# Physics/Essays/Fedosin/Stellar constants

Stellar constants characterize stellar level of matter, describing typical physical quantities inherent in stars and planetary systems of stars. In some cases stellar constants are 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 systems with main-sequence stars

According to similarity of matter levels and SPФ symmetry between corresponding objects and phenomena we can establish similarity relations and predict physical quantities, characterizing them. This allows us to connect different levels of matter in the framework of Theory of Infinite Hierarchical Nesting of Matter.
At the level of stars following similarity coefficients are used between atoms and main-sequence stars:
* Coefficient of similarity in mass: $~\Phi =6.654\cdot 10^{55}$.
* Coefficient of similarity in speed: $~S=S_{0}{\frac {A}{Z}}$, where $~S_{0}={\frac {C_{s}}{c}}=7.34\cdot 10^{-4}$ is coefficient of similarity in speed for hydrogen system, $~C_{s}$ is characteristic speed of matter particles in main-sequence star with minimum mass $~M_{sp}$, $~c$ is speed of light as characteristic speed of matter in proton, $~A$ and $~Z$ are mass and charge numbers of the star, found from correspondence between stars and chemical elements (more on this in article Discreteness of stellar parameters).
* Coefficient of similarity in size: $~P=P_{0}{\frac {Z}{A}}$, where $~P_{0}=5.437\cdot 10^{22}$.
* Coefficient of similarity in time: $~\Pi ={\frac {P}{S}}=\Pi _{0}{\frac {Z^{2}}{A^{2}}}$, where $~\Pi _{0}={\frac {P_{0}}{S_{0}}}=7.41\cdot 10^{25}$.
To determine stellar constants, it is necessary to multiply constants for the level of atoms and elementary particles by similarity coefficients according to dimensionality of corresponding physical quantities. Some stellar constants can also be calculated through other stellar constants.

### Constants of hydrogen system

Stellar hydrogen system consists of a star – analogue of proton and a planet – analogue of electron. The constants, describing these objects and their interaction, equal:
* Minimum mass of main sequence star: $~M_{sp}=M_{p}\Phi =0.056M_{c}=1.11\cdot 10^{29}$ kg, where $~M_{p}$ is proton mass, $~M_{c}$ is mass of the Sun.
* Mass of planet, which is analogue of electron: $~M_{\Pi }=M_{e}\Phi =6.06\cdot 10^{25}$ kg or 10.1 Earth masses, where $~M_{e}$ is mass of electron.
* Stellar speed $~C_{s}=220$ km/s as characteristic speed of matter particles in main-sequence star with minimum mass.
* Stellar Dirac constant for planetary systems of main-sequence stars: $~\hbar _{s}=\hbar \Phi SP=\hbar \Phi S_{0}P_{0}=2.8\cdot 10^{41}$ J∙s, where $~\hbar$ is Dirac constant.
* Stellar Bohr radius in stellar hydrogen system: $~R_{F}={\frac {\hbar _{s}^{2}}{GM_{sp}M_{\Pi }^{2}}}=2.88\cdot 10^{12}$ m = 19.25 a.u., where $~G$ is gravitational constant.
* Orbital speed of planet – analogue of electron in stellar Bohr radius in stellar hydrogen system: $~V_{\Pi }={\frac {GM_{sp}M_{\Pi }}{\hbar _{s}}}=1.6$ km/s.
* Stellar fine structure constant : $~\alpha _{s}={\frac {V_{\Pi }}{C_{s}}}={\frac {GM_{sp}M_{\Pi }}{\hbar _{s}C_{s}}}=0.007297$.
In stellar hydrogen system balance of forces acting on planet, and condition for orbital angular momentum are as follows:
    $~{\frac {GM_{sp}M_{\Pi }}{R^{2}}}={\frac {M_{\Pi }V^{2}}{R}}$,
    $~M_{\Pi }VR=n\hbar _{s}$,
which implies that:
    $~V={\frac {GM_{sp}M_{\Pi }}{n\hbar _{s}}}$,
    $~R={\frac {n^{2}\hbar _{s}^{2}}{GM_{sp}M_{\Pi }^{2}}}$.
With $~n=1$ orbit of planet corresponds to Bohr radius in hydrogen atom, and speed and orbital radius of the planet become equal to $~V_{\Pi }$ and $~R_{F}$.

