Astron. Astrophys. 322, 533-544 (1997)

2. Pre-supernova binary systems

2.1. Evolutionary models

2.1.1. General frame

Starting from an evolved binary system composed of a massive main-sequence star (of mass ) and a neutron star (of mass m), we assume that close NS-NS systems are formed through the following stages (e.g. see the review of Bhattacharya & Van den Heuvel 1991):
(1). Stellar wind mass loss phase from the massive star. If enough wind material is accreted onto the neutron star, binary systems appear as standard/Be-type HMXBs.
(2). Non-conservative mass transfer phase when the evolved star overflows its Roche lobe. The remaining system consists of a helium star and a recycled neutron star.
(3). Stellar wind mass loss phase from the helium star (if still on the helium main-sequence).
(4). Possible second non-conservative mass transfer phase from the helium star when helium is exhausted in the core, resulting in the formation of a carbon-oxygen (CO) star.
(5). Supernova explosion with a kick velocity imparted to the newly born neutron star.

In standard scenarios, the physical process used to describe the first non-conservative mass transfer phase involves the rapid formation of a CE around the system, leading to the spiral-in of the neutron star into the envelope. In this picture, the efficiency parameter of orbital momentum transfer into the CE and the kick velocity are the two parameters which correspond to the uncertainty in the late stages of evolution. In the following, we shall describe each stage (1)-(4) for the formation of pre-supernova binaries and extend the parameter space to when there is a lack of theoretical ground to justify the adoption of some particular physical processes.

In the Roche model, the combination of gravitational and centrifugal forces allows one to define sets of points of the same potential. In the particular case of synchronous rotation and circular orbit, a critical equipotential surface intersects itself at the inner Lagrange point L₁, defining the two Roche lobes, one surrounding each star. An accurate expression for the Roche lobe radius of the component with mass M was found by Eggleton (1983):

where q is the mass ratio defined as and a is the orbital separation.

Evolutionary tracks for the primary are taken from Schaller et al. (1992) with an initial chemical composition X=0.70 and Z=0.02. The grids include stellar wind mass loss in massive stars and core overshooting with =0.20. Following Thorsett et al. (1993), we assume the mass of the neutron star to be . Evolutionary paths for binaries are followed as a function of the initial mass of the main-sequence star and the initial orbital separation . For our purpose, we ignore the time spent by the primary on the main-sequence phase before the formation of the first neutron star. Thus, the massive binary component is assumed to evolve from the ZAMS as a single star as long as the system remains detached, although it may have accreted matter from the progenitor of the neutron star. We shall discuss this point in Sect.4. The first supernova explosion induced a moderate eccentricity into the orbit. As the massive star expands, tidal interaction tends to circularize the orbital motion. Since the eccentricity has only a second-order effect on the tidal evolution of the orbital separation (Zahn 1977), we make the approximation of initially circular binaries.

2.1.2. Stellar wind mass loss phase

In most papers, the change in orbital separation during this phase is assumed to occur according to the Jeans' mode (e.g. Tutukov & Yungelson 1993, 1994; Lipunov et al. 1996, Portegies Zwart & Verbunt 1996). In this mode, matter escapes in a spherically symmetric way from the system at a high velocity with the specific angular momentum of the star considered as a point mass. The change in orbital separation then follows from , where is the total mass of the system. This relationship strictly applies to binaries in which the mass-losing star lies deep inside its Roche lobe. For a Roche lobe filling factor not far from unity (R is the radius of the star), orbital evolution is more likely to be driven by the tidal exchange between rotational and orbital angular momentum. We assume, however, that tidal effects are of secondary importance in view of major uncertainties in the next stage.

2.1.3. Non-conservative mass transfer phase

Due to evolutionary expansion of its envelope, the massive binary component begins to transfer mass to the neutron star through L₁ when it fills its Roche lobe. From grids of stellar models and Eq.(1), we determine the corresponding evolutionary state of the star for each initial set of parameters (, ). Fig. 1 depicts the cases A, B and C of mass transfer as defined by Kippenhahn & Weigert (1967).

Fig. 1. Evolutionary state of the massive star in a binary with a neutron star companion when the primary first fills its Roche lobe. The 3 solid lines delimit the regions where cases A, B and C of mass transfer occur, respectively core-H exhaustion, He-ignition and more advanced nuclear burning stages of the star. From bottom to top, the dotted lines refer to the central helium content of Yc= 0.95, 0.90, 0.85, 0.80 and 0.0 during the core-He burning phase.

