Physics/Essays/Fedosin/Stellar Stefan–Boltzmann constant

Stellar Stefan–Boltzmann constant, denoted as $$~ \Sigma_s $$, is a constant that relates the average luminosity of a sufficiently large stellar system with the area of its outer surface and the average temperature of the kinetic motion of stars in this system.

The definition of the stellar Stefan–Boltzmann constant was made in 1999 in the works of Sergey Fedosin. Using the similarity of matter levels, SPФ symmetry and the theory of Infinite Hierarchical Nesting of Matter, Fedosin calculated the similarity coefficients between the atomic and stellar levels of matter. This allowed finding various stellar constants based on the dimensional equations.

For the main-sequence stars of minimum mass the stellar Stefan–Boltzmann constant is:
 * $$~ \Sigma_s= \frac {\sigma_m \Phi }{ \Pi^3_0 } ,$$

where $$~\sigma_m $$ is the Stefan–Boltzmann constant for the objects at the level of elementary particles, similar in their properties to the main-sequence stars, $$~\Phi = 6.654 \cdot 10^{55}$$ is the coefficient of similarity in mass, and $$~\Pi_0 = 7.41 \cdot 10^{25}$$ is the coefficient of similarity in time.

If we assume that $$~\sigma_m $$ is equal to the Stefan–Boltzmann constant, then we obtain $$~ \Sigma_s = 9.3 \cdot 10^{-30}$$ W/(m2∙K4).

In case if the stellar system consists of more massive stars, the effective stellar Stefan–Boltzmann constant increases by a factor equal to $$~ \frac {A^2} {Z^2}$$, where $$~A$$ and $$~Z$$ are the mass and charge numbers of stars, which characterize the stellar system on the average and are found from the similarity between stars and chemical elements (more about it in the article Discreteness of stellar parameters).

Theory
According to the Stefan–Boltzmann law, the radiation power of a black body is proportional to the surface area and to the fourth power of the body’s temperature:
 * $$~P_b= \sigma S_b \epsilon T^4$$,

where $$~\epsilon$$ is the emissivity (for all substances $$~\epsilon<1$$, for a perfect black body $$~\epsilon=1$$), $$~S_b$$ is the body’s surface area, $$~T$$ is the body’s temperature.

To apply this formula at the level of stars, we need to pass from the atomic systems to the stellar systems, which implies we need to use the constant $$~ \Sigma_s $$ instead of $$~\sigma $$.

Application
The stellar Stefan–Boltzmann constant allows us to relate the luminosity (the radiation power) of the galaxy, its surface area and the average kinetic temperature of stars. There are various methods for estimating the luminosity of galaxies. Similarly, the average kinetic temperature of stars in galaxies can be found in different ways, for example, by the velocities of stars in the galaxy and the stellar Boltzmann constant, or by the total energy and the number of nucleons in the galaxy. If we substitute in the formula:
 * $$~P_g= \Sigma_s S_g T^4,$$

the integral luminosity of our Galaxy, the Milky Way, $$~P_g= 7.6 \cdot 10^{36}$$ W, and the area of the galaxy $$~S_g= 1.3 \cdot 10^{42}$$ m2, with the galaxy’s form of a flat disk with the radius of about 15 kpc, we obtain the estimate of the effective kinetic temperature of the stellar “gas” of the galaxy: $$~T \approx 9 \cdot 10^5 $$ K.

Stellar matter
After the long-time evolution of stars, they must turn into white dwarfs and neutron stars. The latter will cluster into star systems, similar in their properties to atoms and molecules. Thus the stellar matter emerges, the basis of which are neutron stars and magnetars as the stars that carry a strong magnetic field and an electric charge.

The stellar Stefan–Boltzmann constant for neutron stars is:
 * $$~ \Sigma'_s= \frac {\sigma \Phi' }{ \Pi'^3 } = 4.2 \cdot 10^{-10} $$ W/(m2 ∙K4),

where $$~\sigma = 5.67 \cdot 10^{-8}$$ W/(m2 ∙K4) is the Stefan–Boltzmann constant as the constant characterizing the nucleon level of matter, $$~\Phi' = 1.62 \cdot 10^{57}$$ is the coefficient of similarity in mass, $$~\Pi' = P'/S' = 6 \cdot 10^{19}$$ is the coefficient of similarity in time, $$~P' = 1.4 \cdot 10^{19}$$ is the coefficient of similarity in sizes, and $$~ S' = 0.23 $$ is the coefficient of similarity in speeds.

The constant $$~ \Sigma'_s $$ must be included in the formula for the power of radiation from the stellar matter, heated to the temperature $$~T$$. The Stefan–Boltzmann constant $$~\sigma $$ exceeds the value of the stellar constant $$~ \Sigma'_s $$, which reflects the fact that the energy density increases with transition to the lower levels of matter. In the limiting case, the area of the stellar matter must not be less than the surface area of one neutron star $$~ S_s = 4 \pi R^2_s$$. For a neutron star RX J1856.5-3754, radiating approximately as a black body with the temperature of the order of $$~ T = 4.3 \cdot 10^{5}$$ K, at the radius of $$~ R_s= 14$$ km, the formula for luminosity leads to the following:
 * $$~P_s= \Sigma'_s S_s T^4 = 3.5 \cdot 10^{22} $$ W or $$~9.1 \cdot 10^{-4} P_c $$,

where $$~P_c $$ denotes the luminosity of the Sun. In fact, a neutron star radiates almost 300 times more, and to calculate the star’s luminosity we must use the value $$~\sigma $$ instead of $$~ \Sigma'_s $$. This difference is due to the difference in temperatures: if in the Stefan–Boltzmann law the kinetic temperature of the moving particles of the black body’s surface is used, averaged due to a number of interactions, then applying the law to one matter particle leads to inaccuracy, since the temperature of the particle’s surface is related to its internal processes and may not be equal to the kinetic temperature, arising from the motion of a set of particles.