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):
The object |
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 non-radial 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