The Binding Energy Parameter for Common Envelope Evolution
aa r X i v : . [ a s t r o - ph . S R ] M a y Research in Astron. Astrophys. Vol.0 (200x) No.0, 000–000 R esearchin A stronomyand A strophysics The Binding Energy Parameter for Common Envelope Evolution
Chen Wang , , Kun Jia , and Xiang-Dong Li , Department of Astronomy, Nanjing University, Nanjing 210046, China; [email protected] Key laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry ofEducation, Nanjing 210046, China
Abstract
The binding energy parameter λ plays a vital role in common envelope evolu-tion. Though it is well known that λ takes different values for stars with different massesand varies during stellar evolution, it has been erroneously adopted as a constant in mostof the population synthesis calculations. We have systematically calculated the values of λ for stars of masses − M ⊙ by use of an updated stellar evolution code, taking intoaccount contribution from both gravitational energy and internal energy to the bindingenergy of the envelope. We adopt the criterion for the core-envelope boundary advocatedby Ivanova (2011). A new kind of λ with the enthalpy prescription is also investigated.We present fitting formulae for the calculated values of various kinds of λ , which can beused in future population synthesis studies. Key words: binaries: general — stars: evolution— stars: mass-loss
Common envelope (CE) evolution is one of the most important and yet unresolved stages in the forma-tion of various types of binary systems including low-mass X-ray binaries and cataclysmic variables.For semi-detached binaries, mass transfer can be dynamically unstable if the mass ratio is larger than acritical value or the envelope of the donor star is in convective equilibrium. In this case, the accretingstar, usually the less massive star, cannot maintain thermal equilibrium, and the transferred material ac-cumulates on its surface. As a result, both components are expected to overflow their respective Rochelobes (RLs), forming an envelope enshrouding both stars. The accreting star then spirals into the donor’senvelope, using its orbital energy to expel the envelope. This is the so-called common envelope (CE)evolution (see Iben & Livio 1993; Taam & Sandquist 2000; Ivanova et al. 2013 for reviews). The out-come of CE evolution is either a compact binary consisting of the donor’s core and the companion star,or a single object due to merger of the two stars, depending on whether the available orbital energy islarge enough to eject the donor’s envelope. This process can be described by the following equation(Webbink 1984), E bind = α CE (cid:18) GM core M a f − GM M a i (cid:19) , (1)where E bind = − Z M M core GM ( r ) r dm (2)is the binding energy of the envelope, α CE the efficiency parameter that denotes the fraction of theorbital energy used to eject the CE, G the gravitational constant, M and M core the masses of the donor Wang, Jia & Li and its core, M the mass of the companion star, and a i and a f the pre- and post-CE orbital separations,respectively.It has been suggested that the internal energy (including both thermal and recombination energies)in the envelope may also contribute to the binding energy, so a more general form for E bind can bewritten as E bind = Z M M core (cid:20) − GM ( r ) r + U (cid:21) dm, (3)where U is the internal energy (Han et al. 1994; Dewi & Tauris 2000). More recently,Ivanova & Chaichenets (2011) proposed that this canonical energy formalism should be modified withan additional P/ρ term (where P is the pressure and ρ is the density of the gas) by taking into accountthe mass outflows during the spiral-in stage. These authors argue that, the standard form (1) or (2) isbased on the consideration that the envelope of a giant star is dispersed or unstable once its total en-ergy W env > , but neither of the two considerations has to occur when the envelope has quasi-steadyoutflows. For such envelopes, the material obeys the first law of thermodynamics, and the criterion fora mass shell to reach the point of no return in its expansion turns to be that the sum of its kinetic en-ergy, potential energy, and enthalpy, rather than the total energy, becomes positive. This is so-called theenthalpy model (see Ivanova et al. 2013 for a detailed discussion). Assuming that the velocity of gas atinfinity is zero, the binding energy is expressed as E bind = Z M M core (cid:20) − GM ( r ) r + U + Pρ (cid:21) dm. (4)Since the P/ρ term is always non-negative and orders of magnitude larger than U , the absolute valueof E bind decreases substantially in this case. One should be cautious that quasi-stationary mass outflowonly develops when the envelope experiences a slow self-regualted phase during the spiral-in stage, thatis, the spiral-in phase proceeds on a thermal timescale (Ivanova et al. 2013).de Kool (1990) proposed a convenient way to evaluate the binding energy by introducing