Dewi, J.D.M.

Tauris, Th.M.

International Migration, Integration and Social Cohesion online publications

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)On the binding energy of the common envelope evolution. Dependency on thedefinition of the stellar core boundary during spiral-in.Tauris, Th.M.; Dewi, J.D.M.Publication date2001Published inAstronomy & AstrophysicsLink to publicationCitation for published version (APA):Tauris, T. M., & Dewi, J. D. M. (2001). On the binding energy of the common envelopeevolution. Dependency on the definition of the stellar core boundary during spiral-in.Astronomy & Astrophysics, 1, 170-173.General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:11 Mar 2023A&A 369, 170–173 (2001)DOI: 10.1051/0004-6361:20010099c© ESO 2001Astronomy&AstrophysicsResearch NoteOn the binding energy parameter of common envelope evolutionDependency on the definition of the stellar core boundary during spiral-inT. M. Tauris1 and J. D. M. Dewi2,3,41 Nordic Institute for Theoretical Physics (NORDITA), Blegdamsvej 17, 2100 Copenhagen Ø, Denmark2 Astronomical Institute Anton Pannekoek, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam,The Netherlands3 Bosscha Observatory, Lembang 40391, Bandung, Indonesia4 Department of Astronomy, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesiae-mail: jasinta@astro.uva.nlReceived 9 November 2000 / Accepted 9 January 2001Abstract. According to the standard picture for binary interactions, the outcome of binaries surviving theevolution through a common envelope (CE) and spiral-in phase is determined by the internal structure of thedonor star at the onset of the mass transfer, as well as the poorly-known efficiency parameter, ηCE, for the ejectionof the H-envelope of the donor. In this Research Note we discuss the bifurcation point which separates the ejected,unprocessed H-rich material from the inner core region of the donor (the central part of the star which will latercontract to form a compact object). We demonstrate that the exact location of this point is very important forevaluating the binding energy parameter, λ, which is used to determine the post-CE orbital separation. Here wecompare various methods to define the bifurcation point (core/envelope boundary) of evolved stars with masses4, 7, 10 and 20 M. We consider the specific nuclear energy production rate profile, the change in the mass-density gradient (Bisscheroux 1998), the inner region containing less than 10% hydrogen, the method suggestedby Han et al. (1994) and the entropy profile. We also calculated effective polytropic index profiles. The entropyprofile method measures the convective boundary (at the onset of flatness in the specific entropy) which is notequivalent to the core boundary for RGB stars. Hence, this method is not applicable for RGB stars, unless theactual bifurcation point of a CE is located at the bottom of the outer convection zone (resulting in larger values ofλ and larger post-CE orbital separations). On the AGB, where highly degenerate and condensed cores are formed,we find good agreement between the various methods, except for massive (∼20 M) stars.Key words. stars: evolution – stars: mass loss – binaries: general1. IntroductionThe simple formalism introduced by Webbink (1984) andde Kool (1990) to estimate the orbital decay of binariesevolving through a CE and spiral-in evolution requiresknowledge about the core mass, Mcore, inside the bifur-cation point of the donor star as well as the λ-parameter,which is a numerical factor introduced in order to correcttheir simple formula for estimating the binding energy ofthe stellar envelope. It has been common practice to usea constant value of λ = 0.5 for the λ-parameter. We havedemonstrated (Dewi & Tauris 2000) that this is not a goodapproach, since λ varies strongly throughout the evolutionof a star. We concluded that all observations of binary pul-sars originating from a CE evolution are consistent withSend offprint requests to: T. M. Tauris,e-mail: tauris@nordita.dkηCE ≤ 1, as a result of large values