Issue 
A&A
Volume 504, Number 1, September II 2009



Page(s)  161  170  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/200811144  
Published online  09 July 2009 
Structure and evolution of rotationally and tidally distorted stars
H. F. Song^{1,2}  Z. Zhong^{1}  Y. Lu^{1}
1  College of science, Guizhou University, Guiyang, 550025, PR China
2 
Joint Centre for Astronomy, National Astronomical ObservatoriesGuizhou University, Guiyang, 550025, PR China
Received 14 October 2008 / Accepted 2 June 2009
Abstract
Aims. This paper aims to study the configuration of two components caused by rotational and tidal distortions in the model of a binary system.
Methods. The potentials of the two distorted components can be approximated to 2nddegree harmonics. Furthermore, both the accretion luminosity (
)
and the irradiative luminosity are included in stellar structure equations.
Results. The equilibrium structure of rotationally and tidally distorted star is exactly a triaxial ellipsoids. A formula describing the isobars is presented, and the rotational velocity and the gravitational acceleration at the primary surface simulated. The results show the distortion at the outer layers of the primary increases with temporal variation and system evolution. Besides, it was observed that the luminosity accretion is unstable, and the curve of the energygeneration rate fluctuates after the main sequence in rotation sequences. The luminosity in rotation sequences is slightly weaker than that in nonrotation sequences. As a result, the volume expands slowly. Polar ejection is intensified by the tidal effect. The ejection of an equatorial ring may be favoured by both the opacity effect and the
effect in the binary system.
Key words: stars: rotation  stars: binaries: close
1 Introduction
In the conventional model of binary stars, there is no consideration of spin and tidal effects (Eggleton 1971, 1972, 1973; Hofmeister et al. 1964; Kippenhahn et al. 1967; etc.); however, rotation and tide have been regarded as two important physical factors in recent years, so they need to be considered for a better understanding of the evolution of massive close binaries (e.g., Heger et al. 2000a; Meynet & Maeder 2000). The structure and evolution of rotating single stars has been studied by many investigators (Kippenhahn & Thomas 1970; Endal & Sofia 1976; Pinsonneaul et al. 1989; Meynet & Maeder 1997; Langer 1998, 1999; Huang 2004a). However, it is also very important to study the evolution of rotating binary stars (Jackson 1970; Chan & Chau 1979; Langer 2003; Huang 2004b; Petrovic et al. 2005a,b; Yoon et al. 2006). The effect of spin on structure equations has been investigated (e.g. the present Eggleton's stellar evolution code; Li et al. 2004a,b, 2005; K hler 2002). They adopted the lowestorder approximate analysis in which two components were treated as spherical stars. In fact, with the joint effects of spin and tide, the structure of a star changes from spherically symmetric to nonspherically symmetric. Then, the stellar structure equations become three dimensional. Theory distinguishes two components in the tide, namely equilibrium tide (Zahn 1966) and the dynamical tide (Zahn 1975). Then, the dissipation mechanisms acting on those tides, namely the viscous friction for the equilibrium tide and the radiative damping for the dynamical tide, have been identified (Zahn 1966, 1975, 1977). The distortion throughout the outer regions of the two components is not small in shortperiod binary systems. The higherorder terms in the external gravitational field should not be ignored (Jackson 1970).
