Physics/Essays/Fedosin/Quantization of parameters of cosmic systems

Quantization of parameters of cosmic systems is a property of the systems observed in space, which have relatively stable fixed stationary states and in which transitions between these states are possible under the influence of external disturbances or in case of energy loss. The result of the transition between stationary states is the quantized change of energy and characteristic angular momentum of the systems. The typical examples are satellite systems – atoms, planetary star systems, systems of normal and dwarf galaxies. By definition, it is considered that satellites are less massive than the main objects. In extreme cases the masses of the system's components can be equal as in a diatomic molecule consisting of atoms of the same chemical element or as in corresponding binary stars. In systems with numerous objects quantization can acquire dynamic character and is determined by long-range forces between the objects.

Quantization is most clearly revealed in systems containing compact objects with degenerate state of matter, such as atomic nuclei and neutron stars. These objects have discrete physical properties and are usually the main objects in satellite systems. In particular, the mass of the atomic nucleus is proportional to the number of nucleons, and at the level of stars we observe discreteness of stellar parameters and the similarity between atoms and stars, including correspondence between the masses and the abundance in nature.

The charge and the mass of electrons in atoms are not arbitrary values, but to a large extent are determined by the history of electrons' emergence. Analysis of beta decay in the substantial electron model and in the substantial neutron model shows that the properties of electrons are secondary to the properties of nucleons. At the same time there are connections between the mass, the charge and the radius of a proton, which are determined by the properties of the matter and the equation of its state and which lead to discreteness of the proton's properties.

From this it follows that discreteness of the fundamental properties of main objects and their satellites, which arises in the course of co-evolution under action of the fundamental interactions, leads to the repeating structure of satellite systems at different levels of matter and to quantization of their parameters. Manifestation of discreteness of objects' properties is existing in space of hierarchically nested levels of matter, the masses and sizes of the carriers of which are related to each other by the law of geometric progression. According to the similarity of matter levels and SPФ symmetry, similarity relations can be established between the corresponding objects and phenomena and the physical quantities characterizing them can be predicted. This allows connecting different forms of quantization in the framework of Infinite Hierarchical Nesting of Matter.

Quantization in atomic systems
The idea of quanta is fundamental in quantum mechanics that describes the behavior of atoms and elementary particles. Historically, the first quantum model was Bohr model, in which the main role is played by the quantum of action in the form of Dirac constant $$~\hbar=1.054 \cdot 10^{-34}$$ J∙s. The hydrogen atom in Bohr model is the representative of hydrogen system at the level of atoms, in which the electron in the ground state has the orbital angular momentum equal to $$~\hbar$$. In atoms and ions the following quantities are quantized:
 * The number of nucleons in atomic nuclei;
 * The masses and charges of atomic nuclei;
 * The number of electrons;
 * The masses and charges of electron shells;
 * The energy of electrons in different states;
 * The orbital and spin angular momenta of atomic nuclei and electrons and their projections on the selected direction;
 * The magnetic moments of atomic nuclei and electrons and their projections on the selected direction.

Quantization usually takes place in those cases when transitions of a system are possible from one state to another and back, and the states themselves should be somehow fixed. The high stability of the spectral lines during emission of photons from atoms shows that the stationary states of atoms are controlled by special mechanisms. As a result of their action an atom cannot move from one state to another excited state, if the energy of the incident photon or of excitation is not sufficient for transition between the states. The conditions for emerging of stationary states are the following:
 * 1) Coinciding in magnitude of the energy fluxes of electromagnetic and gravitational fields and the kinetic energy flux of the electron cloud matter. In this case, the energy momentum of the field is not transmitted to the momentum of the kinetic energy of the electron matter's rotation around the nucleus and the rotation is stable. At the same time due to the absence of change in rotation of the electron matter, there is no emission from the electron.
 * 2) Axially symmetric shape of electron cloud, due to which there is no electromagnetic emission from the electron cloud despite its rotation.
 * 3) The structure of various electron shells in the form of a set of parallel disks of approximately same size within each shell. This allows the electrons, as charged and magnetized disks consisting of matter of low density, to interact with each other according to the Pauli exclusion principle and to change electron configuration in chemical reactions.
 * 4) The balance of electron clouds in atoms, which is caused by: the electrostatic forces between the nucleus and each cloud, as well as between neighboring electron clouds; intrinsic electrostatic forces inside each charged cloud; strong gravitation, mainly between electrons and the nucleus; the centripetal forces of rotation. The equality of the Coulomb force and the strong gravitation allows us to estimate the strong gravitational constant.

The existence of stationary states of atoms leads to quantization of the energy and the angular momentum of electron states and to discreteness of atomic spectra. Consideration of multielectron atoms shows that as the mass and charge of the nucleus increase, the electron shells get closer to each other, producing reciprocal pressure, and the inner shells get closer to the nucleus while the electrons' energy increases. In electron transition from one energy state to another underlying state, the angular frequency of electromagnetic emission is associated with the rate of energy change $$~E$$ during the change of the angular momentum $$~L$$:
 * $$ \omega = \frac {dE}{dL} $$.

The typical change of the angular momentum of the electron during transition between the energy levels is $$~\hbar$$, due to which it is assumed that the energy of the emitted photon as an emission quantum equals $$ E = \hbar \omega $$.

Quantization of the number of nucleons in atomic nuclei is closely related to the discreteness of the nuclear masses and to Mattauch-Shchukarev isobar rule, according to which after the replacement of a neutron by a proton (or a proton by a neutron) in the nucleus of a stable isotope, a radioactive isotope will appear.

The law of charge conservation holds almost exactly in all known phenomena, and the minimum value of the electric charge at the atomic level is the elementary charge $$~e=1.602\cdot 10^{-19}$$ C. With the help of the Planck constant and the elementary charge we calculate the magnetic flux quantum $$~\Phi_0 = \frac {h}{2e} =2.067\cdot 10^{-15}$$ Wb, which is found in experiments with superconductors. In quantum mechanics this quantum is associated with the motion of charges in the form of Cooper pairs, but it has also classical explanation.

In physics such phenomena are also associated with quantization as Bohr magneton, nuclear magneton, multiplicity (chemistry), radioactive decay, nuclear magnetic resonance, electron paramagnetic resonance, Zeeman effect and many other phenomena.

In chemistry the idea of quanta is revealed as valence (chemistry) and is expressed in the law of matter amount conservation, Faraday's laws of electrolysis and other laws.

Planetary systems
If at the micro-level quantization is found in many phenomena, at the macro-level or at the level of stars it may seem unexpected. However, the conditions for the emergence of quanta can occasionally occur at various levels of matter. For example, the planetary systems of stars and satellite systems of planets which are similar to atoms by the number of objects and the nature of acting forces.