a param-eter λ to characterizing the central concentration of the donor’s envelope, E bind = − GM M env λa i r L , (5)where M env = M − M core is the mass of the envelope, r L = R L /a i is the ratio of the donor’s RLradius and the orbital separation at the onset of CE. Typically, a i r L is taken to be the stellar radius oncea star fills its RL. Thus the post-CE separation can be determined by inserting Eq. (5) into Eq. (1), a f a i = M core M M M + 2 M env /α CE λr L . (6)It should be emphasized that both α CE and λ are variables depending on stellar and binary parame-ters, although they have been treated as constant ( < ) in most of the population synthesis calculations,due to both poor understanding of them and convenience for calculation. However, many studies (e.g.Dewi & Tauris 2000; Podsiadlowski et al. 2003; Webbink 2008; Xu & Li 2010a,b; Wong et al. 2014)have shown that λ varies as the star evolves and can deviate far from a constant value (say, 0.5). Someinvestigations also suggested that α CE may depend on the binary parameters such as the componentmass and the orbital period (e.g., Taam & Sandquist 2000; Podsiadlowski et al. 2003; De Marco et al.2011; Davis et al. 2012).Systematic calculations of the values of λ have been performed by Dewi & Tauris (2000),Podsiadlowski et al. (2003), and Xu & Li (2010a,b). In the latter, fitting formulae for λ have also beenprovided so they can be incorporated into population synthesis investigations. In this work we re-visitthis problem and provide more reliable λ values by taking into account the following factors.First, we adopt the Modules for Experiments in Stellar Astrophysics (MESA) code (Paxton et al.2011, 2013, 2015) to calculate stellar evolution, which is more powerful in probing the stellar structure he Binding Energy Parameter λ for CE Evolution 3 than Eggleton (1971)’s evolution code EV previously adopted by Xu & Li (2010a,b). Employing modernsoftware engineering tools and techniques allows MESA to consistently evolve stellar models throughchallenging phases for stellar evolution codes in the past, for example, the He core flash in low-mass starsand advanced nuclear burning in massive stars (Paxton et al. 2011). It also adopts denser grids for stellarstructure than the EV code. We find that stars appear to be generally less compact after evolving offmain sequence when modeled with MESA compared with the EV code. The structure of the hydrogen-burning shell, which is near the defined core-envelope boundary, plays a vital role in determining thevalue of λ .Second, besides the traditional λ related to gravitational energy and internal energy, we also calcu-late the values of λ in the enthalpy prescription.Third, it is well known that the λ -value is sensitive to the definition of the core-envelope boundary(see Ivanova et al. 2013 for a detailed discussion). It was arbitrarily assumed to be the (10 − hy-drogen layer in Dewi & Tauris (2000) and Xu & Li (2010a). Ivanova (2011) proposed that this boundaryshould be defined in the hydrogen shell which has the maximum local sonic velocity (i.e., the maximalcompression) prior to CE evolution. This criterion comes from the study of the outcome of the CE eventand the fact that a He core would experience a post-CE thermal readjustment phase, and presents a moreself-consistent definition of the core-envelope boundary.This paper is organized as follows. In section 2, we briefly describe the stellar models and assump-tions adopted. We present the calculated results and fitting formulae for λ in section 3. Our conclusionsare in section 4. We adopted an updated version (7624) of the MESA code to calculate the binding energy parameter λ for stars with initial masses in the range of − M ⊙ . We consider Pop. I stars with the chemicalcompositions of X = 0 . and Z = 0 . . Our previous study has shown that there is not significantchange in the values of λ for Pop. I and II stars (Xu & Li 2010a,b).It has been recognized that stellar winds play an important role in determining the λ parameter,especially for massive stars (Podsiadlowski et al. 2003). Here, we adopt two prescriptions for the windmass loss rates. The first one, denoted as Wind1, is same as in Hurley et al. (2000) and Vink (2001) (forO and B stars), and the second, denoted as Wind2, takes the maximum value of the above loss rates inall the evolutionary stages, to be consistent with Xu & Li (2010a,b).The Wind1 prescription is described as follows:(1) Stellar wind mass loss described by Nieuwenhuijzen & de Jager (1990): ˙ M NJ ( M ⊙ yr − ) = 9 . × − R . L . M . , (7)where M , R , and L are the stellar mass, radius, and luminosity in solar units, respectively.(2) Wind loss from giant branch stars by Kudritzki & Reimers (1978): ˙ M R ( M ⊙ yr − ) = 2 × − LRM . (8)(3) Wind loss from AGB stars by Vassiliadis & Wood (1993): log ˙ M VW ( M ⊙ yr − ) = − . . P −
100 max( M − . , . , (9)where log P = min(3 . , − . − . M + 1 .
