- Email: [email protected]
Thermohaline-mixing—binary evolution
New Astronomy 8 (2003) 23–28 www.elsevier.com / locate / newast
Thermohaline-mixing—binary evolution O.M. Bitzaraki a , *, C.A. Tout b , H. Rovithis-Livaniou a a
Section of Astrophysics, Astronomy and Mechanics, Panepistimiopolis, Zografos, GR15784, Athens, Greece b Institute of Astronomy, The Observatories, Madingley Road, Cambridge CB3 0 HA, UK Received 22 August 2002; received in revised form 11 September 2002; accepted 30 September 2002 Communicated by E.P.J. van den Heuvel
Abstract We explore evolutionary scenarios for the formation of the Luminous Supersoft X-Ray Sources and of Low-Mass X-ray binary system (LMXBs). A subclass of the former systems is considered as candidates for Type Ia supernova explosions. We include the effect of thermohaline mixing in the evolutionary calculations of binary systems consisting of a donor of intermediate mass 7–8 M ( and a low companion mass in the range 1.5–3.0 M ( . Case C of mass transfer is followed by a CE phase. A self-consistent treatment of winds was also incorporated in the newer version of Eggleton’s code. We keep track of the post-CE evolution during which the post-CE remnant of this star is examined to fill its Roche lobe during He shell burning. For the above case, if the diffusion time-scale is shorter than the evolutionary time scale, the companion accretes and mixes the transferred matter into its radiative outer layers by means of the so-called thermohaline mixing. This modifies the evolution of the systems with respect to that of systems without He-enrichment and reduces the number of Supersoft Systems by as much as 25–50 per cent. Also it increases the number of systems that will evolve to Type Ia supernova or a LMXB, but it is not (yet) possible to give a quantitative estimate of this increase. 2002 Elsevier Science B.V. All rights reserved. PACS: 97.10.Cv; 97.60.2s; 97.80.Jp; 98.70.Qy Keywords: Binaries: general; Stars: evolution; Stars: interiors; X-rays: binaries
1. Introduction Thermohaline-mixing has so far been studied with application to oceanography. According to the pioneering paper by Kippenhahn et al. (1980), thermohaline instability can also be a very efficient way of mixing the newly added He into the radiative outer layers of a MS star. We study here how this *Corresponding author. E-mail addresses: [email protected] (O.M. Bitzaraki), [email protected] (C.A. Tout), [email protected] (H. RovithisLivaniou).
may affect the evolution of binary systems that are thought to be progenitors of supersoft X-ray sources and LMXBs. We employ the newer version of Eggleton’s code (Eggleton, 1971, 1972, 1973, 1983; Eggleton et al., 1973; Pols et al., 1998), with an improved version of a subroutine for calculating the evolution of the binary orbit (Savonije, 2002) during the evolution of binary systems. We consider two illustrative examples for the evolution of two binary systems: the first system consisting of an intermediate mass star of a 7 M ( and a companion of mass 1.5 M ( ; the second binary has a more massive primary of 8 M ( with the same companion mass.
1384-1076 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S1384-1076( 02 )00199-9
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O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
Fig. 1. Top: Evolution in the H–R diagram of a system with initial component masses 7 1 1.5 M ( and initial orbital period (at ZAMS) of 350 days. Bottom: Evolution in the H–R diagram of a system with initial component masses 8 1 1.5 M ( and initial orbital period (at ZAMS) of 500 days (bottom panel). These cases of evolution involve two phases of Roche-lobe overflow. After the second Roche-lobe overflow the secondary masses have increased to 2.1 M ( and 2.2 M ( , respectively, by accreting helium-enhanced matter. Note the non-solar composition of the secondary star due to the effect of thermohaline mixing of He enhanced matter into its outer radiative layers. For comparison, dotted lines show the evolution of the same mass of secondaries but of solar chemical composition. (Symbols defined in the left-hand side of the figures represent time intervals as indicated.)
