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Book Boris V. Vasiliev Astrophysics

Where U ≈ − GM2 R 0

Where U ≈ − GM2 R 0 is the potential energy of the system, G is the gravitational constant, M and R 0 are the mass and the radius of the star. As the plasma temperature is high enough, the energy of the black radiation cannot be neglected. The full energy of the stellar plasma depending on the particle energy and the black radiation energy E total = − 3 π2 kT Ne + 2 15 at equilibrium state must be minimal, i.e. ) ( ∂Etotal ∂T N,V ( ) 3 kT V kT (2.20) c = 0. (2.21) This condition at Ne = n⋆ gives a possibility to estimate the temperature of the hot V stellar plasma at the steady state: T ⋆ ≈ Z c ka B ≈ 10 7 Z K. (2.22) The last obtained estimation can raise doubts. At ”terrestrial” conditions, the energy of any substance reduces to a minimum at T → 0. It is caused by a positivity of a heat capacity of all of substances. But the steady-state energy of star is negative and its absolute value increases with increasing of temperature (Eq.(2.19)). It is the main property of a star as a thermodynamical object. This effect is a reflection of an influence of the gravitation on a stellar substance and is characterized by a negative effective heat capacity. The own heat capacity of a stellar substance (without gravitation) stays positive. With the increasing of the temperature, the role of the black radiation increases (E br ∼ T 4 ). When its role dominates, the star obtains a positive heat capacity. The energy minimum corresponds to a point between these two branches. 2.2.3 Are accepted assumptions correct? At expansion in series of the full energy of a Fermi-gas, it was supposed that the condition of applicability of Boltzmann-statistics (2.1) is valid. The substitution of obtained values of the equilibrium density n ⋆ (Eq.(2.18)) and equilibrium temperature T ⋆ (Eq.(2.22)) shows that the ratio E F (n ⋆) kT ⋆ ≈ 3.1Zα ≪ 1. (2.23) Where α ≈ 1 is fine structure constant. 137 At appropriate substitution, the condition of expansion in series of the electric potential (2.12) gives r ≈ (n⋆ 1/3 r D) −1 ≈ α 1/2 ≪ 1. (2.24) r D Thus, obtained values of steady-state parameters of plasma are in full agreement with assumptions which was made above. 19


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