It is a very complex process to determine the equilibrium structure of the two components. Therefore, approximate methods have been widely adopted for studying these effects. In 1933, the theory of distorted polytropes was introduced by Chandrasekhar. Kopal (1972, 1974) developed the concept of Roche equipotential and of Roche coordinates to analyse the problem of rotationally and tidally distorted stars in a binary system. Bur a (1989a, 1988) took advantage of the highorder perturbing potential to describe rotational and tidal deformations to discuss the figures and dynamic parameters of synchronously orbiting satellites in the solar system. The equilibrium structure of the two components were treated as two nonsymmetric rotational ellipsoids with two different semimajor axes a_{1} and a_{2} ( a_{1}>a_{2}) by Huang (2004b). It is very important that Kippenhahn & Thomas (1970) introduced a method of simplifying the twodimensional model with conservative rotation and allowed the structure equations for a onedimensional star to incorporate the hydrostatic effect of rotation. This method has been adopted by Endal & Sofia (1976) and Meynet & Maeder (1997), who applied it to the case of shellular rotation law (Zahn 1992). In this case, the rotation rate takes the simplified form of . It was demonstrated that the shape of an isobar in the case of the shellular rotation law is identical to one of the equipotentials in the conservative case of Meynet & Maeder (1997).
At the semidetached stage, both mass transfer between the components and luminosity change of a secondary exist due to the release of accretion energy which is correlative with the external potential of the two components. When the joint effect of rotation and tide are considered, the potential of the two components are different from those in nonrotational cases. Therefore, the luminosity due to the release of accretion energy, as well as irradiation energy, can significantly alter the structure and evolution of the secondary. In a rotating star, meridional circulation and shear turbulence exist, both of which can drive the transport of chemical elements. This effect is stronger and has already been studied by many scholars (Endal & Sofia 1978; Pinsonneaul et al. 1989; Chaboyer & Zahn 1992; Zahn 1992; Meynet & Maeder 1997; Maeder 2000; Meader & Zahn 1998; Maeder & Meynet 2000; Denissenkov et al. 1999; Talon et al. 1997; Decressin et al. 2009). In this paper, the amplitude expression for the radial component of the meridional circulation velocity U(r) considers the effect of tidal force, which may be important in a massive close binary system.
This paper is divided into four main sections. In Sect. 2, the structure equations of rotating binary stars are presented. Material diffusion equations and boundary conditions are provided. Then, the accretion luminosity, including gravitational energy, heat energy, and radiation energy, is deduced. In Sect. 3, the results of numerical calculation are described and discussed in detail. In Sect. 4, conclusions are drawn.
2 Model for rotating binary stars
2.1 Potential of rotating binary stars
It is well known that the rotation of a component is synchronous
with the orbital motion of a system thanks to a strong tidal effect.
Such synchronous rotation also exists inside the component (Giuricin
et al. 1984; Van Hamme & Wilson 1990); therefore,
conventional theories usually assume that two components rotate
synchronously and revolve in circular orbits (Kippenhahn & Weigert
1967; De Loore 1980; Huang & Taam
1990; Vanbeveren 1991; De Greve 1993). A coordinate system rotating with the orbital angular
velocity of the stars is introduced. The mass centre of the primary
is regarded as the origin, and it is presumed that the zaxis is
perpendicular to the orbital plane, and the positive xaxis
penetrates the mass centre of the secondary. The gravitational
potential at any point
of the surface of the
primary can be approximately expressed as
(1) 
where V is the gravitational potential and given by Bur e (1989a, 1988),
(2) 
Here, is the tidal potential (Bur e 1989a)
(3) 
where it is assumed that the mean equatorial radius equals that of the equivalent sphere in the above equation for the convenience of calculation. Both M_{1} and M_{2} are the mass of the primary and the secondary, respectively, and r_{p} represents each equivalent radius inside the star, and are the associated Legendre function ( , ), D is the distance between the two components, and is the orbital angular velocity of the system. It can be represented by
(4) 
where J_{2}^{(0)} and J_{2}^{(2)} are dimensionless stokes parameters. If M_{1} can generally be negligible compared to M_{2}, the stokes parameters can be expressed as (Bur e 1989a, 1988)
(5) 
(6) 
where is the secular Love number, which is expressed as a measure of the bodyyieldtocentrifugal deformation, and is an analogous parameter that is introduced to describe the secular tidal deformations. The response of the body to its centrifugal acceleration and to the tidal perturbing potential is different in the usual case. Therefore, the bodyyieldto centrifugal deformation is not equal to the bodyyieldtotidal deformation. If the subject investigated is regarded as an ideal elastic body, the bodyyieldto centrifugal deformation is equal to the bodyyieldtotidal deformation, . In the ideal static equilibrium, (Bur e 1989a). We assume the ideal static equilibrium in this paper. q is the mass ratio of the secondary to the primary ( ). With Eqs. (2) and (3) being combined with Eq. (1), the potential of the primary can be obtained as
=  (7)  
The potential of the secondary is deduced by substituting M_{2}for M_{1} and for q. The isobar defined by the equation is assumed to be a triaxial ellipsoid with three semimajor axes: a, b, and c. The shortest axis defined by cis identical to its rotational axis and perpendicular to its orbital plane. The longest axis defined by a is identical with its xaxis.
