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Hydrogen system

 

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Hmodel

 

The model of hydrogen system.

 

Hydrogen system is ideal system of two objects held near each other by fundamental forces, with ratio of objects' masses equal to ratio of proton mass to electron mass. The concept of hydrogen system is used to describe similarity of matter levels in Theory of Infinite Hierarchical Nesting of Matter, according to which hydrogen systems characterize simplest and most common systems of two bodies in Universe. Each hydrogen system consists of primary massive object and low-mass satellite rotating around it. At atomic level hydrogen system is hydrogen atom comprising a proton and an electron. Theoretical definition of specific properties of a hydrogen system (mass of primary object, distance to satellite in ground state, etc.) is not univocal and depends on additional assumptions.

Contents

Hydrogen atom

Uniqueness of hydrogen atom is that the most total balance between strong gravitation and electromagnetic forces is achieved in it. [1] Table 1 shows parameters of hydrogen atom, which is standard hydrogen system.

 

 

Table 1. Parameters of hydrogen atom in ground state

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

Radius of electron's orbit

RB = 5.2917706∙10-11 m

 

 

As it is shown in substantial electron model, electron in hydrogen atom in ground state represents a discoidal cloud, with inner edge of disk  ~\frac {R_B}{2} and outer edge  ~\frac {3R_B}{2}. Matter of electron disk rotates around atomic nucleus differentially, that is with different angular velocities, depending on distance from nucleus. Since electron bears electrical charge and charge's rotation is electric current, then magnetic moment of electron takes place which is equal to Bohr magneton: [2]

 

~ P_{me} = \frac {e \hbar }{2 M_e},

 

where ~e   is elementary charge, ~\hbar   is Dirac constant.

Proton also has a magnetic moment, exceeding 2.7928456 times nuclear magneton:

 

~ P_{mp} = 2.7928456 \frac { e \hbar }{2 M_p}.

 

Radius of electron's orbit specified in Table 1 is mean radius of electron cloud and is called Bohr radius. Orbital velocity of electron is rotation velocity of electron matter at Bohr radius, which is found from relation:

 

~ V_e = \alpha c = \frac {e^2 }{4 \pi \varepsilon_0 \hbar},

 

where ~\alpha   is fine structure constant, ~c   is speed of light, ~\varepsilon_0   is electric constant.

Formula for Bohr radius is as follows:

 

~ R_B = \frac {4 \pi \varepsilon_0 \hbar^2 }{ M_e e^2}=\frac {\hbar }{ \alpha c M_e } = \frac {\hbar }{V_e M_e }.

 

From this equation it follows that orbital angular momentum of electron in ground state is equal to ~ L_e = M_e V_e R_B=\hbar . In presented formulas for velocity and radius of electron's orbit small additives are not included which arise when center of electron cloud is shifted relative to proton. In this case, cloud and proton rotate around common centre of mass, electron obtains dynamic spin and loses energy due to electromagnetic emission until it achieves stationary state of matter rotation. In stationary state there is no emission from electron charge due to axially symmetric shape of electron cloud.

Stellar and galactic hydrogen systems

R. Oldershaw model

R. Oldershaw in his model assumes that stars of spectral type M, with mass of order of ~ 0.145 M_c   (where ~ M_c   is t Solar mass), are stellar analogue of hydrogen atom with mass ~M_p . [3]  Then coefficient of similarity in mass equals  X=\frac {0.145 M_c }{M_p}=1.73 \cdot 10^{56}. Mass of object corresponding to electron can be obtained by multiplying electron mass by coefficient of similarity in mass:  ~M_e X=1.58 \cdot 10^{26} kg or 26 Earth masses.

