# Physics/Essays/Fedosin/Natural scale

Natural scale is the scale, that is associated with the mass of electron. The method of constructing of the scale is the same as for Planck scale and Stoney scale.
The natural gravitational coupling constant is:
    $\alpha _{N}={\frac {m_{e}^{2}}{2hc\,\varepsilon _{g}}}=1.7517846\cdot 10^{-45},\$
where
$m_{e}$ is electron mass;
$h$ is Planck constant;
$c$ is speed of light in vacuum;
$\varepsilon _{g}={\frac {1}{4\pi G}}$ is gravitoelectric gravitational constant;
$G$ is gravitational constant.

## Fundamental units of vacuum

The set of primary vacuum constants is: the speed of light $~c$; the electric constant $~\varepsilon _{0}$; the speed of gravity $~c_{g}$ (usually equated to the speed of light); the gravitational constant $~G$.
The set of secondary vacuum constants is: The vacuum permeability: $\mu _{0}={\frac {1}{\varepsilon _{0}c^{2}}}\$;
The electromagnetic impedance of free space:
    $Z_{0}=\mu _{0}c={\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}={\frac {1}{\varepsilon _{0}c}}$;
The gravitoelectric gravitational constant: $~\varepsilon _{g}={\frac {1}{4\pi G}}$;
The gravitomagnetic gravitational constant: $~\mu _{g}={\frac {4\pi G}{c_{g}^{2}}}$;
The gravitational characteristic impedance of free space:
    $~\rho _{g}={\sqrt {\frac {\mu _{g}}{\varepsilon _{g}}}}={\frac {4\pi G}{c_{g}}}.$

## Natural scale units based on electron mass

For the sake of completeness in the Table 1 the main Natural scale units in the form consistent with Tables for the Stoney scale and vacuum constants are presented.
Table 1: Base Natural scale units :
| Name | Dimension | Expressions | SI equivalent |
| Electron mass | Mass (M) | $m_{e}\$ | $9.10938215(45)\cdot 10^{-31}$ kg |
| Electron wavelength | Length (L) | $\lambda _{N}={\frac {h}{m_{e}c}}$ | $2.4263102175(33)\cdot 10^{-12}$ m |
| Natural gravitational coupling constant | Dimensionless | $\alpha _{N}={\frac {m_{e}^{2}}{2hc\varepsilon _{g}}}$ | $1.7517846\cdot 10^{-45}$ |
| Natural fictitious gravitational torsion mass | Dynamic mass (L$^2$T$^{−1}$) | $m_{\Omega }={\frac {h}{m_{e}}}\$ | $7.27339\cdot 10^{-4}$ m$^2$ s$^{−1}$ |
| Natural torsion coupling constant | Dimensionless | $\beta _{N}={\frac {m_{\Omega }^{2}}{2hc\mu _{g}}}={\frac {1}{4\alpha _{N}}}\$ | $1.42757\cdot 10^{44}$ |
| Natural charge | Electric charge (Q) | $q_{N}={\sqrt {2hc\varepsilon _{0}\alpha _{N}}}=e{\sqrt {\frac {\alpha _{N}}{\alpha }}}=e\left({\frac {m_{e}}{m_{S}}}\right)$ | $7.84854579\cdot 10^{-41}$ C |
| Natural electric coupling constant | Dimensionless | $\alpha _{N}={\frac {q_{N}^{2}}{2hc\varepsilon _{0}}}\$ | $1.7517846\cdot 10^{-45}$ |
| Natural fictitious magnetic charge | Magnetic charge (L$^2$MT$^{−1}$Q$^{−1}$) | $q_{m}={\frac {h}{q_{N}}}={\frac {h}{e}}{\sqrt {\frac {\alpha }{\alpha _{N}}}}\$ | {$8.44229\cdot 10^{6}$ Wb |
| Natural magnetic coupling constant | Dimensionless | $\beta _{N}={\frac {q_{m}^{2}}{2hc\mu _{0}}}={\frac {1}{4\alpha _{N}}}$ | $1.42757\cdot 10^{44}$ |
| Natural gravitational impedance quantum | Gravitational impedance (L$^2$M$^{−1}$T$^{−1}$) | $R_{N}={\frac {m_{\Omega }}{m_{e}}}={\frac {h}{m_{e}^{2}}}\$ | $7.98492\cdot 10^{26}$ m$^2$ kg$^{−1}$ s$^{−1}$ |
| Natural electromagnetic impedance quantum | Electrical impedance (L$^2$M$^{−1}$T$^{−1}$Q$^{−2}$) | $R_{e}={\frac {q_{m}}{q_{N}}}={\frac {h}{e^{2}}}{\frac {\alpha }{\alpha _{N}}}\$ | $1.07562\cdot 10^{47}$ Ohm. |
The difference of Natural coupling constant $\alpha _{N}\$ and fine structure constant $\alpha \$ is so high, that the Natural scale now is the base for the Dirac large numbers and different numerology approaches.
Actually, the relationship between fine structure constant and Natural coupling constant yields the Dirac number:
    $\xi _{D}={\frac {\alpha }{\alpha _{N}}}=\left({\frac {e}{m_{e}}}\right)^{2}{\frac {\varepsilon _{g}}{\varepsilon _{0}}}=\left({\frac {m_{S}}{m_{e}}}\right)^{2}=4.167\cdot 10^{42},\$
where $m_{S}\$ is the Stoney mass.