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In [[physics]], '''Weak Natural scale''' is the fundamental ''scale of matter'', 
named after 
 
It defines the [[Weak Natural coupling constant]]:
::<math>\alpha_{WN} = \frac{m_{\nu}^2}{2hc\,\varepsilon_G}, \ </math>
where 
*<math>\!m_{\nu} </math> is the upper limit for [[electron neutrino]] mass;
*<math>\!h </math> is the [[Planck constant]];
*<math>\!c</math> is the [[speed of light]] in vacuum;
*<math>\!\varepsilon_G</math> is the [[gravitational electric-like constant]]. Thus, the Natural scale defines the force interaction dimensionless parameter, something like  ''gravitational fine structure constant''.

== History ==
Usually, the Weak Natural scale now considered for definition of the ''weak interaction'' force and has not appropriate attention that should be fo the scale of matter. However, the real strength of forces is determined by the scale only, but not the metter type: charge or mass. 

== Fundamental units of vacuum ==
[[Dielectric constant]] [1]:
::<math>\varepsilon_E = \varepsilon_0 = 8.854187817\cdot 10^{-12} \ </math> [[Farad|F]] [[metre|m]]<sup>&minus;1</sup>
[[Magnetic constant]]:
::<math>\mu_E = \mu_0 = \frac{1}{\varepsilon_0 c^2} = 1.2566370614\cdot 10^{-6} \ </math> [[Henry|H]] [[metre|m]]<sup>&minus;1</sup>
Electrodynamic [[velocity of light]]:
::<math>c_E = \frac{1}{\sqrt{\varepsilon_E\mu_E}} = 2.99792458\cdot 10^8 \ </math> [[metre|m]] [[second|s]]<sup>&minus;1</sup> 
Electrodynamic [[characteristic impedance of free space|vacuum impedance]]:
::<math>\rho_{E0} = \sqrt{\frac{\mu_E}{\varepsilon_E}} = 376.730313461 \ </math> [[Ohm]]
Dielectric-like gravitational constant:
::<math>\varepsilon_G = \frac{1}{4\pi G} = 1.192708\cdot 10^9 \ </math> [[kilogram|kg]] [[Second|s]]<sup>2</sup> [[metre|m]]<sup>&minus;3</sup> 
Magnetic-like gravitational constant:
::<math>\mu_G = \frac{4\pi G}{c^2} = 9.328772\cdot 10^{-27} \ </math> [[metre|m]] [[kilogram|kg]]<sup>&minus;1</sup>
Gravidynamic [[velocity of light]]:
::<math>c_G = \frac{1}{\sqrt{\varepsilon_G\mu_G}} = 2.9979246\cdot 10^8 \ </math> [[metre|m]] [[second|s]]<sup>&minus;1</sup> 
Gravidynamic vacuum impedance:
::<math>\rho_{G0} = \sqrt{\frac{\mu_G}{\varepsilon_G}} = 2.7966954\cdot 10^{-18} \ </math> [[metre|m]]<sup>2</sup> [[kilogram|kg]]<sup>&minus;1</sup> [[second|s]]<sup>&minus;1</sup> 
Considering that all Natural, Stoney and Planck units are derivatives from the ''vacuum units'', therefore the last are more fundamental that units of any scale.

== Weak interaction Natural scale units ==
The weak scale of Natural units is based on the [[electron neutrino]] mass. As is known, neutrinos are generated during the [[annihilation]] process, which is going through intermediate [[positronium]] atom. 
The effective mass of the positronoum atom is:

:<math>m_{Bp} = \frac{m_em_p}{m_e + m_p} = 0.5m_N, \ </math>

where <math>m_e, m_p = m_N </math> are [[electron]] and [[positron]] mass respectively.
The energy scale for the positronium atom is:

:<math>W_{Bp} =  \frac{\hbar^2}{2m_{Bp}a_{Bp}^2} = \left(\frac{\alpha}{2}\right)^2\cdot m_Nc^2 = m_{WN}c^2, \ </math>

where <math>a_{Bp} = 2a_B = \frac{\lambda_N}{\pi \alpha} </math> is the length scale for positronium, and
<math>m_{WN} = \sqrt{\alpha_W}\cdot m_N</math> is the upper value for the neutrino mass, and 
<math>\alpha_W = \left(\frac{\alpha}{2}\right)^4 = 1.7723168\cdot 10^{-10} </math> is the weak interaction force constant (or weak fine structure constant).