2.2 Considering stellar structure equations with spin and tidal effects
The spin of the two components is rigid rotation, and it belongs to
conservative rotation. The definition of equivalent sphere was
adopted in a practical calculation. Therefore, the triaxial
ellipsoid model is simplified to a onedimensional model. The
structure equations are presented as
(8) 
(9) 
(10) 
where is the energy source per unit mass caused by mass overflow and irradiation. Because accretion luminosity is caused by energy sources in the gainer's outermost layer, there exists
(11) 
where is the photosphere mass of the secondary. The surface temperature of the secondary may be approximated by the formula, , where L_{2} is the luminosity coming to the photosphere from the stellar interior, and is the StefanBoltzmann constant:
(12) 
(13) 
(14) 
(15) 
where and are the mean values of effective gravity and its opposites over the isobar surface, and is the radiative temperature gradient. The factors f_{P} and f_{T} depend on the shape of the isobars.
2.3 Calculation of quantities f_{P} and f_{T}
2.3.1 Shape and gravitational acceleration of triaxial ellipsoid
To obtain the factors f_{P} and f_{T}, the mean values
and
over the isobar surface have to be calculated.
Therefore, the shape of isobars must be given first. The functions
for the semimajor axes a, b, and c to the radius of the
equivalent sphere r_{P} can be obtained from Eq. (7) as
(16) 
(17) 
(18) 
The left hand side of Eq. (17) corresponds to and , while the one of Eq. (18) corresponds to and . The three semimajor axes a, b, and c of a triaxial ellipsoid can be obtained numerically by solving (16), (17), and (18). From (7), the quantities g_{r}, , and at the surface of the two components take the forms of
(19) 
=  
(20) 
=  
=  (21) 
However, the total potential in the stellar interior (to firstorder approximation) can be composed by four parts (Kopal 1959, 1960, 1974; Endal & Sofia 1976; Landin 2009): , the spherical symmetric part of the gravitational potential; , the cylindrically symmetric potential due to rotation; the nonsymmetric potential due to tidal force, and , the nonsymmetric part of the gravitational potential due to the distortion of the component considering the rotational and tidal effects. Therefore, the total potential at is
=  
=  
(22) 
The quantity can be evaluated by numerically integrating the Radau's equation (cf. Kopal 1959)
(23) 
for j=2,3,4, and boundary condition . The quantity r_{0} is the mean radius of the corresponding isobar. The local effective gravity is given by differentiation of the total potential and is written as
(24) 
The integral in above equations and their derivatives must be evaluated numerically. The mean values of and over the surfaces of the triaxial ellipsoids can be obtained as
(25) 
(26) 
According to Eqs. (13) and (14), the values of f_{P} and f_{T} can be obtained when the mean values and are known. r_{1} and r_{2} are the distances between the centre of the components and the surfaces of two triaxial ellipsoids. They are
(27) 
The surface area S_{p} of the isobar can be expressed as
(28) 
2.4 Element diffusion process
The effect of meridian circulation can drive the transport of
chemical elements and angular momentum in rotating stars. For the
components in solidbody rotation, no differential rotation exists
that can cause shear turbulence. According to Endal & Sofia (1978)
and Pinsonneault (1989), the transport of chemical composition is
treated as a diffusion process. The equation takes the form of
(Chaboyer & Zahn 1992)
(29) 
where is a source term from nuclear reactions, and is the relative abundance of nuclide. The diffusion coefficient given by Heger et al. (2000a) can be expressed as
(30) 
where and H_{v,ES} denote the extent of the instability and the velocity scale height, respectively. The expression for the amplitude of the radial component of the meridional circulation velocity U(r) (derived from Kippenhahn & Weight 1990) has been modified to take the effects of radiation pressure and tidal force intoaccount, which are important in a massive close binary system. It is noticed that
U(r)  =  (31)  
where the term is the local ratio of centrifugal force and tidal force to gravity, is the ratio of the specific heats , L_{r} represents the luminosity at radius r, M_{r} is the mass enclosed within a sphere of radius r, is the actual gradient and is adiabatic temperature gradient, gives the mean energy production rate, and is local generation rate of nuclearenergy.
