Close Binary Galaxies: Application to Source of Energy and Expansion in Universe

TL;DR

This paper explores the evolution and energy transformation in binary galaxy systems by applying nuclear fusion concepts to galactic dynamics, revealing how symmetric binaries form and release energy during symmetrization.

Contribution

It introduces a novel application of nuclear physics ideas to galactic systems, analyzing the formation and energy processes of binary galaxies.

Findings

Stable symmetric binary galaxies can form under certain conditions.

Symmetrization converts gravitational energy into internal energy, releasing large amounts of energy.

The process influences galaxy evolution and energy distribution.

Abstract

Applying the microscopic nuclear physics ideas for fusion reactions to macroscopic galactic systems, we study the evolution of the compact binary galaxy in mass asymmetry (transfer) coordinate. The conditions for the formation of stable symmetric binary galaxy are analyzed. The role of symmetrization of asymmetric binary galaxy in the transformation of gravitational energy into internal energy of galaxies accompanied by the release of a large amount of energy during the symmetrization process is revealed.

Tables

Table 1. Table 1: The calculated ( n = 2 / 5 𝑛 2 5 n=2/5 ) | η i | subscript 𝜂 𝑖 |\eta_{i}| , η b subscript 𝜂 𝑏 \eta_{b} , Δ ​ U = U ​ ( η i ) − U ​ ( η = 0 ) Δ 𝑈 𝑈 subscript 𝜂 𝑖 𝑈 𝜂 0 \Delta U=U(\eta_{i})-U(\eta=0) , B η = U ​ ( η b ) − U ​ ( η i ) subscript 𝐵 𝜂 𝑈 subscript 𝜂 𝑏 𝑈 subscript 𝜂 𝑖 B_{\eta}=U(\eta_{b})-U(\eta_{i}) , and observed data for the morphological type T ​ y 𝑇 𝑦 Ty of galaxy according to Hubble classification, the total orbital mass M / M ⊙ 𝑀 subscript 𝑀 direct-product M/M_{\odot} , the projection of the linear distance X 𝑋 X between components, the linear diameters A 25 ​ ( 1 ​ i ) = 2 ​ R 1 ​ i subscript 𝐴 25 1 𝑖 2 subscript 𝑅 1 𝑖 A_{25}(1i)=2R_{1i} and A 25 ​ ( 2 ​ i ) = 2 ​ R 2 ​ i subscript 𝐴 25 2 𝑖 2 subscript 𝑅 2 𝑖 A_{25}(2i)=2R_{2i} of components of the close binary elliptic galaxies indicated by the serial numbers in the catalogue of isolated pairs of galaxies Karachentsev .

Di-galaxy	$Ty$	$\frac{M}{M_{\odot }}$	$X$	$A_{25}\left(1i\right)$	$A_{25}\left(2i\right)$	$\left|{\eta }_i\right|$	${\eta }_b$	$\Delta U$	$B_{\eta }$
			(kpc)	(kpc)	(kpc)			($J$)	($J$)
194	E-E	$2.72\times {10}^{11}$	27.0	36.9	29.3	0.28	0.42	$2\times {10}^{50}$	$2\times {10}^{50}$
279	E-E	$3.76\times {10}^{11}$	20.6	27.2	17.5	0.50	0.59	$3\times {10}^{51}$	$2\times {10}^{50}$
399	E-E	$5.84\times {10}^{11}$	28.2	27.1	26.5	0.03	0.27	$3\times {10}^{48}$	$2\times {10}^{50}$
501	E-E	$1.05\times {10}^{13}$	38.3	36.0	35.7	0.01	0.25	$1\times {10}^{50}$	$3\times {10}^{52}$
554	E-E	$1.05\times {10}^{12}$	57.6	52.2	42.7	0.25	0.26	$2\times {10}^{50}$	$2\times {10}^{48}$
577	E-E	$2.27\times {10}^{11}$	21.3	29.0	25.5	0.16	0.46	$6\times {10}^{49}$	$2\times {10}^{50}$

Table 2. Table 2: The same as in Table I, but for other close binary elliptic-spiral galaxies.