of λ possible when theinternal thermodynamic energy is included.We also raised the important question of how to definethe core mass boundary – i.e. once the spiral-in processis initiated, and frictional torques deposit kinetic energyin the envelope, where is the exact location of the pointof bifurcation in the envelope which separates the ejectedmaterial from the remaining condensed core region? Thisquestion is the key issue in this Research Note. It is neces-sary to knowMcore accurately, since it is used to determinethe exact value of λ, which is a strongly increasing func-tion exactly at the core/envelope transition. Hence, theestimates of the post-CE orbital separation also dependstrongly on the value of Mcore.We refer the reader to Dewi & Tauris (2000) and ref-erences therein for further details on the topic and therelevant energy equations and parameters.T. M. Tauris and J. D. M. Dewi: On the binding energy parameter of common envelope evolution 171Fig. 1. The internal structure of a 10 M star at the tip of the RGB (left) and at the tip of the AGB (right). See text2. Structure profiles of evolved starsIn Fig. 1 we show an example of our calculations of thestellar structure for a 10 M star. The results for a 4, 7and 20 M star are summarized in Table 1. We used achemical composition of X = 0.70 and Z = 0.02, and amixing-length parameter of α = l/Hp = 2.0. Convective172 T. M. Tauris and J. D. M. Dewi: On the binding energy parameter of common envelope evolutionTable 1. The values of Mcore/M, λg and λb estimated from calculations of four different stars – see text for discussionM = 4.0 M M = 7.0 M M = 10.0 M M = 20.0 Mtip of RGB tip of AGB AGB tip of RGB tip of AGB tip of AGBR = 67 R R = 1040 R R = 374 R R = 374 R R = 588 R R = 1040 Rmethod Mcore λg λb Mcore λg λb Mcore λg λb Mcore λg λb Mcore λg λb Mcore λg λbmax εnuc 0.58 0.32 0.62 1.37 0.91 — 1.58 0.10 0.20 1.99 0.09 0.13 2.47 0.06 0.10 6.80 0.02 0.05X < 0.10 0.59 0.36 0.70 1.37 0.91 — 1.80 0.59 2.85 2.08 0.11 0.21 2.84 0.40 1.80 6.80 0.02 0.05∂2 log ρ/∂m2 = 0 0.68 0.66 1.60 1.37 0.91 — 1.80 0.59 2.85 2.54 0.54 1.80 2.85 0.46 2.20 7.50 0.22 0.95Han et al. 0.64 0.59 1.17 1.37 0.91 — 1.80 0.59 2.85 2.52 0.51 1.75 2.85 0.46 2.20 7.60 0.32 1.70entropy profile 0.74 0.73 1.81 1.37 0.91 — 1.80 0.59 2.85 2.62 0.60 2.23 2.86 0.55 3.20 7.80 0.45 3.50overshooting was also taken into account. Here we used anovershooting constant of δov = 0.10 (Pols et al. 1998). Weobtained almost the same core masses using δov = 0.12.For the 10 and 20 M star we assumed a wind mass-loss rate according to de Jager (de Jager et al. 1988;Nieuwenhuijzen & de Jager 1990). Hence, at the tip of theAGB these two stars are estimated to have evaporateddown to 9.59 and 16.2 M, respectively. We assumed nomass loss from the 4 and 7 M stars.We will now briefly discuss the calculations representedin Fig. 1. In the bottom panels the two solid lines showour calculations of λ. We distinguish λb (calculated fromthe total binding energy) from λg (as derived from gravi-tational binding energy alone) – see Dewi & Tauris (2000).2.1. Energy production rate and chemical compositionA very crude definition of the stellar core region is sim-ply as the inner part containing less than a certain massfraction of hydrogen. As an example, we used X < 0.10 inDewi & Tauris (2000) to define the core of the donor star.This is shown in the top panels of Fig. 1 as the area tothe left of the gray shaded region. The dotted line is thecore radius as a function of its mass.A somewhat more physically meaningful approach (al-though not necessarily more correct in practice) is to de-fine the core region below the (outer) shell burning source– for example, below the point of maximum energy pro-duction inside the H-burning shell. In the top panels thesolid line shows the local nuclear energy production rate.The shell burning regions are clearly seen. The most abun-dant local chemical elements are written in the panel. Ascan be seen from the figure and Table 1, this definitionsometimes yields significantly lower values