# Similarity of matter levels

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.  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.

## Large numbers

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.  According to his hypothesis, the following relations should hold:

$~{\frac {T}{t}}={\frac {R}{r}}=({\frac {M}{m}})^{1/2}=\Lambda \approx 10^{38}-10^{41},$

where $~T,R,M$  specify the characteristic time of the process, the size and the mass of the Metagalaxy, $~t,r,m$  specify the same parameters for elementary particles.

The hypothesis of large numbers was also considered by Weyl in 1919,  Eddington in 1931,   Jordan in 1947,  Klein and others.

Weyl considered a hypothetical object with the mass $~M_{H}$  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 $~r_{0}=2.8\cdot 10^{-15}$  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 $~R_{H}$ :

$~M_{H}c^{2}={\frac {GM_{e}^{2}}{r_{0}}}={\frac {e^{2}}{4\pi \varepsilon _{0}R_{H}}}$ ,

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:

$~M_{e}c^{2}={\frac {e^{2}}{4\pi \varepsilon _{0}r_{0}}}$ ,

where $~c$  is the speed of light, $~G$  is the gravitational constant, $~M_{e}$  is the mass of the electron, $~e$  is the elementary charge as the proton charge, $~\varepsilon _{0}$  is the electric constant.

From here it follows that, $~{\frac {R_{H}}{r_{0}}}={\frac {e^{2}}{4\pi \varepsilon _{0}GM_{e}^{2}}}=4\cdot 10^{42}$ , and $~R_{H}=1.2\cdot 10^{28}$  m $~=3.8\cdot 10^{11}$  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 $~M_{H}c^{2}$  can be interpreted as the equality between the gravitational energy of two electrons at the distance $~r_{0}$  from each other, and their electrical energy at the distance $~R_{H}$ . In this case, the large value $~R_{H}$  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:

$~{\frac {F_{e}}{F_{g}}}={\frac {e^{2}}{4\pi \varepsilon _{0}GM_{p}M_{e}}}=2.27\cdot 10^{39}$ ,

where $~M_{p}$  is the proton mass.

At the same time the strong gravitational constant, by its definition, equals:

$\Gamma ={\frac {e^{2}}{4\pi \varepsilon _{0}M_{p}M_{e}}}=1{.}514\cdot 10^{29}$  m3•s–2•kg–1.

Therefore $~{\frac {F_{e}}{F_{g}}}={\frac {\Gamma }{G}}={\frac {R_{H}M_{e}}{r_{0}M_{p}}}$ , 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 $~{\frac {R_{H}}{r_{0}}}$ .

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 $~R_{H}$  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.   However, such combination of quantum mechanics and general theory of relativity is not quite convincing, and therefore the search for other explanations continues.

## Quantum properties of stellar 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 $~n$ . 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.   

The similar results are obtained under the assumption that the planetary orbits are quantized proportionally to the square of the quantum number.   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 $~n=1$  and $~n=2$  for the inner planets, and with $~n=1$  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.

## The models of similarity

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 Dirac-Eddington 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  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 1022. 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.

### Sukhonos and Yun Pyo Jung

"The wave of stability." On the scale axis of the Universe all the main objects and their "cores" are located periodically. Below the periodicity is given of location on this axis of the scale "zones of influence" of the four basic forces of nature.

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 1020 times.  This is well illustrated by the following examples:

1. Normal stars have an average size of the order of 1012 cm and are composed of atoms with the size 10-8 cm.
2. White dwarfs have the average size 1010 cm, are composed of highly compressed atoms with the size 10-10 cm.
3. Neutron stars are highly compressed by gravitation – up to 107 cm, are composed of the nucleons with the size 10-13 cm.

In all cases, the scale "distance" between the system and its components is the same – 1020. Built by Sukhonos the scale similarity in size conforms to the Dirac-Eddington hypothesis of large numbers, since

$~\Lambda \approx 10^{40}=(10^{20})^{2}.$

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.  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 1030. 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∙1015 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∙106 of solar masses with the radius of 45 a.u. or 7∙1012 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 1030. 

### Oldershaw

Robert Oldershaw went further and determined the coincident with each other coefficients of similarity in size and time, equal to Λ = 5.2∙1017, and the coefficient of similarity in mass X = ΛD = 1.7∙1056, 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. 

The hydrogen system at the level of stars, according to Oldershaw, consists of the main sequence star with the mass $~0.145M_{c}$ , 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 Λ2, 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:

$~R={\frac {2G_{N}M}{c^{2}}},$

where $~G_{N}$  is the constant of gravitation acting on the given level of matter, $~N=-1$  for the atomic level, $~N=0$  for the level of the stars, $~N=+1$  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 m3/(kg∙s2). Since the coefficients of similarity in size and time are considered equal, we obtain:

$~{\frac {G_{0}}{G_{-1}}}={\frac {G_{+1}}{G_{0}}}={\frac {\Lambda }{X}}=3\cdot 10^{-39},$

where $~G_{0}$  is the ordinary gravitational constant.