Titius–Bode law
The Titius–Bode law describes the empirical dependence of the mean orbital radius of the planets of the Solar system on the planet's number:
 * $$ ~r_n =0.1 \cdot (4+3\cdot 2^n) $$ a.u.,

where $$~n= - \infty $$ is for Mercury, $$~n= 0{,}1{,}2 $$ is for to Venus, Earth and Mars, $$~n= 3{,}4{,}5{,}6 $$ for the asteroid belt, Jupiter, Saturn and Uranus, respectively. Although the law is violated for Neptune, in which case instead of the distance to Neptune it provides the distance to Pluto with $$~n= 7 $$, the law demonstrates that distances to the planets increase approximately exponentially. Dependency on $$~n $$ can be considered as a form of quantization of permissible orbital radii of planets. One justification for the law is the idea that the boundaries of zones in which the planets once were formed, starting with Venus, are determined by the formula:
 * $$ ~r_k =0.1 \cdot (4+ 2^k) $$ a.u.,

where $$~k=1{,} 2{,} 3 ... $$

Then based on the Titius–Bode law we can find the positions of planets as arithmetic mean values between two adjacent boundaries:
 * $$ ~r_n = \frac {r_k + r_{k+1}}{2} $$,

with $$~n=k-1 $$.

Another explanation involves the influence of gravitational energy fluxes, arising from the static gravitational field of the Sun and its gravitational torsion field from its rotation, on the fluxes of kinetic energy of rotation of the gas-dust matter of protoplanets during their accumulation.

In some orbits the energy fluxes become aligned, which leads to stable stationary states, in which the matter can accumulate more efficiently, subsequently forming planets. This process is equivalent to the emergence of stationary states in atoms, leading to quantization of energy levels and angular momentums of electrons. As a consequence of the states' quantization, orbital resonance is possible between the close planets, in which their orbital periods are related to each other as small integers.

Using Schrödinger equation
Since the states of the hydrogen atom in quantum mechanics are accurately calculated by solving the Schrödinger equation, the same approach was used to model the permissible orbit radii of the planets around the Sun and stars. Nearly for all the planets we were able to choose the appropriate quantum numbers $$~n$$ and $$~l$$, characterizing the energy and angular momentum. Besides, it appears that many planetary systems of different stars have similar structure in respect of distribution of orbital radii of planets.

Approximately the same results were obtained in a series of researches in the solution of the generalized Schrödinger equation. For the orbital radii of planetary systems a formula was found:
 * $$~r_n= \frac {G M n^2}{\omega^2} $$,

where $$~ G $$ is the gravitational constant, $$ ~M $$ is the mass of the central body (a star in the planetary system, a planet in the satellite system), $$~ \omega $$ is a constant with the dimension of velocity, which depends on the properties of the system and is equal to the orbital velocity at the primary (first) orbit. In the Solar system $$ ~\omega =144.7$$ km/s is for the inner planets of Mercury, Venus, Earth and Mars, for which $$~ n =3{,}4{,}5{,}6$$, respectively. For Jupiter, Saturn, Uranus, Neptune and Pluto $$~n=2{,}3{,}4{,}5{,}6$$ at a lower value $$ ~\omega=28$$ km/s. Different values $$~ \omega$$ for inner and large planets in this theory can be explained by the process of protoplanetary matter accumulation – first large accumulation zones were formed and then in the inner zone the secondary division into smaller areas took place, where small planets appeared.

Quantization of the orbital radii of planets
The principle of the angular momentum quantization and its increasing in proportion to the number $$~n$$ is used to show that the inner and outer planets of the Solar system are divided into two separate groups. One of the results is the formula for the orbital radii of planets:
 * $$~r_n= r_1 n^2, \qquad \qquad (1)$$

where $$~ r_1$$ is the radius with $$~n=1$$. For the inner terrestrial planets $$~ r_1\approx 6.36 \cdot 10^9$$ m and the orbits with $$~n=1$$ and $$~n=2$$ are empty due to the assumed influence of the Sun, for Mercury $$~n=3$$, for Venus $$~n=4$$, for the Earth $$~n=5$$, for Mars $$~n=6$$, for the asteroid Ceres $$~n=8$$. For the outer planets, starting with Jupiter, it is assumed that $$~ r_1 \approx 1.75 \cdot 10^{11}$$ m, $$~n=2{,}3{,}4{,}5{,}6$$. If we denote the orbital angular momentum of a planet as $$~L_n= M_n V_n r_n $$, the specific orbital angular momentum as $$~L_{ns}= \frac { L_n}{ M_n }= V_n r_n $$, and if we use the expression for the orbital velocity in the form $$~V_n= \sqrt {\frac {G M_c}{r_n}} $$, where $$~M_c$$ is the mass of the Sun, then in view of (1) we obtain:
 * $$~L_{ns}= V_n r_n = \sqrt {G M_c r_n}= const \times n$$.

This equation means that the relative orbital angular momentum of planets is quantized, as the orbital angular momentum of the electron is quantized in the Bohr model of the hydrogen system. For the period of orbital revolution of planets we find the relation: $$~T^{1/3}_n= const \times n$$.

Planets' satellites
Satellite systems of Jupiter, Saturn and Uranus are populated enough, to be compared with the distribution of planets in the Solar system. For the planets' satellites the Titius–Bode law applies, but with a reduced basic distance, where instead of 0.1 a.u. the value of the order of 60,000 km is used.

To describe quantization of the orbits of satellites the same approach can be used as that for the orbital motion of planets. In this case, the orbital radii are proportional not to the square of the planet's number, but to the square of the satellite's number.

Similarity coefficients
Considering the correspondence between atoms and main-sequence stars, Sergey Fedosin discovered that the Solar system is similar to the isotope of oxygen atom with the mass number $$~A=18$$ and charge number $$~Z=8$$. The ratio of the Sun's mass to the nucleus mass of the isotope of oxygen atom $$~M_O$$ sets the coefficient of similarity in mass:
 * $$~\Phi = \frac {M_c}{M_O} =6.654 \cdot 10^{55}$$.

Multiplying $$~\Phi$$ by the electron mass, we can find the mass of the planet corresponding to the electron: $$~M_{\Pi}=6.06 \cdot 10^{25}$$ kg or 10.1 of Earth masses.

The coefficient of similarity in speed is given by:
 * $$~S= S_0 \frac {A} {Z}$$,

where $$~S_0= \frac {C_s} {c}=7.34 \cdot 10^{-4}$$ is the coefficient of similarity in speed for the hydrogen system, $$~C_s=220$$ km/s is the characteristic speed of the matter particles in the main sequence star with minimum mass $$~M_{sp}=0.056 M_c$$, $$~c$$ is the speed of light as the characteristic speed of matter in a proton.