94 log R ) . (10)The maximum wind loss rate in this prescription is limited to ˙ M VW , max = 1 . × − L M ⊙ yr − . (11) Wang, Jia & Li R (R ⊙ ) l o g λ ⊙ R (R ⊙ ) l o g λ ⊙ R (R ⊙ ) l o g λ ⊙ R (R ⊙ ) l o g λ ⊙ R (R ⊙ ) l o g λ ⊙ R (R ⊙ ) l o g λ ⊙ R (R ⊙ ) l o g λ
10 M ⊙ R (R ⊙ ) l o g λ
12 M ⊙ R (R ⊙ ) l o g λ
15 M ⊙ Fig. 1
Evolution of the binding energy parameters λ with the stellar radius R for − M ⊙ stars. The red, blue and green lines represent λ h , λ b , and λ g , and the solid and dashed linesrepresent the results with the Wind1 and Wind2 prescriptions, respectively.(4) Wolf-Rayet-like mass loss by Hamann et al. (1995) and Hamann & Koesterke (1998): ˙ M WR ( M ⊙ yr − ) = 10 − L . (1 . − µ ) M ⊙ yr − , (12)with µ = ( M − M core M )min { . , max[1 . , ( LL ) κ ] } , (13)where L = 7 . × , and κ = − . .(5) O and B type star’s wind loss according to Vink (2001): log ˙ M OB ( M ⊙ yr − ) = − . ± . . ± . L/ ) − . ± . M/ − . ± . v ∞ /v esc . . ± . T eff / − . ± .
90) log ( T eff / } , with v ∞ /v esc = 2 . for 27500 K < T eff ≤ (where T eff is the effective temperature). log ˙ M OB ( M ⊙ yr − ) = − . ± . . ± . L/ ) − . ± . M/ − . ± . v ∞ /v esc . . ± .
10) log ( T eff / , he Binding Energy Parameter λ for CE Evolution 5 R (R ⊙ ) l o g λ
18 M ⊙ R (R ⊙ ) l o g λ
20 M ⊙ R (R ⊙ ) l o g λ
25 M ⊙ R (R ⊙ ) l o g λ
30 M ⊙ R (R ⊙ ) l o g λ
35 M ⊙ R (R ⊙ ) l o g λ
40 M ⊙ R (R ⊙ ) l o g λ
45 M ⊙ R (R ⊙ ) l o g λ
50 M ⊙ R (R ⊙ ) l o g λ
60 M ⊙ Fig. 2
Same as Fig. 1 but for − M ⊙ stars.with v ∞ /v esc = 1 . for 12500 K ≤ T eff ≤ .The mass loss in the Wind2 prescription is taken to be ˙ M = max( ˙ M NJ , ˙ M R , ˙ M VW , ˙ M WR , ˙ M OB ) . (14)We ignore the effect of stellar rotation in the calculation, because CE evolution usually occurs whenthe donor star has already entered the giant phase with slow rotation. We have calculated the values of different λ s for − M ⊙ stars by combining Eqs. [2]-[5], consideringgravitational binding energy alone, total energy, and total energy plus enthalpy in the stellar envelopeseparately. We denote them as λ g , λ b and λ h , respectively, that is, − GM M env λ g a i r L = − Z M M core GM ( r ) r dm, − GM M env λ b a i r L = Z M M core (cid:20) − GM ( r ) r + U (cid:21) dm, and − GM M env λ h a i r L = Z M M core (cid:20) − GM ( r ) r + U + Pρ (cid:21) dm. Wang, Jia & Li (R ⊙ )1.51.00.50.00.51.01.52.02.53.0 l o g λ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ λ h λ b (R ⊙ )2.52.01.51.00.50.00.5 l o g λ h ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ Wind1Wind2
Fig. 3
