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Astronomical Lviv 1771
On representation
of magnetized gas structures
during mass transfer
in Beta Lyrae system
Skulsky M.Yu.
Lviv Polytechnic National University, Ukraine
E-mail: [email protected]
A little background
Half a century I carry out spectral studies well-known
interacting massive binary system Beta Lyr, the results of
which led to some progress in the interpretation of the
physical nature of the system. First of all, it concerns the
identification lines of accretor and determination of the
masses of both components directly from a spectrum of
systems, detection and investigation of the magnetic field
of the donor, the study of the dynamics and structure of the
accretion disk. The most systematic observations carried
out on a powerful telescopes of that time, particularly on
the 2.6-m telescope of the Crimean and 6-m telescope of
the Special astrophysical observatories (SAO).
Schematic model of the interacting binary system Lyrae (view from
above of the orbit plane): the bright B8III donor with mass of M1 = 2.9
M⊙ and accretor with mass of M2 = 13M⊙ that is wrapped by thick
disk with pseudoatmosphere of A5III type. For the mass ratio of
0.223 the distance between the centers of the two components is A =
58R⊙. With the observer’s eyesight towards the center component
are defined scopes of hot regions on disk in phases 0.40P and 0.80P
(according to [1]) and other directions observation (0.0P phase
corresponds to the center of the main eclipse in the binary system).
The mass transfer from the donor up to the accretor (so-called fan
structure of gas streams is designated by curved arrows) occurs both
in the direction between gravitational centers of the two components
along of the orbital phases 0.5-0.0P and in the direction of the
magnetic field axis "H" along of the phases 0.35-0.85P. This leads to
shock collisions and the appearance of "hot half-arc" on the accretion
disk edge (thick line). The orbital phases 0.05P and 0.95P of the
optimal projections of the disk edges (so-called satellite-disk) on
donor are marked. In these phases in the spectrum of the system is
clearly visible so-called satellite-lines with radial velocities of opposite
signs (up to 250-270 km/s), reflecting the rotational effect of the disk.
The satellite-disk size is marked by two rows of rounded arrows and
the small circle within this satellite-disk is showing the center of mass
of the binary system.
The recent investigation was caused by the publications of
the results of the self-consistent simulations of the light
curves of the well-known massive close binary system Beta
Lyrae. The one major among them is the statement of the
significant contribution of the accretion disk radiation in the
light curve of the system. In particular, there were identified
two hot regions with temperatures that are 10% and 20%
higher than the average on the disk rim. It is assumed that
these shock regions might be formed by gas flows in
regions of collisions on the disk in the mass transfer
between components.
Indeed, the hotter region of the rim disk at the phase 0.40P
is explained naturally by the Coriolis force deflection of the
main gas flow that is directed from a donor through a
Lagrange point to the aссretor's Roche lobe and with a
further collision of this flow with the disk. However, this
explanation can not be suitable to a wide hot region of the
disk rim that is observed at the phase 0.80P.
Our analysis of absolute spectrophotometry, of curves change of
magnetic field, of radial velocities and intensities of spectral
lines along the orbital phases indicates that the hot region on
disk at phases near 0.80P has a special nature. The donor's
magnetic field should be taken into account because at these
phases its dipole axis is actually aimed to the observer and
deflected towards disk of the accretor. This hot region on the
disk can be formed by the collision with this disk of magnetized
gas which is canalized by the donor magnetic field in a certain
way oriented in space. The energy effect of the collision on the
disc is significantly strengthened by the counter rotation of the
outer edges disc towards the falling gas flows. The structure of
the donor's magnetic field is effective at heating and the hotter
region on the disk rim at phases near 0.40P. The specific
configuration of the magnetic field of the donor explains why at
rotation of the donor and the accretor around their common
center of masses on line of sight of the observer these two
hotter regions dominate on the accretion disk.
