Stellar Planck constant

The stellar Planck constant, denoted as hs, is used to describe the characteristic quantities of angular momentum and action inherent in the objects of the stellar level of matter.

This constant first appeared in the works of Sergey Fedosin in 1999.  While developing the theory of Infinite Hierarchical Nesting of Matter Fedosin proved the theorem of SPФ symmetry and determined the similarity coefficients between different levels of matter.

The quantity hs depends on the type of selected objects. For ordinary stars and for planets, revolving around them, hs = 1.8∙1042 J∙s. For degenerate objects, such as neutron stars, the stellar Planck constant is slightly greater in magnitude: h’s = 3,5∙1042 J∙s. Using the stellar Planck constant (by dividing it by 2π) we can obtain the stellar Dirac constant.

The need for the constant

According to the theory of Infinite Hierarchical Nesting of Matter, which was fully confirmed by experimental observations, the matter is distributed in space not uniformly but rather discretely. For example, we may find that dust particles, due to the dependence of their concentration on the mass of dust particles, are located in separate groups, so that at some masses there are almost no dust particles. A similar pattern is observed for practically all space objects, ranging from elementary particles to Metagalaxy. As was determined, the masses and sizes of objects in different groups differ from each other by the law of geometric progression, which allows us to establish the similarity relations between them. The following table provides the coefficients of similarity in mass, size and characteristic speeds between the elementary particles and ordinary stars for hydrogen systems (proton and electron and the corresponding star and planet):

The object Mass, kg Orbit radius, m Orbital speed, m/s
Planetary system 1.11∙1029 2.88∙1012 1.6∙103
Hydrogen atom 1.67∙10-27 5.3∙10-11 2.19∙106
Similarity coefficients Ф = 6.654∙1055 Р0 = 5.437∙1022 S0 = 7.34∙10-4

In order to obtain hs based on the dimensional analysis we need to multiply the Planck constant h by the product of the similarity coefficients Ф∙Р0∙S0. For nucleons the characteristic angular momentum is equal to their spin, which equals the value h/4π = ħ/2, where ħ is the Dirac constant. Similarly for the main sequence stars the value hs/4π = ħs/2 = 1.4∙1041 J∙s (ħ s is the stellar Dirac constant) should be the characteristic quantity reflecting the proper rotation of these stars. For example, the proper angular momentum of the Sun equals 1.6∙1041J∙s. 

The orbital angular momentum of the inner planets of the Solar system (Mercury, Venus, Earth, Mars) and Pluto is an order of magnitude less than hs/4π. At the same time, for the giant planets, such as Jupiter, Saturn, Uranus, and Neptune, the orbital angular moments is an order of magnitude greater than hs/4π. Such a large difference is caused by the fact that the planets’ masses vary significantly. But if we consider the orbital angular momenta of planets relative to their mass unit, it appears that planets are located at orbits similar to the Bohr orbit in an atom. This implies that the specific orbital angular momentum increases in direct proportion to the orbit’s number, while not only the orbital angular momentum but also the proper angular momentum of planets is quantized.  As for the planets’ moons, quantization of their specific orbital momenta is also observed.  This confirms quantization of parameters of cosmic systems.

If we analyze the total angular momenta of the planetary systems of stars with different masses during their formation from gas clouds, in view of the orbital angular momenta of planets and the proper angular momenta of stars, then extrapolating to the planetary system of the star with the lowest possible mass 0.056 Мcc is the Solar mass) it turns out that the total angular momentum of the planetary system of such a star (which is a brown dwarf) equals ħs.

For degenerate objects it is convenient to determine the coefficients of similarity in mass, size and characteristic speeds with the help of the parameters of elementary particles and neutron stars (in particular, the data for the proton and the corresponding neutron star are used): 

The object Mass, kg Radius, m Characteristic speed of particles, m/s
Neutron star 2.7∙1030 1.2∙104 6.8∙107
Proton 1.67∙10-27 8.7∙10-16 2.99∙108
Similarity coefficients Ф’ = 1.62∙1057 Р’ = 1.4∙1019 S’ = 2.3∙10-1

The stellar Planck constant for degenerate objects is h’s = h ∙ Ф’ ∙ S’ ∙ Р’ = 3.5∙1042 J∙s and the stellar Dirac constant is ħ’s = ħ ∙ Ф’ ∙ S’ ∙ Р’ = h’s/2π = 5.5∙1041 J∙s. Based on the analogy with the nucleon, the quantum spin of which is equal to ħ/2, for a neutron star the characteristic angular momentum of the proper rotation equal to ħ’s/2 should be expected. We can take as an example one of the most rapidly rotating pulsars PSR B1937+21, the rotation period of which equals Ts = 1.558 ms.  For neutron stars it is assumed that their typical inertia moment is equal to Js = 1038 kg ∙m2.  For the angular momentum of the pulsar under consideration we obtain: L = 2πJs/Ts = 4∙1041 J∙s, which is close enough to the value ħ’s/2. The proper angular momenta of the white dwarfs do not exceed ħ’s/2 either.