### Other constants

* Acceleration of free fall at surface of main-sequence star of minimum mass: $~g_{s}={\frac {GM_{sp}}{R_{sp}^{2}}}=3.1\cdot 10^{3}$ m/s$^2$, with radius of the star $~R_{sp}=0.07$ solar radii.
* Stellar Boltzmann constant for planetary systems of main-sequence stars: $~K_{s}=AK_{ps}$, where $~A$ is mass number of the star, $~K_{ps}=1.18\cdot 10^{33}$ J/K.
* Stellar mole is defined as amount of matter, consisting of stars equal in number to $~N_{A}=6.022\cdot 10^{23}$ (stellar mole)$^{–1}$, where the number $~N_{A}$ is Avogadro constant.
* Stellar gas constant for gas from stars: $~R_{st}=K_{s}N_{A}=AK_{ps}N_{A}=AR_{pst}$, where $~R_{pst}=K_{ps}N_{A}=7.1\cdot 10^{56}$ J/(K∙stellar mole) is stellar gas constant for main-sequence stars of minimum mass.
* Gyromagnetic ratio for object – analogue of electron: $~{\frac {e}{M_{e}}}{\frac {P_{0}^{0.5}S_{0}}{\Phi ^{0.5}}}=3.69\cdot 10^{-9}$ C/kg (or rad/(T∙s)), where $~e$ is elementary charge.
* Gyromagnetic relation for stellar object – analogue of atomic nucleus: $~{\frac {e}{M_{p}}}{\frac {P_{0}^{0.5}S_{0}}{\Phi ^{0.5}}}=2.01\cdot 10^{-12}$ C/kg (or rad/(T∙s)).
* Stellar Stefan–Boltzmann constant: $~\Sigma _{s}={\frac {\sigma \Phi }{\Pi _{0}^{3}}}=9.3\cdot 10^{-30}$ W/(m$^2$ ∙K$^4$), where $~\sigma$ is Stefan–Boltzmann constant.
* Stellar radiation constant: $~A_{s}={\frac {a\Phi S_{0}^{2}}{P_{0}^{3}}}=1.69\cdot 10^{-34}$ J/(m$^3$∙K$^4$), where $~a={\frac {4\sigma }{c}}$ is radiation constant.
* Absolute value of total energy of main-sequence star with minimum mass in its proper gravitational field: $~E_{s}=M_{sp}C_{s}^{2}=5.4\cdot 10^{39}$ J.
By definition, gyromagnetic ratio (magnetomechanical ratio) is ratio of dipole magnetic moment of object to its proper angular momentum. For electron value of spin as characteristic angular momentum is assumed to be equal to $~{\frac {\hbar }{2}}$, and magnetic moment is equal to Bohr magneton:
    $~\mu _{B}={\frac {e\hbar }{2M_{e}}}$.
The measure of magnetic moment of atomic nuclei is nuclear magneton:
    $~\mu _{N}={\frac {e\hbar }{2M_{p}}}$.
From this it follows that gyromagnetic ratio for Bohr magneton and nuclear magneton is equal to ratio of charge to corresponding mass. If on coordinate plane with coordinate axes for magnetic moment and proper angular momentum, we draw straight lines corresponding to gyromagnetic ratios for object – analogue of electron and for stellar object – analogue of atomic nucleus, it turns out that magnetic moments of cosmic objects, from planets’ moons to galaxies, fall into space between these straight lines (see Summary dependence "magnetic moment – spin" for planets, stars and Galaxy).

## Constants for systems with neutron stars

Coefficients of similarity between atoms and neutron stars are:
* Coefficient of similarity in mass: $~\Phi '=1.62\cdot 10^{57}$.
* Coefficient of similarity in speed: $~S'={\frac {C'_{s}}{c}}=0.23$, where $~C'_{s}$ is characteristic speed of matter particles in typical neutron star.
* Coefficient of similarity in size: $~P'=1.4\cdot 10^{19}$.
* Coefficient of similarity in time: $~\Pi '={\frac {P'}{S'}}=6.1\cdot 10^{19}$.