In binaries of very low mass ratio, mass transfer from the more massive component leads to a rapid decrease of the orbital separation and thereby of the Roche lobe radii ¹. Then, the mass transfer rate depends on the response to mass loss of the donor star (e.g. Hjellming & Webbink 1987). In case A and early case B binaries, the massive star with a radiative envelope out of thermal equilibrium contracts less rapidly than its Roche radius. The mass transfer then proceeds on the thermal time scale . In late case B and case C systems, the deep convective envelope of the evolved star is highly unstable and tends to expand on a dynamical time scale rather than to contract when mass is removed. Therefore, expected mass transfer rates are much larger than in the case of a radiative envelope. In both cases, the accretion rate onto the neutron star is limited by the Eddington rate and any further accretion will be prevented by radiation pressure. The excess overflowing matter is expected to form a CE or an extended thick accretion disk around the neutron star. In these two scenarios, binary systems undergo a phase of extensive angular momentum loss.

The important problem of mass accretion onto the neutron star was addressed by Chevalier (1993) and Brown (1995) who found that the Eddington limit does not apply to spherically symmetrical situations with high accretion rates . In this regime, neutrino losses allow the neutron star in a dense stellar envelope to accrete a substantial amount of matter with the result of black hole formation. However, Chevalier (1996) showed that for a red supergiant companion, rotation of the gas prevents strong accretion onto the neutron star. In this picture, black hole formation should occur only if the spiral-in continues into the stellar core of the primary. Therefore, we assume that survival binaries after the mass transfer phase are composed of a remnant helium core and a neutron star of negligible accreted mass.

Common envelope evolution

This model is the most frequently used to describe the mass transfer phase in binaries of low mass ratio with a neutron star (e.g. Meurs & Van den Heuvel 1989; Rathnasree 1993; Tutukov & Yungelson 1993, 1994; Pols & Marinus 1994; Portegies Zwart & Verbunt 1996). Once the neutron star is embedded with a differential velocity into the envelope of its companion, orbital evolution is driven by frictional forces, resulting in a spiral-in process. Binaries will survive this phase if energy deposition generated by the large frictional drag allows the ejection of the entire CE carrying with it nearly all of the orbital angular momentum. Then, the final orbital parameters can be calculated by assuming that the loss of orbital energy is used to expel the envelope of binding energy according to , where is an efficiency parameter less than or equal to one (Webbink 1984). This prescription leads to the following relation between the original and final orbital separation and (De Kool 1990):

where approximated by 0.5 is a factor depending on the mass distribution in the envelope, is the mass of the remnant core. Equation (2) yields:

For the sake of comparison with previous works, we will take the value of in the range 0.5 to 1.

Jets from the compact star

In non-spherical situations, centrifugal forces keep gas from directly reaching the neutron star surface and matter lost from the massive star may form a supercritical accretion disk around the compact star (Shakura & Sunyaev 1973; Kornilov & Lipunov 1983 a, b; Bhattacharya & Van den Heuvel 1991). Owing to the Eddington limit, the larger part of the transferred matter is expelled from the inner regions of the disk. Following this model, matter is thought to escape from the neutron star in the form of relativistic jets, carrying away its orbital angular momentum.

Van den Heuvel et al. (1980) showed that the occurence of this physical process in SS 433 accounts well for the unusual behaviour of this system. The SS 433 object associated with the X-ray source A1909+04 appears to be an eclipsing binary system with an orbital period of 13.1 days (Crampton et al. 1980). Observational characteristics are usually ascribed to the ejection of collimated jets of matter from a compact object at 0.26c (e.g. see the review of Margon 1984). In most models, the observed periodic rotation of the jet axis is related to the orientation of a thick accretion disk whose plane precesses with a 164 day period. Although the nature of the underlying components are not known exactly, observations suggest that an OB star that has evolved beyond the HMXB phase transfers matter to a compact object through the inner Lagrangean point in a thermal time scale, resulting in a mass transfer rate highly super-Eddington (Cherepashchuck 1981). In addition, Van den Heuvel et al. (1980) and Zealey et al. (1980) estimated a lower limit for the mass loss rate in the beams.