These examples show that at the level of stars the characteristic angular momenta exist that describe the orbital and spin rotation of the typical objects – planets and stars. These angular momenta are the natural units for measuring the angular momenta of all the objects of this level. The same approach can be extended to all other known levels of matter. For example, at the level of galaxies the characteristic angular momentum hg is about 1068 J∙s, at the level of metagalaxies – hm is about 1089 J∙s, and at the level of preons – hp is about 10–46J∙s. 

The methods of estimating the stellar Planck constant

1) In addition to the above, there is also a number of relations, in which the stellar Planck constant is involved. In one of them an approximate formula for the Planck constant is used, which relates it with the proton mass Mp, the proton radius Rp and the speed of light c:

$~h=2M_{p}R_{p}c.\qquad \qquad (1)$ This formula can be obtained, if we assume hypothetical conversion of the proton rest energy into the photon energy with the electromagnetic wave period $T$ equal to $T=2R_{p}/c$ (as if the photon passes the proton diameter at the speed of light):

$~M_{p}c^{2}=h\nu =h/T=hc/2R_{p},$ where $\nu$ is the wave frequency.

Another derivation of formula $(1)$ uses the concept of the material wave or de Broglie wave as a consequence of internal oscillations in the matter of elementary particles.  For a neutron star we can similarly arrive at the value close to the stellar Planck constant:

$~h_{s}^{'}=2M_{s}R_{s}C_{s}=4.4\cdot 10^{42}$ J∙s,

where Ms and Rs are the mass and radius of the neutron star, Cs is the characteristic speed of particles in the neutron star; these parameters correspond to the data in the table of degenerate objects’ parameters (see. above). In turn, the velocity Cs is obtained from the equality between the binding energy (see mass–energy equivalence) and the total energy of the star, which, due to the virial theorem, is approximately equal to half of the star’s gravitational energy:

$~M_{s}C_{s}^{2}\approx {\frac {KGM_{s}^{2}}{2R_{s}}},$ where $K$ is the coefficient of the order of unity, depending on the distribution of matter in the star, and $G$ is the gravitational constant.

2) The statistical angular momentum for black holes. Black holes are hypothetical objects kept from collapse of the gravitation force. The speed of the matter’s motion inside black holes should reach the speed of light. If the matter moves randomly relative to the three coordinate axes, then in order to calculate the statistical angular momentum we should take one third of the total stellar mass Mbs, because on the average only this part will instantaneously rotate relative to the given rotation axis. Assuming that Mbs is equal to the mass of a typical neutron star Ms, the radius of the corresponding Schwarzschild black hole will be equal to:

$~R_{bs}={\frac {2GM_{bs}}{c^{2}}}=4.1$ km.

The angular momentum of a uniform rotating ball is determined by the product of the ball’s mass by its radius, by the speed of rotation at the equator, and by the coefficient equal to 0.4:

$~L_{bs}={\frac {0,4M_{bs}R_{bs}c}{3}}=4,6\cdot 10^{41}$ J∙s.

The obtained value is close to the value ħ’s/2.

3) Estimation of the stellar Planck constant can be made with the help of analysis of the proper non-radial oscillations, assumed to take place in the black hole at its excitation. The energy of such oscillations $\Delta E$ should be related to the oscillation frequency $~\nu$ in the same way as in the photon, with replacement of the Planck constant with the stellar Planck constant:

$~h_{s}^{'}={\frac {\Delta E}{\nu }}.$ The value $~\nu$ for the second spherical harmonics of the Schwarzschild black hole was determined as follows: 

$~\nu ={\frac {0.37367c^{3}}{2\pi GM_{bs}}}.$ As the black hole excitation energy $\Delta E$ we can take the energy, equivalent in its meaning to the energy of the proton’s excitation to the state of the first nucleon resonance $\Delta 1(1232)$ . For the proton the excitation energy equals $0.31M_{p}c^{2}$ . Accordingly, for the black hole we can use the ratio $\Delta E<0.31M_{bs}c^{2}$ . If we assume that the mass of the black hole is equal to the mass of a typical neutron star Ms, we obtain the following:

$~h_{s}^{'}<{\frac {0.31M_{bs}^{2}2\pi G}{0.37367c}}=9\cdot 10^{42}$ J∙s.

4) There are also a number of methods to determine the stellar the Planck constant, associated with the manifestation of the stellar Dirac constant in the world of stars. This could include the characteristic angular momentum of the matter rotating in the form of disks around neutron stars as the analogue of the characteristic orbital angular momentum of the electron in the hydrogen atom multiple of the Dirac constant ħ. In another method, the analogy is drawn between the Regge trajectories for nucleons and the dependence of the angular momentum of neutron stars on the square of their mass, which gives an estimate of ħ’s (more detailed information about it can be found in the article stellar Dirac constant).