{| class="wikitable" style="margin: 1em auto 1em auto; background-color: #ffffff"
|+Table 5: Base weak Natural scale units
! Name
! Dimension
! Expressions
! [[SI]] equivalent with uncertainties [1]
|- align="left"
| '''[[Neutrino mass]]'''
| [[Mass]] (M)
| <math>m_{WN} = \sqrt{\alpha_W}m_N  \ </math>
| <math>1.2127370\cdot 10^{-35} </math> [[kilogram|kg]]
|-
| '''[[Neutrino wavelength]]'''
| [[Length]] (L)
| <math>\lambda_{WN} = \frac{h}{m_{WN}c}</math>
| <math>1.8822505\cdot 10^{-7} </math> [[metre|m]]
|-
| '''Weak interaction force constant'''
| Dimensionless
| <math>\alpha_{W} = \left(\frac{\alpha}{2}\right)^4</math>
| <math>1.7723168\cdot 10^{-10} </math>
|- 
| '''Weak gravity force constant'''
| Dimensionless
| <math>\alpha_{WN} = \alpha_W\cdot \alpha_{GN}</math>
| <math>3.10472\cdot 10^{-55} </math>
|-
| '''Weak Natural "dynamic mass"'''
| [[Dynamic mass]] (L<sup>2</sup>T<sup>&nbsp;&minus;1</sup>)
| <math>\varphi_{WGN} = \frac{h}{m_{WN}} \ </math>
| <math>5.46373\cdot 10^{0} </math> [[metre|m]]<sup>2</sup> [[second|s]]<sup>&minus;1</sup>
|-
| '''Weak Natural "dynamic mass" force constant'''
| Dimensionless
| <math>\beta_{GN} = \frac{\varphi_{WGN}^2}{2hc\mu_G} = \frac{1}{4\alpha_{WN}} \ </math>
| <math>3.10472\cdot 10^{53} </math>
|-
| '''Weak Natural time'''
| [[Time]] (T)
| <math>t_\text{WN} = \frac{\lambda_\text{WN}}{c} </math>
| <math>6.07922\cdot 10^{-16} </math>  [[second|s]]
|}

== Weak Planck scale units ==
The primordial level of matter has two standard scales: [[Physics/Essays/Fedosin/Planck scale|Planck]] (defines the [[Physics/Essays/Fedosin/Planck mass|Planck mass]]) and [[Physics/Essays/Fedosin/Stoney scale|Stoney]] (defines the [[Physics/Essays/Fedosin/Stoney mass|Stoney mass]]). 
However, it has the third primordial scale that could be named as the ''weak interaction scale'', which has the following force constant:
:<math>\alpha_W = (\frac{\alpha}{2})^4, \ </math>
that is the same as in the weak natural scale.

The weak primordial mass will be:
:<math>m_{WP} = \sqrt{\alpha_W}\cdot m_P = 2.897473\cdot 10^{-13} \ </math>kg,
where <math>m_P \ </math> is the [[Physics/Essays/Fedosin/Planck mass|Planck mass]].

The weak primordial wavelength is:
:<math>\lambda_{WP} = \frac{h}{cm_{WP}} = 7.62809\cdot 10^{-30} \ </math>m

The weak primordial time is:
:<math>t_{WP} = \frac{h}{c^2m_{WP}} = 2.544458\cdot 10^{-38} \ </math>s

== Work function and Universe scale ==
The standard definition of the work function in the ''strength field'' is:
:<math>A_{\lambda} = F_{\lambda}\cdot \lambda = \frac{\hbar c}{\lambda} = \frac{m_{\lambda}c^2}{2\pi}. \ </math>
So, the complex weak displacement work in the ''weak natural force'' will be:
:<math>A_{NWW} = F_{WN}\cdot \lambda_W = \alpha_W \alpha_N\cdot \frac{\hbar c}{\lambda_W}, \ </math>
where 
:<math>F_{WN} = \frac{m_{WN}^2}{2hc\epsilon_G} = \alpha_W \alpha_N\cdot \frac{\hbar c}{r^2} \ </math>
is the weak natural force, and <math>\lambda_W </math> is the weak Planck wavelength.

Considering the ''Universe bubble'' as the minimal energy scale:
:<math>W_U = h\nu_U = \frac{\hbar c}{\lambda_U}, \ </math>
where <math>\lambda_U </math> is the ''Universe wavelength'', and equating the above energies, we derive the following fundamental relationship:
:<math>\frac{\lambda_W}{\alpha_W \alpha_N} = \frac{\lambda_U}{2\pi}, \ </math>
from which the [[Universe]] length parameter could be derived:
:<math>\lambda_U = \frac{2\pi \lambda_W}{\alpha_W \alpha_N} = 1.5437\cdot 10^{26} \ </math>m
which value is consistent with the 15 billion years. 

