Discreteness of stellar parameters is the property of distribution of the
observed stars, in which the values of some of their parameters are preferred
and are more common than others.
From the physical point of view, the cause of the
discreteness of stellar parameters is the discreteness associated with the
equations of the state of the stellar matter and with the phase transitions in
this matter. This discreteness leads to dividing all stars into different
types, such as mainsequence stars, subdwarfs,
giants, supergiants, white dwarfs and neutron stars.
For mainsequence stars primary discreteness is also important, arising from
different masses, angular momenta and the temperature of the gas clouds, which
form the stars.
The idea of discreteness of stellar parameters conforms
to the Theory of Infinite Hierarchical
Nesting of Matter, the similarity of
matter levels and the quantization of
parameters of cosmic systems. This is due to the fact that planetary
systems of stars are in many aspects similar to atoms, and at the atomic matter
level the masses of atoms are discrete as well as their other parameters,
including the electric charge and the magnetic moment. The similarity between
the atoms and the stars leads to significant peculiarities of the description
of discreteness of stellar parameters, which specify our understanding of the
evolution of cosmic objects under the influence of the fundamental forces.
Contents

Robert L. Oldershaw since the 70's has been studying the hierarchical
structure of the Universe and the discreteness of the parameters of its
objects. The relations between the sizes, the durations of processes and the
masses of similar objects in his model are as follows: ^{[1]}
where N is the number of the matter level, for example,
the level of stars; N1 is the number of the lower level of matter, for
example, the atomic level of matter; and are dimensionless coefficients, which are subject to be determined.
In the assumption that the Universe is fractal and
consists of selfsimilar objects, the following formula is used: ^{[2]}
where is
the number of objects at the N1 level, which are part of the N level; is the similarity constant or the fractal dimension.
If we assume that , then it
follows that .
Finding the similarity between the atoms and the stars,
Oldershaw faced the question – to which atom does the Solar System correspond?
As a first approximation, he considers Jupiter and the Sun as some analogue of
the hydrogen atom. For the radius of the orbit and the velocity of the electron
in the hydrogen atom in the Bohr theory there are the following relations:
where is
the principal quantum number, is the Bohr radius, is the electron’s velocity on the Bohr radius with .
In the Oldershaw model the velocities of similar objects,
defined as the changes of the corresponding distances per relevant time unit,
have the similarity coefficient equal to unity. This follows from the fact that
in the transition from the lowest level of matter to a higher level, the
distances and the time intervals are multiplied by the same coefficient , and the velocities remain the same.
Hence, for the orbital radius a.u. and the velocity
km/s of Jupiter we must have:
Based on the large value , the Solar system is assumed to be
similar to the Rydberg atom. Although Oldershaw determines by
comparing the Solar system and the hydrogen atom, but to determine the
coefficient of similarity in mass, he does not compare the masses of the Sun
and of the nucleus of the hydrogen atom (or the masses of Jupiter and of the
electron). Instead, he believes that often observed stars of the spectral type
M, with the mass of about (where is the mass of the Sun) are the stellar analogue of the hydrogen atom, with
the mass . Then, the coefficient of similarity in
mass is equal to , .
As an additional argument in favor of such determination
is the fact that the planetary nebula with the white dwarfs located in them
with typical masses are associated as the stellar analogues of
the positive helium ion, which contains four nucleons. With this approach, the
stars of the spectral type K are similar to the helium nuclei, and the Sun must
be the analogue of the nuclide, containing 7 nucleons, like lithium. More
massive mainsequence stars, giants and supergiants are considered as the
stellar analogues of the Rydberg atoms and ions. Since the matter of the
electron, bound in the atom, is assumed to be somehow distributed over the
volume of the atom, the matter of the stellar analogue of the electron in
planetary systems can be in the form of the spherical shell of the star with
small , or can be in the
form of planets with large . Using the radii of the atoms and ions in
ordinary and up to the most highly excited Rydberg states, and multiplying
these radii by , Oldershaw simulates the observed radii
of the mainsequence stars, giants and supergiants. Table 1 shows the masses of
the stars of the lower part of the main sequence, which are expected in the
Oldershaw model.