Comparison of the values of λ for different stars. The left panel the solid and dashedlines represent λ h and λ b for stars less massive than M ⊙ with the Wind1 prescription,respectively. The right panel shows λ h as a function of the stellar radius for stars more massivethan M ⊙ . The solid and dashed lines represent the results with the Wind1 and Wind2prescriptions, respectively.More massive stars may lose most of their envelope through strong winds so CE evolution is not likelyto occur. For each star, we follow its evolution until it ascends the thermally pulsating asymptotic giantbranch (TP-AGB) where the star initiates repeating expansions and contractions (for stars > M ⊙ ), orthe code crushes automatically. The binding energy between the envelope and the maximum compres-sion point in the hydrogen burning shell is calculated once a star evolves to produce such a shell. Here,the maximum compression point is the place with the highest local sonic velocity (i.e., the largest valuefor P/ρ in the shell).Figures 1 and 2 show the evolution of λ s with respect to the stellar radius R for stars with differentmasses. The solid and dashed lines represent the results under the Wind1 and Wind2 prescriptions,respectively. The green, blue, and red lines correspond to λ g , λ b and λ h respectively in each case. Ingeneral they demonstrate similar evolutionary trend, with λ b being roughly twice as large as λ g , whichis a natural consequence of the Viral theorem, and λ h being several times larger than λ b . For stars withmass ∼ M ⊙ or > M ⊙ , λ s decrease constantly along the evolutionary tracks, while for stars withmass in between, λ s decrease with increasing R at first and then increase when they have ascended theAGB and developed a deep convective envelope (see also Podsiadlowski et al. 2003). Most interestingly,for ∼ − M ⊙ stars, λ h (and λ b in some cases) increases drastically in the supergiant phase, andcan reach a “boiling pot” zone (see Han et al. 1994; Ivanova 2011), where the binding energy becomespositive before it expands to the reach the maximum radius.The differences between the solid and dashed lines demonstrate the influence of wind loss on themass and the compactness of the envelope, especially near the core-envelope boundary. We can seethat for stars of mass . M ⊙ (except M ⊙ ), or stars of mass ∼ − M ⊙ but with radius . − R ⊙ , the λ -values in the Wind1 case roughly coincide with those in the Wind2 case,reflecting that the two prescriptions are almost the same in such situations. In other cases, the λ -valuesin the Wind2 case are usually smaller than in the Wind1 case because of steeper density profile in theenvelope (see also Podsiadlowski et al., 2003).To see how the binding energy changes with stellar mass, we compare λ h and λ b as a function of R for stars with different masses in Fig. 3. Generally more massive stars have smaller λ , implying thatejection of the envelope is more difficult. This feature is particularly important for the formation of blackhole low-mass X-ray binaries (Justham et al., 2006; Wang, Jia, & Li, 2016, and references there in). he Binding Energy