Quantization of
parameters of cosmic systems
Quantization of parameters of cosmic systems is a property of the systems observed in space, which have relatively
stable fixed stationary states and in which transitions between these states
are possible under the influence of external disturbances or in case of energy
loss. The result of the transition between stationary states is the quantized
change of energy and characteristic angular momentum of the systems. The
typical examples are satellite systems – atoms, planetary star systems, systems
of normal and dwarf galaxies. By definition, it is considered that satellites
are less massive than the main objects. In extreme cases the masses of the
system's components can be equal as in a diatomic molecule consisting of atoms
of the same chemical element or as in corresponding binary stars. In systems
with numerous objects quantization can acquire dynamic character and is
determined by longrange forces between the objects.
Quantization is
most clearly revealed in systems containing compact objects with degenerate
state of matter, such as atomic nuclei and neutron stars. These objects have
discrete physical properties and are usually the main objects in satellite
systems. In particular, the mass of the atomic nucleus is proportional to the
number of nucleons, and at the level of stars we observe discreteness of stellar parameters and the
similarity between atoms and stars, including correspondence between the masses
and the abundance in nature.
The charge and
the mass of electrons in atoms are not arbitrary values, but to a large extent
are determined by the history of electrons' emergence. Analysis of beta decay
in the substantial electron model and
in the substantial neutron model shows that the properties of electrons are secondary to the properties of
nucleons. ^{[1]} At the same time there are connections between the
mass, the charge and the radius of a proton, which are determined by the
properties of the matter and the equation of its state and which lead to
discreteness of the proton's properties. ^{[2]}
From this it
follows that discreteness of the fundamental properties of main objects and
their satellites, which arises in the course of coevolution under action of
the fundamental interactions, leads to the repeating structure of satellite
systems at different levels of matter and to quantization of their parameters.
Manifestation of discreteness of objects' properties is existing in space of
hierarchically nested levels of matter, the masses and sizes of the carriers of
which are related to each other by the law of geometric progression. According
to the similarity of matter levels and SPФ symmetry, similarity relations can be
established between the corresponding objects and phenomena and the physical
quantities characterizing them can be predicted. This allows connecting
different forms of quantization in the framework of Infinite Hierarchical Nesting of Matter.
Contents
Quantization in atomic systems
The idea of
quanta is fundamental in quantum mechanics that describes the behavior of atoms
and elementary particles. Historically, the first quantum model was Bohr model,
in which the main role is played by the quantum of action in the form of Dirac
constant J∙s. The hydrogen atom in Bohr model is the representative of hydrogen system at the level of atoms, in
which the electron in the ground state has the orbital angular momentum equal
to . In atoms and ions the following quantities are quantized:
Quantization
usually takes place in those cases when transitions of a system are possible
from one state to another and back, and the states themselves should be somehow
fixed. The high stability of the spectral lines during emission of photons from
atoms shows that the stationary states of atoms are controlled by special
mechanisms. As a result of their action an atom cannot move from one state to
another excited state, if the energy of the incident photon or of excitation is
not sufficient for transition between the states. The conditions for emerging
of stationary states are the following: ^{[1]}
The existence
of stationary states of atoms leads to quantization of the energy and the
angular momentum of electron states and to discreteness of atomic spectra.
Consideration of multielectron atoms shows that as the mass and charge of the
nucleus increase, the electron shells get closer to each other, producing
reciprocal pressure, and the inner shells get closer to the nucleus while the
electrons' energy increases. In electron transition from one energy state to
another underlying state, the angular frequency of electromagnetic emission is
associated with the rate of energy change during the change of the angular
momentum : ^{[1]}
.
The typical
change of the angular momentum of the electron during transition between the
energy levels is , due to which it is assumed that the energy of the emitted photon as an
emission quantum equals .
Quantization of
the number of nucleons in atomic nuclei is closely related to the discreteness
of the nuclear masses and to MattauchShchukarev
