The similarity of
matter levels is a principle in the Theory of Infinite Hierarchical Nesting of
Matter, with the help of which connections between the different levels of
matter are described. This principle is a part of the similarity law of
carriers of different scale levels. ^{[1]} The
similarity of matter levels conforms to the SPФ
symmetry and is illustrated by discreteness of stellar parameters, quantization of parameters of cosmic systems,
the existence of hydrogen systems.
Similarity relations allow us to find the parameters
of the objects which are inaccessible for direct observation (the smallest
structural units of the matter of elementary particles, the objects with sizes
greater than the Metagalaxy), including mass, size, spin, electrical charge,
magnetic moment, energy, characteristic
speed of matter, temperature, etc. as well as the values of fundamental
physical constants inherent in the matter levels. At the level of stars the
examples of such constants are stellar
Planck constant, stellar Dirac constant,
stellar Boltzmann constant, and other
stellar constants. Due to the nesting of one matter levels into other, the
massive objects are composed of the particles of lower levels of matter. This
leads to the interrelation of the characteristics of objects and the states of
their matter, as well as to the symmetry between the properties of the matter
particles and the properties of objects, which is manifested through the
relations of similarity. The possibility of the location of cosmic objects at
different levels of matter as on the scale axis gives the idea of the scale dimension, considered as the fifth
dimension of spacetime.
Contents

In 1937 Dirac suggested the hypothesis of large
numbers, according to which the parameters of the Metagalaxy (it was then
called the Universe, although now it is established that the Metagalaxy is only
part of the Universe) can be found through the parameters of elementary
particles by multiplying them by some large numbers. ^{[2]}
According to his hypothesis, the following relations should hold:
where specify the characteristic time of the
process, the size and the mass of the Metagalaxy, specify the same parameters for elementary
particles.
The hypothesis of large numbers was also considered by
Weyl in 1919, ^{[3]} Eddington in 1931, ^{[4]} ^{[5]} Jordan in 1947, ^{[6]} Klein and others.
Weyl considered a hypothetical object with the mass which sets the rest energy equal to the
gravitational energy of the electron, provided that the radius of the electron
is equal to the classical electron radius m, as well as equal to the
electrical energy of the object, provided that the charge of the object is
equal to the electron charge, and the radius of the object is :
,
and the classical electron radius is determined from the
condition of equality of the rest energy of the electron in the form of a
spherical shell and its electrical energy:
,
where is the speed of light, is the gravitational constant, is the mass of the electron, is the elementary charge as the proton charge,
is the electric constant.
From here it follows that, , and m pc, so that the radius of the
hypothetical object exceeds more than by an order of magnitude the observable
part of the Universe.
The above equation for the rest energy can be interpreted as the equality between the
gravitational energy of two electrons at the distance from each other, and their
electrical energy at the distance . In this case, the large value is obtained as the consequence of the weakness
of the gravitational force between the electrons in comparison with their
electrical force and it seems not related to the size of the Metagalaxy.
Indeed, if we divide the electrical force between the proton and the electron
by the absolute value of the force of their gravitational attraction, we shall
obtain the value:
,
where is the proton mass.
At the same time the strong gravitational constant, by its
definition, equals:
m^{3}•s^{–2}•kg^{–1}.
Therefore , that
is the ratio of electrical force to the gravitational force between the proton
and the electron is equal to the ratio of the strong gravitational constant to
the ordinary gravitational constant and is proportional to the ratio of
sizes .
In the gravitational
model of strong interaction the strong gravitation acts between the matter
of hadrons and as well as between the matter of leptons. At the level of atoms,
the strong gravitation is the same as
the ordinary gravitation at the level of planets and stars. In this picture the
distance is not related to the size of the Metagalaxy.
One of the attempts to explain the hypothesis of large
numbers is the use of quantum ideas with consideration of hadrons, compact
stellar objects and the Universe as the objects similar to black holes. ^{[7]} ^{[8]} However, such combination
of quantum mechanics and general theory of relativity is not quite convincing,
and therefore the search for other explanations continues.
Main source: Quantization of parameters of cosmic systems
The Titius–Bode law, which
appeared long before quantum mechanics, was intended to mathematically describe
the smooth dependence of the radii of the orbits of planets in the Solar system
on the number of the planet . At the present time various
schemes are suggested in which the orbits of the planets are described by
quantum numbers for the energy and the orbital momentum, similarly to the way
in which the states of the electrons in the atom are specified. In particular,
for modeling the acceptable radii of the orbits of the planets near the Sun and
the stars the solutions of the Schrödinger equation are used. ^{[9]} ^{[10]} ^{[11]}
The similar results are obtained under the assumption
that the planetary orbits are quantized proportionally to the square of the
quantum number. ^{[12]} ^{[13]}
As the rule it is assumed that the inner terrestrial planets and outer large
planets are independent in the respect that they have different sets of quantum
numbers. In these approaches, there are no planets of the Solar system in the
orbits with and for the inner planets, and with for the outer planets.
In general the transfer of the methods of quantum
mechanics on the level of stellar and planetary objects is the logical
development of the idea of similarity of matter levels, since the quantization
is a universal property of matter.
According to the general opinion, if we take two
similar systems, one – in the microworld and the other – in the macroworld,
then the rate of time, understood as the number of similar events per unit
time, is much higher in the microworld than in the macroworld. This is the
consequence of the fact that the duration of an event in the microworld is
small as compared with the duration of a similar event in the macroworld
because of the difference in sizes. Under the similarity coefficient the
dimensionless quantity is understood, which is equal to the ratio of two
identical physical quantities, which refer to the compared and in some ways
similar to each other objects at different levels of matter. As it follows from
the theory of dimensions, it is sufficient to know only three similarity
coefficients, for example, of similarity in mass, size and time, in order to
find with their help any other similarity coefficients for mechanical
quantities. Large numbers of DiracEddington in fact represent the similarity
coefficients between the Metagalaxy and elementary particles.
One of the problems with the similarity coefficients
in various models of similarity is before their determination first to uniquely
identify the compared matter levels and the objects corresponding to them. For
example, Fournier d'Alba considered ^{[14]} that the ratio of linear sizes of a star and an
atom, as well as the ratio of durations of their similar processes, is
expressed by the number 10^{22}. But the sizes of stars with the same
mass can differ by thousands of times, which makes the estimate of the
coefficient of similarity in size not unique and depending much on the chosen
type of stars. The purpose of using physically justified similarity relations
between different matter levels is to build the model of the Universe, in which
it becomes clear how the large arises from the small, what forces of the nature
are fundamental and inherent in all levels, and how they interact with each
other, giving rise to each other.
Sergey Sukhonos arranged all
the known objects of the microworld, the macroworld and the megaworld
on one scale axis of sizes and found out that the properties of objects are
repeated periodically with increasing of the size approximately 10^{20}
times. ^{[15]} This is well illustrated by the
following examples:
In all cases, the scale "distance" between
the system and its components is the same – 10^{20}. Built by Sukhonos the scale similarity in size conforms to the
DiracEddington hypothesis of large numbers, since
In the regularity of distribution of objects on the
matter levels Sukhonos found bimodality in the sizes
of objects. This is manifested, for example, in the distribution of the atoms’
diameters by sizes, in the similar distribution for the stars and galaxies, as
well as in the distribution of areas of countries, regions, states, provinces,
etc. ^{[16]} In order to combine bimodality and
periodicity of changing of the sizes of the objects of matter levels, the
mechanism of multistage cluster convolution (the theory of centrosymmetric
packing) and global scale standing waves are considered.
For the coefficients of similarity in size and time Yun Pyo Jung
derives the value of the order of 10^{30}. To obtain these
coefficients, he compares the radius of the atomic nuclei (≈ 10^{15}
m) and the radius of the nuclei of galaxies, presumably equal in the case of
the Galaxy (its other name is the Milky Way) 0.33 light years or 0.1 pc, which
is equal to 3∙10^{15} m. However recent studies show that the nuclei of
galaxies do not have any unambiguous definition. Rounded thickening in our
Galaxy, called the bulge, has the radius of 200 pc, and the area in the center
of the Galaxy called Sagittarius A* contains the mass 4.3∙10^{6} of
solar masses with the radius of 45 a.u. or 7∙10^{12}
m. Another way to determine the coefficient of similarity in size is with the
help of the ratio of the radius of the galaxy (≈ 30 kpc) to the radius of the
atom, of the order of 10^{10} m. Yun Pyo
Jung also considers the ratios of the radii or the typical sizes in the objects
similar to each other, such as the molecules and groups of galaxies,
macromolecules and clusters of galaxies, organelles and superclusters of
galaxies, biological cells with the radius of 25 μm
and the observed cosmos with the radius of 15 billion light years, again
obtaining the value of the order of 10^{30}. ^{[17]}
Robert Oldershaw went further and determined the
coincident with each other coefficients of similarity in size and time, equal
to Λ = 5.2∙10^{17}, and the coefficient of similarity in mass X = Λ^{D}
= 1.7∙10^{56}, where the exponent D = 3.174. At the same time Oldershaw
compares the atomic nuclei, stars and galaxies as the corresponding objects at
three levels of matter. ^{[18]}
The hydrogen system
at the level of stars, according to Oldershaw, consists of the main sequence
star with the mass , and of the object – the analogue
of the electron with the mass equal to 26 Earth’s masses. If we convert the
Bohr radius into the corresponding radius at the level of stars by multiplying
by the coefficient of similarity in size Λ, then this object must be located in
the shell of the star. If this object is considered as being in the excited
state, it can take the form of the planet.
