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The hydrogen system is ideal system of two objects held near each other by fundamental forces, with the ratio of the objects' masses equal to the ratio of the proton mass to the electron mass. The concept of hydrogen system is used to describe the similarity of matter levels in the Theory of Infinite Hierarchical Nesting of Matter, according to which the hydrogen systems characterize the simplest and most common in the universe systems of two bodies. Each hydrogen system consists of the primary massive object and the low-mass satellite rotating around it. At the atomic level the hydrogen system is the hydrogen atom comprising a proton and an electron. Theoretical definition of specific properties of a hydrogen system (the mass of the primary object, the distance to the satellite in the ground state, etc.) is not univocal and depends on additional assumptions.
Uniqueness of a hydrogen atom is that the most total balance between the strong gravitation and electromagnetic forces is achieved in it.  Table 1 shows the parameters of a hydrogen atom, which is the standard hydrogen system.
|Proton mass||Mp = 1.6726485∙10-27 kg|
|Electron mass||Me = 9.109534∙10-31 kg|
|Electron's orbital velocity||Ve = 2.187691∙106 m/s|
|The radius of electron's orbit||RB = 5.2917706∙10-11 m|
As it is shown in the substantial electron model, the electron in the hydrogen atom in the ground state represents a discoidal cloud, with the inner edge of the disk and the outer edge . The matter of the electron disk rotates around the atomic nucleus differentially, that is with different angular velocities, depending on the distance from the nucleus. Since the electron bears the electrical charge and the charge's rotation is the electric current, then the magnetic moment of the electron takes place which is equal to the Bohr magneton: 
Proton also has a magnetic moment, exceeding 2.7928456 times the nuclear magneton:
The radius of the electron's orbit specified in Table 1 is the mean radius of the electron cloud and is called the Bohr radius. The orbital velocity of the electron is the rotation velocity of the electron matter at the Bohr radius, which is found from the relation:
The formula for the Bohr radius is as follows:
From this equation it follows that the orbital angular momentum of the electron in the ground state is equal to . In the presented formulas for the velocity and the radius of the electron's orbit small additives are not included which arise when the center of the electron cloud is shifted relative to the proton. In this case, the cloud and the proton rotate around the common center of mass, the electron obtains dynamic spin and loses energy due to electromagnetic emission until it achieves the stationary state of the matter rotation. In the stationary state there is no emission from the electron charge due to the axially symmetric shape of the electron cloud.
Stellar and galactic hydrogen systems
R. Oldershaw model
R. Oldershaw in his model assumes that the stars of spectral type M, with the mass of the order of (where is the Solar mass), are the stellar analogue of the hydrogen atom with the mass .  Then the coefficient of similarity in mass equals . The mass of the object corresponding to the electron can be obtained by multiplying the electron mass by the coefficient of similarity in mass: kg or 26 Earth masses.
The coefficient of similarity in sizes (and in time) according to Oldershaw equals to . Multiplying this quantity by the Bohr radius we can estimate the radius of the stellar hydrogen system: m or 0.039 Solar radii. Since the dwarf stars of main sequence with the mass have the radius about 0.15 Solar radii, the Oldershaw's stellar hydrogen system can be located entirely within the star. Explaining this phenomenon, Oldershaw assumes that as according to quantum mechanics the electron matter is somehow distributed in the atom, so in case of stars, the matter of the object – the electron's analogue can be distributed in the spherical shell of the star. The fact that the radius of the star with the mass exceeds the radius of the stellar hydrogen system in this case is the consequence of the fact that the stellar matter is in an excited state, and the object – the electron's analogue has higher energy levels, which in case of more excitation turn into Rydberg states, in which the object can take the form of separate planets. According to this picture a star with a planet around it is considered as the analogue of a negative hydrogen ion, consisting of a proton and two electrons (one electron corresponds to the planet and the other electron corresponds to the object – the electron's analogue inside the star). Since negative hydrogen ions are rare, Oldershaw predicts sharp minimum in the number of planetary systems with one planet for dwarf stars with the mass around .
In order to obtain the masses of the hydrogen system's objects at the level of galaxies according to Oldershaw it is necessary to multiply the masses of the proton and electron by , which gives kg and kg, respectively. To explain such large masses and the observed interaction at the level of galaxies Oldershaw introduces a new gravitational constant of very small value, which can be found from dimensional equations for this constant. Since the dimension of the gravitational constant is cubic meter divided by kilogram and squared second, and the coefficients of similarity in sizes and time, according to Oldershaw, have equal value, then we obtain:
- m3 /(kg ∙ s2),
where is the gravitational constant.
