1. Introduction
Di-star systems are widespread objects: about a quarter of all stars belong to the di-star family. Compact or close di-stars (binaries) with separation distances of only a few stellar diameters are of interest for stellar evolution, for example, for a merger process of stars. The distinctive feature of close binary systems is matter transfer (and ejection) [
1,
2,
3,
4,
5,
6,
7], which is obviously impossible between two well-separated stars. A spectacular recent case is KIC 9832227, which attracted a lot of interest from the media when this binary star was predicted [
8] to merge in 2022, which would lead to the formation of a red nova. Note that this has not happened yet. The luminous red novae have only recently been identified as a separate class of stellar transients [
9]. They show up by relatively long outbursts with spectral distributions centered in the red, ranging in luminosities as intermediate between classical novae and supernovae. The luminous red nova V1309 Sco has played a key role. The Optical Gravitational Lensing Experiment (OGLE) revealed a shape of the light curve characteristic of a contact di-star with an exponentially decreasing orbital period [
10]. These data well confirmed the previous prediction [
11] that merging contact binary stars lead to luminous red novae.
The merger process is represented as
common-envelope evolution [
12,
13]. Despite the short duration of this process, it has crucial consequences for stellar evolution. The binary components may survive with a reduced orbital separation, leading in some cases to compact binaries consisting of white dwarfs, neutron stars, and black holes. Some of these systems can merge with the gravitational wave emission, as recently observed with the LIGO and Virgo interferometers [
14]. Alternatively, the binary object can merge into a single one having exotic properties. There is a general opinion that the process of merger is an insufficiently studied problem of stellar evolution [
8,
15,
16]. However, there is some consensus on the mechanisms of the important stages of contact binary evolution. Initial separations in a binary are large, with orbital periods of days to months. Interaction with the third star and the tidal friction may reduce the binary period to a day or two, as found in the dynamical simulations [
17] and observations [
18]. Both magnetic braking and internal structure evolution may bring the binary into contact. If this happens, the stellar radii and Roche geometry prevent further evolution of the orbital period, ending in a stellar merger. As found in ref. [
11], red novae with a wide range of luminosities can result from contact di-stars with various initial masses.
As is commonly believed, contact binaries, for example, W UMa, end their evolution by merging into a single star [
7]. The dissipation of orbital energy in the initial phase of merger led to the appearance of the glowing red nova V1309 Sco observed in 2008. However, the details of the matter transfer between the parts of the contact binary object or the mechanism triggering the merger are still unclear. The work [
16] favors a gradual mass transfer from a less massive (with a smaller radius) secondary star to a primary one driven by the energy received from the primary. The contact is supported by the aforementioned magnetic braking or internal structure evolution. This results in an abrupt end when the mass ratio leads to the Darwin instability to start a merger which for a red nova seems to be the triggering mechanism for the outburst [
10,
19]. The Darwin instability occurs when the mass ratio becomes small enough, and the heavy star can no longer keep the light star synchronously rotating via tidal interaction. The orbital angular momentum transferred from the intrinsic spin changes the orbit more than the spin, which leads to a runaway. As found, for massive primary stars of the main sequence, this happens at a mass ratio of 0.09 [
20]. There is also another scenario [
21]: after contact of stars, there is a brief but intense mass transfer in a di-star system, changing the originally more massive star into a less massive one. This process may oscillate until a stable contact configuration is eventually achieved. A dynamic mass transfer without the Darwin instability was also investigated [
22], and mergers triggered by a tidal runaway based on a non-equilibrium response to tidal dissipation [
23] were considered. Thus, more detailed observational and theoretical studies are required.
Overcontact, contact, and near-contact binaries, forming di-star compounds with the average distances between the stars close to the sum of their radii [
4,
5,
6,
7] are of great interest in the study of stellar evolution. Information about the evolution of compact binaries is necessary to understand the processes observed in isolated stars. Compact binary stars are a good laboratory for a wide range of astrophysical phenomena, such as mass transfer between stars. The observations of their evolution verify our understanding of the inner structure of stars.