Assuming that at the level of atoms $~G_{-1}=2{.}18\cdot 10^{28}$  m3•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 Λ2, 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 105, 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 $~G_{+1}=2\cdot 10^{-49}$  m3•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∙1082 kg and 5∙1085 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 108 – 109 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∙1017 times the size of galaxies. With the help of telescopes and different techniques we can see the most distant quasars at the distance only 105 – 106 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 Kerr-Newman 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 $\hbar$  and the speed of light $~c$ , included in Planck units, but instead of the ordinary gravitational constant he uses the strong gravitational constant $~G_{-1}$ . This allowed him to determine the "modernized" Planck values:

• Mass $~={\sqrt {\frac {\hbar c}{G_{-1}}}}=1{.}2\cdot 10^{-27}$  kg.
• Length $~={\sqrt {\frac {G_{-1}\hbar }{c^{3}}}}=2{.}93\cdot 10^{-16}$  m.
• Time $~={\sqrt {\frac {G_{-1}\hbar }{c^{5}}}}=9{.}81\cdot 10^{-25}$  s.

The obtained values are close enough to the parameters of the proton. 

### Fedosin

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.  The basic levels in this range of matter levels include the level of elementary particles and the level of stars. The most stable and long-lived 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 1057 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 $~D_{\Phi }$  and $~D_{P}$ , 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 coefficients of similarity

##### Between atoms and main sequence 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  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 $~M_{ps}=0.056M_{c}$ , where $~M_{c}$  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 $\Phi =6.654\cdot 10^{55}$ . If we multiply the mass of the electron by $~\Phi$ , we obtain the mass of the planet Мpl = 6.06∙1025 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 Мpl corresponds to the radius Rpl = 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. 

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 Mps = 1.11∙1029 RF =2.88∙1012 Vpl = 1.6∙103
The hydrogen atom Mp = 1.67∙10-27 RB =5.3∙10-11 Ve = 2.19∙106
Similarity coefficients Ф = 6.654∙1055 Р0 = 5.437∙1022 S0 = 7.34∙10-4

If we consider a hydrogen-like atom, the speed of the orbital rotation of electron in it is proportional to the nuclear charge $~z$ . 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 $~A$ . It follows that the coefficient of similarity in speed is given by: $S=S_{0}{\frac {A}{z}}$ . For the same reason for the coefficient of similarity in size we obtain: $P=P_{0}{\frac {z}{A}}$ . In order to find the coefficient of similarity in speed $~S_{0}$  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:

$~E_{n}=-M_{n}c^{2},$
$~E_{s}=-M_{s}C_{x}^{2}=-M_{s}C_{s}^{2}({\frac {A}{z}})^{2},$

where the characteristic speed $~C_{x}=C_{s}{\frac {A}{z}}$  of the matter in the star depends on the mass number $~A$  and the charge number $~z$  of the star. The total energy of the star can be calculated by the formula:

$~E_{s}=-{\frac {\delta GM_{s}^{2}}{2R_{s}}},$

here $~G$  is the gravitational constant, $~M_{s}$  and $~R_{s}$  are the mass and the radius of the star, $~\delta$  is the coefficient depending on the distribution of the matter, in case of uniform mass density $~\delta =0.6$ .

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 $~C_{x}$  for them. Since the masses of stars are associated with the corresponding nuclides which have the mass and charge numbers, then from the relation $~C_{x}=C_{s}{\frac {A}{z}}$  we can determine the value of the characteristic stellar speed: $~C_{s}=220$  km/s.  The ratio of the speeds in the form $~S_{0}={\frac {C_{s}}{c}}$  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 $~P_{0}$  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 $~P_{0}$  is close enough to the ratio between the semi-axes 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:

$\Pi _{0}={\frac {P_{0}}{S_{0}}}=7.41\cdot 10^{25}$ .

The analogue of stellar Dirac constant for ordinary stars:

$\hbar _{s}=\hbar \Phi P_{0}S_{0}=2.8\cdot 10^{41}$  J∙s,

where $\hbar$  is the Dirac constant.

##### Between atoms and neutron stars

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.  

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∙1030 Rs = 1.2∙104 C' s = 6.8∙107
Proton Mp = 1.67∙10-27 Rp = 8.7∙10-16 c = 2.99∙108
Similarity coefficients Ф' = 1.62∙1057 Р' = 1.4∙1019 S' = 2.3∙10-1

If we multiply the mass of the electron by $~\Phi '$ , we obtain the mass of the object-the analogue of the electron М d = 1.5∙1027 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:

$~\Pi '={\frac {P'}{S'}}=6.1\cdot 10^{19}$ .

The value of the Dirac stellar constant for the compact degenerate stars is:

$~\hbar '_{s}=\hbar \Phi 'P'S'=5.5\cdot 10^{41}$  J∙s.

Multiplying the Boltzmann constant by the coefficient of similarity in energy, we can find the stellar Boltzmann constant: $K'_{s}=k\Phi 'S'^{2}=1.18\cdot 10^{33}$  J/K.

#### The gravitational fine structure constant

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 $~c$ :

$\alpha ={\frac {V_{e}}{c}}={\frac {e^{2}}{2\varepsilon _{0}hc}}={\frac {1}{137.036}}$ ,

where $~e$  is the elementary charge, $~\varepsilon _{0}$  is the electric constant, $~h$  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 $~\Gamma$ , the proton mass $~M_{p}$  and the electron mass $~M_{e}$ :

$\alpha =\alpha _{pp}{\frac {M_{e}}{M_{p}}}={\frac {\Gamma M_{p}M_{e}}{\hbar c}},$

where $~\alpha _{pp}={\frac {\Gamma M_{p}^{2}}{\hbar c}}=13.4$  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 $~V_{pl}$ :

$~\alpha _{s}={\frac {V_{pl}}{C_{s}}}={\frac {GM_{ps}M_{pl}}{\hbar _{s}C_{s}}}={\frac {1}{137.036}}$ .