However, introduction of the gravitational constant for galaxies does not solve the problem completely. Indeed, let the dwarf galaxy with the mass rotate around the galaxy with the mass . The equality of gravitation and the centripetal force in equilibrium in an ordinary case and from Oldershaw's point of view is as follows:
Assuming the rotation velocity of a dwarf galaxy and the distance from the normal galaxy to be equal in both cases, we obtain:
This equality with reasonable masses of galaxies is not satisfied, that makes the validity of Oldershaw's galactic hydrogen system parameters questionable.
Oldershaw also admits that part of the mass is “transformed” in singularities of black holes, located by him inside of the galaxies. Considering the proton and the electron as black holes, he determines their radii by Schwarzschild formula, and then transfers this approach to the level of stars. In this case, another type of hydrogen systems consists of two black holes, one of which, with the mass and the radius about 400 m, corresponds to the proton, and the other black hole, with the mass which is 1836 times less and the radius of about 20 cm, is the analogue of the electron. As a consequence, it is assumed that these black holes are the basis of the dark matter.
S. Fedosin model
Modeling the hydrogen system consisting of a planet and a main sequence star of minimal mass, Fedosin predetermined the mass of such star. This was done by comparing the multitude of all known atomic nuclei and stars of different masses. As a result discreteness of stellar parameters was discovered as similarity between the nuclides of chemical elements and the stars of corresponding masses, as well as similarity with respect to their abundance in the Universe and to their magnetic properties. The mass of a main sequence star of minimum mass is kg, where is the Sun's mass, is the Jupiter mass. The represents the minimum mass of a brown dwarf with a minimum radius and is in good agreement with the data in the paper.  The mass of the planet – the electron's analogue is 1836 times less than the mass of the star corresponding to situation in the hydrogen atom. The mass of such planet is 10.1 Earth masses and it orbits the star at a distance of the order of 19 a.u.
|The star's mass||Mps = 1.11∙1029 kg|
|The planet's mass||Mп = 6.06∙1025 kg|
|The planet's orbital velocity||Vп = 1.6∙103 m/s|
|The planet's orbital radius||RF = 2.88∙1012 m|
The relation between the masses of the objects of hydrogen systems in Tables 2 and 1 and the relations between the orbital velocities and the orbital radii are set by the corresponding coefficients of similarity in mass, speed and size: 
The coefficient of similarity in time, understood as the ratio of time flow rates between atomic time and ordinary stellar systems, equals:
Between the parameters of the stellar hydrogen system there is a relationship, resulting from the balance of gravitational force and centripetal force on a circular orbit:
Based on this relation we determine the radius of the planet's orbit using the known velocity . In turn, the orbital velocity, as in the hydrogen atom, is given by:
where is the fine structure constant, km/s is the stellar speed, which is the characteristic speed of the matter of the star with the mass .
The speed is found based on similarity with the proton, for which the rest energy equals . From the point of view of the principle of mass–energy equivalence, this energy is equal to the binding energy in the field of strong gravitation. For the main-sequence star with minimum mass the corresponding binding energy will equal . The total energies of stars as their binding energies were studied by many authors, which allowed us to determine the speed and the parameters of the stellar hydrogen system. 
The characteristic angular momentum for planetary systems is the orbital angular momentum of the planet – the electron's analogue J∙s. The fine structure constant has the same value in the atomic hydrogen system and in the analogous system for planetary systems, and it can be expressed not only by electromagnetic but also by gravitational quantities:
where is the strong gravitational constant.