The problem of the origin of binary stars or binary galaxies is still unclear [
4,
5,
6,
7,
24,
25]. As shown in ref. [
26], there is no dissociative equilibrium between single and binary stars in the galaxy. The number of binary stars is many orders of magnitude larger than expected for dissociative equilibrium. So the origin of binary stars is not related to the capture of one star by another into a bound orbit. In addition, there is no sharp difference between close and wide binary stars, and the angular momenta relative to their centers of gravity are extremely large. Angular momenta are in the interval between values close to the angular moments of stars with extremely high rotational velocity (close binary stars) and values exceeding these values by thousands of times (wide binary stars). These facts refute the assumption about the origin of binary stars due to the fission of individual stars. According to the law of conservation, the angular momentum of a binary star does not exceed the angular momentum of individual stars with extremely high rotational speeds. These conclusions are valid only if we ignore external influences on binary star evolution [
26,
27,
28].
Since mass transfer is an important observable for close di-stars and di-galaxies, it is meaningful to study the evolution of the system in the mass asymmetry coordinate
, where
are the masses of the components of the binary system at fixed total mass
and orbital angular momentum
L of the system. Classical Newtonian mechanics can be used to explore the evolution of close binary stars and galaxies in their center-of-mass coordinate system by analyzing the total potential energy as a function of
[
29,
30,
31,
32,
33,
34,
35]. The limits of the formation and evolution of binary systems are of interest. The methods used were tested for similar processes in nuclear systems where mass asymmetry is an important collective coordinate governing the fusion of two nuclei [
36,
37]. A nuclear molecule or a dinuclear system consists of two individual touching nuclei. There are two main collective degrees of freedom in a dinuclear system which govern its dynamics: (i) the relative motion between the clusters leading to quasistationary states in the internuclear potential and to the decay into two fragments which is called quasifission since no compound system is first formed, (ii) the transfer of nucleons or light particles between two clusters of the dinuclear system leading to evolution in mass and charge asymmetries. There is a structural forbiddenness for the motion of the nuclei to smaller internuclear distances during the fusion process. Fusion of heavy nuclei in the internuclear distance
R is impossible and can occur due to the transfer of nucleons, i.e., by a motion in
[
36,
37].
Nuclear dynamics is certainly different from the gravitational interactions of di-stars and di-galaxies. Nuclear reactions are governed by short-range strong interactions, onto which minor contributions of long-range (repulsive) Coulomb and centrifugal forces are superimposed. The dinuclear approach is a key tool for describing the fusion of two heavy nuclei. In the approaching phase and after fusion, there is a mass loss by the emission of protons, neutrons, and light clusters like Alpha particles. Once a critical distance and mass ratio are reached, fusion occurs. A highly excited compound nucleus is formed in thermal equilibrium at temperatures of the order of one to a few MeV, corresponding to
K, cooled down rapidly by ejection of nucleons, nuclear clusters, and
quanta. Hence, dinuclear dynamics covers essentially the same spectrum of phenomena as expected for di-stars and di-galaxies. Thus, it is worth exploring to what extent the method from the femtoscale of microscopic objects is applicable to macroscopic binary galactic and stellar systems [
29,
30,
31,
32,
33,
34,
35].
2. Theoretical Approach
The differential of the total energy of a binary stellar or galactic system is expressed as a function of relative distance
, conjugate canonical momentum
, and mass asymmetry coordinate
as
As we consider the binary system as a closed system, the conservation of the total energy results in
where
is the general solution. In the center of the mass system, the total energy of the binary system is a sum of radial and orbital parts of kinetic energies and potential energy. As seen below, we attach the orbital kinetic energy part to the interaction
V between two components of the binary system. In this case, the expression of the total energy of the di-star system reads as
where
is the radial component of the momentum
and
is the reduced mass.
The total potential energy of a binary stellar or galactic system,
is the sum of the potential energies
of its components (
), and the energy
V of their interaction. The radiation energy is much smaller than the absolute values of
and
V to be disregarded. In Equation (
5),
G,
,
, and
are, respectively, the gravitational constant, the dimensionless structural factor, the mass, and the radius of the component. In general, the value of
is determined by the density profile of a stellar or a galactic object. By employing the relation known from observations, we express the radius of the object in terms of its mass as [
29,
30]
where
n and
g are the constants. So
Since two objects rotate around the common center of mass, the star–star interaction potential contains, together with the gravitational energy of interaction
, the kinetic energy of orbital rotation
:
where
L is the orbital angular momentum of the binary system, which is conserved during the conservative mass transfer. At
and
,
and
respectively [
38]. Here,
is the touching distance. From the conditions
and
, we find the equilibrium relative distance between two objects corresponding to the minimum of
V:
at
and
at
. Finally, one can derive the expression for the object–object interaction potential
at
or
at
. Here,
and
and
are, respectively, the distance between its components and the reduced mass of the initial binary system. Deriving Equations (
12) and (
13), we employ the known relation
Using Equations (
6), (
12) and (
13), we obtain the final expression for the total potential energy (
4) of the binary system
at
and
at
.