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:

$\alpha _{s}={\frac {V_{d}}{C'_{s}}}={\frac {GM_{s}M_{d}}{\hbar '_{s}C'_{s}}}={\frac {1}{137.036}}$ .

Due to the similarity relations the constants $~\alpha$  and $~\alpha _{s}$  are equal to each other.

#### The relations between the coefficients of similarity

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 quantum-mechanical 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: 

${\frac {P_{0}S_{0}^{2}}{\Phi }}={\frac {2\pi GM_{p}M_{e}}{\alpha hc}}={\frac {P'S'^{2}}{\Phi '}}$ .

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 $\Gamma ={\frac {e^{2}}{4\pi \varepsilon _{0}M_{p}M_{e}}}$ , we obtain the following:

${\frac {P_{0}S_{0}^{2}}{\Phi }}={\frac {G}{\Gamma }}={\frac {P'S'^{2}}{\Phi '}}$ .

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.

#### The horizontal dimensionless coefficients

For the hydrogen atom we can determine the dimensionless coefficients associated with the mass, sizes and speeds:

1. The ratio of the proton mass to the electron mass: $\beta ={\frac {M_{p}}{M_{e}}}=1836.15$ .
2. The ratio of the Bohr radius to the radius of the proton: $\delta ={\frac {r_{B}}{R_{p}}}={\frac {h^{2}\varepsilon _{0}}{\pi e^{2}M_{e}R_{p}}}=6.08\cdot 10^{4}\approx {\frac {2M_{p}ch\varepsilon _{0}}{\pi e^{2}M_{e}}}$ , where it is supposed that equality $~h\approx 2M_{p}cR_{p}$  is valid.
3. The ratio of the electron’s speed in the first Bohr orbit to the speed of light (the fine structure constant): $\alpha ={\frac {V_{e}}{c}}={\frac {e^{2}}{2\varepsilon _{0}hc}}={\frac {1}{137.036}}=7.2973525376\cdot 10^{-3}$ .

For these coefficients we obtain the relation:

$~\beta =\pi \alpha \delta$ .

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.

#### Discreteness of the similarity coefficients

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∙1026 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 $~D_{\Phi }$  and $~D_{P}$ .

##### In mass

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:

$D_{\Phi }=\Phi ^{1/10}=3.8222\cdot 10^{5}$ .

Table 3 shows the levels of matter from atoms to stars, obtained by multiplying the electron mass $M_{e}=9.1095\cdot 10^{-31}$  kg by the degree of the similarity coefficient $~D_{\Phi }$ . In the first multiplication we obtain $3.482\cdot 10^{-25}$  kg, that is, the mass of the chemical element which has the mass number $~A$  approximately equal to 210. Such element is lead or bismuth, the most massive of the stable chemical elements. The second multiplication by $~D_{\Phi }$  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 $A=210$  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∙103
Meteorites, comets 2.84∙103 — 1.08∙109
Large meteorites, comets 1.08∙109 — 4.15∙1014
Comets, asteroids, moons 4.15∙1014 — 1.58∙1020
Asteroids, moons, inner planets 1.58∙1020 — 6.06∙1025
Outer planets — normal stars Мpl = 6.06∙1025 — 2.32∙1031 = 11.6 Мc

The mass Мpl = 6.06∙1025 kg in Table 3 corresponds to the mass of the planet, which is the analogue of the electron. The mass 11.6 Мc 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 Мc . The proton corresponds to the stars with the minimum mass $~M_{ps}=0.056M_{c}$ , where $~M_{c}$  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 $~D_{\Phi }$ , 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, Мc
Outer planets — normal stars Мpl = 3.05∙10-5 — 11.6
Massive stars, star clusters, dwarf galaxies 11.6 — 4.43∙106
Dwarf galaxies — normal galaxies 4.43∙106 — 1.7∙1012
Massive galaxies — superclusters of galaxies 1.7∙1012 — 6.51∙1017
Superclusters of galaxies — normal metagalaxies 6.51∙1017 — 2.49∙1023

The dwarf galaxy with the mass 4.43∙106 Мc is the analogue of the electron, and the normal galaxy with the minimum mass 8.15∙109 Мc corresponds to the proton in the hydrogen atom. Our Galaxy is presumably the analogue of the chemical element with the mass number $~A=18-20$ , 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 Мmg = 2.368∙1051 kg or 1.19∙1021 Мc corresponds to the proton.

##### In size

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:

$D_{P}=P_{0}^{1/12}=78.4538$ .
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 Rpl = 2∙107 — 3.85∙109
Asteroids, moons, inner planets 2.55∙105 — 2∙107
Comets, asteroids, moons 3.25∙103 — 2.55∙105
Large meteorites, comets 41.4 — 3.25∙103
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 $~D_{P}$ .