O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
For both systems the initial orbital period is taken to be several hundred days so that the primary fills its Roche lobe on the Early Asymptotic Giant Branch (at e.g t 5 3.65 3 10 7 yrs). The former system is a potential progenitor system of a SSS source and a possible candidate for type Ia SNe, (Branch et al., 1985) while the latter is more likely to form a LMXR binary via the accretion induced collapse scenario of O–Ne–Mg white dwarfs. The primaries of both systems are expected, after filling their Roche lobes, to experience a Common-Envelope (CE) phase and the Post-CE evolution is then followed up to the cooling WD branch. During the whole evolutionary set, wind mass-loss is treated properly for the various areas of the H–R diagram, as described in a separate paper (Bitzaraki et al., 2002). The evolution of both components of both systems in the HR diagram is depicted in upper and lower parts of Fig. 1, respectively. Please note that the masses 2.1 and 2.2 solar masses indicate the masses of the initial 1.5 solar mass secondaries after they accreted the He-rich matter from their companions.
2. Calculations In the above HR diagrams some characteristic phases of the evolution of primaries are represented by letters. Point A 1 denotes the first Roche lobe overflow at the EAGB. A 2 shows the first Roche lobe underflow during Common-Envelope phase (CE), while A 3 represents the termination of CE. A 4 denotes the position in the HR-diagram at which all of the H-rich envelope of the post-CE remnant has been removed by means of winds and H-shell burning. A 5 represents the evolutionary status of the post-CE remnant at which the initial primary overfills its Roche lobe for second time due to He-shell burning. This phase lasts until the remnant reaches point A 6 at which a superwind was to be applied if the remnant is to evolve towards the cooling branch of WDs. The remnant reaches its maximum surface temperature at point A 7 . At the final point A 8 a CO white dwarf has been reached, the end point of the evolution of the primary component. For the case of a 7 M ( star the point at which the post-CE fills its Roche lobe during He-shell burning has surface
25
abundances XH 5 0.09902, XHe 5 0.8815, XC 5 0.00016, XN 5 0.01253, XO 5 0.00107, XNe 5 0.00185, XMg 5 0.00076. The luminosity due to He burning reaches 1.1 3 10 4 L ( . The post-CE remnant has then a mass of 1.63 M ( with a CO core of 0.88 M(. For the case of the 8 M ( primary the corresponding abundances by mass at the same point are approximately equal to the above values. The luminosity due to He burning now reaches higher values and reaches 1.64 3 10 4 L ( . The remnant of the primary is of 1.94 M ( consisting of an O–Ne–Mg core mass of 0.7177 M ( around which there exists a CO mantle of 0.3035 M ( and a He-rich envelope of 0.9189 M ( . It follows from our calculations that in both systems He-rich matter of total mass of 0.6 M ( and 0.7 M ( is transferred and accumulated by the secondary MS component, respectively. If one further assumes that thermohaline instability is an efficient process of mixing material into the radiative outer layers of the secondary then its chemical composition changes. As a result, a solar composition MS star of 1.5 M ( that accretes 0.6 M ( He-rich material, and after mixing is calculated to have homogeneous composition by mass with XH 5 0.50, XHe 5 0.48 and heavy elements abundances unchanged. This corresponds to the 7 M ( primary. The same abundances are also calculated for the case of the primary of 8 M ( . As a consequence, the mixing of He-rich matter into the radiative layers of the unevolved secondary star is expected to increase the mean molecular weight from 0.6 to approximately 0.7 for a fully ionized mixture of H and He and heavy element abundances. The increase of molecular weight causes a rise in the nuclear burning which produces an increased luminosity when H is transformed to He according to homology relations (L ~ to m 4 ). The larger luminosities cause the secondary to evolve faster on a nuclear time scale. We consider now for comparison, the case of a solar chemical composition secondary mass of 2.1 M ( . It takes such a secondary of mass 2.1 M ( , 8.32 3 10 8 yrs to start transferring mass unstably at high rates of the order of 10 27 M ( yr 21 near the end of its MS lifetime. Its He core at the start of mass transfer is 0.19 M ( . The thermal time scale during the mass transfer phase varies between 9.7 3 10 5 yrs and 3.3 3 10 6 yrs. The high mass transfer rates support steady H burning on
26
O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
the white dwarf which lasts for 2.0 3 10 6 yrs. After this time interval the mass transfer becomes stable and the mass transfer rates gradually decrease driving unsteady H burning onto CO or O–Ne–Mg WD.