Table 1. The
expected discreteness of the stellar masses 

Atoms / Isotopes 
Number of nucleons, N 
The atomic mass, Da 
The predicted
mass of the stars, 
H^{1} 
1 
1.008 
0.146 
H^{2} 
2 
2.014 
0.292 
He^{3}, (H^{3}) 
3 
3.016 
0.437 
He^{4} 
4 
4.003 
0.580 
Stable nuclei with 5 nucleons are
absent 
– 
The "gap" in the distribution
of the nuclear masses 
The "gap" in the
distribution of stellar masses with 0.73 
Li^{6}, (He^{6}) 
6 
6.015 
0.872 
Li^{7}, (Be^{7}) 
7 
7.016 
1.017 
Li^{8}, (Be^{8}, B^{8}) 
8 
8.005 
1.160 
Be^{9}, (Li^{9}) 
9 
9.012 
1.306 
Oldershaw also compares the variable stars with the Rydberg atoms. In
particular, the stars of the type RR Lyrae are
considered to be the analogues of the neutral helium atom, in which there are
transitions of electrons between the states with the principal quantum number from 7 to 9. ^{[3]}
In the Rydberg atoms for the motion of the electron, like in the planetary systems
for the motion of planets, the relation holds between the square of the orbital
period of rotation and the cube of the orbit’s radius: . Oldershaw transfers this relation to the
different types of variable stars, recalculating the coefficients into the coefficients for the level of stars
by multiplying by . Thus he connects the oscillation periods
of the brightness of variable stars with their radii.
Until 1985 Oldershaw believed that the objects of any level
of matter are composed mostly of the objects of the lower level of matter with
almost no changes in their state. Then he changed his opinion, attributing to
the black holes the dominant role in the cosmological hierarchy. In this case,
the objects of the lower levels of matter form the observed mass, but these
objects change dramatically in the singularities of black holes. From the
coefficients of similarity in size and time obtained by Oldershaw , it follows that the radius of the star, the
analogue of the proton, can be obtained by multiplying the radius of the proton
by . This gives
approximately the same radius, which the star would have with the mass , if it were a black hole:
m,
where is the gravitational constant, is
the speed of light.
The fact that the dwarf stars of the spectral type M,
considered to be the stellar analogue of the proton, have much larger radii, is
explained by the fact that these stars are in the excited state. Oldershaw also
uses the Schwarzschild formula for the radius of the black hole, in order to
estimate the radius of the proton:
m,
where m^{3}•s^{–2}•kg^{–1}
is the strong gravitational constant,
assumed by Oldershaw, which is found by him through the ordinary gravitational
constant and the similarity coefficients with the help of dimension relations.
Oldershaw assumes the mediummass white dwarfs, based on
their typical masses of about and
, to be the
stellar analogues of the positive helium ions He(3) and He(4) in the ground
state. Multiplying the radius of the helium ion ( ) by he
obtains the value of the order of m as the radius of typical white dwarfs.
The radius of white dwarfs decreases with increasing of their mass, which is
consistent with the decrease of the radii of hydrogen ions with increasing of
their mass and charge. It can be noted that the morphology of planetary
nebulae, surrounding some white dwarfs, is in many ways similar to the
morphology of the electronwave function in the atoms. The rotation periods of
white dwarfs are grouped near the values of seconds and s. If we divide these periods by the
coefficient of similarity in time , we obtain seconds and s, respectively. These periods are close to the oscillation periods of the
electromagnetic emission in the electron transitions in helium ions. ^{[4]} This coincidence can be partly explained by the fact
that the frequency of electron emission is close to the frequency of its
orbital rotation in the atom and the electron motion is regulated by the action
of strong gravitation, which is
approximately equal in magnitude to the electric force. It turns out that the
rotation of the electron in some ways is similar to the rotation of the surface
of white dwarfs.