At the top - the variability with
the orbital phase of the
donor’s effective magnetic
field;
in the middle - the variability
with orbital phase of absolute
radiation flux in H-alpha
emission;
beneath - a 20% increase of
the radiation flux in H-alpha
emission at collisions with a
disk of gas flows in the phase
of 0.40P
Changes of the radial
velocities of red (at the
top) and of violet
(beneath) emission
peaks in the CIII1175
line with the orbital
phase. Poles of the
donor's magnetic field
along the axis of «H» in
the phases 0.35P-0.85P
are marked as lines with
arrows.
The main result of the comparison of different
data methodically following observations: the
Beta Lyrae system shows clear correlation the
variability with orbital phase of absolute
radiation flux in H-alpha emission and the
variability with the orbital phase of the donor’s
effective magnetic field. The figure shows that
both extremes dependencies agree along the
orbital phase.
The behavior of the spectral lines outside the
Lyman limit varies similarly.
The main conclusion is that the mass transfer, the
formation of circumstellar gas structures, their
dynamics and energetics are regulated by magnetic
field of the donor.
But there are many questions primarily to specialists such
direction Conference. For example, thirty years I have no
answers to the 10-second observations in the line Halpha on the 6-m telescope of SAO. It concerns the
unusually large variability of the line H-alpha when
observations are conducted in the orbital phase magnetic
poles on the donor. What is the nature of this
phenomenon? Maybe someone will be surprised and
interested? Those who try to meet this challenge, I
suggest cooperation.
Now, concerning of our understanding of
the new phenomenon according to the
second declared theme
On the spatial structure
and wave resonances
in the Solar planetary system
Skulsky M.Yu.
Lviv Polytechnic National University, Ukraine
E-mail: [email protected]
Our interest in this research coincided with the
discovery of extrasolar planet systems.
Several hundreds studied exoplanets and tens their
systems promoted a research of scenarios of their
dynamical evolution. The main scenarios of formation
of planetary systems are researched on the basis of
the so-called Nice Model.
The principal conclusion is that there is no
universal rule for the ordering of the planets
with scientifically proved physical mechanism.
Neither the Titius-Bode law and its modifications to
the Solar system, or some regularities in the
structures of the exoplanet systems are not
substantiated certain physical mechanism.
At the same time, for the Solar system were
revealed principles the ordering of planets on the
basis the program of magnetic field registration of
the Sun as a star. Our study of these principles
offers a wave algorithm of the ordering of planets
in the Solar system.
The results of its wave structurization are publish in:
Skulsky M.Yu. // Astron. School's Report. 2011, V.
7, N 1, P. 28-35; and: Skulsky M.Yu. On the
wave structure in the spatial organization of the
Solar planetary system // Science and Education
a New Dimension. Natural and Technical
Sciences, III(5), Issue 41, Р. 63-67, 2015.
www.seanewdim.com
The planets in Solar System (Kotov V. and Kuchmi S., 1985)
is followed to two simple laws:
2a  L / n - for circular orbits of the inner planets,
2a  nL - for the sections of the orbits of the outer planets
(here a- semi-major axis of the orbit, n - the first natural
numbers).
For the inner planets (Mercury, Venus, Earth and Mars) n =
8, 4, 3, 2, to asteroids - n = 1.
For outer (Saturn, Uranus, Neptune) and dwarf planets (Pluto
and Eris) this natural numbers is respectively: n = 1, 2, 3, 4, 7.
Only by Jupiter n = 1/2, i.e. fractional.
The physical nature of these laws was not clear.
Since the L - scale has the dimension of a
wave with a length  = c P = 19.24 AU, the
both laws L - resonance transformed us into a
wave notation. They were similar standing
wave equations describing the numerous
physical processes. And the planets in the
Solar system are located on the algorithm of
standing waves with a length sw   / 2
Standing waves
Thus, the radial arrangement of the outer planets corresponds
the definition of the phenomenon of standing waves.
It is known that:
a standing wave is formed by the interference between direct
and reflected waves at opposite their propagation in the same
body; the standing wave is sw   / 2 - half wave; linear
dimension this body is a multiple of a quarter wave  / 4 ;
along with the multiplicity of the body are the vibration nodes
and anti-nodes, respectively, with the double amplitude and
amplitude of zero.
