# Physics/Essays/Fedosin/Similarity of matter levels

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

## Large numbers

In 1937 Dirac suggested hypothesis of large numbers, according to which parameters of Metagalaxy (it was then called Universe, although now it is established that Metagalaxy is only part of Universe) can be found through parameters of elementary particles by multiplying them by some large numbers. According to his hypothesis, 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 characteristic time of a process, size and mass of Metagalaxy, $~t,r,m$ specify the same parameters for elementary particles.
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 mass $~M_{H}$ which sets rest energy equal to gravitational energy of electron, provided that radius of electron is equal to classical electron radius $~r_{0}=2.8\cdot 10^{-15}$ m, as well as equal to electrical energy of the object, provided that charge of the object is equal to electron charge, and 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 classical electron radius is determined from condition of equality of rest energy of electron in 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 speed of light, $~G$ is gravitational constant, $~M_{e}$ is mass of electron, $~e$ is elementary charge as charge of proton, $~\varepsilon _{0}$ is 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 radius of hypothetical object exceeds more than by an order of magnitude observable part of Universe.
The above equation for rest energy $~M_{H}c^{2}$ can be interpreted as equality between gravitational energy of two electrons at distance $~r_{0}$ from each other, and their electrical energy at distance $~R_{H}$. In this case, large value $~R_{H}$ is obtained as consequence of weakness of gravitational force between electrons in comparison with their electrical force and it seems not related to size of Metagalaxy. Indeed, if we divide electrical force between proton and electron by absolute value of force of their gravitational attraction, we 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 proton mass.
At the same time strong gravitational constant, by its definition, equals:
    $\Gamma ={\frac {e^{2}}{4\pi \varepsilon _{0}M_{p}M_{e}}}=1{.}514\cdot 10^{29}$ m$^3$•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 gravitational force between proton and electron is equal to ratio of strong gravitational constant to ordinary gravitational constant and is proportional to ratio of sizes $~{\frac {R_{H}}{r_{0}}}$.
In gravitational model of strong interaction strong gravitation acts between matter of hadrons and as well as between matter of leptons. At the level of atoms, strong gravitation is the same as ordinary gravitation at the level of planets and stars. In this picture distance $~R_{H}$ is not related to size of Metagalaxy.
One of attempts to explain hypothesis of large numbers is use of quantum ideas with consideration of hadrons, compact stellar objects and Universe as objects similar to black holes. However, such combination of quantum mechanics and general theory of relativity is not quite convincing, and therefore search for other explanations continues.

## Quantum properties of stellar systems

Titius–Bode law, which appeared long before quantum mechanics, was intended to mathematically describe smooth dependence of radii of orbits of planets in the Solar system on planet number $~n$. At present time various schemes are suggested in which orbits of planets are described by quantum numbers for energy and orbital momentum, similarly to way in which states of electrons in atom are specified. In particular, for modeling acceptable radii of planetary orbits near the Sun and other stars solutions of Schrödinger equation are used.
Similar results are obtained under assumption that planetary orbits are quantized proportionally to square of quantum number. As a rule it is assumed that 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 Solar system in orbits with $~n=1$ and $~n=2$ for inner planets, and with $~n=1$ for outer planets.
In general transfer of methods of quantum mechanics on the level of stellar and planetary objects is logical development of idea of similarity of matter levels, since quantization is a universal property of matter.

## Models of similarity

According to general opinion, if we take two similar systems, one – in microworld and the other – in macroworld, then rate of time, understood as number of similar events per unit time, is much higher in microworld than in macroworld. This is consequence of the fact that duration of an event in microworld is small as compared with duration of a similar event in macroworld because of difference in sizes. Under similarity coefficient dimensionless quantity is understood, which is equal to ratio of two identical physical quantities, which refer to compared and in some ways similar to each other objects at different levels of matter. As it follows from 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 similarity coefficients between Metagalaxy and elementary particles.
One of problems with similarity coefficients in various models of similarity is before their determination first to uniquely identify compared matter levels and objects corresponding to them. For example, Fournier d'Alba considered that ratio of linear sizes of a star and an atom, as well as ratio of durations of their similar processes, is expressed by the number 10$^{22}$. But sizes of stars with the same mass can differ by thousands of times, which makes estimate of coefficient of similarity in size not unique and depending much on chosen type of stars. Purpose of using physically justified similarity relations between different matter levels is to build the model of Universe, in which it becomes clear how the large arises from the small, what forces of 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

Sergey Sukhonos arranged all known objects of microworld, macroworld and megaworld on one scale axis of sizes and found out that properties of objects are repeated periodically with increasing of size approximately 10$^{20}$ times. This is well illustrated by the following examples:
* Normal stars have an average size of the order of 10$^{12}$ cm and are composed of atoms with the size 10$^{-8}$ cm.
* White dwarfs have the average size 10$^{10}$ cm, are composed of highly compressed atoms with the size 10$^{-10}$ cm.
* Neutron stars are highly compressed by gravitation – up to 10$^7$ cm, are composed of nucleons with the size 10$^{-13}$ cm.
In all cases, the scale "distance" between the system and its components is the same – 10$^{20}$. Built by Sukhonos the scale similarity in size conforms to Dirac-Eddington hypothesis of large numbers, since
    $~\Lambda \approx 10^{40}=(10^{20})^{2}.$
In regularity of distribution of objects on matter levels Sukhonos found bimodality in sizes of objects. This is manifested, for example, in distribution of atoms’ diameters by sizes, in similar distribution for stars and galaxies, as well as in distribution of areas of countries, regions, states, provinces, etc. In order to combine bimodality and periodicity of changing of sizes of objects of matter levels, mechanism of multistage cluster convolution (theory of centrosymmetric packing) and global scale standing waves are considered.
For coefficients of similarity in size and time Yun Pyo Jung derives the value of order of 10$^{30}$. To obtain these coefficients, he compares radius of atomic nuclei (≈ 10$^{-15}$ m) and radius of nuclei of galaxies, presumably equal in case of the Galaxy (its other name is Milky Way) 0.33 light years or 0.1 pc, which is equal to 3∙10$^{15}$ m. However recent studies show that nuclei of galaxies do not have any unambiguous definition. Rounded thickening in our Galaxy, called bulge, has radius of 200 pc, and area in center of the Galaxy called Sagittarius A* contains the mass 4.3∙10$^6$ of solar masses with radius of 45 a.u. or 7∙10$^{12}$ m. Another way to determine coefficient of similarity in size is with the help of ratio of radius of Galaxy (≈ 30 kpc) to radius of atom, of the order of 10$^{-10}$ m. Yun Pyo Jung also considers ratios of radii or typical sizes in objects similar to each other, such as molecules and groups of galaxies, macromolecules and clusters of galaxies, organelles and superclusters of galaxies, biological cells with radius of 25 μm and observed cosmos with radius of 15 billion light years, again obtaining the value of order of 10$^{30}$.

