Synchronization and circularization in early-type binaries on main sequence

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1 Mon. Not. R. Astron. Soc. 401, (2010) doi: /j x Synchronization and circularization in early-type binaries on main sequence Kh. F. Khaliullin and A. I. Khaliullina Sternberg Astronomical Institute, 13 Universitetskij Prospect, Moscow , Russia Accepted 2009 August 28. Received 2009 June 19; in original form 2009 April 1 ABSTRACT We showed in a preceding paper based on an analysis of the observed rates of apsidal motion that synchronization in early-type eclipsing binaries continues on the main sequence, and the observed synchronization times, t syn, agree with the Zahn s theory and are inconsistent with the shorter time-scale proposed by Tassoul. It follows from this that circularization in early-type binaries must also proceed in accordance with the Zahn s theory because the circularization times, t circ, in both theories are rather tightly related to t syn via relation t circ αt syn, where α is the orbital-to-axial momentum ratio. To further investigate this problem, we compile a catalogue of 101 eclipsing binaries with early-type main-sequence components (M 1,2 > 1.6 M ). We determine the ages, t, and circularization time-scales, t circ, for all these systems in terms of the two competing theories by comparing observational data with modern models of stellar evolution of Claret and atmospheric models of Kurucz. We compute t circ with the allowance for the evolutionary variations of the physical parameters of the components and, for the first time in such studies, also take into account the variations of the orbital parameters (P, a, e) in the process of circularization subject to the conservation of the total angular momentum. The results of these computations show that the mechanism of orbital circularization in earlytype close binary systems (CBSs) suggested by Tassoul is, like in the case of synchronization, inconsistent with observational data. At the same time, the Zahn s mechanism, which is based on the dissipation of the energy of dynamic tides in the upper layers of the envelopes of CBSs components due to non-adiabaticity of these layers, agrees satisfactorily with observations. Key words: binaries: close binaries: eclipsing stars: early-type stars: evolution stars: fundamental parameters. 1 INTRODUCTION The studies of orbital synchronization and circularization in close binary systems (CBSs) are a valuable data source on the internal structure of component stars, their evolutionary status, and even about their formation conditions (Zahn 2008). The field is currently characterized by the competition of the two most popular theories of the orbital synchronization and circularization of CBSs with earlytype components. The first theory was developed by Zahn (1975) and Zahn (1977), and is based on the mechanism of energy dissipation via dynamic tides in non-adiabatic surface layers of the component stars. The second theory was developed by Tassoul (1987) and Tassoul (1988), and is based on tidal dissipation of kinetic energy of large-scale meridional flows. Both the circularization (t circ ) and synchronization (t syn ) time-scales implied by these mechanisms hfh@sai.msu.ru differ by almost three (!) orders of magnitude. One would expect that, given such a difference in efficiency, observational data should distinguish between the two tidal energy dissipation mechanisms by favouring one of them. However, until recently the results of a large series of works dedicated to comparing observational data with the predictions of different theories provided no definitive conclusions in favour of a certain theory. For example, in one of the last papers of this series, Claret, Gimenez & Cunha (1995) found that observations agree well with the Tassoul s theory, whereas Claret & Cunha (1997) concluded that the agreement between the Zahn s theory and observations is also satisfactory. The conflicting conclusions of different authors are due to the following main problems: (i) The observed axial rotation velocities, V r, characterize surface layers, whereas inner layers may rotate faster (Zahn 1977; Goldreich & Nicholson 1989; Yildiz 2003, 2005). In both the theories, the surface layers of the components may be pseudosynchronized with orbital rotation at the periastron, and this is what C 2009 The Authors. Journal compilation C 2009 RAS