The spatial organization of the Solar planetary system is
described by two interrelated kinematic algorithms of
the single wave mechanism that is similar to the
phenomenon of standing waves with length sw   / 2
(here:   cP  19.24 AU, c is the speed of light
and P =160 min is a period of global oscillations of
the Sun). The principle of the ordering for outer planets
and dwarf planets is represented in the wave form
as a  n / 2 or a  (2n  1) / 4 (where a is a semimajor axis and n is a whole
number) by arranging
c
these planets at distances from the Sun proportional to
a quarter and a half of the wavelength (Jupiter -  / 4 ,
Saturn -  / 2 , Uranium – 2  / 2 , Neptune – 3  / 2 ,
Pluto – 4  / 2 , Eris – 7  / 2 ). The principle of the orbit
ordering for inner planets is expressed as 2a  m
with the step   (1/ 12) / 2 and m  3, 6, 8, 12 for
orbit lengths from Mercury to Mars that is
commensurable with the length of standing wave  / 2
and its harmonics.
Importantly, it was revealed an explicit
resonance of proper oscillations of the
Sun and planets. Their global periods are
virtually multiples to kP / 2 , where k  1, 2, 3.
These signs of a quantization of the
gravitational interaction of the Sun as a
star and planets are related to the length of
the standing wave  / 2 . Wave and
gravitational resonances put questions
about their origin in the Solar system. Also
such interconnected findings should be
considered as essential on the
background of the current knowledge
about the laws of structuring planets in the
Solar and exoplanet systems.
Objective evidence
Table 1 contains the Solar system
characteristics: masses, semi-major axes
of the planet orbits, semi-major axis ratio
to wavelengths and other. The slides are
the result of the analysis of this table,
figures and other data.
Structural architecture of the outer planets
The regularity of L - resonance 2a  nL0 for
cross sections of the orbits of the outer planets in the
wave notation is a  n / 2 . It is found that the
distances of the outer planets from the Sun are
multiple of a quarter or half wave:
Jupiter -  / 4 , Saturn -  / 2 , Uranus - 2  / 2
Neptune - 3  / 2
Distances well-known dwarf planets are:
Pluto - 4  / 2 and more distant Eris - 7  / 2 see Figure 1.
For Jupiter
a  (2n  1) / 4
and n = 0.
Distances of trans-Neptunian comet families
The distribution of periodic comet aphelions with very
eccentric orbits and direct motion near the plane of the
ecliptic was obtained in the range of 15 - 200 AU
(Kozlov V. - Astron. School's Report, 2009, V.6, N2,
R.163). Three families of comet at average distances 56,
86 and 106 AU, which corresponds to 2.91, 4.47 and
5.51  or in standing waves to 6, 9 and 11  / 2 was
located.
Conclusion: The distance trans-Neptunian comet
families from the Sun are multiples of the standing
wave length sw   / 2 as a parameter of their
location.
The principle of outer planet ordering as the
phenomenon related to phenomenon of standing waves

Standing waves and outer objects in the Solar system
So, the working formula for the distance of external
objects from the Sun: a  n / 2,where sw   / 2
- the length of the standing wave (  = c P = 19.24 AU).
The outer planets (except Jupiter) and the majority of transNeptunian objects, including a family of comets, described
by the relation a  2n / 4 , when n= 1,2,3,…11 (to
a distance of 110 AU). The relation described the distance
from the Sun by one third of trans-Neptunian objects and
Jupiter is: a  ( 2n  1) / 4 (for Jupiter n= 0).
The principal conclusion: in this algorithm of standing
waves Jupiter is located at the shortest distance from
the Sun, and this algorithm can not describe the radial
ordering of the inner planets.
The wave structure of inner planets
The inner planets can not obey the wave algorithm inherent in
the outer planets, as they are located from the Sun within  / 4
And really for them there are the another principle of L –
resonance:
2a  L / n
2am
1
Transforming this principle to the form
sw
we obtaine an equation of standing waves for circles orbits of
the inner planets.
1
Here:
sw  sw / 12   / 24 - “daughter" standing
wave;
numbers m  3, 6, 8, 12 commensurate to the length of the
standing wave and its harmonics quantized the length of orbits
of the inner planets, respectively, from Mercury to Mars.
