Stellar
Planck constant
The stellar
Planck constant, denoted as h_{s}, 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. ^{[1]}
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
h_{s}
depends on the type of selected objects. For ordinary stars and for
planets, revolving around them, h_{s} =
1.8∙10^{42} J∙s. For degenerate objects, such as neutron stars, the
stellar Planck constant is slightly greater in magnitude: h’_{s} =
3,5∙10^{42} J∙s. Using the stellar Planck constant (by dividing it by 2π) we can obtain the stellar
Dirac constant.
Содержание
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):
Mass, kg 
Orbit radius, m 
Orbital speed, m/s 

Planetary system 
1.11∙10^{29 } 
2.88∙10^{12 } 
1.6∙10^{3 } 
Hydrogen atom 
1.67∙10^{27 } 
5.3∙10^{11 } 
2.19∙10^{6 } 
Similarity coefficients 
Ф = 6.654∙10^{55 } 
Р_{0} = 5.437∙10^{22 } 
S_{0} = 7.34∙10^{4 } 
In order to obtain h_{s}
based on the dimensional analysis we need to multiply the Planck constant h by
the product of the similarity coefficients
Ф∙Р_{0}∙S_{0}. 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 h_{s}/4π = ħ_{s}/2 = 1.4∙10^{41}
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∙10^{41}J∙s. ^{[2]}
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 h_{s}/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
h_{s}/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. ^{[3]} As for the planets’
moons, quantization of their specific orbital momenta is also observed. ^{[4]} 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
М_{c}
( М_{c} 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 ħ_{s}. ^{[1]}
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) ^{ [5] }:
The object 
Mass, kg 
Radius, m 
Characteristic
speed of particles, m/s 
Neutron star 
2.7∙10^{30} 
1.2∙10^{4} 
6.8∙10^{7} 
Proton 
1.67∙10^{27} 
8.7∙10^{16} 
2.99∙10^{8} 
Similarity coefficients 
Ф’ = 1.62∙10^{57} 
Р’ = 1.4∙10^{19} 
S’ = 2.3∙10^{1} 
The stellar Planck constant for degenerate
objects is h’_{s} = h ∙ Ф’ ∙ S’ ∙ Р’ = 3.5∙10^{42} J∙s and the stellar Dirac constant is
ħ’_{s} = ħ ∙ Ф’ ∙ S’ ∙ Р’ = h’_{s}/2π = 5.5∙10^{41} 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 T_{s}
= 1.558 ms. ^{[6]} For neutron stars it is
assumed that their typical inertia moment is equal to J_{s} = 10^{38}
kg ∙m^{2}. ^{[7]} For
the angular momentum of the pulsar under consideration we obtain: L = 2πJ_{s}/T_{s}
= 4∙10^{41} 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 h_{g}
is about 10^{68} J∙s, at the level of metagalaxies – h_{m} is
about 10^{89} J∙s, and at the level of preons – h_{p} is about 10^{–46}J∙s. ^{[1]}
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 M_{p}, the proton radius R_{p} and the speed of light c:
This formula can be obtained, if we assume
hypothetical conversion of the proton rest energy into the photon energy with
the electromagnetic wave period equal to
(as if the photon passes the proton diameter
at the speed of light):
where is the wave frequency.
Another derivation of formula uses the concept of the material wave or de Broglie wave as a consequence of internal
oscillations in the matter of elementary particles. ^{[1]} For a neutron star we can similarly arrive at
the value close to the stellar Planck constant:
J∙s,
where M_{s}
and R_{s}_{
} are the mass and radius of the
neutron star, C_{s} 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
C_{s} 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:
where is the coefficient of the order of unity,
depending on the distribution of matter in the star, and 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
M_{bs}, because on the average only
this part will instantaneously rotate relative to the given rotation axis.
Assuming that M_{bs} is equal to the mass of a typical neutron
star M_{s},
the radius of the corresponding Schwarzschild black hole will be equal to:
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:
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 nonradial oscillations,
assumed to take place in the black hole at its excitation. The energy of such
oscillations should be related to the oscillation frequency
in the same way as in the photon,
with replacement of the Planck constant with the stellar Planck constant:
The value for the second spherical harmonics of the
Schwarzschild black hole was determined as follows: ^{[8]}
As the black hole excitation energy 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 . For the proton the excitation
energy equals . Accordingly, for the black hole we
can use the ratio .
If we assume that the mass of the black hole is equal to the mass of a
typical neutron star M_{s},
we obtain the following:
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).
See also
References
External links
Source: http://sergf.ru/sppen.htm