### Oldershaw

Robert L. Oldershaw went further and determined coincident with each other coefficients of similarity in size and time, equal to Λ = 5.2∙10$^{17}$, and coefficient of similarity in mass X = Λ$^D$ = 1.7∙10$^{56}$, where exponent D = 3.174. At the same time Oldershaw compares atomic nuclei, stars and galaxies as corresponding objects at three levels of matter.
Hydrogen system at the level of stars, according to Oldershaw, consists of main sequence star with mass $~0.145M_{c}$, and of object – analogue of electron with mass equal to 26 Earth’s masses. If to convert Bohr radius into corresponding radius at the level of stars by multiplying by coefficient of similarity in size Λ, then this object must be located in shell of the star. If this object is considered as being in excited state, it can take form of a planet.
To obtain radius of ordinary galaxy it is necessary multiply radius of corresponding atomic nucleus by Λ$^2$, which gives range of radii of galaxies from 7 to 75 kpc (similar to sizes of proton and nucleus of lead, respectively). Since Oldershaw believes that coefficients of similarity between levels of matter are the same for all objects and do not depend on type of these objects, he has a problem with obtaining sizes of dwarf and giant galaxies (0.1 kpc and 500 kpc, respectively). To solve this problem, he expands range of objects at the atomic level, adding to atoms and ions 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 Schwarzschild formula:
    $~R={\frac {2G_{N}M}{c^{2}}},$
where $~G_{N}$ is constant of gravitation acting on the given level of matter, $~N=-1$ for the atomic level, $~N=0$ for the level of stars, $~N=+1$ for the level of galaxies.
Constant of gravitation is calculated using coefficients of similarity taking into account dimension of this constant equal to m$^3$/(kg∙s$^2$). Since coefficients of similarity in size and time are considered equal, he obtains:
    $~{\frac {G_{0}}{G_{-1}}}={\frac {G_{+1}}{G_{0}}}={\frac {\Lambda }{X}}=3\cdot 10^{-39},$
where $~G_{0}$ is ordinary gravitational constant.
Assuming that at the level of atoms $~G_{-1}=2{.}18\cdot 10^{28}$ m$^3$•s$^{–2}$•kg$^{–1}$ is strong gravitational constant, Oldershaw finds corresponding radius of electron 4.4∙10$^{-19}$ m, and radius of proton 0.81∙10$^{-15}$ m. If we multiply this radius of electron by Λ$^2$, we obtain radius of 3.9 pc, corresponding to nuclei of globular star clusters. According to Oldershaw, these objects with sizes of globular clusters are analogue of electrons at the level of galaxies. However ratio of minimum galaxy mass of normal galaxy to mass of typical globular cluster has the order of magnitude 10$^5$, which is much greater than the ratio of proton mass to electron mass, which is equal to 1836. Another problem is that number of globular clusters in galaxies is many times greater than number of electrons in atoms. Besides, black holes are only suspected inside globular clusters and galaxies.
At the level of galaxies gravitational constant according to Oldershaw is equal to $~G_{+1}=2\cdot 10^{-49}$ m$^3$•s$^{–2}$•kg$^{–1}$. If to use Schwarzschild formula with this gravitational constant and sizes of galactic objects – the analogues of electron and proton, we obtain very large masses – about 2.7∙10$^{82}$ kg and 5∙10$^{85}$ kg, respectively. Oldershaw believes that we do not notice such masses of galaxies, because at the level of galaxies gravitational constant is extremely small. He also considers Metagalaxy to be the result of explosion, similar to supernova explosion, which explains high effective temperature of galaxies, producing gas similar to hot fully ionized gas. To calculate the temperature, value of average peculiar velocity of galaxies is used, equal to 700 km/s. Atomic nuclei moving at such a velocity, have kinetic temperature of about 10$^8$ – 10$^9$ Kelvin degrees, and the same temperature is attributed to gas from the galaxies.
Oldershaw states that observable Universe is considerably smaller in size than the object that must be at the metagalactic level of matter, exceeding Λ = 5.2∙10$^{17}$ times the size of galaxies. With the help of telescopes and different techniques we can see most distant quasars at distance only 10$^5$ – 10$^6$ greater than radii of typical galaxies. Among other conclusions is the assumption that dark matter consists of black holes; the ether is assumed to consist of charged relativistic particles; electrical force is substantiated as result of emission by large charges of tiny particles, so that the proton and the electron in form of corresponding Kerr-Newman rotating micro black holes must emit smaller charged particles.
As natural units of measuring physical quantities at the atomic level Oldershaw uses a set of Dirac constant $\hbar$ and speed of light $~c$, included in Planck units, but instead of ordinary gravitational constant he uses strong gravitational constant $~G_{-1}$. This allowed him to determine "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.
Obtained values are close enough to parameters of proton.

### Fedosin

In Sergey Fedosin’s monograph on theory of similarity eighteen levels of matter from preons to metagalaxies were divided into basic and intermediate by their masses and sizes. 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 maximum number of constituent particles and having maximum density of matter and energy. Matter of these carriers is degenerate, that is, their constituent particles have approximately the same quantum states, and therefore state of such matter is described by laws of quantum mechanics. Neutron star contains about 10$^{57}$ nucleons, and by induction it is assumed that nucleon contains the same number of quantum particles.
Fedosin’s approach has the following features:
1) He does not support Oldershaw’s idea of hierarchical nesting of black holes as basic structures for considered objects at different levels of matter, due to denying existence of black holes as such.
2) 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 compact stars (to compact galaxies, compact objects of other levels). This means difference between similarity coefficients for different types of objects.
3) Coefficients of similarity in size and time do not coincide in magnitude with each other, in contrast to models of other researchers.
4) Fedosin divides matter levels into basic and intermediate. Basic levels of matter are characterized by the fact that on objects of these levels fundamental forces achieve extreme values. Objects of these levels have highest density of matter and energy, they are most stable, have a spherical shape and form basis of larger objects.
5) Connection between intermediate levels of matter is carried out by discrete coefficients of similarity, so that 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 masses and sizes of carriers at scales of masses and sizes change in geometric progression with the constant factors $~D_{\Phi }$ and $~D_{P}$, respectively.
6) Similarity relations of objects and physical phenomena are performed most accurately with those objects, evolution of which is repeated by the same scenario at different levels of matter. For example, at the level of elementary particles real analogues for 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.

#### Coefficients of similarity
##### Between atoms and main sequence stars

Coefficient of similarity in mass is determined by Fedosin based on accurate data on masses of 446 binary main sequence stars from the Svechnikov’s catalogue and data provided by other authors. Processing of available data leads to conclusion that analogue of the Solar system at the atomic level is isotope of oxygen O(18), and hydrogen corresponds to stars with the minimum mass $~M_{ps}=0.056M_{c}$, where $~M_{c}$ is mass of the Sun. Ratio of mass of the Sun to mass of nuclide O(18) gives coefficient of similarity in mass $\Phi =6.654\cdot 10^{55}$. After multiplying electron mass by $~\Phi$, mass of corresponding planet is determined as M$_{pl}$ = 6.06∙10$^{25}$ kg or 10.1 Earth masses. From dependence of radius of planets on their mass it follows that mass of the planet М$_{pl}$ corresponds to radius R$_{pl}$ = 20000 km or 3.1 Earth’s radii. Measurements of radii of planets in different planetary systems also shows that most of planets have radius from 2 to 4 Earth’s radii.
A convenient model to determine coefficients of similarity is 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 analogue of electron, and at the level of galaxies – of normal and dwarf galaxies. Parameters of objects of hydrogen system for atoms and main sequence stars are given in Table 1.
For a hydrogen-like atom, speed of orbital rotation of electron in it is proportional to nuclear charge $~z$. In corresponding planetary system speed of planet’s rotation around star is proportional to mass of the star, i.e., to its mass number $~A$. It follows that coefficient of similarity in speed is given by: $S=S_{0}{\frac {A}{z}}$. From the same considerations, the following is obtained for coefficient of similarity in size: $P=P_{0}{\frac {z}{A}}$. In order to find coefficient of similarity in speed $~S_{0}$ the following method is used – in assumption that speeds of orbital motion of electron and planet are determined by characteristics of attracting body (atomic nucleus or star), not the orbital motion speeds are considered, but characteristic speeds of matter inside corresponding attracting bodies. Given mass–energy equivalence, total energies of atomic nucleus and 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 characteristic speed $~C_{x}=C_{s}{\frac {A}{z}}$ of matter in the star depends on mass number $~A$ and charge number $~z$ of the star. Total energy of the star can be calculated by formula:
    $~E_{s}=-{\frac {\delta GM_{s}^{2}}{2R_{s}}},$
here $~G$ is gravitational constant, $~M_{s}$ and $~R_{s}$ are mass and radius of the star, $~\delta$ is coefficient depending on distribution of matter, in case of uniform mass density $~\delta =0.6$.
Results of calculating total energy of main sequence stars by various authors make it possible to construct dependence of the energy on mass of the stars and find characteristic speed $~C_{x}$ for them. Since masses of stars are associated with corresponding nuclides, which have mass and charge numbers, then from the relation $~C_{x}=C_{s}{\frac {A}{z}}$ value of characteristic stellar velocity is determined: $~C_{s}=220$ km/s. Ratio of the speeds in form $~S_{0}={\frac {C_{s}}{c}}$ makes it possible to determine coefficient of similarity in speed, indicated in Table 1 for hydrogen system, and to find orbital velocity of the planet. Further, from equilibrium of gravitational force and centripetal force, radius of the planet’s orbit is calculated and coefficient of similarity in size $~P_{0}$, as ratio of radius of planet's orbit to radius of electron’s orbit in hydrogen atom in its ground state. The value $~P_{0}$ is close enough to ratio between semi-axes of orbits of binary stars and bond lengths of corresponding molecules, to ratio of sizes of Solar system and oxygen atom, to ratio of size of Mercury's orbit and sevenfold ionized oxygen ion, to ratio of sizes of stars’ nuclei and sizes of atomic nuclei.
Coefficient of similarity in time, as ratio of time flow speeds between nuclear and ordinary stellar systems, is:
    $\Pi _{0}={\frac {P_{0}}{S_{0}}}=7.41\cdot 10^{25}$.
Value of stellar Dirac constant for ordinary stars is:
    $\hbar _{s}=\hbar \Phi P_{0}S_{0}=2.8\cdot 10^{41}$ J∙s,
where $\hbar$ is Dirac constant.