For the binary systems considered, the velocity , where c is the speed of light; therefore, the relativistic effects can be ignored. Since , the gravitational field is rather weak, and the equations of Newtonian mechanics can be used instead of the equations of general relativity. Note that exotic binary systems with neutron stars, white dwarfs, and black holes are not considered in this section.
2.1. Binary Stars
In order to calculate the factor
, we make use of the single-star model from ref. [
6], because it describes well the observed relations between the temperature, radius, mass, and luminosity of stars; the mass distribution of stars; magnetic fields of stars; spectra of seismic vibrations of the Sun; and other features of stars. Employing the dimensionless structure factor
from [
6], the observed radius-mass relation (
)
and the relation between the star masses
and
in the binary system and the mass asymmetry coordinate
, we find from Equations (
15) and (
16) that
at
and
at
, where
Here, we assume that the orbital angular momentum
and total mass
M are conserved during the evolution of a di-star in the mass asymmetry coordinate
. The orbital angular momentum
is calculated by using the observed masses
of stars and period
of their orbital rotation [
39,
40,
41,
42,
43].
As seen from Equation (
18), the stability of a binary stellar system depends on the orbital-rotation period
or on the value of
and total mass
M. Employing Equation (
18), we can study the evolution of a binary stellar system in
. The extremes of the potential energy are determined from the numerical solution of the equation
As follows,
(symmetric binary system) is the root of Equation (
20) and at
the potential has a minimum if
or
and a maximum if
. The minimum at
lies symmetrically relative to the two barriers at
. Expanding Equation (
20) up to the third-order terms in
and solving it, the barrier positions are obtained as
So at
the potential energy as a function of
has two symmetric maxima at
and the minimum at
. The fusion of two stars with
occurs only by overcoming the barrier at
or
. With decreasing ratio
, the value of
increases and the symmetric di-star system becomes more stable. The evolution of two stars with
to the symmetric di-star configuration is energetically favorable. Thus, an initially asymmetric binary system with
is driven to mass symmetry, implying a flow of mass towards equilibrium and an increase in internal energy of stars by the amount
(
Figure 1a). At
,
and the inverse
U-type potential has a maximum at
. In such a system, the fusion of stars (one star “swallows” another) is the only mode of motion in
transforming the di-star into a mono-star with the release of energy
(
Figure 1b).
If
, then
. In this case, the condition
means that an asymmetric system with the mass ratio
moves to more asymmetric configurations and the relative distances between its components increase in accordance with Equation (
11). These unstable binary stars with
are unlikely to live long. Indeed, close binary stars with a high mass ratio are very rare objects (
Figure 2). Note that this constraint on the mass ratio is independent of the relation used between the mass and the radius of the object.
2.2. Binary Galaxies
Employing
for the dimensionless structural factor, the radius-mass relation (
) [
24]
observed for the galaxies of large mass (
), where
and
are, respectively, the masses and radii of the components of the binary system before transferring the mass, and the relation between the coordinate
and the galaxy masses
and
in the binary system, we write Equations (
15) and (
16) as
at
and
at
, where
In order to calculate
and
, the observed values of
M,
,
, and
are used, where
X is the projection of the linear distance between the components of the binary galaxy from ref. [
24]. The potential energy has an extremum at
; it is the minimum if
or
and the maximum if
. As readily seen, the extreme points of the potential depend only on
and
. Note that for touching binary systems (
) there is a symmetric minimum because the condition
or
always holds. Expanding the equation
to the third-order terms in
inclusive and solving it, we find the barrier positions at
(
). As a result,
. So, in a strongly asymmetric binary system with the mass ratio
, the galaxies should fly apart. Indeed, the binary galaxies with a high mass ratio are rare objects [
24] (
Figure 2).