2 258 Kh. F. Khaliullin and A. I. Khaliullina is actually observed. That is, V r values favour no particular synchronization mechanism. (ii) Unlike wide binaries (visual double stars) with their uniform distribution of eccentricities throughout the entire interval from 0 to 1 (Abt 2005), a large fraction of CBSs have circular orbits, which are impossible to explain in terms of the Zahn s circularization theory. This very fact prompted Tassoul to develop a more efficient (from his viewpoint) circularization theory. (iii) The so far unresolved nature of the problem of the formation of CBSs prevents unambiguous computation of the variation of their orbits and component axial rotation at the pre-main sequence (MS) stage of evolution. In their pioneering work, Zahn & Bouchet (1989) showed that this evolutionary stage plays an important part in orbital circularization in late-type CBSs (M = M ). Unfortunately, certain difficulties prevented such an analysis for massive stars with radiative envelopes. (iv) According to the recent data (Larson 2001), stars usually form in large groups, and single stars and binaries originate from a decay of their parent groups. It is clear that a multiple system may disrupt at any stage of its evolution, resulting in a radical change of the orbit of the CBSs. This explains why some systems may deviate from the overall relationships characterizing the behaviour of other CBSs. In a preceding paper (Khaliullin & Khaliullina 2007, hereafter Paper I), we managed to circumvent the first of the hurdles mentioned above. To this end, we used the fact that the observed period of apsidal rotation, U obs, depends not only on the observed parameters of the eclipsing system, but also on other, unobservable, ones: k 2,i and i,wherek 2,i is the apsidal parameter describing the radial density distribution in ith component and i are the angular axial rotation velocities in deep layers, which are responsible for the apsidal motion. So far, throughout the entire century-long history of the apsidal-motion problem, the researchers used the observed U obs to determine parameters k 2,i at fixed i values inferred for the surface (i.e. observed) layers. In Paper I, we reverse the problem and use U obs to determine i at fixed model k 2,i. As a result, we find axial rotation of components with radiative envelopes in many systems to be unsynchronized with orbital rotation and the process of synchronization to continue on the main-sequence. Moreover, the observed t syn is three orders of magnitude longer than implied by the theory of Tassoul (1987, 1988) and agrees satisfactorily with the predictions of the theory of Zahn (1977). 1 Hence, the circularization time-scales, t circ, must also agree with the Zahn s theory because t circ and t syn are rather tightly linked in both theories via relation t circ /t syn = A α, whereα is the orbital-to-rotational angular momentum ratio and A, a close-to-unity factor. The aim of this paper was to confirm that the observational data agrees with the Zahn s theory also in the case of orbital circularization in early-type CBSs. 1 In both papers of Zahn (1975, 1977) there is a typographical inaccuracy in equations for t syn : the factor 5 2 5/3 is printed as 5 2 5/3. In Paper I, we used the 52 5/3 factor to compute the t syn values in terms of the Zahn s theory. We therefore have to replace this factor by its correct value of 5 2 5/3.The corrected A Z the linear coefficient needed to ensure the agreement between observations and theory (at η = 0) is equal to A Z = 42/52 5/ /3 = 0.92(!), which is indicative of almost exact agreement between observations and the Zahn s synchronization theory. However, in later versions of the equation for t syn Zahn replaced the factor in question by just 5, i.e. he dropped the factor of 2 5/ = 0.5 dex (Zahn 2008). Section 2 gives the main relations that determine t circ in terms of the observed parameters of CBSs and describes the technique of the computation of this quantity by integrating the corresponding differential equations over time. Note that, unlike the authors of the previous works, we compute t/t circ with the allowance for the evolutionary variations of the orbital periods and, hence, the variations of the semimajor axes in the process of the circularization of the elliptical orbit subject to the conservation of the total angular momentum. Section 3 includes Tables 1 and 2, which list the observed and computed physical parameters for detached eclipsing systems with photoelectric light curves and bona fide photometric elements. In Section 4, we briefly describe the technique used to determine the component masses in eclipsing systems with no spectroscopic data or with the radial-velocity curve available only for one of the components. Section 5 compares the observational data from Table 1 and 2 with the theoretically expected values inferred by analysing the dependences of the orbital eccentricity on t/t circ,wheret is the age of the system. In Section 6, we compare our results with the results obtained by other authors. Section 7 discusses the results and summarises the main conclusions of the work. 2 CIRCULARIZATION TIME-SCALE IN EARLY-TYPE CLOSE BINARIES According to the Zahn s (1977) theory, radiative damping of dynamic tides due to non-adiabaticity of the outer layers of the envelopes of components in early-type CBSs implies the following relation for the orbital circularization time-scale, t circ : 1 (s 1 ) = 21 t circ 2 ( GM R 3 ) 1/2 ( ) R 21/2 q(1 + q) 11/6 E 2, (1) a where q = M /M is the mass ratio and M, the mass of the companion. R and M are the radius and mass of the star, respectively, in absolute units. G is the gravitational constant, a is the semimajor axis of the orbit, E 2 is the tidal factor ( R c /R) 8 and R c is the convective core radius. According to the theory of Tassoul (1987, 1988), dissipation of kinetic energy of tidally induced large-scale mass flows implies the following relation: t circ (yr) = [4 (N/4)] (1 + q)2/3 (L/L ) 1/4 rg 2 (M/M ) 23/12 (R/R ) 5 (P ) 49/12, (2) where L, M and R are the luminosity, mass, and radius of the star, respectively. P is the orbital period (in days), N = 0 for radiative envelopes, r g = (J/MR 2 ) 1/2 is the gyration radius and J is the moment of inertia of the star. Both relations above take into account tidal dissipation processes in one of the components of the binary system. The differential equation that describes the variations of the eccentricity and considers the effects of both components is dlne dt = 1 t circ = [ 1 t circ,1 + 1 t circ,2 ], (3) where subscripts 1 and 2 refer to the primary and secondary components, respectively. When computing t circ one must bear in mind that all the parameters appearing in relations (1) and (2) vary with time, and some of them change at an especially high rate. Fig. 1 shows the dependence

3 Synchronization and circularization in CBSs 259 Table 1. Physical and geometric parameters of early-type main-sequence eclipsing binaries (M1,2 > 1.6 M ) derived from observational data. N Name NSB9 P r1 r2 i e M1/M M2/M V B V U B b y m1 c1 J H K Ref. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) 1 σ Aql KP Aql , 60 3 V805 Aql , 88 4 V889 Aql V1182 Aql , 71 6 V539 Ara , 21 7 βaur , 99 8 AR Aur WW Aur ZZ Boo AS Cam , WW Cam CW CMa , GZ CMa SW CMa DW Car EM Car GL Car QX Car AR Cas MU Cas OX Cas PV Cas , V364 Cas V459 Cas V744 Cas V821 Cas KT Cen AH Cep CW Cep , NY Cep SY Cep WX Cep V397 Cep XY Cet , RS Cha , Y Cyg MY Cyg , V453 Cyg , 98

4 260 Kh. F. Khaliullin and A. I. Khaliullina Table 1 continued N Name NSB9 P r1 r2 i e M1/M M2/M V B V U B b y m1 c1 J H K Ref. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) 40 V478 Cyg , V541 Cyg , V909 Cyg , 51, V1147 Cyg DI Her HS Her , RX Her , V624 Her χ 2 Hya , AI Hya , CO Lac , SS Lac V364 Lac GG Lup RU Mon GM Nor TV Nor U Oph V451 Oph , V577 Oph FT Ori GG Ori , OO Peg AG Per , IQ Per V436 Per V615 Per V618 Per V621 Per ζ Phe , KX Pup NO Pup PV Pup VV Pyx V760 Sco V906 Sco AL Scl YY Sgr , V526 Sgr , V1647 Sgr , V2283 Sgr V3903 Sgr HO Tel

5 Synchronization and circularization in CBSs 261 Table 1 continued N Name NSB9 P r1 r2 i e M1/M M2/M V B V U B b y m1 c1 J H K Ref. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) 83 DN UMa , AO Vel , CV Vel , EO Vel PT Vel DR Vul HV NSV NSV SC SC SC SC SC SC SC SC SC SC Notes. Column 1: the running number. Column 2: the GCVS name. Column 3: the number in the SB9 catalogue. Column 4: the orbital period in days. Columns 5 and 6: the fractional radii of the components, r1 and r2, in terms of the semimajor axis a of the relative orbit. Column 7: the inclination of the orbit. Column 8: the eccentricity of the orbit as inferred from the light-curve solution. Columns 9 and 10: the masses of the components as inferred from an analysis of their radial-velocity curve. Columns 11 13: the outside-eclipse visual magnitudes V and the B V and U B colour indices in the UBV photometric system. Columns 14 16: the outside-eclipse m1, c1, and b y colour indices in Stroemgren s uvby photometric system. Columns 17-19: outside-eclipse JHK photometry adopted from the Two Micron All Sky Survey catalogue. Column 20: references to the principal sources of photometric and spectroscopic parameters of the systems. 1. Alencar et al. (1997); 2. Andersen (1975a); 3. Andersen (1975b); 4. Andersen (1983); 5. Andersen & Gimenez (1985); 6. Andersen & Clausen (1989); 7. Andersen, Clausen & Nordstrom (1984); 8. Andersen, Clausen & Gimenez (1993); 9. Andersen et al. (1983); 10. Andersen et al. (1985); 11. Bakış et al. (2008); 12. Barembaum & Etzel (1995); 13. Bell, Hilditch & Adamson (1986); 14. Bell, Hilditch & Adamson (1987); 15. Bulut et al. (2006); 16. Bulut et al. (2005); 17. Burkholder, Massey & Morrell (1997); 18. Cakirli, Ibanoglu & Frasca (2007); 19. Cester et al. (1978); 20. Chaubey (1984); 21. Clausen (1996); 22. Clausen & Gyldenkerne (1976); 23. Clausen & Grønbech (1977); 24. Clausen & Nordström (1978); 25. Clausen & Nordström (1980); 26. Clausen & Gimenez (1991); 