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VbvAstE-001

Book Boris V. Vasiliev Astrophysics

The energy of black

The energy of black radiation inside a star atmosphere is ∫ R0 ( ) π 2 3 ( ) 16 kT⋆ R⋆ E a(br) = 4π R ⋆ 15 kT⋆ r 2 dr = 3 ( ) π 2 3 kT⋆ c r 13 15 kT⋆ V ⋆. (5.15) c The total energy of black radiation inside a star is E(br) = E ⋆(br) + E a(br) = 16 ( ) π 2 3 ( ) 3 kT⋆ 13 15 kT⋆ V ⋆ = 1.23 π2 kT⋆ c 15 kT⋆ V ⋆ (5.16) c 5.2.2 The full energy of a star In accordance with (5.3), the full energy of a star i.e. E star = − 15 7 E star = −E kinetic + E(br) (5.17) ( ) 16 π 2 3 kT⋆ kT⋆N⋆ + 13 15 kT⋆ V ⋆. (5.18) c The steady state of a star is determined by a minimum of its full energy: ( dE star = 0, (5.19) it corresponds to the condition: dT ⋆ )N=const,V=const − 15 64π2 N⋆ + 7 13 · 15 ( ) 3 kT⋆ V ⋆ = 0. (5.20) c Together with Eq.(2.18) it defines the equilibrium temperature of a star core: ( ) 1/3 ( ) 25 · 13 c T ⋆ = Z ≈ Z · 2.13 · 10 7 K (5.21) 28π 4 ka B 5.3 Main star parameters 5.3.1 The star mass The virial theorem connect kinetic energy of a system with its potential energy. In accordance with Eqs.(5.13) and (5.6) Introducing the non-dimensional parameter 9 GM 2 ⋆ = 30 kT⋆N⋆. (5.22) 14 R ⋆ 7 η = GM⋆ A Z mp R ⋆kT ⋆ , (5.23) 31

we obtain η = 20 = 6.67, (5.24) 3 and at taking into account Eqs.(2.18) and (5.21), the core mass is M ⋆ = [ 20 3 ( 25 · 13 28 ) ] 1/3 3/2 3 MCh ( 4 · 3.14 A ) 2 = 6.84 M Ch ( A ) 2 (5.25) Z Z The obtained equation plays a very important role, because together with Eq.(4.20), it gives a possibility to predict the total mass of a star: M = 2M ⋆ = 13.68M Ch ( A ) 2 ≈ 25.34M⊙ ( A ) 2 . (5.26) Z Z The comparison of obtained prediction Eq.(5.26) with measuring data gives a method to check our theory. Although there is no way to determine chemical composition of cores of far stars, some predictions can be made in this way. At first, there must be no stars which masses exceed the mass of the Sun by more than one and a half orders, because it accords to limiting mass of stars consisting from hydrogen with A/Z = 1. Secondly, the action of a specific mechanism (see. Sec.10) can make neutronexcess nuclei stable, but it don’t give a base to suppose that stars with A/Z > 10 (and with mass in hundred times less than hydrogen stars) can exist. Thus, the theory predicts that the whole mass spectrum must be placed in the interval from 0.25 up to approximately 25 solar masses. These predications are verified by measurements quite exactly. The mass distribution of binary stars 1 is shown in Fig.5.1 [10]. The mass spectrum of close binary stars 2 is shown in Fig.5.2. The very important element of the direct and clear confirmation of the theory is obviously visible on these figures - both spectra is dropped near the value A/Z = 1. Beside it, one can easy see, that the mass spectrum of binary stars (Fig.(5.1)) consists of series of well-isolated lines which are representing the stars with integer values of ratios A/Z = 3, 4, 5..., corresponding hydrogen-3,4,5 ... or helium-6,8,10 ... (also line with the half-integer ratio A/Z = 3/2, corresponding, probably, to helium- 3, Be-6, C-9...). The existence of stable stars with ratios A/Z ≥ 3 raises questions. It is generally assumed that stars are composed of hydrogen-1, deuterium, helium-4 and other heavier elements with A/Z ≈ 2. Nuclei with A/Z ≥ 3 are the neutron-excess and so short-lived, that they can not build a long-lived stars. Neutron-excess nuclei can become stable under the action of mechanism of neutronization, which is acting inside the dwarfs. It is accepted to think that this mechanism must not work into 1 The use of these data is caused by the fact that only the measurement of parameters of binary star rotation gives a possibility to determine their masses with satisfactory accuracy. 2 The data of these measurements were obtained in different observatories of the world. The last time the summary table with these data was gathered by Khaliulilin Kh.F. (Sternberg Astronomical Institute) [11] in his dissertation (in Russian) which has unfortunately has a restricted access. With his consent and for readers convenience, we place that table in Appendix. 32

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