VbvAstE-001

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Chapter 7 Magnetic fields and magnetic moments of stars 7.1 Magnetic moments of celestial bodies A thin spherical surface with radius r carrying an electric charge q at the rotation around its axis with frequency Ω obtains the magnetic moment m = r2 qΩ. (7.1) 3c The rotation of a ball charged at density ϱ(r) will induce the magnetic moment µ = Ω 3c ∫ R 0 r 2 ϱ(r) 4πr 2 dr. (7.2) Thus the positively charged core of a star induces the magnetic moment √ GM⋆R 2 ⋆ m + = Ω. (7.3) 5c A negative charge will be concentrated in the star atmosphere. The absolute value of atmospheric charge is equal to the positive charge of a core. As the atmospheric charge is placed near the surface of a star, its magnetic moment will be more than the core magnetic moment. The calculation shows that as a result, the total magnetic moment of the star will have the same order of magnitude as the core but it will be negative: √ G m Σ ≈ − c M⋆R2 ⋆Ω. (7.4) 55
Simultaneously, the torque of a ball with mass M and radius R is L ≈ M ⋆R 2 ⋆Ω. (7.5) As a result, for celestial bodies where the force of their gravity induces the electric polarization according to Eq.(4.2), the giromagnetic ratio will depend on world constants only: m Σ L ≈ − √ G c . (7.6) This relation was obtained for the first time by P.M.S.Blackett [6]. He shows that giromagnetic ratios of the Earth, the Sun and the star 78 Vir are really near to √ G/c. By now the magnetic fields, masses, radii and velocities of rotation are known for all planets of the Solar system and for a some stars [18]. These measuring data are shown in Fig.(7.1), which is taken from [18]. It is possible to see that these data are in satisfactory agreement with Blackett’s ratio. At some assumption, the same parameters can be calculated for pulsars. All measured masses of pulsars are equal by the order of magnitude [21]. It is in satisfactory agreement with the condition of equilibrium of relativistic matter (see ). It gives a possibility to consider that masses and radii of pulsars are determined. According to generally accepted point of view, pulsar radiation is related with its rotation, and it gives their rotation velocity. These assumptions permit to calculate the giromagnetic ratios for three pulsars with known magnetic fields on their poles [5]. It is possible to see from Fig.(7.1), the giromagnetic ratios of these pulsars are in agreement with Blackett’s ratio. 7.2 Magnetic fields of hot stars At the estimation of the magnetic field on the star pole, it is necessary to find the field which is induced by stellar atmosphere. The field which is induced by stellar core is small because R ⋆ ≪ R 0. The field of atmosphere m − = Ω 3c ∫ R0 R ⋆ 4π divP 3 r4 dr. (7.7) can be calculated numerically. But, for our purpose it is enough to estimate this field in order of value: H ≈ 2m− . (7.8) R0 3 As the field on the star pole √ G2M⋆R0 2 m − ≈ Ω (7.9) c H ≈ −4 √ GM⋆ cR 0 Ω. (7.10) 56
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Boris V.Vasiliev ASTROPHYSICS and a
Page 5 and 6: 4
Page 7 and 8: 5 The main parameters of star 29 5.
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Page 11 and 12: theories of stars that are not purs
Page 13 and 14: But plasma is an electrically polar
Page 15 and 16: can be described by definite ration
Page 17 and 18: But even at so high temperatures, a
Page 19 and 20: 2.2 The energy-preferable state of
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Page 23 and 24: 3.2 The equilibrium of a nucleus in
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Page 27 and 28: As a result, the electric force app
Page 29 and 30: and in accordance with Eq.(4.11), t
Page 31 and 32: The total kinetic energy of plasma
Page 33 and 34: we obtain η = 20 = 6.67, (5.24) 3
Page 35 and 36: 40 N 35 30 10 9 8 7 6 5 4 3 2 1 A/Z
Page 37 and 38: One can show it easily, that in thi
Page 39 and 40: 2.10 log (TR/R o T o ) 1.55 1.00 me
Page 41 and 42: averaged temperature and averaged p
Page 43 and 44: log R/R o 1.5 1.3 1.1 0.9 measured
Page 45 and 46: The Table(6.2). The relations of ma
Page 47 and 48: The Table(6.2)(continuation). N Sta
Page 49 and 50: The Table(6.2)(continuation). N Sta
Page 51 and 52: log T/T o 1.0 0.9 0.8 0.7 0.6 measu
Page 53 and 54: Figure 6.4: The schematic of the st
Page 55: 54
Page 59 and 60: At taking into account above obtain
Page 61 and 62: 60
Page 63 and 64: of periastron rotation of close bin
Page 65 and 66: 8.3 The angular velocity of the aps
Page 67 and 68: N -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
Page 69 and 70: Figure 9.1: (a) The power spectrum
Page 71 and 72: Figure 9.2: (a) The measured power
Page 73 and 74: 9.3 The basic elastic oscillation o
Page 75 and 76: 9.5 The spectrum of solar core osci
Page 77 and 78: star mass spectrum. At first, we ca
Page 79 and 80: where ˜ λ C = m ec is the Compto
Page 81 and 82: This energy is many orders of magni
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Page 85 and 86: Figure 11.1: The steady states of s
Page 87 and 88: It opens a way to estimate the mass
Page 89 and 90: N 10 9 8 7 6 5 4 3 2 1 0 1.0 1.1 1.
Page 91 and 92: This makes it possible to estimate
Page 93 and 94: to p F ≈ mc, its energy and press
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Page 97 and 98: approach to the separation of the E
Page 99 and 100: is evident as soon as in this proce
Page 101 and 102: Figure 12.1: The dependence of the
Page 103 and 104: Figure 12.2: a) The external radius
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It gives a possibility to conclude
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This allows us to explain the obser
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[14] Landau L.D. and Lifshits E.M.:
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N Name of star U P M 1 /M ⊙ M 2 /
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[9] Gemenez A. and Clausen J.V., Fo
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[30] Hill G. and Holmgern D.E. Stud
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[52] Semeniuk I. Apsidal motion in
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[75] Andersen J. and Gimenes A. Abs
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