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 Rп = 2∙107 m — 3.85∙109 m
Subgiants, giants, supergiants 3.85∙109 m — 3.02∙1011 m
Planetary systems of stars 3.02∙1011 m — 2.37∙1013 m
Binary and multiple stars 2.37∙1013 m — 1.86∙1015 m = 0.06 pc
Compact O-B groups and T-associations 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  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 $D_{P}^{6}$ , 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 $D_{P}^{9}$  we obtain the estimate of the radius of the metagalaxy corresponding to the proton: Rmg = 1.8∙108 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 $D_{P}=P_{0}^{1/12}=78.4538$  is large enough, since the change of the mass of the objects $D_{\Phi }=\Phi ^{1/10}=3.8222\cdot 10^{5}$  times corresponds to it. It is convenient to pass to logarithmic units: $lg(D_{P})=lg78.4538=1.895$ . A quarter of this value equals: $~0.25lg(D_{P})=K=0.474$ , which corresponds to the change of sizes approximately $~10^{K}=10^{0.474}\approx 3$  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 $~K$ , or of its integer parts.  For the blocks of the Earth’s crust also there is a correlation with the value $~K$ .  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.

##### In speed

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:

$D_{S}=S_{0}^{1/5}=0.2361$ .

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 $~D_{S}$ .

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 $~C_{x}$  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: 

$~E=MC_{x}^{2}={\frac {\delta GM^{2}}{2R}},\qquad \qquad (1)$

where $~\delta =0.6$  with the uniform matter distribution in the object, $~\delta =0.62$  for the objects of the type of nucleons and neutron stars, $~G$  is the gravitational constant, $~M$  and $~R$  are the mass and the radius of the object.

With the help of relation (1) we can determine the characteristic speed $~C_{x}$  of each object at the level of stars. In particular, with $~\delta =0.6$  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 $~C_{s}=220$  km/s is the characteristic speed of matter of the main sequence star with the minimum mass $~M_{ps}=0.056M_{c}$ . 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.

#### The correspondence between the atomic, stellar and galactic systems

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 B8-B9, 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 non-metal 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 non-metal elements and also metal-poor. 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 L-dwarfs). These stars (for example the star MOA-2007-BLG-192L) 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∙106 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.  

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. 

#### Explanation of large numbers

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:

1. The coefficient of similarity in mass: $~{\frac {M_{mg}}{M_{p}}}=1.4\cdot 10^{79}$ .
2. The coefficient of similarity in size: $~{\frac {R_{mg}}{R_{p}}}=6.3\cdot 10^{39}$ .

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:

$~{\frac {R}{r}}\approx {\sqrt {\frac {M}{m}}}\approx \Lambda \approx 10^{38}-10^{41}$ .

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.

#### The magnetic properties

In addition to the fact that chemical elements can be set in mutual one-to-one 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 $~\Phi$ . 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.  The magnetic moment of the electron and the nuclear magneton are given by:

$~\mu _{B}={\frac {e}{M_{e}}}{\frac {\hbar }{2}}=K_{e}{\frac {\hbar }{2}}$ ,
$~\mu _{n}={\frac {e}{M_{p}}}{\frac {\hbar }{2}}=K_{n}{\frac {\hbar }{2}}$ ,

where $~e$  is the elementary charge, $~M_{e}$  and $~M_{p}$  are the masses of the electron and the proton, $~{\frac {\hbar }{2}}$  is quantum spin of the electron and the proton, $~K_{e}$  and $~K_{n}$  are the corresponding gyromagnetic ratios equal to the ratio of the charge to the mass.

For relation between the magnetic moments $~P_{m}$  and the spin $~I$  of stellar objects similar to electrons and atomic nuclei, we can write down:

$~P_{m\Pi }=K_{\Pi }I_{\Pi }$ ,
$~P_{ms}=K_{s}I_{s}$ ,

where $~K_{\Pi }$  and $~K_{s}$  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:

$~K_{\Pi }=K_{e}{\frac {P_{0}^{0.5}S_{0}}{\Phi ^{0.5}}}=3.69\cdot 10^{-9}$  C/kg,
$~K_{s}=K_{n}{\frac {P_{0}^{0.5}S_{0}}{\Phi ^{0.5}}}=2.01\cdot 10^{-12}$  C/kg.

Another expression for gyromagnetic ratios at the level of stars has the form:

$~K_{\Pi }={\sqrt {\frac {4\pi \varepsilon _{0}GM_{p}}{M_{e}}}}=3.69\cdot 10^{-9}$  C/kg,
$~K_{s}={\sqrt {\frac {4\pi \varepsilon _{0}GM_{e}}{M_{p}}}}=2.01\cdot 10^{-12}$  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 X-ray 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 $~P_{m}\sim I^{0.7}$ , while for the planets and stars there is a linear dependence $~P_{m}\sim I$ . Non-coincidence 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 $~I\sim M^{1.5}$  and $~P_{m}\sim I^{0.7}\sim M$ , where $~M$  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 $~I\sim M^{5/3}$  and $~P_{m}\sim I\sim M^{5/3}$ . 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: 

$~P_{m}\sim \omega R^{4}{\sqrt {\rho }}$ ,

where $~\rho$  is the mass density of the body with the radius $~R$ , $~\omega$  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 $~P_{m}\sim M^{4/3}$ .