 

Coefficient of similarity in sizes (and in time) according to Oldershaw equals to \Lambda=5.2 \cdot 10^{17}. Multiplying this quantity by Bohr radius he estimates radius of stellar hydrogen system: ~ \Lambda R_B =2.75 \cdot 10^{7} m or 0.039 Solar radii. Since dwarf stars of main sequence with mass ~0.145 M_c   have radius about 0.15 Solar radii, Oldershaw's stellar hydrogen system can be located entirely within the star. Explaining this phenomenon, Oldershaw assumes that as according to quantum mechanics electron matter is somehow distributed in atom, so in case of stars, matter of object –electron's analogue can be distributed in spherical shell of the star. The fact that radius of the star with mass ~ 0.145 M_c   exceeds radius of stellar hydrogen system in this case is consequence of the fact that stellar matter is in an excited state, and object – electron's analogue has higher energy levels, which in case of more excitation turn into Rydberg states, in which the object can take form of separate planets. According to this picture a star with a planet around it is considered as analogue of a negative hydrogen ion, consisting of a proton and two electrons (one electron corresponds to planet and other electron corresponds to object – electron's analogue inside the star). Since negative hydrogen ions are rare, Oldershaw predicts sharp minimum in number of planetary systems with one planet for dwarf stars with mass around ~0.145 M_c .

 

In order to obtain masses of hydrogen system's objects at the level of galaxies according to Oldershaw it is necessary to multiply masses of proton and electron by ~ X^2, which gives ~M_{gp}=5 \cdot 10^{85} kg and ~M_{ge}=2.7 \cdot 10^{82} kg, respectively. To explain such large masses and observed interaction at the level of galaxies Oldershaw introduces a new gravitational constant ~ G_g   of very small value, which can be found from dimensional equations for this constant. Since dimension of gravitational constant is cubic meter divided by kilogram and squared second, and coefficients of similarity in sizes and time, according to Oldershaw, have equal value, then he obtain:

 

~ G_g = G \frac {\Lambda }{X} =2 \cdot 10^{-49} m3 /(kg ∙ s2),

 

where ~ G   is gravitational constant.

However, introduction of gravitational constant ~ G_g   for galaxies does not solve the problem completely. Indeed, let dwarf galaxy with mass ~M_{gd}  rotate around galaxy with mass ~M_{g}. Equality of gravitation and centripetal force in equilibrium in an ordinary case and from Oldershaw's point of view is as follows:

 

~ \frac { G M_{g} M_{gd}}{R^2} = \frac { M_{gd} V^2 }{R},

 

 

~ \frac { G_g M_{gp} M_{ge}}{R^2} = \frac { M_{ge} V^2 }{R}.

 

Assuming rotation velocity ~ V  of a dwarf galaxy and distance ~ R  from normal galaxy to be equal in both cases, we obtain:

 

~ G M_{g} = G_g M_{gp}.

This equality with reasonable masses of galaxies ~ M_g   is not satisfied, that makes validity of Oldershaw's galactic hydrogen system parameters questionable.

 

Oldershaw also admits that part of mass is “transformed” in singularities of black holes, located by him inside of galaxies. Considering proton and 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 mass ~ 0.145 M_c   and radius about 400 m, corresponds to proton, and other black hole, with mass which is 1836 times less and radius of about 20 cm, is analogue of electron. As a consequence, it is assumed that these black holes are basis of dark matter.

 

S. Fedosin model

Planetary systems

Modeling hydrogen system consisting of a planet and a main sequence star of minimal mass, Fedosin predetermined mass of such star. This was done by comparing multitude of all known atomic nuclei and stars of different masses. As a result discreteness of stellar parameters was discovered as similarity between nuclides of chemical elements and stars of corresponding masses, as well as similarity with respect to their abundance in Universe and to their magnetic properties. Mass of a main sequence star of minimum mass is ~M_{{ps}}=0.056M_{c}=58.5M_{j}=1.11\cdot 10^{{29}} kg, where ~M_{c} is the Sun's mass, ~M_{j} is the Jupiter mass. Mass ~M_{{ps}} represents minimum mass of brown dwarf with a minimum radius and is in good agreement with data in the paper. [4] Mass ~ M_{\Pi}   of planet – electron's analogue is 1836 times less than mass of the star ~M_{{ps}}. The mass of such planet is 10.1 Earth masses and it orbits the star at a distance ~R_{F}  of the order of 19 a.u. Importance of hydrogen system for planetary systems is due to the fact that most stars of minimum mass contain at least one planet. Thus, studies show that for every 100 brown dwarfs there are on average 120 planets with masses in range of 0.75 to 3 Earth masses and about 60 more massive planets with masses from 3 to 30 Earth masses. [5]

 

 