##### Between atoms and neutron stars

For the level of quantum and compact objects – elementary particles and neutron stars, coefficients of similarity in mass, size and characteristic speed are slightly different from similarity coefficients for atoms and ordinary stars. Table 2 uses data for proton and neutron star.
After multiplying electron mass by $~\Phi '$, mass of object- analogue of electron is determined as М $_d$ = 1.5∙10$^{27}$ kg, which equals 250 of Earth’s mass, or 0.78 of Jupiter’s mass. Сoefficient of similarity in time, as ratio of rate of time between elementary particles and neutron stars, is:
    $~\Pi '={\frac {P'}{S'}}=6.1\cdot 10^{19}$.
Value of Dirac stellar constant for compact degenerate stars is:
    $~\hbar '_{s}=\hbar \Phi 'P'S'=5.5\cdot 10^{41}$ J∙s.
Multiplying Boltzmann constant by coefficient of similarity in energy, we can find stellar Boltzmann constant: $K'_{s}=k\Phi 'S'^{2}=1.18\cdot 10^{33}$ J/K.

#### Gravitational fine structure constant

Electromagnetic fine structure constant is defined as ratio of speed of electron in 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 elementary charge, $~\varepsilon _{0}$ is electric constant, $~h$ is Planck constant. It is easy to prove that for stationary circular orbits of electron in hydrogen atom fine structure constant is equal to ratio of total energy of the electron to energy of a photon, wavelength of which is equal to twice circumference of electron’s rotation.
Fine structure constant can be expressed through strong gravitational constant $~\Gamma$, proton mass $~M_{p}$ and 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 coupling constant of strong gravitational interaction, used without taking into account absorption of gravitons in matter of two interacting nucleons.
Similarly to this, gravitational fine structure constant is calculated for planet orbiting the star – analogue of proton at the speed $~V_{pl}$:
    $~\alpha _{s}={\frac {V_{pl}}{C_{s}}}={\frac {GM_{ps}M_{pl}}{\hbar _{s}C_{s}}}={\frac {1}{137.036}}$.
The same ratio for object – analogue of electron orbiting 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 similarity relations constants $~\alpha$ and $~\alpha _{s}$ are equal to each other.

#### Relations between coefficients of similarity

Using virial theorem, it is possible determine total energy of a star through its mass, radius, and gravitational constant. On the other hand, energy of a star can be calculated as sum of quantum-mechanical energies of cells of atomic sizes, through total number of nucleons in the star, Planck constant, proton mass and cell size (obtained through radius of the star and number of nucleons). Given that ratio of mass (radius) of the star to mass (radius) of proton is coefficient of similarity in mass (size), and ratio of characteristic speeds of matter in the star and in proton gives coefficient of similarity in speed, then for coefficients of similarity the following relation is found:
    ${\frac {P_{0}S_{0}^{2}}{\Phi }}={\frac {2\pi GM_{p}M_{e}}{\alpha hc}}={\frac {P'S'^{2}}{\Phi '}}$.
Left side of the equality contains similarity coefficients for systems with main sequence star of minimum mass, and the right side – for systems with neutron stars, taken with respect to hydrogen atom and proton, respectively. The relations between coefficients of similarity show that not all of these coefficients are independent on each other. Taking into account expressions for fine structure constant and for strong gravitational constant $\Gamma ={\frac {e^{2}}{4\pi \varepsilon _{0}M_{p}M_{e}}}$ gives the following:
    ${\frac {P_{0}S_{0}^{2}}{\Phi }}={\frac {G}{\Gamma }}={\frac {P'S'^{2}}{\Phi '}}$.
This expression for ratio of gravitational constants conforms to ratio of corresponding coefficients of similarity, as it follows from dimension of gravitational constant. Thus, coefficients of similarity between basic levels of matter can not be arbitrary, they are limited by ratio of gravitational constants at these levels of matter.

#### Horizontal dimensionless coefficients

For hydrogen atom we can determine dimensionless coefficients associated with mass, sizes and speeds:
* Ratio of proton mass to electron mass: $\beta ={\frac {M_{p}}{M_{e}}}=1836.15$.
* Ratio of Bohr radius to radius of 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.
* Ratio of electron’s speed in first Bohr orbit to speed of light (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 (hydrogen atom, planetary system, etc.), the horizontal dimensionless coefficients are the same, so the above relation between the coefficients does not change.

#### Discreteness of similarity coefficients

Analyzing similarity of levels of matter Fedosin considers characteristic masses of carriers that are in range from 10$^{-38}$ kg to 5∙10$^{26}$ of the Sun’s mass. Sizes of the carriers vary from 10$^{-19}$ m to 372 Gpc. Total number of matter levels is 18, at the lowest level preons are located, and at the highest – metagalaxies and superclusters of metagalaxies. Due to the fact that in nature matter carriers are not uniformly distributed, but are concentrated in certain groups, in which difference in sizes and masses of the carriers is not so large in comparison with difference between the groups, it becomes possible to determine coefficients of similarity not only between basic, but also between intermediate levels of matter. It turns out that masses and 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 arbitrarily chosen level of matter. This makes it possible to estimate masses and sizes of carriers of any other level of matter by appropriate multiplication by the factors $~D_{\Phi }$ and $~D_{P}$.

##### In mass

Between such matter levels as elementary particles and ordinary stars, there is nine intermediate levels of matter. To find coefficient of similarity in mass between adjacent intermediate levels, it is necessary to extract tenth root of coefficient of similarity in mass between atoms and 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 electron mass $M_{e}=9.1095\cdot 10^{-31}$ kg by degree of similarity coefficient $~D_{\Phi }$. First multiplication gives $3.482\cdot 10^{-25}$ kg, that is, mass of chemical element which has mass number $~A$ approximately equal to 210. Such element is lead or bismuth, the most massive of stable chemical elements. The second multiplication by $~D_{\Phi }$ gives mass of the largest stable molecular complexes, and so on.
Mass М$_{pl}$ = 6.06∙10$^{25}$ kg in Table 3 corresponds to mass of the planet, which is analogue of electron. Mass 11.6 М$_c$ is mass of the main sequence star of spectral type B1, which is analogue of nuclide of the type of lead or bismuth. Under normal stars such main sequence stars are understood, masses of which do not exceed 11.6 М$_c$. Proton corresponds to stars with the minimum mass $~M_{ps}=0.056M_{c}$, where $~M_{c}$ is mass of the Sun.
Preons correspond by their masses to comets, asteroids, moons of planets; partons correspond to large asteroids, moons, and inner planets; atoms are similar to planetary systems of stars, and tiny specks of dust by number of atoms of which they consist are analogues of galaxies. To estimate masses of preons and partons, it should be taken into account that direct analogy for atoms and elementary particles are systems with neutron stars, and not systems with main sequence stars. Since partons are similar to asteroids and inner planets, the masses of which are less than masses of neutron stars, then masses of partons must be less than masses of nucleons in the same proportion. Preons are one scale level lower and have less masses than partons. Hence, 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.
Subsequent multiplication of masses of similar objects at matter levels by similarity coefficient $~D_{\Phi }$ makes it possible to determine masses of objects from stars to metagalaxies according to Table 4.
Dwarf galaxy with mass 4.43∙10$^6$ М$_c$ is analogue of electron, and normal galaxy with minimum mass 8.15∙10$^9$ М$_c$ corresponds to proton in hydrogen atom. Our Galaxy is presumably analogue of chemical element with mass number $~A=18-20$, and forms with the Large and Small Magellanic Clouds, which are galaxies of small size, an association similar to water molecule. At the level of metagalaxies normal metagalaxy with mass М$_{mg}$ = 2.368∙10$^{51}$ kg or 1.19∙10$^{21}$ М$_c$ corresponds to proton.

