Physics/Essays/Fedosin/Stellar constants

Stellar constants characterize the stellar level of matter, describing the typical physical quantities inherent in stars and planetary systems of stars. In some cases stellar constants are the natural units, in which physical quantities can be measured at the level of stars. A considerable part of stellar constants was introduced by Sergey Fedosin in 1999.

Constants for the systems with main-sequence stars
According to the similarity of matter levels and SPФ symmetry between the corresponding objects and phenomena we can establish similarity relations and predict the physical quantities, characterizing them. This allows us to connect different levels of matter in the framework of the Theory of Infinite Hierarchical Nesting of Matter.

At the level of stars the following similarity coefficients are used between atoms and main-sequence stars:
 * 1) The coefficient of similarity in mass: $$~\Phi = 6.654 \cdot 10^{55}$$.
 * 2) The coefficient of similarity in speed: $$~S= S_0 \frac {A} {Z}$$, where $$~S_0= \frac {C_s} {c}=7.34 \cdot 10^{-4}$$ is the coefficient of similarity in speed for the hydrogen system, $$~C_s$$ is the characteristic speed of the matter particles in the main-sequence star with the minimum mass $$~M_{sp}$$, $$~c$$ is the speed of light as the characteristic speed of the matter in the proton, $$~A$$ and $$~Z$$ are the mass and charge numbers of the star, found from the correspondence between stars and chemical elements (more on this in the article Discreteness of stellar parameters).
 * 3) The coefficient of similarity in size: $$~P= P_0 \frac {Z} {A}$$, where $$~P_0 = 5.437 \cdot 10^{22}$$.
 * 4) The coefficient of similarity in time: $$~\Pi= \frac {P}{S}= \Pi_0 \frac {Z^2} {A^2}$$, where $$~\Pi_0 = \frac { P_0}{ S_0}=7.41 \cdot 10^{25}$$.

In determining the stellar constants the constants for the level of atoms and elementary particles are commonly used, which are multiplied by the coefficients of similarity according to the dimension of physical quantities. Some stellar constants can also be calculated through other stellar constants.

Constants of the hydrogen system
The stellar hydrogen system consists of the star – the analogue of the proton and the planet – the analogue of the electron. The constants, describing these objects and their interaction, equal:
 * 1) The minimum mass of the main sequence star: $$~M_{sp}=M_p \Phi = 0.056 M_c=1.11\cdot 10^{29}$$  kg, where $$~ M_p$$ is the proton mass, $$~ M_c $$ is the mass of the Sun.
 * 2) The mass of the planet, which is the analogue of the electron: $$~M_{\Pi}=M_e \Phi =6.06 \cdot 10^{25}$$  kg or 10.1 Earth masses, where $$~ M_ e $$ is the mass of the electron.
 * 3) The stellar speed $$~C_s=220$$ km/s as the characteristic speed of the matter particles in the main-sequence star with the minimum mass.
 * 4) The stellar Dirac constant for planetary systems of main-sequence stars: $$~\hbar_s =\hbar \Phi S P=\hbar \Phi S_0 P_0=2.8 \cdot 10^{41} $$ J∙s, where $$~\hbar$$ is the Dirac constant.
 * 5)  The stellar Bohr radius in the stellar hydrogen system: $$~R_F=\frac {\hbar^2_s  }{G M_{sp} M^2_{\Pi} }=2.88 \cdot 10^{12}$$ m = 19.25  a.u., where $$~ G$$  is the gravitational constant.
 * 6) The orbital velocity of the planet – the analogue of the electron in the stellar Bohr radius in the stellar hydrogen system: $$~ V_{\Pi} = \frac {G M_{sp} M_{\Pi} }{\hbar_s }=1.6 $$ km/s.
 * 7) The stellar  fine structure constant : $$~ \alpha_s =\frac { V_{\Pi}}{ C_s } =\frac {G M_{sp} M_{\Pi} }{\hbar_s C_s }=0.007297 $$.

In the stellar hydrogen system the balance of forces acting on the planet, and the condition for the orbital angular momentum are as follows:
 * $$~ \frac {G M_{sp} M_{\Pi} }{R^2} = \frac {M_{\Pi} V^2 }{R} $$,
 * $$~ M_{\Pi} V R = n \hbar_s $$,

which implies that:
 * $$~ V= \frac {G M_{sp} M_{\Pi} }{ n \hbar_s }$$,
 * $$~ R= \frac { n^2 \hbar^2_s }{G M_{sp} M^2_{\Pi} }$$.