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: 

$~P_{m}=\omega R_{2}^{4}{\sqrt {\rho _{2}}}{\sqrt {\frac {32\pi ^{2}k_{3}}{225\mu _{0}}}}$ ,

where $~R_{2}$  and $~\rho _{2}$  are the radius and the mass density of the core of the planet, $~k_{3}\approx 3\cdot 10^{-9}$  is the coefficient of proportionality between the density of the magnetic force and the Coriolis force, $~\mu _{0}$  is the vacuum permeability. One of the consequences of this is that the density $~U$  of the magnetic energy is proportional to the density $~\epsilon$  of the kinetic energy of rotation of the conductive and magnetized matter: $~U={\frac {2k_{3}\epsilon }{9}}$ . 

#### Planetary and moon systems

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: 

$~L_{ns}={\frac {L_{n}}{M_{n}}}=V_{n}R_{n}=K_{1}n{\frac {\hbar _{s}}{M_{pl}}}$ ,
$~R_{n}={\frac {K_{1}^{2}n^{2}\hbar _{s}^{2}}{GM_{c}M_{pl}^{2}}}$ ,

where $~L_{n}$  is the orbital angular momentum of the planet in the orbit with the number $~n$ ; $~M_{n}$ , $~V_{n}$  and $~R_{n}$  are the mass of the planet, its orbital velocity and the average radius of the orbit; $~\hbar _{s}=2.8\cdot 10^{41}$  J∙s is the stellar Dirac constant for ordinary stars; $~M_{pl}=6.06\cdot 10^{25}$  kg is the mass of the planet corresponding to the electron by the theory of similarity; $~G$  is the gravitational constant; $~M_{c}$  is the mass of the Sun; $~K_{1}=0.5$  from the correspondence with the empirical data.

For planetary moons the corresponding quantization of the specific orbital angular momenta is also observed.  In addition, it is shown that the specific spin mechanical moments of the proper rotation of planets in the Solar system are quantized. 

#### The similarity of objects

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 high-energy 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 + νμ.

In turn, muon decays into electron (positron) and electron and muon neutrinos in the reaction:

μ → е + ν е + νμ.

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 hydrogen-helium composition,  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.  The object LP 40-365 is considered as a white dwarf with a mass of 0.14 Solar masses and it has a high speed of the proper motion.  The matter of such objects is unstable and therefore such stars must undergo catastrophic changes of their state in the periods of time 105 – 107 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. It is possible that due to this emission the object LP 40-365 achieved its extraordinarily high speed. 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 particles-resonances, 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 109 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 X-ray pulsars, which are the main candidates for magnetars.  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 $~R$  and the mass $~M$  of the hadron: 

$~R=R_{p}({\frac {M_{p}}{M}})^{1/3},$

where $~R_{p}$  and $~M_{p}$  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 Mass-energy, MeV Mass,
10–27 kg
10–16 m
Binding energy $~W$ , MeV Characteristic
spin, 10–35 J•s
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 mass-energy, converted from MeV to Joules, by the squared speed of light. The characteristic angular momentum of the particle is given by:

$~L_{x}=MC_{x}R,\qquad \qquad (2)$

and the characteristic speed $~C_{x}$  of the particle’s matter is calculated by the formula (1):

$~W=MC_{x}^{2}={\frac {\delta \Gamma M^{2}}{2R}},$

here $~\delta =0.62$  for the objects of the type of nucleons and neutron stars, $~\Gamma$  is the strong gravitational constant.

For the proton there is an approximate formula $~h=2M_{p}cR_{p}$ , from which for the characteristic spin of the proton we obtain: $~{\frac {\hbar }{2}}={\frac {M_{p}cR_{p}}{2\pi }}$ , where $~c$  is the speed of light and the characteristic speed of the proton matter, $~h$  is the Planck constant. If we apply the same approach for the characteristic spin of muon, we shall obtain the following:

$~L_{\mu }={\frac {M_{\mu }C_{\mu }R_{\mu }}{2\pi }}={\frac {M_{\mu }R_{\mu }}{2\pi }}{\sqrt {\frac {\delta \Gamma M_{\mu }}{2R_{\mu }}}}=9.1\cdot 10^{-35}$  J•s with $~\delta =0.6$ .

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•1011 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•1067 J•s. This is close enough to the value 9.7•1066 J•s according to the known data. 

#### Gravitational constants

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 $~\Gamma$  is significantly different from the ordinary gravitational constant $~G$ . 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:

${\frac {G}{\Gamma }}={\frac {P_{0}S_{0}^{2}}{\Phi }}={\frac {P'S'^{2}}{\Phi '}}$ ,

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 $\Phi _{g}=D_{\Phi }^{2}=1.46\cdot 10^{11}$ . Similarly, from Table 6 the coefficient of similarity in size equals $P_{g}=D_{P}^{6}=2.33\cdot 10^{11}$ . The average velocities $~V_{s}$  of the motion of stars in spiral galaxies of low mass apparently do not exceed the characteristic speed $~C_{s}=220$  km/s of the motion of matter in the star of minimum mass. Hence, the coefficient of similarity in speed $~S_{g}={\frac {V_{s}}{C_{s}}}$  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: 

$G_{g}={\frac {GP_{g}S_{g}^{2}}{\Phi _{g}}}\approx G$ .