27. Clausen, Gyldenkerne & Gronbech (1977); 28. Clausen, Gimenez & Scarfe (1986); 29. Clausen, Gimenez & van Houten (1995); 30. Clausen et al. (2003); 31. Clausen et al. (2007); 32. Crinklaw & Etzel (1989); 33. Cristaldi (1970); 34. Degirmenci et al. (2007); 35. Garcia & Gimenez (1986); 36. Gimenez & Clausen (1986); 37. Gimenez & Clausen (1994); 38. Giuricin & Mardirossian (1981); 39. González et al. (2006); 40. Graczyk (2003); 41. Grønbech (1976); 42. Gülmen, Seser & Güdür (1988); 43. Haefner, Skillen & degroot (1987); 44. Harmanec et al. (1997); 45. Hilditch (1972); 46. Ibanoglu & Gülmen (1974); 47. Jeffreeys (1980); 48. Joergensen & Gronbech (1978); 49. Kardopolov, Lavrov & Filip ev (1986); 50. Khaliullin (1985); 51. Khaliullin (private communication); 52. Khaliullin & Kozyreva (1983); 53. Khaliullin & Khaliullina (1989); 54. Khaliullin & Khaliullina (2006); 55. Khaliullin, Antipin & Khaliullina (2006); 56. Khaliullina & Khaliullin (1988); 57. Khaliullina et al. (1985); 58. Krylov et al. (2003); 59. Lacy (1982); 60. Lacy (1987); 61. Lacy (1993a); 62. Lacy (1993b); 63. Lacy (1997a); 64. Lacy (1997b); 65. Lacy (1997c); 66. Lacy (1998); 67. Lacy & Frueh (1985); 68. Lacy, Claret & Sabby (2004a); 69. Lacy, Claret & Sabby (2004b); 70. Lacy et al. (2002); 71. Mayer, Drechsel & Lorenz (2005); 72. McDonald (1949); 73. Munari et al. (2001); 74. Nordström & Johansen (1994); 75. North et al. (1997); 76. O Connell (1974); 77. Popper (1959); 78. Popper (1970); 79. Popper (1971); 80. Popper (1974); 81. Popper (1981); 82. Popper (1982); 83. Popper (1983); 84. Popper (1984); 85. Popper (1986); 86. Popper (1987); 87. Popper (1988); 88. Popper & Etzel (1981); 89. Popper & Hill (1991); 90. Popper et al. (1985); 91. Semeniuk (1967); 92. Simon, Sturm & Fiedler (1994); 93. Smak (1967); 94. Smith (1948); 95. Söderhjelm (1975); 96. Southworth & Clausen (2007); 97 Southworth, Maxted & Smalley (2004a); 98. Southworth, Maxted & Smalley (2004b); 99. Southworth et al. (2007); 100. Southworth et al. (2004); 101. Southworth et al. (2005); 102. Srivastava (1987); 103. Torres & Stefanik (2000); 104. Torres & Stefanik (2000); 105. Torreset al. (1999); 106. Vaz & Andersen (1984); 107. Vaz, Andersen & Claret (2007); 108. Vaz et al. (1997); 109. Volkov & Khaliullin (2002); 110. Volkov (private communication); 111. Wetterer, Bloomer & Caton (2006); 112. Williamon (1976); 113. Young (1992).

6 262 Kh. F. Khaliullin and A. I. Khaliullina Table 2. Physical and geometric parameters of the eclipsing binaries from Table 1 inferred by comparing observational data with the stellar evolutionary models of Claret (2004), and the t/t circ ratios for these systems computed in terms of different theories of circularization with the allowance for the evolutionary variations of the orbital parameters and physical characteristics of the components. N Name M 1 /M M 2 /M R 1 /R R 2 /R a/r log T 1 log T 2 log t log(t/t circ ) log(t/t circ ) log(t/t circ ) α obs = Zahn Zahn(mod) Tassoul L orb /L spin (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) 1 σ Aql KP Aql V805 Aql V889 Aql V1182 Aql V539 Ara βaur AR Aur WW Aur ZZ Boo AS Cam WW Cam CW CMa GZ CMa SW CMa DW Car EM Car GL Car QX Car AR Cas MU Cas OX Cas PV Cas V364 Cas V459 Cas V744 Cas V821 Cas KT Cen AH Cep CW Cep NY Cep SY Cep WX Cep V397 Cep XY Cet RS Cha Y Cyg MY Cyg V453 Cyg V478 Cyg V541 Cyg V909 Cyg V1147 Cyg DI Her HS Her RX Her V624 Her χ 2 Hya AI Hya CO Lac SS Lac V364 Lac GG Lup RU Mon GM Nor TV Nor U Oph V451 Oph

7 Synchronization and circularization in CBSs 263 Table 2 continued N Name M 1 /M M 2 /M R 1 /R R 2 /R a/r log T 1 log T 2 log t log(t/t circ ) log(t/t circ ) log(t/t circ ) α obs = Zahn Zahn(mod) Tassoul L orb /L spin (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) 59 V577 Oph FT Ori GG Ori OO Peg AG Per IQ Per V436 Per V615 Per V618 Per V621 Per ζ Phe KX Pup NO Pup PV Pup VV Pyx V760 Sco V906 Sco AL Sql YY Sgr V526 Sgr V1647 Sgr V2283 Sgr V3903 Sgr HO Tel DN UMa AO Vel CV Vel EO Vel PT Vel DR Vul HV NSV NSV SC SC SC SC SC SC SC SC SC SC Notes. Column 1: the running number. Column 2: the GCVS name. Columns 3 and 4: the component masses as inferred by comparing all observational data for the binary with the evolutionary models of Claret. Columns 5 and 6: the component radii. Column 7: the semimajor axis of the orbit. Columns 8 and 9: the logarithms of effective temperatures of the components, log T 1 and log T 2. Column 10: log t, where the age t of the binary is in years. Column 11: log(t/t circ ) computed in terms of the Zahn s theory. Column 12: log(t/t circ ) computed using modified equation (14) for Zahn with extra coefficients F 212 (e) anda Z = 2. Column 13: log(t/t circ ) computed in terms of the Tassoul s theory. Column 14: the observed-orbital-to-axial momentum ratio. of the tidal factor E 2 on age t for stars of different