##### In size

According to substantial electron model, the electron charge is so high that strong gravitation of its matter is not able to counteract electrical force of repulsion of charged matter units. However, in atom mass and charge of nucleus are sufficient to keep electron in form of some axisymmetric figure in which matter of electron is rotating around the nucleus. Thus, electron radius as radius of an independent elementary particle is not determined. In connection with this, in Table 5 determining of sizes of objects at the intermediate matter levels is done not from the radius of electron in direction of larger sizes, but in opposite direction. Starting point is not the radius of electron but radius R$_{pl}$ = 2∙10$^7$ m of planet with mass М$_{pl}$ = 6.06∙10$^{25}$ kg, which is the analogue of electron. Radius R$_{pl}$ is determined from dependence of radius of planets of the Solar system on mass. In the first row of Table 5 radius 3.85∙10$^9$ m is given, which corresponds to radius of a star with mass 11.6 М$_c$. Radius of the star – analogue of proton is assumed to be 0.07 of the Solar radius, or 4.9∙10$^7$ m according to recent measurements.
The exponent of progression for coefficient of similarity in size is 12, because in contrast to similarity in mass between the level of elementary particles and the level of stars there are two additional levels associated with sizes of atoms (this is accompanied by the fact that in transition from sizes of atomic nuclei to sizes of atoms mass of objects is almost unchanged). Hence, coefficient of similarity in size between adjacent intermediate levels is determined as twelfth root of coefficient similarity in size between atoms and planetary systems of main sequence stars:
    $D_{P}=P_{0}^{1/12}=78.4538$.
For comparison, one of the largest covalent atomic radii with value of 2.25∙10$^{-10}$ m belongs to cesium atom, and radius of uranium nucleus is of the order of 0.8∙10$^{-14}$ m. Data in Tables 3 and 5 are connected because object masses are proportional to mass density and cube of radius. Comparison of different models of objects from stars to elementary particles, their densities and observed masses and sizes shows that characteristic sizes in Table 5 differ not more than 2 – 3 times from observed values. Estimating sizes of partons and preons is done similarly to estimating their masses. In particular, relations between sizes of neutron star and planets (moons of planets, asteroids) are considered, and similar relations between nucleons and partons. Hence, objects of the level of partons must have radii in the range from 1.1∙10$^{-14}$ m to 9∙10$^{-13}$ m, and objects of the level of preons – from 1.5∙10$^{-16}$ m to 1.1∙10$^{-14}$ m.
Sizes of objects from stars to metagalaxies in Table 6 are determined by multiplying by degrees of coefficient of similarity in size $~D_{P}$.
Optical radii of galaxies, corresponding to electron and proton by mass, are found from observations of galaxies and on average are equal to 350 pc and 2.5 kpc. However, multiplying radius of the star – analogue of proton by the size similarity coefficient $D_{P}^{6}$ gives only 370 pc. Difference from the optical radius of corresponding galaxy 2.5 kpc is connected with the fact that normal galaxies with minimum mass are rather flat spiral systems and the radius 2.5 kpc is the largest radius of disc, and the radius 370 pc is a radius averaged over volume of the galaxy. Galaxies with 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 observed, radius of which can reach 38 kpc.
By multiplying radius of star – analogue of proton by coefficient of similarity in size $D_{P}^{9}$ estimate of radius of metagalaxy corresponding to proton is found: R$_{mg}$ = 1.8∙10$^8$ pc. For metagalaxy similar to heavy nuclei such as lead, radius would be about 14 Gpc. Observable Universe at present has the same radius.
Coefficient of similarity in size $D_{P}=P_{0}^{1/12}=78.4538$ is large enough, since change of mass of 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 change of sizes approximately $~10^{K}=10^{0.474}\approx 3$ times. There are researches in which it is found that distribution of sizes of various organisms in flora and fauna, from viruses and to largest organisms, corresponds to change of typical sizes that are multiple on logarithmic scale either of the value $~K$, or of its integer parts. For blocks of the Earth’s crust also there is a correlation with the value $~K$. These data conform to results of Sergey Sukhonos’ researches and confirm universality of discrete coefficients of similarity, which can be applied to objects of both living and inanimate nature.

##### In speed

Discrete coefficient of similarity in speed is determined as fifth root of coefficient of similarity in speed between atoms and planetary systems of main sequence stars:
    $D_{S}=S_{0}^{1/5}=0.2361$.
Characteristic speeds in Table 7 are obtained by successive multiplication of speed of light c = 299792 km/s by degrees of coefficient $~D_{S}$.
Characteristic speed $~C_{x}$ of particles of an object is associated with absolute value of total energy of the object or its binding energy in field of ordinary (or strong) gravitation:
    $~E=MC_{x}^{2}={\frac {\delta GM^{2}}{2R}},\qquad \qquad (1)$
where $~\delta =0.6$ for uniform matter distribution in the object, $~\delta =0.62$ for objects of the type of nucleons and neutron stars, $~G$ is gravitational constant, $~M$ and $~R$ are mass and radius of the object.
With the help of relation (1) it is possible determine characteristic speed $~C_{x}$ of each object at the level of stars. In particular, with $~\delta =0.6$ characteristic speed of dwarf planet Ceres is about 0.2 km/s, of Mercury – 1.64 km/s, of Mars – 1.94 km/s, of 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. Speed $~C_{s}=220$ km/s is characteristic speed of matter of main sequence star with minimum mass $~M_{ps}=0.056M_{c}$. In such stars, thermonuclear reactions occur mainly at stage of formation of the stars, and then they slowly weaken. These stars at the same time can be considered hydrogen white dwarfs, since the main mass of hydrogen will never turn into helium, and internal pressure in the star is maintained by gas of degenerate electrons.
In fact, speeds in Table 7 differentiate objects by their state of matter and position in hierarchy of stars and planets. Transitional states in the range of characteristic speeds 3946 — 16711 km/s occur in collisions of stars of type of white dwarfs and main sequence stars. The result is either ejection of excessive matter from white dwarfs, or the state of white dwarf is transformed into state of neutron star. Exotic objects can appear for a short time as a result of collisions of neutron stars with other objects. According to model of quark quasiparticles quarks are quasiparticles rather than real particles, so quark stars, as well as black holes are hypothetical objects from point of view of the theory.

#### Correspondence between atomic, stellar and galactic systems

With the help of coefficients of similarity in time, mass and size, based on theory of similarity and dimensions of physical quantities it becomes possible to predict physical parameters of carriers of matter at any level. In particular, it was shown that the Solar system is similar by its properties to atom with mass number 18, and mass of electron corresponds to a planet with mass of order of mass of Uranus. Discreteness of stellar parameters was also discovered similar to 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 elements of periodic table of chemical elements, inaccuracy is only 10$^{–6}$ %. Besides abundance of corresponding atoms and stars in nature significantly coincided. For example, 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 chemical elements N, F, Na, P, which are also significantly deficient compared to adjacent chemical elements in chemical composition of the Sun and in nebulae. At the same time, iron peak, observed in abundance of chemical elements, is repeated in rise of number of stars of spectral classes B8-B9, with the masses about 3.2 solar masses.
Among other similar properties of atoms and stars, following can be noted: atoms combine into molecules and, similarly, stellar pairs and multiple stars corresponding to these atoms in mass are found; distribution of magnetic moments by intensity in atoms and their stellar analogues coincides, 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 center of specks of dust chemical elements – metals are dominating, and on periphery – non-metal elements. Similarly, it turns out that in central parts of galaxies stars have an increased number of metals, and in halo of galaxies the stars dominate which are analogues of non-metal elements and also metal-poor. For 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), one can determine corresponding characteristic mass equal to 4.4∙10$^6$ solar masses, and radius of about 371 pc. Modern estimates of masses and sizes of dwarf galaxies are really close to these values.
It is interesting that total energy of stars, consisting of their gravitational and internal thermal energy, can be calculated very accurately using Einstein formula, generalized for all objects. More precisely, total energy of a star is obtained by multiplying stellar mass by square of characteristic speed of 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 data from Table 2 for proton and data from Tables 4 and 6 one can determine coefficients of similarity between proton and Metagalaxy – analogue of the proton:
* Coefficient of similarity in mass: $~{\frac {M_{mg}}{M_{p}}}=1.4\cdot 10^{79}$.
* 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 ratios of sizes and masses between elementary particles and 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 consequence of the fact that masses and sizes of objects in 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 similarity coefficients the correlation for large numbers occurred. Connection between parameters of Metagalaxy and elementary particles is not accidental – it is mediated by hierarchical structure of Universe, when any object is similar to other objects at different levels of matter, and includes objects of lower levels of matter.

