Issue 
A&A
Volume 600, April 2017



Article Number  A42  
Number of page(s)  9  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201629784  
Published online  28 March 2017 
The structure of criticallyrotating accreting stars
^{1} College of Physics, Guizhou University, Guiyang, 550025 Guizhou Province, PR China
email: hfsong@gzu.edu.cn; songhanfeng@163.com
^{2} Geneva Observatory, University of Geneva, 1290 Sauverny, Switzerland
^{3} School of Physics and Electronic Engineering, Kaili University, Kaili, 556011 Guizhou Province, PR China
^{4} College of Science, Jimei University, Xiamen city, 361021 Fujian Province, PR China
^{5} Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 650011 Kunming, PR China
Received: 25 September 2016
Accepted: 17 December 2016
Context. The structure characteristics of the criticallyrotating accretor in binaries are investigated in this paper, on the basis of the potential function including rotational and tidal distortions.
Aims. Our aim is to investigate the structure of the accretor when the accreting star reaches the critical velocity.
Methods. In this paper, we have implemented the prescription described by Kippenhahn & Thomas (1970, Proc. IAU Colloq., 4, 20) and Landin et al. (2009, A&A, 494, 209).
Results. The traditional model merely included the hydrodynamical effect of rotation. When comparing this model with ours, we find that it is very necessary for the rapidly rotating accreting star to include the gravitational potentials from tides Ψ_{tide}, and the distortions of the star resulting from rotation Ψ_{dis,rot}. Furthermore, we find that the mean effective gravitational acceleration can be decreased in the distort model, and the star shifts towards low temperature and low luminosity. Rotation and tides can extend the convection zone below the surface, and reduce the convective core in the center of stars due to the SolbergHoiland criterion. Rotational distortions derived from Ψ_{dis,rot} can intensify the critical velocity whereas the tide force derived from Ψ_{tide} tends to reduce the critical velocity. Rapid rotation induced by mass transfer also causes the central temperature to decrease, and triggers efficient mixing which can significantly modify the Hprofile.
Key words: binaries : close / stars: rotation / stars: evolution / stars: massive
© ESO, 2017
1. Introduction
It is well known that rotation is regarded as a very important factor that needs to be considered to investigate the evolution of the massive star (Heger et al. 2000; Meynet & Maeder 2000; Maeder & Meynet 2012; Langer 2012). The centrifugal force changes the stellar shape from sphere to spheroid. The equatorial radius can be about 1.5 times larger than the polar radius according to Roche approximation when the stars approach the breakup rotation (Georgy et al. 2011). As a result of the deformation of the star, the radiative energy flux and the effective temperature vary with the latitude (von Zeipel 1924). Rotation can drive various mixing processes in the stellar interior and rotational mixing is probably the most important effect in massive stars (Meynet et al. 2006, 2010; Mathis & Zahn 2004; Mathis et al. 2004; Zahn 1992). Meridional circulations and shear turbulence dominate rotationally induced mixing in massive stars, and transfer both angular momentum and chemical species (Maeder & Zahn 1998; Talon & Zahn 1997; Maeder 2003). Fresh nitrogen thus appears at the surface of the star and becomes continuously more enriched as function of time during corehydrogen burning (Maeder & Meynet 2000b).
In a rotating binary system, the joint effect of rotation and tides modifies the shapes of two components from the perfect sphere to triaxial ellipsoids (Landin et al. 2009; Song et al. 2009, 2011; Huang 2004). Eggleton & Kiseleva (1998) have presented the equations governing the quadrupole tensor of a star distorted both by rotation and by the presence of a companion in a possibly eccentric orbit. During the evolution of massive binaries, mass and angular momentum are transferred from the donor to the recipient during Roche lobe overflow (RLOF). Rapid mass transfer can spin up an accreting star to critical rotation (Packet 1981; Langer et al. 2003; Langer 2012). In the present work, we aim to investigate for the first time how the deformation of stars can be induced by both rotation and tide at critical rotation. In particular, we aim to discover how the critical rotation modifies the structure and evolution of the accreting star.
The paper is organized as follows: the potential function which included the combined effects of rotational and tidal distortions, and the structure equations of rotating binaries are given in Sect. 2. Various torques that affect the accreting star are introduced in Sect. 3. In Sect. 4, the result of numerical calculations are described and discussed in detail. And finally in Sect. 5, conclusions are made.