Since the stellar matter is held by gravitational forces, the characteristic speed of matter $$~C_x$$ is determined by half of the gravitational energy, the absolute value of which defines the full energy of the star with the mass $$~M_s$$:
 * $$~E_s=M_s C^2_x=\frac {\delta G M^2_s}{2R_s},\qquad \qquad (2)$$

where $$~R_s$$ is the radius of the star, $$~\delta $$ depends on the matter distribution in the star and is equal to 0.6 in a uniform case.

Multiplying the rest energy of the atomic nucleus $$~m c^2 $$, where $$~m $$ is the mass of the nucleus, by the similarity coefficients it is possible to find the absolute value of full energy of the main-sequence star:
 * $$~E_s= m c^2 \Phi S^2= M_s c^2 S^2_0 \frac {A^2} {Z^2}= M_s C^2_s \frac {A^2} {Z^2}$$.

The coefficient of similarity in energy between the oxygen atom and the Solar system is defined by the product $$~\Phi S^2 $$. Comparing the ionization energies of electrons in the oxygen atom with the specific energy of planets in the Solar system due to their attraction to the Sun gives the ratio of these energies, which differs not more than three times from the coefficient of similarity in energy.

The coefficient of similarity in size is given by:
 * $$~P= P_0 \frac {Z} {A}$$,

where $$~P_0 = 5.437 \cdot 10^{22}$$ is calculated from the ratios of the hydrogen system. If we multiply the Bohr radius in the hydrogen atom by $$~P_0$$, we obtain the value 19.25 a.u., which is close enough to the Uranus orbit. There are several ways to estimate the coefficient of similarity in size, which give similar results: 1) Comparing the semi-axes of the orbits of binary stars and the bond lengths in molecules; 2) Comparing the dimensions in the Solar system and in the oxygen atom; 3) Comparing the orbits of Mercury and of the hydrogen-like ion of oxygen; 4) Comparing the dimensions of atomic nuclei and stars.

The coefficient of similarity in time is:
 * $$~\Pi= \frac {P}{S}= \Pi_0 \frac {Z^2} {A^2}$$, where $$~\Pi_0 =\frac { P_0}{ S_0} =7.41 \cdot 10^{25} $$.

By means of similarity coefficients the stellar Dirac constant is determined for planetary systems of main-sequence stars:
 * $$~\hbar_s =\hbar \Phi S P=\hbar \Phi S_0 P_0=2.8 \cdot 10^{41} $$ J∙s,

where $$~\hbar$$ is the Dirac constant. The quantity $$~\frac {\hbar_s}{2}$$ almost coincides with the intrinsic angular momentum of the Solar rotation.

The orbital motion of planets
In quantum mechanics it is considered that the oxygen atom has two electron layers. K-layer includes 1s-shell and two electrons. L-layer includes 2s-shell with two electrons and 2p-shell with four electrons. It is assumed that electrons in s-states do not have the orbital angular momentum. From formal comparison of the oxygen atom and the Solar system it follows that Mercury and Venus are equivalent to 1s-shell, Earth and Mars – to 2s-shell and the large planets, the orbital angular momenta of which are much greater than those of the terrestrial planets, are similar to 2p-shell of the oxygen atom. In the Bohr model of hydrogen atom the orbital angular momentum of electron is quantized:
 * $$~L= n \hbar$$.

If we assume that planets in their rotation around the Sun have little influence on each other and they are located in orbits that are allowed in the hydrogen-like atom, then for specific orbital angular momenta of planets we obtain quantum relation with accuracy up to 25 %:
 * $$~L_{ns}= \frac { L_n}{ M_n }= V_n R_n = K_1 n \frac {\hbar_s }{ M_{\Pi}}$$,

where $$~M_{\Pi}=6.06 \cdot 10^{25}$$ kg is the mass of the planet corresponding to electron, $$~ K_1= 0.5$$ follows from correspondence with the empirical data.

If we substitute here the expressions for the orbital velocity in the form $$~V_n= \sqrt {\frac {G M_c}{R_n}} $$, we could determine the orbital radii of planets:
 * $$~R_n= \frac { K^2_1 n^2 \hbar^2_s } {G M_c M^2_{\Pi} }$$.

The quantity $$~ K_1 \frac {\hbar_s }{ M_{\Pi}}=2.31 \cdot 10^{15}$$ m2/s differentiates planets from planetary satellites. For example, the dwarf planet Ceres, with the orbit between Mars and Jupiter, has the specific orbital angular momentum equal to $$~ 7.4 \cdot 10^{15}$$ m2/s, while for Jupiter's satellite Callisto this quantity is much less: $$~ 1.7 \cdot 10^{13}$$ m2/s.

The orbital angular momenta of planetary satellites
In the satellite systems of Jupiter, Saturn and Uranus, by analogy with specific orbital angular momenta of planets, quantum Bohr formula is used as follows:
 * $$~\frac {L}{M}= V R =  \frac {2 \pi R^2 }{T}= n K$$,

where $$~L$$ and $$~M$$ are the orbital angular momentum and the mass of the satellite in a circular orbit, $$~ V$$ and $$~ R$$ are the average velocity in orbit and the average orbital radius of the satellite, $$~ T$$ is the revolution period in orbit with the number $$~ n$$, $$~ K$$ is a constant, which depends on the satellite system.

In all satellite systems there is no satellite with the number $$~ n=1$$, its role is played by the rings, which are located near each of the planets.

In Jupiter's system there are eight regular satellites, starting with Metis ($$~ n=2$$) and ending with Callisto $$~ n=8$$. The rest eight outer satellites from Leda to Sinope are small in size, have significant eccentricities and inclinations to Jupiter's equator, and the last 4 satellites rotate in the opposite direction. They can be considered asteroids captured by Jupiter. The quantity $$~ K_1=1.65 \cdot 10^{12}$$ m2/s in general characterizes the regular satellite system of Jupiter, which appeared during the period of the planet's formation.

A similar situation takes place for Saturn (eleven regular satellites from Atlas, Prometheus and Pandora with $$~ n=2$$ to Iapetus with $$~ n=11$$), and for Uranus ( regular satellites from Cordelia (1986 U7) with $$~ n=2$$ to Oberon with $$~ n=9$$), for which $$~ K_2=0.74 \cdot 10^{12}$$ m2/s and $$~ K_3=1.97 \cdot 10^{11}$$ m2/s, respectively.