With $$~ n=1$$ the orbit of the planet corresponds to the Bohr radius in the hydrogen atom, and the velocity and the orbital radius of the planet become equal to $$~ V_{\Pi} $$ and $$~ R_F $$.

Other constants

 * 1) The acceleration of free fall at the surface of the main-sequence star of minimum mass: $$~g_s = \frac {G M_{sp}}{R^2_{sp}}= 3.1 \cdot 10^3$$ m/s2, with the radius of the star $$~ R_{sp}= 0.07 $$ solar radii.
 * 2) The stellar Boltzmann constant for planetary systems of main-sequence stars: $$~K_s = A K_{ps} $$, where $$~A$$ is the mass number of the star, $$~K_{ps}= 1.18 \cdot 10^{33} $$ J/K.
 * 3) The stellar mole is defined as the amount of matter, consisting of stars, the number of which is $$~ N_A = 6.022 \cdot 10^{23}$$ (stellar mole)–1, where the number $$~ N_A$$ is the Avogadro constant.
 * 4) The stellar gas constant for the gas from stars: $$~ R_{st} = K_s N_A = A K_{ps} N_A =A R_{pst}$$, where $$~ R_{pst}= K_{ps} N_A = 7.1 \cdot 10^{56}$$  J/(K∙stellar mole) is the stellar gas constant for main-sequence stars of minimum mass.
 * 5) The gyromagnetic ratio for the object – the analogue of the electron: $$~ \frac {e }{ M_e } \frac { P^{0.5}_0 S_0}{\Phi^{0.5}}=3.69 \cdot 10^{-9}$$ C/kg (or rad/(T∙s)), where $$~e$$ is the elementary charge.
 * 6) The gyromagnetic relation for the stellar object – the analogue of the atomic nucleus: $$~ \frac {e }{ M_p } \frac { P^{0.5}_0 S_0}{\Phi^{0.5}}=2.01 \cdot 10^{-12}$$ C/kg (or rad/(T∙s)).
 * 7) The stellar Stefan–Boltzmann constant: $$~ \Sigma_s= \frac {\sigma \Phi }{ \Pi^3_0 } =9.3 \cdot 10^{-30}$$ W/(m2 ∙K4), where $$~\sigma $$ is the Stefan–Boltzmann constant.
 * 8) The stellar radiation constant: $$~ A_s= \frac {a \Phi S^2_0}{P^3_0 } =1.69 \cdot 10^{-34}$$ J/(m3∙K4), where $$~a = \frac {4 \sigma}{c}$$ is the radiation constant.
 * 9) The absolute value of full energy of the main-sequence star with the minimum mass in its proper gravitational field: $$~ E_s= M_{sp} C^2_s =5.4 \cdot 10^{39}$$ J.



By definition, the gyromagnetic ratio (magnetomechanical ratio) is the ratio of the dipole magnetic moment of the object to its proper angular momentum. For the electron the value of the spin as the characteristic angular momentum is assumed to be equal to $$~\frac {\hbar}{2}$$, and the magnetic moment is equal to the Bohr magneton:
 * $$~ \mu_B = \frac {e \hbar }{ 2M_e } $$.

The measure of the magnetic moment of atomic nuclei is the nuclear magneton:
 * $$~ \mu_N = \frac {e \hbar }{ 2M_p } $$.

From this it follows that the gyromagnetic ratio for the Bohr magneton and the nuclear magneton is equal to the ratio of the charge to the corresponding mass. If on the coordinate plane with the coordinate axes, equal to the magnetic moment and the proper angular momentum, we draw straight lines corresponding to the gyromagnetic ratios for the object – the analogue of the electron and for the stellar object – the analogue of the atomic nucleus, it turns out that the magnetic moments of cosmic objects, from the planets’ moons to galaxies, fall into the space between these straight lines (see the figure).

Constants for the systems with neutron stars
The coefficients of similarity between atoms and neutron stars:
 * 1) The coefficient of similarity in mass: $$~\Phi' = 1.62 \cdot 10^{57}$$.
 * 2) The coefficient of similarity in speed: $$~S'= \frac {C'_s} {c}=0.23$$,  where $$~C'_s$$ is the characteristic speed of the matter particles in a typical neutron star.
 * 3) The coefficient of similarity in size: $$~P' = 1.4 \cdot 10^{19}$$.
 * 4) The coefficient of similarity in time: $$~\Pi'= \frac {P'}{S'}= 6.1 \cdot 10^{19}$$.

