# Origin of the orbital period change in contact binary stars

V.V. Sargsyan1,2, H. Lenske2, G.G. Adamian1, and N.V. Antonenko1 1Joint
Institute for Nuclear Research, 141980 Dubna, Russia,
2Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany

###### Abstract

The evolution of contact binary star systems in mass asymmetry (transfer)
coordinate is considered. The orbital period changes is explained by an
evolution in mass asymmetry towards the symmetry (symmetrization of binary
system). It is predicted that a decreasing and an increasing orbital periods
are related, respectively, with the non-overlapping and overlapping stage of
the binary star during its symmetrization. A huge amount of energy $\Delta
U\approx 10^{41}$ J is converted from the the potential energy into internal
energy of the stars during the symmetrization. As shown, the merger of stars
in the binary systems, including KIC 9832227, is energetically an unfavorable
process. The sensitivity of the calculated results to the values of total mass
and orbital angular momentum is analyzed.

###### pacs:

26.90.+n, 95.30.-k
Keywords: close binary stars, mass transfer, mass asymmetry

##  I Introduction

Overcontact, contact, and near-contact binaries, forming di-star compounds
with the average distances between the stars of the same order as the sum of
their radii Boya:2002 ; Eggleton:2006 ; Vasil:2012 ; Cher:2013  are of great
interest for stellar evolution. The compact binary stars are a good laboratory
for a wide variety of astrophysical phenomena, for example, the mass transfer
between stars. The information on formation and evolution of compact binaries
is required to understand the processes observed in isolated stars. The
observations of the stages of evolution in these binaries provide a
verification of our understanding of the inner structure and dynamical
interaction of stars.

It has been found that the orbital period of W-type overcontact binary GW Cep
($q=M_{1i}/M_{2i}=0.37$, $\eta_{i}=(M_{1i}-M_{2i})/(M_{1i}+M_{2i})=0.46$,
$M_{1i}$ and $M_{2i}$ are the masses of stars in the binary) is decreasing
with time Qian . For the overcontact binaries VY Cet ($q=0.67$,
$\eta_{i}=0.20$) and V700 Cyg ($q=0.65$, $\eta_{i}=0.21$), a cyclic
oscillations have been found to be superimposed on a secular period increase.
This effect has been explained either by the strong external perturbation,
i.e. by a close-by third object, or by the magnetic activity cycles. For the
EM Lac ($q=0.63$, $\eta_{i}=0.23$) and AW Vir ($q=0.76$, $\eta_{i}=0.14$)
binaries, the periods show a secular increase. It has been concluded that the
period variations of a W UMa-type binary star is correlated with the mass
ratio $q$ and the mass $M_{1i}$ of the primary component Qian . The low mass
ratio $q$ in binaries usually results in a decreasing period, while the
periods of larger-$q$ systems are increasing.

Because mass transfer is an important observable for close binary systems in
which the two stars are nearly in contact Boya:2002 ; Eggleton:2006 ;
Vasil:2012 ; Cher:2013 ; Qian ; IJMPE ; IJMPE2 , it is necessary to study the
influence of evolution of these stellar systems in the mass asymmetry
coordinate $\eta=(M_{1}-M_{2})/M$ [$M_{k}$ ($k$ =1,2) are the stellar masses
and $M=M_{1}+M_{2}$] on the orbital period variations. This is the main aim of
the present article. As in our previous works IJMPE ; IJMPE2 , we analyze the
total potential energy $U(\eta)$ and the orbital period as functions of $\eta$
at fixed total mass $M=M_{1}+M_{2}$ and orbital angular momentum $L=L_{i}$ of
the system.

##  II Theoretical Method

The total potential energy of the di-star system

$\displaystyle U=U_{1}+U_{2}+V$ (1)

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

$\displaystyle U_{k}=-\omega_{k}\frac{GM_{k}^{2}}{2R_{k}},$ (2)

where $G$, $M_{k}$, and $R_{k}$ are the gravitational constant, mass, and
radius of the star, respectively. Employing the values of the dimensionless
structural factor