To obtain the radius of the ordinary galaxy we must
multiply the radius of the corresponding atomic nucleus by Λ^{2}, which
gives the range of radii of galaxies from 7 to 75 kpc (similar to the proton
and the nucleus of lead, respectively). Since Oldershaw believes that the
coefficients of similarity between the levels of matter are the same for all
objects and do not depend on the type of these objects, he has a problem with
obtaining the sizes of dwarf and giant galaxies (0.1 kpc and 500 kpc,
respectively). To solve this problem, he expands the range of objects at the
atomic level, adding to the atoms and ions the separate nucleons, hadrons,
mesons and leptons. Assuming that all objects at the subnuclear
level are similar to black holes, to estimate their radius Oldershaw applies
the Schwarzschild formula:
where is the constant of gravitation acting on the
given level of matter, for the atomic level, for the level of the stars, for the level of galaxies.
The constant of gravitation is calculated using the
coefficients of similarity taking into account the dimension of this constant
equal to m^{3}/(kg∙s^{2}). Since the coefficients of similarity
in size and time are considered equal, we obtain:
where is the ordinary gravitational constant.
Assuming that at the level of atoms m^{3}•s^{–2}•kg^{–1} is
the strong gravitational constant,
Oldershaw finds the corresponding radius of the electron 4.4∙10^{19}
m, and the radius of the proton 0.81∙10^{15} m. If we multiply this
radius of the electron by Λ^{2}, we shall obtain the radius of 3.9 pc,
corresponding to the nuclei of globular star clusters. According to Oldershaw,
these objects with the sizes of the globular clusters are the analogue of
electrons at the level of galaxies. However the ratio of the minimum galaxy
mass of a normal galaxy to the mass of a typical globular cluster has the order
of magnitude 10^{5}, which is much greater than the ratio of the proton
mass to the electron mass, which is equal to 1836. Another problem is that the
number of globular clusters in galaxies is many times greater than the number
of electrons in atoms. Besides, black holes are only suspected inside the
globular clusters and galaxies.
At the level of galaxies the gravitational constant
according to Oldershaw is equal to m^{3}•s^{–2}•kg^{–1}. If we use the
Schwarzschild formula with this gravitational constant and the sizes of
galactic objects – the analogues of the electron and the proton, we obtain very
large masses – about 2.7∙10^{82} kg and 5∙10^{85} kg,
respectively. Oldershaw believes that we do not notice such masses of galaxies,
because at the level of galaxies the gravitational constant is extremely small.
He also considers the Metagalaxy to be the result of explosion, similar to the
supernova explosion, which explains the high effective temperature of galaxies,
producing gas similar to the hot fully ionized gas. To calculate the
temperature, the value of the average peculiar velocity of galaxies is used,
equal to 700 km/s. Atomic nuclei moving at such a velocity, have the kinetic
temperature of about 10^{8} – 10^{9} Kelvin degrees, and the
same temperature is attributed to the gas from the galaxies.
Oldershaw states that the observable Universe is
considerably smaller in size than the object that must be at the metagalactic
level of matter, exceeding Λ = 5.2∙10^{17} times the size of galaxies.
With the help of telescopes and different techniques we can see the most
distant quasars at the distance only 10^{5} – 10^{6} greater
than the radii of typical galaxies. Among other conclusions is the assumption
that the dark matter consists of black holes; the ether is assumed to consist
of charged relativistic particles; the electrical force is substantiated as the
result of emission by large charges of tiny particles, so that the proton and
the electron in the form of the corresponding KerrNewman rotating black microholes must emit smaller charged particles.
As the natural units of measuring the physical
quantities at the atomic level Oldershaw uses a set of Dirac constant and the speed of light , included in Planck units, but instead
of the ordinary gravitational constant he uses the strong gravitational
constant . This allowed him to determine the
"modernized" Planck values:
The obtained values are close enough to the parameters
of the proton. ^{[19]}
In Sergey Fedosin’s monograph
on the theory of similarity the eighteen levels of matter from preons to
metagalaxies were divided into basic and intermediate by their masses and
sizes. ^{[20]} The basic levels in this range
of matter levels include the level of elementary particles and the level of
stars. The most stable and longlived carriers are located at these levels,
such as nucleons and neutron stars, containing the maximum number of
constituent particles and having the maximum density of matter and energy. The
matter of these carriers is degenerate, that is, their constituent particles
have approximately the same quantum states, and therefore the state of such
matter is described by the laws of quantum mechanics. The neutron star contains
about 10^{57} nucleons, and by induction we assume that the nucleon
contains the same number of quantum particles.
Fedosin’s approach has the following
features:
1) He does not support the Oldershaw’s idea of the
hierarchical nesting of black holes as the basic structures for the considered
objects at different levels of matter, due to denying the existence of black
holes as such.
2) The similarity coefficients between the level of
atoms and elementary particles and the level of stars (or level of galaxies,
other levels) are changed if we transit from main sequence stars (or from
normal galaxies and standard objects) to the compact stars (to compact
galaxies, compact objects of other levels). This means the difference between
the similarity coefficients for different types of objects.
3) The coefficients of similarity in size and time do
not coincide in magnitude with each other, in contrast to the models of other
researchers.
4) Fedosin divides the matter levels into basic and
intermediate. The basic levels of matter are characterized by the fact that on
the objects of these levels the fundamental forces achieve the extreme values.
The objects of these levels have the highest density of matter and energy, they
are most stable, have a spherical shape and form the basis of larger objects.
5) The connection between the intermediate levels of
the matter is carried out by discrete coefficients of similarity, so that the
ratios of masses, sizes and characteristic speeds of processes between any
similar objects at the adjacent levels of matter remain the same. This leads to
the fact that the masses and the sizes of carriers at the scale of the masses
and sizes change in a
geometric progression
with the constant factors and , respectively.
6) The similarity relations of objects and physical
phenomena are performed most accurately with those objects, the evolution of
which is repeated by the same scenario at the different levels of matter. For
example, at the level of elementary particles the real analogues for the main
sequence stars should be the objects which give rise to nuons and nucleons. In turn, at the level
of stars the exact analogues of nuons and nucleons are considered white dwarfs
and neutron stars.
The coefficient of similarity in mass is determined by
Fedosin based on the accurate data on the masses of 446 binary main sequence
stars from the Svechnikov’s catalogue ^{[21]} and the data provided by other authors. Processing
of the available data leads to the conclusion that the analogue of the Solar
system at the atomic level is the isotope of oxygen O(18), and the hydrogen
corresponds to the stars with the minimum mass , where is the mass of the Sun. The ratio of the mass
of the Sun to the mass of the nuclide O(18) gives the coefficient of similarity
in mass . If we multiply the mass of the electron by , we obtain the mass of the planet М_{pl}
= 6.06∙10^{25} kg or 10.1 Earth’s masses. From the dependence of the
radius of the planets on their mass it follows that the mass of the planet М_{pl}
corresponds to the radius R_{pl}
= 20000 km or 3.1 Earth’s radii. Measurements of the radii of planets in
different planetary systems also shows that most of the planets have the radius
from 2 to 4 Earth’s radii. ^{[22]}
The convenient model to determine the coefficients of
similarity is the hydrogen system,
consisting at the atomic level of the proton and the electron, at the level of
stars – of the star with minimum mass and of the planet as the analogue of the electron,
and at the level of galaxies – of normal and dwarf galaxies. The parameters of
the objects of the hydrogen system for atoms and the main sequence stars are
given in Table 1.
Table 1. The hydrogen system for atoms and main
sequence stars 

Object 
Mass, kg 
Radius of the
orbit, m 
Speed at the
orbit, m/s 
Planetary system 
M_{ps} = 1.11∙10^{29} 
R_{F} =2.88∙10^{12} 
V_{pl} = 1.6∙10^{3} 
The hydrogen atom 
M_{p} = 1.67∙10^{27} 
R_{B} =5.3∙10^{11} 
V_{e} = 2.19∙10^{6} 
Similarity coefficients 
Ф = 6.654∙10^{55} 
Р_{0} = 5.437∙10^{22} 
S_{0} = 7.34∙10^{4} 
If we consider a hydrogenlike atom, the speed of the
orbital rotation of electron in it is proportional to the nuclear charge . In the corresponding planetary
system the speed of the planet’s rotation around the star is proportional to
the mass of the star, i.e., to its mass number . It follows that the coefficient of
similarity in speed is given by: . For the same reason for the
coefficient of similarity in size we obtain:
. In order to find the coefficient of
similarity in speed the following method is used – in the assumption
that the speeds of the orbital motion of the electron and the planet are
determined by the characteristics of the attracting body (the atomic nucleus or
the star), not the orbital motion speeds are considered, but the characteristic
speeds of the matter inside the corresponding attracting bodies. Given the
mass–energy equivalence, the total energies of the atomic nucleus and the star
are:
where the characteristic speed of the matter in the star
depends on the mass number and the charge number of the star. The total energy of the star can
be calculated by the formula:
here is the gravitational constant, and are the mass and the radius of the star, is the coefficient depending on the
distribution of the matter, in case of uniform mass density .