Systems with neutron stars
From the point of view of the density of energy and matter, neutron stars are much closer to nucleons than main-sequence stars. Therefore, the similarity between atoms and neutron stars is more exact. Most of the known masses of neutron stars are close to the value , and this mass is taken as the mass of the star – the proton's analogue.  Dividing this mass by 1836 (this number is the ratio of the proton's mass to the electron's mass) we estimate the mass of the object – the electron's analogue. It is equal to 250 Earth masses or 0.78 Jupiter masses.
|The star's mass||M' ps = 2.7∙1030 kg|
|The mass of the object – the electron's analogue||M' п = 1.5∙1027 kg|
|The orbital velocity||V' п = 4.96∙105 m/s|
|The orbital radius||R' F = 7.4∙108 m|
Using the expression for the binding energy of the neutron star as the absolute value of its total energy in the form:
where , km is the star's radius,   Fedosin estimates the characteristic speed of the stellar matter m/s. From this using the fine structure constant, the orbital velocity of the object – the electron's analogue in Table 3 is determined, and using relation (1) we determine the orbital radius:
As the objects – the electron's analogues we assume magnetized disks with a high content of iron, which are discovered near the X-ray pulsars – which are the main candidates to magnetars.  The mean radii of the disks are close to the radius , as well as to the Roche radius, at which the planets are disintegrated due to strong star's gravitation. If we compare with the Solar system, in which the Sun's mass is 1.35 times less than the mass of a neutron star, than the radius turns out larger than the Solar radius and less than the orbital radius of Mercury.
The ratios of objects' parameters in Table 3 and Table 1 give the coefficients of similarity between atoms and neutron stars:
For the coefficient of similarity in time and the characteristic angular momentum for neutron stars we obtain: 
The quantity sets the stellar Dirac constant for compact stars. Using the coefficients of similarity and the dimensional relations for physical quantities we determine the electric charge and the magnetic moment of the magnetar, which is the proton's analogue:
where and are the elementary charge and the magnetic moment of the proton , respectively.
Magnetic field at the pole of the magnetar is equal to T,
where is the vacuum permeability.
Similarity relations also lead to the following formula:
Due to electrical neutrality of the hydrogen system, the disks near the positively charged magnetars must have a charge, opposite in sign and equal in magnitude to . Rotation of disks as well as rotation of the electron in the atom, creates a magnetic moment, which is found by the formula:
where is the magnetic moment of the electron.
Estimating the parameters of the hydrogen system at the level of galaxies, Fedosin takes into account the discreteness of similarity coefficients, arising from the similarity of matter levels and the existence of basic and intermediate levels of matter. Atoms and stars belong to the basic levels of matter, while galaxies belong to the intermediate level of matter.
Since masses and sizes of objects increase exponentially from one level to another, it allows to estimate masses and sizes of carriers at any level of matter by means of respective multiplication by the factors and . Between atoms and stars there are nine more intermediate levels of matter. Hence the coefficient of similarity in mass between the adjacent intermediate levels is found as the tenth root of the coefficient of similarity in mass between atoms and main-sequence stars:
On the other hand, between atoms and stars there are eleven scale levels, nine of which are associated with the sizes of the objects of intermediate levels, and two additional levels take place during transition from the sizes of atomic nuclei to the sizes of atoms. As a result, the coefficient of similarity in size between the adjacent intermediate levels is determined as the twelfth root of the coefficient of similarity in size between atoms and planetary systems of main-sequence stars:
In terms of masses, galaxies are located two levels higher than stars, but in terms of sizes they are six levels higher. This results in the following relations for the masses of galaxies and the orbital radius of a dwarf galaxy in Table 4 :
where is the mass of the main sequence stars of minimum mass, and are the planet's mass and its orbital radius in Table 2 .
|The mass of a normal galaxy||Mpg = 8.15∙109 Mc|
|The mass of a dwarf galaxy||Mgd = 4.43∙106 Mc|
|The orbital velocity||Vgd = 1.3∙103 m/s|
|The orbital radius||Rgd = 6.7∙1023 m|
The mass of is consistent with the mass of a normal dwarf galaxy with minimum radius and minimum luminosity in the article. 
The orbital speed of a dwarf galaxy is estimated with the help of the orbital radius and the mass of the galaxy from the relation similar to (1):
Table 4 shows that Mpc, which is much more than the ordinary distances between galaxies. At the same time the velocity of orbital rotation of a dwarf galaxy is too little compared with the ordinary velocities of galaxies.
Estimation of the characteristic speed of stars in a normal galaxy of minimum mass is made with the help of formula (2) with :
where the volume-averaged radius of the galaxy is determined by multiplying the radius of the main sequence star of minimum mass Solar radii by the sixth degree of discrete coefficient of similarity in size : m = 520 pc. From here the characteristic speed of stars in the galaxy is km/s. In the hydrogen system obtained in Table 4, the ratio of the orbital velocity of a dwarf galaxy to the characteristic speed of the stars in a normal galaxy is approximately equal to the fine structure constant, just as it is in the hydrogen atom and in planetary systems.