of Mcore.2.2. The binding energy profileThe solid lines in the second panel are sinh−1 (∆W ),where ∆W = Egrav + Eint or ∆W = Egrav is the bind-ing energy of the envelope to the core (with or withoutthe inclusion of the internal thermodynamic energy, re-spectively). Han et al. (1994) introduced this function todefine the core mass boundary. The location is assumed tobe at the transition between the strongly increasing ∆Wand the outer region where ∆W varies slowly with mass.We used the intersection of straight-line fits to find Mcore.2.3. The mass-density gradientThe dotted line in the second panel shows the mass-density profile. The arrow indicates the point where∂2 log ρ/∂m2 = 0. This criterion was used by Bisscheroux(1998) to locate the core boundary. The disadvantage ofthis method, from a technical point of view, is that thereis not always a unique solution to the equation (for smallmass steps).2.4. The entropy profile and effective polytropic indexThe entropy profile for the 10 M star is plotted as asolid line in the third panel. Here “entropy” refers to thelocal specific entropy per baryon in units of the Boltzmannconstant. The arrow indicates the sharp onset of the flatentropy gradient – another criterion for determining thebifurcation point. The core masses at the tip of the RGB,defined by this method, are always larger than those de-rived from other criteria. The clear discrepancy betweenthe entropy profile method and the other alternatives onthe RGB, leading to different estimates of the remainingmass after spiral-in, is easy to understand. The bifurca-tion point obtained from the entropy method is expectedto be located further out, since it is based on the transi-tion between the convective and the radiative layer in thestellar envelope. Unlike evolved low-mass (1 M) stars,these more massive stars do not have an outer convectionzone which penetrates all the way down to the H-shellnear the standard core boundary region (e.g. Kippenhahn& Weigert 1990). Hence, the entropy method results in(too) large a core mass on the RGB.The dotted line in the third panel is the “effective poly-tropic index” defined by: n ≡ 1/(γ − 1) with the “adia-batic” index: γ = ∂ lnP/∂ ln ρ. Giant stars can be con-sidered as a condensed polytrope with a core point masssurrounded by an extended isentropic n = 3/2, γ = 5/3envelope. Less evolved stars are more realistically de-scribed as composite polytropes consisting of n = 3, γ =4/3 cores with n = 3/2, γ = 5/3 envelopes (Hjellming& Webbink 1987). Here, we are mostly interested in theT. M. Tauris and J. D. M. Dewi: On the binding energy parameter of common envelope evolution 173clear discontinuous behaviour of n at the core boundary inorder to determine its exact location. This discontinuityrepresents an important transition in the stellar structureand could be interpreted as the critical point of bifurca-tion at the end of the spiral-in and envelope ejection phase(cf. Sect. 3.2).3. Discussion3.1. Mass transfer initiated on the RGB vs. AGBWe have now demonstrated how different methods resultin different core masses on the RGB and hence differentvalues of λ. As an example, one can consider a 4 M starat the tip of the RGB. Here the maximum energy produc-tion rate and the 10% hydrogen criterion yield core massesof 0.58 and 0.59 M, respectively. The mass-density gra-dient method results in Mcore = 0.68M and the outcomeof applying Han et al.’s method is approximately 0.64M.Using the entropy profile and the effective polytropic in-dex method yields Mcore = 0.74 M. The spread in Mcorefrom the different approaches results in: 0.6 < λb < 1.8and 0.3 < λg < 0.75 (with and without internal thermo-dynamic energy). If we now assume the simple CE energyequations (Webbink 1984; de Kool 1990) to be approxi-mately valid, and this 4 M (R = 67 R) star is in acommon envelope with an in-spiralling 1.3 M neutronstar, the evolution would then result in a final post-CEorbital separation of 0.70 or 2.75 R, for Mcore = 0.58 and0.74M, respectively. Here we assumed λ = λb, ηCE = 1.0and no significant