The hydrogen system with a magnetar
For degenerate stellar objects the stellar hydrogen system consists of the magnetar – the analogue of the proton and the disc (discon) – the analogue of the electron. These objects are characterized by the following constants:
 * 1) The magnetar mass $$~M_{s}=M_p \Phi' = 1.35 M_c=2.7\cdot 10^{30}$$ kg.
 * 2) The mass of the discon – the analogue of the electron: $$~ M_d=M_e \Phi' =1.5 \cdot 10^{27}$$ kg or 250 Earth masses, or 0.78 Jupiter masses.
 * 3) The stellar speed $$~ C'_s= 6.8 \cdot 10^{7}$$ m/s as the characteristic speed of the matter particles in a typical neutron star.
 * 4) The stellar Dirac constant for the system with the magnetar: $$~\hbar'_s= \hbar \Phi' P' S' =5.5 \cdot 10^{41} $$ J∙s.
 * 5) The stellar Bohr radius in the stellar hydrogen system: $$~ {R'}_F = \frac {{\hbar'}^2_s }{GM_s M^2_d }=7.4 \cdot 10^8$$ m.
 * 6) The orbital velocity of the matter of the discon on the stellar Bohr radius in the stellar hydrogen system: $$~ V_d= \frac {G M_{s} M_d }{\hbar'_s }=496 $$ km/s.
 * 7) The stellar charge: $$ Q_s  = e (\Phi' P')^{0.5} S' = 5.5 \cdot 10^{18}$$ C.
 * 8) The stellar  fine structure constant : $$~ \alpha_s =\frac { V_d}{ C'_s } = \frac{ Q_s^2}{4 \pi \varepsilon_0 \hbar'_s C'_s }= \frac {G M_{s} M_d}{\hbar'_s C'_s }=0.007297 $$, where $$~ \varepsilon_0 $$ is the electric constant.
 * 9) The measure of the magnetic moment for neutron stars (stellar magneton): $$~ \mu_s = \frac {Q_s \hbar'_s }{ 2M_s } = 5.6 \cdot 10^{29}$$ J/T.
 * 10) The magnetic moment of the magnetar: $$ P_{ms}  = P_{mp} {\Phi'}^{0.5} {P'}^{1.5} {S'}^2 =2.79\mu_s = 1.6 \cdot 10^{30}$$ J/T, where $$~P_{mp}$$ is the magnetic moment of the proton.
 * 11) The magnetic moment of the discon – the analogue of the electron: $$ P_{md }  = P_{me} {\Phi'}^{0.5} {P'}^{1.5} {S'}^2  =\frac {Q_s \hbar'_s }{ 2 M_d }=1.03 \cdot 10^{33}$$ J/T, where $$~P_{me}$$ is the magnetic moment of the electron.
 * 12) The free fall acceleration at the surface of the magnetar: $$~g_m = \frac {G M_{s}}{R^2_{s}}= 1.2 \cdot 10^{12}$$ m/s2, with the radius of the star $$~ R_{s}= 12$$ km.
 * 13) The absolute value of full energy of the magnetar in the proper gravitational field: $$~ E_m= M_{s} {C'}^2_s =1.2 \cdot 10^{46}$$ J.
 * 14) The stellar gravitational torsion flux quantum, as the velocity circulation quantum, is: $$ \Phi_s = \frac{ \pi \hbar'_s }{M_s} = 6.4 \cdot 10^{11}$$ m2/s.
 * 15) The stellar magnetic flux quantum : $$ \Phi_m = \frac{\pi \hbar'_s } { Q_s } = 3.1\cdot 10^{23} $$ J/A.

Gravitational constants
At the level of stars the ordinary gravitation is acting with the gravitational constant $$~ G= 6.67428 \cdot 10^{-11}$$ m3/(kg∙ s2). In the framework of Le Sage's theory of gravitation the gravitational constant is associated with other physical quantities, characterizing the fluxes of gravitons:
 * 1) The cross-section of interaction of gravitons with the matter: $$~ \sigma_N= 7 \cdot 10^{-50}$$ m².
 * 2) The power of the energy flux of gravitons per unit area from the unit solid angle: $$~ U= p c B_0=\frac { c G M^2_p}{4\sigma^2_N } = 1 \cdot 10^{42}$$ W/(sr∙m2), where $$~p$$ is the momentum of the graviton, moving at the speed of light $$~c$$, $$~ B_0$$ is the flux of gravitons, crossing per unit time the unit area perpendicular to the flux from the unit solid angle, $$~ M_p$$ is the mass of the nucleon.
 * 3) The maximum gravitational pressure from gravitons: $$~P_g= 4 \pi p B_0= 4 \cdot 10^{34}$$ Pa, which is approximately equal to the density of gravitational energy of the fluxes of gravitons.