Thus, observational ground supports a scenario in which orbital evolution is driven by the ejection of matter carrying with it the specific angular momentum of the neutron star as follows. Logarithmic differentiation of total orbital angular momentum yields:

The contribution of angular momentum loss due to mass outflow from the inner edge of the disk is given by:

where is the fraction of matter accreted onto the compact star. Substituting this expression in Eq.(4), we obtain

Integration of this equation gives the change of the orbital separation:

where and . In the limit , the orbital evolution when accretion of matter onto the neutron star is thought to be negligible is given by:

The exponential term in Eq.(8) makes the shrinkage factor extremely sensitive to the mass ratio. Systems with initial low mass ratio are unlikely to survive this mass transfer phase if all of the hydrogen-rich envelope is ejected through this process. From Eq.(6), the timescale for orbital decay can be approximated by:

Thus, this timescale in binaries containing a neutron star is roughly an order of magnitude shorter than the timescale for mass loss .

An intermediate model

No hydrodynamical calculation has yet been done which computes the initial stage of the non-conservative mass transfer phase in high-mass binaries containing a compact star. In particular, the spiral-in process in the CE scenario implies that the neutron star is already embedded in the outer layers of the massive companion. The existence of a thick accretion disk with two opposed collimated jets in SS 433 suggests that a substage of fast mass transfer ² occurs when the massive star begins overflowing its Roche lobe (Lipunov et al. 1996). The further evolution of the system is hard to predict since the thick disk may not remain stable with increasing large mass-transfer rate. It is useful to define a parameter which is the fraction of mass lost during this substage of mass transfer: , where corresponds to the amount of matter leaving the system in the form of jets and is the mass of the envelope. Orbital evolution is first computed according to Eq.(8) in which . When the transferred matter leads to the formation of a common envelope, Eq.(3) is used with .

To give an upper limit to the value of , we consider a binary with an orbital separation of 500 . In this system, a 16 star has a radius of 300 at the onset of mass transfer and a core mass of 5.1 . According to Eq.(8), the amount of mass loss required to bring the neutron star near the surface of the primary is only of 0.4 . The distribution of mass for an evolved star with respect to radius favors the rapid formation of a CE as soon as the neutron star encounters the outer layers of its companion. For instance, if one assumes that the ejection of matter still continues in the form of jets, the orbit would decay from the stellar surface to half-radius for an amount of mass ejected of 0.5 . This is clearly inconsistent with the structure of the 16 star since more than approximately 4 is located in the outer half of the star (cf. Fig. 1 in Terman et al. 1995). Therefore, we will take as a conservative limit for the first mass transfer phase.

2.1.4. Stellar wind mass loss from the helium star

Very close binaries are left after the non-conservative mass transfer phase, consisting of the neutron star and the helium core of the massive star. Vanbeveren & De Greve (1979) found that the mass at the end of a case B mass transfer corresponds to the mass of the convective core during core-H burning when of the remnant model, where the atmospherical hydrogen abundance is typically 0.25 for Galactic models. Using this criterion, De Loore & Vanbeveren (1994) performed evolutionary computations for massive close binaries, providing a best fit relation between the remnant mass and ZAMS mass :

Comparatively few studies have dealt with the third type of binary evolution, case C mass transfer. We extend the previous relationship for the core mass stripped of its hydrogen-rich envelope during the core-He burning stage (CHeB) of the primary. For more advanced nuclear burning stages, we use the prescription of Portegies Zwart & Verbunt (1996):

where is a factor taking into account effects of core overshooting (Shore et al. 1994). The outer radius of a helium star on the He main-sequence phase is taken from Langer (1989a):

When the hydrogen-rich envelope has been removed by strong stellar wind mass loss, the further evolution of the helium star remnant is thought to be affected by WR stellar wind mass loss during the CHeB phase. For hydrogenless WR stars, Langer (1989b) introduced mass dependent mass loss rates:

where , and yr^-1 for hydrogenless WN stars and yr^-1 for WC/WO stars.

As the surface conditions do not show significant physical difference, helium cores revealed by mass transfer in a binary or by mass loss from a single star, are expected to experience the same stellar wind mass loss. Woosley & al. (1995) have considered the time evolution of mass-losing helium stars with initial masses between 4 and 20 according to Eq.(13). An important feature raised from their results (see also Woosley & al. 1993) is the convergence of all helium stars to small final masses in the range 2.26 - 3.55 . However, they noted that there was weak observational justification in the adopted mass loss rate for helium stars below 5 .