The results of calculating the energy of main sequence
stars by various authors allow us to build the dependence of the total energy
on the mass of stars and to find the characteristic speed for them. Since the masses of stars are
associated with the corresponding nuclides which have the mass and charge
numbers, then from the relation we can determine the value of
the characteristic stellar speed: km/s. ^{[20]}
The ratio of the speeds in the form allows us to determine the coefficient
of similarity in speed listed in Table 1 for the hydrogen system, and to find
the orbital speed of the planet. Further, from the equation of equilibrium of
the gravitational force and the centripetal force we calculate the radius of
the planet’s orbit and the coefficient of similarity in size as the ratio of the radius of the planet's orbit
to the radius of the electron’s orbit in the hydrogen atom in its ground state.
The value is close enough to the ratio between the
semiaxes of the orbits of binary stars and the bond lengths of the
corresponding molecules, to the ratio of the sizes of the Solar system and the
oxygen atom, to the ratio of the size of Mercury's orbit and sevenfold ionized
oxygen ion, to the ratio of the sizes of the stars’ nuclei and the sizes of the
atomic nuclei.
The coefficient of similarity in time, as the ratio of
time flow speeds between the nuclear and ordinary stellar systems, is:
.
The analogue of stellar
Dirac constant for ordinary stars:
J∙s,
where is the Dirac constant.
For the level of quantum and compact objects – the
elementary particles and neutron stars, the coefficients of similarity in mass,
size and characteristic speed are slightly different from the similarity
coefficients for atoms and ordinary stars. In Table 2 the data for the proton
and neutron star are used. ^{[20]} ^{[23]}
Table 2. The
coefficients of similarity between the proton and neutron star 

Object 
Mass, kg 
Radius, m 
The
characteristic speed of matter, m/s 
Neutron star 
M _{s} = 2.7∙10^{30} 
R_{s} = 1.2∙10^{4} 
C' _{s} = 6.8∙10^{7} 
Proton 
M_{p} = 1.67∙10^{27} 
R_{p} = 8.7∙10^{16} 
c = 2.99∙10^{8} 
Similarity coefficients 
Ф' = 1.62∙10^{57} 
Р' = 1.4∙10^{19} 
S' = 2.3∙10^{1} 
If we multiply the mass of the electron by , we obtain the mass of the
objectthe analogue of the electron М
_{d} = 1.5∙10^{27} kg, which equals 250 of the Earth’s mass, or
0.78 of the Jupiter’s mass. The coefficient of similarity in time, as the ratio
of the rate of time between elementary particles and neutron stars, is:
.
The value of the Dirac stellar constant for the
compact degenerate stars is:
J∙s.
Multiplying the Boltzmann constant by the coefficient of similarity in energy, we can find the stellar
Boltzmann constant: J/K.
The electromagnetic fine
structure constant is defined as the ratio of the velocity of the electron
in the hydrogen atom at the Bohr orbit to the speed of light :
,
where is the elementary charge, is the electric
constant, is the Planck constant. It is easy to prove
that for stationary circular orbits of the electron in the hydrogen atom the
fine structure constant is equal to the ratio of the total energy of the
electron to the photon’s energy, the wavelength of which is equal to twice the
circumference of the electron’s rotation.
The fine structure constant can be expressed through
the strong gravitational constant , the proton mass and the electron mass :
where is the coupling constant of strong
gravitational interaction, taken without taking into account the absorption of
gravitons in the matter of the two interacting nucleons.
Similarly to this, the gravitational fine structure
constant is calculated for the planet orbiting the star – the analogue of the
proton at the velocity : ^{[20]}
.
The same ratio for the object – the analogue of the
electron orbiting the neutron star in the form of a disk (discon),
has the form:
.
Due to the similarity relations the constants and are equal to each other.
With the help of the virial theorem we can determine
the total energy of the star through its mass, radius and gravitational
constant. On the other hand, the energy of the star can be calculated as the
sum of the quantummechanical energies of cells of atomic sizes, through the
total number of nucleons in the star, the Planck constant, the proton mass and
the cell size (obtained through the radius of the star and the number of
nucleons). Given that the ratio of the mass (radius) of the star to the mass
(radius) of the proton is the coefficient of similarity in mass (size), and the
ratio of the characteristic speeds of the matter in the star and in the proton
gives the coefficient of similarity in speed, then for the coefficients of
similarity we obtain the following relation: ^{[20]}
.
The left side of the equality contains the similarity
coefficients for the systems with the main sequence star of minimum mass, and
the right side – for the systems with neutron stars, taken with respect to the
hydrogen atom and the proton, respectively. The relations between the
coefficients of similarity show that not all of these coefficients are
independent on each other. If we use the expressions for the fine structure
constant and for the strong gravitational constant , we obtain the following:
.
This expression for the ratio of the gravitational
constants conforms to the ratio of the corresponding coefficients of
similarity, as it follows from the dimension of the gravitational constant.
Thus, the coefficients of similarity between the basic levels of matter can not be arbitrary, they are limited by the ratio of the
gravitational constants at these levels of matter.
For the hydrogen atom we can determine the
dimensionless coefficients associated with the mass, sizes and speeds:
For these coefficients we obtain the relation:
.
In each hydrogen system, regardless of its components
(the hydrogen atom, the planetary system, etc.), the horizontal dimensionless
coefficients are the same, so the above relation between the coefficients does
not change.
Analyzing the similarity of the levels of matter
Fedosin considers the characteristic masses of the carriers that are in the
range from 10^{38} kg to 5∙10^{26} of the Sun’s masses. The
sizes of the carriers vary from 10^{19} m to 372 Gpc.
The total number of matter levels is 18, at the lowest level the preons are
located, and at the highest – the metagalaxies and superclusters of metagalaxies.
Due to the fact that in the nature the matter carriers are not uniformly
distributed, but are concentrated in certain groups, in which the difference in
sizes and masses of the carriers is not so large in comparison with the
difference between the groups, it becomes possible to determine the
coefficients of similarity not only between the basic, but also between the
intermediate levels of matter. It turns out that the masses and the sizes of
objects increase in a
geometric progression
from one level to another, if we start counting from a certain group of objects
belonging to the arbitrarily chosen level of matter. This allows us to estimate
the masses and sizes of the carriers of any other level of matter by means of
the corresponding multiplication by the factors and .
Between such levels matter as elementary particles and
ordinary stars, we find nine intermediate levels of matter. To find the
coefficient of similarity in mass between the adjacent intermediate levels, it
is necessary to extract the tenth root of the coefficient of similarity in mass
between the atoms and the main sequence stars:
.
Table 3 shows the levels of matter from atoms to
stars, obtained by multiplying the electron mass kg by
the degree of the similarity coefficient . In the first multiplication we
obtain kg, that is, the mass of the
chemical element which has the mass number approximately equal to 210. Such element is
lead or bismuth, the most massive of the stable chemical elements. The second
multiplication by gives the mass of the largest stable molecular
complexes, and so on.
Table 3. The
distribution by mass of the objects of the matter levels from atoms to stars 

The level of
matter 
Mass, kg 
Electron — the chemical element 
9.1095∙10^{31} — 3.482∙10^{25} 
Large nuclei — molecular complexes 
3.482∙10^{25} — 1.33∙10^{19} 
Cosmic dust 
1.33∙10^{19} — 5.09∙10^{14} 
Micrometeorites 
5.09∙10^{14} — 1.94∙10^{8} 
Small meteorites 
1.94∙10^{8} — 7.43∙10^{3} 
Meteorites 
7.43∙10^{3} — 2.84∙10^{3} 
Meteorites, comets 
2.84∙10^{3} — 1.08∙10^{9} 
Large meteorites, comets 
1.08∙10^{9} — 4.15∙10^{14} 
Comets, asteroids, moons 
4.15∙10^{14} — 1.58∙10^{20} 
Asteroids, moons, inner planets 
1.58∙10^{20} — 6.06∙10^{25} 
Outer planets — normal stars 
М_{pl} = 6.06∙10^{25} — 2.32∙10^{31} = 11.6 М_{c} 
The mass М_{pl} = 6.06∙10^{25} kg in Table 3
corresponds to the mass of the planet, which is the analogue of the electron. The
mass 11.6 М_{c}
is the mass of the main sequence star of the spectral type B1, which is the
analogue of the nuclide of the type of lead or bismuth. Under normal stars such
main sequence stars are understood, the masses of which do not exceed 11.6 М_{c}
. The proton corresponds to the stars with the minimum mass , where is the mass of the Sun.
