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 longrange 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. ^{[1]} 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. ^{[2]}
From this it follows that discreteness of the
fundamental properties of main objects and their satellites, which arises in
the course of coevolution 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.
Contents
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 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 . In atoms and ions the following quantities are quantized:
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]}
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 during the change of the angular
momentum : ^{[1]}
.
The typical change of the angular momentum of the
electron during transition between the energy levels is , due to which it is assumed that the energy of the emitted photon as an
emission quantum equals .
Quantization of the number of nucleons in atomic nuclei
is closely related to the discreteness of the nuclear masses and to MattauchShchukarev 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 C. With the help of the Planck
constant and the elementary charge we calculate the magnetic flux quantum 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. ^{[2]}
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 microlevel quantization is found in many
phenomena, at the macrolevel 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. ^{[3]} 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:
a.u.,
where is for Mercury,
is for to Venus, Earth and Mars, 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 , the law demonstrates that distances to the planets increase approximately
exponentially. Dependency on 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: ^{[4]}
a.u.,
where
Then based on the Titius–Bode
law we can find the positions of planets as arithmetic mean values between two
adjacent boundaries:
,
with .
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 gasdust
matter of protoplanets during their accumulation. ^{[2]}
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. ^{[5]} Nearly for all the planets we were
able to choose the appropriate quantum numbers and
, 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. ^{[6]}
^{[7]} For the orbital radii of planetary systems a formula was found:
,
where is the gravitational
constant, is the mass of the central body (a
star in the planetary system, a planet in the satellite system), 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 km/s is for the inner planets of
Mercury, Venus, Earth and Mars, for which , respectively. For Jupiter, Saturn,
Uranus, Neptune and Pluto at a lower value km/s. Different values 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 is used to show that the inner and outer planets of the Solar system are
divided into two separate groups. ^{[8]} ^{[9]} One of the
results is the formula for the orbital radii of planets:
where is the radius with . For the inner terrestrial planets m and the orbits with and are empty due to the assumed
influence of the Sun, for Mercury , for Venus , for the Earth , for Mars , for the asteroid Ceres . For the outer planets, starting with Jupiter, it is assumed that m, .
If we denote the orbital angular momentum of a
planet as , the specific orbital angular momentum as , and if we use the expression
for the orbital velocity in the form , where is the mass of the Sun, then in
view of (1) we obtain:
.
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: .
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. ^{[4]}
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. ^{[8]}
Similarity to atoms
Similarity coefficients
Considering the correspondence between atoms and
mainsequence stars, Sergey Fedosin discovered that the Solar system is similar
to the isotope of oxygen atom with the mass number and charge number . ^{[10]} The ratio of the Sun's mass to the nucleus mass of the
isotope of oxygen atom sets the coefficient of similarity
in mass:
.
Multiplying by the electron mass, we can find
the mass of the planet corresponding to the electron: kg or 10.1 of Earth masses.
The coefficient of similarity in speed is given by:
,
where is the coefficient of similarity
in speed for the hydrogen system, km/s is the characteristic speed of the matter particles in the main sequence star with minimum mass , 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 is determined by half of the
gravitational energy, the absolute value of which defines the full energy of
the star with the mass :
where is the radius of the star, 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 , where is the mass of the nucleus, by the
similarity coefficients it is possible to find the absolute value of full