For mass transfer rates in a narrow region ~ max # M ~ #M ~ max ~ max | 8.5 3 0.4M where M 10 27 (MWD 2 0.52) M ( yr 21 (Nomoto, 1982), steady nuclear burning of hydrogen can occur onto the WD
Fig. 2. Porb and MWD as a function of the secondary mass (thick full and dashed lines, respectively, in the upper panel) and the evolution of ~ as a function of the secondary mass (thick full lines in the lower panel) for the systems with He-enriched secondaries. The companion is a M CO WD of mass 0.98 M ( in an initial orbital period of | 1.4 day. The above system is a candidate system for a Luminous SSS, leading possibly to type Ia SN as the white dwarf mass increases to 1.4 M ( . For comparison, thin lines show the evolution of Porb and MWD (full ~ (full line in bottom panel) but for a 2.1 M ( donor of solar composition. Here the WD and dashed line, respectively, in top panel) and M mass does not grow sufficiently to reach type Ia SN (see text).
O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
surface increasing its mass. This mass transfer rates are shown with dotted lines in Figs. 2 and 3 (lower panel). For mass transfer rates below the lower rate for steady burning and higher than 10 28 M ( yr 21 dependent also on the WD mass as given in Prialnik and Kovetz (1995) and Kato and Hachisu (1999),
27
weak H flashes occur onto the WD surface leading to a partial increase in the WD mass; the rest of it being lost from the surface of the WD. And finally for ~ # 10 28 M ( yr 21 explosive H flashes occur onto M the surface of the white dwarf leading to nova explosion during which a part of the WD may be also ejected (Prialnik and Kovetz, 1995). We use
Fig. 3. Similar to Fig. 2 but for an initial system of 8 1 2.2 M ( . The companion is an O–Ne–Mg WD of 1.07 M ( in a initial orbital period of |1.4 day. The system with the He-enriched secondary can thus produce a LMXB. In the system with a normal composition secondary the WD does not grow to collapse to a neutron star.
28
O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
these facts for calculating the growth of the WD mass during the mass-transfer phase. By comparison a secondary of 2.1 M ( with homogeneous He-enriched chemical composition, by means of thermohaline mixing, evolves much faster. ~ rises rapidly to After 2.6 3 10 8 yrs from ZAMS M values higher than 10 26 M ( yr 21 , the secondary having developed a more massive He core of 0.29 M ( . The thermal time scale ranges from 2.55 3 10 5 yrs to 1.1 3 10 6 yrs when the WD has accreted mass and grows up to the Chandrasekhar limit. The total time interval growth for mass transfer rates in excess of 10 27 M ( yr 21 is 1.55 3 10 6 yrs. We therefore obtain, a shorter duration of mass-transfer rates above the upper limit of 10 27 M ( yr 21 by a factor of 25% for a SSSs with enhanced He-abundances in its envelope. For the case of the system of the 2.2 M ( star and the O–Ne–Mg WD, the time duration of mass transfer rates higher than the lower boundary for steady H burning in the case of a donor having experienced a thermohaline instability until the formation of a NS (when the O–Ne–Mg has grown up to 1.4 M ( ), is found to be 1.04 3 10 6 yrs. For a solar composition donor of 2.2 M ( star the duration of the thermal time-scale mass-transfer rates is 2.0 3 10 6 yrs. However, here the WD does not grow to the Chandrasekhar limit. But in any case here the Supersoft Source stage lasts twice as long as with the He-enriched donor.