There is no source or sink at the inner and the outer boundaries of
the two components. Therefore, the boundary conditions are used as
(32) 
where the subscript i denotes different layers inside stars. The initial abundance equals the one at the zeroage main sequence. Therefore, the initial condition is
(33) 
Table 1: Parameters at different evolutionary points a, b, c, d, e, and f in sequences of cases 1 and 2.
2.5 Luminosity accretion
In the case where the joint effect of rotation and tide is ignored, the two components are spherically symmetric. The star fills its Roche lobe and begins to transfer matter to the companion. However, in the case with the effects of rotation and tide being considered, the components are triaxial ellipsoids. The condition for the mass overflow through Roche lobe flow should be revised as (Huang 2004b). It is assumed that the transferred mass is distributed within a thin shell at the surface of the primary before the transfer, and within a thin shell at the surface of the secondary after the transfer. Three forms of energy (including potential energy, heat energy, and radiative energy) are transferred to the secondary. The mass transfer rate is . Two different cases are considered:
 a)
 If the joint effect of rotation and tide is ignored, the
accretion luminosity can be expressed directly in terms of the Roche
lobe potential at the inner Lagrangian point,
,
and at
the surface of the secondary
(Han & Webbink 1999):
= (34)
where X_{L1} is the distance between the primary and L_{1}, and R_{2} is the radius of the secondary.  b)
 If the joint effect of rotation and tide is considered, the
equilibrium structure of the two components will be treated as
triaxial ellipsoids. The release of potential energy because of the
accretion of a mass rate
to the secondary is given by
= (35)
where is the potential of the secondary. Similarly, as the two components have different temperatures, the transmitted thermal energy will be
(36)
where and represent the effective temperature of the primary and the secondary, respectively, and and are the mean molecular weights of the primary and the secondary, respectively. refers to proton mass. Because of the irradiation, energy accumulated by the primary and the secondary can be given by (Huang & Taam 1990)
(37)
where R_{1} and R_{2} are the radii of the primary and the secondary, and L_{1} and L_{2} are the luminosities of the primary and the secondary, respectively. The total accretion luminosity is
(38)
Because a part of the total energy may be dissipated dynamically, is assumed to range from 0.1 to 0.5 (Huang 1993). A value is adopted.