== Weak Planck scale and Solar planatery system ==
=== Planet resonator characteristics ===
Geometrical parameters of any planetary object determine the following resonance frequency:
:<math>\omega_{p} = \frac{c}{R_{p}} = 4.7055793\cdot 10^1 rad/s \ </math>,
here is used the Earth as an example.
This resonance frequency could be connected with the "minimal mass":
:<math>m_{pmin} = \frac{\hbar \omega_{p}}{c^2} = 5.521382\cdot 10^{-50} kg \ </math>,
where <math>\hbar </math> is the reduced Planck constant, and <math>c </math> is the speed of light.

Considering that gravitational resonator has its oscillations on the surface, therefore it is interesting to determine the minimal surface radius connected with the "minimal mass":
:<math>r_{pSmin} = R_{p}\cdot \sqrt{\frac{m_{pmin}}{M_{p}}} = 6.123873\cdot 10^{-31} m \ </math>.
The relationship between minimal radius and the Weak Planck length is:
:<math>\frac{r_{pmin}}{l_{WP}} = 1.98248 \ </math>,
where <math>l_{WP} = \frac{l_P}{\sqrt{\alpha_{WP}}} = 1.21405\cdot 10^{-30}m \ </math> is the Weak Planck length. 
Thus, considering the Solar Planetary System, all objects as gravitational resonators, we will have the small surface area about the Weak Planck scale, where the minimal resonator energy quant <math>\hbar \omega_{p} </math> replaced. 

The full sets of the planetary data are presented in the Table 2.

{| class="wikitable" style="margin: 1em auto 1em auto; background-color: #ffffff"
|+Table 2: Solar planatery system
! Object
! Radius,  [[metre|m]]
! Mass, [[kilogram|kg]]
! Minimal Mass, [[kilogram|kg]]
! Minimal Radius,  [[metre|m]]
! <math>l_W/r_{min}</math>
|- align="left"
| '''[[Stars/Sun|Sun]]'''
| <math>6.96\cdot 10^8</math>
| <math>1.989\cdot 10^{30}</math>
| <math>5.054\cdot 10^{-52}</math>
| <math>1.1095\cdot 10^{-32}</math>
| <math>109.4</math>
|-
| '''[[Jupiter]]'''
| <math>7.13\cdot 10^7</math>
| <math>1.899\cdot 10^{27}</math>
|<math>4.934\cdot 10^{-51}</math>
| <math>1.149\cdot 10^{-31}</math>
| <math>10.56</math>
|-
| '''[[Saturn]]'''
| <math>6.01\cdot 10^7</math>
| <math>5.686\cdot 10^{26}</math>
| <math>5.853\cdot 10^{-51}</math>
| <math>1.928\cdot 10^{-31}</math>
| <math>6.3</math>
|-
| '''[[Neptun]]'''
| <math>2.51\cdot 10^7</math>
| <math>1.03\cdot 10^{26}</math>
| <math>1.402\cdot 10^{-50}</math>
| <math>2.928\cdot 10^{-31}</math>
| <math>4.15</math>
|-
| '''[[Uran]]'''
| <math>2.45\cdot 10^7</math>
| <math>8.689\cdot 10^{25}</math>
| <math>1.436\cdot 10^{-50}</math>
| <math>3.149\cdot 10^{-31}</math>
| <math>3.86</math>
|-
| '''[[Earth]]'''
| <math>6.371\cdot 10^6</math>
| <math>5.976\cdot 10^{24}</math>
| <math>5.521\cdot 10^{-50}</math>
| <math>6.124\cdot 10^{-31}</math>
| <math>1.98</math>
|-
| '''[[Venus]]'''
| <math>6.07\cdot 10^6</math>
| <math>4.87\cdot 10^{24}</math>
| <math>5.795\cdot 10^{-50}</math>
| <math>6.622\cdot 10^{-31}</math>
| <math>1.83</math>
|-
| '''[[Mars]]'''
| <math>3.395\cdot 10^6</math>
| <math>6.424\cdot 10^{23}</math>
| <math>1.036\cdot 10^{-49}</math>
| <math>1.364\cdot 10^{-30}</math>
| <math>0.89</math>
|-
| '''[[Mercury]]'''
| <math>2.425\cdot 10^6</math>
| <math>3.311\cdot 10^{23}</math>
| <math>1.451\cdot 10^{-49}</math>
| <math>2.185\cdot 10^{-30}</math>
| <math>0.756</math>
|}

== See also ==
* [[Physics/Essays/Fedosin/Planck scale|Planck scale]]
* [[Physics/Essays/Fedosin/Stoney scale|Stoney scale]]
* [[Physics/Essays/Fedosin/Natural scale|Natural scale]]

== References ==

* 1. Latest (2006) values of the constants [http://physics.nist.gov/cuu/Constants/index.html]

[[Category:Physical constants]]
[[Category:Gravitation]]
[[Category:Cosmology]]