# Physics/Essays/Fedosin/Stellar Dirac constant

The stellar Dirac constant, denoted as ħ$_s$, is a physical constant, a natural unit of angular momentum and action for the objects of the stellar level of matter.

## Origin

The introduction of the stellar Dirac constant was one of the consequences of the development of the theory of Infinite Hierarchical Nesting of Matter. In 1999, while Sergey Fedosin studied similarity of matter levels and SPФ symmetry, he determined the values of the stellar Planck constant h$_s$, which was related to the stellar Dirac constant by a factor 2π : h$_s$ = 2π ħ$_s$.
At each level of matter we can distinguish objects that have similar mass, but have different sizes and matter densities. This is possible if the stability of matter is maintained by various mechanisms. Thus, in the main sequence stars the stability is maintained by the pressure of non-relativistic plasma, in white dwarfs – by the pressure of electrons, and in neutron stars – by the pressure of degenerate nucleon gas. Hence it follows that in order to establish similarity between the stars and elementary particles, depending on the types of stars, different sets of similarity coefficients can be used. In addition, the stars of different types must have noncoincident values of characteristic angular momentum.
For ordinary stars and for planets, revolving around them, it is assumed that ħ$_s$ = 2.8∙10$^{41}$ J∙s. For degenerate objects, such as neutron stars, the stellar Dirac constant is greater in magnitude: ħ’$_s$ = 5.5∙10$^{41}$ J∙s.
The values of the corresponding stellar Dirac constant can be obtained with the help of the known coefficients of similarity between the levels of stars and elementary particles. At the level of elementary particles the standard unit of angular momentum is the Dirac constant ħ. Taking into account the dimensional analysis, in order to determine ħ$_s$ and ħ’$_s$ we must multiply ħ by the corresponding coefficients of similarity in mass, size and speeds (more detailed information about it can be found in the articles similarity of matter levels, discreteness of stellar parameters, stellar constants, hydrogen system).
In the hydrogen atom the orbital angular momentum of the electron is quantized and is proportional to ħ, and the nuclear spin is assumed to be equal to the value ħ/2. Similarly, the value ħ$_s$ for planetary systems specifies the characteristic orbital angular momentum of a typical planet, and the value ħ$_s$/2 is close to the limiting angular momentum of the low-mass main-sequence stars. At the same time, the value ħ’$_s$/2 describes the angular momentum of rapidly rotating neutron stars, such as PSR 1937+214, for which the angular momentum, as the product of their inertia moment by the angular speed of rotation, can reach L = 4∙10$^{41}$ J∙s. In white dwarfs the proper angular momenta also do not exceed the value ħ’$_s$/2.
The analysis of the orbital rotation of moons near planets shows that their angular momentum was determined by the angular momentum of protoplanets during the formation of the Solar system. The same applies to the orbital angular momentum of planets, which have obtained their angular momentum from the rapidly rotating shell of the Sun at the stage of compression of a gas-dust cloud into a star. The discovered quantization of the specific orbital and spin angular momenta of planets and planets’ moons supports the fact that quantization in atomic and stellar systems has the same mechanism that is associated with equilibrium of energy fluxes in the matter of electrons and protoplanetary clouds, respectively, at certain distances from the central objects.

## The stellar Dirac constant in various relations

1) For elementary particles, the Chew-Frautschi plots are known, which correspond to Regge trajectories in quantum mechanics and relate the spin of particles in units of Dirac constant and the squared mass-energy of these particles. Passing from the nucleon Chew-Frautschi trajectory to the corresponding trajectory for neutron stars, taking into account the data on the limiting rotation of neutron stars, we obtain the following estimate: ħ’$_s$ < 1.2∙10$^{42}$ Дж∙с.
2) The coefficient of similarity in size Р’ can be found as the ratio of the neutron star radius to the proton radius. If we now multiply the Bohr radius (this is the most probable location of the electron in the hydrogen atom) by Р’, we will obtain the value of the order of 10$^9$ m. The Roche limit (the distance, within which any planet near a neutron star must disintegrate due to the gravitational force gradient) has the same value. Observations show that at the given radius disks of scattered matter are found near a number of neutron stars. In the theory of Infinite Hierarchical Nesting of Matter, such disks are assumed to be the analogues of electrons in atoms. If we calculate the angular momentum of these disks, it appears to be close in value to the stellar Dirac constant ħ’$_s$, similarly to the angular momentum of the electron in the hydrogen atom, which is equal to ħ.
3) The stellar Planck constant and the stellar Dirac constant are related by a numerical factor, therefore, in order to estimate the latter constant the methods can be used, which are described in the article stellar Planck constant. They include:
a) The ratio based on the de Broglie waves:
    $~h_{s}^{'}=2M_{s}R_{s}C_{s}=4.4\cdot 10^{42}$ J∙s,
where M$_s$ and R$_s$ are the mass and radius of the neutron star, C$_s$ is the characteristic speed of the particles in the neutron star.
b) The statistical angular momentum for a black hole as a measure of ħ’$_s$/2.
c) Calculation of ħ’$_s$ as a coefficient of proportionality between the natural oscillation frequency and the excitation energy in the black hole.