The inner planets: structural architecture
The relation for lengths of planet orbits (see Table 1)
from Mercury to Mars is represented in the form:
or
(1 / 4)( / 2) : (1 / 2)( / 2) : (2 / 3)( / 2) : ( / 2)

/8
:
/4
:
/3
:
/2
The length of the orbit of Mars is directly equal to the
length of the standing wave sw   / 2 , in the
lengths of the orbits of Venus and Mercury are
embedded on one of its first and third harmonic, and in
the length of the orbit of the Earth - its two second
harmonics.
Fragments of schematic orbits of the inner planets are
presented in the fractions of the wavelength  .
Distances to the planets are normalized to the semimajor axis of Earth's orbit
Therefore, the spatial organization of the Solar
planetary system could be formed in one
physical process but in two interrelated
kinematic algorithms of the one wave
mechanism.
The length of the standing wave seen as the
structural factor in the Solar planetary system.
These results are unusual but they are
interconnected on the basis of the simple
physical equations for standing waves that are
identical to those in many physical processes
What is the nature of the phenomenon?
Phenomenon of standing waves and resonance of the
proper oscillations of the Sun and the planets
It is known that the academician B. A. Severny
drew attention to the surprising proximity of the
detected 160-minute oscillations of the Sun and
a "global" proper oscillations of the Sun, equal
to 167 minutes.
In connection with the phenomenon of standing
waves we are interested in this and discovered
the resonance of such proper oscillations of the
Sun and planets...
Periods of proper oscillations of the Sun and the planets
The formula used to estimate the "global" periods of proper
oscillations of the Sun and the planets of mass M and radius R,
is known:
3
T
2
(R/GM
)
It is valid for spherical gravitating objects with mass M,
assuming a homogeneous gravitational field of a spherically
symmetric distribution of mass. It is true and for a period in
which a particle of mass m execute harmonic oscillations at a
distance X from the center of the sphere (and at its surface,
where X = R):
T  2 (m / k )  2 ( X / g )  2 ( R / g )  2 ( R / GM )
3
2
g
GM
/R
Here:
and G - the gravitational constant.
The resonance of proper oscillations of the Sun and the
planets
Proper period of the Sun oscillation is 167.3 minutes
and close to the period of global pulsations (P = 160
minutes). Calculated T-periods are, respectively for the
inner planets from Mercury to Mars 85, 90, 84 and 100
minutes, and for the outer planets from Jupiter to
Neptune - 172, 236, 177 and 158 minutes. Proper
oscillations of the inner planets (with a average
deviation of about 5%) are in resonance with the global
oscillations of the Sun at a ratio of 0.5: 1, and outer
planets - as a 1: 1 (for Saturn - 1: 1.5), that are its first
harmonics, reflecting in 0.5 P comprehensive resonance
of proper oscillations of the Sun and the planets -Table1
The resonance of proper oscillations of the Sun and
the planets
This can be described by the simple equation T  kP / 2 . It
obeys the rules of integers k = 1, 2, 3 and here P/2 as the
first harmonic of P-pulsations of the Sun is the least common
multiple for these T-periods (with an average relative error in
5%). It is recognized that there is a common resonance of
global oscillations of planets and Sun. It is the first unusual
result: the whole “planetary orchestra” is tuned to the
frequency of global pulsations of the Sun. Table 1 shows
also for all of these objects that there is the second simple
equation cT / 2  sw   / 2 , where k = 1, 2, 3. Consequently,
global oscillations of planets are determined of this standing
wave. There is the second unusual result: periods of global
oscillations of planets and the Sun are directly associated
with the standing wave sw   / 2 as with the structural
factor in the Solar system.
The resonance of proper oscillations
of the Sun and the planets
•So, the relative positions of both types of
planets, their sizes and weight as related
phenomena are immediately related to the length
of the standing wave sw   / 2 as a structural factor
•In general, these simple calculations gave
unexpected results about the unusual aspects of
gravitational interactions between the Sun and the
planets. These data can be be seen as proof the
the existence of global “enigmatic” P-pulsations
of the Sun.