accretion onto the neutron star duringspiral-in. However, the core radius of the stripped star is0.1 R (Mcore = 0.58 M) or 3.5 R (Mcore = 0.74 M).Hence, in the latter case, the large core would then havea radius larger than the predicted post-CE separation.Therefore this system would coalesce during the spiral-in– perhaps leaving behind a black hole. Assuming λ = λg(i.e. considering gravitational binding energy alone), orchoosing a small efficiency parameter (e.g. ηCE = 0.3),would decrease the predicted post-CE separation still fur-ther and lead to an earlier coalescense. We conclude thatknowledge of the accurate bifurcation boundary is crucialto forecast the outcome of a CE evolution on the RGB.Many binaries come into contact while the donor staris expanding enormously on the AGB. If we consider thesame 4 M star as before, but now at the tip of the AGB,all methods yield exactly the same core mass. For the 7and 10 M stars on the AGB, the different methods alsoyield the same core mass (except for the location of themaximum energy production rate in the H-shell, which isbelow this boundary at the outer edge of the shell source).It is important to notice the fact that the envelope bind-ing energy is a strongly increasing quantity near/at thecore boundary on the AGB. Hence, the resulting values ofλ are also increasing sharply at this transition (see Fig. 1).Since the material of the donor star is peeled off from theoutside, we suggest using the (larger) value of lambda atthe mesh point, just outside the bifurcation point.3.2. Envelope ejection processWhile the location of the bifurcation point, separatingthe ejected envelope from the condensed core region, isstraightforward to define on the AGB, it is important toknow for RGB stars if the envelope of the evolved donor isejected at the bottom of the convection zone, or closer tothe stellar center at the core boundary near/at the outershell source (surrounding the inner part of the star). Forconvective material in an isentropic envelope, R ∝M−1/3and thus removal of envelope mass causes the star to ex-pand. Therefore, in a CE system, the envelope is easilypeeled off until the entropy profile increases outwards (inthe deeper radiative layers). From this point on the starwill no longer expand and the envelope may already beseparated from the inner region, depending on the detailsof the ejection process. It would be interesting to see ifmulti-dimensional hydrodynamical calculations could an-swer this question.Binaries which come into Roche-lobe contact early inthe stellar evolution of the donor will not survive thespiral-in process at all (there is simply not enough orbitalenergy available to expel the envelope).The values of λ presented in Table 1 in Dewi & Tauris(2000) were calculated from the 10% hydrogen criterion.From the work presented here we can see that these val-ues should therefore be taken as minimum values of λ,supporting our previous conclusion that ηCE need not belarge since the value of λ itself is expected to be large.Acknowledgements. We thank the referee for improving (butsubstantially shortening) this paper. T. M. T. acknowledgesthe receipt of a NORDITA fellowship. This work was sponsoredby NWO Spinoza Grant 08-0 to E. P. J. van den Heuvel.ReferencesBisscheroux, B. 1998, M.Sc. Thesis, Univ. Amsterdamde Jager, C., Nieuwenhuijzen, H., & van der Hucht, K. A. 1988,A&AS, 72, 259de Kool, M. 1990, ApJ, 358, 189Dewi, J. D. M., & Tauris, T. M. 2000, A&A, 360, 1043Han, Z., Podsiadlowski, P., & Eggleton, P. P. 1994, MNRAS,270, 121Hjellming, M. S., & Webbink, R. F. 1987, ApJ, 318, 794Kippenhahn, R., & Weigert, A. 1990, Stellar Structure andEvolution, A&A Library (Springer-Verlag)Nieuwenhuijzen, H., & de Jager, C. 1990, A&A, 231, 134Pols, O. R., Schröder, K.-P., Hurley, J. R., Tout, C. A., &Eggleton, P. P. 1998, MNRAS, 298, 525Webbink, R. F. 1984, ApJ, 277, 355

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On the binding energy of the common envelope evolution. Dependency on the definition of the stellar core boundary during spiral-in.