The gravitational characteristic impedance of free space is: $$\rho_{g0} = \frac{4\pi G}{c}= 2.796696\cdot 10^{-18} \mathrm {m^2/(s\cdot kg)}. \ $$

It is assumed that strong gravitation is responsible for the integrity of objects with the sizes of elementary particles, and the strong gravitational constant is $$~\Gamma=1.514 \cdot 10^{29} $$ m3/(kg ∙s2). In the gravitational model of strong interaction the strong gravitation, the gravitational torsion fields, emerging during the rotation and the motion of elementary particles, and the electromagnetic forces are responsible for the strong interaction.

Dimensionless constants
In the hydrogen system we can determine the dimensionless constants associated with the mass, sizes and speeds:
 * 1) The ratio of the proton mass to the electron mass: $$\beta= \frac {M_p}{M_e}= 1836.15$$.
 * 2) The ratio of the Bohr radius to the radius of the proton $$~R_p$$ :  $$ \delta= \frac {r_B}{R_p}= \frac { h^2}{4\pi^2 \Gamma M_p M^2_e R_p } = \frac {h^2 \varepsilon_0}{\pi e^2 M_e R_p } = 6.08 \cdot 10^4 \approx \frac {2 M_p c h \varepsilon_0}{\pi e^2 M_e }$$, where the approximate equality for the Planck constant is used $$~h = 2\pi \hbar \approx 2 M_p c R_p$$.
 * 3) The ratio of the electron velocity on the first Bohr orbit to the speed of light (the fine structure constant): $$\alpha= \frac {V_e}{c}=\frac {e^2}{2\varepsilon_0 h c}= \frac {2 \pi \Gamma M_p M_e }{h c }=\frac {1}{137.036}= 7.2973525376  \cdot 10^{-3}$$.

For these coefficients we obtain the relation:
 * $$~\beta= \pi \alpha \delta $$.

In the hydrogen system, including the main-sequence star and the planet (or the magnetar and the discon around it), after substituting in the formulas for the dimensionless constants of the constants atomic quantities with the corresponding stellar quantities, the values of these constants remain the same, as a result the above relation between the dimensionless constants does not change. In particular, for the system with the magnetar and the discon we obtain:
 * 1) The ratio of the mass of the magnetar to the mass of the discon: $$\beta= \frac {M_s}{M_d}= 1836.15$$.
 * 2) The ratio of the stellar Bohr radius to the radius of the magnetar: $$\delta= \frac {{R'}_F }{R_s}=\frac { {h'}^2_s }{4\pi^2 G M_s M^2_d R_s } = \frac {{h'}^2_s \varepsilon_0}{\pi Q^2_s M_d R_s } =6.08 \cdot 10^4 \approx  \frac {2 M_s C'_s h'_s \varepsilon_0}{\pi Q^2_s M_d }$$, where the approximate equality for stellar Planck constant is used: $$~ h'_s = 2\pi \hbar'_s \approx 2 M_s C'_s R_s$$.
 * 3) The ratio of the velocity of the discon matter on the stellar Bohr radius to the stellar speed (the stellar fine structure constant): $$\alpha= \frac {V_d}{ C'_s }=\frac { Q^2_s }{2\varepsilon_0 h'_s C'_s }= \frac {2 \pi G M_s M_d }{ h'_s C'_s }=\frac {1}{137.036}= 7.2973525376  \cdot 10^{-3}.$$

Then again $$~\beta= \pi \alpha \delta $$.

Another type of the dimensionless constant is the constant of gravitational interaction, which shows the relative force of the interaction between two magnetars. This constant is calculated as the ratio of the gravitational interaction energy of two magnetars to the energy, associated with the stellar Dirac constant $$~\hbar'_s $$ and with the stellar speed $$~C'_s $$ :
 * $$\alpha_{mm}= \frac{\beta G M^2_s }{\hbar'_s C'_s }=13{.}4 \beta $$,

where the coefficient $$\beta =0.26$$ for the interaction of two neutron stars as the consequence of the exponential decay of the flux of gravitons in the matter according to the Le Sage's theory of gravitation, and for the less dense bodies $$\beta $$ tends to unity.

The obtained value of the dimensionless constant $$~\alpha_{mm}$$ is of the same order of magnitude as the coupling constant of interaction for two protons in the field of strong gravitation, which follows from SPФ symmetry and the similarity of matter levels of atoms and stars.