$\displaystyle\omega_{k}=1.644\left(\frac{M_{\odot}}{M_{k}}\right)^{1/4}$ (3)

and radius

$R_{k}=R_{\odot}\left(\frac{M_{k}}{M_{\odot}}\right)^{2/3}$

of the star from the model of Ref. Vasil:2012 , we obtain

$\displaystyle U_{k}$ $\displaystyle=$
$\displaystyle-\omega_{0}GM_{k}^{13/12}/2,$ $\displaystyle\omega_{0}$
$\displaystyle=$ $\displaystyle 1.644\frac{M_{\odot}^{11/12}}{R_{\odot}},$ (4)

where $M_{\odot}$ and $R_{\odot}$ are mass and radius of the Sun,
respectively. Because the average density
$\rho_{k}=M_{\odot}\rho_{\odot}/M_{k}$ ($\rho_{\odot}$ is the average density
of the Sun) increase with decreasing the mass $M_{k}$ of star Vasil:2012 , the
structural factor $\omega_{k}$ depends on $M_{k}$ in Eq. (3). The change of
$\eta$ from 0 to 1 leads to the change of $\omega_{1}$ by about of 16%. Note
that, in general, the dimensionless structural factor $\omega_{i}$ is
determined by the density profile of the star. In the present paper, we employ
the values of the structural factor of the stars from the model of Ref.
Vasil:2012 . This model well describes the observable temperature-radius-mass-
luminosity relations of stars, especially binary stars, the spectra of seismic
oscillations of the Sun, distribution of stars on their masses, magnetic
fields of stars and etc. The stellar radii, masses, and temperatures are
expressed by the corresponding ratios of the fundamental constants, and the
individuality of stars is determined by two parameters - by the charge and
mass numbers of nuclei, from which the star is composed Vasil:2012 .

Because the two stars rotate with respect to each other around the common
center of mass, the star-star interaction potential contains, together with
the gravitational energy of interaction $V_{G}$ of two stars, the kinetic
energy of orbital rotation $V_{R}$:

$\displaystyle V(R)=V_{G}+V_{R}=V_{G}+\frac{L^{2}}{2\mu R^{2}},$ (5)

where $L$ is the orbital angular momentum of the di-star which is conserved
during the conservative mass transfer and $\mu=\frac{M_{1}M_{2}}{M}$ is the
reduced mass. At $R\geq R_{t}=R_{1}+R_{2}$ and $R\leq R_{t}$,

$\displaystyle V_{G}(R)=-\frac{GM_{1}M_{2}}{R}$ (6)

and

$\displaystyle
V_{G}(R)=-\frac{GM_{1}M_{2}}{2R_{t}}\left[3-\frac{R^{2}}{R_{t}^{2}}\right],$
(7)

respectively NPA . Here, $R_{t}$ is the touching distance. From the fix point
conditions $\partial V/\partial R|_{R=R_{m}}=0$ and $\partial^{2}V/\partial
R^{2}|_{R=R_{m}}>0$, we find the relative equilibrium distance between two
stars corresponding to the minimum of $V$:

$\displaystyle R_{m}=\frac{L^{2}}{G\mu^{2}M}$ (8)

at $R_{m}\geq R_{t}$ or

$\displaystyle R_{m}=\left(\frac{L^{2}R_{t}^{3}}{G\mu^{2}M}\right)^{1/4}$ (9)

at $R_{m}\leq R_{t}$. Finally, one can derive the expression for the star-star
interaction potential

$\displaystyle V(R_{m})=-\frac{GM_{1}M_{2}}{2R_{m}}$ (10)

at $R_{m}\geq R_{t}$ or

$\displaystyle
V(R_{m})=-\frac{GM_{1}M_{2}}{R_{t}}\left[\frac{3}{2}-\frac{R_{m}^{2}}{R_{t}^{2}}\right]$
(11)

at $R_{m}\leq R_{t}$. In Eqs. (10) and (11), $R_{m}$ is the semi-major axis of
the elliptical relative orbit. Note that for the di-star systems considered,
$v(R_{m})=(GM/R_{m})^{1/2}\ll c$, where $v$ and $c$ is the velocities of
orbital motion and light, respectively, 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 mass asymmetry coordinate $\eta$ instead of masses
$M_{1}=\frac{M}{2}(1+\eta)$ and $M_{2}=\frac{M}{2}(1-\eta)$, we rewrite the
final expression (1) for the total potential energy as