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)On the binding energy of the common envelope evolution. Dependency on thedefinition of the stellar core boundary during spiral-in.Tauris, Th.M.; Dewi, J.D.M.Publication date2001Published inAstronomy & AstrophysicsLink to publicationCitation for published version (APA):Tauris, T. M., & Dewi, J. D. M. (2001). On the binding energy of the common envelopeevolution. Dependency on the definition of the stellar core boundary during spiral-in.Astronomy & Astrophysics, 1, 170-173.General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:11 Nov 2022A&A 369, 170–173 (2001)DOI: 10.1051/0004-6361:20010099c© ESO 2001Astronomy&AstrophysicsResearch NoteOn the binding energy parameter of common envelope evolutionDependency on the definition of the stellar core boundary during spiral-inT. M. Tauris1 and J. D. M. Dewi2,3,41 Nordic Institute for Theoretical Physics (NORDITA), Blegdamsvej 17, 2100 Copenhagen Ø, Denmark2 Astronomical Institute Anton Pannekoek, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam,The Netherlands3 Bosscha Observatory, Lembang 40391, Bandung, Indonesia4 Department of Astronomy, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesiae-mail: jasinta@astro.uva.nlReceived 9 November 2000 / Accepted 9 January 2001Abstract. According to the standard picture for binary interactions, the outcome of binaries surviving theevolution through a common envelope (CE) and spiral-in phase is determined by the internal structure of thedonor star at the onset of the mass transfer, as well as the poorly-known efficiency parameter, ηCE, for the ejectionof the H-envelope of the donor. In this Research Note we discuss the bifurcation point which separates the ejected,unprocessed H-rich material from the inner core region of the donor (the central part of the star which will latercontract to form a compact object). We demonstrate that the exact location of this point is very important forevaluating the binding energy parameter, λ, which is used to determine the post-CE orbital separation. Here wecompare various methods to define the bifurcation point (core/envelope boundary) of evolved stars with masses4, 7, 10 and 20 M. We consider the specific nuclear energy production rate profile, the change in the mass-density gradient (Bisscheroux 1998), the inner region containing less than 10% hydrogen, the method suggestedby Han et al. (1994) and the entropy profile. We also calculated effective polytropic index profiles. The entropyprofile method measures the convective boundary (at the onset of flatness in the specific entropy) which is notequivalent to the core boundary for RGB stars. Hence, this method is not applicable for RGB stars, unless theactual bifurcation point of a CE is located at the bottom of the outer convection zone (resulting in larger values ofλ and larger post-CE orbital separations). On the AGB, where highly degenerate and condensed cores are formed,we find good agreement between the various methods, except for massive (∼20 M) stars.Key words. stars: evolution – stars: mass loss – binaries: general1. IntroductionThe simple formalism introduced by Webbink (1984) andde Kool (1990) to estimate the orbital decay of binariesevolving through a CE and spiral-in evolution requiresknowledge about the core mass, Mcore, inside the bifur-cation point of the donor star as well as the λ-parameter,which is a numerical factor introduced in order to correcttheir simple formula for estimating the binding energy ofthe stellar envelope. It has been common practice to usea constant value of λ = 0.5 for the λ-parameter. We havedemonstrated (Dewi & Tauris 2000) that this is not a goodapproach, since λ varies strongly throughout the evolutionof a star. We concluded that all observations of binary pul-sars originating from a CE evolution are consistent withSend offprint requests to: T. M. Tauris,e-mail: tauris@nordita.dkηCE ≤ 1, as a result of large values of λ possible when theinternal thermodynamic energy is included.We also raised the important