Parameter λ for CE Evolution 7 Our calculated binding energy parameters λ g and λ b evolve in a way in general accord with in pre-vious studies. However, there are some remarkable differences in specific circumstances. (1) Comparingwith Xu & Li (2010a,b), we find that for − M ⊙ stars the values of λ g and λ b increase more rapidlywith radius during the AGB stage. For example, for a M ⊙ star, λ b ≃ at R ⊙ in Xu & Li(2010a,b), but ≃ in our case. (2) Our calculations show that the λ -values increase with radius at thevery end of the evolutionary stages for stars less massive than ∼ M ⊙ (except for M ⊙ star), whilethis upper mass limit becomes lower, i.e., ∼ M ⊙ in Xu & Li (2010a,b) and Podsiadlowski et al.(2003). (3) The λ -values do not show a significant decline at the very end of the evolution for stars moremassive than ∼ M ⊙ as observed by Podsiadlowski et al. (2003).Finally, similar as Xu & Li (2010a,b) we perform polynomial fitting for the calculated λ -values, log λ = a + a x + a x + a x + a x + a x + a x , (15)where x = R/R ⊙ . For − M ⊙ stars that have “hook”-like features in λ , we divide its post-main-sequence evolution into three stages, and fit the λ -values separately. Stage 1 begins at the exhaustion ofcentral hydrogen and ends when the star starts to shrink (i.e., near the ignition of central He). Stage 2is the following shrinking phase, and in stage 3 the star expands again, until the end of the evolution.In Table 1, we list the fitting parameters. We use the coefficient of determination R (i.e., the ratioof the regression sum of squares to the total sum of squares) to evaluate the goodness of fit: R = 1 corresponds to perfect fit, while R = 0 indicates that the equation does not fit the data at all. In ourfitting results, the values of R in all the cases are above 0.95.Table 1: Fitting parameters for λ . Mass ( M ⊙ ) Wind loss stage λ a a a a a a a λ h λ b λ g -0.229504942 -0.006642966 -7.90E-05 3.36E-06 -4.09E-08 2.18E-10 -4.30E-131 Wind2 λ h λ b λ g -0.226240947 -0.007668935 -3.70E-05 2.17E-06 -2.67E-08 1.37E-10 -2.60E-132 Wind1 1 λ h λ b -0.055628327 -0.020278881 0.006650167 -0.000473348 1.49E-05 -2.18E-07 1.23E-09 λ g -0.365205393 -0.008048466 0.004718683 -0.000366995 1.19E-05 -1.79E-07 1.02E-092 λ h λ b λ g -0.318594414 0.01353053 -0.001715174 6.52E-05 -7.02E-07 -6.60E-09 1.24E-103 λ h λ b λ g -0.278858406 -0.011175318 0.000408553 -6.72E-06 5.31E-08 -1.95E-10 2.66E-133 Wind1 1 λ h λ b λ g λ h -34.47232944 8.284562942 -0.792027265 0.039213067 -0.001063886 1.50E-05 -8.66E-08 λ b -21.26420026 4.998641287 -0.47435718 0.023385099 -0.000633095 8.94E-06 -5.15E-08 λ g -17.03794141 3.912297317 -0.369651011 0.018176049 -0.000491475 6.94E-06 -4.00E-083 λ h λ b -0.304912365 0.016425641 -0.000214159 9.87E-07 1.30E-09 -2.05E-11 3.98E-14 λ g -0.556787574 0.013693598 -0.000219799 1.53E-06 -4.85E-09 5.74E-12 5.16E-174 Wind1 1 λ h λ b λ g λ h -158.2144537 21.3665151 -1.175629306 0.033828851 -0.000537603 4.48E-06 -1.53E-08 λ b -113.5126224 15.15018357 -0.827530643 0.023676471 -0.000374536 3.11E-06 -1.06E-08 λ g -95.30490909 12.61279533 -0.686025008 0.019562306 -0.000308629 2.56E-06 -8.69E-093 λ h λ b -0.865298169 0.023395321 -0.000167028 2.40E-07 2.51E-09 -1.07E-11 1.22E-14 λ g -1.318602631 0.035303846 -0.000452023 2.89E-06 -9.71E-09 1.63E-11 -1.08E-146 Wind1 1 λ h -0.080220619 0.018717693 -0.001599316 4.43E-05 -5.53E-07 3.21E-09 -7.08E-12 λ b -0.181116753 -0.007930443 -0.000669357 2.93E-05 -4.24E-07 2.63E-09 -6.00E-12 λ g -0.333386602 -0.022017403 -0.00015568 2.03E-05 -3.38E-07 2.21E-09 -5.17E-122 λ h λ b λ g λ h Wang, Jia & Li
Table 1: Fitting parameters for λ (continued). Mass ( M ⊙ ) Wind loss stage λ a a a a a a a λ b λ g λ h -0.035692021 0.005572222 -0.000485211 8.19E-06 -5.94E-08 1.97E-10 -2.46E-13 λ b -0.167246715 -0.00963629 -0.000147715 4.83E-06 -4.17E-08 1.50E-10 -1.95E-13 λ g -0.332644809 -0.019069101 6.84E-05 2.52E-06 -2.87E-08 1.12E-10 -1.52E-132 λ h -2.365068424 0.124227402 -0.002735286 2.83E-05 -1.50E-07 3.93E-10 -4.04E-13 λ b -4.533738779 0.212107478 -0.004366164 4.29E-05 -2.17E-07 5.44E-10 -5.37E-13 λ g -5.641282203 0.250488572 -0.005065902 4.90E-05 -2.44E-07 6.02E-10 -5.86E-133 λ h λ b λ g λ h -0.018057871 -0.000377557 -0.000140871 1.79E-06 -9.01E-09 2.03E-11 -1.70E-14 λ b -0.190593228 -0.008930875 8.17E-06 6.41E-07 -4.49E-09 1.14E-11 -1.01E-14 λ g -0.380447742 -0.015346001 0.000118835 -2.39E-07 -8.97E-10 4.20E-12 -4.47E-152 λ h -3.620269125 0.112281354 -0.001478372 9.22E-06 -2.96E-08 4.74E-11 -2.99E-14 λ b -4.030432414 0.107086292 -0.001308216 7.39E-06 -2.07E-08 2.77E-11 -1.36E-14 λ g -4.196331979 0.09768744 -0.001127814 5.80E-06 -1.38E-08 1.33E-11 -2.25E-153 λ h λ b -1.908657748 0.037327062 -0.000439988 2.22E-06 -5.37E-09 6.26E-12 -2.83E-15 λ g -2.592139763 0.048484157 -0.000573382 2.98E-06 -7.57E-09 9.32E-12 -4.48E-1512 Wind1 1 λ h λ b -0.179897401 -0.007866116 1.32E-05 2.14E-07 -1.20E-09 2.31E-12 -1.53E-15 λ g -0.382469592 -0.01241696 6.94E-05 -1.11E-07 -2.31E-10 8.69E-13 -6.91E-162 λ h -3527.277445 49.22217981 -0.28437734 0.000870754 -1.49E-06 1.35E-09 -5.09E-13 λ b -4066.073674 56.0114003 -0.319954463 0.000969959 -1.65E-06 1.48E-09 -5.54E-13 λ g -4223.887667 57.81457007 -0.328432188 0.000990866 -1.67E-06 1.50E-09 -5.60E-133 λ h -1123.405701 13.39173279 -0.065527774 0.000168289 -2.39E-07 1.79E-10 -5.48E-14 λ b -650.9073393 7.754985137 -0.037961302 9.75E-05 -1.39E-07 1.03E-10 -3.17E-14 λ g -408.154727 4.861180309 -0.02384045 6.14E-05 -8.74E-08 6.54E-11 -2.01E-1415 Wind1 1 λ h -0.028115873 -0.002865229 -2.06E-05 1.88E-07 -5.34E-10 6.45E-13 -2.85E-16 λ b -0.189058198 -0.006845831 1.38E-05 6.09E-08 -2.87E-10 4.01E-13 -1.89E-16 λ g -0.384916072 -0.0111889 5.15E-05 -8.88E-08 