Oldershaw notes that many stellar systems demonstrate the
dependence of the angular momentum on the square of the mass of the form , and the atomic systems – the dependence
of the form , where the coefficients and have the dimension . The ratio can be found using the similarity
coefficients and the dimension theory:
The logarithm of the ratio is
equal to 38.51, which is consistent with the average empirical estimates,
giving the value . Similarly, the correlation is derived
for the dependences between the magnetic moment and the spin of stellar and
atomic systems: , . From the similarity coefficients it
follows that , and the
observations give .
The atomic masses are almost entirely determined by the
masses of their nuclei, ranging from 1.00794 Da
= 1.6737∙10^{−27} kg for hydrogen up to about 207.9766521 Da for the
heaviest stable isotope – lead Pb(208). More massive
atoms contain radioactive nuclei and decay with time. The masses of
mainsequence stars are usually not more than 50 solar masses , and can be less than .
To determine the analogue of the Solar system at the
atomic level Sergey Fedosin applied the mathematical procedure based on two
ideas: 1) The number of planets in the Solar System is 8, if we do not assume
Pluto to be a real planet due to its small mass and size (almost like the asteroid
Ceres), very large orbital inclination to the ecliptic, the significant
eccentricity, slow proper rotation, the direction of its intrinsic angular
momentum which is not perpendicular to the ecliptic plane (like in the case of
most of other planets) but is in parallel to the ecliptic. 2) The masses of
stars, including the Sun, are discrete almost the same way as it happens in the
case of atoms.
Hence it follows that the Solar system by the number of
planets, as the analogues of electrons, can be similar to the atom of the
isotope of oxygen or fluorine, with the corresponding atomic number . To verify this assumption we make the
proportion:
where is
the exactly known mass of some star, is the mass number for this star.
Equation (1) is similar to the equation for the mass ratio of the two
nuclides and the ratio of their mass numbers. From (1) we can determine the
mass number of the star through its mass: . If the discreteness of the stellar
masses were exact enough, then in case of the correct choice of the mass number
for the Sun, the mass numbers of various stars would be almost integer numbers, and the condition would be satisfied:
where is
the integer part of the number for the ith
star, and the function with some choice of has the minimum.
The exact masses of the stars were found from the Svechnikov’s catalog, ^{[5]} and
from the data of other authors, and the function was calculated for various in
the range from 15 to 21. As a result it turned out that the minimum of the
function is
reached at , so that the Solar system is the analogue
of the stable isotope of oxygen O(18). This implies the following: 1) The
nucleus of the hydrogen atom corresponds to the star with the minimum mass of
about ,
which is 58 Jupiter masses. Such stars are now discovered and are called brown
dwarfs. 2) The coefficient of similarity in mass between the atoms and the
mainsequence stars is equal to the ratio of the Sun’s mass to the mass of the
nuclide with the atomic number equal to 18. This gives the value of the
similarity coefficient 3) The electron corresponds to the
planet with the mass equal to 10.1 Earth masses, which is less than the mass of
Uranus. ^{[6]}
The results of studies of various parameters of stars,
averaged over the set of wellstudied mainsequence stars, are given in Table
2. ^{[6]} For the stars with masses and less the more accurate average radii are
additionally given in brackets, measured by the longbaseline interferometer. ^{[7]}
Table 2. The
characteristics of mainsequence stars 

Mass, 
Spectral 
Radius, 
Luminosity, 
Temperature, 
Density, 
Bolometric 
26 
O8 
8.7 
180000 
40400 
56 
−8.4 
20.5 
O9 
7.4 
93600 
37200 
71 
−7.7 
13.3 
B0 
6 
22400 
28800 