### Hydrogen system with magnetar

For degenerate stellar objects stellar hydrogen system consists of magnetar – analogue of proton and disc (discon) – analogue of electron. These objects are characterized by following constants:
* Magnetar mass $~M_{s}=M_{p}\Phi '=1.35M_{c}=2.7\cdot 10^{30}$ kg.
* Mass of discon – analogue of electron: $~M_{d}=M_{e}\Phi '=1.5\cdot 10^{27}$ kg or 250 Earth masses, or 0.78 Jupiter masses.
* Stellar speed $~C'_{s}=6.8\cdot 10^{7}$ m/s as characteristic speed of matter particles in typical neutron star.
* Stellar Dirac constant for system with magnetar: $~\hbar '_{s}=\hbar \Phi 'P'S'=5.5\cdot 10^{41}$ J∙s.
* Stellar Bohr radius in stellar hydrogen system: $~{R'}_{F}={\frac {{\hbar '}_{s}^{2}}{GM_{s}M_{d}^{2}}}=7.4\cdot 10^{8}$ m.
* Orbital speed of matter of discon on stellar Bohr radius in stellar hydrogen system: $~V_{d}={\frac {GM_{s}M_{d}}{\hbar '_{s}}}=496$ km/s.
* Stellar charge: $Q_{s}=e(\Phi 'P')^{0.5}S'=5.5\cdot 10^{18}$ C.
* Stellar fine structure constant : $~\alpha _{s}={\frac {V_{d}}{C'_{s}}}={\frac {Q_{s}^{2}}{4\pi \varepsilon _{0}\hbar '_{s}C'_{s}}}={\frac {GM_{s}M_{d}}{\hbar '_{s}C'_{s}}}=0.007297$, where $~\varepsilon _{0}$ is electric constant.
* Measure of magnetic moment for neutron stars (stellar magneton): $~\mu _{s}={\frac {Q_{s}\hbar '_{s}}{2M_{s}}}=5.6\cdot 10^{29}$ J/T.
* Magnetic moment of 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 magnetic moment of proton.
* Magnetic moment of discon – analogue of electron: $P_{md}=P_{me}{\Phi '}^{0.5}{P'}^{1.5}{S'}^{2}={\frac {Q_{s}\hbar '_{s}}{2M_{d}}}=1.03\cdot 10^{33}$ J/T, where $~P_{me}$ is magnetic moment of electron.
* Free fall acceleration at surface of magnetar: $~g_{m}={\frac {GM_{s}}{R_{s}^{2}}}=1.2\cdot 10^{12}$ m/s$^2$, with radius of star $~R_{s}=12$ km.
* Absolute value of total energy of magnetar in proper gravitational field: $~E_{m}=M_{s}{C'}_{s}^{2}=1.2\cdot 10^{46}$ J.
* Stellar gravitational torsion flux quantum, as velocity circulation quantum, is: $\Phi _{s}={\frac {\pi \hbar '_{s}}{M_{s}}}=6.4\cdot 10^{11}$ m$^2$/s.
* 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 ordinary gravitation is acting with gravitational constant $~G=6.67428\cdot 10^{-11}$ m$^3$/(kg∙ s$^2$). In framework of Le Sage's theory of gravitation gravitational constant is associated with other physical quantities, characterizing fluxes of gravitons:
* Cross-section of interaction of gravitons with matter: $~\sigma _{N}=7\cdot 10^{-50}$ m².
* Power of energy flux of gravitons per unit area from unit solid angle: $~U=pcB_{0}={\frac {cGM_{p}^{2}}{4\sigma _{N}^{2}}}=1\cdot 10^{42}$ W/(sr∙m$^2$), where $~p$ is momentum of graviton, moving at speed of light $~c$, $~B_{0}$ is flux of gravitons, crossing per unit time unit area perpendicular to flux from unit solid angle, $~M_{p}$ is mass of nucleon .
* Maximum gravitational pressure from gravitons: $~P_{g}=4\pi pB_{0}=4\cdot 10^{34}$ Pa, which is approximately equal to density of gravitational energy of fluxes of gravitons.
* Maximum gravitational force acting on a body: $~F_{g}={\frac {c^{4}}{16k^{2}G}}=2.1\cdot 10^{43}$ N, where $~k=0.6$ for a homogeneous spherical body.
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 integrity of objects with sizes of elementary particles, and strong gravitational constant is $~\Gamma =1.514\cdot 10^{29}$ m$^3$/(kg ∙s$^2$). In gravitational model of strong interaction strong gravitation, gravitational torsion fields, emerging during rotation and motion of elementary particles, and electromagnetic forces are responsible for strong interaction.