As to a criticism of standing waves phenomenon and
the Solar system structure -1
The problem consists in that the wave algorithm in the
structure of the Solar system confirms existence of Ppulsations of the Sun, while theoretical studies of internal
structure of the Sun do not confirm the existence of these
pulsations (Appourchaux and Palle 2013). They note the
fast damping of low-frequency g-modes at their transfer to
surface of the Sun. However, it should be noted that one of
possible variants in searches of an answer can consist in
consideration of the interaction of gravitational and
magnetodynamic processes. First, we consider here the
own pulsation of the Sun as a whole (it is the global
period). Indeed, the surface P-pulsations of the Sun
(Severny et al. 1976, Kotov and Khanejchuk 2011) were
revealed by the method of registration the magnetic field of
the Sun as a star.
As to a criticism of standing waves phenomenon
and the Solar system structure -1 (continuation)
Second, it is known that tachocline region which is
responsible for the enhancement of the magnetic field of
the Sun and plays a key role in nature solar cycle lies on
the bottom of the convection zone. The global g-modes
should reach this zone without significant damping. It
allows to suppose the modulation of general magnetic field
by g-modes and their transfer together with this field to
surface of the Sun. Then the impact on the structure of
planets will be carried out the magnetic field of the Sun
which is modulated by the low-frequency oscillations of
global g-modes. In this aspect, such observation method is
a promising in the study of the internal structure of the Sun
and solar-planetary interactions. In particular, one can
expect to detect the variability of the P-pulsations of the
Sun within the 22-year cycle.
As to a criticism of standing waves phenomenon and the
Solar system structure - 2
This is a question about the origin and nature of this
phenomenon. Empiric data offer a suggestion that global
P-pulsations of the Sun, regardless of reason and time of
their origin, were able to synchronize and save the wave
structure of the Solar system in the process of its evolution
up to the present tense, although now these pulsations can
be observed in relict form. In the terms of the current
knowledge, it is not difficult to represent the scenario of
forming of the Solar system by the mechanism of
interference of coherent waves within a protoplanetary disk
but without considering the physical nature of these waves
(Skulsky 2011, 2015). These waves have the velocity of
light, but it is difficult to study them with the help of
hypotheses of the electromagnetic or gravitational nature.
As to a criticism of standing waves phenomenon and the
Solar system structure – 2 (continuation)
The energy of electromagnetic waves with 19.24 AU is very small.
Nevertheless, it would be interesting to model the formation of
planets in the protoplanetary disk taking into account the influence of
the Schumann-like resonances (Schuman 1952) in the cavity
between the surface of the young Sun and ionized gravitational
shafts. A possibility of origin and interference of coherent waves from
spherical objects interacting gravitationally is problematic because
this is forbidden in the general theory of relativity. However, there are
other possible interpretations of gravitation. For example, the
gravitational field can have not only the tensor component (as in the
general relativity) but also the scalar one. The scalar component may
be emitted in spherically symmetric oscillations of any source of
gravitation, including the Sun. The phenomenon of wave
structurization of the Solar system can be represented as a
relativistic delay of scalar part of the gravitational field or a
disturbance on the newtonian potential. In any case, the hypothesis
of the possible existence of gravitational waves (including those with
length in 19.24 A.U.) and their interaction can have the right to
existence although it is not easy to interpret in terms of the accepted
modern concepts.
P0
• The task outlined by the Crimea AO in 1960s for
registration the magnetic field of the Sun as a star
advanced afterwards systematic measurements of
global photospheric oscillations. They based on
measurements of the Doppler effect in iron line on
512.37 nm with zero Lande factor. This program led to
discovery of “enigmatic” pulsations of the Sun with a
period 160 min (Severny et al. 1976) and the ordering of
planets in the Solar system (Kotov and Kuchmi 1985,
Kotov and Khanejchuk 2011).
Severny A., Kotov V., Tsap. T. Nature 1976, V. 259, P. 87; Scherrer
P.H, Wilcox J.M. Solar Phys. 1983, V.82, P.37
Kotov V., Kuchmi S. Izv. Crim. Astr. Obs.1985, V. 72, P. 199;
Kotov V., Khanejchuk V. Izv. Crim. Astr. Obs. 2011, V.107, P. 99.