2. The potential function and the rotating stellar structure equations
2.1. The potential function including the rotational and tidal distortions
We consider a close binary consisting of two stars with masses M_{1} and M_{2} in a circular orbit with orbital period P_{orb}, let a be the orbital separation. A coordinate system ℛ_{E} rotating with the spin angular velocity of the secondary star M_{2} is introduced (cf. Fig. 1). The mass center of the secondary star M_{2} is regarded as the origin. It is presumed that the zaxis is perpendicular to the orbital plane, and the positive xaxis penetrates the mass center of the primary star M_{1}. The two stars rotate with angular velocity Ω_{1} and Ω_{2} around an axis perpendicular to the orbital plane.
If the coordinates of the point P are the radius r, the polar angular θ, and the azimuthal angle φ, then the total potential in the stellar interior (to firstorder approximation) consists of five parts (Kopal 1960, 1974; Endal & Sofia 1976; Landin et al. 2009): the spherical symmetric part of the gravitational potential, ψ_{grav}, the cylindrically symmetric potential due to rotation, ψ_{rot}, the potential due to tides ψ_{tide}, and the nonsymmetric part of the gravitational potential due to the distortion of the star resulting from rotation, ψ_{dis,rot} and tides ψ_{dis,tide}. This total potential is written as (1)where r and Ω_{2} are respectively the radius and the spin angular velocity of the rotating star 2, λ = cosϕsinθ, and P_{j}(λ) are the jthorder Legendre polynomials. ψ_{r} and ψ_{t} are, respectively, the potential functions which are induced by rotation (ψ_{r} = ψ_{rot} + ψ_{dis,rot}), and tide (ψ_{t} = ψ_{tide} + ψ_{dis,tide}). In Eq. (1) above, the quantities η_{j} are of particular importance for our research. They can be derived from the Radau′s differential equation with a 4thorder RungeKutta method (Press et al. 1992) (2)where η_{j}(0) = j−2(j = 2,3,4), ρ(r) is the local density at a distance r from the center, and is the mean density within inner sphere of radius r.
Fig. 1
Spherical coordinates system associated with the equatorial reference frame ℛ_{E}:O,X,Y, Z of an extended body M_{2}; We have r_{p} = (r,θ,ϕ). The spin of M_{2} is perpendicular to the orbital plane. The dashed line illustrates the orbit of M_{1}. 
2.2. The correction factors in the rotating stellar model
Adopting the Kippenhahn and Thomas method, we can define the mean quantities as ⟨ g_{e} ⟩ and . These are the values of effective gravity and its opposite values over the isobar surface (Kippenhahn & Thomas 1970), where , the mass inside the equipotential surfaces is M_{Ψ} and luminosity is L_{Ψ}. The equipotential surface has the area of S_{Ψ}, and r_{Ψ} is the radius of the topological equivalent sphere with the same volume V_{Ψ}. We note that the local effective gravity g_{e} has three components which have been derived from Eqs. (A.7)−(A.9). The quantity dσ can be derived from Eq. (A.26).
The correction factors in the rotating stellar model are written as In the case of isolated and nonrotating stars, we have used f_{p} = f_{t} = 1. Using the lowest approximation, the correction factors are treated as and f_{t} = 1 in the traditional rotating model which merely included the hydrodynamical effect of rotation (Kähler 1986).
2.3. The temperature gradients and SolbergHoiland criterion for stability
The effect of rotation and tides on the temperature gradient has been included in our computations. They are given by where ∇_{rad} and ∇_{ad} are the radiative temperature gradient and the adiabatic gradient, respectively, the quantity is the ratio of gas pressure to the total pressure. A fluid element displaced in a rotating star is also subject to the restoring effect of angular momentum conservation (Kippenhahn & Weigert 1990). This leads to the SolbergHoiland criterion for stability, in which (9)with (10)for constant angular velocity (Maeder et al. 2008). Here g_{grav} is the self gravity and H_{p} designates as the pressure scale height. In onedimensional stellar model, the quantity ⟨ sinθ ⟩ has a value of ~0.75.