3. Conclusions We summarise our results for the effect of thermohaline mixing to binary evolution: Secondaries that experience thermohaline instability of He-rich material, evolve faster than stars of typical solar composition. Such stars develop heavier He cores, and higher luminosities. Since their thermal time scale also becomes shorter they can achieve higher mass transfer rates during the first unstable mass transfer episode from the more massive primary to the less massive secondary. As a result of the high
mass-transfer rates it proves to be easier for the secondary to sustain the required mass-transfer rates for steady H-burning onto the surface of a CO or O–Ne–Mg WD, making its mass grow up to the Chandrasekhar limit. Since white dwarfs accreting mass up to 1.4 M ( may be observed as Luminous SSSs, we expect decrease in the number of luminous SSSs that have so far been predicted by theory (for donors of solar composition) by | 25% in the case of the 7 M ( primary and by | 50% in the case of the 8 M ( primary. On the other hand, the above scenarios favour the formation of a larger number of type Ia supernovae and LMXBs, as in the here considered systems with He-enriched secondaries the White Dwarfs are able to grow to the Chandrasekhar limit, whereas in systems with normal composition secondaries they are not. We studied here only the evolution of a few systems. In order to see by how much the obtained results would affect the Type Ia SN rate and the LMXB formation rate, one would have to carry out a full population synthesis calculation. This is, however, beyond the scope of this paper.
References Bitzaraki, O.M., Rovithis-Livaniou, H., Tout, C.A., van den Heuvel, E.P.J., 2002. A&A (submitted). Branch, D., Doggett, J.B., Nomoto, K., Thielemann, F.-K., 1985. ApJ 294, 619–625. Eggleton, P.P., 1971. MNRAS 151, 351. Eggleton, P.P., 1972. MNRAS 156, 361. Eggleton, P.P., Faulkner, J., Flannery, B.P., 1973. A&A 23, 325. Eggleton, P.P., 1973. MNRAS 163, 279. Eggleton, P.P., 1983. MNRAS 268, 368. Kato, M., Hachisu, I., 1999. ApJ 513, L41–44. Kippenhahn, R., Ruschenplatt, G., Thomas, H.-C., 1980. A&A 91, 175. Nomoto, K., 1982. ApJ 253, 798. Pols, O.R., Schroder, K.-P., Hurley, J.R., Tout, C.A., Eggleton, P.P., 1998. MNRAS 298, 525. Prialnik, D., Kovetz, A., 1995. ApJ 445, 789. Savonije, G.J., 2002. Private communication.
Thermohaline-mixing—binary evolution O.M. Bitzaraki a , *, C.A. Tout b , H. Rovithis-Livaniou a a
Section of Astrophysics, Astronomy and Mechanics, Panepistimiopolis, Zografos, GR15784, Athens, Greece b Institute of Astronomy, The Observatories, Madingley Road, Cambridge CB3 0 HA, UK Received 22 August 2002; received in revised form 11 September 2002; accepted 30 September 2002 Communicated by E.P.J. van den Heuvel
Abstract We explore evolutionary scenarios for the formation of the Luminous Supersoft X-Ray Sources and of Low-Mass X-ray binary system (LMXBs). A subclass of the former systems is considered as candidates for Type Ia supernova explosions. We include the effect of thermohaline mixing in the evolutionary calculations of binary systems consisting of a donor of intermediate mass 7–8 M ( and a low companion mass in the range 1.5–3.0 M ( . Case C of mass transfer is followed by a CE phase. A self-consistent treatment of winds was also incorporated in the newer version of Eggleton’s code. We keep track of the post-CE evolution during which the post-CE remnant of this star is examined to fill its Roche lobe during He shell burning. For the above case, if the diffusion time-scale is shorter than the evolutionary time scale, the companion accretes and mixes the transferred matter into its radiative outer layers by means of the so-called thermohaline mixing. This modifies the evolution of the systems with respect to that of systems without He-enrichment and reduces the number of Supersoft Systems by as much as 25–50 per cent. Also it increases the number of systems that will evolve to Type Ia supernova or a LMXB, but it is not (yet) possible to give a quantitative estimate of this increase. 2002 Elsevier Science B.V. All rights reserved. PACS: 97.10.Cv; 97.60.2s; 97.80.Jp; 98.70.Qy Keywords: Binaries: general; Stars: evolution; Stars: interiors; X-rays: binaries
1. Introduction Thermohaline-mixing has so far been studied with application to oceanography. According to the pioneering paper by Kippenhahn et al. (1980), thermohaline instability can also be a very efficient way of mixing the newly added He into the radiative outer layers of a MS star. We study here how this *Corresponding author. E-mail addresses: [email protected] (O.M. Bitzaraki), [email protected] (C.A. Tout), [email protected] (H. RovithisLivaniou).