energy of the mainsequence star:
.
The coefficient of similarity in energy between the
oxygen atom and the Solar system is defined by the product . 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 energiy. ^{[10]}
The coefficient of similarity in size is given by:
,
where is calculated from the ratios of the
hydrogen system. If we multiply the Bohr
radius in the hydrogen atom by , 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 semiaxes 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 hydrogenlike ion of oxygen; 4) Comparing the dimensions of atomic
nuclei and stars.
The coefficient of similarity in time is:
,
where .
By means of similarity coefficients the stellar Dirac constant is determined for
planetary systems of mainsequence stars:
J∙s,
where is the Dirac constant. The quantity
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. Klayer includes 1sshell and two
electrons. Llayer includes 2sshell with two electrons and 2pshell with four
electrons. It is assumed that electrons in sstates 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 1sshell, Earth and
Mars – to 2sshell and the large planets, the orbital angular momenta of which
are much greater than those of the terrestrial planets, are similar to 2pshell
of the oxygen atom. In the Bohr model of hydrogen atom the orbital angular
momentum of electron is quantized:
.
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 hydrogenlike atom, then for specific orbital angular
momenta of planets we obtain quantum relation with accuracy up to 25 %: ^{[10]}
,
where kg is the mass of the planet corresponding to electron, follows from correspondence with
the empirical data.
If we substitute here the expressions for the
orbital velocity in the form , we could
determine the orbital radii of planets:
.
The quantity
m^{2}/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 m^{2}/s, while for Jupiter's
satellite Callisto this quantity is much less: m^{2}/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: ^{[1]}
,
where and are the orbital
angular momentum and the mass of the satellite in a circular orbit, and are the average velocity in orbit
and the average orbital radius of the satellite, is the
revolution period in orbit with the number , is a constant,
which depends on the satellite system.
In all satellite systems there is no satellite with
the number , 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 () and ending with Callisto . 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 m^{2}/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 to Iapetus with ), and for Uranus (regular satellites from Cordelia (1986 U7) with to Oberon with ), for which m^{2}/s and m^{2}/s, respectively.
The meaning of the quantities , and 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 , and , 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 gasdust
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 are equal to as in Bohr
theory for the electron, then the total orbital angular momentum is:
,
where 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 and the total orbital angular
momentum of planets: . The quantities 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 and the charge numbers of the stars, determined
on the basis of their similarity with the chemical elements (see discreteness of stellar parameters). ^{[10]}
Since the planetary systems are formed from gasdust clouds, then must be equal to the angular
momentum of such clouds. For a cloud with the mass and the radius the estimate of the total rate of
rotation at the equator is . Then the angular
momentum of the cloud is equal to:
,
where the coefficient reflects the change in the angular
momentum due to nonuniformity of the mass density and the differential
rotation of the cloud.
Between the mass and the radius of the cloud there
is a dependence: , where 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: . Taking it into account we obtain:
The value J∙s is determined from the diagram
of dependence of the angular momentum of the
planetary system on the mass number and it appears to be close to the
value of the Dirac stellar constant , which is correct for a planetary system of one planet and a mainsequence
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. ^{[11]}
For each gravitationally bound object with mass and radius we can introduce their
characteristic angular momentum according to the following formula:
where the characteristic speed is found from the ratio of energies
similar to (2).
The electron spin is usually determined by the
Planck constant in the form . Similarly for planets the intrinsic angular momenta are proportional to
the quantity and to the
planet number as the quantum number :
where 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.^{[10]}
Table 1. 