The meaning of the quantities $$~ K_1$$,$$~ K_2$$ and $$~ K_3$$ lies in the fact that they reflect the specific orbital angular momentum of that part of a cloud, from which formation of one or another planetary system begins. This is confirmed by the fact that products of equatorial velocities of Jupiter, Saturn and Uranus and the corresponding radii of the planets are close to the values $$~ K_1$$,$$~ K_2$$ and $$~ K_3$$, respectively.

The total angular momentum of planetary systems
With the help of the quantum approach we can determine if the planetary systems around stars are formed from weakly rotating gas-dust clouds. If in a first approximation we assume that the mass distribution of planets in planetary systems is of such kind that the orbital angular momenta of planets in orbit $$~ n$$ are equal to $$~ L_n=n \hbar_s $$ as in Bohr theory for the electron, then the total orbital angular momentum is:
 * $$~L= \sum {L_n} = \hbar_s \sum^Z_{n=1} {n} = \hbar_s \frac {Z (Z+1)}{2}$$,

where $$~ Z$$ is the charge number of the star, specifying the number of planets.

The total angular momentum of the planetary system is composed of the proper angular momentum of the star $$~ I_s$$ and the total orbital angular momentum of planets: $$~ L_0=I_s+L$$. The quantities $$~ L_0$$ were calculated for planetary systems of main sequence stars, based on the observed typical velocities of stars' rotation around their axes, their masses and radii, taking into account the mass numbers $$~A$$ and the charge numbers $$~Z$$ of the stars, determined on the basis of their similarity with the chemical elements (see discreteness of stellar parameters). Since the planetary systems are formed from gas-dust clouds, then $$~ L_0$$ must be equal to the angular momentum of such clouds. For a cloud with the mass $$~ M_s$$ and the radius $$~ R$$ the estimate of the total rate of rotation at the equator is $$~ V= \sqrt {\frac {G M_s}{R}}$$. Then the angular momentum of the cloud is equal to:
 * $$~ I= 0.4 D M_s V R =0.4 D M_s \sqrt {G M_s R}$$,

where the coefficient $$~ D $$ reflects the change in the angular momentum due to non-uniformity of the mass density and the differential rotation of the cloud.

Between the mass and the radius of the cloud there is a dependence: $$~ M_s= \frac {4 \pi R^3 \rho}{3}$$, where $$~ \rho $$ is the mean mass density of the cloud. In addition, by analogy with the mass of atomic nuclei, the stellar mass is associated with the mass number and the mass of a star with the minimum mass: $$~ M_s=A M_{sp}$$. Taking it into account we obtain:
 * $$~ I= 0.4 D G^{1/2} (\frac {3}{4 \pi \rho})^{1/6} M^{5/3}_{sp} A^{5/3}= B A^{5/3} = L_0. \qquad \qquad (3)$$

The value $$~ B=3 \cdot 10^{41} $$ J∙s is determined from the diagram of dependence of the angular momentum $$~ L_0$$ of the planetary system on the mass number $$~ A$$ and it appears to be close to the value of the Dirac stellar constant $$~\hbar_s$$, which is correct for a planetary system of one planet and a main-sequence star with minimum mass.

Intrinsic angular momenta of planets
By analogy with quantization of orbital and spin angular momenta of electrons in atoms, we make an assumption about quantization of intrinsic angular momenta of planets.

For each gravitationally bound object with mass $$~ M$$ and radius $$~ R$$ we can introduce their characteristic angular momentum according to the following formula:
 * $$~L_x=M R C_x = M R \sqrt {\frac {\delta G M}{2R}}, \qquad \qquad (4)$$

where the characteristic speed $$~ C_x$$ is found from the ratio of energies similar to (2).

The electron spin is usually determined by the Planck constant in the form $$~\frac {h}{4 \pi} $$. Similarly for planets the intrinsic angular momenta are proportional to the quantity $$~\frac {L_x}{4 \pi} $$ and to the planet number as the quantum number $$~ n$$:
 * $$~I_{\Pi}=\frac {K_2}{4 \pi } \sqrt {\frac {\delta G M}{2R}} M R n, \qquad \qquad (5)$$

where $$~ K_2= 0.25$$ is obtained from correspondence to the empirical data.

Table 1 compares the intrinsic angular momenta of the planets with the values calculated by relation (5); we see that only inhibited planets Mercury and Venus noticeably deviate from this dependence.

Large and small planets
Quantization of masses of atomic nuclei is associated with mass discreteness of nucleons, which are part of nuclei. For planets mass discreteness can also be proved, including the existence of the minimum and maximum masses. In order to calculate the maximum mass, the equality of the planet's gravitational energy and the electrostatic energy per atom is used. If the gravitational energy is too high, the electron shells of the matter atoms start getting compressed and the planet can turn into a white dwarf, in which the gravitational force is opposed by the pressure of degenerate electrons. For the matter of a planet, composed of hydrogen, the maximum electrostatic energy approximately corresponds to the energy of an electron in a hydrogen atom in the ground state. The equality of energy has the following form:
 * $$ ~\frac {\delta G M^2} {NR} = \frac {e^2}{4 \pi \varepsilon_0 r_B } $$,

where $$ ~M$$ and $$~ R$$ are the mass and the radius of the planet, $$ ~N= \frac {M}{M_p}$$ is the number of nucleons in the planet's matter, $$~ M_p$$ is the proton mass, $$~ \varepsilon_0$$ is the electric constant, $$ ~r_B$$ is the Bohr radius.

The planet's volume can be calculated by the radius as well as by the total amount of hydrogen atoms:
 * $$~ \frac {4 \pi R^3} {3} = \frac {4 \pi N r^3_B} {3} $$.

Taking it into account, at $$~ \delta =0.6$$ we determine the mass and the radius of a massive planet, exceeding to some extent the mass and the radius of Jupiter:
 * $$~ M=\frac {e^3} {M^2_p (4 \pi \varepsilon_0 \delta G)^{1.5} } = 4.9 \cdot 10^{27}$$ kg $$~= 2.6 M_J $$.
 * $$~ R=\frac {e r_B} {M_p (4 \pi \varepsilon_0 \delta G)^{0.5} } = 7.6 \cdot 10^{7}$$ m $$~= 1.06 R_J $$.

To estimate the minimum mass of a planet we use the equality of half of the gravitational energy and the internal thermal energy according to the virial theorem:
 * $$ ~\frac {\delta G M^2} {2R} = \frac {3N kT}{2} $$,

where $$ ~k$$ is the Boltzmann constant, $$~ T$$ is the average internal temperature. Expressing the planet mass by the density and the radius in the form $$~ M=\frac {4 \pi \rho R^3 }{3} $$, and using the equation $$ ~N= \frac {M}{M_p}$$, we obtain the mass and the radius:
 * $$~ M=\frac {9 \pi \rho } {2} (\frac {kT}{\pi \delta G \rho M_p })^{1.5} $$.
 * $$~ R=\frac {3 } {2} (\frac {kT}{\pi \delta G \rho M_p })^{0.5} $$.