#### 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 magnetic properties. There are not many magnetic nuclei with large magnetic moments, and the same holds for magnetic stars. In this case there is a correlation between masses of magnetic stars and masses of magnetic nuclei, which are related to each other by coefficient of similarity in mass $~\Phi$. Distribution of magnetic stars and their connection with magnetic nuclei is described in discreteness of stellar parameters.
Knowing magnetic moment of electron and nuclear magneton, as well as Dirac constant as a characteristic value of angular momentum of microparticles, using similarity coefficients, one can calculate corresponding values for planets and stellar objects.. Magnetic moment of electron and 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 elementary charge, $~M_{e}$ and $~M_{p}$ are masses of electron and proton, $~{\frac {\hbar }{2}}$ is quantum spin of electron and proton, $~K_{e}$ and $~K_{n}$ are corresponding gyromagnetic ratios equal to ratio of charge to mass.
Similarly to this for relationship between magnetic moments $~P_{m}$ and spin $~I$ of stellar objects, the following is obtained:
    $~P_{m\Pi }=K_{\Pi }I_{\Pi }$,
    $~P_{ms}=K_{s}I_{s}$,
where $~K_{\Pi }$ and $~K_{s}$ are corresponding gyromagnetic ratios.
According to theory of dimensions, gyromagnetic ratios for stellar objects can be found through similarity coefficients:
    $~K_{\Pi }=K_{e}{\frac {P_{0}^{0.5}S_{0}}{\Phi ^{0.5}}}=3.69\cdot 10^{-9}$ 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.
On the summary dependence "magnetic moment – spin" for planets, stars and our Galaxy, magnetic moments of the Moon, Mercury, Earth, Jupiter and the Sun are given for two values of their spins: spin of nucleus and total spin. Crosses are usual nonmagnetic stars; the rectangle Ар is magnetic stars of spectral class A. Positions are indicated of magnetic and nonmagnetic white dwarfs, radio and X-ray pulsars, extreme black hole BH (indicated by a big point) with mass 1.414 solar masses, and bulge and Galaxy as a whole, taking into account possible spread of values. Almost all of objects are located within or on the border of a stripe, cut off by the line of stellar Bohr magneton (upper) and the line of stellar nuclear magneton (lower).
The fact that value of gravitational constant does not change much at the level of galaxies, as it follows from similarity coefficients, leads to the fact that magnetic moments of galaxies correspond to dependences between magnetic moment and 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. On a plane with logarithmic coordinates "magnetic moment - spin" one can draw lines between points for magnetic moments of electron and nucleon, and corresponding points for magnetic moments of planets and stars. This gives a slope of these lines equal to 0.7. This means dependence of the form $~P_{m}\sim I^{0.7}$, while for planets and stars there is a linear dependence $~P_{m}\sim I$.
Non-coincidence of dependence arises from different mechanisms of generating magnetic field. Let there be objects with highest rotating, having largest magnetic fields, and consisting of small particles of matter held from disintegration by molecular forces of constant magnitude. For such objects, relations $~I\sim M^{1.5}$ and $~P_{m}\sim I^{0.7}\sim M$ are obtained, where $~M$ is mass of the object. Here increase of magnetic moment is associated just with increase of mass and matter quantity. For stellar objects attractive force of matter depends on mass and radius, which gives $~I\sim M^{5/3}$ and $~P_{m}\sim I\sim M^{5/3}$. With increase of mass magnetic moment of stellar objects increases faster than of separate particles of matter. Within dynamo theory there is a formula:
    $~P_{m}\sim \omega R^{4}{\sqrt {\rho }}$,
where $~\rho$ is mass density of a body with radius $~R$, $~\omega$ is angular velocity of rotation of the body.
This formula in case of limiting rotation, on condition of equality of gravitational attraction and centripetal force, gives $~P_{m}\sim M^{4/3}$.
In electrokinetic model, in which magnetism of cosmic bodies is consequence of rotation and separation of electrical charges within a 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 radius and mass density of core of a planet, $~k_{3}\approx 3\cdot 10^{-9}$ is coefficient of proportionality between density of magnetic force and Coriolis force, $~\mu _{0}$ is vacuum permeability. One of the consequences of this is that density $~U$ of magnetic energy is proportional to density $~\epsilon$ of kinetic energy of rotation of conductive and magnetized matter: $~U={\frac {2k_{3}\epsilon }{9}}$.

#### Planetary and moon systems

At the level of planetary systems quantization of parameters of cosmic systems is manifested in applicability of Bohr atom model for calculating parameters of orbits of planets. As a result, there are formulas for specific orbital angular momenta and orbital radii of 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 orbital angular momentum of planet in orbit with number $~n$; $~M_{n}$, $~V_{n}$ and $~R_{n}$ are mass of the planet, its orbital velocity and average radius of the orbit; $~\hbar _{s}=2.8\cdot 10^{41}$ J∙s is stellar Dirac constant for ordinary stars; $~M_{pl}=6.06\cdot 10^{25}$ kg is mass of the planet corresponding to electron by theory of similarity; $~G$ is gravitational constant; $~M_{c}$ is mass of the Sun; $~K_{1}=0.5$ from correspondence with empirical data.
For planetary moons corresponding quantization of specific orbital angular momenta is also observed. In addition, it is shown that specific spin mechanical moments of proper rotation of planets in Solar system are quantized.

#### Similarity of objects

Similarity relations work most accurately between corresponding levels of matter, for example, between levels of elementary particles and stars with degenerate state of matter such as white dwarfs and neutron stars. In collisions of high-energy particles mesons often appear, which, like overwhelming majority of elementary particles, are unstable and decay. Meson of minimum mass is pion, which is 6,8 times lighter than nucleon and decays into muon and muon neutrino (antineutrino) in reaction:
π → μ + ν$_μ$.
In turn, muon decays into electron (positron) and electron and muon neutrinos in the reaction:
μ → е + ν$_е$ + ν$_μ$.
From point of view of similarity, pion corresponds to neutron star with mass 0.2 solar masses, and muon – to charged stellar object with mass 0.16 solar masses. The mass 0.16 of Solar mass is exactly equal to Chandrasekhar limit for white dwarfs of hydrogen-helium composition, at lower masses, a star that is a white dwarf is unstable. From observations one of the least massive white dwarfs SDSS J0917 +46 has 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.
Matter of such objects is unstable and therefore such stars must undergo catastrophic changes of their state in periods of time 10$^5$ – 10$^7$ years. First, low massive neutron star decays in an explosive way with formation of charged and magnetized object and with emission, which is 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 neutron star undergoes new transformation, with ejection of charged shell, which is analogue of electron.
In described picture hadrons are similar to neutron stars in unstable, stable or excited states. The latter refers mainly to the particles-resonances, which by their short lifetime correspond to massive, very hot and unstable neutron stars. In substantial neutron model it is assumed that analogues of neutrons are neutron stars with the masses about 1.4 solar masses, and according to the substantial proton model analogue of protons are magnetars.
In the world of compact stars electrons also have their analogue. In hydrogen atom the most probable location of electron in ground state is Bohr radius. When multiplying the Bohr radius by coefficient of similarity in size Р' , the obtained value is of the order of 10$^9$ m. This value is exactly equal to distance from neutron star at which planets decay near stars due to strong gravitational field. This distance is called Roche limit. Based on the foregoing, nucleons become similar to neutron stars, while electrons in atom correspond to discs discovered near X-ray pulsars, which are main candidates for magnetars. In this case sizes of discs coincide with Roche radius near the neutron star. Electrons in form of discs are considered in the substantial electron model, which allows us to explain the origin of the electron spin.
Using similarity relations, one can estimate radii of elementary particles, their binding energies, characteristic angular momentum, and characteristic spin. For hadrons, based on analogy of their matter structure with neutron stars, ratio is used between radius $~R$ and mass $~M$ of hadron:
    $~R=R_{p}({\frac {M_{p}}{M}})^{1/3},$
where $~R_{p}$ and $~M_{p}$ are radius and mass of proton.
Table 8 shows masses and radii of proton, pion and muon. Radius of muon is found based on radius of white dwarf corresponding to muon.
Table 8. Characteristics of proton, pion and muon :
| Particle | Mass-energy, MeV | Mass, 10$^{–27}$ kg | Radius, 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 |
Masses of particles in Table 8 are obtained by dividing mass-energy, converted from MeV to Joules, by squared speed of light. Characteristic angular momentum of a particle is given by:
    $~L_{x}=MC_{x}R,\qquad \qquad (2)$
and characteristic speed $~C_{x}$ of particle’s matter is calculated by the formula (1):
    $~W=MC_{x}^{2}={\frac {\delta \Gamma M^{2}}{2R}},$
here $~\delta =0.62$ for objects of the type of nucleons and neutron stars, $~\Gamma$ is strong gravitational constant.
For proton there is an approximate formula $~h=2M_{p}cR_{p}$, from which for characteristic spin of proton it follows: $~{\frac {\hbar }{2}}={\frac {M_{p}cR_{p}}{2\pi }}$, where $~c$ is speed of light and characteristic speed of proton matter, $~h$ is Planck constant. A similar approach for characteristic spin of muon yields 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 $~\delta =0.6$.
Сharacteristic spin of muon exceeds the value of quantum spin ħ/2, accepted for fermions and leptons. For pion with its radius according to Table 8, spin is equal to 0.05 ħ, i.e. considerably less than minimum spin of fermion, equal to ħ/2. As a consequence, quantum spin of pion is assumed to be zero, and the pion is considered as boson.
With the help of relation (2) one can estimate characteristic angular momentum to our Galaxy Milky Way. Assuming that mass of the galaxy is 1.6•10$^{11}$ solar masses, radius is 15 kpc, and characteristic speed of stars is 220 km/s, the angular momentum value obtained is 3.3•10$^{67}$ J•s. This is close enough to value 9.7•10$^{66}$ J•s according to known data.

