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VbvAstE-001

Book Boris V. Vasiliev Astrophysics

measured spectrum. The

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

VbvNeuE-001
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