Di-galaxy	$Ty$	$\frac{M}{M_{\odot }}$	$X$	$A_{25}\left(1i\right)$	$A_{25}\left(2i\right)$	$\left|{\eta }_i\right|$	${\eta }_b$	$\Delta U$	$B_{\eta }$
			(kpc)	(kpc)	(kpc)			($J$)	($J$)
144	E-Sa	$1.43\times {10}^{11}$	17.1	18.3	16.0	0.17	0.35	$3\times {10}^{49}$	$3\times {10}^{49}$
254	E-Sb	$5.16\times {10}^{11}$	47.7	48.9	43.3	0.15	0.32	$6\times {10}^{49}$	$8\times {10}^{49}$
331	SO-E	$8.15\times {10}^{10}$	17.4	23.4	10.1	0.78	0.80	$2\times {10}^{51}$	$3\times {10}^{48}$
552	Sa-E	$5.01\times {10}^{11}$	38.8	39.5	26.2	0.47	0.49	$3\times {10}^{50}$	$6\times {10}^{48}$

Table 3. Table 3: The same as in Table I, but for close binary spiral galaxies. The values of Δ ​ U Δ 𝑈 \Delta U and B η subscript 𝐵 𝜂 B_{\eta} are calculated if η b > | η i | subscript 𝜂 𝑏 subscript 𝜂 𝑖 \eta_{b}>|\eta_{i}| .

Di-galaxy	$Ty$	$\frac{M}{M_{\odot }}$	$X$	$A_{25}\left(1i\right)$	$A_{25}\left(2i\right)$	$\left|{\eta }_i\right|$	${\eta }_b$	$\Delta U$	$B_{\eta }$
			(kpc)	(kpc)	(kpc)			($J$)	($J$)
1	Sb-Sb	$3.70\times {10}^{11}$	9.3	12.1	11.8	0.28	0.44	$2\times {10}^{50}$	$6\times {10}^{50}$
64	Sc-Sc	$6.20\times {10}^{11}$	39.5	55.8	27.4	0.71	0.75	$2\times {10}^{52}$	$2\times {10}^{50}$
105	Sb-Sb	$1.89\times {10}^{11}$	23.5	28.8	21.2	0.37	0.49	$2\times {10}^{50}$	$5\times {10}^{49}$
165	Sa-Sb	$2.23\times {10}^{11}$	10.0	12.9	9.9	0.32	0.49	$5\times {10}^{50}$	$2\times {10}^{50}$
201	Sa-Sb	$1.05\times {10}^{11}$	32.5	34.8	22.8	0.48	0.52	$9\times {10}^{49}$	$1\times {10}^{48}$
206	Sb-Sb	$1.28\times {10}^{11}$	16.8	13.2	9.0	0.45	0.34
237	Sb-Sb	$4.20\times {10}^{12}$	46.3	42.9	35.7	0.23	0.26	$5\times {10}^{51}$	$4\times {10}^{50}$
243	Sb-SO	$3.02\times {10}^{11}$	37.6	28.1	21.7	0.31	0.07
272	Sc-Sc	$1.11\times {10}^{12}$	83.5	32.7	23.6	0.39	0.00
297	Sc-Sc	$1.27\times {10}^{11}$	29.4	29.9	18.7	0.53	0.54	$2\times {10}^{50}$	$1\times {10}^{47}$
439	Sa-SO	$4.95\times {10}^{11}$	29.4	23.4	13.0	0.63	0.56
474	Sa-Sb	$4.77\times {10}^{11}$	21.7	27.1	23.3	0.19	0.43	$3\times {10}^{50}$	$6\times {10}^{50}$
490	Sa-Sa	$4.90\times {10}^{11}$	30.0	29.6	19.6	0.47	0.48	$2\times {10}^{51}$	$6\times {10}^{47}$
507	Sb-Sa	$1.60\times {10}^{12}$	32.2	35.5	25.1	0.41	0.47	$1\times {10}^{52}$	$8\times {10}^{50}$
516	Sc-Sb	$2.59\times {10}^{11}$	20.0	21.1	20.3	0.05	0.33	$4\times {10}^{48}$	$9\times {10}^{49}$
524	Sb-Sc	$4.83\times {10}^{11}$	15.9	17.4	14.6	0.22	0.37	$4\times {10}^{50}$	$3\times {10}^{50}$
539	SO-Sa	$9.68\times {10}^{10}$	17.8	25.8	17.0	0.48	0.60	$2\times {10}^{50}$	$3\times {10}^{49}$