#### Gravitational constants

As stated above, based on principle of similarity at the level of elementary particles strong gravitation is introduced into consideration, and strong gravitational constant $~\Gamma$ is significantly different from ordinary gravitational constant $~G$. Action of strong gravitation and gravitational torsion fields of elementary particles can explain strong interaction based on gravitational model of strong interaction. For ratio of gravitational constants the following formula is valid:
    ${\frac {G}{\Gamma }}={\frac {P_{0}S_{0}^{2}}{\Phi }}={\frac {P'S'^{2}}{\Phi '}}$,
which contains coefficients of similarity in size, speed and mass for normal and neutron stars, respectively, taken with respect to hydrogen system.
This formula should be understood in following way, that in transition from one matter level to another, effective gravitational constant changes in law of gravitation between objects. As an example, effective gravitational constant for galaxies is estimated. From Table 4, coefficient of similarity in mass between normal galaxies and main sequence stars is $\Phi _{g}=D_{\Phi }^{2}=1.46\cdot 10^{11}$. Similarly, from Table 6 coefficient of similarity in size equals $P_{g}=D_{P}^{6}=2.33\cdot 10^{11}$. Average speeds $~V_{s}$ of motion of stars in spiral galaxies of low mass apparently do not exceed characteristic speed $~C_{s}=220$ km/s of motion of matter in the star of minimum mass. Hence, coefficient of similarity in speed $~S_{g}={\frac {V_{s}}{C_{s}}}$ is close to unity and for effective gravitational constant at the level of galaxies with accuracy up to a coefficient of the order unity there is 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 rapid decrease of gravitational constant at the level of galaxies, obtained by R. Oldershaw.
In general, in transition to a higher scale level of matter decrease of effective gravitational constant is predicted, based on Le Sage's theory of gravitation and nesting of matter levels into each other.

#### Natural units

It is known that with the help of three independent physical quantities it is possible calculate characteristic parameters of mechanical system. For example, Planck units of mass, length, time, energy, momentum, etc. are based on Dirac constant $~\hbar$, speed of light $~c$ and 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 International System of Units includes Boltzmann constant $~k$ and factor $~{\frac {1}{4\pi \varepsilon _{0}}}$, where $~\varepsilon _{0}$ is electric constant. Planck units are used in quantum physics, where $~\hbar$ is characteristic angular momentum, but since ordinary gravitation constant $~G$ in microworld must be replaced by strong gravitational constant, Planck units do not uniquely characterize any level of matter and only formally refer to natural units of physical quantities. Only Planck charge, which does not contain gravitational constant, is close to 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 fine structure constant.
At the same time, using at the level of main sequence stars stellar Planck constant $h_{s}=h\Phi P_{0}S_{0}=1.76\cdot 10^{42}$ J∙s, stellar velocity $~C_{s}=220$ km/s, gravitational constant, as well as proportionality coefficients of the order of unity associated with geometry of shape of sphere and distribution of matter, then with their help it is possible to obtain values that are quite close to parameters of a star of minimum mass:
* Mass $~M_{s}={\sqrt {\frac {h_{s}C_{s}}{G}}}=7{.}62\cdot 10^{28}$ kg.
* Radius $~R_{s}={\sqrt {\frac {Gh_{s}}{C_{s}^{3}}}}=1{.}05\cdot 10^{8}$ m.
* Characteristic time $~t_{s}={\sqrt {\frac {Gh_{s}}{C_{s}^{5}}}}=477$ s.
* Characteristic angular momentum $~L_{s}=M_{s}R_{s}C_{s}=h_{s}$.
* 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/m$^3$.
* 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.
* Gravitational acceleration $~g_{s}={\frac {GM_{s}}{R_{s}^{2}}}={\sqrt {\frac {C_{s}^{7}}{Gh_{s}}}}=461$ m/s$^2$.
* Absolute value of 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.
* Maximum luminosity (energy emission power) $~W_{s}={\frac {E_{s}}{t_{s}}}={\frac {C_{s}^{5}}{G}}=7{.}7\cdot 10^{36}$ W.
* 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 stellar Boltzmann constant.
Time $~t_{s}$ here characterizes the time required to cross radius of the star at the speed $~C_{s}$, and this stellar speed is characteristic speed of matter inside the star. Substituting expression for average density in formula for characteristic time gives approximate relation for time of fall of the matter in gravitational field : $~t_{s}\approx {\sqrt {\frac {3}{4G\rho _{s}}}}$. Product of absolute value of total energy and characteristic time gives relation similar to Heisenberg uncertainty relation: $~E_{s}t_{s}=h_{s}$. Maximum luminosity of the star is close to luminosity of Galaxy, as well as to luminosity of supernova.
As independent quantities for natural units, which characterize objects of different matter levels, it is possible also take characteristic mass, speed, and angular momentum. For example, assuming as primary mass $~M_{s}$, speed $~C_{s}$, and angular momentum $~h_{s}$, gravitational constant is expressed in the form: $~G\approx {\frac {h_{s}C_{s}}{M_{s}^{2}}}$, and then this expression is used in the formulas above. Thus, it is possible to estimate parameters of a main sequence star through its mass, characteristic speed of matter, and characteristic spin of this star.
Passing from stars to atoms, and using as basic values Planck constant $~h$, speed of light $~c$, Boltzmann constant $~k$, multiplier $~{\frac {1}{4\pi \varepsilon _{0}}}$ and strong gravitational constant in the form
    $\Gamma ={\frac {e^{2}}{4\pi \varepsilon _{0}M_{p}M_{e}}}={\frac {\alpha hc}{2\pi M_{p}M_{e}}}=1{.}514\cdot 10^{29}$ m$^3$•s$^{–2}$•kg$^{–1}$,
where $~M_{p}$ is proton mass, $~M_{e}$ is electron mass, there is estimation parameters of proton as main object at the level of elementary particles:
* Mass $~M'_{p}\approx {\sqrt {\frac {hc}{\Gamma }}}=1{.}15\cdot 10^{-27}$ kg.
* Radius $~r'_{p}\approx {\frac {h}{2M'_{p}c}}={\sqrt {\frac {\Gamma h}{4c^{3}}}}=0{.}96\cdot 10^{-15}$ m.
* Characteristic time $~t'_{p}={\sqrt {\frac {\Gamma h}{4c^{5}}}}=3{.}2\cdot 10^{-24}$ s.
* Characteristic angular momentum $~L'_{p}=2M'_{p}r'_{p}c=h$.
* 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/m$^3$.
* 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.
* Gravitational acceleration $~g'_{p}={\frac {\Gamma M'_{p}}{{r'}_{p}^{2}}}={\sqrt {\frac {16c^{7}}{\Gamma h}}}=1{.}9\cdot 10^{32}$ m/s$^2$.
* Absolute value of total energy $~E'_{p}=M'_{p}c^{2}={\sqrt {\frac {c^{5}h}{\Gamma }}}=1\cdot 10^{-10}$ J.
* Maximum luminosity (energy emission power) $~W'_{p}={\frac {E_{p}}{t_{p}}}={\frac {2c^{5}}{\Gamma }}=3{.}2\cdot 10^{13}$ W.
* Maximum temperature $~T'_{p}\approx {\frac {E'_{p}}{3k}}={\frac {1}{3k}}{\sqrt {\frac {c^{5}h}{\Gamma }}}=2{.}5\cdot 10^{12}$ K.
* 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.
Using natural units, similar to main sequence stars, parameters of galaxies and even metagalaxies can be obtained. For example, taking from Table 4 mass of metagalaxy 2.49∙10$^{23}$ М$_c$, and from Table 6 its radius 14.05 Gpc, average mass density $\rho _{M}=1.45\cdot 10^{-27}$ kg/m$^3$ is found, and characteristic time of matter relaxation in the field of regular forces and time of free fall under 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 Universe according to Big Bang model. In addition, such arguments in favor of Big Bang, as cosmic microwave background radiation and Hubble's law can be understood without using the idea of Big Bang. All other arguments in favor of Big Bang can have other explanations, which subjects to well-grounded and many-sided criticism the concept of explosion of Universe.