Substituting here the same mass density as that of the dwarf planet Ceres $$ ~2.1\cdot 10^{3} $$ kg/m3, and the temperature $$ ~T=2.7$$ K (the temperature of background radiation), we can estimate the minimum mass and the radius of the planet: $$ ~M=7.3\cdot 10^{20} $$ kg, $$ ~R=440 $$ km, which is slightly less than Ceres' parameters.

Stellar constants
For planetary systems of main sequence stars the stellar constants are:
 * 1) The minimum mass of a main sequence star, which is the analog of a proton: $$~M_{sp}=0.056 M_c=1.11\cdot 10^{29}$$ kg.
 * 2) The mass of a planet, which is the analog of an electron: $$~M_{\Pi}=6.06 \cdot 10^{25}$$ kg or 10.1 Earth masses.
 * 3) The stellar speed $$~C_s=220$$ km/s as the characteristic speed of the matter particles in a main sequence star with minimum mass.
 * 4) The stellar Bohr radius in the stellar hydrogen system: $$~R_F=2.88 \cdot 10^{12}$$ m = 19.25 a.u.
 * 5) The orbital velocity of a planet which is the analogue of an electron on the stellar Bohr radius in the stellar hydrogen system: $$~ V_{\Pi}=1.6 $$ km/s.
 * 6) The stellar  fine structure constant  $$~ \alpha_s =\frac { V_{\Pi}}{ C_s }=0.007297 $$.
 * 7) The stellar Dirac constant for planetary systems of main-sequence stars: $$~\hbar_s =\hbar \Phi S P=\hbar \Phi S_0 P_0=2.8 \cdot 10^{41} $$ J∙s.
 * 8) The stellar Boltzmann constant: $$~K_s = K_{ps} A $$, where $$~A$$ is the mass number of a star, $$~K_{ps}= 1.18 \cdot 10^{33} $$ J/K.
 * 9) The stellar mole is defined as the amount of matter, consisting of stars, the number of which equals $$~ N_A = 6.022 \cdot 10^{23}$$ (stellar mole)–1,  where $$~ N_A$$ is the Avogadro number.
 * 10) The stellar gas constant for the gas of stars: $$~ R_{st} = K_s N_A = A K_{ps} N_A =A R_{pst}$$, 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.
 * 11) The gyromagnetic ratio for the object which is the analog of an electron: $$~ \frac {e }{ M_e } \frac { P^{0.5}_0 S_0}{\Phi^{0.5}}=3.69 \cdot 10^{-9}$$ C/kg (or 1/(T∙s ), where $$~e$$ is the elementary charge.
 * 12) The gyromagnetic ratio for the stellar object which is the analogue of an atomic nucleus: $$~ \frac {e }{ M_p } \frac { P^{0.5}_0 S_0}{\Phi^{0.5}}=2.01 \cdot 10^{-12}$$ C/kg (or 1/(T∙s)).
 * 13) The stellar Stefan–Boltzmann constant: $$~ \Sigma_s= \frac {\sigma \Phi }{ \Pi^3_0 } =9.3 \cdot 10^{-30}$$ W/(m2 ∙K4), where $$~\sigma $$ is the Stefan–Boltzmann constant.
 * 14) The stellar radiation density constant: $$~ A_s= \frac {a \Phi S^2_0}{P^3_0} =1.69 \cdot 10^{-34}$$ J/(m3∙K4), where $$~a = \frac {4 \sigma}{c}$$ is the radiation density constant.
 * 15) The free fall acceleration at the surface of a main sequence star of minimum mass: $$~g_s = \frac {GM_{sp}}{R^2_{sp}}= 3.1 \cdot 10^3$$ m/s2, with the stellar radius $$~ R_{sp}= 0.07 $$ Solar radii.
 * 16) The absolute value of full energy of a main sequence star with minimum mass in its proper gravitational field: $$~ E_s= M_{sp} C^2_s =5.4 \cdot 10^{39}$$ J.

Similarity coefficients in Fedosin's model
The ratios of the mass (radius) of a typical neutron star to the mass (radius) of a proton determine the coefficients of similarity in mass and size, respectively:
 * $$~\Phi' = \frac {M_s}{M_p}=1.62 \cdot 10^{57} $$.
 * $$~P' = \frac {R_s}{R_p}=1.4 \cdot 10^{19} $$.

From relation (2) we can estimate the characteristic speed of the matter particles of the neutron star under consideration:
 * $$~E_s=M_s {C'}^2_s=\frac {\delta G M^2_s}{2R_s}, \qquad C'_s= 6.8 \cdot 10^{7} $$ m/s.

The coefficient of similarity in speed equals:
 * $$~S' = \frac { C'_s}{c}=0.23 $$ ,

here $$~c $$ is the speed of light.

The coefficient of similarity in time as the ratio of time flow rates between the elementary particles and neutron stars:
 * $$~\Pi' = \frac {P'}{S'}=6.1 \cdot 10^{19} $$.

The value of the stellar Dirac constant for neutron stars:
 * $$~\hbar'_s= \hbar \Phi' P' S' =5.5 \cdot 10^{41} $$ J∙s.

This quantity is close to the limit angular momentum of intrinsic rotation of neutron stars. Multiplying the Bohr radius in a hydrogen atom by $$~P'$$ we obtain the value $$~R'_F =7.4 \cdot 10^{8}$$ m. This distance almost coincides with the Roche limit, at which planets disintegrate in the strong gravitational field of a neutron star. The disks discovered near neutron stars also have the characteristic radius $$~R_d $$ of the order of $$~ R'_F $$. From the point of view of the similarity of matter levels and the substantial electron model, the disks around neutron stars are similar to electron disks in atoms. If we multiply the electron mass by $$~\Phi' $$, we can estimate the mass of the disk: $$~ M_d= 1.5 \cdot 10^{27}$$ kg. The model of an electron in the form of a disk can explain the origin of the electron spin in the atom.

Magnetars
Based on the similarity between atoms and stars, magnetars, as strongly magnetized neutron stars, are considered as the analogues of a proton. The similarity of these objects is shown in the substantial proton model, and the evolution – in the substantial neutron model. Magnetars and the disks rotating around them form the hydrogen system for degenerate objects at the level of stars, and the angular momentum of disks in the ground state is equal to the stellar Dirac constant $$~\hbar'_s $$. It is expected that the long-term evolution of stars will lead eventually to transformation of all stars into white dwarfs, neutron stars and magnetars, and the latter will form groups of stars similar to atomic nuclei and will be the matter basis at the level of stars. In this case the mass of each portion of stellar matter will be quantized with accuracy up to the mass of one magnetar.