Equations

$$dE(\mathbf{R},\mathbf{P},\eta )=\frac{\partial E}{\partial t}dt+\frac{\partial E}{\partial \mathbf{R}}d\mathbf{R}+\frac{\partial E}{\partial \mathbf{P}}d\mathbf{P}+\frac{\partial E}{\partial \eta }d\eta .$$

$$E=\frac{P_R^2}{2\mu }+U,$$

$$U=U_1+U_2+V$$

$$U_k=-\frac{G{M}_k^2}{2R_k},$$

$$R_k=\frac{1}{g}{M}_k^n$$

$$U_k=-\frac{Gg{M}_k^{2-n}}{2}.$$

$$g=\frac{M^n}{{\left(R_{1i}^{1/n}+R_{2i}^{1/n}\right)}^n}$$

$$V(R)=V_{\mathrm{grav}}(R)+V_{\mathrm{rot}}(R)=-\frac{G{M}_1{M}_2}{2R_m}$$

$$R_m={\left(\frac{{\mu }_i}{\mu }\right)}^2{R}_{mi},$$

$$V(R_m)=-{\omega }_V\frac{G{M}_1^3{M}_2^3}{2},$$

$${\omega }_V={\left(M^2{\mu }_i^2{R}_{mi}\right)}^{-1}.$$

$$U=-\frac{G}{2}\left(g[M_1^{2-n}+M_2^{2-n}]+{\omega }_V{M}_1^3{M}_2^3\right).$$

$$U=-\alpha [(1+\eta {)}^{2-n}+(1-\eta {)}^{2-n}]-\beta [1-{\eta }^2{]}^3,$$

$$\alpha =\frac{Gg}{2}{\left(\frac{M}{2}\right)}^{2-n}$$

$$\beta =\frac{G{M}^2}{128}\frac{(R_{1i}^{1/n}+R_{2i}^{1/n}{)}^4}{R_{mi}{R}_{1i}^{2/n}{R}_{2i}^{2/n}}.$$

$$\frac{\partial U}{\partial \eta }=-\alpha [2-n][(1+\eta {)}^{1-n}-(1-\eta {)}^{1-n}]+6\beta \eta [1-{\eta }^2{]}^2=0,$$

$$\alpha <\frac{3}{(1-n)(2-n)}\beta$$

$$R_{mi}<\frac{3{\left(R_{1i}^{1/n}+R_{2i}^{1/n}\right)}^{4+n}}{2^{4+n}(1-n)(2-n)R_{1i}^{2/n}{R}_{2i}^{2/n}}$$

$$\alpha >\frac{3}{(1-n)(2-n)}\beta .$$

$${\alpha }_{cr}=\alpha =\frac{3}{(1-n)(2-n)}\beta .$$

$${\eta }_b$$

$$R_{mi}>\frac{0.15{\left(R_{1i}^{5/2}+R_{2i}^{5/2}\right)}^{22/5}}{R_{1i}^5{R}_{2i}^5}.$$

