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Astron. Astrophys. 318, 812-818 (1997)

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2. The population synthesis model

We start with a population consisting of a single star population and a binary population. In order to estimate the number of stars in different evolutionary phases, we need

a. an MCB evolutionary model.

b. a single star evolutionary model.

c. the lifetimes of the different evolutionary phases.

d. star and binary parameter distributions.

We use the MCB evolutionary model which is discussed in detail in Paper I and summarized in figure 1.

[FIGURE] Fig. 1a. The intermediate mass or massive close binary evolutionary model up to the end of core helium burning of the original primary. We consider three mass intervals; M1 is the initial mass of the primary, [FORMULA] is the minimum initial mass a primary of a close binary must have in order to evolve into a WR star after Roche lobe overflow (RLOF); [FORMULA] is the minimum initial mass of a star above which stellar wind mass loss during the RSG phase is sufficient in order to remove most of the hydrogen rich layers, i.e. for a star to evolve into a WR star without the explicit help of the RLOF process (from Vanbeveren et al., 1996).
[FIGURE] Fig. 1b. The evolutionary model of a massive close binary after the supernova explosion of the original primary; [FORMULA] is the minimum mass a primary must have in order for a supernova to occur; [FORMULA] has the same meaning as in figure 1a but applied to the rejuvenated mass gainer of the binary; [FORMULA] is the mass of the hydrogen deficient core helium burning star after the spiral-in phase (from Vanbeveren et al., 1996).

The model accounts for the following processes:

[FORMULA] The LBV scenario for binaries with primary masses larger than 40-50 [FORMULA].

[FORMULA] The RSG scenario for case Bc/case C binaries with [FORMULA] [FORMULA] M1 [FORMULA] 40-50 [FORMULA] (M1 = primary mass). The value of [FORMULA] ranges between 20 [FORMULA] and 30 [FORMULA] and depends on the adopted evolutionary scenario for single stars. For the evolution of massive single stars we use the 'standard single star scenario' and the 'alternative single star scenario' discussed in Paper I.

[FORMULA] MCBs with an initial mass ratio q [FORMULA] 0.2-0.3 do not perform Roche lobe overflow (RLOF) but the low mass secondary is dragged into the envelope of the primary and spirals-in. As was outlined in Paper I, most of them merge. Their further evolution is uncertain. One possibility is that the evolution of the merger resembles the evolution of a star who has accreted an amount of mass (= mass of the low mass component). This model is used here although the final numbers do only marginally depend on this class of MCBs.

[FORMULA] Case Bc and case C binaries evolve through a common envelope phase for which we use the spiral-in prescription of Webbinck (1984). It is characterized by a parameter [FORMULA] describing the efficiency by which orbital energy is converted by friction into potential energy of the mass that needs to be removed from the system.

[FORMULA] By far the most important class of MCBs for population synthesis is the case Br class (and to a lesser extend the case A class). When the mass ratio of the binary is larger than some value [FORMULA], their evolution is quasi-conservative and we use the parameter prescription of Vanbeveren et al. (1979) characterized by the parameters [FORMULA] (= the fraction of mass lost by the primary due to RLOF which is accreted by the secondary), and the parameter [FORMULA] (= the specific angular momentum removed from the orbit by the mass leaving the binary). If during the RLOF phase significant mass loss from the system occurs on the Kelvin-Helmholtz timescale (= the RLOF timescale), this can to our knowledge only happen if matter leaves the binary through the second Lagrangian point L2 forming a ring around the binary or if material lost by the primary gains sufficient energy from the orbit by dynamical friction in order to be pushed out of the binary. Both processes imply very large [FORMULA] values, i.e. [FORMULA] (Paper I).

[FORMULA] The post-RLOF evolution during core helium burning (CHeB) is determined by WR like stellar wind mass loss rates.

[FORMULA] The evolution of an MCB through the first SN explosion is followed using the presciption of asymmetric SN explosions of Sutantyo (1978). The distribution of kick velocities has been studied by Lyne and Lorimer (1994) and can be very well described by Eq. 1.

[FORMULA] When after the SN explosion the binary was disrupted, the further evolution of the OB single star is followed by using the appropriate single star scenario. When the binary was not disrupted, as the OB star evolves and expands, the CC spirals in possibly removing the outer layers of the OB star. If the CC does not merge, a WR+CC binary can be formed. If the OB+CC system merges, its further evolution is followed using two limiting models (see also Paper I):

a. during the merging process the OB star has lost most of its hydrogen rich layers and thus a WR star is formed with a CC in its centre: we call this a 'weird' WR star. In order to compute their number we adopt the same WR timescales as for ordinary WR stars in close binaries.

b. if the OB+CC binary merges, the evolution of the OB-type star is that of a normal single star which possibly evolves into a WR star after mass loss by an efficient stellar wind during the red supergiant phase. Also in this case the WR star is a 'weird' WR star.

[FORMULA] The evolution of a CHeB star + CC binary is then followed through the second SN explosion, similarly as through the first one, using the distribution of kick velocities given by Eq. 1.

[FORMULA] The runaway velocity of the OB+CC binary (if the SN did not disrupt the binary) or of the OB single star (if the SN did disrupt the binary) after SN, is computed as explained in Paper I.

A population synthesis model contains the following parameters:

[FORMULA] IMF [FORMULA] [FORMULA] with [FORMULA] = 2.7 and [FORMULA] = 2,

[FORMULA] The mass ratio distribution of MCBs: we use [FORMULA] (q) = flat distribution, [FORMULA] (q) = the mass ratio distribution of Hogeveen (1991) or [FORMULA] which peaks moderately towards q = 1.

[FORMULA] The period distribution of MCBs: [FORMULA] [FORMULA].

[FORMULA] The efficiency factor during the different spiral-in phases in MCBs, i.e. [FORMULA] for non-evolved binaries with an initial mass ratio q [FORMULA] 0.2, [FORMULA] for case Bc and case C binaries and [FORMULA] for OB + CC binaries.

[FORMULA] The value of [FORMULA] for case A/case Br binaries above which [FORMULA] is assumed to be constant (= [FORMULA]). To start with, we compute the results with [FORMULA] = 1 and [FORMULA] = 0.5 independent from the period of the case Br binary. This period independency is probably not very realistic. Once the period of the binary is large enough (P [FORMULA] 1000 days) for a system to evolve as a case Bc, [FORMULA] = 0. For periods small enough, the gasstream from the first Lagrangian point hits the companion directly. It may therefore be tempting to propose a relation where [FORMULA] decreases with increasing period, i.e. [FORMULA] = 1 - 10-3 P (P in days).

[FORMULA] The angular momentum loss during the non-conservative RLOF of case A and case Br binaries; we use [FORMULA] = 3 and [FORMULA] = 6.

[FORMULA] The minimum initial mass [FORMULA] for a single star to become a WR star and the corresponding single star evolutionary model.

[FORMULA] The final fate of OB+CC binaries who merge during their ensuing spiral-in phase.

[FORMULA] The fraction f of MCBs with periods up to 10 years in the population. Remember that the period of interacting massive binaries ranges between 1000 and 2000 days so that the fraction of interacting binaries is smaller than f.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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