#### Galaxy as thermodynamic system

From 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 concentration of stars increases rapidly in direction towards center of the Galaxy, average density $~\rho$, understood as average mass of stars per unit volume, also increases. Dependence of density on current radius in International System of Units is given by:
    $~\rho =4.4\cdot 10^{14}R^{-1.71}$,
where galactic radius $~R$ is substituted in meters.
According to this dependence it is found that air under normal conditions has the same concentration of molecules, which is equal to concentration of stars near galactic radius 6.4∙10$^{16}$ m or 2.1 pc. Almost entire volume of the Galaxy is similar to collisionless and very rarified gas. In the center, with the radius 0.047 pc concentration of stars reaches concentration of such light and solid substance as coke. Average gas pressure from the stars in the Galaxy is given by:
    $~p={\frac {\rho V^{2}}{3}}$,
where $~V$ is average velocity of stars.
If take into account data on speeds of stars depending on galactic radius in the range from 200 pc to 10 kpc (average speed is about 235 km/s), for pressure approximate formula in SI units is:
    $~p=1.8\cdot 10^{10}\rho$.
Linear dependence of pressure on mass density means that the state of stellar gas is isothermal. Despite formation of stars and pressure of Galaxy, its temperature changes little, as all the excess energy is carried away by electromagnetic emission. Temperature $~T_{g}$ of Galaxy can be estimated in different ways:
* By formula for internal energy of Galaxy as energy of motion of stars in the form $~E={\frac {3kNT_{g}}{2\mu }}$, where $~E=2.5\cdot 10^{52}$ J is estimate of energy by virial theorem, $~k$ is Boltzmann constant, $~N$ is total number of nucleons, $~\mu =0.64$ is number of nucleons per gas particle as it is accepted for the Sun.
* By formula for luminosity of Galaxy of the form $~L=\Sigma _{s}A_{g}T_{g}^{4}$, where $~L=7.6\cdot 10^{36}$ W, $~A_{g}$ is area of Galaxy disc, $~\Sigma _{s}={\sigma \Phi S_{0}^{3}}{P_{0}^{3}}=9.3\cdot 10^{-30}$ W/(m$^2$∙K$^4$) is stellar Stefan–Boltzmann constant, found through Stefan–Boltzmann constant $~\sigma$ and coefficients of similarity in mass, speed and size for main sequence stars.
* By formula for pressure of stellar gas of the form $~p=nK_{s}T_{g}$, where $~n$ is concentration of stars, $~K_{s}=AK_{ps}$, where $~K_{ps}=1.18\cdot 10^{33}$ J/K is stellar Boltzmann constant, $~A$ is mass number of typical stars, characterizing Galaxy on the average.
Averaged temperature of stellar gas in Galaxy is about $~T_{g}=2\cdot 10^{6}$ K. Another way to determine the temperature of Galaxy is associated with generalized gas law for stellar gas:
    $~pV_{g}={\frac {M_{g}R_{st}T_{g}}{M_{sm}}}$,
where $~V_{g}$ and $~M_{g}$ are volume and mass of Galaxy, $~R_{st}$ is stellar gas constant, $~M_{sm}$ is mass of one stellar mole of substance, consisting of stars.
For ordinary gas constant there is a relation: $~R=kN_{A}$, where $~N_{A}=6.022\cdot 10^{23}$ mole$^{–1}$ is 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 main sequence stars of minimum mass.
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 mass of one stellar mole of substance, consisting of main sequence stars of minimum mass.
Typical stars in our Galaxy are stars with mass equal to half of mass of the Sun, and with mass number $~A=9$. Left part of the generalized gas law for stellar gas can be expressed through energy of Galaxy in the following form:
    $~pV_{g}={\frac {2E}{3}}={\frac {M_{g}V^{2}}{3}}$.
After substituting the quantities into right side of the generalized gas law for stellar gas there is found the next:
    $~{\frac {M_{g}R_{st}T_{g}}{M_{sm}}}={\frac {M_{g}AR_{pst}T_{g}}{AM_{psm}}}={\frac {M_{g}R_{pst}T_{g}}{M_{psm}}}$.
Kinetic temperature of stellar gas of Galaxy is found from comparison of left and right sides of the generalized gas law with average rotation speed of stars in 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 ratio between energy of Galaxy, energy of stars and their velocity, principle of locality of stellar speed is formulated: "Average speed of stars relative to system in which they were formed, does not exceed stellar speed $~C_{s}{\frac {A}{Z}}$, where $~A$ and $~Z$ are mass and charge numbers, corresponding to typical main-sequence stars in this system."

