A&A 422, 981–986 (2004) Astronomy DOI: 10.1051/0004-6361:20034041 & c ESO 2004 Astrophysics Evolution of rotating binary stars R. Q. Huang National Astronomical Observatories/Yunnan Observatory, the Chinese Academy of Sciences, Kunmin 650011, PR China e-mail: [email protected] Received 2 July 2003 / Accepted 16 March 2004 Abstract. A model for the evolution of rotating binary stars is presented. The evolution of a binary system consisting of a 9 M and a 6 M star is studied in the mass transfer phase Case A. Specific attention is focused on the differences between the evolutions with and without the effects of rotation. The results indicate that the effects of rotation prolong the lifetime of the primary, and causes its evolutionary track in the HR diagram to shift towards lower luminosity. The mass and the surface helium composition of the primary become lower and the orbital period of the system becomes significantly longer during the later stage of the evolution. Key words. stars: rotation – stars: evolution 1. Introduction 2. Model for rotating binary stars A series of observations indicate that rotational mixing ex- 2.1. Equipotential and the equivalent sphere ists in rotating stars (Herrero et al. 1992; Charbonnel 1994, Observations (Giuricin et al. 1984; Van Hamme & Wilson 1995; Walborn 1976; Fransson et al. 1989). The evolution 1990; Pan & Tan 1998) show that most detached binary sys- of rotating single stars has been studied by many investi- tems are synchronized systems; only a few systems are non- gations (Kippenhahn & Thomas 1970; Endal & Sofia 1976; synchronized. The conventional theories for the evolution of Pinsonneaul et al. 1989; Meynet & Maeder 1997). It is impor- close binaries usually assume that the synchronism of the ro- tant to study the evolution of rotating binary stars. To construct tation with the orbital motion is always reached (Kippenhahn a model for the evolution of binary systems with the effect of & Weigert 1967; De Loore 1980; Huang & Taam 1990; rotation, we have to take into account a series of special proper- Vanbeveren 1991; De Greve 1993). The potential for a syn- ties occurring in the rotating binaries: (1) The Roche potential chronous rotating component in a Roche model is given by for a rotating component consists of four terms expressing the ff e ects of gravitation, the rotation of the component, the rota- Ψ=GM1 + 1 2 2 2 + 1 2 − 2 ω ri sin θ ω (Xω ri sin θ) (1) tion of the system and the tide, respectively. This results in the ri 2 2 differences between the equipotential for the rotating compo- FGM + 2 , nents and for the rotating single stars. (2) Owing to the effect (A − r sin θ)2 + r2 cos2 θ of tide, the rotation of the component is solid-body rotation and i i synchronous with the orbital motion. (3) In the case of binary where M and M are the masses of the primary and the sec- systems, we have to study not only the structure and evolu- 1 2 ondary, A is the distance between the two components, X is tion of the components, but also the evolution of the system ω the distance between the primary and the axis through the cen- caused by the loss of mass and angular momentum via stellar ter of mass (Fig. 1, X = AM /(M + M )), and ω is the or- winds, the transfer of mass and angular momentum between ω 2 1 2 bital angular velocity of the system (ω2 = G(M + M )/A3). the components and the changes in the moments of inertia of 1 2 The first and second terms on the right hand side of Eq. (1) the components. correspond to the contributions of the gravitation and the ro- In Sect. 2, we discuss the special properties occurring in tation of the component, respectively. The third and the fourth rotating binaries, and introduce a model for the binary systems terms correspond to the contributions of the rotation of the sys- with the effects of rotation. In Sect. 3, the evolution of a binary tem and the tidal action, respectively. The basic assumption of system consisting of a 9 M and a 6 M star is studied in the the Roche model is that the mass of the component is concen- mass transfer phase Case A, and the differences between the re- trated in the mass center. This assumption will deviate when sults for the evolutions with and without the effects of rotation the configurations of the components become non-symmetric are discussed in detail. rotational ellipsoids. In such cases the value of