isobar rule, according to which after the replacement of a neutron by a proton
(or a proton by a neutron) in the nucleus of a stable isotope, a radioactive
isotope will appear.
The law of
charge conservation holds almost exactly in all known phenomena, and the
minimum value of the electric charge at the atomic level is the elementary
charge C. With the help of the Planck
constant and the elementary charge we calculate the magnetic flux quantum Wb, which is found in experiments
with superconductors. In quantum mechanics this quantum is associated with the
motion of charges in the form of Cooper pairs, but it has also classical
explanation. ^{[2]}
In physics such
phenomena are also associated with quantization as Bohr magneton, nuclear
magneton, multiplicity (chemistry), radioactive decay, nuclear magnetic
resonance, electron paramagnetic resonance, Zeeman effect and many other
phenomena.
In chemistry
the idea of quanta is revealed as valence (chemistry) and is expressed in the
law of matter amount conservation, Faraday's laws of electrolysis and other
laws.
Planetary systems
If at the
microlevel quantization is found in many phenomena, at the macrolevel or at
the level of stars it may seem unexpected. However, the conditions for the
emergence of quanta can occasionally occur at various levels of matter. ^{[3]}
For example, the planetary systems of stars and satellite systems of planets
which are similar to atoms by the number of objects and the nature of acting
forces.
Titius–Bode law
The Titius–Bode law describes the empirical dependence of the
mean orbital radius of the planets of the Solar system on the planet's number:
a.u.,
where is for Mercury, is for to Venus, Earth and Mars, for the asteroid belt, Jupiter,
Saturn and Uranus, respectively. Although the law is violated for Neptune, in
which case instead of the distance to Neptune it provides the distance to Pluto
with , the law demonstrates that distances to the planets increase approximately
exponentially. Dependency on can be considered as a form of
quantization of permissible orbital radii of planets. One justification for the
law is the idea that the boundaries of zones in which the planets once were
formed, starting with Venus, are determined by the formula: ^{[4]}
a.u.,
where
Then based on
the Titius–Bode law we can find the positions of planets
as arithmetic mean values between two adjacent boundaries:
,
with .
Another
explanation involves the influence of gravitational energy fluxes, arising from
the static gravitational field of the Sun and its gravitational torsion field from its
rotation, on the fluxes of kinetic energy of rotation of the gasdust matter of
protoplanets during their accumulation. ^{[2]}
In some orbits
the energy fluxes become aligned, which leads to stable stationary states, in
which the matter can accumulate more efficiently, subsequently forming planets.
This process is equivalent to the emergence of stationary states in atoms,
leading to quantization of energy levels and angular momentums of electrons. As
a consequence of the states' quantization, orbital resonance is possible
between the close planets, in which their orbital periods are related to each
other as small integers.
Using Schrödinger equation
Since the
states of the hydrogen atom in quantum mechanics are accurately calculated by
solving the Schrödinger equation, the same approach was used to model the
permissible orbit radii of the planets around the Sun and stars. ^{[5]}
Nearly for all the planets we were able to choose the appropriate quantum
numbers and
, characterizing the energy and angular momentum. Besides, it appears that
many planetary systems of different stars have similar structure in respect of
distribution of orbital radii of planets.
Approximately
the same results were obtained in a series of researches in the solution of the
generalized Schrödinger equation. ^{[6]} ^{[7]} For the orbital
radii of planetary systems a formula was found:
,
where is the gravitational constant, is the mass of the central body (a
star in the planetary system, a planet in the satellite system), is a constant with the dimension of velocity, which depends on the
properties of the system and is equal to the orbital velocity at the primary
(first) orbit. In the Solar system km/s is for the inner planets of
Mercury, Venus, Earth and Mars, for which , respectively. For Jupiter, Saturn,
Uranus, Neptune and Pluto at a lower value km/s. Different values for inner and large planets in this theory can be explained by the process
of protoplanetary matter accumulation – first large accumulation zones were
formed and then in the inner zone the secondary division into smaller areas
took place, where small planets appeared.
Quantization of the orbital radii
of planets
The principle
of the angular momentum quantization and its increasing in proportion to the
number is used to show that the inner and outer planets of the Solar system are
divided into two separate groups. ^{[8]} ^{[9]} One of the
results is the formula for the orbital radii of planets:
where is the radius with . For the inner terrestrial planets m and the orbits with and are empty due to the assumed influence of the Sun, for Mercury , for Venus , for the Earth , for Mars , for the asteroid Ceres . For the outer planets, starting with Jupiter, it is assumed that m, .
If we denote
the orbital angular momentum of a planet as , the specific orbital angular momentum as , and if we use
the expression for the orbital velocity in the form , where is the mass of the Sun, then in
view of (1) we obtain:
.
This equation
means that the relative orbital angular momentum of planets is quantized, as
the orbital angular momentum of the electron is quantized in the Bohr model of
the hydrogen system. For the period of
orbital revolution of planets we find the relation: .
Planets' satellites
Satellite
systems of Jupiter, Saturn and Uranus are populated enough, to be compared with
the distribution of planets in the Solar system. For the planets' satellites
the Titius–Bode law applies, but with a reduced basic
distance, where instead of 0.1 a.u. the value of the order of 60,000 km is
used. ^{[4]}
To describe
quantization of the orbits of satellites the same approach can be used as that
for the orbital motion of planets. In this case, the orbital radii are
proportional not to the square of the planet's number, but to the square of the
satellite's number. ^{[8]}
Similarity to atoms
Similarity coefficients
Considering the
correspondence between atoms and mainsequence stars, Sergey Fedosin discovered
that the Solar system is similar to the isotope of oxygen atom with the mass
number and charge number . ^{[10]} The ratio of the Sun's mass to the nucleus mass of the
isotope of oxygen atom sets the coefficient of similarity
in mass:
.
Multiplying by the electron mass, we can find
the mass of the planet corresponding to the electron: kg or 10.1 of Earth masses.
The coefficient
of similarity in speed is given by:
,
where is the coefficient of similarity in speed for the hydrogen system, km/s is the characteristic speed of the matter particles in the
main sequence star with minimum mass , is the speed of light as the
characteristic speed of matter in a
proton.
Since the
stellar matter is held by gravitational forces, the characteristic speed of matter is determined by half of the
gravitational energy, the absolute value of which defines the full energy of
the star with the mass :
where is the radius of the star, depends on the matter distribution
in the star and is equal to 0.6 in a uniform case .
Multiplying the
rest energy of the atomic nucleus , where is the mass of the nucleus, by the
similarity coefficients it is possible to find the absolute value of full
energy of the mainsequence star:
.
The coefficient
of similarity in energy between the oxygen atom and the Solar system is defined
by the product . Comparing the ionization energies of electrons in the oxygen atom with
the specific energy of planets in the Solar system due to their attraction to
the Sun gives the ratio of these energies, which differs not more than three
times from the coefficient of similarity in energiy. ^{[10]}
The coefficient
of similarity in size is given by:
,
where is calculated from the ratios of the
hydrogen system. If we multiply the Bohr
radius in the hydrogen atom by , we obtain the value 19.25 a.u., which is close enough to the Uranus
orbit. There are several ways to estimate the coefficient of similarity in
size, which give similar results: 1) Comparing the semiaxes of the orbits of
binary stars and the bond lengths in molecules; 2) Comparing the dimensions in
the Solar system and in the oxygen atom; 3) Comparing the orbits of Mercury and
of the hydrogenlike ion of oxygen; 4) Comparing the dimensions of atomic
nuclei and stars.
The coefficient
of similarity in time is:
, where
.
By means of
similarity coefficients the stellar Dirac
constant is determined for planetary systems of mainsequence stars:
J∙s,
where is the Dirac constant. The quantity
almost coincides with the intrinsic
angular momentum of the Solar rotation.
The orbital motion of planets
In quantum
mechanics it is considered that the oxygen atom has two electron layers.
Klayer includes 1sshell and two electrons. Llayer includes 2sshell with two
electrons and 2pshell with four electrons. It is assumed that electrons in
sstates do not have the orbital angular momentum. From formal comparison of
the oxygen atom and the Solar system it follows that Mercury and Venus are
equivalent to 1sshell, Earth and Mars – to 2sshell and the large planets, the
orbital angular momenta of which are much greater than those of the terrestrial
planets, are similar to 2pshell of the oxygen atom. In the Bohr model of hydrogen
atom the orbital angular momentum of electron is quantized:
.
If we assume
that planets in their rotation around the Sun have little influence on each
other and they are located in orbits that are allowed in the hydrogenlike
atom, then for specific orbital angular momenta of planets we obtain quantum
relation with accuracy up to 25 %: ^{[10]}
,
where kg is the mass of the planet corresponding to electron, follows from correspondence with
the empirical data.
If we
substitute here the expressions for the orbital velocity in the form , we could determine the orbital radii of planets:
.
The
quantity m^{2}/s differentiates planets from planetary satellites. For
example, the dwarf planet Ceres, with the orbit between Mars and Jupiter, has
the specific orbital angular momentum equal to m^{2}/s, while for Jupiter's
satellite Callisto this quantity is much less: m^{2}/s.
The orbital angular momenta of
planetary satellites
In the
satellite systems of Jupiter, Saturn and Uranus, by analogy with specific
orbital angular momenta of planets, quantum Bohr formula is used as follows: ^{[1]}
,
where and are the orbital angular momentum and the mass of the satellite in a
circular orbit, and are the average velocity in orbit
and the average orbital radius of the satellite, is the revolution period in orbit with the number , is a constant, which depends on the satellite system.
In all
satellite systems there is no satellite with the number , its role is played by the rings, which are located near each of the
planets.
In Jupiter's
system there are eight regular satellites, starting with Metis () and ending with Callisto . The rest eight outer satellites from Leda to Sinope are small in size,
have significant eccentricities and inclinations to Jupiter's equator, and the
last 4 satellites rotate in the opposite direction. They can be considered
asteroids captured by Jupiter. The quantity m^{2}/s in general
characterizes the regular satellite system of Jupiter, which appeared during
the period of the planet's formation.
A similar
situation takes place for Saturn (eleven regular satellites from Atlas,
Prometheus and Pandora with to Iapetus with ), and for Uranus (regular satellites from Cordelia (1986 U7) with to Oberon with ), for which m^{2}/s and m^{2}/s, respectively.
The meaning of
the quantities , and lies in the fact that they reflect
the specific orbital angular momentum of that part of a cloud, from which
formation of one or another planetary system begins. This is confirmed by the
fact that products of equatorial velocities of Jupiter, Saturn and Uranus and
the corresponding radii of the planets are close to the values , and , respectively.
The total angular momentum of
planetary systems
With the help
of the quantum approach we can determine if the planetary systems around stars
are formed from weakly rotating gasdust clouds. If in a first approximation we
assume that the mass distribution of planets in planetary systems is of such
kind that the orbital angular momenta of planets in orbit are equal to as in Bohr theory for the electron, then the total orbital angular momentum
is:
,
where is the charge number of the star,
specifying the number of planets.
The total
angular momentum of the planetary system is composed of the proper angular
momentum of the star and the total orbital angular
momentum of planets: . The quantities were calculated for planetary
systems of main sequence stars, based on the observed typical velocities of
stars' rotation around their axes, their masses and radii, taking into account
the mass numbers and the charge numbers of the stars, determined on the basis of their similarity with the chemical
elements (see discreteness of stellar
parameters). ^{[10]} Since the planetary systems are formed from
gasdust clouds, then must be equal to the angular
momentum of such clouds. For a cloud with the mass and the radius the estimate of the total rate of
rotation at the equator is . Then the angular momentum of the cloud is equal
to:
,
where the
coefficient reflects the change in the angular
momentum due to nonuniformity of the mass density and the differential
rotation of the cloud.
Between the
mass and the radius of the cloud there is a dependence: , where is the mean mass density of the cloud.
In addition, by analogy with the mass of atomic nuclei, the stellar mass is
associated with the mass number and the mass of a star with the minimum mass: . Taking it into account we obtain:
The value J∙s is determined from the diagram
of dependence of the angular momentum of the planetary system on the mass number and it appears to be close to the
value of the Dirac stellar constant , which is correct for a planetary system of one planet and a mainsequence
star with minimum mass.
Intrinsic angular momenta of
planets
By analogy with
quantization of orbital and spin angular momenta of electrons in atoms, we make
an assumption about quantization of intrinsic angular momenta of planets. ^{[11]}
For each
gravitationally bound object with mass and radius we can introduce their
characteristic angular momentum according to the following formula:
where the
characteristic speed is found from the ratio of energies
similar to (2).
The electron
spin is usually determined by the Planck constant in the form . Similarly for planets the intrinsic angular momenta are proportional to
the quantity and to the planet number as the quantum number :
where is obtained from correspondence to
the empirical data.
Table 1
compares the intrinsic angular momenta of the planets with the values
calculated by relation (5); we see that only inhibited planets Mercury and
Venus noticeably deviate from this dependence.^{[10]}
Table 1. 