2.4. Expressions of the critical velocity
The high distortion which is induced by rapid rotation and tides has an important impact on the critical velocity. When a rotating star of mass M_{2}, and spin angular velocity Ω_{2} is in the binary system, the total gravity is the sum of the gravitational, centrifugal, and tidal accelerations: (11)The critical angular velocity Ω_{c,1} corresponds to the angular velocity at equator of the star such that the centrifugal force balances the gravity exactly. The classical critical angular velocity or the Ω − limit (to distinguish it from the ΩΓ limit as defined in Maeder & Meynet 2000) can be given by the equation, (12)where the quantity R_{eb} denotes the equatorial radius at the breakup velocity, the critical velocity is v_{crit,1} = R_{eb}Ω_{crit}. The quantity Ω_{crit} is the critical angular velocity.
2.5. The transport equation for angular momentum
The transport of the angular momentum can be treated as a diffusion process (Endal & Sofia 1978; Pinsonneault et al. 1989). The equation can be written as (Heger et al. 2000) (13)where ν is the turbulent viscosity which includes various instabilities (ν = D_{conv} + D_{sem} + D_{DSI} + D_{SHI} + D_{SSI} + D_{ES} + D_{GSF}), and i is the specific angular momentum of a shell at mass coordinate m. The final term in the above equation (an advection term) accounts for contraction or expansion of the layers at constant mass coordinate.
2.6. The transport of the chemical elements
The diffusion equation of the chemical elements can be given by (Heger et al. 2000) (14)where is a source term from nuclear reactions, and y_{α} is the relative abundance of αth nuclide. Meridional circulations are the main physical process which can mix the chemical elements in a solidbody rotating star. The diffusion coefficient due to meridional circulations can be used as D_{dif} ≡ min { d_{inst},H_{v,ES} } U_{2}(r). d_{inst} and H_{v,ES} denote the extent of the instability and the velocity scale height, respectively. We currently use an approximate formula for meridional circulations by Maeder & Zahn (1998), (15)where , the quantity ϵ = E_{nuc} + E_{grav} is the total local energy emission, the quantity ϵ_{m} is L/m, C_{p} is the specific heat capacity at constant pressure. The thermodynamic ratio used by Maeder & Zahn (1998) is set to unity. This is correct for a perfect gas. We have also approximated the factor of Zahn (1992) by . The description of angular momentum transport and rotational mixing in massive stars is given in the references (Maeder & Meynet 2000b; Rieutord 2006; Decressin et al. 2009). The diffusive approach is only a first step for such complex modeling.
3. The spinup and spindown mechanisms of the accreting star
The masstransfer rate from star 1 to star 2 during RLOF is (16)where the quantity C is set to 1.0 × 10^{4}M_{⊙}/yr. For Rochelobe radius R_{L}, we use Eggleton’s (1983) analytic expression. During the evolution of massive binaries, mass and angular momentum are transferred from the donor to the recipient during RLOF. Rapid mass transfer can spin up an accreting star to critical rotation (Packet 1981; Langer 2012).
By comparing the radius of the accreting star R_{A} to the minimum distance R_{min} of the stream from the centre of the gainer, we determined whether disk accretion, when R_{A}<R_{min} or direct impact accretion, when R_{A}>R_{min}, occurs. According to Ulrich & Burger (1976), the quantity R_{min} can be calculated by R_{min} ≈ 0.0425a [q(1 + q)] ^{1/4}. The quantity q is the mass ratio of the companion to the considered star. The variation of spin angular momentum of the accreting star can be expressed as (17)where J_{2} is the spin angular momentum of the accreting star, is the torque due to mass transfer, is the wind torque, is the tide torque. The spinningup of the gainer due to RLOF is characterized by an enhancement of its rotational angular momentum , simulating the direct hit scenario which is given by de Mink et al. (2013) as (18)The change in the accreting star’s spin angular momentum due to tidal interaction is expressed as (19)where the quantity I_{2} is the moment of inertia of the star 2, ω_{orb} is the orbital angular velocity and is , s_{22} is tidal frequency and is defined as , and E_{2} is a secondorder tidal coefficient which depends on the stellar structure (Zahn 1975). It is expressed by (Yoon et al. 2010). The quantity R_{conv,2} is the convectivecore radius. Tidal interactions spin the star down when Ω_{2}> Ω_{orb} and up Ω_{2}< Ω_{orb} (Song et al. 2013, 2016). The tidal torque in the star with a convective envelop is given by (20)where the friction time is , the quantity λ_{22} is same as that in Zahn (1989). In Eq. (17), the loss of the spin angular momentum through stellar winds is given by (21)where we take ξ = 0.43, Ṁ_{w} to be the massloss rate by stellar winds and it is given by de Jager et al. (1988). We have adopted the enhanced massloss rate of Langer (1998).