Full text

Keywords: close binary galaxy, mass transfer, mass asymmetry

I Introduction

The observations of binary galaxies provide us the most direct and, therefore, the most reliable data on the masses and evolution of galaxies Karachentsev ; Karachentsev2 ; Pipino . The mass of a galaxy can be estimated with the virial theorem (the virial method), which is based on the assumption of the statistical balance of the kinetic and potential energy in the system. The mass of di-galaxies can be calculated using the simple premise of closed Keplerian motions of galaxies as two point-like masses (so-called the orbital motion method) Karachentsev . This orbital motion method seems to be more reliable because of lower uncertainties Karachentsev . The observations of binary galaxies show that the bulk of mass of galaxies complies with their standard optical (visible) boundaries. Strong support for this approach is obtained by the observation that the di-galaxy systems have a low ratio of the average orbital mass to luminosity, that is inconsistent with the hypothesis of an invisible (virial) mass Karachentsev .

In studies of binary galaxies one can test the concepts of their formation and evolution. The evolutionary path of the binary galaxy is not equivalent to that of isolated single galaxies. The mutual tidal interaction favors strong emission features in interacting galaxies. Many evidences have been gathered that the star formation processes in binary galaxy occur more frequently than in single galaxies or members of galactic associations. In comparison with single galaxies, the binary ones contain a higher percentage of objects of early structural types, the Seyfert objects, blue Markarian galaxies, and quasars Karachentsev . The galaxies in binary systems reveal mutual correlation by a number of integral features: luminosity, linear diameter, structural type, mass to luminosity ratio, the value of its own spin. As follows from the observations, the orbits of binary galaxies are close to circular, implying large angular momenta. For example, in the binary spiral (elliptical) galaxies, the ratio between the value of orbital angular momentum and the sum of spins of its components is about 1.3–2.5 (10) Karachentsev .

The compact or close binaries, which form di-galaxy compounds with the average distances between the galaxies of the same order as the sum of their radii, are of great interest for galactic evolution Karachentsev ; Karachentsev2 ; Pipino . Because mass transfer is an important observable in close binaries, it is necessary to study the evolution of the system in the mass asymmetry (transfer) coordinate $\eta=(M_{1}-M_{2})/(M_{1}+M_{2})$ where $M_{k}$ $(k=1,2)$ are the galactic masses at fixed total mass $M=M_{1}+M_{2}$ of the system IJMPE ; IJMPE2 . In our previous works IJMPE ; IJMPE2 , we used classical Newtonian mechanics and studied the evolution of close binary stars in the mass asymmetry (transfer) coordinate. For close binary stars, there are a lot of experimental and theoretical literatures about the mass transfer between two constituents Kopal:1978 ; Shore:1994 ; Hild:2001 ; Boya:2002 ; Eggleton:2006 ; Cher:2013 ; Tutuk ; Qian . In Refs. IJMPE ; IJMPE2 , we applied theoretical methods which are successfully applied to the corresponding processes in nuclear systems where the mass asymmetry plays an important role as the collective coordinate governing fusion of two heavy nuclei Adamian:2012 ; Adamian:2014 . Nuclear dynamics, of course, is quite different from the gravitational interactions in di-stars. Nuclear reactions are dominated by short-ranged strong interactions, to which minor contributions of long-range (repulsive) Coulomb and centrifugal forces are superimposed. However, extending the methods and results from the femto-scale of microscopic nuclear physics to macroscopic binary stellar systems, we obtained that the driving potentials for the di-star systems resembles in many aspects the driving potentials for the microscopic dinuclear systems IJMPE ; IJMPE2 ; Adamian:2012 ; Adamian:2014 . In the present paper, we explore the landscape of total potential energy of the close binary galaxy as a function of mass asymmetry coordinate, searching specifically for the evolution paths in the mass asymmetry coordinate and mass transfer source of the transformation of gravitational energy to other types of energy in the universe.

