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Book Boris V. Vasiliev Astrophysics

Using the definition of

Using the definition of a chemical potential of ideal gas (of particles with spin=1/2) [12] [ ( ) Ne 2π 2 3/2 ] µ e = kT log (2.8) 2V m ekT we obtain the full energy of the hot electron gas [ E e ≈ 3 ( ) 2 kT Ne 1 + π3/2 aBe 2 3/2 ] n e , (2.9) 4 kT where a B = 2 m ee 2 is the Bohr radius. 2.1.3 The correction for correlation of charged particles in a hot plasma At high temperatures, the plasma particles are uniformly distributed in space. At this limit, the energy of ion-electron interaction tends to zero. Some correlation in space distribution of particles arises as the positively charged particle groups around itself preferably particles with negative charges and vice versa. It is accepted to estimate the energy of this correlation by the method developed by Debye-Hükkel for strong electrolytes [12]. The energy of a charged particle inside plasma is equal to eϕ, where e is the charge of a particle, and ϕ is the electric potential induced by other particles on the considered particle. This potential inside plasma is determined by the Debye law [12]: where the Debye radius is r D = ϕ(r) = e r e− r r D (2.10) ( 4πe 2 kT −1/2 ∑ n aZa) 2 (2.11) For small values of ratio r r D , the potential can be expanded into a series ϕ(r) = e r − e + ... (2.12) r D The following terms are converted into zero at r → 0. The first term of this series is the potential of the considered particle. The second term √ ( 3/2 E = −e 3 π ∑ N aZa) 2 (2.13) kT V a is a potential induced by other particles of plasma on the charge under consideration. And so the correlation energy of plasma consisting of N e electrons and (N e/Z) nuclei with charge Z in volume V is [12] a E corr = −e 3 √ πne kT Z3/2 N e (2.14) 17

2.2 The energy-preferable state of a hot plasma 2.2.1 The energy-preferable density of a hot plasma Finally, under consideration of both main corrections taking into account the interparticle interaction, the full energy of plasma is given by E ≈ 3 [ ( 2 kT Ne 1 + π3/2 aBe 2 4 kT ) 3/2 n e − 2π1/2 3 ( Z kT ) 3/2 ] e 3 n 1/2 e (2.15) The plasma into a star has a feature. A star generates the energy into its inner region and radiates it from the surface. At the steady state of a star, its substance must be in the equilibrium state with a minimum of its energy. The radiation is not in equilibrium of course and can be considered as a star environment. The equilibrium state of a body in an environment is related to the minimum of the function ([12]20): E − T oS + P oV, (2.16) where T o and P o are the temperature and the pressure of an environment. At taking in to account that the star radiation is going away into vacuum, the two last items can be neglected and one can obtain the equilibrium equation of a star substance as the minimum of its full energy: dE plasma dn e = 0. (2.17) Now taking into account Eq.(2.15), one obtains that an equilibrium condition corresponds to the equilibrium density of the electron gas of a hot plasma n equilibrium e ≡ n ⋆ = 16 Z 3 ≈ 1.2 · 10 24 Z 3 cm −3 , (2.18) 9π a 3 B It gives the electron density ≈ 3 · 10 25 cm −3 for the equilibrium state of the hot plasma of helium. 2.2.2 The estimation of temperature of energy-preferable state of a hot stellar plasma As the steady-state value of the density of a hot non-relativistic plasma is known, we can obtain an energy-preferable temperature of a hot non-relativistic plasma. The virial theorem [12, 22] claims that the full energy of particles E, if they form a stable system with the Coulomb law interaction, must be equal to their kinetic energy T with a negative sign. Neglecting small corrections at a high temperature, one can write the full energy of a hot dense plasma as E plasma = U + 3 2 kT Ne = − 3 kT Ne. (2.19) 2 18

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