Fig. 2
Panel a): variation of the force f_{dis,rot} which is derived form the potential function Ψ_{dis,rot} versus coordinates θ and ϕ at the surface of accretors in model M1. Panel b): variation of the radial component of tidal forces which is derived from the potential function Ψ_{tide} at the surface of accretor versus coordinates θ and ϕ in model M1. Panel c): variation of the gravitational acceleration  g_{eff}  at the secondary star’s surface under coordinates θ and ϕ in Roche model. The radius is treated as the constant value. Panel d): variation of the gravitational acceleration  g_{eff}  at the secondary star’s surface under coordinates θ and ϕ in model M1. The effective gravitational acceleration can be derived from Eq. (1). 
4. Results of numerical calculation
4.1. The depiction of models
We used the most recent stellar evolution code TWIN, originally written by Eggleton (1971, 1972, 1973) and Eggleton et al. (1973). The nuclear reaction rates were from Caughlan & Fowler (1988) and were updated by Pols et al. (1995) and Stancliffe et al. (2005). The procedures used have been more recently updated by Han et al. (1994). The conservative binarystar evolution has been discussed by Nelson & Eggleton (2001). The convection is treated by mixinglength theory (BohmVitense 1958) and a model for convective overshooting is included (Schroder et al. 1997). We have adopted the massloss rates of Vink et al. (2001) for massive stars. The two component stars in the binary system were calculated simultaneously. The initial mass of the system components was set at 20 M_{⊙} and 16 M_{⊙} with the metallicity of the Sun X = 0.7 and Z = 0.02. We took the mixing length parameter α = l/H_{P} to be 2.0. We note that in this study, we focus on the structure of the accreting star at the critical rotation. We have calculated two different types of model. They are as follows:

MKE: the configurations of the two components are assumed tobe spherical. In this case,, f_{t} = 1, the effect of rotation on stellar structure is only taken into account as a reduction of the effective gravity. This method is referred to as the KE method (Kähler 1986). Moreover, rotational mixing has been included in this model. The initial spin periods P_{s,1,2} for two components are taken the values of 1.5 days while the initial orbital period is 2.65 days.

M1: the effect of distortions and mixing processes have been included. The accreting star can be spun up during rapid mass transfer. The initial spin periods P_{s,1,2} for two components and the initial orbital period are the same as the ones in model MKE.
4.2. Impact of the structure of the rotating star
4.2.1. The distribution of the force derived from the quadrupolar potential Ψ_{dis,rot}
In order to investigate the structure of the critically rotating accretor, we have chosen the evolutionary time of t = 6.3323 Myr. The velocity of the secondary star reaches ~0.95v_{cri,1} due to rapid mass transfer in model M1, and the spin period is 0.729 d. The orbital period is 2.315 d and the mass of the secondary star is 16.153 M_{⊙}. The mass of the primary star is 19.481 M_{⊙}.
Panel a in Fig. 2 shows the variations of the force f_{dis,rot} () which are derived from the rotational distortion potential Ψ_{dis,rot}. The positive value indicates that the force pulls the accretor outward (away from the accretor) whereas the negative value denotes that the force pushes the accretor inward. This force arrives at 1.776 × 10^{2}cm/s^{2} at two polar points E and F (cf. Fig. 1). It goes down to −18.116 cm/s^{2} and contributes −3.02% to the radial centrifugal force at the point A. The primary minimum has a value of −22.790 cm/s^{2} at points B and D and dominates a fraction of about −4.14% for the radial centrifugal force. The secondary minimum is −19.438 cm/s^{2} at the position of point C and contributes −3.39% to the radial centrifugal force. The result implies that the force f_{dis,rot} can lead to a reduction of the self gravity () at the two poles, and a increase of the gravity at the equator.
The force f_{dis,rot} can significantly affect the critical velocity and oblateness (cf. Fig. 3) of accretors. The quadrupolar correction due to tides Ψ_{dis,tide} is smaller by a factor of ten than the one of rotational distortion Ψ_{dis,rot}. Therefore, it may be neglected because of long orbital period of 2.315 d.