Preons correspond by their masses to comets,
asteroids, moons of the planets; partons correspond to large asteroids, moons,
and the inner planets; atoms are similar to planetary systems of stars, and
tiny specks of dust by the number of atoms of which they consist are the
analogues of the galaxies. To estimate the masses of preons and partons we
should take into account that the direct analogy for the atoms and elementary
particles are the systems with neutron stars, not the systems with main
sequence stars. Since the partons are similar to asteroids and inner planets,
the masses of which are less than the masses of neutron stars, then the masses
of the partons must be less than the masses of the nucleons in the same
proportion. Preons are one scale level lower and have less masses than partons.
Hence, the objects of the parton level must have
masses in the range from 9.4∙10^{38} kg to 3.6∙10^{32} kg,
and the level of preons – from 2.5∙10^{43} kg to 9.4∙10^{38}
kg.
If we continue to multiply the masses of the similar
objects at the levels of matter by the coefficient of similarity , we can determine the masses of
objects from stars to metagalaxies according to Table 4.
Table 4. The mass
distribution of objects at the matter levels from the stars to metagalaxies 

The level of
matter 
Mass, М_{c} 
Outer planets — normal stars 
М_{pl} = 3.05∙10^{5} — 11.6 
Massive stars, star clusters, dwarf galaxies 
11.6 — 4.43∙10^{6} 
Dwarf galaxies — normal galaxies 
4.43∙10^{6} — 1.7∙10^{12} 
Massive galaxies — superclusters of galaxies 
1.7∙10^{12} — 6.51∙10^{17} 
Superclusters of galaxies — normal metagalaxies 
6.51∙10^{17} — 2.49∙10^{23} 
The dwarf galaxy with the mass 4.43∙10^{6} М_{c}
is the analogue of the electron, and the normal galaxy with the minimum mass
8.15∙10^{9} М_{c}
corresponds to the proton in the hydrogen atom. Our Galaxy is presumably the
analogue of the chemical element with the mass number , and forms with the Large and Small
Magellanic Clouds, which are the galaxies of small
size, an association similar to the water molecule. At the level of
metagalaxies the normal metagalaxy with the mass М_{mg} = 2.368∙10^{51}
kg or 1.19∙10^{21} М_{c}
corresponds to the proton.
According to the substantial
electron model, the electron charge is so high that the strong gravitation of its matter is not able
to counteract the electrical force of repulsion of the charged matter units.
However, in the atom the mass and the charge of the nucleus are sufficient to
keep the electron in the form of some axisymmetric figure in which the matter
of the electron is rotating around the nucleus. ^{[24]}
Thus, the electron radius as the radius of an independent elementary particle
is not determined. In connection with this, in Table 5 determining of the sizes
of the objects at the intermediate matter levels is done not from the radius of
the electron in the direction of larger sizes, but in the opposite direction.
The starting point is not the radius of the electron but the radius R_{pl}
= 2∙10^{7} m of the planet with the mass М_{pl} = 6.06∙10^{25}
kg, which is the analogue of the electron. The radius R_{pl} is determined from
the dependence of the radius of the planets of the Solar system on the mass. In
the first row of Table 5 the radius 3.85∙10^{9} m is given, which
corresponds to the radius of the star with the mass 11.6 М_{c} . The radius of the
star – the analogue of the proton is assumed to be 0.07 of the Solar radius, or
4.9∙10^{7} m according to recent measurements. ^{[25]}
The exponent of progression for the coefficient of
similarity in size is 12, because in contrast to the similarity in mass between
the level of elementary particles and the level of stars there are two
additional levels associated with the sizes of atoms (this is accompanied by
the fact that in the transition from the sizes of the atomic nuclei to the
sizes of atoms the mass of the objects is almost unchanged). Hence, the
coefficient of similarity in size between the adjacent intermediate levels is
determined as the twelfth root of the coefficient similarity in size between
atoms and planetary systems of main sequence stars:
.
Table 5. The size distribution of objects at the
matter levels from elementary particles to stars 

The level of
matter 
The average
radius, m 
Outer planets — normal stars 
R_{pl} = 2∙10^{7} — 3.85∙10^{9} 
Asteroids, moons, inner planets 
2.55∙10^{5} — 2∙10^{7} 
Comets, asteroids, moons 
3.25∙10^{3} — 2.55∙10^{5} 
Large meteorites, comets 
41.4 — 3.25∙10^{3} 
Meteorites, comets 
0.528 — 41.4 
Meteorites 
6.73∙10^{3} — 0.528 
Small meteorites 
8.58∙10^{5} — 6.73∙10^{3} 
Micrometeorites 
1.09∙10^{6} — 8.58∙10^{5} 
Cosmic dust 
1.39∙10^{8} — 1.09∙10^{6} 
Molecular complexes 
1.78∙10^{10} — 1.39∙10^{8} 
The sizes of ions and atoms 
2.26∙10^{12} — 1.78∙10^{10} 
The transition from the sizes of atoms to the sizes of nuclei 
2.88∙10^{14} — 2.26∙10^{12} 
Particles — atomic nuclei 
3.68∙10^{16} — 2.88∙10^{14} 
For comparison, one of the largest covalent atomic
radii with the value of 2.25∙10^{10} m belongs to the cesium atom, and
the radius of the uranium nucleus is of the order of 0.8∙10^{14} m.
The data in Tables 3 and 5 are connected because the object masses are
proportional to the mass density and the cube of the radius. Comparison of
different models of objects from stars to elementary particles, their densities
and the observed masses and sizes shows that the characteristic sizes in Table
5 differ not more than 2 – 3 times from the observed values. Estimating the
sizes of partons and preons is done similarly to estimating their masses. In
particular, we consider the relations between the sizes of the neutron star and
planets (moons of planets, asteroids) and the similar relations between
nucleons and partons. Hence, the objects of the level of partons must have
radii in the range from 1.1∙10^{14} m to 9∙10^{13} m, and the
objects of the level of preons – from 1.5∙10^{16} m to 1.1∙10^{14}
m.
The sizes of objects from stars to metagalaxies in
Table 6 are determined by multiplying by the degrees of the coefficient of
similarity in size .
Table 6. The size distribution of objects at the
matter levels from stars to metagalaxies 

The level of
matter 
The average
radius 
Outer planets — normal stars 
R_{п} = 2∙10^{7} m — 3.85∙10^{9} m 
Subgiants, giants, supergiants 
3.85∙10^{9} m — 3.02∙10^{11}
m 
Planetary systems of stars 
3.02∙10^{11} m — 2.37∙10^{13}
m 
Binary and multiple stars 
2.37∙10^{13} m — 1.86∙10^{15}
m = 0.06 pc 
Compact OB groups and Tassociations 
0.06 pc — 4.73 pc 
Open and globular clusters, stellar associations and aggregates 
4.73 pc — 371 pc 
Dwarf galaxies — normal galaxies 
371 pc — 29.1 kpc 
Clusters of galaxies 
29.1 kpc — 2.28 Мпк 
Superclusters of galaxies 
2.28 Mpc
— 179 Mpc 
Superclusters of galaxies — normal metagalaxies 
179 Mpc
— 14.05 Gpc 
The optical radii of galaxies, corresponding to the
electron and the proton by mass, are found from the observations of galaxies ^{[26]} and on the average are equal to 350 pc and 2.5 kpc.
If we multiply the radius of the star – the analogue of the proton by the
coefficient of similarity in size , we shall obtain only 370 pc. The
difference from the optical radius of the galaxy 2.5 kpc is connected with the
fact that the normal galaxies with the minimum mass are rather flat spiral
systems and the radius 2.5 kpc is the largest radius of the disc, and the
radius 370 pc is the radius averaged over the volume of the galaxy. The
galaxies with the radius 29.1 kpc in Table 6 at the level of atoms correspond
by mass to nuclides such as lead or bismuth; there are also very large
galaxies, the radius of which can reach 38 kpc.
By multiplying the radius of the star – the analogue
of the proton by the coefficient of similarity in size we obtain the estimate of the radius of the
metagalaxy corresponding to the proton: R_{mg} = 1.8∙10^{8}
pc. For the metagalaxy similar to the heavy nuclei such as lead, the radius
would be about 14 Gpc. The observable Universe at
present has the same radius.
The coefficient of similarity in size is large
enough, since the change of the mass of the objects times
corresponds to it. It is convenient to pass to logarithmic units: . A quarter of this value equals: , which corresponds to the change
of sizes approximately times. There are researches in
which it is found that the distribution of sizes of various organisms in flora
and fauna, from viruses and to the largest organisms, corresponds to the change
of typical sizes that are multiple on the logarithmic scale either of the value
, or of its
integer parts. ^{[27]} For the blocks of the Earth’s
crust also there is a correlation with the value . ^{[28]}
These data conform to the results of Sergey Sukhonos’
researches and confirm the universality of the discrete coefficients of
similarity, which can be applied to the objects of both animate and inanimate
nature.