4. Origin of Orbital Period Change in Contact Binary Stars
As found, the orbital period of W-type overcontact binary GW Cep (
,
) decreases with time [
39,
40,
41]. For the overcontact binaries VY Cet (
,
) and V700 Cyg (
,
), cyclic oscillations were found to be superimposed on an increase in the secular period. This effect was explained either by strong external perturbation, i.e., by a close-by third object, or by the magnetic activity cycles. For the EM Lac (
,
) and AW Vir (
,
) binaries, the periods demonstrate secular increase. As concluded, the period variations in a W UMa-type binary star correlate with the mass ratio
q and the mass
of the primary component [
39,
40,
41]. The lower the mass ratio
q in binaries, the shorter the period.
Using the relation between
L and
, we obtain the period
of orbital rotation with frequency
. At
and
as
and
respectively. As seen, at initial
and
(
), the system moves towards the symmetric configuration and, correspondingly,
decreases,
decreases (
increases), and finally,
decreases (
increases) [
31,
32].
For the KIC 9832227 system (
), matter is transferred from a heavy star to a light one, the relative distance
between two stars and the period
of the orbital rotation decrease. The evolution in
pushes the system to the touching configuration (
) at some critical mass asymmetry
(
Figure 10). Further evolution in
leads to a configuration with partial overlap (
) of stars. So at
the period
slightly increases because
and
increases with decreasing
. Thus, if the system reaches the point
and partial overlap, the period abruptly changes. A similar period behavior is shown in
Figure 10 for the other contact binaries considered.
For the binary GW Cep (), , the system moves towards the symmetry and the orbital period decreases. For almost symmetric EM Lac () and AW Vir () binaries, and their periods increase. As found, the variations in the period of W UMa-type binary star correlate with the evolution towards a global symmetric minimum. At the low mass ratio q or the large , the binaries usually demonstrate a decreasing period because (), while the periods in systems with larger q (small ) increase because ().
5. Stability of Macroscopic Binary Systems
The Regge theory turned out to be very influential in the development of elementary-particle physics [
45,
46,
47,
48,
49]. As shown in the application of Regge’s ideas to astrophysics, the spins of planets and stars are well described by the Regge-like law for a sphere (
), while the spins of galaxies and clusters of galaxies obey the Regge-like law for a disk (
) [
25,
34,
50,
51,
52,
53]. Unlike semi-phenomenological approaches, these expressions contain only fundamental constants and do not depend on any established empirical quantities [
25,
34,
50,
51,
52,
53]. In Refs. [
25,
34,
50,
51,
52,
53], a cosmic analog of the Chew–Frautschi plot with two important cosmological Eddington and Chandrasekhar points was also constructed.
The Darwin instability effect can be studied in a binary star or galaxy by using the well-known model of Refs. [
25,
34,
50,
51,
52,
53] based on the Regge theory. The total angular momentum
of a binary system is the sum of the orbital angular momentum
and the spins
(
1, 2) of individual components:
The values of
and
are expressed using the Regge-like law for stars and planets (
) or galaxies (
):
and
where
ℏ,
,
(
1,2), and
are the Planck constant, masses of proton and astrophysical objects (planets, stars, or galaxies), and the total mass of the system, respectively. The maximum (
L and
are antiparallel) and minimum (
L and
are parallel) orbital angular momenta are
and
respectively. Using the coordinate
instead of masses
and
[
33] and Equations (
28)–(
30), we derive
At
(the symmetric binary system), we have
and
For the symmetric ( ) binary star (planet) and binary galaxy, 0.44 and 0.41, respectively.
As follows from these expressions, for very asymmetric binaries, the ratios
are almost independent of
l. According to ref. [
20], the Darwin instability occurs when the binary mass ratio is very small [
] or the mass asymmetry is very large [
]. As seen in
Figure 11, the ratios
and
continuously increase with
. Since these ratios are larger than 1/3, all possible binary stars (planets) or binary galaxies, independently of their mass asymmetry
, have the Darwin instability (
) [
19] and, thus, should merge. However, the observations do not confirm this, which probably means that there is no the Darwin instability effect in such binary systems and, accordingly, the mechanism of merger has a different origin and should be revealed. Since the spins of planets, stars, galaxies, and clusters of galaxies are well described by the Regge theory [
25,
34,
50,
51,
52,
53], we can be sure of the correctness of this conclusion.