$\displaystyle
U=-\frac{GM_{\odot}^{2}}{2R_{\odot}}\left(\alpha[(1+\eta)^{13/12}+(1-\eta)^{13/12}]+\beta_{1}[1-\eta^{2}]^{3}\right)$
(12)

at $R_{m}\geq R_{t}$ or

$\displaystyle U$ $\displaystyle=$
$\displaystyle-\frac{GM_{\odot}^{2}}{2R_{\odot}}\left(\alpha[(1+\eta)^{13/12}+(1-\eta)^{13/12}]\right.$
(13) $\displaystyle+$
$\displaystyle\left.\beta_{2}\frac{1-\eta^{2}}{(1+\eta)^{2/3}+(1-\eta)^{2/3}}\left[\frac{3}{2}-\frac{\gamma}{[1-\eta^{2}][(1+\eta)^{2/3}+(1-\eta)^{2/3}]^{1/2}}\right]\right)$

at $R\leq R_{t}$, where

$\alpha=1.644\left(\frac{M}{2M_{\odot}}\right)^{13/12},$
$\beta_{1}=\frac{GM^{5}}{64L^{2}}=\frac{GM_{\odot}^{5}}{2L^{2}}\left(\frac{M}{2M_{\odot}}\right)^{5},$
$\beta_{2}=2\left(\frac{M}{2M_{\odot}}\right)^{4/3},$

and

$\gamma=\frac{2^{7/3}LM_{\odot}^{1/3}}{\left(GR_{\odot}M^{11/3}\right)^{1/2}}.$

Here, we assume that the orbital angular momentum $L$ and the total mass $M$
are conserved during the conservative evolution of the di-star in the mass
asymmetry coordinate $\eta$. The orbital angular momentum $L$ is calculated by
using the experimental masses $M_{k,i}$ of stars and period $P_{orb,i}$ of
their orbital rotation Qian ; Egg . As seen from Eq. (13), the stability of
the binary star system depends on the orbital angular momentum $L$ the total
mass $M$, and the structural factor.

To obtain the period $P_{orb}=\frac{2\pi}{\omega_{orb}}=\frac{2\pi\mu
R_{m}^{2}}{L}$ of orbital rotation with frequency $\omega_{orb}$, we use the
relation between $L$ and $R_{m}$. At $R_{m}>R_{t}$ and $R_{m}\leq R_{t}$, we
have the periods

$\displaystyle P^{>}_{orb}=2\pi\left(\frac{R_{m}^{3}}{GM}\right)^{1/2}$ (14)

and

$\displaystyle P^{<}_{orb}=2\pi\left(\frac{R_{t}^{3}}{GM}\right)^{1/2},$ (15)

respectively. As seen from formulas, at initial $|\eta_{i}|$ smaller than the
position $|\eta_{b}|$ of barrier of the potential energy $U$ (see Sect. III)
and $R_{m}>R_{t}$ ($R_{m}\leq R_{t}$), the system moves towards the symmetric
configuration and, respectively, $\eta$ decreases, $R_{m}$ decreases ($R_{t}$
increases), and, finally, the $P^{>}_{orb}$ decreases ($P^{<}_{orb}$
increases).

##  III Application to Close Binaries

Many di-star systems have different $M$ and $L$ (Table I) Qian ; Egg  and,
correspondingly, the potential energy shapes. The potential energies (driving
potentials) $U(\eta)$ of the close di-star systems versus $\eta$ are presented
in Figs. 1 and 2. In the calculations, we assume that the orbital angular
momentum and the total mass are conserved during the conservative evolution of
the di-star in mass asymmetry coordinate $\eta$. For all binary systems shown,
the potential energies have the barriers at $\eta=\pm\eta_{b}$ and the minimum
at $\eta=\eta_{m}=0$. The barrier in $\eta$ appears as a result of the
interplay between the total gravitational energy $U_{1}+U_{2}$ of the stars
and the star-star interaction potential $V$. These values have different
behavior as a function of mass asymmetry: $U_{1}+U_{2}$ decreases and $V$
increases with changing $\eta$ from $0$ to $\pm 1$. One should stress that the
driving potentials $U(\eta)$ for the di-star systems looks like the driving
potentials for the microscopic dinuclear systems Adamian:2012 ; Adamian:2014 .
The collective coordinate $\eta$, treated a separate dynamical degree of
freedom, plays a comparable important role in macroscopic object as well as in
microscopic dinuclear systems. Note that the same conclusion was drawn in
Refs. IJMPE ; IJMPE2 .