87 
−6.1 
11.1 
B1 
5.4 
12900 
26400 
99 
−5.5 
9.4 
B2.5 
4.9 
7850 
24600 
113 
−5 
6.55 
B2.9 
4.05 
1990 
19200 
139 
−3.5 
6.11 
B3 
3.85 
1484 
18300 
150 
−3.2 
4.89 
B4 
3.35 
593 
15600 
183 
−2.2 
4.39 
B5 
3.1 
344 
14150 
208 
−1.6 
4 
B7 
2.9 
238 
13350 
230 
−1.2 
3.2 
B8− B9 
2.5 
103 
11650 
289 
−0.3 
2.8 
A0 
2.25 
66 
11000 
346 
0.2 
2 
A5 
1.75 
20 
9230 
526 
1.5 
1.8 
F0 
1.6 
13.7 
8800 
620 
1.9 
1.5 
F5 
1.4 
6.54 
7820 
770 
2.7 
1.33 
G0 
1.28 
4.1 
7280 
840 
3.2 
1.07 
G5 
1.05 
1.49 
6240 
1300 
4.3 
0.85 
K0 
0.88 (0.8) 
0,545 
5300 
1760 
5.4 
0.65 
K5 
0.72 (0.61) 
0.22 
4650 
2480 
6.4 
0.52 
M0 
0.6 (0.48) 
0.0944 
4140 
3400 
7.3 
0.22 
M5 
0.3 (0.23) 
0.0065 
3000 
11500 
10.2 
0.11 
M7.25 
0.17 (0.13) 
0.001 
2520 
31600 
12.2 
0.056 
M8.5 
0.128 (0.07) 
0.0001 
1630 
38000 
14.7 
In Table 2 the characteristics of stars are given in relation to the mass , the radius and the luminosity of the Sun; the effective temperatures of the stellar surface are given in Kelvins, the average densities of the stellar
matter and their bolometric stellar magnitudes are given, which take into
account the total emission from the stars. With this data we can plot various
smooth curves, for example, the dependence of the stellar radius on their mass.
Based on the characteristics of the stars from Table 2
and the assumption that the Solar system is the analogue of the oxygen isotope
O(18), we can first build the average dependence of the spectral class of stars
on their mass, and then with the help of relation (1) we can find the mass
numbers of
the stars as the function of the mass . Since the mass numbers of stars and of
the atomic nuclei similar to them coincide, then it becomes possible to find the
correspondence between the spectral classes of stars and the chemical elements
according to Table 3. ^{[6]}
Table 3.
Correspondence between the spectral classes of stars and chemical elements 

Spectral class 
Chemical elements 
B0 
Fr, Ra, Ac, Actinides, Ku, Nc, etc. 
B1 
Au, Hg, Tl, Pb,
Bi, Po, At, Rn. 
B2.5 
Cs, Ba, La, Lanthanides, Hf, Ta, W, Re, Os, Ir, Pt. 
B2.9 
Ag, Cd, In, Sn, Sb, Te, I, Xe. 
B3 
Ru, Rh, Pd. 
B4 
Rb, Sr, Y, Zr, Nb,
Mo, Tc. 
B5 
As, Se, Br,
Cr. 
B7 
Cu, Zn, Ga,
Ge. 
B8−B9 
Fe, Co, Ni. 
A0 
Sc, Ti, V, Cr,
Mn. 
A1 
Ca 
A2−A4 
Ar, K. 
A5−A6 
Cl 
A7−F0 
S 
F2 
P 
F3−F8 
Si 
F8.5 
Al 
F9−G0 
Mg 
G1 
Na 
G2−G4 
Ne 
G5 
F 
G7−K1 
O 
K2 
N 
K2.5−K7 
C 
M0 
Be, B. 
M4 
Li 
M5 
He 
M7.25 
D (deuterium) 
M8.5 
H 
Table 3 shows that almost all mainsequence stars correspond to the chemical
elements of the periodic table. The stars of the spectral type O, which are supergiants, and the superheavy
chemical elements are absent in the Table 3 due to their extremely small
number. In particular, the estimate of the number of the stars of the spectral
type O in the galaxy Large Magellanic Cloud gives the
number not more than 1000, with the total number of stars in the galaxy of the
order of 10^{10}. ^{[8]}
Discreteness of parameters of mainsequence stars reveals
in the fact that the stars with some values of masses are much more prevalent
in number than the stars with other values of masses. This is demonstrated by
catalogs of stars, containing tens or even hundreds of thousands of stars which
can be placed on the plane in the coordinates "the absolute stellar
magnitude – the spectral class", with indication of their number in each
point of the plane. For example, in the Michigan spectral catalogue of stars ^{[9]} we can clearly see that in the spectral classes near
A0 and F5 there are local maxima of stars. With the help of Table 2 we can turn
from the spectral classes of stars to the masses of stars, and from relation
(1) we can calculate the atomic masses of
these stars. The same result is obtained from Table 3, connecting the spectral
classes of stars and the corresponding chemical elements. This allows us to
build on the basis of the Michigan spectral catalogue the dependence of the
relative abundance of stars on their mass number.