Note that in the cases of antiparallel spins with
and
(
Figure 11), the ratios
and
are larger than
in the asymmetric binaries with
.
As seen in
Figure 12, the dependencies of
,
,
, and
on
are different. The matter transfer can increase or decrease the orbital angular momentum. For example, at
the binary system has smaller
. The observations of
L versus
may be useful to distinguish the difference between the orientations of orbital and spin momenta.
Employing Equations (
26)–(
28) and the results of refs. [
29,
31,
32], we obtain new analytical formulas for the relative distance between the components of the binary
at
and
at
. One can similarly derive the formulas for the orbital rotation period
at
and
at
[
31,
32]. The values
and
correspond to the cases of antiparallel and parallel, respectively, orbital and spin angular momenta. In the cases of antiparallel spins,
and
. The observation data result in the relationship
between the radius and mass of the star, where
and
[
6] and the galaxy, where the value of
n depending on mass is in the interval
[
24,
30]. As seen in
Figure 13, at
,
or
decreases with
and, finally,
or
decreases. At
(
Figure 13), the dependence of
as a function of mass asymmetry has similar behavior at antiparallel and parallel orbital and spin angular momenta. In the case of antiparallel spins with
(
) and
, the values of
and
decrease (increase ) with
decreasing from 1 to 0 (
Figure 14). At
(
Figure 15), the value of
or
increases with decreasing
and does not depend on mutual orientations of orbital momentum and spins. In contrast, the distance
or
depends on the orientations of orbital momentum and spins. As seen in
Figure 15, the dependencies of
on
at different spins have various behaviors.
7. On Evolution of Compact Binary Black Holes
The general view is that compact binary systems consisting of white dwarfs, neutron stars, and black holes eventually merge as a result of gravitational wave radiation, as recently observed with the LIGO and Virgo interferometers [
14]. The process of merging is still insufficiently studied. We are going to add a few aspects highlighting the role of matter transfer in binary black holes (BBH) [
34], which are certainly a special class of objects. The presence of the event horizon restrains the flow of mass between two cores of BBH. At a speculative level, quantum tunneling as the Hawking radiation [
54] may play a faint, most likely insignificant role due to not yet received firm confirmation. However, a BBH is usually embedded in a cloud of remnant matter left over from the progenitor stars. Therefore, a BBH should always be considered together with the surrounding accretion disk. The material of that disk may serve as the matter transfer between two cores by the so-called
sloshing effect, as illustrated by the hydrodynamical calculations in ref. [
55].
In order to understand the possibility of matter exchange in a BBH system, we consider the potential energy of a BBH as a function of
. The possibility of matter transfer based on the potential energy of a BBH system should always be understood together with its accretion disk. In addition to the BBH’s own extremely strong gravitational fields, the potential energy contains the interaction potential between two black holes. Being well aware of the strong gravitational fields, we use Newtonian mechanics because, at a distance, post-Newtonian effects can alter the binary potential by no more than 25% [
55], which will not affect the overall BBH properties. General relativity becomes of course, essential at separations close to touching.
The total potential energy
of a BBH is given by the sum of the potential
(
) and rotational energies
(
) of two nonzero spin black holes, and the black hole–black hole interaction potential
V. The energy of the black hole “
k” is
where
G,
, and
are the gravitational constant, mass, and radius of the black hole, respectively. The rotation energy of a black hole is
where
and
are the spin and rotational frequency of the black hole “
k”, respectively.