The discrete coefficient of similarity in speed is
determined as the fifth root of the coefficient of similarity in speed between
the atoms and planetary systems of main sequence stars:
.
The characteristic speeds in Table 7 are obtained by
successive multiplication of the speed of light c = 299792 km/s by the degrees
of the coefficient .
Table 7. The speed distribution of gravitationally bound
objects at the level of planets and stars 

The typical
objects 
The range of
characteristic speeds, km/s 
Exotic objects: quark stars and black holes 
70781 — 299792 
Neutron stars 
16711 — 70781 
The transitional states 
3946 — 16711 
White dwarfs 
931 — 3946 
Main sequence stars, subgiants, red giants 
220 — 931 
Brown dwarfs 
51,9 — 220 
Massive planets 
12.25 — 51.9 
Medium planets 
2.89 — 12.25 
Moons of planets, minor planets 
0.68 — 2.89 
Asteroids, dwarf planets 
0.16 — 0.68 
The characteristic
speed of the particles of the object is associated
with the absolute value of the total energy of the object or its binding energy
in the field of the ordinary (or strong) gravitation: ^{[20]}
where with the uniform matter distribution in the
object, for the objects of the type of nucleons and
neutron stars, is the gravitational constant, and are the mass and the radius of the object.
With the help of relation (1) we can determine the
characteristic speed of each object at the level of stars. In
particular, with the characteristic speed of the dwarf planet
Ceres is about 0.2 km/s, of Mercury – 1.64 km/s, of Mars – 1.94 km/s, of the
Earth – 4.3 km/s, of Uranus – 8.2 km/s, of Jupiter – 23 km/s. In large
asteroids and dwarf planets gravitation can form a rounded shape of these
bodies. The speed km/s is the characteristic speed of matter of
the main sequence star with the minimum mass . In such stars thermonuclear
reactions occur mainly at the stage of formation of the stars, and then they
slowly weaken. These stars at the same time can be considered the hydrogen
white dwarfs, since the main mass of the hydrogen will never turn into helium,
and the internal pressure in the star is maintained by the gas of degenerate
electrons.
In fact, the speeds in Table 7 differentiate objects
by their state of matter and the position in the hierarchy of stars and
planets. The transitional states in the range of the characteristic speeds 3946
— 16711 km/s occur in collisions of stars of the type of white dwarfs and main
sequence stars. The result is either ejection of the excessive matter from
white dwarfs, or the state of the white dwarf is transformed into the state of
the neutron star. Exotic objects can appear for a short time as a result of
collisions of neutron stars with other objects. From the point of view of the model of quark quasiparticles the quarks
are the quasiparticles rather than real particles, so the quark stars, as well
as the black holes are hypothetical objects from the point of view of the theory.
With the help of the coefficients of similarity in
time, mass and size, based on the theory of similarity and dimensions of
physical quantities it becomes possible to predict the physical parameters of
the carriers of matter at any level. In particular, it was shown that the Solar
system is similar by the properties to the atom with the mass number 18, and
the mass of the electron corresponds to the planet with the mass of order of the
mass of Uranus. Discreteness of stellar parameters was also discovered similar
to the division of all known atoms to chemical elements and their isotopes.
Almost all main sequence stars by their mass turned out to be corresponding to
the elements of the periodic table of chemical elements, the inaccuracy is only
10^{–6} %. Besides the abundance of the corresponding atoms and
stars in the nature significantly coincided. For example, the stars with
spectral classes K2, G5, G1, F2, with respective masses about 0.75, 1.07, 1.3
and 1.7 solar masses are very rare. These stars correspond to the chemical
elements N, F, Na, P, which are also significantly deficient compared to
adjacent chemical elements in the chemical composition of the Sun and in the
nebulae. At the same time, the iron peak, observed in the abundance of chemical
elements, is repeated in the rise of the number of stars of spectral classes
B8B9, with the masses about 3.2 solar masses.
Among other similarity properties of atoms and stars
we can note the properties of atoms to gather in molecules and in star pairs
and multiple stars similar to them by masses, the similar in intensity magnetic
moments of atoms and the stars – their analogues, etc. Thus, up to 70 % of
stars similar to the Sun, are part of binary and multiple star systems,
producing stellar gas similar to molecular oxygen. In the center of specks of
dust the chemical elements – metals are dominating, and on the periphery – the
nonmetal elements. Similarly, it turns out that in the central parts of
galaxies the stars have an increased number of metals, and in the halo of
galaxies the stars dominate which are the analogues of nonmetal elements and
also metalpoor. For the minimum mass of stars the value 0.056 solar masses was
predicted, and such stars are really discovered (now referred to as brown
dwarfs or Ldwarfs). These stars (for example the star MOA2007BLG192L) in terms of similarity correspond
to hydrogen.
For dwarf galaxies surrounding normal galaxies (like
electrons in atoms), we can determine the corresponding characteristic mass
equal to 4.4∙10^{6} solar masses, and the radius of the order of 371
pc. Modern estimates of masses and sizes of dwarf galaxies are really close to
these values. ^{[29]} ^{[30]}
It is interesting that the total energy of stars,
consisting of their gravitational and internal thermal energy, can be
calculated very accurately using the Einstein formula, generalized for all
objects. More precisely, the total energy of the star is obtained by
multiplying the stellar mass by the square of the characteristic speed of the
particles inside the star (see the equivalence of mass and energy). This
approach is valid not only for stars, but also for galaxies. ^{[20]}
With the help of the data from Table 2 for the proton
and the data from Tables 4 and 6 we can determine the coefficients of
similarity between the proton and the Metagalaxy – the analogue of the proton:
These coefficients correlate well with the hypothesis
of large numbers, according to which for the ratios of sizes and masses between
the elementary particles and the Metagalaxy the following equation is assumed:
.
This means that Dirac large numbers are the
consequence of the fact that the masses and sizes of objects in the transition
from one matter level to another change in a geometric progression with different coefficients. In particular,
between elementary particles and metagalaxies there are so many intermediate
levels of matter, that as a result between the similarity coefficients the
correlation for large numbers occured. The connection
between the parameters of the Metagalaxy and elementary particles is not
accidental – it is mediated by the hierarchical structure of the Universe, when
any object is similar to other objects at different levels of matter, and
includes the objects of lower levels of the matter.
In addition to the fact that chemical elements can be
set in mutual onetoone correspondence with main sequence star, with almost
coinciding abundance in nature, between nuclides and stars there is also close
correspondence in the magnetic properties. There are not many magnetic nuclei
with large magnetic moments, and the same holds for the magnetic stars. In this
case there is a correlation between the masses of magnetic stars and the masses
of magnetic nuclei, which are related to each other by the coefficient of
similarity in mass . The distribution of magnetic stars
and their connection with magnetic nuclei is described in the article
discreteness of stellar parameters.
If we proceed from the magnetic moment of the electron
and the nuclear magneton, and also from the Dirac constant as the characteristic
value of the angular momentum of microparticles, with the help of the
coefficients of similarity we can calculate the corresponding values for
planets and stellar objects. ^{[20]} The
magnetic moment of the electron and the nuclear magneton are given by:
,
,
where is the elementary charge, and are the masses of the electron and the proton,
is quantum spin of the electron
and the proton, and are the corresponding gyromagnetic ratios
equal to the ratio of the charge to the mass.
For relation between the magnetic moments and the spin of stellar objects similar to electrons and
atomic nuclei, we can write down:
,
,
where and are the corresponding gyromagnetic ratios.
The theory of dimensions allows us to find the
gyromagnetic ratios for the stellar objects through the coefficients of
similarity:
C/kg,
C/kg.
Another
expression for gyromagnetic ratios at the level of stars has the form:
C/kg,
C/kg.
The Picture shows the summary dependence
"magnetic moment – spin" for planets, stars and our Galaxy. The
magnetic moments of the Moon, Mercury, Earth, Jupiter and the Sun are given for
two values of spins: the spin of the nucleus and the total spin. Crosses are
the usual nonmagnetic stars; the rectangle Ар is the
magnetic stars of the spectral class A. The positions are indicated of magnetic
and nonmagnetic white dwarfs, radio and Xray pulsars, the extreme black hole
BH (indicated by a big point) with the mass 1.414 solar masses, and the bulge
and the Galaxy as a whole, taking into account the possible spread of values.
Almost all of the objects are located within or on the border of the stripe, cut
off by the line of the stellar Bohr magneton (upper) and the line of the
stellar nuclear magneton (lower).