In this case it is necessary to make a correction that
the observed abundance of stars differs from the actual due to different
luminosity of stars (bright stars are visible from far away at the distances,
at which the existing faint stars stop being detected). If is the visible number of stars with the
luminosity , then the real number of these stars in the first approximation is given by: ^{[6]}
The actual distribution of stars is shown in Figure 1,
and it can be compared with the distribution of chemical elements. There are
two main distributions of chemical elements: the first – for meteorites and the
Earth's crust, and the second – for the Sun, planetary nebulae and stars. It
turns out that there is close similarity between the distribution of the
abundance of stars in the Galaxy, and the distribution of chemical elements in
the Sun and stars in Figure 2. The similarity of both distributions is also
stressed by the fact that in the range of mass numbers from 35 to 55 in both
figures there is a dip, after which in Figure 2 the socalled iron peak begins.
The figures show that in the Universe the lowmass chemical elements dominate
and the stars corresponding to them.
Over 70 % of the observed stars are part of binary
and multiple systems, ^{[11]} just as atoms are
combined in molecules. With the help of determining the elements of orbits of
the visual binary stars we can very accurately find the masses of the
components. The study of the catalogues of binary stars allows us to show that
most of the stars in the pairs are connected the same way as the atoms, corresponding
to them by mass, form the chemical molecules. With the distances between the
components of pairs less than 50 a.u. there are
extremums in the distribution of the angular separation of the components,
similar to the distribution of the bond lengths in diatomic molecules. The
ratio of the distances between the components of stars to the bond length of
the corresponding molecule gives the estimate of the coefficient of similarity
in size, which is close to the coefficient of the similarity in size between
the hydrogen atom and the corresponding planetary system: . ^{[6]}
The distribution of the orbital planes of binary stars in
the Galaxy Milky Way is rather chaotic, but the longperiod stellar pairs
mostly have the same direction of rotation as the Galaxy as a whole. ^{[12]} For the shortperiod binaries the situation is the
opposite, which is the consequence of the differential rotation and the
dynamics of interaction in approaching of the stars. In the Galaxy the stars
are grouped together in tight groups, open and globular clusters, are part of
the disc and the spherical component. If we count the stars to be similar to
atoms, then from the point of view of the similarity
of matter levels all the known galaxies, by the number of their component
stars, are similar to the dust particles of the corresponding chemical
composition, and the masses and sizes of galaxies and dust particles are
connected by the similarity coefficients. The concentration of stars in the
Galaxy is of such kind that it corresponds to sufficiently rarified gas of
complex chemical composition, and only with the radius of less than 0.047 pc
the "solid substance" will appear, which is similar by its density to
coke, and is rotating as a solid body relative to the center of mass of the
Galaxy. In the central part of the Galaxy and in the disc the massive stars
dominate, corresponding to the atoms of metals and heavy nonmetals, and the
lighter stars in the spherical component of the Galaxy are the analogues of
volatile gases such as oxygen, nitrogen, hydrogen, etc. In addition, if we move
from the Galactic center and consider the amount of metals in the stars come
across, then it will continuously decrease, reflecting the logical evolution of
stars in galaxies.
Discreteness of parameters of cosmic objects does not
stop at the stars, it is found at the level of galaxies. For example, our
Galaxy has the mass number and approximately corresponds to oxygen. This
follows from the coefficients of similarity and the number of dwarf galaxies surrounding
the Galaxy like the electrons in the oxygen atom. The tight group of galaxies,
consisting of the Galaxy and the Large and Small Magellanic
Clouds, can be considered as the water molecule. ^{[6]}
The large neighboring galaxy, the Andromeda Galaxy, has the mass number up to , and forms a kind of molecule with the
Triangulum Galaxy ().