Fig. 3
Time evolution of the critical velocities for three models. The critical velocity v_{1,crit} is derived from the total potential function Ψ = Ψ_{grav} + Ψ_{rot} + Ψ_{tide} + Ψ_{dis,rot} + Ψ_{dis,tide} in model M1(Solid line). The critical velocity v_{2,crit} is derived from potential function Ψ = Ψ_{grav} + Ψ_{rot} + Ψ_{tide} in model M1 (Dotdashed line). The critical velocity v_{3,crit} is derived from the potential function Ψ_{r} = Ψ_{grav} + Ψ_{rot} + Ψ_{dis,rot} in model M1 (Dashed line). 
4.2.2. The distribution of tidal forces
Panel b shows variations of the radial component of the tidal force which is derived from Ψ_{t} = Ψ_{tide} + Ψ_{dis,tide} at the surface of the secondary star under coordinate θ and ϕ in the model M1. It is shown that the radial component of tidal forces goes up from −2.555 × 10^{3} cm/s^{2} at two polar points to 5.466 × 10^{2} cm/s^{2} at the point A. The tidal force can reach 2.59787 × 10^{2} cm/s^{2} at the point C. The difference in the tidal force between points A and C comes from the term ψ_{tide} (j = 3) in Eq. (1) which counteracts the self gravity. Tidal forces greatly shorten the polar radius of accretors, causing the polar radius to become smaller than the equatorial radius. Tides and centrifugal forces pull the accretor outwards at points A and C and cause the equatorial radii to increase. The equatorial radius r_{e1} which confronts the companion star is the biggest one because the tidal force which originates form the term ψ_{tide} (j = 3) at point A can strengthen the centrifugal force. The radial component of tidal forces has a value of −2.144 × 10^{2} cm/s^{2} at the points B and D. For close binaries, the symmetry around the rotation axis is broken by tidal forces and this might produce additional possibilities for internal mixing due to thermal inequalities in the equatorial circle.
These facts show that tidal forces can produce an increase in the effective gravitational acceleration at the two poles. The opposite effect is located at points A and C. At points B and D, tidal forces counteracts with the centrifugal force, leading to a increase in self gravities. Therefore, the shortest equatorial radius is r_{e3}.
The critical oblateness (the ratio of the equatorial radius to the polar one) can be increased when the quadrupolar moment is taken into account. The reason is that if ψ_{dis,rot} is increased, a test particle at equator will experience a stronger self gravity. The star needs to attain a high spin angular velocity and the new location of test particles is situated closer to the companion star to reestablish a zeroacceleration. At critical rotation, a maximum oblateness can attain a value of according to Roche model. Zahn (2010) shows that the critical flattening can attain in single rotating star when the quadrupolar moment has been included. The quantity k_{2} is given by and is typically of the order of 10^{3}−10^{2} for mainsequence stars. In our binary model, the critical flattening can be approximately estimated by the value of at this evolutionary point. Due to tide forces, the critical flattening can exceed the value in the model of single rotating star, produced by Zahn’s model. Furthermore, we find that the critical flattening generally decreases in time due to the star’s nuclear evolution. The reason is that the value of k_{2} generally decreases in time due to the increasing of the mean molecular weight in the stellar core.
4.2.3. Comparing the effective gravities derived from Roche model and distorted model
The surface gravity  g_{eff}  for the accretor M_{2} which is derived from Roche model in Eq. (A.35) is shown in panel c. The effective gravity  g_{eff} , derived from Eq. (1) is illustrated in panel d. Comparing panel c with d, we find that the effective gravity increases ~48.6% at the two polar points and decreases by ~87.5% at point A for the distorted model. The surface effective gravity can decrease by a factor of ~49.3% at points B and point D. Therefore, the effective gravity has a smaller value at the equator and a higher value at two polar points in distorted models. The main reason for this is that the distribution of the effective gravity depends strongly on the distance between the stellar center and the surface. The deformed model treats the star as the triaxial ellipsoid with different radii, whereas the Roche model treats the stellar surface as a sphere.
4.3. The critical velocity
The time evolution of the critical velocity for three models is displayed in Fig. 3. Comparing v_{2,crit} with v_{3,crit}, we find that tidal forces reduce the critical velocity. The reason for this is that tidal forces can offset the gravity at the points confronting the companion (i.e. at point A). Compared v_{2,crit} with v_{1,crit}, we find that distortions induced by rotation and tide can enhance the critical velocity. This can be understood as the force which is derived from ψ_{dis,rot} can help the self gravity to offset the centrifugal force at the point facing towards the companion.