3 Results of numerical calculation
The structure and evolution of binary system was traced with the modified version of a stellar structure program, which was developed by Kippenhahn et al. (1967) and has been updated to include mass and energy transfer processes. The calculation method is based on the technique of Kippenhahn & Thomas (1970) and takes advantage of the concept of isobar (Zahn 1992; Meynet & Maeder 1997). Both components of the binary are calculated simultaneously. The initial mass of the system components is set at 9 and 6 . The initial chemical composition X equals X=0.70, and Z=0.02 is adopted for the two components. Similarly, the initial orbital separation between the two components for all sequences is defined as 20.771 , so mass transfer via Roche lobe occurs in case A (at the central hydrogenburning phase of the primary). Two evolutionary sequences corresponding to the evolution with the joint effect of rotation and tide being considered or ignored are calculated. The sequence denoted by case 1 represents the evolution without the effects of rotation and tide being considered, while the sequence denoted by case 2 represents the evolution with the effects of rotation and tide being considered. The calculation of Roche lobe is taken from the study by Huang & Taam (1990). The nonconservative evolution in the two cases was considered. Because the local flux at colatitude is proportional to the effective gravity according to Von Zeipel theorem (Maeder 1998), the massloss rate due to the stellar winds intensified by tidal, rotational, and irradiative effects is obtained according to Huang & Taam (1990; cf. Table 1). The angular velocity of the system and the orbital separation between the two components change due to a number of factors: changes in physical processes as the binary system evolves, including the loss of mass and angular momentum via stellar winds, mass transfer via Roche lobe overflow, exchange of angular momentum between component rotation and the orbital motion of the system caused by tidal effect, and changes in moments of inertia of the components. The changes in the angular velocity of the system and the orbital separation between the two components can be calculated according to Huang & Taam (1990), and the results are listed in Table 1. Other parameters are treated in the same way for two sequences.
Figure 1: Surface rotating velocity distribution of primary varying with time. Four panels a), b), c), and d) correspond to periods: 2.776, 2.760, 2.746, and 2.628 days, and corresponding evolutive time is 0, , , yrs, respectively. 

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The evolution of the binary system proceeded as follows (cf. Table 1). Evolutionary time, orbital period, mass of two stars, luminosities and effective temperature of two stars, central and surface helium mass fraction of the primary, and mean equatorial rotational velocities of two stars are listed in Table 1. Points a, b, c, d, e, and f denote the zeroage main sequence, the beginning of the mass transfer stage, the beginning of Hshell burning, the end of central hydrogenburning, the beginning of the central heliumburning stage, and the end of calculation, respectively. At the beginning of mass exchange, the luminosity and effective temperature of the primary component decrease rapidly. The secondary accretes for case 1 and for case 2 during the mass transfer in case A. Because of this mass gain, the luminosity and the temperature of the secondary go up. When the mass is transferred from the more massive star to the less massive one, the separation between the centres of the two components as well as the orbital period of the system decrease. Some orbital angular momentum is transformed into the spin angular momentum of both components, and this process is crucial to model the spinup of the accretion star. With mass overflow, the mass of the primary will be less than that of the secondary. When the mass is transferred from the less massive star to the more massive one, the separation between the centres of the two components as well as the orbital period of the system increases. Some spin angular momenta in both of the components are transformed into orbital angular momentum. This physical process results in a longer epilogue after mass transfer.
Figure 2: Variation of relative gravitational accelerations at the surface of primary under coordinate and as mass overflow begins. The quantities g_{r}, , and are the three components of the gravitational acceleration . Quantity g equals the gravitational acceleration of the corresponding equivalent sphere ( ). 