2.15E-11 8.41E-14 -6.07E-172 λ h -43390.80866 415.5416274 -1.652720808 0.003494469 -4.14E-06 2.61E-09 -6.84E-13 λ b -38341.9421 366.9224492 -1.458469908 0.00308221 -3.65E-06 2.30E-09 -6.03E-13 λ g -30256.13593 289.1537559 -1.14796062 0.002423327 -2.87E-06 1.81E-09 -4.72E-133 λ h λ b λ g λ h -0.011555316 -0.00112965 -3.41E-05 1.73E-07 -3.37E-10 2.89E-13 -9.07E-17 λ b -1.93E-01 -0.005828186 1.91E-06 5.64E-08 -1.57E-10 1.58E-13 -5.41E-17 λ g -4.28E-01 -0.010179044 3.42E-05 -4.69E-08 4.25E-12 3.87E-14 -2.07E-17Wind2 λ h -0.085914616 0.000704026 -4.89E-05 2.11E-07 -3.67E-10 2.83E-13 -7.97E-17 λ b -0.218521071 -0.004292671 -1.54E-05 1.15E-07 -2.30E-10 1.90E-13 -5.53E-17 λ g -0.40399817 -0.009339995 1.81E-05 1.74E-08 -9.16E-11 9.45E-14 -3.03E-1720 Wind1 λ h λ b -0.236292531 -4.92E-03 -6.45E-07 4.58E-08 -1.07E-10 9.55E-14 -2.93E-17 λ g -0.57266423 -0.006998902 1.41E-05 1.46E-09 -4.40E-11 5.30E-14 -1.86E-17Wind2 λ h -0.084425289 -0.000248147 -3.41E-05 1.34E-07 -2.05E-10 1.36E-13 -3.30E-17 λ b -0.352025741 -0.001677298 -2.49E-05 1.09E-07 -1.72E-10 1.16E-13 -2.83E-17 λ g -0.684632138 -0.00347904 -1.36E-05 7.81E-08 -1.30E-10 9.01E-14 -2.21E-1725 Wind1 λ h -0.079805635 -0.001539374 -1.62E-05 6.42E-08 -9.17E-11 5.65E-14 -1.26E-17 λ b -0.217137659 -0.005291635 4.59E-06 1.57E-08 -3.82E-11 2.89E-14 -7.22E-18 λ g -0.445989682 -0.009416417 2.67E-05 -3.53E-08 1.83E-11 -7.71E-16 -1.27E-18Wind2 λ h -1.21E-01 -7.73E-04 -2.46E-05 8.75E-08 -1.17E-10 6.73E-14 -1.40E-17 λ b -2.26E-01 -4.80E-03 -2.55E-06 3.71E-08 -6.20E-11 3.94E-14 -8.66E-18 λ g -4.07E-01 -9.64E-03 2.31E-05 -2.12E-08 1.77E-12 6.23E-15 -2.05E-1830 Wind1 λ h -0.068721052 -0.004439298 7.66E-06 -5.86E-09 -9.70E-13 3.75E-15 -1.41E-18 λ b -0.168597281 -0.008185857 2.91E-05 -5.99E-08 6.65E-11 -3.71E-14 8.18E-18 λ g -0.360619104 -0.013289713 5.80E-05 -1.33E-07 1.57E-10 -9.18E-14 2.10E-17Wind2 λ h -2.64E-02 -8.04E-03 3.08E-05 -6.72E-08 7.72E-11 -4.41E-14 9.86E-18 λ b -1.16E-01 -1.10E-02 4.60E-05 -1.02E-07 1.18E-10 -6.72E-14 1.50E-17 λ g -2.79E-01 -1.57E-02 7.00E-05 -1.58E-07 1.83E-10 -1.04E-13 2.30E-1735 Wind1 λ h -0.086834231 -0.005683759 1.61E-05 -2.79E-08 2.67E-11 -1.29E-14 2.48E-18 λ b -0.178466739 -0.008221884 2.75E-05 -5.09E-08 4.98E-11 -2.42E-14 4.64E-18 λ g -0.368625486 -0.012306112 4.58E-05 -8.80E-08 8.71E-11 -4.25E-14 8.12E-18Wind2 λ h λ b -0.082606521 -1.23E-02 4.72E-05 -9.18E-08 9.13E-11 -4.46E-14 8.48E-18 λ g -0.244193403 -0.01566844 6.06E-05 -1.17E-07 1.15E-10 -5.57E-14 1.05E-1740 Wind1 λ h -0.113572735 -0.006236406 1.80E-05 -2.96E-08 2.58E-11 -1.13E-14 1.94E-18 λ b -0.183810999 -0.008309639 2.59E-05 -4.35E-08 3.81E-11 -1.66E-14 2.83E-18 he Binding Energy Parameter λ for CE Evolution 9 Table 1: Fitting parameters for λ (continued). Mass ( M ⊙ ) Wind loss stage λ a a a a a a a λ g -0.349966999 -0.012034168 4.03E-05 -6.88E-08 6.05E-11 -2.62E-14 4.45E-18Wind2 