Kotov V. and Kuchmi S.
(Izv. Crim. Astr. Obs.1985, V. 72, P. 199)
discovered resonance proportionality orbital sizes of
the planets of the Solar system with "scale”
L = c P = 19.24 AU, where c - speed of light
and P = 160 min - the pulsation period of the Sun.
Their "resonance spectrum" (similar to the power
spectrum) showed a single strong peak
corresponding to L - scale. The statistical
significance of the peak (16 objects for the system) is
now estimated (Kotov V., Khanejchuk V. Izv. Crim.
Astr. Obs. 2011, V.107, P. 99) at around 4 sigma
pointing to a common “L - resonance" of the planets
in Solar System (SS).
What is the nature of the phenomenon?
Indeed, according to estimates made by Molchanov
(Molchanov А.М., Icarus, 11, 95, 1969) the probability of
casual formation of the planetary system with properties
of the Solar system is about from 10-10 to 10-11.
The length of the standing wave seen as the structural
factor in the Solar planetary system. These results
are unusual but they are interconnected on the basis
of the simple physical equations.
Therefore, let us consider the prehistory of this
phenomenon at a well-known critical attitude to the
pulsations of the Sun with a period of 160 min
The principle of outer planet ordering as the phenomenon
related to phenomenon of standing waves
Standing waves and transneptunians objects
•Outside the orbit of Neptune (Edgeworth-Kuiper area) near 20
objects with the apparent size from 600 to 2600 km and determined
kinematic parameters of their orbits are known
•1) semi-major axes of the majority of these TNOs are
commensurated to even number  / 4
•Pluto, Orcus and Ixion - 4.1  / 2 - 4-th node
•2002AW
- 4.9  / 2
- 5-th node
•GK147, SM331, VK305, XR190, YW134 - 6  / 2 - 6-th
node
•Eris and 2007OR - 7  / 2 - 7- th node
•2) semi-major axes of these TNOs are multiple to odd
numbers  / 4
• Haumea, Quaoar and Varuna - 4.5  / 2 or 9  / 4
•2007UK126 и CP105 - 7.5 and 8.5  / 2
•Conclusion: The distances between the Sun and the
largest TNOs are multiple to  / 4
/4
/4
/4
The inner planets: the structure and resonances
A new form of equation of standing waves for circles orbits of
the inner planets follows from simple resonant relations of
length orbits for the planets Mercury - Venus, Venus - Earth,
Earth - Mars. They are close to 1: 2, 3: 4, 2: 3.
Number 12 as the common denominator of these resonances
is denoted a discrete set of "daughter" of standing
1
1
 sw /2or sw  sw / 12   / 24 embedded
waves 12
sw
into sw - the basic standing wave. In consideration of this,
1
from the equation 2am
sw in a natural way
(the number 24 is evenly divisible by the number m = 3, 6, 8,
12) we determine the relation for orbit lengths of four inner
planet at wavelengths.
  
Fragments of schematic orbits of the inner planets are presented in the
fractions of the wavelength
. Distances to the planets are normalized
to the semi-major axis of Earth's orbit

Limitedness of the inner planets in the Solar system
1
•It is logical to expect as follows from relations 2am
sw
where m = 3, 6, 8, 12, that a wave train /8
of
: /4
: /3
: /2
the inner planet orbits have terminated the planet with the length of
the orbit is equal  (by m = 24). The absence of the inner planet
from behind Mars (in place of the asteroid belt) can be explained
not only conventional "pumping out" planetesimals Jupiter, but also
the negative role of a powerful resonance 1: 1, what arises through
equality the wavelength of the length  and the orbit of a
hypothetical planet.
•The absence of the planets Mercury front with a minimum length
of the orbits (m = 1, 2) can probably be explained by the role of the
resonance wavelength
of
such
orbits
with
the
length
of
a
"daughter
" standing wave 1sw (for m = 1 the distance a planet from the Sun
is 0.127 AU - in the exoplanet systems at such distances and up to
0.02 AU from the central star locate, as a rule, several planets - the
Solar system is unique).