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The equilibrium configuration deviates from spherical symmetry
because of the centrifugal forces and tidal forces. And the deviated
region mainly lies in the outer layer of a star. In fact, the
distorted stellar surface forms the shape of a triaxial ellipsoid. A
distorted isobar surface can be expressed as
(39) 
which corresponds to the form of the disturbing potential (Zahn 1992). The coefficients f(r) and g(r) can be defined as and . The quantity is the mean density of a star with the mass of M_{1}. It was noticed that at the central hydrogenburning phase, two parameters C_{1} and C_{2} in Eq. (39) gain the values of and , respectively. This formula indicates that the shapes of the two components vary with the potentials of the centrifugal force and the tidal force. The radial deformation is inversely proportional to the mean density of the component. In order to describe the distortion, the distribution of the surface rotating velocities of the primary is illustrated in Fig. 1. The four panels (a), (b), (c), and (d) correspond to the evolutive time of 0, , , years and corresponding periods of 2.776, 2.760, 2.746, and 2.628 days, respectively. The rotational velocity rates for the peaks of the semimajor axes b and a are and 0.8664 in four panels, respectively. The results show that the surface deformation is intensified with the evolution and volumeexpansion of the primary. The distortion throughout the outer region of the primary is considerable. The detailed theoretical models that focus on investigation of the outer regions have somewhat deviated from the Roche model. The highorder perturbed potential is required for studying the structure and evolution of shortperiod binary systems. Matthews & Mathieu (1992) examined 62 spectroscopic binaries with Atype primaries and orbital periods less than 100 days. They concluded that all systems with orbital periods less than or equal to three days have circular orbits or nearly circular orbits. Zahn (1977) and Rieutord & Zahn (1997) have shown how binary synchronization and circularization result from tidal dissipation. Based on smoothed particle hydrodynamics (SPH) simulation, Renvoiz et al. (2002) have quantified the geometrical distortion effect due to the tidal and rotational forces acted on the polytropic secondaries of semidetached binaries. They suggest that the tidal and rotational distortion on the secondary may not be negligible, for it may reach observable levels of on the radius in specific cases of polytropic index and mass ratio. Georgy et al. (2008) display that various effects of the rotation on the surface of a 20 star at a metallicity of 10^{5} and at of the critical rotation velocity. They point out that the star becomes oblate with an equatorialtopolar radius ratio . These results agree closely with ours.
The variation relative gravitational accelerations, the tidal force, and the ratio of on the surface of the primary under the coordinate and at the beginning of mass overflow are shown in Fig. 2. The quantities g_{r}, , and are the three components of gravitational acceleration. The six panels (a), (b), (c), (d), (e), and (f) represent the distribution of g_{r}/g, , , , , and , respectively. The quantity g equals the gravitational acceleration of the corresponding equivalent sphere ( ). When the joint effect of rotation and tide is considered, the gravitational accelerations are different from those in the conventional model. Gravitational acceleration generally has three components.
Figure 3: Time variation of relative accretion luminosity at semidetached stage. Panel a) represents case 1 and panel b) represents case 2. The solid, dotted, dashed and dotteddashed curves correspond to the relative accretion luminosity with respect to total, thermal, potential and irradiative energies, respectively. 

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According to the Von Zeipel theorem, the mass loss due to stellar winds should be proportional to local effective gravity. Polar ejection is intensified by the tidal effect. The higher gravity at the peak of the axis b makes it hotter. The ejection of an equatorial ring may be favoured by both the opacity effect and the higher temperature at the peak of the semiaxis b. This effect is called the effect in this paper. It is predicted that the effect is as important as the effect suggested by Maeder (1998) and Maeder & Desjacques (2001). The shapes of planetary nebulae that deviate from spherical symmetry (axisymmetrical one in particular) are often ascribed to rotation or tidal interaction (Soker 1997). Frankowski & Tylenda (2001) suggest that a masslosing star can be noticeably distorted by tidal forces, thus the wind will exhibit an intrinsic directivity and may be globally intensified. Interestingly enough, the group of the B[e] stars shows a twocomponent stellar wind with a hot, highly ionized, fast wind at the poles and a slow, dense, disklike wind at the equator (Zickgraf 1989). Maeder & Desjacques (2001) have noticed that the polar lobes and skirt in Carinae and other LBV stars may naturally result from the and effects. Langer et al. (1999) have shown that giant LBV outbursts depend on the initial rotation rate. Tout and Eggleton (1988) proposed a formula according to which the tidal torque would enhance the massloss rate by a factor of , where B is a parameter free to be adjusted (ranging from to 10^{4}). Mass loss and associated loss of angular momentum are anisotropic in rotating binary stars. The theories for describing the mass loss and angular momentum loss from stellar winds should be altered partly in future work.