λ h λ b -1.10E-01 -1.12E-02 3.43E-05 -5.22E-08 4.05E-11 -1.55E-14 2.30E-18 λ g -3.19E-01 -1.28E-02 3.90E-05 -5.91E-08 4.58E-11 -1.75E-14 2.62E-1845 Wind1 λ h -0.09033098 -0.0078733 2.16E-05 -3.12E-08 2.35E-11 -8.79E-15 1.29E-18 λ b -0.165033235 -0.00897789 2.51E-05 -3.64E-08 2.74E-11 -1.03E-14 1.50E-18 λ g -0.334563352 -0.011618318 3.36E-05 -4.94E-08 3.75E-11 -1.40E-14 2.06E-18Wind2 λ h -5.84E-02 -9.31E-03 2.27E-05 -2.86E-08 1.87E-11 -6.06E-15 7.70E-19 λ b -1.76E-01 -9.24E-03 2.22E-05 -2.76E-08 1.79E-11 -5.77E-15 7.29E-19 λ g -4.28E-01 -9.41E-03 2.14E-05 -2.57E-08 1.62E-11 -5.09E-15 6.30E-1950 Wind1 λ h -0.077847412 -0.008264901 2.00E-05 -2.52E-08 1.65E-11 -5.39E-15 6.94E-19 λ b -0.173570298 -0.008589709 2.07E-05 -2.60E-08 1.71E-11 -5.58E-15 7.20E-19 λ g -0.387442471 -0.009858782 2.40E-05 -3.02E-08 1.99E-11 -6.55E-15 8.51E-19Wind2 λ h -7.61E-01 -3.18E-03 4.45E-06 -4.66E-09 3.00E-12 -9.92E-16 1.29E-19 λ b -8.68E-01 -2.98E-03 3.68E-06 -3.56E-09 2.26E-12 -7.58E-16 1.00E-19 λ g -1.18E+00 -2.19E-03 5.71E-07 8.88E-10 -7.72E-13 2.32E-16 -2.38E-2060 Wind1 λ h -0.137506172 -0.008191471 2.72E-05 -5.69E-08 6.76E-11 -4.18E-14 1.04E-17 λ b -0.232113077 -0.008216679 2.75E-05 -5.78E-08 6.90E-11 -4.28E-14 1.07E-17 λ g -0.475332579 -0.008678247 3.00E-05 -6.42E-08 7.76E-11 -4.86E-14 1.22E-17Wind2 λ h -0.777181069 -0.002255006 -2.51E-06 9.08E-09 -8.60E-12 3.46E-15 -5.13E-19 λ b -0.843092017 -0.002433256 -2.10E-06 8.63E-09 -8.34E-12 3.39E-15 -5.04E-19 λ g -1.055314774 -0.00279245 -2.09E-06 9.46E-09 -9.29E-12 3.80E-15 -5.66E-19 The binding energy parameter λ is a key parameter in the formation and evolution of close binarysystems. This work is an updated version of Xu & Li (2010a,b), with more self-consistent treatments instellar modeling. The main results are summarized as follows.1. The λ -values vary when a star evolves and strongly depends on the star’s initial mass. It generallydecreases with the increasing stellar radius, but rises at the very end of the evolution for stars lessmassive than ∼ M ⊙ .2. More massive stars tend to have smaller λ . For massive stars ( & M ⊙ ) the λ -values are sub-stantially influenced by the wind mass loss.3. Generally, λ h is several times larger than λ b and λ g , which can assist the ejection of the CE. Forstars in the mass range of ∼ − M ⊙ , the λ h -values can be very large ( > ) and even negativebefore the star reaches its maximum size.4. Our fitting formulae for λ s can serve as useful input parameters in population synthesis investi-gations. Acknowledgements
We are grateful to an anonymous referee for helpful comments. This work wasfunded by the Natural Science Foundation of China under grant numbers 11133001 and 11333004, andthe Strategic Priority Research Program of CAS (under grant number XDB09000000).
References
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