II Theoretical Method

The differential of total energy of di-galaxy system as a function of relative distance ${\bf R}$ of two galaxies, conjugate canonical momentum ${\bf P}$, and mass asymmetry coordinate $\eta$ reads as

[TABLE]

The kinetic energy in $\eta$ is assumed to be small and can be disregarded. In the center of mass system, the total energy of di-galaxy system is a sum of radial and orbital part of kinetic energies and the potential energy. For the reasons indicated below, we attach the orbital kinetic energy part to the galaxy-galaxy interaction $V$. In this case,

[TABLE]

where $P_{R}$ is the radial component of momentum ${\bf P}$ and $\mu=\mu(\eta)=\frac{M_{1}M_{2}}{M}=\frac{M}{4}(1-\eta^{2})$ is the reduced mass.

The total potential energy

[TABLE]

of the di-galaxy system is given by the sum of the potential energies $U_{k}$ ($k=1,2$) of two galaxies and galaxy-galaxy interaction potential $V$. The radiation energy is neglected because the absolute values of the gravitational energy and the intrinsic kinetic energy are much larger. The energy of the galaxy ”$k$” is

[TABLE]

where $G$, $M_{k}$, and $R_{k}$ are the gravitational constant, mass, and radius of the galaxy, respectively. The observational data result in the relationship

[TABLE]

between radius and mass of the galaxy, where the value of constant $n$ is in the interval $\left[\frac{2}{5},\frac{2}{3}\right]$ Karachentsev . Employing this expression, we obtain

[TABLE]

The value of the constant

[TABLE]

is determined by the observed total mass $M=M_{1}+M_{2}=M_{1i}+M_{2i}$ and radii $R_{1i,2i}=\frac{1}{g}M_{1i,2i}^{n}$ of two constituents.

Because the two galaxies rotate with respect to each other around the common center of mass, the galaxy-galaxy interaction potential

[TABLE]

contains, together with the gravitational energy $V_{\rm grav}(R)=-GM_{1}M_{2}/R$ of the interaction of two stars, the energy $V_{\rm rot}(R)=\mu v^{2}/2$ of orbital rotation. In Eq. (6), $v(R)=(GM[2/R-1/R_{m}])^{1/2}$ and $R_{m}$ are the speed and the semi-major axis of the elliptical relative orbit, respectively. For the di-galaxies considered, $v(R_{m})\ll c$, where $c$ is the velocity of light, and one can neglect the relativistic effects. Since $GM_{k}/R_{m}\ll c^{2}$, the gravitational field can be considered weak. Because of these facts, we use the Newtonian law of gravity. Using the Kepler’s laws, we obtain

[TABLE]

where the index ”i” denotes the observed reduced mass $\mu_{i}=\mu(\eta_{i})=\frac{M_{1i}M_{2i}}{M}=\frac{M}{4}(1-\eta_{i}^{2})$ and distance $R_{mi}$ between the galaxies of the initial binary system with mass asymmetry $\eta=\eta_{i}$. Equation (6) is rewritten as

[TABLE]

where

[TABLE]

The final expression for the total potential energy (3) of the di-galaxy system is

[TABLE]

Expressing the masses in terms of the mass asymmetry coordinate $\eta$, $M_{1}=\frac{M}{2}(1+\eta)$ and $M_{2}=\frac{M}{2}(1-\eta)$, we rewrite Eq. (9):

[TABLE]

where

[TABLE]

and

[TABLE]

To obtain $\alpha$ and $\beta$, we use the observed values of $M$, $R_{1i}$, $R_{2i}$, and $R_{mi}=\frac{\pi}{4}X$, where $X$ is projection of the linear distance between the components of binary galaxy, from the catalogue of isolated pairs of galaxies Karachentsev .