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.

#### Natural units

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 $~\hbar$ , the speed of light $~c$  and the gravitational constant $~G$ :

• Planck mass $~M_{Pl}={\sqrt {\frac {\hbar c}{G}}}=2{.}17644(11)\cdot 10^{-8}$  kg.
• Planck length $~l_{Pl}={\frac {\hbar }{M_{Pl}c}}={\sqrt {\frac {G\hbar }{c^{3}}}}=1{.}616252(81)\cdot 10^{-35}$  m.
• Planck time $~t_{Pl}={\frac {l_{Pl}}{c}}={\sqrt {\frac {G\hbar }{c^{5}}}}=5{.}39124(27)\cdot 10^{-44}$  s.

A more complete set of Planck units in the International System of Units includes the Boltzmann constant $~k$  and the factor $~{\frac {1}{4\pi \varepsilon _{0}}}$ , where $~\varepsilon _{0}$  is the electric constant. Planck units are used in quantum physics, where $~\hbar$  is the characteristic angular momentum, but since the ordinary gravitation constant $~G$  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 $e~$ , exceeding it approximately 11.7 times:

$~q_{Pl}={\sqrt {4\pi \varepsilon _{0}\hbar c}}={\sqrt {2ch\varepsilon _{0}}}={\frac {e}{\sqrt {\alpha }}}=1{.}8755459\cdot 10^{-18}$  C,

where $~\alpha$  is the fine structure constant.

At the same time, if we use at the level of the main sequence stars the stellar Planck constant $h_{s}=h\Phi P_{0}S_{0}=1.76\cdot 10^{42}$  J∙s, the stellar speed $~C_{s}=220$  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: 

• The mass $~M_{s}={\sqrt {\frac {h_{s}C_{s}}{G}}}=7{.}62\cdot 10^{28}$  kg.
• The radius $~R_{s}={\sqrt {\frac {Gh_{s}}{C_{s}^{3}}}}=1{.}05\cdot 10^{8}$  m.
• The characteristic time $~t_{s}={\sqrt {\frac {Gh_{s}}{C_{s}^{5}}}}=477$  s.
• The characteristic angular momentum $~L_{s}=M_{s}R_{s}C_{s}=h_{s}$ .
• The average mass density $~\rho _{s}={\frac {3M_{s}}{4\pi R_{s}^{3}}}={\frac {3C_{s}^{5}}{4\pi G^{2}h_{s}}}=1.6\cdot 10^{4}$  kg/m3.
• The average pressure $~p_{s}\approx {\frac {\rho _{s}C_{s}^{2}}{3}}={\frac {C_{s}^{7}}{4\pi G^{2}h_{s}}}=2{.}5\cdot 10^{14}$  Pa.
• The gravitational acceleration $~g_{s}={\frac {GM_{s}}{R_{s}^{2}}}={\sqrt {\frac {C_{s}^{7}}{Gh_{s}}}}=461$  m/s2.
• The absolute value of the total energy $~E_{s}\approx {\frac {GM_{s}^{2}}{R_{s}}}={\sqrt {\frac {C_{s}^{5}h_{s}}{G}}}=3{.}7\cdot 10^{39}$  J.
• The maximum luminosity (energy emission power) $~W_{s}={\frac {E_{s}}{t_{s}}}={\frac {C_{s}^{5}}{G}}=7{.}7\cdot 10^{36}$  W.
• The maximum temperature $~T_{s}\approx {\frac {E_{s}}{3K_{ps}}}={\frac {1}{3K_{ps}}}{\sqrt {\frac {C_{s}^{5}h_{s}}{G}}}=1\cdot 10^{6}$  K,

where $~K_{ps}=1.18\cdot 10^{33}$  J/K is the stellar Boltzmann constant.

The time $~t_{s}$  here characterizes the time required to cross the radius of the star at the speed $~C_{s}$ , 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: $~t_{s}\approx {\sqrt {\frac {3}{4G\rho _{s}}}}$ . The product of the absolute value of the total energy and the characteristic time gives the relation similar to the Heisenberg uncertainty relation: $~E_{s}t_{s}=h_{s}$ . 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 $~M_{s}$ , the speed $~C_{s}$ , and the angular momentum $~h_{s}$ , we can express the gravitational constant in the form: $~G\approx {\frac {h_{s}C_{s}}{M_{s}^{2}}}$ , 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 $~h$ , the speed of light $~c$ , the Boltzmann constant $~k$ , the multiplier $~{\frac {1}{4\pi \varepsilon _{0}}}$  and the strong gravitational constant in the form

$\Gamma ={\frac {e^{2}}{4\pi \varepsilon _{0}M_{p}M_{e}}}={\frac {\alpha hc}{2\pi M_{p}M_{e}}}=1{.}514\cdot 10^{29}$  m3•s–2•kg–1,

where $~M_{p}$  is the mass of the proton, $~M_{e}$  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:

• The mass $~M'_{p}\approx {\sqrt {\frac {hc}{\Gamma }}}=1{.}15\cdot 10^{-27}$  kg.
• The radius $~r'_{p}\approx {\frac {h}{2M'_{p}c}}={\sqrt {\frac {\Gamma h}{4c^{3}}}}=0{.}96\cdot 10^{-15}$  m.
• The characteristic time $~t'_{p}={\sqrt {\frac {\Gamma h}{4c^{5}}}}=3{.}2\cdot 10^{-24}$  s.
• The characteristic angular momentum $~L'_{p}=2M'_{p}r'_{p}c=h$ .
• The average mass density $~{\rho '}_{p}={\frac {3M'_{p}}{4\pi {r'}_{p}^{3}}}={\frac {3c^{5}}{2\pi \Gamma ^{2}h}}=7.6\cdot 10^{16}$  kg/m3.
• The average pressure $~p'_{p}\approx {\frac {\rho '_{p}c^{2}}{3}}={\frac {c^{7}}{2\pi \Gamma ^{2}h}}=2{.}3\cdot 10^{33}$  Pa.
• The gravitational acceleration $~g'_{p}={\frac {\Gamma M'_{p}}{{r'}_{p}^{2}}}={\sqrt {\frac {16c^{7}}{\Gamma h}}}=1{.}9\cdot 10^{32}$  m/s2.
• The absolute value of the total energy $~E'_{p}=M'_{p}c^{2}={\sqrt {\frac {c^{5}h}{\Gamma }}}=1\cdot 10^{-10}$  J.
• The maximum luminosity (energy emission power) $~W'_{p}={\frac {E_{p}}{t_{p}}}={\frac {2c^{5}}{\Gamma }}=3{.}2\cdot 10^{13}$  W.
• The maximum temperature $~T'_{p}\approx {\frac {E'_{p}}{3k}}={\frac {1}{3k}}{\sqrt {\frac {c^{5}h}{\Gamma }}}=2{.}5\cdot 10^{12}$  K.
• The electrical charge $~e={\sqrt {4\pi \varepsilon _{0}\Gamma M_{p}M_{e}}}={\sqrt {2\varepsilon _{0}\alpha ch}}=1{.}602\cdot 10^{-19}$  C.

With the help of natural units similarly to the main sequence stars we can obtain the parameters of galaxies and even metagalaxies.  For example, taking from Table 4 the mass of the metagalaxy 2.49∙1023 Мc , and from Table 6 its radius 14.05 Gpc, we can estimate the average mass density $\rho _{M}=1.45\cdot 10^{-27}$  kg/m3, 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:

$~t_{M}={\sqrt {\frac {3}{4\pi G\rho _{M}}}}=5\cdot 10^{10}$  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.  All the other arguments in favor of the Big Bang can have other explanations, which subjects to well-grounded and many-sided criticism the concept of the explosion of the Universe.

#### The Galaxy as the thermodynamic system

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 $~\rho$ , 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: 

$~\rho =4.4\cdot 10^{14}R^{-1.71}$ ,

where the galactic radius $~R$  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∙1016 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:

$~p={\frac {\rho V^{2}}{3}}$ ,

where $~V$  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:

$~p=1.8\cdot 10^{10}\rho$ .

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 $~T_{g}$  of the Galaxy can be estimated in different ways:

1. By the formula for the internal energy of the Galaxy as the energy of the motion of stars in the form $~E={\frac {3kNT_{g}}{2\mu }}$ , where $~E=2.5\cdot 10^{52}$  J is the estimate of the energy by the virial theorem,  $~k$  is the Boltzmann constant, $~N$  is the total number of nucleons, $~\mu =0.64$  is the number of nucleons per gas particle as it is accepted for the Sun.
2. By the formula for the luminosity of the Galaxy of the form $~L=\Sigma _{s}A_{g}T_{g}^{4}$ , where $~L=7.6\cdot 10^{36}$  W, $~A_{g}$  is the area of the Galaxy disc, $~\Sigma _{s}={\sigma \Phi S_{0}^{3}}{P_{0}^{3}}=9.3\cdot 10^{-30}$  W/(m2∙K4) is the stellar Stefan–Boltzmann constant, found through the Stefan–Boltzmann constant $~\sigma$  and the coefficients of similarity in mass, speed and size for main sequence stars.
3. By the formula for the pressure of the stellar gas of the form $~p=nK_{s}T_{g}$ , where $~n$  is the concentration of stars, $~K_{s}=AK_{ps}$ , where $~K_{ps}=1.18\cdot 10^{33}$  J/K is the stellar Boltzmann constant, $~A$  is the mass number of typical stars, characterizing on the average the Galaxy.

On the average the temperature of the stellar gas in the Galaxy is about $~T_{g}=2\cdot 10^{6}$  K. Another way to determine the temperature of the Galaxy is associated with the generalized gas law for the stellar gas:

$~pV_{g}={\frac {M_{g}R_{st}T_{g}}{M_{sm}}}$ ,

where $~V_{g}$  and $~M_{g}$  are the volume and the mass of the Galaxy, $~R_{st}$  is stellar gas constant, $~M_{sm}$  is the mass of one stellar mole of the substance, consisting of stars.