The evolution of di-star system depends on the initial mass asymmetry
$\eta=\eta_{i}$ at its formation. The original di-star is asymmetric and
$|\eta_{i}|<|\eta_{b}|$, then it is energetically favorable to evolve in
$\eta$ towards a configuration at $\eta=0$, that is, to form a symmetric di-
star system. The matter of a heavy star can move to an adjacent light star
enforcing the symmetrization of di-star without additional driving energy. The
symmetrization of asymmetric binary star leads to the decrease of potential
energy $U$, i. e. the transformation of the potential energy into internal
energy of the stars. A huge amount of energy $\Delta U\approx 10^{41}$ J is
released and converted during the symmetrization (see Figs. 1 and 2). The
resulting symmetric di-star is created at large excitation energy. So, the
binary star systems are the sources of energy in the Universe. Note that for
all binary stars considered the energy of one single star ($|\eta|=1$) is
larger than the energy of the symmetric binary ($\eta=0$) which means that the
merger of stars in the binary system is energetically an unfavorable process.

A spectacular recent case is KIC 9832227 which was predicted Molnar:2017  to
be merge in 2022, enlightening the sky as a red nova. For the KIC 9832227
($|\eta_{i}|=0.63$, $|\eta_{b}|=0.84$), we predict that a fast merger is
excluded (see Fig. 2). This di-star is driven instead towards the mass
symmetry. It should be stressed that the observational data of Ref. Socia
negate the merger of KIC 9832227 in 2022.

For the KIC 9832227, we studied the change of potential energy surface with
the variations of the total mass $M$ and orbital angular momentum $L$ of
system (Figs. 2 and 3). In this way, we take effectively into consideration
the losses of mass and angular momentum of the binary star during its
evolution. The mass or angular momentum may be removed from the system via
stellar wind or gravitational radiation. As seen, a loss of the orbital
angular momentum at a fixed mass $M$ increases the symmetrization rate. The
simultaneous losses of $M$ and $L$ ($\Delta M\sim\Delta L$) by 20% or 50%
weakly influence the shape of potential energy. The decrease of $M$ at fixed
$L$ has a more pronounced effect (the depth of the minimum in $U(\eta)$
decreases) but the evolution to the global symmetric minimum is still
energetically favored. So, the realistic simultaneous losses of $M$ and $L$
almost do not influence the rate of symmetrization of the system. Note that a
loss of the orbital angular momentum at the fixed mass increase the rate of
symmetrization of the system.

For KIC 9832227 system ($|\eta_{i}|=0.63$), the mass is transferring from the
heavy star to light companion, and the relative distance
($R_{m}>R_{t}=R_{1}+R_{2}$) between two stars and the period $P^{>}_{orb}$ of
the orbital rotation are decreasing. The evolution in $\eta$ leads to the
touching configuration ($R_{m}=R_{t}=R_{1}+R_{2}$) of stars at some critical
mass asymmetry $|\eta|=|\eta_{t}|\approx 0.45$ (Fig. 4). Further evolution in
$\eta$ leads to a configuration with partial overlap ($R_{m}<R_{t}$) of the
stars. So, at $|\eta|\leq|\eta_{t}|$ the period $P^{<}_{orb}$ of the orbital
rotation is slightly increasing because $P^{<}_{orb}\sim R_{t}^{3/2}$ and
$R_{t}$ increases with decreasing $\eta$. Thus, once the system has crossed
the point $\eta=\eta_{t}$ and the phase of partial overlap is entered, the
evolution of period changes abruptly. The other contact binaries considered in
Fig. 4 show a similar behavior of period.

For the binary GW Cep ($|\eta_{i}|=0.46$), $|\eta_{i}|>|\eta_{t}|$, the system
moves towards mass symmetry and the orbital period is decreasing with time.
For the almost symmetric EM Lac ($|\eta_{i}|=0.23$) and AW Vir
($|\eta_{i}|=0.14$) binaries, $|\eta_{i}|<|\eta_{t}|$ and the periods show the
secular increase. So, one can conclude that the period variations of a W UMa-
type binary star is correlated with the mass asymmetry evolution towards the
global symmetric minimum. At low mass ratio $q$, i. e. or the large mass
asymmetry $\eta_{i}$, binaries usually show a decreasing period because
$|\eta_{i}|>|\eta_{t}|$ ($R_{m}>R_{t}$), while the periods in systems with
high $q$ (or small $\eta_{i}$) are increasing because $|\eta_{i}|<|\eta_{t}|$
($R_{m}<R_{t}$).