The characteristic
speed of
the matter particles of the object, held in the gravitational field, is given
by: ^{[6]}
where and are the mass and the radius of the object, is the gravitational
constant, is
the coefficient depending on the matter distribution, in the case of the
uniform mass density .
This equation is the relation between the internal energy
of the object, as the kinetic energy of its matter particles, and the energy of
the object in the gravitational field. The absolute value of full energy is
proportional to the mass, which is revealed as the mass–energy equivalence. From equality
(2) we can find the characteristic speeds through the masses of objects, in
this case the discreteness of masses of mainsequence stars implies the
discreteness of their characteristic speeds and of the total energies of stars.
For the stars with the minimum mass the characteristic speed of its matter is km/s. The analogue of this star is the proton, the characteristic speed of
the matter of which is equal to the speed of light . The ratio of these speeds specifies the coefficient
of similarity in speed for the hydrogen
system.
If we consider the hydrogenlike atoms and the stellarplanetary
systems corresponding to them, then the velocity of motion of the electron in
the atom is proportional to the charge of
the atomic nucleus, and the velocity of the planet is proportional to the mass
number . It follows that the coefficient of
similarity in speed between atoms and stars is proportional to the ratio : . Assuming that the same holds for the speed of
the matter in the star, for the characteristic speed and the total energy of
mainsequence stars we obtain the expressions:
These ratios well approximate the results of numerous
calculations of the total energies of stars, made in different ways by various
authors (see references in Fedosin SG, 1999. ^{[6]}).
For hydrogenlike systems the coefficient of similarity in size has the form: .
The speeds are boundary for the maximum speeds of
stellar surfaces’ rotation, as well as for the average velocities of the motion
of stars relative to the stellar systems in which these stars have been formed
(the principle of local stellar velocity).
The observed discreteness of masses, typical sizes and
angular velocities of rotation of stars leads to the discreteness of the
angular momentum of the proper rotation of stars. To estimate the value of the
characteristic angular momentum for mainsequence stars, we should multiply the
Dirac constant by the coefficients of similarity:
J∙s.
The quantity specifies the orbital angular momentum of the
planet, the analogue of the electron, during its rotation in the hydrogen
system around the star of minimum mass, which is the analog of the proton. On
the other hand, is
almost exactly equal to the angular momentum of the proper rotation of the Sun,
which is equal to J∙s. ^{[13]} The
characteristic angular momentum can be also compared with the maximum angular
momentum of rotation of the star of minimum mass. In case of the limiting
rotation of the star at its equator, the acceleration of gravitation and the
centripetal acceleration are equal:
where is
the equatorial velocity.
In view of this and the parameters of the star of minimum
mass from Table 2 the limiting spin of the star, the analogue of the proton, is
equal to:
J∙s,
where is
the coefficient depending on the distribution of matter in the star.
The similar formula for the limiting spin of the proton
gives:
J∙s,
where is
the coefficient depending on the distribution of matter in the proton, is
the strong gravitational constant.
The Heisenberg uncertainty principle for changing the
energy of the quantum process and the time interval of this process sets the
limiting connection with the Dirac constant:
The similar in the meaning relation for the stellar level
is obtained if in the free fall of the matter with the mass to
the volume with the radius during the time , the total gravitational binding energy is released and the star of minimum mass is formed:
J∙s,
where the gravitational acceleration is estimated by the
formula , and the radius of the fall – by the formula
.