question of how to definethe core mass boundary – i.e. once the spiral-in processis initiated, and frictional torques deposit kinetic energyin the envelope, where is the exact location of the pointof bifurcation in the envelope which separates the ejectedmaterial from the remaining condensed core region? Thisquestion is the key issue in this Research Note. It is neces-sary to knowMcore accurately, since it is used to determinethe exact value of λ, which is a strongly increasing func-tion exactly at the core/envelope transition. Hence, theestimates of the post-CE orbital separation also dependstrongly on the value of Mcore.We refer the reader to Dewi & Tauris (2000) and ref-erences therein for further details on the topic and therelevant energy equations and parameters.T. M. Tauris and J. D. M. Dewi: On the binding energy parameter of common envelope evolution 171Fig. 1. The internal structure of a 10 M star at the tip of the RGB (left) and at the tip of the AGB (right). See text2. Structure profiles of evolved starsIn Fig. 1 we show an example of our calculations of thestellar structure for a 10 M star. The results for a 4, 7and 20 M star are summarized in Table 1. We used achemical composition of X = 0.70 and Z = 0.02, and amixing-length parameter of α = l/Hp = 2.0. Convective172 T. M. Tauris and J. D. M. Dewi: On the binding energy parameter of common envelope evolutionTable 1. The values of Mcore/M, λg and λb estimated from calculations of four different stars – see text for discussionM = 4.0 M M = 7.0 M M = 10.0 M M = 20.0 Mtip of RGB tip of AGB AGB tip of RGB tip of AGB tip of AGBR = 67 R R = 1040 R R = 374 R R = 374 R R = 588 R R = 1040 Rmethod Mcore λg λb Mcore λg λb Mcore λg λb Mcore λg λb Mcore λg λb Mcore λg λbmax εnuc 0.58 0.32 0.62 1.37 0.91 — 1.58 0.10 0.20 1.99 0.09 0.13 2.47 0.06 0.10 6.80 0.02 0.05X < 0.10 0.59 0.36 0.70 1.37 0.91 — 1.80 0.59 2.85 2.08 0.11 0.21 2.84 0.40 1.80 6.80 0.02 0.05∂2 log ρ/∂m2 = 0 0.68 0.66 1.60 1.37 0.91 — 1.80 0.59 2.85 2.54 0.54 1.80 2.85 0.46 2.20 7.50 0.22 0.95Han et al. 0.64 0.59 1.17 1.37 0.91 — 1.80 0.59 2.85 2.52 0.51 1.75 2.85 0.46 2.20 7.60 0.32 1.70entropy profile 0.74 0.73 1.81 1.37 0.91 — 1.80 0.59 2.85 2.62 0.60 2.23 2.86 0.55 3.20 7.80 0.45 3.50overshooting was also taken into account. Here we used anovershooting constant of δov = 0.10 (Pols et al. 1998). Weobtained almost the same core masses using δov = 0.12.For the 10 and 20 M star we assumed a wind mass-loss rate according to de Jager (de Jager et al. 1988;Nieuwenhuijzen & de Jager 1990). Hence, at the tip of theAGB these two stars are estimated to have evaporateddown to 9.59 and 16.2 M, respectively. We assumed nomass loss from the 4 and 7 M stars.We will now briefly discuss the calculations representedin Fig. 1. In the bottom panels the two solid lines showour calculations of λ. We distinguish λb (calculated fromthe total binding energy) from λg (as derived from gravi-tational binding energy alone) – see Dewi & Tauris (2000).2.1. Energy production rate and chemical compositionA very crude definition of the stellar core region is sim-ply as the inner part containing less than a certain massfraction of hydrogen. As an example, we used X < 0.10 inDewi & Tauris (2000) to define the core of the donor star.This is shown in the top panels of Fig. 1 as the area tothe left of the gray shaded region. The dotted line is thecore radius as a function of its mass.A somewhat more physically meaningful approach (al-though not necessarily more correct in practice) is to de-fine the core region below the (outer) shell burning source– for example, below the point of maximum energy pro-duction inside the H-burning shell. In the top panels thesolid line shows the local nuclear energy production rate.The shell burning regions are clearly seen. The most abun-dant local chemical elements are written in the panel. Ascan be seen from the figure and Table 1, this definitionsometimes yields significantly lower values of Mcore.2.2. The binding energy profileThe solid lines in the second panel are