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VbvAstE-001

Book Boris V. Vasiliev Astrophysics

Chapter 5 The virial

Chapter 5 The virial theorem and main parameters of a star 5.1 The energy of a star The virial theorem [12, 22] is applicable to a system of particles if they have a finite moving into a volume V . If their interaction obeys to the Coulomb’s law, their potential energy E potential , their kinetic energy E kinetic and pressure P are in the ratio: 2E kinetic + E potential = 3P V. (5.1) On the star surface, the pressure is absent and for the particle system as a whole: and the full energy of plasma particles into a star 2E kinetic = −E potential (5.2) E(plasma) = E kinetic + E potential = −E kinetic . (5.3) Let us calculate the separate items composing the full energy of a star. 5.1.1 The kinetic energy of plasma The kinetic energy of plasma into a core: E⋆ kinitic = 3 kT⋆N⋆. (5.4) 2 The kinetic energy of atmosphere: E kinetic a = 4π ∫ R0 ( ) 10 3 R⋆ R ⋆ 2 kT⋆n⋆ r 2 dr ≈ 3 ( ) 3 r 7 2 kT⋆N⋆ (5.5) 29

The total kinetic energy of plasma particles E kinetic = E kinetic ⋆ 5.1.2 The potential energy of star plasma + Ea kinetic = 15 kT⋆N⋆ (5.6) 7 Inside a star core, the gravity force is balanced by the force of electric nature. Correspondingly, the energy of electric polarization can be considered as balanced by the gravitational energy of plasma. As a result, the potential energy of a core can be considered as equal to zero. In a star atmosphere, this balance is absent. The gravitational energy of an atmosphere ∫ [ Ea G A R0 ( ) ] 3 ( ) 6 1 R⋆ R⋆ = −4πGM ⋆ Z mpn⋆ 2 − rdr (5.7) R ⋆ 2 r r or where and Ea G = 3 ( 1 2 7 − 1 2 The electric energy of atmosphere is E E a = −4π ) GM 2 ⋆ R ⋆ = − 15 28 GM 2 ⋆ R ⋆ (5.8) ∫ R0 R ⋆ 1 2 ϱϕr2 dr, (5.9) ˜ϱ = 1 dPr 2 3r 2 dr (5.10) ˜ϕ = 4π Pr. (5.11) 3 The electric energy: Ea E = − 3 GM 2 ⋆ , (5.12) 28 R ⋆ and total potential energy of atmosphere: Ea potential = Ea G + Ea E = − 9 GM 2 ⋆ . (5.13) 14 R ⋆ The equilibrium in a star depends both on plasma energy and energy of radiation. 5.2 The temperature of a star core 5.2.1 The energy of the black radiation The energy of black radiation inside a star core is ( ) 3 E ⋆(br) = π2 kT⋆ 15 kT⋆ V ⋆. (5.14) c 30

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