may affect the evolution of binary systems that are thought to be progenitors of supersoft X-ray sources and LMXBs. We employ the newer version of Eggleton’s code (Eggleton, 1971, 1972, 1973, 1983; Eggleton et al., 1973; Pols et al., 1998), with an improved version of a subroutine for calculating the evolution of the binary orbit (Savonije, 2002) during the evolution of binary systems. We consider two illustrative examples for the evolution of two binary systems: the first system consisting of an intermediate mass star of a 7 M ( and a companion of mass 1.5 M ( ; the second binary has a more massive primary of 8 M ( with the same companion mass.
1384-1076 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S1384-1076( 02 )00199-9
24
O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
Fig. 1. Top: Evolution in the H–R diagram of a system with initial component masses 7 1 1.5 M ( and initial orbital period (at ZAMS) of 350 days. Bottom: Evolution in the H–R diagram of a system with initial component masses 8 1 1.5 M ( and initial orbital period (at ZAMS) of 500 days (bottom panel). These cases of evolution involve two phases of Roche-lobe overflow. After the second Roche-lobe overflow the secondary masses have increased to 2.1 M ( and 2.2 M ( , respectively, by accreting helium-enhanced matter. Note the non-solar composition of the secondary star due to the effect of thermohaline mixing of He enhanced matter into its outer radiative layers. For comparison, dotted lines show the evolution of the same mass of secondaries but of solar chemical composition. (Symbols defined in the left-hand side of the figures represent time intervals as indicated.)
O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
For both systems the initial orbital period is taken to be several hundred days so that the primary fills its Roche lobe on the Early Asymptotic Giant Branch (at e.g t 5 3.65 3 10 7 yrs). The former system is a potential progenitor system of a SSS source and a possible candidate for type Ia SNe, (Branch et al., 1985) while the latter is more likely to form a LMXR binary via the accretion induced collapse scenario of O–Ne–Mg white dwarfs. The primaries of both systems are expected, after filling their Roche lobes, to experience a Common-Envelope (CE) phase and the Post-CE evolution is then followed up to the cooling WD branch. During the whole evolutionary set, wind mass-loss is treated properly for the various areas of the H–R diagram, as described in a separate paper (Bitzaraki et al., 2002). The evolution of both components of both systems in the HR diagram is depicted in upper and lower parts of Fig. 1, respectively. Please note that the masses 2.1 and 2.2 solar masses indicate the masses of the initial 1.5 solar mass secondaries after they accreted the He-rich matter from their companions.
2. Calculations In the above HR diagrams some characteristic phases of the evolution of primaries are represented by letters. Point A 1 denotes the first Roche lobe overflow at the EAGB. A 2 shows the first Roche lobe underflow during Common-Envelope phase (CE), while A 3 represents the termination of CE. A 4 denotes the position in the HR-diagram at which all of the H-rich envelope of the post-CE remnant has been removed by means of winds and H-shell burning. A 5 represents the evolutionary status of the post-CE remnant at which the initial primary overfills its Roche lobe for second time due to He-shell burning. This phase lasts until the remnant reaches point A 6 at which a superwind was to be applied if the remnant is to evolve towards the cooling branch of WDs. The remnant reaches its maximum surface temperature at point A 7 . At the final point A 8 a CO white dwarf has been reached, the end point of the evolution of the primary component. For the case of a 7 M ( star the point at which the post-CE fills its Roche lobe during He-shell burning has surface
25