sinh−1 (∆W ),where ∆W = Egrav + Eint or ∆W = Egrav is the bind-ing energy of the envelope to the core (with or withoutthe inclusion of the internal thermodynamic energy, re-spectively). Han et al. (1994) introduced this function todefine the core mass boundary. The location is assumed tobe at the transition between the strongly increasing ∆Wand the outer region where ∆W varies slowly with mass.We used the intersection of straight-line fits to find Mcore.2.3. The mass-density gradientThe dotted line in the second panel shows the mass-density profile. The arrow indicates the point where∂2 log ρ/∂m2 = 0. This criterion was used by Bisscheroux(1998) to locate the core boundary. The disadvantage ofthis method, from a technical point of view, is that thereis not always a unique solution to the equation (for smallmass steps).2.4. The entropy profile and effective polytropic indexThe entropy profile for the 10 M star is plotted as asolid line in the third panel. Here “entropy” refers to thelocal specific entropy per baryon in units of the Boltzmannconstant. The arrow indicates the sharp onset of the flatentropy gradient – another criterion for determining thebifurcation point. The core masses at the tip of the RGB,defined by this method, are always larger than those de-rived from other criteria. The clear discrepancy betweenthe entropy profile method and the other alternatives onthe RGB, leading to different estimates of the remainingmass after spiral-in, is easy to understand. The bifurca-tion point obtained from the entropy method is expectedto be located further out, since it is based on the transi-tion between the convective and the radiative layer in thestellar envelope. Unlike evolved low-mass (1 M) stars,these more massive stars do not have an outer convectionzone which penetrates all the way down to the H-shellnear the standard core boundary region (e.g. Kippenhahn& Weigert 1990). Hence, the entropy method results in(too) large a core mass on the RGB.The dotted line in the third panel is the “effective poly-tropic index” defined by: n ≡ 1/(γ − 1) with the “adia-batic” index: γ = ∂ lnP/∂ ln ρ. Giant stars can be con-sidered as a condensed polytrope with a core point masssurrounded by an extended isentropic n = 3/2, γ = 5/3envelope. Less evolved stars are more realistically de-scribed as composite polytropes consisting of n = 3, γ =4/3 cores with n = 3/2, γ = 5/3 envelopes (Hjellming& Webbink 1987). Here, we are mostly interested in theT. M. Tauris and J. D. M. Dewi: On the binding energy parameter of common envelope evolution 173clear discontinuous behaviour of n at the core boundary inorder to determine its exact location. This discontinuityrepresents an important transition in the stellar structureand could be interpreted as the critical point of bifurca-tion at the end of the spiral-in and envelope ejection phase(cf. Sect. 3.2).3. Discussion3.1. Mass transfer initiated on the RGB vs. AGBWe have now demonstrated how different methods resultin different core masses on the RGB and hence differentvalues of λ. As an example, one can consider a 4 M starat the tip of the RGB. Here the maximum energy produc-tion rate and the 10% hydrogen criterion yield core massesof 0.58 and 0.59 M, respectively. The mass-density gra-dient method results in Mcore = 0.68M and the outcomeof applying Han et al.’s method is approximately 0.64M.Using the entropy profile and the effective polytropic in-dex method yields Mcore = 0.74 M. The spread in Mcorefrom the different approaches results in: 0.6 < λb < 1.8and 0.3 < λg < 0.75 (with and without internal thermo-dynamic energy). If we now assume the simple CE energyequations (Webbink 1984; de Kool 1990) to be approxi-mately valid, and this 4 M (R = 67 R) star is in acommon envelope with an in-spiralling 1.3 M neutronstar, the evolution would then result in a final post-CEorbital separation of 0.70 or 2.75 R, for Mcore = 0.58 and0.74M, respectively. Here we assumed λ = λb, ηCE = 1.0and no significant accretion onto the neutron star duringspiral-in. However, the