Figure 4: Panel a): variation in total Hburning generation energy rate in two cases. Panel b): time variation in total Hburning generation energy rate in case 2 after main sequence. The solid curve represents case 2 and the dashed curve represents case 1. 

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Figure 5: Timedependent variation in luminosity and equivalent radius of primary in two cases. The solid and dotted curves have the same meaning as in Fig. 4. 

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The total Hburning energygeneration rates of the primary in the two cases are shown in Fig. 4. Panel (b) shows the Hburning energygeneration rate in case 2 after the main sequence. From the difference between curves in panel (a), it is noticed that the effect of rotation causes the total Hburning energygeneration rate lower. As a result, the evolutive time in the mainsequence stage gets longer (cf. Table 1). Moreover, the larger fuel supply and lower initial luminosity of the rotating stars help to prolong the time which they spend on the main sequence (Heger & Langer 2000b). The lifetime extension in rotating binary star at the mainsequence stage can also be illustrated according to Suchkov (2001). Their results show that the agevelocity relation (AVR) between F stars in the binary system is different from the one between ``truly single'' F stars. The discrepancy between the two AVRs indicates that the putative binaries are, on average, older than similar normal single F stars at the same effective temperature and luminosity. It is speculated that this peculiarity comes from the impact of the interaction of components in a tight pair on stellar evolution, which results in the prolonged mainsequence lifetime of the primary F star. Moreover, no central heliumburning stage exists for case 2 (cf. Table 1). From panel (b), it can be seen that the energygeneration rate of the primary vibrates at the Hshell burning stage in case 2. These facts suggest that the burning of Hshell is unstable in case 2. The reason lies in the centrifugal force reducing the effective gravity at the stellar envelope. The luminosity and surface temperature there decrease (Kippenhahn 1977; Langer 1998; Meynet & Maeder 1997). Thus, the shell source becomes cooler, thinner, and more degenerated as the He core mass increases. As the hydrogen shell becomes instable, the thickness and surface temperature are 0.203 and K, respectively. This physical condition leads to thermal instability (Yoon et al. 2004), and the Hshell source experiences slight oscillation. It is well known that the energygeneration rate is proportional to temperature and density ( ); therefore, the curve of the Hshell energygeneration rate fluctuates.
The timedependent variation in the luminosity and the equivalent radius of the primary in the two cases are illustrated in Fig. 5. Because the rotating star has a lower energygeneration rate, the luminosity of the primary is lower, which is the consequence of decreased central temperature in rotating models due to decreased effective gravity (Meynet & Maeder 1997). Then, the primary expands slowly in case 2. It is observed that case 1 reaches point b at yr, while case 2 reaches point b at yr. The initiation time of mass transfer for case 2 is advanced by about . Similarly, numerical calculation by Petrovic et al. (2005b) shows the radius of the rotating primary increases faster than that of the nonrotating primary due to the influence of centrifugal forces. Their results also show that mass transfer of case A starts earlier in rotating binary system, which is consistent with ours. If the rotating star is still treated as a spherical star, the initiation time of mass overflow should be later than that in the nonrotational case. Actually, because of the distortion by rotation and tide, the time for mass overflow may be extended. Therefore, it is very important to investigate distortion in close binary systems.
The timedependent variation in the helium compositions at the surface of the primary is illustrated in Fig. 6. The Hshell burning begins at yr in case 1 while at yr in case 2 (cf. Table 1). Therefore, the initiation time of Hshell burning is advanced by yr. Moreover, the helium composition at the surface of the primary is 0.280051 at point c, suggesting that the diffusion process progresses slowly in a rotating star. Cantiello et al. (2007) also indicate that rotationally induced mixing before the onset of mass transfer is negligible, in contrast to typical O stars evolving separately; hence, the alteration of surface compositions depends on both initial mass and rotation rates. The sample of the OBtype binaries with orbital periods ranging from one to five days by Hilditch et al. (2005) shows enhanced N abundance up to 0.4 dex. Langer et al. (2008) have discovered that for the same binary system, but with the initial period of six days instead of three days, its mass gainer is accelerated to a rotational velocity of nearly 500 km s^{1}, which produces an extra nitrogen enrichment from more than a factor two to about 1 dex in total. Because there is no central heliumburning phase for case 2, the diffusion process can be neglected in the interior region of the primary after the main sequences.