##  IV Summary

For all contact di-star systems considered, the potential energies have
symmetric barriers at $\eta=\pm|\eta_{b}|$ and the minimum at
$\eta=\eta_{m}=0$. The di-star system is initially formed with $\eta=\eta_{i}$
and $0<|\eta_{i}|<|\eta_{b}|$. The two stars start to exchange matter and the
system is driven to the symmetric di-star configuration (towards a global
minimum of the potential landscape). The mass asymmetry coordinate is
governing the symmetrization driven by the mass transfer process of two stars.
The losses of the total mass and orbital angular momentum weakly influence the
symmetrization of system. Note that the merger of the binary star systems
considered is energetically not favored.

The orbital period changes can be plausibly explained by an evolution in mass
asymmetry $\eta$ towards the symmetry. We predicted that a decreasing and an
increasing orbital periods are related, respectively, with the non-overlapping
($|\eta_{i}|>|\eta_{t}|$, $R_{m}>R_{1}+R_{2}$) and overlapping
($|\eta_{i}|<|\eta_{t}|$, $R_{m}<R_{1}+R_{2}$) stage of the binary star during
its symmetrization. Thus, the observations of changing periods allows us to
distinguish between these two stages of the binary star systems.

##  V Acknowledgements

This work was partially supported by Russian Foundation for Basic Research
(Moscow), grant number 17-52-12015, and DFG (Bonn), contract Le439/16.

## References

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Table 1:  The observed data $M_{1}/M_{\odot}$, $M_{2}/M_{\odot}$, and $P_{orb}$ are from Refs. Qian ; Egg ; Molnar:2017 .  Di-star | $\frac{M_{1}}{M_{\odot}}$ | $\frac{M_{2}}{M_{\odot}}$ | $P_{orb}$
---|---|---|---
|  |  | days
AH Aur | 1.68 | 0.28 | 0.4941
AP Aur | 2.05 | 0.50 | 0.5694
DN Aur | 1.44 | 0.30 | 0.6169
AW Vir | 1.11 | 0.84 | 0.3540
AW UMa | 1.38 | 0.14 | 0.4387
HV UMa | 2.84 | 0.54 | 0.7108
KIC 9832227 | 1.40 | 0.32 | 0.4583
HV Aqr | 1.31 | 0.19 | 0.3734
GX And | 1.23 | 0.29 | 0.4122
RR Cen | 2.09 | 0.45 | 0.6060
EM Lac | 1.06 | 0.67 | 0.3891
GW Cep | 1.06 | 0.39 | 0.3188
V700 Cyg | 0.92 | 0.60 | 0.3400
V870 Ara | 1.34 | 0.11 | 0.3997

Figure 1: The calculated total potential energies $U$ vs $\eta$ for the
indicated overcontact binary star systems. The arrow on $x$-axis shows the
corresponding initial $\eta_{i}$ for binary star.

Figure 2: The same as in Fig. 1, but for other indicated close binary star
systems.

Figure 3:  The calculated total potential energies $U$, the relative $R_{m}$
(solid line) and touching $R_{t}=R_{1}+R_{2}$ (dotted line) distances between
components in the units of the Sun radius $R_{sun}=R_{\odot}$ vs $\eta$ for
the binary star KIC 9832227. The notations ($0.8M,1L$), ($0.5M,1L$),
($0.8M,0.8L$), and ($0.5M,0.5L$) mean that the calculations are performed with
the losses of total mass $M$ and orbital angular momentum $L=L_{i}$ by (20%,
0%), (50%, 0%), (20%, 20%), and (50%, 50%), respectively. The arrow on
$x$-axis shows the corresponding initial $\eta_{i}$ for binary star.

Figure 4:  The calculated relative $R_{m}$ (solid line) and touching
$R_{t}=R_{1}+R_{2}$ (dotted line) distances between components in the units of
the Sun radius $R_{sun}=R_{\odot}$ vs $\eta$ for the indicated binary star
systems.