For the ordinary gas constant there is a relation: $~R=kN_{A}$ , where $~N_{A}=6.022\cdot 10^{23}$  mole–1 is the Avogadro number. Since in the stellar mole the number $~N_{A}$  of stars is also assumed, so the stellar gas constant equals:

$~R_{st}=K_{s}N_{A}=AK_{ps}N_{A}=AR_{pst}$ ,

where $~R_{pst}=K_{ps}N_{A}=7.1\cdot 10^{56}$  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:

$~M_{sm}=M_{s}N_{A}=AM_{ps}N_{A}=AM_{psm}$ ,

where $~M_{psm}=M_{ps}N_{A}=6.68\cdot 10^{52}$  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 $~A=9$ . 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:

$~pV_{g}={\frac {2E}{3}}={\frac {M_{g}V^{2}}{3}}$ .

After substituting the quantities into the right side of the generalized gas law for the stellar gas we obtain:

$~{\frac {M_{g}R_{st}T_{g}}{M_{sm}}}={\frac {M_{g}AR_{pst}T_{g}}{AM_{psm}}}={\frac {M_{g}R_{pst}T_{g}}{M_{psm}}}$ .

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:

$~{\frac {M_{g}V^{2}}{3}}={\frac {M_{g}R_{pst}T_{g}}{M_{psm}}}$ ,
$~T_{g}={\frac {M_{psm}V^{2}}{3R_{pst}}}=1.7\cdot 10^{6}$  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 $~C_{s}{\frac {A}{Z}}$ , where $~A$  and $~Z$  are the mass and the charge numbers, corresponding to the main sequence stars" .

## The similarity of forms and energies of the phenomena

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 1020. 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.  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 gamma-ray bursts in the energy range 1043 – 1044 J, Oldershaw compares with the gamma-radioactive nuclei. The energies of gamma-ray 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 = ΛD = 1.7∙1056, he obtains the energy range from 2.72∙1041 J to 1.87∙1044 J, where the gamma-ray bursters GRB also fall.  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 $~p^{2}=kr^{3}$  and the relation for electrons in Rydberg states. By recalculation of the coefficient $~k$ , 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.  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 ≤ n-1, 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∙1016 с (3∙108 years) 2.5∙1014 с (8∙106 years)

The characteristic rotation periods of active galaxies are about 108 years, and the oscillations are determined by the periods of recurrence of significant ejections of matter from their nuclei, equal about 107 years.  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∙1017.

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: ${\mathcal {2}}=\Phi S_{0}^{2}=3.6\cdot 10^{49}$ , and for the compact objects, such as neutron stars, the coefficient of similarity in energy is equal to ${\mathcal {2}}'={\Phi }'{S'}^{2}=8.6\cdot 10^{55}$ .

The gamma-ray 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:

$t={\frac {h}{W}}=4.1\cdot 10^{-19}-8.3\cdot 10^{-22}$  s.

Multiplying the energies and the oscillation periods of the gamma-ray quanta by the coefficient of similarity in energy ${\mathcal {2}}$  and the coefficient of similarity in time $\Pi _{0}={\frac {P_{0}}{S_{0}}}=7.41\cdot 10^{25}$ , respectively, we can find the energies and the periods at the level of stars: the energies – from 5.7∙1034 J to 2.8∙1037 J, the periods – from 352 days to 17 hours.

These energies and periods conform to the values characteristic of long-period 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, short-period 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. 

If we multiply the energies and the oscillation periods of gamma-ray quanta from atomic nuclei by the similarity coefficients ${\mathcal {2}}'$  and $\Pi '={\frac {P'}{S'}}=6.1\cdot 10^{19}$ , we shall obtain the corresponding energies and periods for the objects of the type of neutron stars:  the energies – from 1.4∙1041 J to 6.9∙1043 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 gamma-ray bursters. The energy of gamma-ray burst from the magnetar SGR 1806-20, recorded on December 27, 2004, is estimated by the value 4∙1039 J.  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 gamma-ray burster GRB 080319B the total energy of the outburst in all the emission ranges was equal up to 1040 J.  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 1013 m contains the mass up to 109 solar masses, producing the emission with the luminosity about 2∙1040 W.  If we divide this luminosity by the number of stars, we shall obtain the value 2∙1031 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 1013 m to the time interval of one day gives the velocity 108 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 108 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 $~\Delta E$  and the time $~\Delta t$  of its change has the form:

$~\Delta E\Delta t\geq L_{x}$ ,

where $~L_{x}$  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 $~L_{x}=4\pi I$  is assumed, and for the orbital angular momenta L the relation $~L_{x}=2\pi L$  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 $~E$  and the time of the formation of stars (the Kelvin-Helmholtz time $~t_{KH}$ ) from separate gas clouds leads to the fact that for a typical star the following relation holds: 

$~Et_{KH}\approx h_{o}=2.1\cdot 10^{57}$  J/s,

where $~h_{o}$  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 $~t_{KH}$ , which can be explained by the time of the stellar core growth due to the thermonuclear reactions in which the mass-energy is released with the value up to 1/130 of the rest energy of the matter. The relation for $~h_{o}$  also reflects the change of the energy in the process of cooling of neutron stars. If instead of $~h_{o}$  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∙1046 J) and the time of release of this energy (several seconds) will be valid.

## Combined scale symmetry

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.

## The philosophical justification

A detailed philosophical analysis of the Theory of Infinite Hierarchical Nesting of Matter and the similarity of matter levels was carried out in 2003.  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.  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.  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.