the fourth term Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20034041 982 R. Q. Huang: Evolution of rotating binary stars 2.2. The interior stellar structure equations Due to the effect of tide, the rotation of the component is syn- chronous with the orbital motion of the system. Such syn- chronous rotation exists also in the interior of the compo- nent. Thus, the rotation of the component is solid-body rotation and conservative. Kippenhahn & Thomas (1970) introduced a method to simplify the two-dimensional model with conserva- tive rotation to a one dimensional model, and gave the structure equations as follows: drΨ 1 = , 2 (7) dMΨ 4πrΨρ dP GMΨ = − f , 2 P (8) dMψ 4πrΨ Fig. 1. Geomety of the Roche potential. dL = εN − εν + εg, (9) on the right hand side of Eq. (1) will decrease. Hence we intro- dMΨ duce a factor F to the fourth term in Eq. (1), and the factor F may have the values between 1 and a small value during the dlnT ∇R fT / fP = (10) phase of Roche lob overflow. The lowest value of this range is, dlnP ∇con however, related to the mass ratio and the distance between the two components, and is quite difficult to determine. where Introducing a quantity q = M2/M1, the dimensionless 4πr4 Roche potential can be obtained from (1) as: = Ψ 1 fP − , (11) GMΨS Ψ 1 2 geff GM 1 1 ri Ψ= 1 + (1 + q) sin2 θ A ri 2 A A 2 2 4πrΨ 1 2 f = , (12) 1 q ri sin θ T −1 + (1 + q) − (2) S Ψ geff g ff 2 1 + q A e 3κLP + Fq · ∇ = , (13) R 4 2 r2 cos2 θ 4acGMΨT − ri sin θ + i 1 A A2 −1 g ff, g ff The equipotential defined by the function Ψ=const. are non- here, e eff are the mean values of the e ective gravity and its inverse over the equipotential surface, ∇R the radiative symmetric rotational ellipsoids with two semimajor axes a1 temperature gradient. and a2 (a1 > a2 ) and one semiminor axis b (Fig. 1). The vol- ume Vψ inside the equipotential and the surface area S ψ of the equipotential of the primary can be expressed as 2.3. Calculations of the quantities fP and fT = 2πb 2 + 2 Vψ a1 a2 , (3) According to Eqs. (11) and (12), the key to get the values of fP 3 −1 and f is to calculate the mean values g ff and g . In prac- T e eff 4π tical calculations, the two-dimensional model for the rotating S = a2 + a2 + b2 . (4) ψ 3 1 2 component is simplified to a one-dimensional model. Thus, the For any quantity, f , which is not constant over the equipotential stratification of the component with non-spherical equipoten- surface, the mean value is defined as tial is replaced by the stratification with equivalent spheres. −1 The calculation of the mean values of g ff and g over the 1 e eff f = f dσ, (5) equipotential surface is divided, therefore, into two steps: the S ψ ψ=const. first step is to get the equipotential surface by giving the corre- where dσ is an infinitesimal area of the surface. sponding equivalent sphere, the second step is to calculate the −1 We define a sphere with its volume equal to that of an mean values of geff and geff over the equipotential surface. equipotential ellipsoid. The radius rψ of the sphere is given by 1/3 3Vψ 2.3.1. Replacement of equipotential by the equivalent r = · (6) ψ 4π sphere This sphere is called the volume equivalent sphere, and its ra- Assuming mΨ is the mass in the equivalent sphere of radius rΨ, mΨ dius’s called the equivalent radius of the equipotential ellipsoid. and the quantity qΨ = , the functions for the semi-axis a1, a2 M1 R. Q. Huang: Evolution of rotating binary stars 983 and b to the rψ of the equipotential can be obtained from Eqs. (1), (3) and (4) as 3 2πb 4πrΨ a2 + a2 = , (14) 3 1 2 3 GM qψ 1 a 2 1 + (1 + q) 1 (15) A a1/A 2 A 1 q a 2 Fq + (1 + q) − 1 + + − a1 Fig. 2. Geometry of the equipotential. The solid curve is a non- 2 1 q A 1 A symmetric rotational ellipsoid with two semimajor axes a and a and 1 2 2 b GM qΨ 1 q Fq one semiminor axis . The dot-dashed curve is a half of a symmet- = 1 + + , ric rotational ellipsoid with semimajor axis a1 and semiminor axis b. A b/A 2 1 + q 2 + b The dotted curve is also a half of a symmetric rotational ellipsoid with 1 A semimajor axis a2 and semiminor axis b. q 2 GM1 ψ + 1 + a2 (1 q) (16) A a2/A 2 A 2 GM1 qψ 1 r2i 2 Ψ = + + 2 2 r (1 q) sin θ 1 q a2 Fq A 2i A + (1 + q) + + A 2 + + a2 2 1 q A 1 A 2 1 q r2i sin θ + (1 + q) + (18) 2 + q A GM1 qΨ 1 q Fq 2 1 = + + , + 2 A b/A 2 1 q + b Fq 1 A + , 2 r2 cos2 θ + r2i sin θ + 12 1 A A2 where q = M2/M1.