4.4. Two rotational factors f_{p} and f_{t}
Figure 4 shows two correction factors f_{p} and f_{t} as a function of the dimensionless radius R/R_{⊙} according to the MKE model and to the present method with model M1. We see that f_{p} declines from 1.0 to 0.652 in the MKE model whereas it declines from 1.0 to 0.363 in present method. The factor f_{p} in model M1 is same as that in the MKE model in the central region because stellar core has a high density. We suggest that the MKE model should be applied only to the inner layers where the local density is greater than the mean density.
Compared MKE model with M1, we find that the maximum correction for the factor f_{p} can attain 44.3% at the stellar surface. The result implies that the surface effective gravity ⟨ g_{e} ⟩ in model M1 has a lower value in distorted model than the one in MKE. The main reason is that great distortions can enhance the centrifugal force. The contribution of this effect to the effective gravity can reach the first order effect ~20% (cf. Fig. 4) and can not be neglected. Distortions induced by rotation and tides can contribute a significant fractional importance to the effective gravity in the outer stellar layers.
The small factor f_{p} in model M1 implies that the accreting star will have a bigger radius and that matter may be evaded easily on the surface. Moreover, the lower factor f_{p} can cause the star to shift dramatically towards low temperature and low luminosity due to gravity darkening. The dynamical timescale may be extended because of larger effective radii and cooler surface temperatures. It is more difficult for a rapidly rotating star to attain hydrostatic equilibrium.
Also, we can see that f_{t} keeps a constant value of 1.0 through the whole star in the MKE method whereas it reduces from 1.0 to 0.585 in the present method. It is illustrated that the maximum correction for the factor f_{t} reaches 41.5% at the stellar surface. The ratio of f_{t} to f_{p} in the MKE model is smaller than the one in M1 below stellar surface.
Fig. 4
Correction factors f_{p}, f_{t}, and, f_{t}/f_{p} versus the dimensionless radius R/R_{⊙} according to model MKE (blue curve) and to model M1 (pink curves). 
4.5. The expansion of convective zones
It is shown in panel a of Fig. 5 that various temperature gradients are functions of the dimensionless radius R_{2}/R_{⊙}. Without rotation, there is a thin outer convective zone which is close to the stellar surface. It extends from 6.74 R_{⊙} to 6.80R_{⊙} – only 0.86% of the radius – and contains a very small fraction of the total stellar mass (~5.0 × 10^{7}). With a ratio in model M1, convective zones of rapid rotating accretors extend according to SolbergHoiland criterion. The outer convective zones extend from 6.641 R_{⊙} to 6.937 R_{⊙} in model MKE whereas they extend from 6.590 R_{⊙} to 6.937 R_{⊙} in the distorted model M1. The outer convective layer is deeper in model M1 than the one in model MKE. The reason is that the quantity in model M1 has a high value in the outer envelope. Therefore, great deformations that are induced by rotation and tides are in favor of convective motion.
Fig. 5
Left panel: temperature gradients ∇_{r}, ∇_{ad}, and ∇_{r}−∇_{Ω} as function of the dimensionless radius R/R_{⊙} according to the MKE model (blue curves) and the M1 model (pink curves) below the stellar surface. The black line denotes the radiative temperature gradient ∇_{r} in the model without rotation. The green line denotes the temperature gradients ∇_{ad}. The red curve indicates the quantity ∇_{Ω}. Right panel: various temperature gradients located at the border of the convective core. The point where black lines cut green lines implies the border of convective core in nonrotating model according to the Schwarzschild criterion. The point where blue lines cut green lines indicates the border of convective core in rotating model according to the Schwarzschild criterion. 
Fig. 6
Left panel: variations of the energy generation by nuclear reactions as a function of the Lagrangian mass at four different evolution points at critical rotation. The recipient approaches the critical rotation at the time of ~6.33 Myr. Right panel: variation of internal distribution of hydrogen mass fraction as a function of the Lagrangian mass at four different evolution points. 
It is shown in panel b that rapidly rotating accretors have a bigger convective core of ~1.338 R_{⊙} according to the Schwarzschild criterion compared with the nonrotational model with a convective core of 1.331 R_{⊙}. If the SolbergHoiland criterion for stability is adopted, rotating accretors will have a small convective core of 1.29 R_{⊙}. Therefore, the SolbergHoiland criterion tends to reduce the central convective region and contribute to a low stellar luminosity.
