of the magnetic fields on stellar radius to be constructed, both the magnetic fields and the radii of stars must be known. For this purpose, we obtained the radii R∗ for all of the stars included in the catalog [11]. We took the radii approximately for 100 objects from the catalog of stellar radii [28] and found R∗ for the stars absent in this catalog from the standard stellar radius–spectral type relation [12]. The derived dependence of the magnetic fields of stars on their radii is shown in Fig. 4b. We see a large spread in  for stars with similar radii, which is obviously attributable to the difference between their magnetic fluxes. The positions of isolated white dwarfs (WD) are also plotted in the same figure. The radii of the white dwarfs and their surface magnetic fields Bs were taken from the catalog [27]. To pass from Bs to the rms magnetic fields , we used the relation [14]



(16) B s = 2 πR * and found  from Eq. (15). In addition, Fig. 4b shows the positions of neutron stars (NS) with super strong magnetic fields (magnetars). We adopted a standard radius of 10 km for all neutron stars and took their magnetic field strengths from the McGill SGR/AXP Online Catalog [http://www.phys ics.mcgill.ca/sim pulsar/magnetar/main.html]. KINEMATICS AND PHYSICS OF CELESTIAL BODIES

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(a)

N

189

(b)

160 24 120 22

80 40

20 1

10

0 100 18 19 20 21 22 23 24 25 26 27 28 log(R夹/R䉺) logF(F, T cm2)

Fig. 5. (a) Dependence of the magnetic fluxes from stars on their radii for objects from the catalog [11] and (b) magnetic flux distribution function for stars in the catalog [11].

The positions of white dwarfs in Fig. 4 are consistent with the assumption about a fossil nature of their magnetic fields. We also see that the magnetic fluxes existing in mainsequence stars are quite sufficient to provide superstrong neutron star magnetic fields if the hypothesis [14] that the magnetic fluxes from massive stars are constant during stellar evolution from the main sequence to the formation of a neutron star is valid. To ascertain to what extent the magnetic fluxes for stars with similar radii can differ, let us analyze the dependence of the magnetic fluxes  on stellar radius presented in Fig. 5a. We see that, although the mag netic fluxes for stars with the same radius can differ by three or more orders of magnitude, the concentra tion of log  in a narrower range (21.4–23.2) is noticeable. The median value of log  is 26.4. Note an appreciable increase in the magnetic fluxes with stellar radius. This dependence may be related to obser vational selection effects, because magnetic fluxes  < 1022 T cm2 in the range of large stellar radii (for giant stars) correspond to magnetic field strengths B < 2 mT, which are hard to detect by the currently available methods of magnetic field measurements. The distribution (histogram) of log  presented in Fig. 5b shows a noticeable asymmetry in the flux distribution and a deficit of stars with low values of log  compared to the number of stars with fluxes higher than the mean value of log  = 22.4. The mean mag netic flux for the white dwarfs presented in Fig. 4 is 21.6, i.e., the mean magnetic flux for the white dwarfs with the data on their surface magnetic fields available in the catalog [27] are a factor of 6 lower than that for mainsequence stars. In particular, this means that even if the magnetic flux for intermediatemass stars decreases during their postmainsequence evolution, the remaining flux may turn out to be suffi cient to explain the measured magnetic field strengths of white dwarfs. One of the reasons for the large spread in magnetic fluxes for mainsequence stars of the same spectral subtype with similar radii can be the assumption that the magnetic flux from a star can change significantly during its mainsequence evolution. To clarify this question, consider a sample of B4–B9 stars with similar masses of (3–5)M䉺 for which the absolute and relative mainsequence lifetimes were determined in [19]. In Fig. 6, log  is plotted against the relative lifetime of a star τ (in fractions of the total mainsequence lifetime). We averaged log  in intervals Δτ = 0.2. We see that log (τ) decreases regularly with increasing τ, consistent with the conclusion in [19]. The rate of decrease in magnetic fluxes can be fitted by an expo nential function F ( τ ) = F0 e