Table 2. Hydrogen system for planets and main sequence stars

Star's mass

Mps = 1.11∙1029 kg

Planet's mass

Mп = 6.06∙1025 kg

Planet's orbital velocity

Vп = 1.6∙103 m/s

Planet's orbital radius

RF = 2.88∙1012 m

 

Relation between masses of objects of hydrogen systems in Tables 2 and 1 and relations between orbital velocities and orbital radii are set by corresponding coefficients of similarity in mass, speed and size: [6]

 

\Phi = \frac {M_{ps}}{M_p}=6.654 \cdot 10^{55},

 

S_0 = \frac {V_{\Pi} }{V_e}=7.34 \cdot 10^{-4},

 

P_0 = \frac {R_{F} }{R_B}=5.437 \cdot 10^{22}.

 

Coefficient of similarity in time, understood as ratio of rates of time flow between atomic and ordinary stellar systems, is equal to:

 

\Pi_0= \frac {P_0}{S_0}=7.41 \cdot 10^{25} .

 

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:

 

~ \frac { G M_{ps} M_{\Pi}}{R^2_{F}} = \frac { M_{\Pi} V^2_{\Pi}}{R_{F }}.\qquad\qquad (1)

 

Based on this relation and using known velocity ~V_{{\Pi }}, radius of planet's orbit ~R_{{F}} is determined. In turn, orbital velocity, as in hydrogen atom, is given by:

~V_{{\Pi }}=\alpha C_{s},

where  ~\alpha = \frac {e^2 }{4 \pi \varepsilon_0 \hbar c}   is fine structure constant, ~ C_s = 220  km/s is stellar speed, which is characteristic speed of matter of star with mass ~ M_{ps} .

 

Speed ~ C_s   is found based on similarity with proton, for which rest energy equals ~ E_0 = M_p c^2 . From point of view of principle of mass–energy equivalence, this energy is equal to binding energy in field of strong gravitation. For main-sequence star with minimum mass ~ M_{ps}   corresponding binding energy equal ~ E_s = M_{ps} C^2_s . Total energies of stars as their binding energies were studied by many authors, so it is possible to determine speed ~ C_s   and parameters of stellar hydrogen system. [6]

 

Characteristic angular momentum for planetary systems is orbital angular momentum of planet – electron's analogue \hbar_s= \hbar \Phi S_0 P_0= M_{\Pi} V_{\Pi} R_{F } =2.8 \cdot 10^{41} J∙s. Fine structure constant has the same value in atomic hydrogen system and in analogous system for planetary systems, and it can be expressed not only by electromagnetic but also by gravitational quantities:

 

~\alpha = \frac { V_{\Pi}}{ C_s }= \frac { G M_{ps} M_{\Pi } }{\hbar_s C_s}=\frac{\Gamma M_p M_e}{\hbar c}=\frac {1}{137.036},

 

where ~\Gamma   is strong gravitational constant.

 

Systems with neutron stars

From point of view of density of energy and matter, neutron stars are much closer to nucleons than main-sequence stars. Therefore, similarity between atoms and neutron stars is more exact. Most of known masses of neutron stars are close to value ~ M'_{ps}=1.35 M_c , [7]  and this mass is taken as mass of star – proton's analogue. [1] Dividing this mass by 1836 (this number is ratio of proton's mass to electron's mass) mass of object – electron's analogue is found. It is equal to 250 Earth masses or 0.78 Jupiter masses. [8]

 

Table 3. Hydrogen system for neutron star

Star's mass

M' ps = 2.7∙1030 kg

Mass of object – electron's analogue

M' п = 1.5∙1027 kg

Orbital velocity

V' п = 4.96∙105 m/s

Orbital radius

R' F = 7.4∙108 m

 

Using expression for binding energy of neutron star as absolute value of its total energy in form:

 

~ E'_s = M'_{ps} {C'}^2_s = \frac {\delta G {M'}^2_{ps}}{2R_s},\qquad\qquad (2)

 

where ~ \delta \approx 0.62 ,  ~R_{s}=12 km is star's radius, 9] [10]  Fedosin estimates characteristic speed of stellar matter  ~ C'_s = 6.8 \cdot 10^7m/s. From this using fine structure constant, orbital velocity of object – electron's analogue in Table 3 is determined, and using relation (1) he determines orbital radius:

 

~ V'_{\Pi} = \alpha C'_s ,

 

~ R'_{F} = \frac { G M'_{ps}}{ {V'}^2_{\Pi}} .