4.6. The energy generation and hydrogen profiles
Variations of the energy generation by nuclear reactions as a function of the Lagrangian mass at four different evolutionary points are shown in left panel of Fig. 6. We find that the region and the intensity of the energy generation through nuclear reactions decrease with the evolution. The reason for this is that centrifugal forces compensate most of the gravity; pressure and temperature in the central parts of accretor are smaller, consequently the energy production rate is also decreasing. (cf. panel b in Fig. 6).
Internal distribution of hydrogen mass fraction of the Lagrangian mass in the model M1 is shown in right panel of Fig. 6. There is a distinct boundary between the convective core and the radiative envelope before mass transfer (at the time of t = 3.5458 Myr). At the time of ~6.320 Myr, the accreting star approaches the critical rotation due to RLOF. From then on, the gradient of internal distribution of hydrogen become smaller and smaller. The reason fo this is that the amplitude of the radial component of the meridional velocity U which scales with Ω is the main driving mechanism for mixing of chemical elements. The result implies that rotationally induced chemical mixing is more efficient and occurs faster than the buildup of chemical gradients at the cutting edge of nuclear fusion (Maeder 1987). Moreover, more massive fuel hydrogen is transported from the outer envelope to the core and these physical processes may prolong the time the accretor spends on the main sequence.
5. Conclusions and summary
The configuration of accretors is one of triaxial ellipsoids which are deformed by tides. The symmetry around the rotation axis is broken by tides and this might produce additional possibilities for internal processes to enhance mixing due to thermal inequalities in the equatorial circle.
The self gravity is balanced by the combined effect of centrifugal forces, rotational distortions, and tidal forces at equator. Therefore, the critical velocity is reduced due to tidal force. The force which is derived from the potential Ψ_{dis,rot} can help the self gravity to offset the centrifugal force at the equator. We can derive a higher critical velocity from the distorted model which includes rotational distortion Ψ_{dis,rot}.
The mean effective gravity ⟨ g_{e} ⟩ can be reduced dramatically in the distorted model due to a small factor f_{p}. The ratio of f_{t} to f_{p} is bigger in model M1 than that in the MKE model. The star appears a deeper convective zone below stellar surface. The accreting star has a bigger convective core according to the Schwarzschild criterion but it will allow a smaller convective core according to the SolbergHoiland criterion.
The critical flattening (defined as the ratio between the equatorial and polar radii) can be increased to 1.5469 and exceed the 1.5 value of Roche model as a result of considering the quadrupolar moment and tidal forces.
The gradient of hydrogen profile becomes smaller and smaller due to rapid meridional circulation and the accretor is mixed efficiently. Both the rate of energy generations and convective core decrease due to the lower temperature produced by the combined effect of rotation and tide. The lifetime of the accreting star during the main sequence can be extended due to efficient mixing and lower rate of energy generations.
Acknowledgments
We thank an anonymous referee. Also, we are very grateful to Professor Peter P. Eggleton for sharing his binary evolution code. This work was sponsored by the National Natural Science Foundation of China (Grant Nos. 11463002 and 11373020), the Open Foundation of key Laboratory for the Structure and evolution of Celestial Objects, Chinese Academy of Science (Grant Nos. OP201405, OP201404(B615015)), the Science Foundation of Jimei University (Grant No. C613030), and Science and Technology Foundation of Guizhou Province (Grant LKK[2013] No. 20).