The fact that the value of the gravitational constant
does not change much at the level of galaxies, as it follows from the
similarity coefficients, leads to the fact that the magnetic moments of
galaxies correspond to the dependences between the magnetic moment and the
spin, determined for stellar objects. However such gravitationally bound
objects as planets, stars, star clusters and galaxies are not direct analogues
of electrons and atomic nuclei, in contrast to neutron stars similar to
nucleons. If on the plane with the logarithmic coordinates "magnetic
moment – spin" we draw a line between the points for the magnetic moments
of the electron and the nucleon, and the corresponding points for the magnetic
moments of planets and stars, the slope of these lines will be equal to
0.7.This means the dependence of the form , while for the planets and stars
there is a linear dependence . Noncoincidence of the dependence
arises from different mechanisms of generating the magnetic field. If we
consider the objects rotating at limiting angular velocity, which have the
largest magnetic fields, then for small particles of matter, which are held from
decay by the molecular forces of constant magnitude, we obtain the relations and , where is the mass of the object. Here the increase
of the magnetic moment is associated just with the increase of the mass and the
matter quantity. For the stellar objects the attractive force of the matter
depends on the mass and the radius, which gives and . With the increase of the mass the magnetic moment of stellar objects
increases faster than of the separate particles of the matter. Within the
dynamo theory there is a formula: ^{[31]}
,
where is the mass density of the body with the
radius , is the angular velocity of rotation of the
body.
This formula in case of the limiting rotation, on
condition of the equality of gravitational attraction and the centripetal
force, gives .
In the electrokinetic model,
in which the magnetism of cosmic bodies is the consequence of rotation and the
separation of electrical charges within the body, Fedosin arrives at the
similar formula:^{ [48]}
,
where and are the radius and the substance density of
the core of the planet, is the coefficient of
proportionality between the density of the magnetic force and the Coriolis
force, is the vacuum permeability. One of the
consequences of this is that the density of the magnetic energy is proportional to the density of the kinetic energy of rotation of the
conductive and magnetized substance: . ^{[4}^{7}^{]}
At the level of planetary systems the quantization of parameters of cosmic systems
is manifested in the applicability of the Bohr atom model for calculating the
parameters of the orbits of planets. As a result, there are formulas for the
specific orbital angular momenta and the orbital radii of the planets in the
Solar system: ^{[20]}
,
,
where is the orbital angular momentum of the planet
in the orbit with the number ; , and are the mass of the planet, its orbital
velocity and the average radius of the orbit; J∙s is the stellar Dirac constant for ordinary stars;
kg is the mass of the planet
corresponding to the electron by the theory of similarity; is the gravitational constant; is the mass of the Sun; from the correspondence with the empirical
data.
For planetary moons the corresponding quantization of
the specific orbital angular momenta is also observed. ^{[24]} In addition, it is shown that the specific
spin mechanical moments of the proper rotation of planets in the Solar system
are quantized. ^{[32]}
Similarity relations work most accurately between the
corresponding levels of matter, for example, between the levels of elementary
particles and stars with the degenerate state of matter such as white dwarfs
and neutron stars. In collisions of highenergy particles mesons often appear,
which, like the overwhelming majority of elementary particles, are unstable and
decay. The meson of the minimum mass is the pion, which is 6,8 times lighter
than the nucleon and decays into muon and muon neutrino (antineutrino) in the
reaction:
π → μ + ν_{μ}.
In turn, muon decays into electron (positron) and
electron and muon neutrinos in the reaction:
μ → е + ν _{е} + ν_{μ}.
From the point of view of similarity, the pion corresponds
to the neutron star with the mass 0.2 solar masses, and the muon – to the
charged stellar object with the mass 0.16 solar masses. The mass 0.16 of the
Solar mass is exactly equal to the Chandrasekhar limit for white dwarfs of the
hydrogenhelium composition, ^{[33]} at lower masses
the star as the white dwarf is unstable. From the observations one of the least
massive white dwarfs SDSS J0917 +46 has the mass 0.17 solar masses. ^{[34]} The matter of such objects is unstable and
therefore such stars must undergo catastrophic changes of their state in the
periods of time 10^{5} – 10^{7} years. First, low massive
neutron star decays in an explosive way with the formation of the charged and
magnetized object and with emission, which is the analogue of muon neutrino. Then
the decay product of the neutron star undergoes new transformation, with the
ejection of the charged shell, which is the analogue of the electron. In the
described picture hadrons are similar to the neutron stars in unstable, stable
or excited states. The latter refers mainly to the particlesresonances, which
by their short lifetime correspond to the massive, very hot and unstable
neutron stars. In the substantial neutron model it is assumed that the analogues of
neutrons are the neutron stars with the masses about 1.4 solar masses, and
according to the substantial proton model
the analogue of protons are magnetars.
In the world of compact stars electrons also have
their analogue. In the hydrogen atom the most probable location of the electron
in the ground state is the Bohr radius. Multiplying the Bohr radius by the
coefficient of similarity in size Р'
we obtain the value of the order of 10^{9} m. This value is exactly
equal to the distance from the neutron star at which the planets decay near
stars due to the strong gravitational field. This distance is called the Roche
limit. Based on the foregoing, nucleons become similar to neutron stars, while
electrons in the atom correspond to discs discovered near the Xray pulsars,
which are the main candidates for magnetars. ^{[35]}
In this case the sizes of discs coincide with the Roche radius near the neutron
star. Electrons in the form of discs are considered in the substantial electron model, which allows
us to explain the origin of the electron spin.
With the help of similarity relations we can estimate
the radii of elementary particles, their binding energies, the characteristic angular
momentum and the characteristic spin. For hadrons, based on the analogy of
their matter structure with neutron stars, the ratio is used between the radius
and the mass of the hadron: ^{[23]}
where and are the radius and the mass of the proton.
Table 8 shows the masses and the radii of proton, pion
and muon. The radius of muon is found based on the radius of the white dwarf
corresponding to muon.
Table 8. The
characteristics of proton, pion and muon 

Particle 
Massenergy, MeV 
Mass, 
Radius, 
Binding energy , MeV 
Characteristic 
Quantum spin 
Proton p^{+} 
938.272029 
1.6726 
8.7 
938.272 
5.34 
ħ/2 
Pion π^{+} 
139.567 
0.249 
16.4 
11 
0.54 
0 
Muon μ^{+} 
105.658 
0.188 
10900 
0.095 
9.1 
ħ/2 
The masses of the particles in Table 8 are obtained by
dividing the massenergy, converted from MeV to Joules, by the squared speed of
light. The characteristic angular momentum of the particle is given by:
and the characteristic speed of the particle’s matter is calculated by the
formula (1):
here for the objects of the type of nucleons and
neutron stars, is the strong
gravitational constant.
For the proton there is an approximate formula , from which for the characteristic
spin of the proton we obtain: , where is the speed of light and the characteristic
speed of the proton matter, is the Planck constant. If we apply the same approach
for the characteristic spin of muon, we shall obtain the following:
J•s with .
The characteristic spin of the muon exceeds the value
of the quantum spin ħ/2, accepted for fermions and leptons. For the pion with
its radius according to Table 8, the spin is equal to 0.05 ħ, i.e. considerably
less than the minimum spin of the fermion, equal to ħ/2. As a consequence, the
quantum spin of the pion is assumed to be zero, and the pion is considered as
boson.
With the help of relation (2) we can estimate the
characteristic angular momentum to our Galaxy Milky Way. Assuming that the mass
of the galaxy is 1.6•10^{11} solar masses, the radius is 15 kpc, the
characteristic speed of the stars is 220 km/s, for the angular momentum we
obtain the value 3.3•10^{67} J•s. This is close enough to the value
9.7•10^{66} J•s according to the known data. ^{[36]}
As stated above, based on the principle of similarity
at the level of elementary particles the strong
gravitation is introduced into consideration, and the strong gravitational constant is significantly different from the ordinary gravitational constant . The action of strong gravitation
and the gravitational torsion fields of
elementary particles can explain the strong interaction based on the gravitational model of strong interaction.
For the ratio of the gravitational constants the following formula is valid:
,
which contains the coefficients of similarity in size,
speed and mass for normal and neutron stars, respectively, taken with respect
to hydrogen.
This formula should be understood in the following
way, that in the transition from one matter level to another, the effective gravitational
constant changes in the law of gravitation between the objects. As the example,
we can estimate the effective gravitational constant for galaxies. From Table
4, the coefficient of similarity in mass between normal galaxies and the main
sequence stars is . Similarly, from Table 6 the coefficient of
similarity in size equals . The average velocities of the motion of stars in spiral galaxies of
low mass apparently do not exceed the characteristic speed km/s of the motion of matter in the star of
minimum mass. Hence, the coefficient of similarity in speeds is close to unity and for the
effective gravitational constant at the level of galaxies with accuracy up to a
coefficient of the order unity we obtain the same value as at the level of
stars: ^{[20]}
.
This result differs substantially from the rapid decrease
of the gravitational constant at the level of galaxies, obtained by R.
Oldershaw.
In general, in the transition to a higher scale level
of matter the decrease of the effective gravitational constant is predicted,
based on the Le Sage's theory of gravitation and the
nesting of matter levels into each other.