Planet 
Intrinsic angular momentum,
J∙s 
Calculated value according
to (5), J∙s 
Mercury 


Venus 


Earth 


Mars 


Jupiter 


Saturn 


Uranus 


Neptune 


Pluto 


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:
,
where and are the mass and the radius of the
planet, is the number of nucleons in the
planet's matter, is the proton mass, is the vacuum
permittivity, is the Bohr radius.
The planet's volume can be calculated by the radius
as well as by the total amount of hydrogen atoms:
.
Taking it into account, at we determine the mass and the
radius of a massive planet, exceeding to some extent the mass and the radius of
Jupiter: ^{[11]}
kg .
m .
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:
,
where is the Boltzmann constant, is the average internal temperature. Expressing the planet mass by the
density and the radius in the form , and using the equation , we obtain the mass and the radius:
.
.
Substituting here the same mass density as that of
the dwarf planet Ceres kg/m^{3}, and the
temperature K (the temperature of background radiation), we can estimate the minimum
mass and the radius of the planet:
kg, km, which is slightly less than
Ceres' parameters.
Stellar constants
For planetary systems of main sequence stars the stellar constants are: ^{[10]}
The systems with neutron stars
Similarity between neutron stars
and nucleons
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:
.
.
From relation (2) we can estimate the
characteristic speed of the matter particles of the neutron star under consideration:
m/s.
The coefficient of similarity in speed equals:
,
here 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:
.
The value of the stellar
Dirac constant for neutron stars:
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 we obtain the value 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 of the order of . ^{[12]} 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 , we can estimate the mass of the disk: 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 . It is expected that the longterm 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:
C,
J/T,
where and 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: ^{[1]}
V,
T,
here 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: ^{[10]}
,
where and 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:
J/T,
where 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 10^{3}
– 10^{5} years, which gives, taking into account the age of the Galaxy
equal to more than 13 billion years, 10^{5} – 10^{7} 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. ^{[13]}
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 negativelycharged 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:
Variable stars
There are several types of stars that reveal
themselves by the fact that their brightness periodically changes. They are:
During each pulsation period stars emit the
following energy quanta: Mira emits during a period of 331 days up to J, the stars of Delta Scuti type emit during 3 hours about J. The reasons for periodic changes
in brightness of variable stars usually are radial and nonradial 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 radioemission
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, ^{[14]} which is comparable
to the accuracy of the best atomic frequency standards.
Galactic systems
Similarity between the
mainsequence stars and galaxies
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 . ^{[10]} The ratio of the masses of galaxies to the masses of
corresponding main sequence stars is , where 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 :
,
where is the Sun's mass.
Repeated multiplication by gives the mass of normal metagalaxy
with minimum mass:
.
The planet which is the analogue of an electron
corresponds to a dwarf galaxy with the mass .
The ratio of the radii of galaxies to the
corresponding radii of main sequence stars equals , where 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 we obtain the radius of the normal
galaxy with minimum mass: pc. This radius is the
volumeaverage 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
planetthe analogue of an electron m by , we make an estimate of the average radius of the dwarf galaxy: 151 pc.
The value of the galaxy's characteristic speed can be found from relation
(2): km/s.
The ratio of galaxy's speed to the star's
speed gives the
coefficient of similarity in speed: .
By multiplying the stellar Dirac constant by the similarity
coefficients we determine the galactic Dirac constant:
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: ^{[10]}
.
From this at the mass we find the angular momentum of the
normal galaxy with the minimum mass: J∙s. According to other sources, ^{[15]}
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 , which they would have in case of a spherical shape, according to the similarity
theory.
Formation of galaxies from
gasdust clouds
Assuming the formation processes of stars and
galaxies from gasdust 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:
,
which gives the relation kg/m^{3}.
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 and the Galaxy mass and we will obtain:
= 30 kpc/.
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 kg/m^{3}.
Heisenberg relation
Heisenberg uncertainty principle relates the characteristic
changes of atomic energy and the energy change intervals by the formula:
,
where is the Planck constant.
For stellar and galactic systems similarly we can
write: ^{[10]}
,
where is the characteristic angular
momentum, which is associated either with the object's intrinsic angular
momentum by relation of the form , or with the orbital momentum by relation of the form .
If we assume that for the Galaxy the energy change
is equal to the total energy in the gravitational field: J, and the time interval is equal to
the relaxation time in the field of regular forces years, where is the mass
density, then it allows us to make an estimate of the Galaxy's characteristic
angular momentum:
J∙s.
This value can be compared with the Galaxy's
angular momentum of about J∙s. ^{[16]}
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 , and assuming the time period to be equal to the time of the star
formation (the KelvinHelmholtz time) , we obtain the following:
,
where 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 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 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 widelyspread lowmass
stars 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. ^{[10]} In the stationary case the Poisson's equation
holds for the gravitational potential :
, ,
where is the mass density which depends
on the radius .
The mass density can be considered in the form , where is a little additive to the main
mass density . Similarly the potential will consist of two parts: . Due to the potential addition, the star with the mass will have additional energy . The stars orbiting the Galactic nucleus produce stellar gas, which is
characterized by the temperature . We can assume that a statistical formula holds for the dependence of
density distribution deviation from the mean:
,
where is the density change amplitude, is the stellar
Boltzmann constant.
Expanding the exponent for the density in the form:
,
and solving the Poisson equation only for , we find the potential additive:
,
where and
are some constants.
Given this potential additive, the density additive
is:
.
Density fluctuations against the average density
background look like the rings in the Galactic disk. Given the sine
periodicity, for the change of the radius between the adjacent maxima we
obtain:
.
With the value J/K, the kinetic temperature of the stars' motion K,
kg/m^{3} 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 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.
References
See
also
External
links
Source:
http://sergf.ru/kpken.htm