masses in the age interval from t = 0 (Zero Age Main Sequence; ZAMS) to the end of the main-sequence evolution (t = t MS ), and Fig. 2 shows the log(r t /R 0 ) 9 dependences on age for different masses according to modern evolutionary models of Claret (2004) for Z = Even within the MS, E 2 and (R t /R 0 ) 9 vary by up to over several orders of magnitude and hence the t circ estimates may differ by a comparable factor if the age of the binary systems and the time dependence of these and other parameters are not exactly known. There was an opinion (Zahn 1977), which was based on earlier stellar evolution models by Stothers & Chin (1975), that the product of parameters (E 2 R 21/2 /R 3/2 ), which appears in relation (1) for t circ, decreases with age, implying that circularization occurs mostly during the early stages of the main-sequence evolution. However, modern evolutionary models by Claret exhibit the opposite pattern, as it is evident from Fig. 3. This figure shows log(e 2 R 9 ) t log(e 2 R 9 ) 0 plotted as a function of age t. The log(e 2 R 9 ) quantity normalized to ZAMS increases with time for all masses and hence circularization must be even faster during later evolutionary stages. Therefore, to quantitatively compare the observational data with theoretical predictions, (i) we need bona fide observational parameters for CBS (t, M, R, L, P, e, etc.) and (ii) we must know how these

8 264 Kh. F. Khaliullin and A. I. Khaliullina P = 2 ḍ 65; a = 17.4 R and e = Such radical changes of the orbital parameters have a significant effect on circularization time estimates, t circ, for many systems. Therefore, t circ estimates made assuming P = const and, thereby, a = const, albeit being the most common until recently, cannot be considered suitable for comparing observational data with theoretical predictions. In the case of varying physical parameters of the components and ever-decreasing total energy due to its tidal dissipation, the only constant quantity (for an isolated binary) is its total angular momentum (Hut 1981), L = i=1,2 M i R 2 i r2 g,i i + G 1/2 M 1 M 2 a1/2 (M 1 + M 2 ) 1/2 (1 e 2 ) 1/2 const, (4) Figure 1. Variation of tidal coefficient E 2 with age t for stars of different mass from t = 0 (ZAMS) until the end of the main-sequence stage (t = t MS ) according the stellar evolution models of Claret (2004) for Z = Figure 2. Variation of log(r t /R 0 ) 9 with time. where subscript i indicates whether the corresponding quantity refers to the primary (1) or secondary (2) component; i, the angular velocities of axial rotation of the components, and the remaining designations were introduced above. The first term in this equation gives the sum of the angular momenta of the components, L spin = L spin,1 + L spin,2, and the second term, the orbital angular momentum L orb. This expression is written for the case of coincident directions of the vectors of the orbital and axial rotation. The law of conservation of L allows us to establish an evolutionary relation between eccentricity e and semimajor axis a. To this end, consider expression (4) in more detail. The variations of the physical parameters of the components with age M i (t), R i (t) and r g,i (t) are known from stellar evolutionary models. Four parameters a, e, 1 and 2 appear in equation (4) in addition to the quantities just mentioned. To reveal the evolutionary relation, a(e), we must eliminate the latter two parameters from our list of unknowns. This task is facilitated by the fact that t circ αt syn in the theories of synchronization and circularization considered here, where α is the orbital-to-rotation angular momentum ratio. The t circ /t i ratio must be of about the same value ( α), where t i is the time-scale of collinearization of the vectors of axial and orbital rotation (Hut 1981). The ratio α 50 for all eccentric eclipsing systems listed in Table 1 (see Section 3). It is therefore safe to say that systems in the process of circularization must be pseudosynchronized, the vectors of their L spin and L orb must be collinear, and hence equation (4) must apply to them. According to Hut (1981), pseudo-synchronization in an elliptical orbit occurs at 1 = 2 = ps = 2π P (1 + e)1/2 (1 e) 3/2 1 + (15/2)e2 + (45/8)e 4 + (5/16)e 6 [ 1 + 3e2 + (3/8)e 4], (5) (1 + e) 2 Figure 3. Variation of log(e 2 R 9 ) with time. parameters vary with time from ZAMS and until the observed age t. The most reliable observational data are those provided by eclipsing binaries, which we list in the next section along with their known parameters. The model parameters (r g, E 2 ) and the variations of the physical parameters of the components with age are provided by stellar evolutionary models. In the process of orbital circularization, not only the physical parameters of the components, but also the orbital parameters