It is known that with the help of three independent
physical quantities we can calculate the characteristic parameters of the
mechanical system. For example, the Planck units of mass, length, time, energy,
momentum, etc. are based on Dirac constant , the speed of light and the gravitational constant :
A more complete set of Planck units in the
International System of Units includes the Boltzmann constant and the factor
, where is the electric
constant. Planck
units are used in quantum physics, where is the characteristic angular momentum, but
since the ordinary gravitation constant in the microworld must be replaced by the
strong gravitational constant, the Planck units do not uniquely characterize
any level of matter and only formally refer to the natural units of physical
quantities. Only the Planck charge, which does not contain the gravitational
constant, is close to the electrical elementary charge , exceeding it approximately 11.7
times:
C,
where is the fine structure constant.
At the same time, if we use at the level of the main
sequence stars the stellar Planck constant J∙s, the
stellar speed km/s, the gravitational constant and the
coefficients of proportionality of the order of unity, related to the geometry
of the ball shape and the distribution of matter, then with their help we can
obtain the values which are sufficiently close to the parameters of the star of
minimum mass: ^{[20]}
where J/K is the stellar
Boltzmann constant.
The time here characterizes the time required to cross
the radius of the star at the speed , and this stellar speed is the
characteristic speed of the matter inside the star. Substituting the expression
for the average density in the formula for the characteristic time we find the
approximate relation for the time of the fall of the matter in the
gravitational field: . The product of the absolute value
of the total energy and the characteristic time gives the relation similar to
the Heisenberg uncertainty relation: . The maximum luminosity of the star
is close to the luminosity of the Galaxy, as well as to the luminosity of the
supernova.
As the independent quantities for the natural units,
which characterize the objects of different matter levels, we can also take the
characteristic mass, speed, and angular momentum. For example, assuming as
primary the mass , the speed , and the angular momentum , we can express the gravitational
constant in the form: , and
then substitute this expression into the formulas above. This allows us to
estimate the parameters of the main sequence star through its mass,
characteristic speed of the matter and the characteristic spin of this star.
Passing from stars to atoms, and using as the basic
values the Planck constant , the speed of light , the Boltzmann constant , the multiplier and the strong gravitational constant in the form
m^{3}•s^{–2}•kg^{–1},
where is the mass of the proton, is the electron mass, we can estimate in the
first approximation the parameters of the proton as the main subject at the
level of elementary particles:
With the help of natural units similarly to the main
sequence stars we can obtain the parameters of galaxies and even metagalaxies. ^{[20]} For example, taking from Table 4 the mass
of the metagalaxy 2.49∙10^{23} М_{c} , and from Table 6
its radius 14.05 Gpc, we can estimate the average
mass density kg/m^{3},
and the characteristic time of the matter relaxation in the field of the
regular forces and the time of free fall under the influence of gravitational
forces:
years.
This time almost four times exceeds the time 13.7
billion years of existence of the Universe according to the Big Bang model. In
addition, such arguments in favor of the Big Bang, as the cosmic microwave
background radiation and Hubble's law can be understood without using the idea
of the Big Bang. ^{[37]} All the other arguments in
favor of the Big Bang can have other explanations, which subjects to
wellgrounded and manysided criticism the concept of the explosion of the
Universe.
From the point of view of similarity, the Milky Way
galaxy resembles a gas cluster, rotating about its axis; the role of atoms is
played by stars. Since the concentration of stars increases rapidly in the
direction towards the center of the Galaxy, the average density , understood as the average mass of
stars per unit volume, also increases. The dependence of the density on the
current radius in the International System of Units is given by: ^{[20]}
,
where the galactic radius is substituted in meters.
According to this dependence we can estimate that the
air under normal conditions has the same concentration of molecules, which is
equal to the concentration of stars near the galactic radius 6.4∙10^{16}
m or 2.1 pc. Almost the entire volume of the Galaxy is similar to the collisionless and very rarified gas. In the center, with
the radius 0.047 pc the concentration of stars reaches the concentration of
such light and solid substance as coke. The average gas pressure from the stars
in the Galaxy is given by:
,
where is the average velocity of stars.
If we take into account the data on the velocities of
stars depending on the galactic radius in the range from 200 pc to 10 kpc (the
average velocity is about 235 km/s), for the pressure the approximate formula
in SI units is:
.
The linear dependence of the pressure on the mass
density means that the state of the stellar gas is isothermal. Despite the formation
of stars and the compression of the Galaxy, its temperature changes little, as
all the excess energy is carried away by electromagnetic emission. The
temperature of the Galaxy can be estimated in different
ways:
On the average the temperature of the stellar gas in
the Galaxy is about K. Another way to determine the
temperature of the Galaxy is associated with the generalized gas law for the
stellar gas:
,
where and are the volume and the mass of the Galaxy, is stellar gas constant, is the mass of one stellar mole of the substance,
consisting of stars.
For the ordinary gas constant there is a relation: , where mole^{–1} is the Avogadro
number. Since in the stellar mole the number of stars is also assumed, so the stellar gas
constant equals:
,
where J/(K∙stellar mole) is the stellar gas constant for the main
sequence stars of minimum mass.
The mass of one stellar mole of the substance,
consisting of stars, is equal to:
,
where kg/(stellar mole) is the mass of one stellar
mole of the substance from the main sequence stars of minimum mass.
The typical stars in our Galaxy are the stars with the
mass equal to half of the mass of the Sun, and with the mass number . The left part of the generalized
gas law for the stellar gas can be expressed through the energy of the Galaxy
in the following form:
.
After substituting the quantities into the right side
of the generalized gas law for the stellar gas we obtain:
.
The kinetic temperature of the stellar gas of the
Galaxy is found from the comparison of left and right sides of the generalized
gas law with the average rotation velocity of stars in the Galaxy 235 km/s:
,
K.
Based on the ratio between the energy of the Galaxy,
the energy of stars and their velocity, the principle of locality of the
stellar velocity is formulated: "The average velocity of the stars
relative to the system in which they were formed, does not exceed the stellar
speed , where and are the mass and the charge numbers,
corresponding to the main sequence stars" .
The similarity of matter levels is evident in the
coincidence of forms inherent in the objects and the phenomena at different
scale levels. Depending on the characteristics of the accepted model of
similarity, different researchers explain in their own way the occurrence and
recurrence of the same forms.
Sergey Sukhonos in his works
gives the examples of fractality when the shape of even small parts of the
object to a large extent coincides with the shape of the object itself. He also
lists the manifestations observed in the space of the dual mutually
complementary structures: spiral (flat) and elliptical (round) galaxies; subdwarfs as the primary stars of the Galaxy with a deficit
of heavy elements, and ordinary main sequence stars; large outer and small
inner planets of the Solar system; the monocentric and polycentric structures
at different levels of matter, emerging in the processes of synthesis and
division. Located on the scale axis of sizes, the shapes of objects are
repeated periodically, with the ratio of the sizes of the order of 10^{20}.
This allows us to simulate the dominant shapes with the periodic function in
the form of a wave. The reason of the periodicity is assumed the existence of
the fourth spatial dimension (see the scale
dimension). The latter can be interpreted as the fact that the objects can
move not only in the three ordinary directions in space (as well as move in
time), but also by changing their sizes and mass can move from one matter level
to another. At the same time the situation will periodically arise, when due to
the environmental conditions the initial shape of the object will be retained
due to the minimum of the factors changing the shape.
Robert Oldershaw draws attention to the distribution of matter in space, where
the main mass of the matter consists of hydrogen and light elements. The same
is observed for the stars – according to the initial function of masses, the
most common stars are dwarf stars. Among the galaxies the small galaxies are
also dominating. Another observation is associated with the coincidence of the
geometrical forms of the functions of the electron density in the atom for
different energy levels with the corresponding orbital angular momenta of the
electron and their projections on the preferred direction on the one hand, and
the shapes of stellar objects on the other hand. ^{[39]}
The examples are the symmetrical conical jets and the equatorial ejections from
the star Eta Carina, the ring planetary nebula Shapley 1, the spherical
planetary nebula Abell 39 and other similar objects.
Oldershaw considers the planetary nebulae to be the analogues of fully ionized
atoms.
Neutron stars, such as GRB, producing short and
powerful gammaray bursts in the energy range 10^{43} – 10^{44}
J, Oldershaw compares with the gammaradioactive nuclei. The energies of
gammaray quanta from the nuclei lie in the range from 10 keV
to 7 MeV. Applying the multiplication by the coefficient of the similarity in energy,
which coincides with the coefficient of similarity in mass X = Λ^{D} =
1.7∙10^{56}, he obtains the energy range from 2.72∙10^{41} J to
1.87∙10^{44} J, where the gammaray bursters
GRB also fall. ^{[40]} For the variable stars, such as
RR Lyrae, Oldershaw finds correspondence between the
oscillation period of their brightness and the radius of stars which is similar
by the form to the third Kepler law for the planets of the Solar system and the relation for electrons in Rydberg
states. By recalculation of the coefficient , with the help of the coefficients
of similarity, he makes these stars similar to the excited states of the helium
atom He(4), in which electron transitions occur between the levels 7 ≤ n ≤ 10
and l ≤ 1. ^{[41]} Similarly, the variable stars such
as Delta Scuti (δ Scuti)
are considered to be the analogues of the excited atoms of carbon, oxygen and
nitrogen in the states with 3 ≤ n ≤ 6 and 0 ≤ l ≤ n1, and the stars such as ZZ
Cetis – the analogues of the excited states of ions
from helium to boron.