In this ratio the product of the energy change and the
time of the change coincides by the order of magnitude with . In the Galaxy during its formation the
separation of gas clouds into fragments takes place, of which the stars are
formed. The mass of the matter, of which this or that star is formed, is not
isolated, since it is influenced by the forces of gravitation from the other
fragments. As a result, the real time of the formation of stars with the masses
less is determined by the time of the accretion of
the shell , and with large masses – by the
KelvinHelmholtz time . The product of stellar energy and the
real time of their formation in the Galaxy sets the new characteristic angular
momentum . This angular momentum is close by its
value to the average orbital angular momentum of the stars in the Galaxy. For
the Sun the orbital angular momentum is equal to J∙s, and using the coefficients of
similarity for the galaxies we obtain J∙s. ^{[6]}
All the known stars can be divided into two great classes
– nonmagnetic and magnetic. This division is to some extent conventional,
since nonmagnetic stars have small total dipole magnetic field, but in some
points of the surface they can have significant local magnetic fields. Magnetic
stars have the dependence between the angular momentum of the proper rotation
and the total magnetic moment, and we can observe the change of sign and
overturn of the dipole magnetic moment like on the Sun. Magnetic stars usually
rotate 2 – 4 times more slowly than nonmagnetic, and their matter composition
has excess of elements of the type of iron and rareearth elements.
Figure 3 shows the distribution of magnetic stars by
spectral classes and the field strengths at the surface, showing clear
discreteness of the magnetic properties. The stars of the spectral class A0
have the largest magnetic fields. If using Table 3 we find the analogues of
these stars at the level of atoms, we shall obtain the atomic nuclei of the type
Sc (45), Ti (47), Ti (49), V (50), V (51), Cr (53), Mn
(55), Co (59). Indeed, among the atomic nuclei these nuclei have the largest
magnetic moments, except the extremely rare nuclides Nb
(93), Tc (99), In (113). The magnetic stars are seen in the spectral classes
near A2, A3, F0, F2, F5, which correspond to the magnetic nuclides Ca (43), K
(3941), Cl (3537), S (3335), P ( 31), Si (29), Al (27). Finally, to the
spectral class M such magnetic nuclides correspond as He (3), Li (7), Be (9), B
(1011). How can we explain such similar distributions by the magnetic
properties of the stars and the atomic nuclei similar to them? According to one
of the assumptions, the similarity of stars and the atomic nuclei by their
masses is supplemented by the fact that the stars contain increased
concentrations of the atoms to which these stars are similar.
There are two main hypotheses describing the magnetic
fields of stars. The dynamo theory suggests that the selfsustaining
magnetic field is possible due to the convective fluxes of the conductive
matter in the interior of stars and planets, the effect of electromagnetic
induction and the centripetal forces of rotation. However, there is still no
consistent mechanism of dynamo even for the Sun, which would accurately take
into account the change of sign of its dipole magnetic field. ^{[15]} In another hypothesis the origin of the magnetic
field of stars is associated with their rotation. It is noted that the magnetic
moments of planets, stars and even galaxies in the dependence "magnetic
moment – spin" are located within two parallel lines. ^{[6]} The upper line corresponds in view of the
similarity coefficients to the Bohr magneton, and the lower line corresponds to
the nuclear magneton. The inclination of the lines is equal to unity, so that
the magnetic moments are directly proportional to the spin (for the planets
they are proportional to the spin of the planets’ nuclei). In his
electrokinetic model Sergey Fedosin justifies the emergence of the magnetic
field based on the concept of separation of charges in the matter of cosmic
bodies. In this model, the magnitude of the magnetic field is proportional to
the angular velocity of the body’s rotation and the radius of the convective
layer. At the same time the periods of change of the polarity of the magnetic
field of the Earth and the Sun are calculated through the sizes of the
convective layer and the convection speed of the matter. The solar activity is
the consequence of the periodic conversion of the thermal energy into the
electromagnetic form of energy. ^{[16] [17]}
In contrast to mainsequence stars, the mass density of white
dwarfs and neutron stars is much higher, equal by the order of magnitude to 10^{9}
kg/m^{3} and 10^{17} kg/m^{3}, respectively. If in
ordinary stars the gravitational pressure is balanced by the pressure of
thermally ionized plasma, in white dwarfs – by the pressure of electrons, then
in neutron stars the force of gravitation is counteracted by the pressure of
the degenerate neutron gas. White dwarfs are essentially the cores of ordinary
stars, in which thermonuclear reactions reach the final stage with ejection of
the stellar shell at the stage of red giant. It is considered that all stars
with masses up to must turn into white dwarfs. There is the
Chandrasekhar limit, equal to approximately , above which the white dwarf can become a
neutron star. The chemical composition of white dwarfs is determined by the
initial mass of stars, of which they are formed. Depending on the initial mass,
thermonuclear reactions occur with hydrogen burning and its transformation into
helium, helium also can burn, giving carbon and more massive nuclei of oxygen,
neon, magnesium. As a result dense hydrogen stars of low mass can appear, as
well as helium, carbon and more complex composition white dwarfs, and the
discreteness of the mass of mainsequence stars is supplemented by additional
discreteness associated with the evolution and the chemical composition of the
emerging white dwarfs.