abundances XH 5 0.09902, XHe 5 0.8815, XC 5 0.00016, XN 5 0.01253, XO 5 0.00107, XNe 5 0.00185, XMg 5 0.00076. The luminosity due to He burning reaches 1.1 3 10 4 L ( . The post-CE remnant has then a mass of 1.63 M ( with a CO core of 0.88 M(. For the case of the 8 M ( primary the corresponding abundances by mass at the same point are approximately equal to the above values. The luminosity due to He burning now reaches higher values and reaches 1.64 3 10 4 L ( . The remnant of the primary is of 1.94 M ( consisting of an O–Ne–Mg core mass of 0.7177 M ( around which there exists a CO mantle of 0.3035 M ( and a He-rich envelope of 0.9189 M ( . It follows from our calculations that in both systems He-rich matter of total mass of 0.6 M ( and 0.7 M ( is transferred and accumulated by the secondary MS component, respectively. If one further assumes that thermohaline instability is an efficient process of mixing material into the radiative outer layers of the secondary then its chemical composition changes. As a result, a solar composition MS star of 1.5 M ( that accretes 0.6 M ( He-rich material, and after mixing is calculated to have homogeneous composition by mass with XH 5 0.50, XHe 5 0.48 and heavy elements abundances unchanged. This corresponds to the 7 M ( primary. The same abundances are also calculated for the case of the primary of 8 M ( . As a consequence, the mixing of He-rich matter into the radiative layers of the unevolved secondary star is expected to increase the mean molecular weight from 0.6 to approximately 0.7 for a fully ionized mixture of H and He and heavy element abundances. The increase of molecular weight causes a rise in the nuclear burning which produces an increased luminosity when H is transformed to He according to homology relations (L ~ to m 4 ). The larger luminosities cause the secondary to evolve faster on a nuclear time scale. We consider now for comparison, the case of a solar chemical composition secondary mass of 2.1 M ( . It takes such a secondary of mass 2.1 M ( , 8.32 3 10 8 yrs to start transferring mass unstably at high rates of the order of 10 27 M ( yr 21 near the end of its MS lifetime. Its He core at the start of mass transfer is 0.19 M ( . The thermal time scale during the mass transfer phase varies between 9.7 3 10 5 yrs and 3.3 3 10 6 yrs. The high mass transfer rates support steady H burning on
26
O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
the white dwarf which lasts for 2.0 3 10 6 yrs. After this time interval the mass transfer becomes stable and the mass transfer rates gradually decrease driving unsteady H burning onto CO or O–Ne–Mg WD.
For mass transfer rates in a narrow region ~ max # M ~ #M ~ max ~ max | 8.5 3 0.4M where M 10 27 (MWD 2 0.52) M ( yr 21 (Nomoto, 1982), steady nuclear burning of hydrogen can occur onto the WD
Fig. 2. Porb and MWD as a function of the secondary mass (thick full and dashed lines, respectively, in the upper panel) and the evolution of ~ as a function of the secondary mass (thick full lines in the lower panel) for the systems with He-enriched secondaries. The companion is a M CO WD of mass 0.98 M ( in an initial orbital period of | 1.4 day. The above system is a candidate system for a Luminous SSS, leading possibly to type Ia SN as the white dwarf mass increases to 1.4 M ( . For comparison, thin lines show the evolution of Porb and MWD (full ~ (full line in bottom panel) but for a 2.1 M ( donor of solar composition. Here the WD and dashed line, respectively, in top panel) and M mass does not grow sufficiently to reach type Ia SN (see text).
O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
surface increasing its mass. This mass transfer rates are shown with dotted lines in Figs. 2 and 3 (lower panel). For mass transfer rates below the lower rate for steady burning and higher than 10 28 M ( yr 21 dependent also on the WD mass as given in Prialnik and Kovetz (1995) and Kato and Hachisu (1999),
27
weak H flashes occur onto the WD surface leading to a partial increase in the WD mass; the rest of it being lost from the surface of the WD. And finally for ~ # 10 28 M ( yr 21 explosive H flashes occur onto M the surface of the white dwarf leading to nova explosion during which a part of the WD may be also ejected (Prialnik and Kovetz, 1995). We use
Fig. 3. Similar to Fig. 2 but for an initial system of 8 1 2.2 M ( . The companion is an O–Ne–Mg WD of 1.07 M ( in a initial orbital period of |1.4 day. The system with the He-enriched secondary can thus produce a LMXB. In the system with a normal composition secondary the WD does not grow to collapse to a neutron star.