In Table 9 Oldershaw compares the axial rotation
periods and the natural oscillation periods of typical objects at the levels of
atoms, stars and galaxies.
Table 9. The
characteristic rotation periods and natural oscillation periods of the
objects at the levels of atoms, stars and galaxies 

Object 
Rotation period 
Natural
oscillation period 
Atomic nucleus 
5∙10^{–20} s 
1∙10^{–21} s 
Neutron star 
3∙10^{–2} s 
5∙10^{–4} s 
Active galaxy 
1∙10^{16} с (3∙10^{8}
years) 
2.5∙10^{14} с (8∙10^{6}
years) 
The characteristic rotation periods of active galaxies
are about 10^{8} years, and the oscillations are determined by the
periods of recurrence of significant ejections of matter from their nuclei,
equal about 10^{7} years. ^{[42]} The period
of natural oscillations for neutron stars is associated with the periods of
pulsations of the waves propagating in the stellar matter after collision with
other bodies. These times for the various objects are related by the
coefficient of similarity in time equal to Λ = 5.2∙10^{17}.
Sergey Fedosin describes at all the levels of matter,
where gravitational forces dominate, the hydrogen systems, consisting of the
main object and the moon (satellite), with the same difference in their masses
as between the proton and the electron. Hydrogen systems are as numerous and
widely spread in the Universe, as the hydrogen atoms. The values of the
similarity coefficients according to Fedosin derived from the similarity of
hydrogen systems are different from the values of the coefficients according to
Oldershaw. In particular, the coefficient of similarity in energy for the main
sequence stars equals the product of the coefficient of similarity in mass and
the square of the coefficient of similarity in speed: , and for the compact objects, such
as neutron stars, the coefficient of similarity in energy is equal to .
The gammaray quanta, emitted by the atomic nuclei
under radioactivity, have ordinary energies W from 10 keV to 5
MeV, with the period of the electromagnetic wave in the range:
s.
Multiplying the energies and the oscillation periods
of the gammaray quanta by the coefficient of similarity in energy and the coefficient of similarity in time , respectively, we can find the energies and
the periods at the level of stars: the energies – from 5.7∙10^{34} J to
2.8∙10^{37} J, the periods – from 352 days to 17 hours.
These energies and periods conform to the values
characteristic of longperiod variable stars such as Mira (o Ceti), semiregular variables such as SR, variables such as
RV Taurus, classical Cepheids such as δ Cepheid, δ Scutids and W Virginids,
shortperiod Cepheids such as RR Lyrae.
The energy of expansion of planetary nebulae correspond by the energy to the
alpha decay, and the nova outbursts – to the beta decay of atomic nuclei. ^{[20]}
If we multiply the energies and the oscillation
periods of gammaray quanta from atomic nuclei by the similarity coefficients and , we shall obtain the corresponding energies
and periods for the objects of the type of neutron stars: ^{[24]} the energies – from 1.4∙10^{41} J to
6.9∙10^{43} J, the periods – from 25 s to 0.05 s.
These energies and periods of outbursts are quite close
to the values characteristic of the gammaray bursters.
The energy of gammaray burst from the magnetar SGR 180620, recorded on
December 27, 2004, is estimated by the value 4∙10^{39} J. ^{[43]} Following the outburst the radio emission was
observed from the expanding matter at the velocity about 0.2 of the speed of
light. In the gammaray burster GRB 080319B the total
energy of the outburst in all the emission ranges was equal up to 10^{40}
J. ^{[44]} Although the nature of atomic nuclei and
stars differs significantly, the given examples with the energies of periodic
processes show another aspect of similarity of these matter levels.
The active galactic nuclei and the processes occurring
in them are considered by Fedosin as the consequence of the large number of neutron
stars in the centers of galaxies. For the nucleus of the quasar 3C 273 it is
assumed that the volume with the radius about 10^{13} m contains the
mass up to 10^{9} solar masses, producing the emission with the
luminosity about 2∙10^{40} W. ^{[45]} If we
divide this luminosity by the number of stars, we shall obtain the value 2∙10^{31}
W, which is close to the critical luminosity of neutron stars with the
accretion of matter to their surface. In this case, the phenomenon of quasars
and active galactic nuclei can be explained by the accumulation of a large
number of neutron stars. These stars have strong magnetic fields and can have
magnetic moments, aligned in one direction, creating the regular overall
magnetic field. Due to this field the powerful jets of ionized matter are
possible, which are often observed near active nuclei. The luminosity of 3C 273
can vary significantly during the time of one day or more. The ratio of the
size of the active nucleus 10^{13} m to the time interval of one day
gives the velocity 10^{8} m/s. This velocity can be interpreted as the
velocity of the outburst propagation in the nucleus which occurs as the result
of the interaction of large amounts of relativistic plasma with neutron stars.
The plasma can fall on the active nucleus at high velocities under the
influence of gravitational forces. On the other hand, if neutron stars in the
active nucleus are retained by the proper gravitation and centripetal forces,
they must rotate at the velocities almost up to 10^{8} m/s.
The example of similarity is the use of the Heisenberg
uncertainty principle not only at the level of elementary particles, but also
at the level of stars and even galaxies. The uncertainty relation for the
change of the process energy and the time of its change has the form:
,
where is the characteristic angular momentum of the
object.
In order to conform to the quantities accepted in
quantum mechanics, for the spin angular momentum I the relation is assumed, and for the orbital angular
momenta L the relation is used. In the Galaxy the total energy of
stars in the gravitational field of each other, taking into account the orbital
galactic rotation, are approximately equal to the total energies of stars in
their proper gravitational field, without taking into account the fields of
other stars. Considering these energies and the time of the formation of stars (the
KelvinHelmholtz time ) from separate gas clouds leads to
the fact that for a typical star the following relation holds: ^{[20]}
J/s,
where is the stellar orbital angular momentum.
In addition, the lifetime of the star of the main
sequence on the average exceeds 122 times the time , which can be explained by the time
of the stellar core growth due to the thermonuclear reactions in which the
massenergy is released with the value up to 1/130 of the rest energy of the
matter. The relation for also reflects the change of the energy in the
process of cooling of neutron stars. If instead of we substitute the characteristic spin angular
momentum of the star, in the supernova explosion of which a neutron star is
formed, then for this angular momentum the uncertainty relation for the total
energy of the neutron star (about 2∙10^{46} J) and the time of release
of this energy (several seconds) will be valid.
Transition from one matter level to another can be
made directly in the equations describing the interaction and the motion of the
carriers or the state of matter. It turns out that the simultaneous
substitution in these equations of masses, sizes and velocities of the carriers
of one level with the masses, sizes and velocities of the carriers of another
level of matter, leaves the equation invariant with respect to this
substitution. Thus new combined symmetry is revealed, which follows from the
theory of similarity and is called SPФ
symmetry. The SPФ transformations, as well as the transformations of CPT
symmetry, leave the laws of bodies’ motion unchanged.
A detailed philosophical analysis of the Theory of Infinite Hierarchical Nesting of Matter and
the similarity of matter levels was carried out in 2003. ^{[1]}
At each matter level we can distinguish the characteristic main carriers and
the boundary points of measure. Transitions from one matter level to another
are carried by the law of transition from quantity to quality, when the number
of carriers in the object exceeds the permissible limits of measure, typical
for this object. At different spatial levels of matter the similar fractal
structures, carriers of matter and field quanta are found. These objects as the
elements are included in the hierarchical structure of the Universe, repeating
in similar natural phenomena, ensuring the unity and integrity of the Universe,
revealing the symmetry of similarity.
The laws of similarity and hierarchy of the matter
levels are valid for living systems. It is proved that the masses and the sizes
of all known living organisms correlate with the masses and the sizes of the
carriers of the corresponding levels of matter, repeating them. ^{[47]} Thus the complementarity of the animate and
inanimate is manifested, the conclusion is made of the eternity of life as part
of the eternity of the Universe, the question of the origin of life is solved.
In addition, the infinite nesting of the living is
discovered – inside the autonomous living organisms at each level there must be
living structures of smaller sizes and of lower scale levels. They are the true
builders and creators of large organisms, controlling their reactions and vital
functions as huge complex systems. The presence of nesting of different types
of the living is illustrated in the typical example – in the human body there
are so many bacteria that the total mass can reach two kilograms. [1] The cells in multicellular
organisms and the bacteria are approximately equal in size, but the bacteria
can exist in the environment autonomously for a long time. Viruses and smallest
prions can cause various diseases, when their programs of development are
contrary to the vital functions of the multicellular organism. Prions contain a
certain number of atoms, but the life at a deeper level exists not on the atoms
and molecules, but on smaller physical entities. It is assumed that these
carriers of life, which are not yet directly recorded by the modern
observational facilities, control all the living beings exceeding them in size
and set the programs of their existence.
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