P, a, e, i etc. must vary with time. These changes may be rather important. Our computations show that a model binary with the initial (ZAMS) parameters equal to M 1 = M 2 = 5M ; P = 120 d ; a = 220 R,ande = 0.96 develops, as a result of the circularization on the MS, into a very close system with M 1 = M 2 = 5M ; where orbital period P is related to the semimajor axis via the Kepler s law, P = a 3/2 2π. (6) G 1/2 (M 1 + M 2 ) 1/2 We thus unambiguously link a and e via the condition of angular momentum conservation by expressing i in terms of eccentricity and semimajor axis of the orbit, and can therefore compute the variations (in the process of circularization) of the orbital parameters (P, a) needed for computing t circ. When comparing observational data to the theoretically expected values (see Section 5), we use the (t/t circ ; e) diagrams, where e and t are the orbital eccentricity and the age of the system, respectively. In accordance with the above, to determine t/t circ, we must integrate

9 Synchronization and circularization in CBSs 265 equation (3) over time from t = 0 (ZAMS) to t, t t dt t dt = +. (7) t circ 0 t circ,t 0 t circ,2 Here, we must stress a few points. First, we do not know the orbital parameters at t = 0, and therefore start our integration at t and proceed back in time until t = 0. We sort of reconstruct the past state of the system. Secondly, Claret s ( ) stellar models for non-rotating stars are computed on rather coarse grids in terms of mass (from 0.8 to 125 M ) and metallicities (Z = 0.002, 0.004, 0.007, 0.010, 0.020, and 0.100), and on a very uneven age grid. One of the main advantages of these models, compared to other similar grids, is that they provide all the necessary parameters for computing t circ. For convenience, and to speed up operations with the model grids, we first transformed them via non-linear interpolation into finer grids, which are uniform in age ( t = t MS /1000, which is equal to the integration step) and mass logarithms ( log M = 0.002). Such a fine and uniform grid (within the main-sequence) allows using these models without complex interpolation procedures, which is important in the case of the minimization of non-linear functional depending on many parameters (see Section 4). To make possible visualization and control at every stage of computations, we adopt a very simple and demonstrative procedure of numerical integration, which can be subdivided into the following stages. (i) We use the observed parameters of the system, stellar evolutionary models and equations (4) (6) to compute the constant value L of the total angular momentum of the system. (ii) The integration step over age is t = t MS /1000. We compute the component parameter values (R i, M i, r g,i, E 2,i ) at the middle of each such age interval from 0 to t based on the observed parameters of the system and stellar evolutionary models of Claret. (iii) We then use these parameters to compute t/t circ for each given jth step by equations (1) and (2) in terms of the Zahn s and Tassoul s theories, respectively. We adopt the required a and P values from the previous, (j 1)-th step, and use the observed values from Table 2 for the first step. (iv) We use the relation (Zahn 1977) 1 = 1 de (8) t circ e dt to compute the new eccentricity value for the next, (j + 1)-th step (via backward integration): e j+1 = e j e t/tcirc, (9) where the second e on the right-hand side is the base of natural logarithms. (v) We then use the new eccentricity value, e j+1, to compute i quantities by equation (5). (vi) We numerically solve equation (4) to determine the only unknown quantity the semimajor axis a, which, via relation (6), also determines P. (vii) We pass to the next integration step. Computations end at t = 0. We sum up the t/t circ values: t/t circ = N ( t/t circ,j )and store them in a separate file. To make the reconstruction of the dynamic orbit possible, we also store the orbital parameter values (e j, a j, P j, L spin,i,j, L orb,j, i,j ) obtained at each jth step. We describe here numerical integration in much detail for the following reasons. First, an analytical solution of the problem is impossible to derive in practice because of the complex evolutionary j=1 variations of all parameters of the binary. Secondly, the fact that some of the parameters used in integration (M, R, r g, E 2 )referto the current, jth integration step, while the values of other parameters (P, a, e) refer to the previous, (j 1)th step, requires the use of an iterative technique. Computations showed that for the sufficiently small step ( t = t MS /1000) adopted in this work two iterations are enough. Control forward-and-backward integration of different binary systems demonstrated excellent convergence of the results to the initial data. Note in conclusion that relations (1) and (2) were derived for the case of small eccentricities and pseudo-synchronization of axial and orbital rotation. As it is said above, the systems (from Table 1) in the process of circularization must be pseudo-synchronized. We discuss the need for corrections to the equations for circularization in the case of large eccentricities in Section 5. Here we make the following point. Relation (8), which links 1/t circ and de/dt, is valid only if 1,2 = ω, whereω is the mean orbital angular velocity. In reality, de/dt depends on ω and Zahn (1977, equation 5.10) gives this dependence in the form: 1 de e dt = 1 1 t circ 14 [ (1 + ζ )8/3 + ζ 8/ (1 ζ )8/3 4 ], (10) where ζ = 2( ω)/ω, and1/t circ is given by equation (1). Zahn plots de/dt as a function of /ω and shows that at /ω > the eccentricity increases with time, i.e. de/dt >0. In the case of pseudo-synchronization ps /ω > for an elliptical orbit even for eccentricities e>0.4. This, however, does not imply the increase of eccentricity in this case. Zahn s relation (10) is valid only at e 0. Our analysis of the results of a more rigorous and detailed averaging of tidal effects over the orbital period performed by Hut (1981) for the equilibrium tides showed that at = ps quantity de/dt < 0 for all values of eccentricity, and de/dt depends only slightly on ( ps ω). Therefore we can assume, without significant loss of accuracy, that (1/e) de/dt = 1/t circ, as we adopt in this paper. We will address this problem in more detail in our next paper, to be dedicated to the circularization of low-massive systems. 3 OBSERVATIONAL DATA FOR THE ANALYSIS OF ORBITAL CIRCULARIZATION IN EARLY-TYPE CBS As it is evident from the previous section and as Claret & Cunha (1997) reasonably point out in their paper, only eclipsing binaries are suitable for quantitative comparison of observational data and theory. Only for such systems we can determine the radii, masses, and other parameters of the components with sufficient accuracy (to within 2 3 per cent for R i and M i ) and, based on these data, determine the ages and circularization time-scales. Table 1 lists the observational data for 101 eclipsing binaries with early-type components. We selected these systems from among several thousand currently known eclipsing binaries based on the following criteria. (i) The system must be detached and must not exhibit features typical of Algol-type eclipsing binaries, i.e. no evidence for either past or present mass exchange. Both components must be on the MS. (ii) Photometric elements inferred from an analysis of a photoelectric light curve must be available for the system. (iii) Radial-velocity curves must be available for the system, or, in the absence thereof, multicolour (UBVR or ubvy) light curves and

10 266 Kh. F. Khaliullin and A. I. Khaliullina the corresponding photometric elements. We describe the technique used to determine the component masses for such systems in the next section. (iv) M 1 and M 2 > 1.6 M. We believe our list to be currently the most complete one and contain the overwhelming majority of known eclipsing binaries meeting the above selection criteria. 4 DETERMINATION OF THE COMPONENT MASSES BY CONFRONTING OBSERVATIONAL DATA WITH THE STELLAR AND ATMOSPHERIC MODELS Claret et al. (1995) and Claret & Cunha (1997) based their investigation of the circularization and synchronization on the data for 45 double-lined eclipsing binaries from Andersen (1991). For all these systems, reliable radial-velocity curves are available for both components and therefore the observed masses, M obs 1 and M obs 2,are also known for both components. However, only 28 of the systems consist of early-type stars (M 1,2 > 1.6 M ), and three of these (SZ Cen, TZ For and V1031 Ori) are not main-sequence stars. The other 25 systems we included into our Table 1. Although the list of such double-lined eclipsing binaries has somewhat expanded since then, observed masses are still unavailable for many of the systems listed in Table 1. We have to infer sufficiently reliable estimates for these masses based on the known parameters of the system. In many practical cases, this problem can be solved using one-, or at most two-parameter relations like M(Sp),M(Sp, L), etc (Khaliullin 1985; Harmanec 1988). However, for early-type stars such relations yield masses that are accurate to 8 10 per cent at best, which is utterly inadequate for addressing the problem of circularization. That is why we developed and apply for the first time in this paper a technique for estimating the component masses in eclipsing systems simultaneously with their ages by confronting all bona fide observational data for the star considered with the theoretical stellar models of Claret ( ) and Kurucz s (1993) model atmospheres. We will describe in detail our technique of estimating