The theoretical range of changing of the masses of
neutron stars is from to
. The lower
mass limit is associated with the instability of the matter in the form of neutron
liquid due to the small gravitational pressure in the star, which can lead to
the destruction of lowmass star in explosive way. The upper mass limit is
called the Tolman–Oppenheimer–Volkoff limit. It is assumed that with large masses the
gravitational pressure overcomes the internucleon repulsive forces in the star and
it collapses into a denser object of the type of hypothetical quark stars or
black holes. Most of the observed masses of neutron stars, found accurately
enough in binary systems with pulsars, differ little from the Chandrasekhar
limit and are equal to . Probably, some neutron stars reach the
mass and more. Such discreteness of masses is
explained by the equation of the state of matter and the standard way of
formation of stars in supernova outbursts, when the excess mass is expelled
from the surface of the emerging neutron star. The radii of the stars are in
the range from 11 to 15 km, the uncertainties here arise from inexact knowledge
of the equations of state in the theoretical modeling, and from the ambiguity
of interpretation of the observed radius of the photosphere relative to the
radius of the star.
Using relation (2) we can calculate that the
characteristic speed of the matter particles for white dwarfs is in the range from 930 to
4000 km/s, and for neutron stars – from 17000 to 71000 km/s. For planets the
speed does not exceed 52 km/s (for the Earth it is
4.3 km/s).
If in the Oldershaw hierarchical model the similarity
coefficients between the atoms and the stars do not depend on the type of
stars, then in the Fedosin model it is not so. As the model of the proton the
neutron star is considered with the mass kg and the radius m and the characteristic speed m/s. The corresponding parameters of the proton: the mass kg, the radius m, the characteristic speed m/s (the speed of light). Hence, we find the coefficients of similarity: in
mass , in sizes , in speeds . The coefficient of similarity in the
time of processes’ duration has the form , and it is not equal to the coefficient of similarity in size. ^{[6]} ^{[17]}
With the help of the coefficients of similarity and the relations
of dimension of physical quantities we can calculate the characteristic angular
momentum of compact objects in the form of the stellar Dirac constant, the electric
charge and the magnetic moment of the star, the analogue of the proton:
J∙s,
C,
J/T.
Here and are the elementary charge and the magnetic moment of the proton.
According to the substantial neutron model and the substantial proton model it is assumed that
the neutron’s analogue is the ordinary neutron star, and the proton’s analogue
is the magnetar
carrying the electric charge and the magnetic moment . Due to its large charge the magnetar is
able to generate highenergy cosmic rays. ^{[17]}
For comparison, in the Oldershaw model, based on his similarity coefficients
for the stars, the significant electrical charge of the order of C
is admitted.
The pion is the hadron of the smallest mass, among the
compact objects its analogue is the neutron star with the mass . The white dwarf with the mass and the radius m corresponds to muon. Since the charged
pion turns into the muon, then it is expected that the neutron star, the analogue
of the pion, also turns into the white dwarf with time, due to the reactions of
weak interaction in the matter of the star. The atomic nuclei as the compound
of nucleons correspond at the level of stars the groups of neutron stars that
are closely bound by gravitational forces and gravitational
torsion fields. Such groups of stars can be located in massive Xray
systems and in the centers of galaxies. Neutron star, proton, as well as
Metagalaxy, considered as relativistic
uniform system, turn out to be extreme objects in terms of the dependence
of their gravitational field on the radius. ^{[19]}