Using the similarity coefficients and the relations of physical quantities' dimensions we can determine the electric charge and magnetic moment of the magnetar, which is the proton's analogue:
 * $$ Q_s = e (\Phi' P')^{0.5} S' = 5.5 \cdot 10^{18}$$ C,
 * $$ P_{ms} = P_{mp} {\Phi'}^{0.5} {P'}^{1.5} {S'}^2 = 1.6 \cdot 10^{30}$$ J/T,

where $$~e$$ and $$~P_{mp}$$ are the elementary charge and the magnetic moment of the proton, respectively.

The values of the electric potential and the magnetic field induction at the magnetar's pole:
 * $$ \psi_s = \frac { Q_s }{4 \pi \varepsilon_0 R_s}= 4.2\cdot 10^{24}$$ V,
 * $$ B_s = \frac { \mu_0 P_{ms} }{2 \pi R^3_s}= 1.8\cdot 10^{11}$$ T,

here $$~\mu_0$$ is the vacuum permeability.

The ratio of the magnetar's charge to its mass according to the theory of similarity is given by the formula:
 * $$ \frac {Q_s}{ M_s} = \sqrt {\frac {4 \pi \varepsilon_0 G M_e}{M_p}   }$$,

where $$~M_e$$ and $$~M_p$$ are the masses of the electron and the proton.

In the hydrogen system the magnetar and the disk have equal charge but opposite in sign, as a proton and an electron. Disk's rotation creates a magnetic moment, which can be found by the formula:
 * $$ P_{md } = P_{me} {\Phi'}^{0.5} {P'}^{1.5} {S'}^2 = \frac {Q_s \hbar'_s }{2 M_d}= \sqrt {\frac {4 \pi \varepsilon_0 G M_p}{M_e}   } \frac {\hbar'_s }{2} =1.03 \cdot 10^{33}$$ J/T,

where $$~P_{me}$$ is the magnetic moment of the electron.

If we proceed from the estimates of appearance of magnetars and neutron stars in our galaxy: one magnetar can appear in 103 – 105 years, which gives, taking into account the age of the Galaxy equal to more than 13 billion years, 105 – 107 magnetars and an order of magnitude more neutron stars, it allows us to explain the reason of high energy cosmic rays. According to one assumption, protons and light nuclei, that make up cosmic rays, are accelerated by electric fields arising due to rapid rotation of the dipole magnetic field of magnetars. In contrast, the presence of the intrinsic electric charge and the constant electric field in magnetars, according to similarity to proton, means that magnetars can accelerate the particles of cosmic rays almost without expenses by their rotational or magnetic energy. Emerging of negatively-charged disks near magnetars leads to electrical neutrality of the system and to decrease of the number of cosmic rays emitted. The presence of a large number of magnetars and neutron stars in the centers of most galaxies also allows us not to use the hypothesis of massive black holes to explain the effects of active galactic nuclei.

The values of stellar constants
For the systems with neutron stars, the stellar constants are as follows:
 * 1) The mass of a typical neutron star, which is the analogue of a proton: $$~M_s=1.35 M_c=2.7\cdot 10^{30}$$ kg.
 * 2) The mass of a disk, which is the analogue of an electron: $$~ M_d= 1.5 \cdot 10^{27}$$ kg, which is equal to 250 Earth masses or 0.78 Jupiter masses.
 * 3) The stellar speed $$~ C'_s= 6.8 \cdot 10^{7}$$ m/s as the characteristic speed of the matter particles in a typical neutron star.
 * 4) The stellar Bohr radius in the stellar hydrogen system: $$~ R'_F = \frac {{\hbar'}^2_s }{G M_s M^2_d }=7.4 \cdot 10^{8}$$ m.
 * 5) The orbital velocity of the disk matter on the stellar Bohr radius in the stellar hydrogen system: $$~ V_d= \frac {G M_s M_d }{\hbar'_s }=496 $$ km/s.
 * 6) The stellar fine structure constant: $$~ \alpha_s =\frac { V_d}{ C'_s } = \frac {G M_s M_d }{\hbar'_s C'_s }=0.007297 $$.
 * 7) The stellar Dirac constant: $$~\hbar'_s= \hbar \Phi' P' S' =5.5 \cdot 10^{41} $$ J∙s.
 * 8) The star charge: $$ Q_s  = e (\Phi' P')^{0.5} S' = 5.5 \cdot 10^{18}$$ C.
 * 9) The magnetic moment of a magnetar: $$ P_{ms}  = P_{mp} {\Phi'}^{0.5} {P'}^{1.5} {S'}^2 = 1.6 \cdot 10^{30}$$ J/T.
 * 10) The magnetic moment of a disk, which is the analogue of an electron: $$ P_{md }  = P_{me} {\Phi'}^{0.5} {P'}^{1.5} {S'}^2  =\frac {Q_s \hbar'_s }{ 2 M_d }=1.03 \cdot 10^{33}$$ J/T.
 * 11) The free fall acceleration at the surface of a magnetar: $$~g_m = \frac {G M_{s}}{R^2_{s}}= 1.2 \cdot 10^{12}$$ m/s2, with the star radius $$~ R_{s}= 12$$ km.
 * 12) The absolute value of full energy of a magnetar in its gravitational field: $$~ E_m= M_s {C'}^2_s =1.2 \cdot 10^{46}$$ J.