## Dimensionless constants

In hydrogen system we can determine dimensionless constants associated with mass, sizes and speeds:
* Ratio of proton mass to electron mass: $\beta ={\frac {M_{p}}{M_{e}}}=1836.15$.
* Ratio of Bohr radius to radius of proton $~R_{p}$ : $\delta ={\frac {r_{B}}{R_{p}}}={\frac {h^{2}}{4\pi ^{2}\Gamma M_{p}M_{e}^{2}R_{p}}}={\frac {h^{2}\varepsilon _{0}}{\pi e^{2}M_{e}R_{p}}}=6.08\cdot 10^{4}\approx {\frac {2M_{p}ch\varepsilon _{0}}{\pi e^{2}M_{e}}}$, where approximate equality for Planck constant is used $~h=2\pi \hbar \approx 2M_{p}cR_{p}$.
* Ratio of electron speed on first Bohr orbit to speed of light (fine structure constant): $\alpha ={\frac {V_{e}}{c}}={\frac {e^{2}}{2\varepsilon _{0}hc}}={\frac {2\pi \Gamma M_{p}M_{e}}{hc}}={\frac {1}{137.036}}=7.2973525376\cdot 10^{-3}$.
For these coefficients we obtain the relation:
    $~\beta =\pi \alpha \delta$.
In hydrogen system including main sequence star and a planet (or magnetar and a discon around it), after replacing atomic quantities in formulas for dimensionless constants with corresponding stellar quantities, values of these constants remain the same. In particular, for system with magnetar and discon we obtain:
* Ratio of mass of magnetar to mass of discon: $\beta ={\frac {M_{s}}{M_{d}}}=1836.15$.
* Ratio of stellar Bohr radius to radius of magnetar: $\delta ={\frac {{R'}_{F}}{R_{s}}}={\frac {{h'}_{s}^{2}}{4\pi ^{2}GM_{s}M_{d}^{2}R_{s}}}={\frac {{h'}_{s}^{2}\varepsilon _{0}}{\pi Q_{s}^{2}M_{d}R_{s}}}=6.08\cdot 10^{4}\approx {\frac {2M_{s}C'_{s}h'_{s}\varepsilon _{0}}{\pi Q_{s}^{2}M_{d}}}$, where approximate equality for stellar Planck constant is used: $~h'_{s}=2\pi \hbar '_{s}\approx 2M_{s}C'_{s}R_{s}$.
* Ratio of speed of discon matter on stellar Bohr radius to stellar speed (stellar fine structure constant): $\alpha ={\frac {V_{d}}{C'_{s}}}={\frac {Q_{s}^{2}}{2\varepsilon _{0}h'_{s}C'_{s}}}={\frac {2\pi GM_{s}M_{d}}{h'_{s}C'_{s}}}={\frac {1}{137.036}}=7.2973525376\cdot 10^{-3}.$
This results in $~\beta =\pi \alpha \delta$.
Another type of dimensionless constant is constant of gravitational interaction, which shows relative force of the interaction between two magnetars. This constant is calculated as ratio of gravitational interaction energy of two magnetars to energy, associated with stellar Dirac constant $~\hbar '_{s}$ and with stellar speed $~C'_{s}$ :
    $\alpha _{mm}={\frac {\beta GM_{s}^{2}}{\hbar '_{s}C'_{s}}}=13{.}4\beta$,
where coefficient $\beta =0.26$ for interaction of two neutron stars as consequence of exponential decay of flux of gravitons in matter according to Le Sage's theory of gravitation, and for less dense bodies $\beta$ tends to unity.
Obtained value of dimensionless constant $~\alpha _{mm}$ is of the same order of magnitude as coupling constant of interaction for two protons in field of strong gravitation, which follows from SPФ symmetry and similarity of matter levels of atoms and stars.