the component masses depending on the number of parameters involved and their accuracy, as well as the results obtained, in our next paper, which is now in preparation. Here, we only emphasise the following points. In the process of optimization, which is based on minimizing χ 2 (n) statistics, we determine the values of four parameters M 1, M 2, t and A V,whereA V is the total visual interstellar extinction and n, the number of parameters (which is be as high as 26 for some of the systems listed in Table 1). The main observed and theoretical parameters were: r 1, r 2, r 1 + r 2, L 1, L 2, J 1, J 2, V, B, U, u, v, b, y, J, H, K, etc. Note that we compute the fractional luminosities L i and surface brightness s J i of the components based on published results of light-curve analyses in all photometric bands. We use the following minimization procedure, which is based on simple enumeration of possible variants of M 1, M 2 (with a step of log M = 0.002), t (with a step of t = t MS /1000) and episodic refinements of A V. For each pair of M and t Claret s models yield T e, L,logg, which can, via Kurucz s model atmospheres, be easily transformed into the theoretical colour indices and fractional component luminosities in various photometric systems, whereas, given P, M 1, M 2 and g values, one can determine the absolute radii R 1 and R 2, the semimajor axis of the orbit and, consequently, the fractional radii r 1 and r 2. We then minimize the sum of the squared deviations of the corresponding observed parameters from their theoretical values normalized to their standard errors. We now make the following points. First, for systems with available bona fide radial velocity curve for only one of components, we introduce the mass function f (M) as one of the parameters, and for the systems with accurately measured parallaxes we treat the π values as an extra parameter. For most of the systems, we use Claret s (2004) solar composition (Z = 0.02) evolutionary grids. For the systems with known Z, we use the grids corresponding to this chemical composition. Our minimization procedure is adapted to the different completeness and great variety of observational data for different systems. Secondly, not all of the above parameters are fully independent. The photometric elements for different components and different photometric bands (r i, L i, J i ) are intricately intercorrelated. For example, the k = r 2 /r 1 ratio for many eclipsing systems can be varied over a wide range without degrading appreciably the accuracy of the approximation of observational data by the theoretical light curve, whereas the sum of the radii, r 1 + r 2, is inferred with much higher accuracy (Popper 1982; Khaliullina, Khaliullin & Martynov 1985; Andersen 1991; North, Studer & Kunzli 1997). The same is true of the fractional component luminosities L i, which are inferred with even lower accuracy than surface brightness values J i. On the one hand, this justifies treating r 1 + r 2 as one of the parameters along with r 1 and r 2 and J i. On the other hand, the inter-correlation between the parameters reduces substantially their effective number and increases the possible scatter of our χ 2 (n) values compared to the standard distribution of this quantity (Bolshev & Smirnov 1983). Table 2 lists the results of the computations. The estimated errors of our inferred model masses do not exceed 3 per cent, which is only a factor of 1.5 greater than the errors of the spectroscopic masses for the 25 systems in Anderson s (1991) catalogue. We therefore believe that the systems listed in Table 1 can be used as a reliable observational base for testing the existing theories of circularization. Here, it is necessary to note the following. Masses of components of some systems from Tables 1 and 2 differ up to 15 per cent. For example, in case of AR Aur spectroscopic mass, M 1 (Table 1) = 2.48 M, and mass derived from models, M 1 (Table 2) = 2.79 M. The existing radial velocity measurements of AR Aur are by Wyse (1936) and Harper (1938). Nordström & Johansen (1994) re-analysed the radial-velocity curves of the above authors and obtained the results shown in Table 3. The later Harper s data agree better with our results. However, Nordström & Johansen (1994) prefer the data of Wyse (1936), and therefore everywhere Wyse s masses are quoted. Meanwhile Rachkovskaia (1985) found q = M 1 /M 2 = 1 for AR Aur. Moreover, AR Aur is a triple system (Chochol et al. 1988) and the radial-velocity curve can be perturbed due to the dynamical and spectroscopic influence of the third body. If inexplicable contradictions between the spectroscopic and photometric data were found, we did not include such systems in our sample. Table 3. Comparison of mass ratios. Wyse (1936) Harper (1938) Our inferred model masses (Table 2) M 1 /M M 2 /M