Variable stars
There are several types of stars that reveal themselves by the fact that their brightness periodically changes. They are:
 * 1) Long-period variable stars such as Mira (Omicron Ceti), the brightness oscillation periods of which are from 90 to 1000 days, mostly about 260 – 340 days. These stars are red giants and supergiants of spectral classes M, R, N, S, C and they have the mass of order of the Solar mass.
 * 2) Semiregular variable stars of SR type, their minimum oscillation periods are less than 50 days, the maximum periods are up to 1000 days. Spectral classes of these stars are F, G, K, M, and according to their luminosity they are giants and supergiants.
 * 3) Variable stars such as Alpha² Canum Venaticorum variable, which have strong magnetic fields and because of it have different surface brightness, change their brightness with periods ranging from 0.5 to 160 days due to their rotation.
 * 4) RV Tauri variable stars, yellow supergiants with periods ranging from 30 to 150 days and the spectral classes F, G, K, M.
 * 5) Classical or long-period Cepheid variable stars are characterized by brightness oscillations which are very stable in amplitude and period. Periods last from 1 to 70 days (also there are Cepheids with a period of more than 100 days in the Magellanic Clouds and the Andromeda Nebula). Typical representatives of Cepheids are Delta Cephei (belongs to population I) and W Virginis (representative of population II). The stars like Delta Cephei are supergiants, mainly belong to the luminosity class Ib and the spectral classes from F5 to K0. The radii of these stars are great – from $$~ 5 \cdot 10^{6}$$ to $$ ~ 10^{8}$$ km, with the mass from 3 to 16 $$ M_c $$. The masses of such stars as W Virginis are about $$0.55  M_c $$.
 * 6) Variable stars of Alpha Cygni type have periods ranging from several days to several weeks. These stars are supergiants of the spectral classes B and A, brightness variations occur due to non-radial pulsations.
 * 7) Short-period Cepheids of RR Lyrae type with periods ranging from 0.2 to 1 day (with maximum of 0.5 days) belong to the spherical component (population II). The stars of RRab type with periods more than 0.44 days have a radius $$5.5 R_c $$ and mass $$ 0.5M_c $$,  and the stars of RRc-type (characterized by a smoother shape of the light curve) have periods > 0.36 days and a radius $$4.5 R_c $$ and mass $$0.6 M_c $$.
 * 8) Beta Cephei variables (as well as Beta Canis Majoris) with luminosity class IV or III (subgiants, giants), spectral classes B0.5 – B2, masses more than $$ 10M_c $$ and pulsation periods of 3 – 7 hours.
 * 9) The stars of Delta Scuti type (dwarf Cepheids) of the spectral classes A or F with pulsation period of 1.3 – 7.2 hours. They belong to population I. At the period of 3.36 hours the radius of these stars reaches $$ 3R_c $$ and the mass – $$2 M_c $$.
 * 10) The stars of Chi Centauri type belong to spectra B2 IV or III, have periods of 29 – 43 minutes, with very small amplitude.
 * 11) Pulsating white dwarfs of ZZ Ceti type with periods ranging from hundreds to thousands seconds.

During each pulsation period stars emit the following energy quanta: Mira emits during a period of 331 days up to $$ 10^{38}$$ J, the stars of Delta Scuti type emit during 3 hours about $$ 10^{33}$$ J. The reasons for periodic changes in brightness of variable stars usually are radial and non-radial pulsations, chromospheric activity, periodic eclipses of stars in a close binary system. Accordingly, all variable stars are divided into three large classes: pulsating variables, eruptive variables and eclipsing variables. These classes are subdivided into different types, some types – into subtypes. The pulsation mechanism of Cepheids is associated with the strong dependence of the opacity of helium and hydrogen stellar layers on their degree of ionization. During compression of a star, the matter ionization increases, the emission is retained more in the matter, heats it and stops the starting compression. During expansion of a star, on the contrary, ionization decreases, and the star is cooled by the outgoing emission. At some point, the star begins to compress due to the gravity forces, passes the equilibrium state and then a new cycle is repeated.

Neutron stars – pulsars have the highest accuracy of rotation periods repeatability, which is detected by the radio-emission pulses from their active zones, which are probably located near the magnetic poles and are rotating together with the star. In particular, the pulse period of the pulsar PSR B1937+21 is equal to 0.0015578064488724 seconds and is known with an accuracy up to 13 significant digits, which is comparable to the accuracy of the best atomic frequency standards.

Galactic similarity coefficients
Taking into account the division of matter levels to main and intermediate and discreteness of similarity coefficients (see the corresponding section in the article Similarity of matter levels), when masses and sizes of objects increase exponentially, Sergey Fedosin determined that our Galaxy is similar to an atom isotope with the mass number $$~A=18-20$$. The ratio of the masses of galaxies to the masses of corresponding main sequence stars is $$~D^2_{\Phi}$$, where $$D_{\Phi } = \Phi^{1/10} =3.8222 \cdot 10^{5} $$ is the coefficient of similarity in mass between the adjacent levels of matter.

The mass of a normal galaxy with minimum mass corresponding to the minimum mass of a main sequence star is obtained by multiplying the star mass by $$~D^2_{\Phi}$$:
 * $$~M_{gp}= M_{sp} D^2_{\Phi} =8.15 \cdot 10^{9} M_c$$,

where $$~ M_c $$ is the Sun's mass.

Repeated multiplication by $$~D^2_{\Phi}$$ gives the mass of normal metagalaxy with minimum mass:
 * $$~M_{mp}= M_{gp} D^2_{\Phi} =1.19 \cdot 10^{21} M_c$$.

The planet which is the analogue of an electron corresponds to a dwarf galaxy with the mass $$~M_{ge}= M_{\Pi} D^2_{\Phi} =4.43 \cdot 10^{6} M_c$$.

The ratio of the radii of galaxies to the corresponding radii of main sequence stars equals $$~D^6_P$$, where $$D_P = P^{1/12}_0 =78.4538 $$ is the coefficient of similarity in size between the adjacent levels of matter. By multiplying the radius of the main sequence star with minimum mass 0.07 solar radii by $$~D^6_P$$ we obtain the radius of the normal galaxy with minimum mass: $$~R_{gp}=370$$ pc. This radius is the volume-average radius, but since normal galaxies are generally plane spiral systems, the radius of the disk of the galaxy with minimum mass reaches 2.5 kpc. In the same way, multiplying the radius of the planet-the analogue of an electron $$~ 2 \cdot 10^{7} $$ m by $$~D^6_P$$, we make an estimate of the average radius of the dwarf galaxy: 151 pc. The value of the galaxy's characteristic speed $$~C_g $$ can be found from relation (2): $$~C_g= \sqrt {\frac {\delta G M_{gp}}{2R_{gp}}}=169$$ km/s. The ratio of galaxy's speed $$~C_g$$ to the star's speed $$~C_s$$ gives the coefficient of similarity in speed: $$~ D_s =\frac {C_g}{ C_s}=0.77 $$.

By multiplying the stellar Dirac constant by the similarity coefficients we determine the galactic Dirac constant:
 * $$~\hbar_g =\hbar_s D^2_{\Phi} D_s D^6_P =6.1 \cdot 10^{63} $$ J∙s.

The intrinsic angular momentum of spiral galaxies depending on their masses in the system of physical units SI can be approximated by the expression:
 * $$~I_g= 1.2 \cdot 10^{-2} M^{5/3}_g $$.

From this at the mass $$~M_{gp}$$ we find the angular momentum of the normal galaxy with the minimum mass: $$~I_{gp}=1.2 \cdot 10^{65}$$ J∙s. According to other sources, $$~I_{gp}=7 \cdot 10^{64}$$ J∙s. Thus due to their flattened shape, spiral galaxies have angular momentum which is more than an order of magnitude larger than the angular momentum $$~\hbar_g /2$$, which they would have in case of a spherical shape, according to the similarity theory.

