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Book Boris V. Vasiliev Astrophysics

6.1.2 The Poisson’s

6.1.2 The Poisson’s adiabat The thermodynamical potential of a system consisting of N molecules of ideal gas at temperature T and pressure P can be written as [12]: The entropy of this system Φ = const · N + NT lnP − Nc P T lnT. (6.10) S = const · N − NlnP + Nc P lnT. (6.11) As at adiabatic process, the entropy remains constant −NT lnP + Nc P T lnT = const, (6.12) we can write the equation for relation of averaged pressure in a system with its volume (The Poisson’s adiabat) [12]: ̂P V ˜γ = const, (6.13) where ˜γ = c P cV is the exponent of adiabatic constant. In considered case taking into account of Eqs.(6.6) and (6.5), we obtain As V 1/3 ∼ R 0, we have for equilibrium condition ˜γ = cP c V = 1 3 . (6.14) ̂P R 0 = const. (6.15) 6.2 The mass-radius ratio Using Eq.(6.1) from Eq.(6.15), we use the equation for dependence of masses of stars on their radii: M 2 = const (6.16) R 3 0 This equation shows the existence of internal constraint of chemical parameters of equilibrium state of a star. Indeed, the substitution of obtained determinations Eq.(5.37) and (5.38)) into Eq.(6.16) gives: Z ∼ (A/Z) 5/6 (6.17) Simultaneously the observational data of masses, of radii and their temperatures was obtained by astronomers for close binary stars [11]. The dependence of radii of these stars over these masses is shown in Fig.6.1 on double logarithmic scale. The solid line shows the result of fitting of measurement data R 0 ∼ M 0.68 . It is close to theoretical dependence R 0 ∼ M 2/3 (Eq.6.16) which is shown by dotted line. 41

log R/R o 1.5 1.3 1.1 0.9 measured R~M 0.68 0.7 0.5 0.3 0.1 theory R~M 2/3 -0.1 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 log M/M o Figure 6.1: The dependence of radii of stars over the star mass [11]. Here the radius of stars is normalized to the sunny radius, the stars masses are normalized to the mass of the Sum. The data are shown on double logarithmic scale. The solid line shows the result of fitting of measurement data R 0 ∼ M 0.68 . The theoretical dependence R 0 ∼ M 2/3 (6.16) is shown by the dotted line. 42

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