Figure 6: Timedependent variation in surface helium in two cases. The solid and dotted curves have the same meaning as in Fig. 4. 

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4 Conclusions
The main achievements of this study may be summarised as follows.
 (a)
 The distortion throughout the outer layer of the primary is
considerable. The detailed theoretical models that investigate the
outer regions of the two components have deviated somewhat from the
lowest approximation of the Roche model. The highorder perturbing
potential is required especially in the investigation of the
evolution of shortperiod binary system.
 (b)
 The equilibrium structures of distorted stars are actually
triaxial ellipsoids. A formula describing rotationally and tidally
distorted stars is presented. The shape of the ellipsoid is related
to the mean density of the component and the potentials of
centrifugal and tidal force.
 (c)
 The radial components of the centrifugal force and the tidal
force cause the variation in gravitation. The tangent components of
the centrifugal force and the tidal force cannot be equalized and,
instead, they change the shapes of the components from perfect
spheres to triaxial ellipsoids. Mass loss and associated angular
momentum loss are anisotropic in rotating binary stars. Ejection is
intensified by tidal effect. The ejection of an equatorial ring may
be favoured by both the opacity effect and the higher temperature at
the peak of semiaxis b. This effect is called the
effect in this paper.
 (d)
 The rotating star has an unstable Hburning shell after the main sequence. The components expand slowly due to their lower luminosity. If the components are still treated as spherical stars, some important physical processes can be ignored.
Acknowledgements
We are grateful to Professor Norbert Langer and Dr. Stéphane Mathis for their valuable suggestions and insightful remarks, which have improved this paper greatly. Also we thank Professor Norbert Langer for his kind help in improving our English.
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All Tables
Table 1: Parameters at different evolutionary points a, b, c, d, e, and f in sequences of cases 1 and 2.
All Figures
Figure 1: Surface rotating velocity distribution of primary varying with time. Four panels a), b), c), and d) correspond to periods: 2.776, 2.760, 2.746, and 2.628 days, and corresponding evolutive time is 0, , , yrs, respectively. 

Open with DEXTER  
In the text 
Figure 2: Variation of relative gravitational accelerations at the surface of primary under coordinate and as mass overflow begins. The quantities g_{r}, , and are the three components of the gravitational acceleration . Quantity g equals the gravitational acceleration of the corresponding equivalent sphere ( ). 

Open with DEXTER  
In the text 
Figure 3: Time variation of relative accretion luminosity at semidetached stage. Panel a) represents case 1 and panel b) represents case 2. The solid, dotted, dashed and dotteddashed curves correspond to the relative accretion luminosity with respect to total, thermal, potential and irradiative energies, respectively. 

Open with DEXTER  
In the text 
Figure 4: Panel a): variation in total Hburning generation energy rate in two cases. Panel b): time variation in total Hburning generation energy rate in case 2 after main sequence. The solid curve represents case 2 and the dashed curve represents case 1. 

Open with DEXTER  
In the text 
Figure 5: Timedependent variation in luminosity and equivalent radius of primary in two cases. The solid and dotted curves have the same meaning as in Fig. 4. 

Open with DEXTER  
In the text 
Figure 6: Timedependent variation in surface helium in two cases. The solid and dotted curves have the same meaning as in Fig. 4. 

Open with DEXTER  
In the text 
Copyright ESO 2009
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