## Similarity of forms and energies of phenomena

Similarity of matter levels is evident in coincidence of forms inherent in objects and phenomena at different scale levels. Depending on characteristics of accepted model of similarity, different researchers explain in their own way occurrence and recurrence of the same forms.
Sergey Sukhonos in his works gives examples of fractality when shape of even small parts of an object to a large extent coincides with shape of the object itself. He also lists manifestations observed in space of dual mutually complementary structures: spiral (flat) and elliptical (round) galaxies; subdwarfs as primary stars of Galaxy with a deficit of heavy elements, and ordinary main sequence stars; large outer and small inner planets of Solar system; monocentric and polycentric structures at different levels of matter, emerging in processes of synthesis and division. Objects located on scale axis of sizes have a shape that is periodically repeated, with ratio of sizes of these objects being of the order of 10$^{20}$. This allows the dominant forms to be modeled by a periodic function in the form of some wave. The reason of periodicity is assumed existence of fourth spatial dimension (see scale dimension). The latter can be interpreted as the fact that objects can move not only in 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, situation is periodically arisen when, due to surrounding conditions, initial form of corresponding objects is preserved despite the action of shape-changing factors.
Robert L. Oldershaw draws attention to distribution of matter in space, where main mass of matter consists of hydrogen and light elements. The same is observed for stars – according to initial function of masses, the most common stars are dwarf stars. Among galaxies small galaxies are also dominating. Another observation is associated with coincidence of geometrical forms of functions of electron density in atom for different energy levels with corresponding orbital angular momenta of electron and their projections on preferred direction on one hand, and shapes of stellar objects on the other hand. The examples are symmetrical conical jets and equatorial ejections from the star Eta Carina, ring planetary nebula Shapley 1, spherical planetary nebula Abell 39 and other similar objects. Oldershaw considers planetary nebulae to be the analogues of fully ionized atoms.
Neutron stars, such as GRB, producing short and powerful gamma-ray bursts in energy range 10$^{43}$ – 10$^{44}$ J, Oldershaw compares with gamma-radioactive nuclei. Energies of gamma-ray quanta from the nuclei lie in range from 10 keV to 7 MeV. Applying multiplication by coefficient of similarity in energy, which coincides with coefficient of similarity in mass X = Λ$^D$ = 1.7∙10$^{56}$, he obtains energy range from 2.72∙10$^{41}$ J to 1.87∙10$^{44}$ J, where gamma-ray bursters GRB also fall. For variable stars, such as RR Lyrae, Oldershaw finds correspondence between oscillation period of their brightness and radius of stars which is similar by the form to third Kepler law for planets of Solar system $~p^{2}=kr^{3}$ and relation for electrons in Rydberg states. By recalculation of the coefficient $~k$, with the help of coefficients of similarity, he makes these stars similar to excited states of helium atom He(4), in which electron transitions occur between levels 7 ≤ n ≤ 10 and l ≤ 1. Similarly, variable stars such as Delta Scuti (δ Scuti) are considered to be analogues of excited atoms of carbon, oxygen and nitrogen in states with 3 ≤ n ≤ 6 and 0 ≤ l ≤ n-1, and stars such as ZZ Cetis – analogues of excited states of ions from helium to boron.
In Table 9 Oldershaw compares axial rotation periods and natural oscillation periods of typical objects at the levels of atoms, stars and galaxies.
Table 9. Characteristic rotation periods and natural oscillation periods of objects at levels of atoms, stars and galaxies :
| Object | Rotation period | Natural oscillation period |
| Atomic nucleus | 5∙10$^{–20}$ s | 1∙10$^{–21}$ s |
| Neutron star | 3∙10$^{–2}$ s | 5∙10$^{–4}$ s |
| Active galaxy | 1∙10$^{16}$ с (3∙10$^8$ years) | 2.5∙10$^{14}$ с (8∙10$^6$ years) |
Characteristic rotation periods of active galaxies are about 10$^8$ years, and oscillations are determined by periods of recurrence of significant ejections of matter from their nuclei, equal about 10$^7$ years. Period of natural oscillations for neutron stars is associated with periods of pulsations of waves propagating in stellar matter after collision with other bodies. These times for various objects are related by coefficient of similarity in time equal to Λ = 5.2∙10$^{17}$.
Sergey Fedosin describes at all levels of matter, where gravitational forces dominate, hydrogen systems, consisting of main object and a moon (satellite), with the same difference in their masses as between proton and electron. Hydrogen systems are as numerous and widely spread in Universe, as hydrogen atoms. Values of similarity coefficients according to Fedosin derived from similarity of hydrogen systems are different from values of coefficients according to Oldershaw. In particular, coefficient of similarity in energy for main sequence stars equals product of coefficient of similarity in mass and square of coefficient of similarity in speed: $K_{E}=\Phi S_{0}^{2}=3.6\cdot 10^{49}$, and for compact objects, such as neutron stars, coefficient of similarity in energy is equal to ${\in }'={K_{E}}'={S'}^{2}=8.6\cdot 10^{55}$.
Gamma-ray quanta, emitted by atomic nuclei under radioactivity, have ordinary energies W from 10 keV to 5 MeV, with period of electromagnetic oscillation in range:
    $t={\frac {h}{W}}=4.1\cdot 10^{-19}-8.3\cdot 10^{-22}$ s.
To find corresponding energies and periods at the level of stars, it is necessary to multiply energies and periods of gamma-quanta oscillations by energy similarity coefficient $K_{E}$ and by time similarity coefficient $\Pi _{0}={\frac {P_{0}}{S_{0}}}=7.41\cdot 10^{25}$, respectively: energies – from 5.7∙10$^{34}$ J to 2.8∙10$^{37}$ J, periods – from 352 days to 17 hours.
These energies and periods conform to 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. Energy of expansion of planetary nebulae correspond by energy to alpha decay, and nova outbursts – to beta decay of atomic nuclei.
If energies and periods of oscillations of gamma quanta from atomic nuclei are multiplied by similarity coefficients ${K_{E}}'$ and $\Pi '={\frac {P'}{S'}}=6.1\cdot 10^{19}$, corresponding energies and periods for objects such as neutron stars are found: energies – from 1.4∙10$^{41}$ J to 6.9∙10$^{43}$ J, periods – from 25 s to 0.05 s.
These energies and periods of outbursts are quite close to values characteristic of gamma-ray bursters. The energy of gamma-ray burst from magnetar SGR 1806-20, recorded on December 27, 2004, is estimated by the value 4∙10$^{39}$ J. Following the outburst radio emission was observed from expanding matter at the speed about 0.2 of speed of light. In gamma-ray burster GRB 080319B total energy of outburst in all emission ranges was equal up to 10$^{40}$ J. Although nature of atomic nuclei and stars differs significantly, given examples with energies of periodic processes show another aspect of similarity of these matter levels.
Active galactic nuclei and processes occurring in them are considered by Fedosin as consequence of large number of neutron stars in centers of galaxies. For nucleus of quasar 3C 273 it is assumed that volume with radius about 10$^{13}$ m contains mass up to 10$^9$ solar masses, producing emission with luminosity about 2∙10$^{40}$ W. If divide this luminosity by number of stars, the value 2∙10$^{31}$ W is obtained, which is close to critical luminosity of neutron stars with accretion of matter to their surface. In this case, the phenomenon of quasars and active galactic nuclei can be explained by accumulation of a large number of neutron stars. These stars have strong magnetic fields and can have magnetic moments, aligned in one direction, creating regular overall magnetic field. Due to this field powerful jets of ionized matter are possible, which are often observed near active nuclei. Luminosity of 3C 273 can vary significantly during of one day or more time. Ratio of size of active nucleus 10$^{13}$ m to time interval of one day gives the speed 10$^8$ m/s. This speed can be interpreted as the speed of outburst propagation in the nucleus which occurs as a result of interaction of large amounts of relativistic plasma with neutron stars. The plasma can fall on active nucleus at high velocities under influence of gravitational forces. On the other hand, if neutron stars in the active nucleus are retained by proper gravitation and centripetal forces, they must rotate at speeds almost up to 10$^8$ m/s.
Another example of similarity is use of Heisenberg uncertainty principle not only at the level of elementary particles, but also at the level of stars and even galaxies. Uncertainty relation for change in a process energy $~\Delta E$ and time $~\Delta t$ of its change has the form:
    $~\Delta E\Delta t\geq L_{x}$,
where $~L_{x}$ is characteristic angular momentum of object in the process.
In order to conform to quantities accepted in quantum mechanics, for spin angular momentum I relation $~L_{x}=4\pi I$ is assumed, and for orbital angular momenta L relation $~L_{x}=2\pi L$ is used. In Galaxy total energy of stars in common gravitational field, taking into account orbital galactic rotation, are approximately equal to total energies of stars in their proper gravitational field, without taking into account fields of other stars. Considering these energies $~E$ and time of formation of stars (Kelvin-Helmholtz time $~t_{KH}$) from separate gas clouds leads to the fact that for a typical star following relation holds:
    $~Et_{KH}\approx h_{o}=2.1\cdot 10^{57}$ J/s,
where $~h_{o}$ is stellar orbital angular momentum.
In addition, lifetime of a star of main sequence on the average exceeds 122 times time $~t_{KH}$, which can be explained by time of stellar core growth due to thermonuclear reactions in which mass-energy is released with the value up to 1/130 of rest energy of matter. Relation for $~h_{o}$ also reflects change of energy in process of cooling of neutron stars. Instead of $~h_{o}$ it is possible substitute characteristic spin angular momentum of a star, in supernova explosion of which a neutron star is formed, then for this angular momentum uncertainty relation for total energy of the neutron star (of the order of 2∙10$^{46}$ J) and time of emission of this energy (several seconds) will be valid.

## Combined scale symmetry

Transition from one matter level to another can be made directly in equations describing interaction and motion of carriers or state of matter. It turns out that simultaneous substitution in these equations of masses, sizes and speeds of carriers of one level with masses, sizes and speeds of carriers of another level of matter, leaves the equation invariant with respect to this substitution. Thus, new combined symmetry is revealed, which follows from theory of similarity and is called SPФ symmetry. The SPФ transformations, as well as transformations of CPT symmetry, leave laws of bodies’ motion unchanged.

## Philosophical justification

Detailed philosophical analysis of Theory of Infinite Hierarchical Nesting of Matter and similarity of matter levels was carried out in 2003. At each level of matter, characteristic main carriers and boundary points of measure can be identified. Transitions from one matter level to another are carried by the law of transition from quantity to quality, when number of carriers in an object exceeds permissible limits of measure, typical for this object. At different spatial levels of matter similar fractal structures, carriers of matter and field quanta are found. These objects as elements are included in hierarchical structure of Universe, repeating in similar natural phenomena, ensuring unity and integrity of Universe, revealing symmetry of similarity.
From Le Sage's theory of gravitation follows existence in electrogravitational vacuum of a multitude of invisible relativistic particles with high energies, penetrating all bodies and creating electromagnetic and gravitational forces. Parameters of these particles are determined by similarity of matter levels, while motion and interaction of particles with matter and fields is described by standard physical laws. This is confirmed by calculations, according to which energy and momentum of vacuum particles can be used to create thrust in new-generation engines for spaceships.
Laws of similarity and hierarchy of matter levels are valid for living systems too. It is proved that masses and sizes of all known living organisms correlate with masses and sizes of carriers of corresponding levels of matter, repeating them. This demonstrates complementarity of living and non-living, draws a conclusion about eternity of life as an integral part of eternity of Universe, and clarifies the question of origin of life.
In addition, infinite nesting of living is discovered – inside autonomous living organisms at each level there must be living structures of smaller sizes and of lower scale levels. They are true builders and creators of large organisms, controlling their reactions and vital functions as huge complex systems. Presence of nesting of different types of living is illustrated in typical example – in human body there are so many bacteria that its total mass can reach 0.2 kilograms. Cells in multicellular organisms and bacteria are approximately equal in size, but the bacteria can exist in environment autonomously for a long time. Viruses and smallest prions can cause various diseases, when their programs of development are contrary to vital functions of multicellular organism. Prions contain a certain number of atoms, but 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 modern observational facilities, control all living beings exceeding them in size and set programs of their existence.