– ζτ

,

(17)

where 0 = 1.09 × 1023 T cm2 and ζ = 2.04. The dependence of the mean magnetic field  on the mainsequence lifetime of a star was analyzed in [4]. It turned out that the dependence (τ) could also be fitted by an exponential function with an exponent ζ = 2.01. Thus, the rates of decrease in both magnetic field and magnetic flux for Btype stars are identical, which is obviously explained by small variations in the radii of stars during their main sequence evolution. The mean effective magnetic field B was found to decrease with increasing τ in [17]. The dependence of the rms magnetic field  and magnetic flux  for A and Btype stars on their mainsequence lifetime was analyzed in [21]. It was established that for early A and late B stars with masses of (3–5)M䉺, the vari KINEMATICS AND PHYSICS OF CELESTIAL BODIES

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KHOLTYGIN et al. logF(F, T cm2) 23.2 22.8 22.4 22.0

0

0.2

0.4

0.6

0.8

τ

Fig. 6. Mean magnetic fluxes  for Btype stars versus relative mainsequence lifetime τ of a star. The triangles represent the values of  averaged over intervals Δτ = 0.2; the dotted line indicates an exponential fit to the dependence (τ).

ations in both  and  are small, except for the first 15–20% of the star’s mainsequence lifetime. According to [21], the mean magnetic field and magnetic flux for these stars in the interval 0 < τ < 0.2 decrease by a factor of 3–4 and then remain almost constant, in agreement with the rates of decrease in these quantities we obtained. The decrease in mean stellar magnetic field B shown in Fig. 6 may be related to the dissipation of weak magnetic fields noted above. CONCLUSIONS Ninety years after the formulation of spectral analysis principles by Kirchhoff, it has become possible to determine stellar magnetic fields on their basis. In more than six decades elapsed since the discovery of stellar magnetic fields, the magnetic fields have been measured for more than a thousand stars of various spectral types. Based on a statistical analysis of the catalogs of magnetic fields by Bychkov et al. (2009) and Romanyuk and Kudryavtsev (2008), we analyzed the statistical properties of the mean magnetic fields for OBA stars. We constructed the distribution function of the mean magnetic field for B and Atype stars and found its sharp decrease in the range of weak magnetic fields. The statistical properties of the magnetic fluxes for mainsequence stars, white dwarfs, and neutron stars were analyzed. We established that the magnetic fluxes for white dwarfs and neutron stars are consistent with the assumption about the origin of their magnetic fields as the fossils of the magnetic fields of mainsequence stars. ACKNOWLEDGMENTS This study was supported by the Program of the Russian President for support of leading scientific schools (NSh1318.2008.2). REFERENCES 1. D. N. Monin, S. N. Fabrika, and G. G. Valyavin, “Magnetic Survey of Bright Northern Stars. New Results,” Preprint SAO No. 150 (Nizhn. Arkhyz, 2002). 2. I. I. Romanyuk and D. O. Kudryavtsev, “Magnetic Fields of Chemically Peculiar Stars. I. The Catalog of Mag netic CPStars,” Astrofiz. Byull. 63, 148–165 (2008). 3. S. N. Fabrika, V. G. Shtol’, G. G. Valyavin, and V. D. Bychkov, “Measurements of Magnetic Fields of White Dwarfs,” Pis’ma Astron. Zh. 23, 43–47 (1997) [Astron. Lett. 23, 43 (1997)]. 4. A. F. Kholtygin, S. N. Fabrika, N. A. Drake, et al., “Statistics of Magnetic Fields of OBStars,” Pis’ma Astron. Zh. 36, 389 (2010) [Astron. Lett. 36, 370 (2010)]. 5. M. Auriere, G. A. Wade, J. Silvester, et al., “Weak Magnetic Fields in Ap/Bp Stars. Evidence for a Dipole Field Lower Limit and a Tentative Interpretation of the Magnetic Dichotomy,” Astron. Astrophys. 475, 1053–1065 (2007). 6. H. W. Babcock, “Zeeman Effect in Stellar Spectra,” Astrophys. J. 105, 105–119 (1947). 7. S. Bagnulo, J. D. Landstreet, E. Mason, et al., “Searching for Links Between Magnetic Fields and Stellar Evo lution. I. A Survey of Magnetic Fields in Open Cluster A and BType Stars with FORS1,” Astron. Astrophys. 450, 777–791 (2006). 8. D. A. Bohlender, J. D. Landstreet, and I. B. Thompson, “A Study of Magnetic Fields in Ap Si and He Weak Stars,” Astron. Astrophys. 269, 355–376 (1993). KINEMATICS AND PHYSICS OF CELESTIAL BODIES

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