The radius of the event horizon (distance from the gravitating mass
at which the particle velocity is equal to
c) [
25,
34,
50,
51,
52,
53]
is derived from the energy conservation law for a particle with mass
m:
, where
is the sum of the kinetic
and rotation
energies. The derivation of the radius
is based on classical mechanics and the Newtonian law of gravity. In ref. [
56], the expression
was also obtained for a non-rotating black hole. Using Equations (
39) and (
41), we obtain
Due to the rotation of two black holes around the common center of mass, the black hole–black hole interaction potential
contains, together with the gravitational potential
, the kinetic energy of orbital rotation
:
where
L,
and
are the orbital angular momentum of a BBH, the speed and the semi-major axis of an elliptical relative orbit, respectively. At
, the black hole–black hole interaction potential is defined as in Equation (
8). From the conditions
and
, we find the relative equilibrium distance between two black holes corresponding to the minimum of
V (see Equation (
10)). The total angular momentum
of a BBH [
25,
34,
50,
51,
52,
53] is assumed to be conserved during the conservative matter transfer and the orbital angular momentum of a BBH is
Here the orbital momentum and spins are parallel and the value of
L is minimal. Employing Equations (10) and (
44), we derive the expression
As seen, the larger
M, the larger
. Since
, the separation of the components increases at
. From Equations (8), (
43) and (
45) we obtain the simple formula
for the interaction potential. The value of
depends only on the reduced mass
and the velocity of light.
So using Equations (
40), (
42) and (
46), we derive the final expression
for the total potential energy of a BBH.
For the BBH considered, and with good accuracy, one can disregard relativistic effects and use the Newtonian law of gravity.
Employing the coordinate
instead of masses
and
, we rewrite expressions (
44) and (
47) for the orbital angular momentum
and the total potential energy
Since the solution of the equation
leads to
and
the potential landscape has a global minimum at
for an arbitrary total mass
M of a BBH (
Figure 16). So the transfer of matter between the black holes in the BBH is energetically favorable and can occur to reach a global minimum. The initial asymmetric system is easily driven to the symmetric BBH. This conclusion does not depend on the choice of parameters. The losses of the total mass and orbital angular momentum do not affect symmetrization of a BBH. The transfer of matter between two black holes becomes possible because they interact with their own extreme gravitational fields. However, the evolution of a BBH in
depends also on the parameter of inertia in this coordinate. Since the surfaces of two black holes are spaced
from each other, the parameter of inertia in
is expected to be very large and, accordingly, prohibits symmetrization of a BBH.
Asymmetrization (the transfer of matter from a lighter component to a heavy one) of a BBH considered is energetically unfavorable, and, correspondingly, the merger channel in
is strongly suppressed. Thus, the question of the mechanism of the merger of two black holes and the origin of gravitational waves remains open [
14].
At antiparallel orbital momentum and spins, the value of
is maximal. For a BBH with this
L,
and
,
,
,
, and all conclusions given above are also valid in this case. Comparing
Figure 16 and
Figure 17, we can draw the same conclusions in the case when black holes in a BBH spin in the opposite directions.
8. Summary
The evolution of isolated binary stellar and galactic systems was considered in the framework of the approach originally formulated for the study of nuclear fusion reactions, thus combining scales that differ by many orders of magnitude. Despite the differences in details, the common aspects of macroscopic objects (di-star, di-galaxy, BBH) and microscopic dinuclear systems prevail. Both types of systems evolve along well-defined trajectories in the classical phase space. The conserved quantity is the total energy, that is, the Hamilton function, of a binary system. Using general arguments, we have shown that the conservation of energy is sufficient to fix the trajectory of a binary system in the landscape of potential energy determined by the masses of the objects and their interaction. Exploiting the stationarity of the total energy, the stability conditions were derived and investigated as a function of mass asymmetry. We emphasized that the interpretation of mass asymmetry as a collective coordinate has been successfully used to describe nuclear reactions, the cluster structure of nuclei, the decay of dinuclear systems, and the fusion of two heavy nuclei [
36,
37]. Here we have shown that this collective degree of freedom plays a comparably important role in space objects. In close di-star or di-galaxy systems, the coordinate
can govern the merger and symmetrization (due to the matter transfer) processes. An interesting aspect is that once
is determined, for example, by observation, it is possible to draw a conclusion about the stellar or galactic structure.
The new theoretical interpretation of a di-star or di-galaxy system is based on the fact that, after the formation, the lifetime of a binary system is long enough to reach equilibrium conditions in the mass asymmetry coordinate. Therefore, we could conclude that the system will be included in the sample of all binary and single configurations with the probability depending on the potential energy of this configuration. For the systems considered, and there are potential barriers at and minimum at . So two distinct evolution scenarios arise: if , the system is driven to the symmetric configuration (towards a global minimum of the potential landscape). However, if , the binary system evolves towards the mono-object system. All the considered asymmetric close binary systems, with the exception of the di-star Cr B and di-galaxies 206, 243, 272, 439, satisfy the condition and a symmetrization process occurs in these systems. The loss of the total mass and orbital angular momentum has little effect on the process of symmetrization. The merger of the binary stars, including the KIC 9832227, and binary galaxies considered is energetically unfavorable. The formation of a single object from the binary system () by thermal diffusion in the mass asymmetry coordinate is strongly suppressed.
