Physics/Essays/Fedosin/Stellar Boltzmann constant

Stellar Boltzmann constant, denoted as Kps, is a physical constant that connects the average energy of motion in a certain set of typical objects of the stellar level of matter with the kinetic temperature characterizing this set. This constant was introduced by Sergey Fedosin in 1999. In the framework of the theory of Infinite Hierarchical Nesting of Matter, Fedosin proved the theorem of SPФ symmetry and determined the similarity coefficients between different levels of matter. This allows us to move from the values of the fundamental physical constants, inherent in the level of elementary particles, to the physical constants of other scale levels.

Since the temperature does not undergo scale transformation, the Boltzmann constant is transformed between the matter levels in the same way as energy. From the theory of dimensions of physical quantities it follows that the Boltzmann constant for the main-sequence stars is Kps = km ∙ Ф ∙ S2, where Ф = 6.654∙1055 is the coefficient of similarity in mass,  S  = 7.34∙10-4 is the coefficient of similarity in speeds, km is the Boltzmann's constant for the objects at the level of elementary particles, similar in their properties to the main-sequence stars. For neutron stars K' s = k ∙ Ф'  ∙ S' 2 = 1.18∙1033 J/K, where k is the Boltzmann constant as the constant for the nucleon matter, Ф'  = 1.62∙1057 is the coefficient of similarity in mass,  S'  = 0.23 is the coefficient of similarity in speeds.

In the first approximation we can assume that Kps and K' s are equal to each other.

Application
In space the most common chemical element is hydrogen. The abundance of other elements is much less, for example, the number of silicon atoms is less than that of hydrogen atoms tens of thousands of times. The same holds true for the abundance of stars of different masses. In accordance with the discreteness of stellar parameters, the abundance of stars fully copies the abundance of chemical elements, and low-mass stars prevail in our Galaxy. In this case, the typical objects characterizing the Galaxy in general must be the stars with minimum mass Мps = 0.056 Мc (Мc is the Solar mass), which are  brown dwarfs and correspond to hydrogen according to the similarity of matter levels. For the set of these stars, the value of the stellar Boltzmann constant is Kps. If the Galaxy consisted only of identical more massive main-sequence stars, the stellar Boltzmann constant for them would be Ks = А Kps, where A would be the mass number corresponding to these stars.

For the average kinetic energy of the stars’ motion in space, we can write the following:
 * $$~ E_k = \frac {M_s v^2}{2}=\frac {3K_s T_k}{2},$$

where $$ M_s = A M_{ps}$$ and $$ v $$ are the mass and root-mean-square velocity of the stars’ motion, $$ T_k $$ is the kinetic temperature.

Hence, taking into account the relation $$ K_{ps} = K'_s $$ at the velocity $$ v =235 $$ km/s we can determine the effective temperature of our Galaxy: $$ T_k = \frac { M_{ps} v^2}{3 K_{ps} }=1.7 \cdot 10^6 $$ K.

The effective temperature of the Galaxy can also be estimated using other methods, for example, with its integral luminosity using the Stefan–Boltzmann law for the radiation from a perfect black body, in this case the stellar Stefan–Boltzmann constant should be used. Another method involves calculation of the gravitational energy of the Galaxy and its total internal kinetic energy, which is approximately equal to $$ E_{gk} = 2.5\cdot 10^{52}$$ J. If we calculate the total number of nucleons N in all the stars of the Galaxy, then from the formula:
 * $$~ E_{gk}= \frac {3kN T}{2\mu },$$

where $$ \mu $$ is the number of nucleons per gas particle, we can find the temperature $$ T = 4\cdot 10^6 $$ K.

The effective pressure of the gas, consisting of a set of stars, is calculated by the formula:
 * $$~ P_g = K_s G_s T_k,$$

where $$ G_s $$ is the concentration of stars found from observations.

The stellar Boltzmann constant can be introduced into the equation of the state of matter inside the star (the matter is considered as a gas, consisting of nuclei, ions and electrons, which is held by the proper gravitational force; the thermal energy of the gas is approximately equal to half the gravitational energy according to the virial theorem). In this case the formula is valid:
 * $$~ PV = \frac { 2E_h }{ 3}= \frac { K_s T_s }{ \mu},$$

where $$ \mu $$ is the number of nucleons per gas particle, $$ V $$, $$ P $$ and $$ T_s $$ are the volume of the star, its average pressure and internal temperature, $$ E_h $$ is the internal thermal energy of the star. This formula holds with an accuracy up to a coefficient of the order of unity, because the star cannot be uniform, the pressure and temperature increase in its interior. We can take the Sun as an example, in which the average pressure reaches almost 1014 Pa, and the average temperature is about 8 million degrees at $$ \mu =0.64$$.

Hence it follows that the stellar Boltzmann constant represents both the intrinsic properties of the stellar objects, describing the relationship between the energy and temperature of the matter, and the relationship between the energy and temperature of the interacting stellar objects in the aggregate. The same conclusion can be made about the physical meaning of the ordinary Boltzmann constant, with substitution of the stellar objects by elementary particles.