Employing Eq. (10), we study the evolution of the di-galaxy system in the mass asymmetry coordinate $\eta$. The extremal points of the potential energy as a function of $\eta$ are found by solving numerically the equation

[TABLE]

and the solutions are the fixed points of the evolution equation for the mass asymmetry. As seen, Eq. (11) is solved for $\eta=\eta_{m}=0$. At this value the potential has an extremum which is a minimum if

[TABLE]

or

[TABLE]

and a maximum if

[TABLE]

The transition point is

[TABLE]

If there is a minimum at $\eta=0$ ($\alpha<\alpha_{cr}$), it is engulfed symmetrically by two barriers at $\eta=\pm\eta_{b}$. As seen, the extremal points depend only on $R_{mi}$, $R_{1i,2i}$, and $n$. Note that for the touching binary system ($R_{mi}\approx R_{1i}+R_{2i}$), a minimum at $\eta=0$ exists because the condition $\alpha<\frac{3}{(1-n)(2-n)}\beta$ or $\frac{3\left(R_{1i}^{1/n}+R_{2i}^{1/n}\right)^{4+n}}{2(1-n)(2-n)\left(4R_{1i}^{1/n}R_{2i}^{1/n}\right)^{\frac{4+n}{2}}}>1$ is always satisfied since $\frac{3}{2(1-n)(2-n)}>1$ and $\frac{\left(R_{1i}^{1/n}+R_{2i}^{1/n}\right)^{4+n}}{\left(4R_{1i}^{1/n}R_{2i}^{1/n}\right)^{\frac{4+n}{2}}}>1$.

Expanding Eq. (11) up to the third order in $\eta$ and solving it, we obtain the position of these barriers at $\eta=\pm\eta_{b}$, where

[TABLE]

So, at $\alpha<\alpha_{cr}$ the potential energy as a function of $\eta$ has two symmetric maxima at $\eta=\pm\eta_{b}$ and the minimum at $\eta=\eta_{m}=0$. As seen from Eq. (12), $\eta_{b}<2^{-1/2}\approx 0.71$. This condition means that in the asymmetric binary system with mass ratio $M_{1i}/M_{2i}>(1+2^{1/2})^{2}\approx 6$ the galaxies move away from each other. Thus, di-galaxies with $|\eta|>\eta_{b}$ are the unstable with respect to the separation and unlikely to exist as a correlated, bound pair for sufficiently long time. Indeed, the binary galaxies with a large mass ratio are very rare objects Karachentsev .

III Application to Close Binaries

In the calculations, we use the observational data for the galactic linear diameters $A_{25}(1i,2i)=2R_{1i,2i}$, the projection of the linear distance $X$ between the components of the binary galaxy, and the total orbital mass $M$ of the pair from the catalogue of the isolated pairs of galaxies Karachentsev . For the close binary galaxies considered, the average relative distance $R_{mi}=\frac{4}{\pi}X$ between galaxies is comparable with the sum of radii of galaxies Karachentsev . The mass transfer between galaxies in a binary system is closely related to their radii and the relative distance. From the observational data for galaxies with large mass $M_{1i,2i}>10^{10}M_{\odot}$, where $M_{\odot}$ is the mass of the Sun, the dependence between the mass and radius of the galaxy is extracted as $R_{k}=\frac{1}{g}M_{k}^{n}$, where $n=2/5$ and $g=M^{2/5}/\left(R_{1i}^{5/2}+R_{2i}^{5/2}\right)^{2/5}$ Karachentsev . The masses of components of the binary system are defined as $M_{1i,2i}=MR^{1/n}_{1i,2i}/\left(R^{1/n}_{1i}+R^{1/n}_{2i}\right)$. Note that the initial mass asymmetry $\eta=\eta_{i}$ depends on $n$: $\eta_{i}=\left(R^{1/n}_{1i}-R^{1/n}_{2i}\right)/\left(R^{1/n}_{1i}+R^{1/n}_{2i}\right)$. The elliptic, spiral, and mixed elliptic-spiral binary galaxies are considered below.