Planet 
Intrinsic angular momentum, J∙s 
Calculated value according to (5), J∙s 
Mercury 


Venus 


Earth 


Mars 


Jupiter 


Saturn 


Uranus 


Neptune 


Pluto 


Large and small planets
Quantization of
masses of atomic nuclei is associated with mass discreteness of nucleons, which
are part of nuclei. For planets mass discreteness can also be proved, including
the existence of the minimum and maximum masses. In order to calculate the
maximum mass, the equality of the planet's gravitational energy and the
electrostatic energy per atom is used. If the gravitational energy is too high,
the electron shells of the matter atoms start getting compressed and the planet
can turn into a white dwarf, in which the gravitational force is opposed by the
pressure of degenerate electrons. For the matter of a planet, composed of
hydrogen, the maximum electrostatic energy approximately corresponds to the
energy of an electron in a hydrogen atom in the ground state. The equality of
energy has the following form:
,
where and are the mass and the radius of the
planet, is the number of nucleons in the
planet's matter, is the proton mass, is the vacuum permittivity, is the Bohr radius.
The planet's
volume can be calculated by the radius as well as by the total amount of
hydrogen atoms:
.
Taking it into
account, at we determine the mass and the
radius of a massive planet, exceeding to some extent the mass and the radius of
Jupiter: ^{[11]}
kg .
m .
To estimate the
minimum mass of a planet we use the equality of half of the gravitational
energy and the internal thermal energy according to the virial theorem:
,
where is the Boltzmann constant, is the average
internal temperature. Expressing the planet mass by the density and the radius
in the form , and using the equation , we obtain the mass and the radius:
.
.
Substituting
here the same mass density as that of the dwarf planet Ceres kg/m^{3}, and the
temperature K (the temperature of background radiation), we can estimate the minimum
mass and the radius of the planet:
kg, km, which is slightly less than
Ceres' parameters.
Stellar constants
For planetary
systems of main sequence stars the stellar
constants are: ^{[10]}
The systems with neutron stars
Similarity between neutron stars
and nucleons
Similarity coefficients in Fedosin's model
The ratios of
the mass (radius) of a typical neutron star to the mass (radius) of a proton
determine the coefficients of similarity in mass and size, respectively:
.
.
From relation
(2) we can estimate the characteristic speed of the matter particles of the
neutron star under consideration:
m/s.
The coefficient
of similarity in speed equals:
,
here is the speed of light.
The coefficient
of similarity in time as the ratio of time flow rates between the elementary
particles and neutron stars:
.
The value of
the stellar Dirac constant for neutron
stars:
J∙s.
This quantity
is close to the limit angular momentum of intrinsic rotation of neutron stars.
Multiplying the Bohr radius in a hydrogen atom by we obtain the value m. This distance almost coincides with the Roche limit, at which planets
disintegrate in the strong gravitational field of a neutron star. The disks
discovered near neutron stars also have the characteristic radius of the order of . ^{[12]} From the point of view of the similarity of matter levels and the substantial electron model, the disks
around neutron stars are similar to electron disks in atoms. If we multiply the
electron mass by , we can estimate the mass of the disk: kg. The model of an electron in the
form of a disk can explain the origin of the electron spin in the atom.
Magnetars
Based on the
similarity between atoms and stars, magnetars, as strongly magnetized neutron
stars, are considered as the analogues of a proton. The similarity of these
objects is shown in the substantial proton
model, and the evolution – in the substantial neutron model. Magnetars and the disks rotating around them form the hydrogen system for degenerate objects at
the level of stars, and the angular momentum of disks in the ground state is
equal to the stellar Dirac constant . It is expected that the longterm evolution of stars will lead eventually
to transformation of all stars into white dwarfs, neutron stars and magnetars,
and the latter will form groups of stars similar to atomic nuclei and will be
the matter basis at the level of stars. In this case the mass of each portion
of stellar matter will be quantized with accuracy up to the mass of one
magnetar.
Using the
similarity coefficients and the relations of physical quantities' dimensions we
can determine the electric charge and magnetic moment of the magnetar, which is
the proton's analogue:
C,
J/T,
where and are the elementary charge and the magnetic moment of the proton,
respectively.
The values of
the electric potential and the magnetic field induction at the magnetar's pole:
^{[1]}
V,
T,
here is the vacuum permeability.
The ratio of
the magnetar's charge to its mass according to the theory of similarity is
given by the formula: ^{[10]}
,