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Appendix A
Appendix A.1 The potential including rotational and tidal distortions and the gravitational acceleration
At the position P(r,θ,ϕ), the total potential for the combined effect of rotation and tide can be expressed in Cartesian spherical coordinate systems. It is given by (Landin et al. 2009; Song et al. 2009) (A.1)The equipotential surface of the distorted star can be described by (Landin et al. 2009) (A.2)where Y_{rot} is a measure of the deviation from sphericity caused by rotation and given by . The quantity Y_{j} is a measure of the deviation from sphericity caused by the tide and given by . r_{0} is the radius of the equipotential surface at the angles (θ_{0},ϕ_{0}), defined such that (A.3)where the quantities A(r_{0}), B(r_{0}), C(r_{0}), and D(r_{0}) are defined by (A.4)The equipotential surface of the stars can be described as (A.5)By integrating Eq. (26) from 0 to r(r_{0},θ,ϕ), we obtain that the volume V_{ψ} of the topologically equivalent sphere with the radius r_{ψ}, (A.6)Differenting the potential function, we can derive three components of the effective gravitational acceleration g_{r}, g_{θ}, and, g_{ϕ}. They are where where
The local effective gravity is written as (A.24)
Appendix A.2 The expression of the surface element dσ on an equipotential of a rotating star
On the surface of the distorted star, the normal to the surface does not coincide with the direction of the vector radius. There is an angle ε between the directions of r and of −g_{eff}, (A.25)Therefore, the surface element dσ can be expressed as (A.26)where
Appendix A.3 Roche potential and the gravitational acceleration
Interactions in a closed binary system are generally treated in the framework of the Roche model (Kopal 1959). We adopt the same coordinate system as the deform model. The gravitational potential which is experienced at the position P(x,y,z) is the sum of the two point mass potential and the rotational potential (Huang & Taam 1990), namely, (A.30)where a be their mutual separation. The coordinates of the position P(x,y,z) can be expressed in Cartesian spherical coordinate systems. Finally, we obtain (A.31)where χ is the angle between a and r. Three components of the gravitational acceleration g_{r}, g_{θ}, and g_{ϕ} take the forms of The local effective gravity is written as (A.35)The potential of the primary star with respect to the center of the primary star can be deduced by substituting M_{1} for M_{2}. The corresponding effective gravity can be derived with the same method. The stellar radii of two components are treated as the constant value in the Roche model. The radius is not treated as the constant value in the distorted model M1.
All Figures
Fig. 1
Spherical coordinates system associated with the equatorial reference frame ℛ_{E}:O,X,Y, Z of an extended body M_{2}; We have r_{p} = (r,θ,ϕ). The spin of M_{2} is perpendicular to the orbital plane. The dashed line illustrates the orbit of M_{1}. 

In the text 
Fig. 2
Panel a): variation of the force f_{dis,rot} which is derived form the potential function Ψ_{dis,rot} versus coordinates θ and ϕ at the surface of accretors in model M1. Panel b): variation of the radial component of tidal forces which is derived from the potential function Ψ_{tide} at the surface of accretor versus coordinates θ and ϕ in model M1. Panel c): variation of the gravitational acceleration  g_{eff}  at the secondary star’s surface under coordinates θ and ϕ in Roche model. The radius is treated as the constant value. Panel d): variation of the gravitational acceleration  g_{eff}  at the secondary star’s surface under coordinates θ and ϕ in model M1. The effective gravitational acceleration can be derived from Eq. (1). 

In the text 
Fig. 3
Time evolution of the critical velocities for three models. The critical velocity v_{1,crit} is derived from the total potential function Ψ = Ψ_{grav} + Ψ_{rot} + Ψ_{tide} + Ψ_{dis,rot} + Ψ_{dis,tide} in model M1(Solid line). The critical velocity v_{2,crit} is derived from potential function Ψ = Ψ_{grav} + Ψ_{rot} + Ψ_{tide} in model M1 (Dotdashed line). The critical velocity v_{3,crit} is derived from the potential function Ψ_{r} = Ψ_{grav} + Ψ_{rot} + Ψ_{dis,rot} in model M1 (Dashed line). 

In the text 
Fig. 4
Correction factors f_{p}, f_{t}, and, f_{t}/f_{p} versus the dimensionless radius R/R_{⊙} according to model MKE (blue curve) and to model M1 (pink curves). 

In the text 
Fig. 5
Left panel: temperature gradients ∇_{r}, ∇_{ad}, and ∇_{r}−∇_{Ω} as function of the dimensionless radius R/R_{⊙} according to the MKE model (blue curves) and the M1 model (pink curves) below the stellar surface. The black line denotes the radiative temperature gradient ∇_{r} in the model without rotation. The green line denotes the temperature gradients ∇_{ad}. The red curve indicates the quantity ∇_{Ω}. Right panel: various temperature gradients located at the border of the convective core. The point where black lines cut green lines implies the border of convective core in nonrotating model according to the Schwarzschild criterion. The point where blue lines cut green lines indicates the border of convective core in rotating model according to the Schwarzschild criterion. 

In the text 
Fig. 6
Left panel: variations of the energy generation by nuclear reactions as a function of the Lagrangian mass at four different evolution points at critical rotation. The recipient approaches the critical rotation at the time of ~6.33 Myr. Right panel: variation of internal distribution of hydrogen mass fraction as a function of the Lagrangian mass at four different evolution points. 

In the text 
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