Various di-galaxies have different $R_{mi}$ and $R_{1i,2i}$, and, correspondingly, the potential energy shapes are varying. The potential energies (driving potentials) $U(\eta)$ of the close elliptic, spiral, and elliptic-spiral di-galaxy systems versus $\eta$ are presented in Figs. 1-5. For all systems shown, except the binary galaxy 272 Karachentsev , $\alpha<\alpha_{cr}=\frac{25}{8}\beta$ and, thus, the potential energies have symmetric barriers at $\eta=\pm\eta_{b}$ and the minimum at $\eta=\eta_{m}=0$. The barrier in $\eta$ appears from the interplay between the total gravitational energy $U_{1}+U_{2}$ of the galaxies and the galaxy-galaxy interaction potential $V$. These energies behave differently as a function of mass asymmetry: $U_{1}+U_{2}$ decreases and $V$ increases with changing $\eta$ from $\eta=0$ to $\eta=\pm 1$. One should stress that the driving potentials $U(\eta)$ for the di-galaxy systems look like the driving potentials for the microscopic dinuclear systems Adamian:2012 ; Adamian:2014 . For the close binary star systems, the same conclusion was drawn in Refs. IJMPE ; IJMPE2 .

An important question is what extent the results depend on the choice of parameters. In Fig. 2, we investigate the effect of varying the mass-radius power-law relation by using $n=2/3$. As seen by comparison to Fig. 1, obtained with $n=2/5$, the shapes of the potentials in both calculations are qualitatively the same, except that the ratio $U(\eta=\eta_{b})/U(\eta=0)$ increases with the value of $n$. The central result of these investigations is that all binary systems evolve to the symmetric configurations with $\eta=0$.

The evolution path of close binary galaxy depends on the initial mass asymmetry $\eta=\eta_{i}$ at its formation. If the original di-galaxy is asymmetric, but $|\eta_{i}|<\eta_{b}$, then it is energetically favorable to evolve in $\eta$ to a configuration in the global minimum at $\eta=0$, that is, to form a symmetric di-galaxy system. The matter of the heavy partner can move freely to an adjacent light galaxy enforcing the symmetrization of di-galaxy without additional driving energy. The symmetrization of an initially asymmetric binary galaxy leads to the decrease of potential energy $U$, thus transforming of the potential energy into internal kinetic energy. For example, for the close elliptic binary galaxies 194 ($\eta_{i}$=0.28), 279 ($\eta_{i}$=0.50), and 554 ($\eta_{i}$=0.25) Karachentsev , the internal energies of galaxies will increase during symmetrization by the amount $\Delta U=U(\eta_{i})-U(\eta=0)=$ $2\times 10^{50}$, $3\times 10^{51}$, and $2\times 10^{50}$ J, respectively (see Fig. 1 and Table I). Because most of close binary galaxies are asymmetric ones, the symmetrization process leads to the release of a large amount of energy in these systems (see Tables I–III).

If $|\eta_{i}|>\eta_{b}$ or $\eta_{b}=0$, the di-galaxy system is unstable and evolves towards more asymmetric system, thus, enforcing the asymmetrization of the di-galaxy (one galaxy ”swallows” the other galaxy). The matter is flowing freely from the light galaxy to the heavy one. That process is sustained without implementing additional external energy. This mode of evolution also leads to energy release. Representative examples for this mode of evolution are the three cases of close spiral binary galaxies, namely the galaxies 206 ($|\eta_{i}|=0.45$, $\eta_{b}=0.34$), 243 ($|\eta_{i}|=0.31$, $\eta_{b}=0.07$), and 439 ($|\eta_{i}|=0.63$, $\eta_{b}=0.56$) Karachentsev , respectively, for which $|\eta_{i}|>\eta_{b}$ (Fig. 5).