Symmetrization of a binary system () due to the matter transfer is one of the important sources of conversion of gravitational energy into other types of energy in the universe. For example, in the cases of compact binary stars and compact binary galaxies, the released energies are about and J, respectively. Symmetrization of a binary system will lead to close objects with equal masses, temperatures, luminosities, and radii, which are observable quantities. During the process of mass-symmetrization, the channels of binary decay in the relative coordinate are closed. The central result of the approach reviewed is that stable binary systems exist only at () because otherwise, the stars are getting closer to merging. Thus, di-stars or di-galaxies with can not exist for long enough. Indeed, binary systems with a large ratio are very rare objects in the universe.
Asymmetrization (the transfer of matter from a lighter component to a heavy one) is equivalent to incomplete merging. Asymmetrization is also the origin of the expansion of a binary galaxy. The separation of components from each other is represented as an analog of the expansion of the universe within a binary system. The condition under which the asymmetrization process is realized depends mainly on the relative distance between the components of binary systems and their linear dimensions.
For contact di-stars, changes in the orbital period can be well explained by evolution in towards symmetry. We predicted that the decrease and increase in orbital periods are associated, respectively, with the non-overlapping (, ) and overlapping (, ) stages of a binary star during its symmetrization. Thus, observing the change of periods allows us to distinguish between these two stages of a di-star.
Based on calculations of potential energy, we have demonstrated that the mass asymmetry coordinate (matter transfer) is useful for analyzing the transformation of a single star (galaxy) into a binary star (galaxy). Mass ejection from a single star could create a binary star in the mass asymmetry coordinate. However, the barrier preventing symmetrization in
is quite high. So the formation of asymmetric and almost symmetric di-star or di-galaxy systems from the respective mono-star or mono-galaxy by thermal overcoming of barriers in the driving potential (the diffusion process in
) is hardly possible. Binary stars (galaxies) with a mass ratio
evolve to more asymmetric configurations and are rather unstable with respect to the decay in the relative distance. Thus, matter ejection from a single star is either captured back or escapes, which is consistent with the findings of refs. [
26,
28]. Thus, it is possible to assume a common origin of the components of a binary star (galaxy) from the pre-stellar (pre-galactic) state of matter.
Based on the Regge-like laws [
25,
34,
50,
51,
52,
53], we demonstrated that all possible binary stars (planets) or binary galaxies, regardless of their mass asymmetry, satisfy the Darwin instability condition (
), which contradicts observations. This output is not sensitive to model parameters. Therefore, we should look for another mechanism that triggers the merger of binary contact components. Employing the Regge-like laws, we derived new analytical formulas for the relative distance and orbital rotation period of the binary system, which depend on the total mass
M, mass asymmetry
, and the fundamental constants
G,
ℏ, and
.
Employing Newton’s law of gravity and considering the BBH potential energy as a function of , we have shown the possibility of transferring matter between black holes in a BBH. The evolution of an asymmetric BBH to a symmetric one () is energetically favorable. Although black holes have their own strong gravitational fields, the transfer of matter in BBH occurs due to the energy of interaction between two black holes. A BBH does not send any signals during its evolution. Perhaps the result of this evolution can be indirectly observed.
Symmetrization of a BBH leads to a decrease of U, thus converting the potential energy into internal kinetic energy. For example, for the BBH () and (), the internal energies of black holes increase during symmetrization by the amount J. So the BBH is also the source of thermal energy. The transfer of matter from a lighter component to a heavier one ( the merger of black holes in a BBH) is not an energetically advantageous process, and, thus, the question of the origin of gravitational waves remains open.
Within the presented model, we can also perform dynamic calculations of the evolution of a binary system in . For example, we can calculate the relaxation time (symmetrization) and the asymmetrization time. In addition to the total potential energy, it is also necessary to calculate the mass parameter and the coefficient of friction for this coordinate. However, this extension of the model is the subject of future research.