28
O.M. Bitzaraki et al. / New Astronomy 8 (2003) 23–28
these facts for calculating the growth of the WD mass during the mass-transfer phase. By comparison a secondary of 2.1 M ( with homogeneous He-enriched chemical composition, by means of thermohaline mixing, evolves much faster. ~ rises rapidly to After 2.6 3 10 8 yrs from ZAMS M values higher than 10 26 M ( yr 21 , the secondary having developed a more massive He core of 0.29 M ( . The thermal time scale ranges from 2.55 3 10 5 yrs to 1.1 3 10 6 yrs when the WD has accreted mass and grows up to the Chandrasekhar limit. The total time interval growth for mass transfer rates in excess of 10 27 M ( yr 21 is 1.55 3 10 6 yrs. We therefore obtain, a shorter duration of mass-transfer rates above the upper limit of 10 27 M ( yr 21 by a factor of 25% for a SSSs with enhanced He-abundances in its envelope. For the case of the system of the 2.2 M ( star and the O–Ne–Mg WD, the time duration of mass transfer rates higher than the lower boundary for steady H burning in the case of a donor having experienced a thermohaline instability until the formation of a NS (when the O–Ne–Mg has grown up to 1.4 M ( ), is found to be 1.04 3 10 6 yrs. For a solar composition donor of 2.2 M ( star the duration of the thermal time-scale mass-transfer rates is 2.0 3 10 6 yrs. However, here the WD does not grow to the Chandrasekhar limit. But in any case here the Supersoft Source stage lasts twice as long as with the He-enriched donor.
3. Conclusions We summarise our results for the effect of thermohaline mixing to binary evolution: Secondaries that experience thermohaline instability of He-rich material, evolve faster than stars of typical solar composition. Such stars develop heavier He cores, and higher luminosities. Since their thermal time scale also becomes shorter they can achieve higher mass transfer rates during the first unstable mass transfer episode from the more massive primary to the less massive secondary. As a result of the high
mass-transfer rates it proves to be easier for the secondary to sustain the required mass-transfer rates for steady H-burning onto the surface of a CO or O–Ne–Mg WD, making its mass grow up to the Chandrasekhar limit. Since white dwarfs accreting mass up to 1.4 M ( may be observed as Luminous SSSs, we expect decrease in the number of luminous SSSs that have so far been predicted by theory (for donors of solar composition) by | 25% in the case of the 7 M ( primary and by | 50% in the case of the 8 M ( primary. On the other hand, the above scenarios favour the formation of a larger number of type Ia supernovae and LMXBs, as in the here considered systems with He-enriched secondaries the White Dwarfs are able to grow to the Chandrasekhar limit, whereas in systems with normal composition secondaries they are not. We studied here only the evolution of a few systems. In order to see by how much the obtained results would affect the Type Ia SN rate and the LMXB formation rate, one would have to carry out a full population synthesis calculation. This is, however, beyond the scope of this paper.
References Bitzaraki, O.M., Rovithis-Livaniou, H., Tout, C.A., van den Heuvel, E.P.J., 2002. A&A (submitted). Branch, D., Doggett, J.B., Nomoto, K., Thielemann, F.-K., 1985. ApJ 294, 619–625. Eggleton, P.P., 1971. MNRAS 151, 351. Eggleton, P.P., 1972. MNRAS 156, 361. Eggleton, P.P., Faulkner, J., Flannery, B.P., 1973. A&A 23, 325. Eggleton, P.P., 1973. MNRAS 163, 279. Eggleton, P.P., 1983. MNRAS 268, 368. Kato, M., Hachisu, I., 1999. ApJ 513, L41–44. Kippenhahn, R., Ruschenplatt, G., Thomas, H.-C., 1980. A&A 91, 175. Nomoto, K., 1982. ApJ 253, 798. Pols, O.R., Schroder, K.-P., Hurley, J.R., Tout, C.A., Eggleton, P.P., 1998. MNRAS 298, 525. Prialnik, D., Kovetz, A., 1995. ApJ 445, 789. Savonije, G.J., 2002. Private communication.