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VbvAstE-001

Book Boris V. Vasiliev Astrophysics

6.3 The mass-temperature

6.3 The mass-temperature and mass-luminosity relations Taking into account Eqs.(4.18), (2.22) and (4.8) one can obtain the relation between surface temperature and the radius of a star or accounting for (6.16) T 0 ∼ R 7/8 0 , (6.19) T 0 ∼ M 7/12 (6.20) The dependence of the temperature on the star surface over the star mass of close binary stars [11] is shown in Fig.(6.2). Here the temperatures of stars are normalized to the sunny surface temperature (5875 C), 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 (T 0 ∼ M 0.59 ). The theoretical dependence T 0 ∼ M 7/12 (Eq.6.20) is shown by dotted line. If parameters of the star are expressed through corresponding solar values τ ≡ T 0 T ⊙ and µ ≡ M M ⊙ , that Eq.(6.20) can be rewritten as τ = 1. (6.21) µ 7/12 τ Numerical values of relations for close binary stars [11] are shown in the µ 7/12 Table(6.2). The analysis of these data leads to few conclusions. The averaging over all tabulated stars gives < τ >= 1.007 ± 0.07. (6.22) µ 7/12 and we can conclude that the variability of measured data of surface temperatures and stellar masses has statistical character. Secondly, Eq.(6.21) is valid for all hot stars (exactly for all stars which are gathered in Tab.(6.2)). ρ The problem with the averaging of looks different. There are a few of giants µ 2/3 ρ and super-giants in this Table. The values of ratio are more than 2 for them. It µ 2/3 seems that, if to exclude these stars from consideration, the averaging over stars of the main sequence gives value close to 1. Evidently, it needs in more detail consideration. The luminosity of a star L 0 ∼ R 2 0T 4 0. (6.23) at taking into account (Eq.6.16) and (Eq.6.20) can be expressed as L 0 ∼ M 11/3 ∼ M 3.67 (6.24) This dependence is shown in Fig.(6.3) It can be seen that all calculated interdependencies R(M),T(M) and L(M) show a good qualitative agreement with the measuring data. At that it is important, that the quantitative explanation of mass-luminosity dependence discovered at the beginning of 20th century is obtained. 49

log T/T o 1.0 0.9 0.8 0.7 0.6 measured T~M 0.59 0.5 0.4 0.3 0.2 theory T~M 7/12 0.1 0.0 -0.1 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 log M/M o Figure 6.2: The dependence of the temperature on the star surface over the star mass of close binary stars [11]. Here the temperatures of stars are normalized to surface temperature of the Sun (5875 C), the stars masses are normalized to the mass of Sum. The data are shown on double logarithmic scale. The solid line shows the result of fitting of measurement data (T 0 ∼ M 0.59 ). The theoretical dependence T 0 ∼ M 7/12 (Eq.6.20) is shown by dotted line. 50

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