 

It is assumed that at the level of star objects – electron's analogues are magnetized disks with a high content of iron, which are discovered near X-ray pulsars – which are main candidates to |magnetars. [11] Mean radii of the disks are close to radius ~ R'_{F}  , as well as to the Roche radius, at which planets are disintegrated due to strong star's gravitation. For the Solar system, in which the Sun's mass is 1.35 times less than mass of a neutron star, radius ~R'_{{F}} turns out larger than Solar radius and less than orbital radius of Mercury.

 

Ratios of objects' parameters in Table 3 and Table 1 give coefficients of similarity between atoms and neutron stars:

 

\Phi' = \frac {M'_{ps}}{M_p}=1.62 \cdot 10^{57},

 

S' = \frac {V'_{\Pi} }{V_e}=2.3 \cdot 10^{-1},

 

P' = \frac {R'_{F} }{R_B}=1.4 \cdot 10^{19}.

 

For coefficient of similarity in time and characteristic angular momentum for neutron stars, the following was obtained: [6]

 

\Pi'= \frac { P' }{ S'}=6.1 \cdot 10^{19} ,

 

\hbar'_s= \hbar \Phi' S' P'= M'_{\Pi} V'_{\Pi} R'_{F} =5.5 \cdot 10^{41} J∙s.

 

The quantity \hbar'_s  sets stellar Dirac constant for compact stars. To determine electric charge and magnetic moment of magnetar, which is the proton's analogue, coefficients of similarity and dimensional relations for physical quantities were used:

 

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 elementary charge and magnetic moment of proton, respectively.

 

Magnetic field at the pole of magnetar is equal to ~B_{s}={\frac {\mu _{0}P_{{ms}}}{2\pi R_{s}^{3}}}=1.8\cdot 10^{{11}} T,

where ~\mu _{0}  is  vacuum permeability.

 

Similarity relations also lead to following formula:

 

\frac {Q_s}{ M'_{ps}} = \sqrt {\frac {4 \pi \varepsilon_0 G M_e}{M_p} }.

Due to electrical neutrality of hydrogen system, disks near positively charged magnetars must have a charge, opposite in sign and equal in magnitude to ~Q_s. Rotation of disks as well as rotation of electron in atom, creates magnetic moment, which is found by formula:

 

P_{m\Pi } = P_{me} {\Phi'}^{0.5} {P'}^{1.5} {S'}^2 = \frac {Q_s \hbar'_s }{2 M'_{\Pi}}= \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 magnetic moment of electron.

 

Galactic systems

Estimating parameters of hydrogen system at the level of galaxies, Fedosin takes into account discreteness of similarity coefficients, arising from similarity of matter levels and existence of basic and intermediate levels of matter. Atoms and stars belong to the basic levels of matter, while galaxies belong to 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 factors ~D_{\Phi }  and ~D_{P}. Between atoms and stars there are nine more intermediate levels of matter. Hence coefficient of similarity in mass between adjacent intermediate levels is found as tenth root of coefficient of similarity in mass between atoms and main-sequence stars:

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

 

On the other hand, between atoms and stars there are eleven scale levels, nine of which are associated with sizes of objects of intermediate levels, and two additional levels take place during transition from sizes of atomic nuclei to sizes of atoms. As a result, coefficient of similarity in size between adjacent intermediate levels is determined as twelfth root of coefficient of similarity in size between atoms and planetary systems of main-sequence stars:

D_{P} = P^{1/12}_0 =78.4538 .

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 following relations for masses of galaxies and orbital radius of a dwarf galaxy in Table 4 :

 

M_{pg} =M_{ps} D^2_{\Phi } ,

 

M_{gd} = M_{\Pi} D^2_{\Phi } ,

 

R_{gd} = R_{F} D^6_{P} ,

 

where ~ M_{ps}  is mass of main sequence stars of minimum mass, ~ M_{\Pi}  and ~ R_{F}  are planet's mass and its orbital radius in Table 2 .

