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measured spectrum. The fundamental mode of this oscillation must be determined by its spherical mode when the Sun radius oscillates without changing of the spherical form of the core. It gives a most low-lying mode with frequency: Ω s ≈ cs R ⋆ , (9.5) where c s is sound velocity in the core. It is not difficult to obtain the numerical estimation of this frequency by order of magnitude. Supposing that the sound velocity in dense matter is 10 7 cm/c and radius is close to 1 10 of external radius of a star, i.e. about 1010 cm, one can obtain as a result F = Ωs 2π ≈ 10−3 Hz (9.6) It gives possibility to conclude that this estimation is in agreement with measured frequencies. Let us consider this mechanism in more detail. 9.2 The sound speed in hot plasma The pressure of high temperature plasma is a sum of the plasma pressure (ideal gas pressure) and the pressure of black radiation: and its entropy is S = 1 A P = n ekT + Z mp ln π2 45 3 c 3 (kT )4 . (9.7) (kT )3/2 n e + 4π2 45 3 c 3 n e (kT ) 3 , (9.8) The sound speed c s can be expressed by Jacobian [12]: ( ) D(P,S) c 2 D(n D(P, S) e,T ) s = D(ρ, S) = ( ) (9.9) or { [ 5 kT c s = 1 + 9 A/Zm p For T = T ⋆ and n e = n ⋆ we have: D(ρ,S) D(n e,T ) ( ) 4π 2 2 2 45 3 c (kT ) 6 3 5n e[n e + 8π2 (kT ) 45 3 c 3 ] 3 ]} 1/2 (9.10) 4π 2 (kT ⋆) 3 45 3 c 3 n ⋆ =≈ 0.18 . (9.11) Finally we obtain: { ) 1/2 ( ) 1/2 5 T ⋆ c s = [1.01] ≈ 3.14 10 7 Z cm/s . (9.12) 9 (A/Z)m p A/Z 71
9.3 The basic elastic oscillation of a spherical core Star cores consist of dense high temperature plasma which is a compressible matter. The basic mode of elastic vibrations of a spherical core is related with its radius oscillation. For the description of this type of oscillation, the potential φ of displacement velocities v r = ∂ψ can be introduced and the motion equation can be reduced to the ∂r wave equation expressed through φ [12]: c 2 s∆φ = ¨φ, (9.13) and a spherical derivative for periodical in time oscillations (∼ e −iΩst ) is: ∆φ = 1 r 2 ( ) ∂ r 2 ∂φ ∂r ∂r = − Ω2 s φ . (9.14) c 2 s It has the finite solution for the full core volume including its center φ = A r sin Ωsr c s , (9.15) where A is a constant. For small oscillations, when displacements on the surface u R are small (u R/R = v R/Ω sR → 0) we obtain the equation: tg ΩsR c s = ΩsR c s (9.16) which has the solution: Ω sR ≈ 4.49. (9.17) c s Taking into account Eq.(9.12)), the main frequency of the core radial elastic oscillation is { [ ] ( 3 } 1/2 Gmp A Ω s = 4.49 1.4 Z + 1) . (9.18) Z r 3 B It can be seen that this frequency depends on Z and A/Z only. Some values of frequencies of radial sound oscillations F = Ω s/2π calculated from this equation for selected A/Z and Z are shown in third column of Table (9.3). Table (9.3) F, mHz F, mHz Z A/Z (calculated star (9.18)) measured 1 1 0.23 ξ Hydrae ∼ 0.1 1 2 0.32 ν Indus 0.3 2 2 0.9 η Bootis 0.85 The Procion(Aα CMi) 1.04 2 3 1.12 β Hydrae 1.08 3 4 2.38 α Cen A 2.37 3 5 2.66 3.4 5 3.24 The Sun 3.23 4 5 4.1 72
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Boris V.Vasiliev ASTROPHYSICS and a
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5 The main parameters of star 29 5.
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theories of stars that are not purs
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But plasma is an electrically polar
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can be described by definite ration
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But even at so high temperatures, a
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2.2 The energy-preferable state of
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Page 23 and 24: 3.2 The equilibrium of a nucleus in
Page 25 and 26: 24
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 and 56: 54
Page 57 and 58: Simultaneously, the torque of a bal
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: Figure 9.2: (a) The measured power
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
Page 83 and 84: 82
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
Page 95 and 96: 94
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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Page 107 and 108: It gives a possibility to conclude
Page 109 and 110: This allows us to explain the obser
Page 111 and 112: [14] Landau L.D. and Lifshits E.M.:
Page 113 and 114: N Name of star U P M 1 /M ⊙ M 2 /
Page 115 and 116: [9] Gemenez A. and Clausen J.V., Fo
Page 117 and 118: [30] Hill G. and Holmgern D.E. Stud
Page 119 and 120: [52] Semeniuk I. Apsidal motion in
Page 121: [75] Andersen J. and Gimenes A. Abs
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