   
Standing waves and the structure of the Solar system:
Conclusions
The results of the wave represent in the two variants structuring of
the Solar system are consistent in a united physical mechanism
similar to the phenomenon of standing waves. The ordering of
all the planets obeys the laws of integers or quantized,
corresponding to a clear resonance relations. Deviations of the
planetary system objects from the calculated position are within
2-5% what could be explained long evolution of the Solar
system. Factor of standing waves become a new aspect of the
wave planetary physics.
The mechanism of standing waves is simply to explain the basic
questions of planetary cosmogony: why the planets' orbits are
nearly circular and coplanar, why the distances of the planets
from the Sun are arranged in a certain way, why the space of the
Solar system is divided into two parts, and there are two groups
of planets, the inner group of which consists of four planets.
Lovis et al. AAMan N HD10180,2010
As for the structure of the known planetary systems,
first of all find out what the interaction between the
planets moving, usually in orbits with large
eccentricity, may lead to their migration. We also
note that the first planet Mercury is a much greater
distance than in the first planets in other systems
(they have the same distance from the star filled
several planets). And among the exoplanet systems
that have planets with a mass of Jupiter, our Jupiter
is at the greatest distance from the Sun as a star. In
this respect, the structure of the Solar system is
unique.
Objective evidence
Table 1 contains the Solar system
characteristics: masses, semi-major axes
of the planet orbits, semi-major axis ratio
to wavelengths, calculated T –periods,
specifically in parts of P-pulsations of the
Sun. The slides are the result of the
analysis of table, figures and other data.
Conclusions: phenomenon of standing waves
and the Solar system structure - 1
The results of the wave investigation of the Solar system
structure could be explained by existence of a physical
mechanism similar to the phenomenon of standing waves
presented in two variants but with the same length of the
standing wave. The ordering of all planets obeys the rules
of integers according to distinct resonance relations. These
results can be considered as empirical and are quite
accurate. Their average deviations from the calculated
positions are within 3-5% which could be explained as a
result of long-term evolution of the Solar system. It is
reasonable to suppose that the formation of both groups of
planets in the Solar system could be realized by means of
a specific wave process. And their movements in a circle
on the orbits corresponded to these wave processes.
Conclusions: phenomenon of standing waves
and the Solar system structure - 2
Secondly, the analysis of the standing waves phenomenon
evinced the common resonance of global oscillations of the Sun
and planets. This resonance irrespective of the inner structure of
these objects showed the definite connection between
gravitational and waves processes. We should take into account
that the gravitational interactions between the planets and the Sun
as a star can be characterized without considering their detailed
3
internal structure (the formula T

2
(R
/GM
) for the global
period reflects the averaged densities for all objects). Their
interactions are described by certain quantized parameters. It
should be considered as an accomplished fact the good
coincidence of effective radii of all objects according to the
equation T  kP / 2 and their observed radii.
Conclusions: phenomenon of standing waves
and the Solar system structure - 3
It is obvious that the Sun as a star and the planets are
showing signs a quantization of their gravitational
interaction and this is associated with the length of the
standing wave as the structural factor in the Solar
planetary system. These results are unusual but they are
interconnected on the basis of the simple physical
equations. In general, the discovered findings of this
research are related to the basic foundations of our
worldview and represent problem in their interpretation.
Such studies should be continued because they raise
questions about the formation of Solar and exoplanet
systems. Thus, it is worth paying attention to the need for a
more complete analysis of this physical problem.
Стоячие волны в Солнечной системе
sw   / 2
 / 2    L  cP sw
c
1
a

n

/
2
 / 4 2amsw P a
kP / 2
  sw / 12   / 24
1
sw
   / 24
1
sw
k
  cP 
T
/8:/4:/3:/2
cT / 2  sw   / 2
sw   / 2
a  2n / 4
a  (2n  1) / 4
n  1,2,3,...11
n  0,1,2,...
cT / 2  sw 2a  m
T  kP / 2
  (1/ 12) / 2
T  kP / 2