In reality the systems containing normal and dwarf galaxies are closer to each other and rotate faster near each other. One explanation of this situation lies in the fact that galaxies do not belong to the basic matter level. A neutron star contains about nucleons, and the same number of particles is supposed in a proton. Meanwhile, in a normal galaxy of minimum mass, usually it is a galaxy of the spiral type, the number of stars does not exceed the value . This number is much less than the number of nucleons in a star. From the point of view of similarity, galaxies contain the same number of stars, as the number of atoms in microscopic dust particles. In contrast to ordinary solid dust particles, the concentration of stars in galaxies is of such kind, that they are similar to strongly rarefied gas clouds, only in the center of which there is solid substance.  If in the hydrogen atom in the ground state the electron's orbital angular momentum is , and the proton's quantum spin has the value , then the orbital angular momentum of a dwarf galaxy can be significantly less than the spin of a normal galaxy. This leads to increase in the orbital velocity of the dwarf galaxy and to a smaller radius of its orbital rotation around the normal galaxy. Probably the loss of the orbital angular momentum by dwarf galaxies is associated with the evolution of galaxies and their formation from large hydrogen clouds, in which the angular momentum is lost due to the friction between the adjacent clouds.
- Comments to the book: Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).
- Sergey Fedosin, The physical theories and infinite hierarchical nesting of matter, Volume 1, LAP LAMBERT Academic Publishing, pages: 580, ISBN 978-3-659-57301-9.
- Robert L. Oldershaw. Critical Test of the Self-Similar Cosmological Paradigm: Anomalously Few Planets Orbiting Low-Mass Red Dwarf Stars. New Adv. Phys., 2009, Vol. 3(2), P. 55-59.
- Theron W. Carmichael. Improved radius determinations for the transiting brown dwarf population in the era of Gaia and TESS. arXiv:2212.02502.
- Fedosin S.G. (1999), Fizika i filosofiia podobiia ot preonov do metagalaktik, Perm, pages 544, ISBN 5-8131-0012-1
- M. de Sá et al. Quantifying the Evidence Against a Mass Gap between Black Holes and Neutron Stars. The Astrophysical Journal, Vol. 941, Number 2, pp. 130 (2022). https://doi.org/10.3847/1538-4357/aca076.
- Fedosin S.G. Cosmic Red Shift, Microwave Background, and New Particles. Galilean Electrodynamics, Vol. 23, Special Issues No. 1, pp. 3-13 (2012). http://dx.doi.org/10.5281/zenodo.890806.
- B.P. Abbott et al. (The LIGO Scientific Collaboration and the Virgo Collaboration). GW170817: Measurements of Neutron Star Radii and Equation of State. Physical Review Letters, Vol. 121, 161101 (2018). http://dx.doi.org/10.1103/PhysRevLett.121.161101. https://arxiv.org/abs/1805.11581.
- Yeunhwan Lim and Jeremy W. Holt. Neutron Star Radii, Deformabilities, and Moments of Inertia from Experimental and Ab Initio Theory Constraints of the 208Pb Neutron Skin Thickness. Galaxies, Vol. 10 (5), Art. 99 (2022). https://doi.org/10.3390/galaxies10050099.
- Wang Zhongxiang, Chakrabarty Deepto, Kaplan David L. A Debris Disk Around An Isolated Young Neutron Star. arXiv: astro-ph / 0604076 v1, 4 Apr 2006.
- Joe Wolf at al. Accurate masses for dispersion-supported galaxies. Monthly Notices of the Royal Astronomical Society, Vol. 406, Issue 2, pp. 1220–1237 (2010). https://doi.org/10.1111/j.1365-2966.2010.16753.x.
- Discreteness of stellar parameters
- Infinite Hierarchical Nesting of Matter
- Mass–energy equivalence
- Quantization of parameters of cosmic systems
- Similarity of matter levels
- Substantial electron model
- Substantial neutron model
- Substantial proton model
- SPФ symmetry
- Stellar constants
- Stellar Dirac constant
- Stellar Planck constant
- Strong gravitation
- Strong gravitational constant
- Systems science
- Systems theory