core radius of the stripped star is0.1 R (Mcore = 0.58 M) or 3.5 R (Mcore = 0.74 M).Hence, in the latter case, the large core would then havea radius larger than the predicted post-CE separation.Therefore this system would coalesce during the spiral-in– perhaps leaving behind a black hole. Assuming λ = λg(i.e. considering gravitational binding energy alone), orchoosing a small efficiency parameter (e.g. ηCE = 0.3),would decrease the predicted post-CE separation still fur-ther and lead to an earlier coalescense. We conclude thatknowledge of the accurate bifurcation boundary is crucialto forecast the outcome of a CE evolution on the RGB.Many binaries come into contact while the donor staris expanding enormously on the AGB. If we consider thesame 4 M star as before, but now at the tip of the AGB,all methods yield exactly the same core mass. For the 7and 10 M stars on the AGB, the different methods alsoyield the same core mass (except for the location of themaximum energy production rate in the H-shell, which isbelow this boundary at the outer edge of the shell source).It is important to notice the fact that the envelope bind-ing energy is a strongly increasing quantity near/at thecore boundary on the AGB. Hence, the resulting values ofλ are also increasing sharply at this transition (see Fig. 1).Since the material of the donor star is peeled off from theoutside, we suggest using the (larger) value of lambda atthe mesh point, just outside the bifurcation point.3.2. Envelope ejection processWhile the location of the bifurcation point, separatingthe ejected envelope from the condensed core region, isstraightforward to define on the AGB, it is important toknow for RGB stars if the envelope of the evolved donor isejected at the bottom of the convection zone, or closer tothe stellar center at the core boundary near/at the outershell source (surrounding the inner part of the star). Forconvective material in an isentropic envelope, R ∝M−1/3and thus removal of envelope mass causes the star to ex-pand. Therefore, in a CE system, the envelope is easilypeeled off until the entropy profile increases outwards (inthe deeper radiative layers). From this point on the starwill no longer expand and the envelope may already beseparated from the inner region, depending on the detailsof the ejection process. It would be interesting to see ifmulti-dimensional hydrodynamical calculations could an-swer this question.Binaries which come into Roche-lobe contact early inthe stellar evolution of the donor will not survive thespiral-in process at all (there is simply not enough orbitalenergy available to expel the envelope).The values of λ presented in Table 1 in Dewi & Tauris(2000) were calculated from the 10% hydrogen criterion.From the work presented here we can see that these val-ues should therefore be taken as minimum values of λ,supporting our previous conclusion that ηCE need not belarge since the value of λ itself is expected to be large.Acknowledgements. We thank the referee for improving (butsubstantially shortening) this paper. T. M. T. acknowledgesthe receipt of a NORDITA fellowship. This work was sponsoredby NWO Spinoza Grant 08-0 to E. P. J. van den Heuvel.ReferencesBisscheroux, B. 1998, M.Sc. Thesis, Univ. Amsterdamde Jager, C., Nieuwenhuijzen, H., & van der Hucht, K. A. 1988,A&AS, 72, 259de Kool, M. 1990, ApJ, 358, 189Dewi, J. D. M., & Tauris, T. M. 2000, A&A, 360, 1043Han, Z., Podsiadlowski, P., & Eggleton, P. P. 1994, MNRAS,270, 121Hjellming, M. S., & Webbink, R. F. 1987, ApJ, 318, 794Kippenhahn, R., & Weigert, A. 1990, Stellar Structure andEvolution, A&A Library (Springer-Verlag)Nieuwenhuijzen, H., & de Jager, C. 1990, A&A, 231, 134Pols, O. R., Schröder, K.-P., Hurley, J. R., Tout, C. A., &Eggleton, P. P. 1998, MNRAS, 298, 525Webbink, R. F. 1984, ApJ, 277, 355

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On the binding energy of the common envelope evolution. Dependency on the definition of the stellar core boundary during spiral-in.

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