Formation of galaxies from gas-dust clouds
Assuming the formation processes of stars and galaxies from gas-dust hydrogen clouds to be similar, we can equate the relation (3), applied to the angular momentum of galaxies, with the empirical dependence of the galaxies' spin on the mass:
 * $$~ I= 0.4 D G^{1/2} (\frac {3}{4 \pi \rho})^{1/6} M^{5/3}_g= 1.2 \cdot 10^{-2} M^{5/3}_g$$,

which gives the relation $$~ \rho = D^6 \cdot 9.2 \cdot 10^{-23}$$ kg/m3.

Between the radius of the parent cloud of our Galaxy, the mass density and the mass there is a standard relation, in which we can substitute the mass density $$~ \rho $$ and the Galaxy mass $$~ M_g $$  and we will obtain:
 * $$~ R = (\frac {3 M_g}{4 \pi \rho })^{1/3} $$ = 30 kpc/$$D^2$$.

The radius of Galaxy corona reaches 40 kpc, and old stars are discovered at distances up to 46 kpc. If we assume the latter radius as the radius of star formation in the primary cloud, then the average density of the cloud at this moment is about $$~ 2.5 \cdot 10^{-23}$$ kg/m3.

Heisenberg relation
Heisenberg uncertainty principle relates the characteristic changes of atomic energy and the energy change intervals by the formula:
 * $$~ \Delta E \Delta t \geq h $$,

where $$~h$$ is the Planck constant.

For stellar and galactic systems similarly we can write:
 * $$~ \Delta E \Delta t \geq L_x $$,

where $$ ~L_x $$ is the characteristic angular momentum, which is associated either with the object's intrinsic angular momentum $$~ I $$ by relation of the form $$ ~L_x =4 \pi I $$, or with the orbital  momentum by relation of the form $$ ~L_x =2 \pi L $$.

If we assume that for the Galaxy the energy change is equal to the total energy in the gravitational field: $$~ \Delta E=2 \cdot 10^{52}$$ J, and the time interval is equal to the relaxation time in the field of regular forces $$~ \Delta t=\frac {2}{\sqrt {G \rho} }=3 \cdot 10^8 - 10^9$$ years, where $$~ \rho $$ is the mass density, then it allows us to make an estimate of the Galaxy's characteristic angular momentum:
 * $$~ L_x \approx \Delta E \Delta t = (2-6) \cdot 10^{68} $$ J∙s.

This value can be compared with the Galaxy's angular momentum of about $$~ I = 1.8 \cdot 10^{67} $$ J∙s.

From the moment of the Galaxy's separation as an independent object, formation of stars started in it, for which the Heisenberg relation holds as well. Substituting in it the energy change with the total energy of the star $$~ \Delta E=E_s$$, and assuming the time period to be equal to the time of the star formation (the Kelvin-Helmholtz time) $$~ \Delta t=t_{KH}$$, we obtain the following:
 * $$~ E_s t_{KH} \approx h_o $$,

where $$~ h_o $$ is the characteristic orbital angular momentum of the star in the Galaxy. For the Sun's rotation in the Galaxy the orbital angular momentum is approximately equal to $$~ L_c = (1.15-1.53) \cdot 10^{56} $$ J∙s. The characteristic orbital angular momentum of the star exceeds 2π times its orbital angular momentum during rotation in the Galaxy, and in this case the greater is the mass of the stars, the less is the value $$~h_o $$ obtained for them. This is due to the fact that massive stars gravitate towards the center of the Galaxy, where the orbital angular momentum is less on the one hand, but on the other hand massive stars are formed and evolve faster. For the most widely-spread low-mass stars $$~h_o =2.1 \cdot 10^{57}$$ J/s.

Density wave
A large number of the observed galaxies are substantially plane spiral systems. For example, the ratio of the diameter of our Galaxy to its thickness is about 30. If we consider the Galaxy as a flat disk, then with the help of a statistical approach we can qualitatively understand the emergence of the spiral arms in it as some form of spatial quantization. In the stationary case the Poisson's equation holds for the gravitational potential $$~\varphi $$ :
 * $$~ \Delta\varphi = \frac {d^2 \varphi }{dr^2} =4 \pi G \rho $$,

where $$~\rho $$ is the mass density which depends on the radius $$~r $$.

The mass density can be considered in the form $$~\rho = \rho_0 + \rho' $$, where $$~\rho' $$ is a little additive to the main mass density $$~\rho_0 $$. Similarly the potential will consist of two parts: $$~\varphi = \varphi_0 + \varphi' $$. Due to the potential addition, the star with the mass $$~M $$ will have additional energy $$~U=M \varphi'$$. The stars orbiting the Galactic nucleus produce stellar gas, which is characterized by the temperature $$~T $$. We can assume that a statistical formula holds for the dependence of density distribution deviation from the mean:
 * $$~\rho' = \varrho \exp {(- \frac {U}{K_s T})} =\varrho \exp {(- \frac { M \varphi'}{K_s T})}$$,

where $$~\varrho$$ is the density change  amplitude, $$~ K_s $$ is the stellar Boltzmann constant.

Expanding the exponent for the density in the form:
 * $$~\rho' = \varrho - \frac { \varrho M \varphi'}{K_s T}$$,

and solving the Poisson equation only for $$~ \rho' $$, we find the potential additive:
 * $$~\varphi' = \frac { K_s T }{M} + A \sin {(r \sqrt {\frac {4 \pi G \varrho M}{ K_s T }  } + \beta)}$$,

where $$~ A $$ and $$~ \beta $$ are some constants.

Given this potential additive, the density additive is:
 * $$\rho' = -\frac { A \varrho M }{ K_s T } \sin {(r \sqrt {\frac {4 \pi G \varrho  M}{ K_s T }  } + \beta )}$$.

Density fluctuations $$~\rho' $$ against the average density background look like the rings in the Galactic disk. Given the sine periodicity, for the change of the radius $$~\delta r $$ between the adjacent maxima we obtain:
 * $$\delta r = \sqrt {\frac {\pi K_s T }{G \varrho  M }  }$$.

With the value $$~ K_s \approx 10^{33} $$ J/K, the kinetic temperature of the stars' motion $$~T \approx 10^{6}$$ K, $$~\varrho = 5 \cdot 10^{-21}$$ kg/m3 as the average mass density on the Sun's galactic orbit and the average stellar mass of about half of the Solar mass, we will obtain $$\delta r = 3$$ pc. This coincides with the observed step value between the spirals, in the form of which the rings are extended due to the Galaxy's rotation.