Mergers of binary galaxy with $|\eta_{i}|<\eta_{b}$ can occur only by overcoming the barrier at $\eta=+\eta_{b}$ or $\eta=-\eta_{b}$. With decreasing ratio $\alpha/\beta$, the value of this barrier $B_{\eta}=U(\eta_{b})-U(\eta_{i})$ increases. Because the barriers $B_{\eta}$ in $\eta$ are quite large for the systems with $|\eta_{i}|<\eta_{b}$ in Tables I–III, the asymmetrization of the di-star system by the thermal diffusion in mass asymmetry coordinate is strongly suppressed. Most likely, the barrier in $\eta$ is the reason why very asymmetric close binary galaxies with $|\eta_{i}|>\eta_{b}$ are rarely observed. This imposes restrictions on the asymmetric configurations with $|\eta|>\eta_{b}$ of the di-galaxy systems. There can not be a stable di-galactical system with a very light galaxy, with only a fraction of the mass of the heavy partner.

If the value of $\alpha$ becomes larger than $\alpha_{cr}$, the minimum in $U(\eta)$ disappears, the inverted $U$-shaped potential shape with maximum at $\eta=0$ forms, and the di-galaxy asymmetrization drives the system finally apart towards separation. Hence, with increasing mass asymmetry, two constituents of the binary galaxy move away from each other and finally separate into single galaxies. The spiral binary galaxy 272 (see Fig. 4 and Table III) is a good candidate for such evolution.

Our calculations show that the double galaxy will undergo either symmetrization or asymmetrization depending critically on the mass asymmetry. Asymmetrization may start as a merger process, but once the critical mass asymmetry is reached, the system is driven apart and ends up by two isolated galaxies of a changed mass ratio. Hence, this scenario corresponds to an incomplete merger.

So, the source of expansion of a binary galaxy is the transfer of mass from the lighter component to the heavy one. A necessary and sufficient condition for this is the fulfillment of the inequality

[TABLE]

This mechanism presented can be generalized for multiple galaxies, groups of galaxies and galaxy associations.

IV Summary

We have shown that the mass asymmetry (transfer) collective degree of freedom plays a comparable important role in macroscopic object as well as in microscopic dinuclear systems. In close binary galaxy, the mass asymmetry coordinate can govern the asymmetrization (the transfer of mass from the lighter component to the heavy one) and symmetrization (the transfer of mass from the heavier component to the light one) of the system. The symmetrization of binary galaxies leads to the release of a large amount of energy about $10^{48-52}$ J, thus reaching the energy release of novae or even close to supernovae events. Thus, the symmetrization of close binary galaxy due to the mass transfer is one of the important sources of the transformation of the gravitational energy to other types of energy, like radiation energy, in the universe. The symmetrization of binary system will lead to $M_{1}/M_{2}\to 1$, $L_{1}/L_{2}\to 1$ ($L_{i}$ are the luminosities of galaxies), $R_{1}/R_{2}\to 1$ which are observable quantities. Asymmetrization is equivalent to the incomplete merging of components of a binary system. The asymmetrization is also the source of expansion of a binary galaxy. The separation of components from each other can be represented as an analogue of the expansion of the universe within the framework of a binary system. The conditions under which either asymmetrization or symmetrization processes are realized have been determined and investigated. These conditions depend mainly on the relative distance between the galaxies and their linear sizes. The limitation of the binary system existence at the mass ratios $M_{1i}/M_{2i}>6$ has been derived.

In the frame of our model, we can also perform dynamic calculations of the evolution of binary system in the mass asymmetry coordinate. For example, we can calculate the relaxation (symmetrization) time and the asymmetrization time. In addition to the total potential energy, it is also necessary to calculate the mass parameter for this coordinate. But this extension of our model is beyond the scope of the present paper.

V Acknowledgements

V.V.S. acknowledge the partial supports from the Alexander von Humboldt-Stiftung (Bonn). This work was partially supported by Russian Foundation for Basic Research (Moscow) and DFG (Bonn), contract Le439/16.