 

 

Table 4. Hydrogen system for galactic systems

Mass of normal galaxy

Mpg = 8.15∙109 Mc

Mass of dwarf galaxy

Mgd = 4.43∙106 Mc

Orbital velocity

Vgd = 1.3∙103 m/s

Orbital radius

Rgd = 6.7∙1023 m

 

Mass M_{{gd}}  is consistent with mass of normal dwarf galaxy with minimum radius and minimum luminosity in the article. [12]

Orbital speed of dwarf galaxy is estimated with the help of orbital radius ~R_{gd}  and mass of galaxy ~ M_{pg}  from relation similar to (1):

 

~ V_{gd} = \sqrt {\frac { G M_{pg} }{R_{gd}}} .

 

Table 4 shows that ~R_{gd} = 22  Mpc, which is much more than the ordinary distances between galaxies. At the same time, orbital rotation speed of dwarf galaxy ~V_{{gd}} is too small compared to usual velocities of galaxies.

 

Estimation of characteristic speed of stars in normal galaxy of minimum mass is made with the help of formula (2) with ~ \delta=0.6 :

 

~ E_g = M_{pg} {C}^2_g = \frac {\delta G {M}^2_{pg}}{2R_g} ,

 

where volume-averaged radius of galaxy is determined by multiplying radius of main sequence star of minimum mass ~ R_s \approx 0.1  Solar radii by sixth degree of discrete coefficient of similarity in size ~D_{P}: ~ R_g = R_s D^6_{P} = 1.6 \cdot 10^{19} m = 520 pc. From here characteristic speed of stars in galaxy is ~ C_g \approx 200  km/s. In hydrogen system obtained in Table 4, ratio of orbital velocity ~V_{gd}  of dwarf galaxy to characteristic speed ~ C_g   of stars in normal galaxy is approximately equal to fine structure constant, just as it is in hydrogen atom and in planetary systems.

 

In reality 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 basic matter level. A neutron star contains about \Phi' = 1.62 \cdot 10^{57} nucleons, and the same number of particles is supposed in a proton. Meanwhile, in normal galaxy of minimum mass, usually it is a galaxy of spiral type, number of stars does not exceed value D^2_{\Phi } = 1.46 \cdot 10^{11}. This number is much less than number of nucleons in a star. From point of view of similarity, galaxies contain the same number of stars, as number of atoms in microscopic dust particles. In contrast to ordinary solid dust particles, 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. [6] If in hydrogen atom in ground state electron's orbital angular momentum is  ~ \hbar , and proton's quantum spin has value ~ \hbar/2 , then orbital angular momentum of dwarf galaxy can be significantly less than spin of normal galaxy. This leads to increase in orbital velocity of dwarf galaxy and to smaller radius of its orbital rotation around normal galaxy. Probably loss of orbital angular momentum by dwarf galaxies is associated with evolution of galaxies and their formation from large hydrogen clouds, in which angular momentum is lost due to friction between adjacent clouds.

References

  1. 1.0 1.1 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).
  2.  Sergey Fedosin, The physical theories and infinite hierarchical nesting of matter, Volume 1, LAP LAMBERT Academic Publishing, pages: 580, ISBN-13: 978-3-659-57301-9.
  3. 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.
  4. Theron W. Carmichael. Improved radius determinations for the transiting brown dwarf population in the era of Gaia and TESS. arXiv:2212.02502.
  5. L. Mignon et all. Radial velocity homogeneous analysis of M dwarfs observed with HARPS. II. Detection limits and planetary occurrence statistics. ArXiv astro-ph.EP. https://doi.org/10.48550/arXiv.2502.06553.
  6. 6.0 6.1 6.2 6.3 Fedosin S.G. (1999), written at Perm, pages 544, Fizika i filosofiia podobiia ot preonov do metagalaktik, ISBN 5-8131-0012-1.
  7. 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.
  8. 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. // Красное смещение и космическое микроволновое фоновое излучение как следствие взаимодействия фотонов с новыми частицами.
  9. 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.
  10. 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.
  11. Wang Zhongxiang, Chakrabarty Deepto, Kaplan David L. A Debris Disk Around An Isolated Young Neutron Star. arXiv: astro-ph / 0604076 v1, 4 Apr 2006.
  12. 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.

See also

External links

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