Recently, Kiriakidis et al. (1993) found that very massive main-sequence stars experience violent pulsational instabilities. These authors reported that the same process also occurs on the helium main-sequence phase (Glatzel et al. 1993). In particular, it was shown that helium stars above 4.5 are unstable with respect to radial pulsations. Such instabilities operating on dynamical time-scales are expected to drive a stellar wind with mass-loss rates as observed in WR stars (see also Langer et al. 1994). This lower mass limit above which dynamical unstable modes appear in helium stars suggests a pile up of final masses near this threshold. Furthermore, these instabilities were found to increase with the mass of the helium stars. According to this picture, the power-law index introduced by Langer can be consistently related to the growing strength of pulsational instability.

In high-mass binaries, mass transfer can lead to initial helium star mass as small as 2.5 . In order to take into account the effects of pulsational instability, we introduce a lower mass limit above which a helium star experiences stellar wind mass loss. Equation (13) was constructed with the necessary condition . The new necessary condition allows us to rewrite the mass-dependent mass loss rate as:

where k and have the same values as in Eq.(13) and .

Fig. 2 shows that well-established high mass loss rates for massive helium stars are also provided when incorporating in Eq.(13).

Fig. 2. Stellar wind mass loss rates as a function of helium star mass according to Eqs.(13) and (14) for hydrogenless WN stars (dotted and solid lines, respectively).

As outlined in Langer (1994), the final outcome of WR models depends mainly on the value of the central helium content at the beginning of the WR stage. Single massive stars spend a large part of their helium burning lifetime on the red supergiant branch and reach the WR regime at a much smaller (e.g. for a single 40 star) than binary stars experiencing a case B mass transfer. Fig. 1 shows that helium cores stripped of their hydrogen-rich envelope during CHeB turn into the WR stage at . This is due to the fact that the maximal radius of a massive star during CHeB is reached at the early stages of this phase. For these stars and case B primaries, final helium star masses are computed according to Eq.(14) with the approximation of at the beginning of the WR stage. He-burning lifetimes and fractions of He-burning spent in WNE and WC/WO substages are taken from Woosley et al. (1993). On the other hand, helium stars revealed by binary interaction when helium is exhausted in the core lose a negligible part of their mass, resulting in pre-supernova He-NS systems with a helium star much more massive than in case B of mass transfer.

The wind-driven secular evolution of He-NS systems is followed according to the Jeans' mode.

2.1.5. Post-helium burning evolution

The advanced evolution of helium stars was first computed by Paczynski (1971) and more recently by Habets (1986b, 1987). In the lower mass range (up to about 3 ), it was found that the outer layers of helium stars evolving at constant mass undergo considerable expansion. Since He-NS systems produced through non-conservative mass transfer phase appear to be very tight, the Roche lobe of low-mass helium stars is reached in almost all of these systems. Furthermore, the outer radii of helium stars between 3 5 show significant expansion so that the mass exchange phase for the closest systems seems unavoidable. Such a mass transfer phase (defined as case BB mass transfer by Delgado & Thomas 1981) can start before or after carbon ignition in the core (cf. Habets 1986a and Avila Reese 1993). In our calculations, a mass exchange sets in if the Roche lobe is smaller than the maximum radius reached at the end of core carbon burning (i.e. point D in the grids of Habets).

Nomoto et al. (1994) proposed that type Ic supernovae such as SN 1994I result from the explosion of bare CO cores that have lost their helium envelope by mass transfer to a close binary companion. It seems unlikely however that CO cores are completely uncovered after mass exchange (Biermann & Kippenhahn 1971, Woosley et al. 1995). As the helium layer is removed, the contribution of the helium burning shell to the luminosity decreases. Once the helium burning shell is extinguished, the star rapidly shrinks and further mass transfer is avoided. We take arbitrarily an average mass of 0.3 for the thin helium layer still covering the CO core. The CO core mass is found from a linear approximation of Habets' evolutionary tracks: .

A helium star overflowing its Roche lobe in close He-NS systems, loses its helium envelope in a thermal time-scale. For instance, the corresponding mass transfer rate ranges from to for a 2.2 helium star with an outer radius at the onset of mass transfer ranging from a few solar radii up to red-giant dimensions. As in the first non-conservative mass transfer phase, the outcome of this phase is subject to uncertainties due to the high mass transfer rates involved. Thus, we model this stage with the same spiral-in prescriptions. However, the mass ratio of He-NS systems implies a decrease in orbital separation by about a factor of 2 if nearly all of the helium envelope is ejected in the form of jets from the neutron star (cf. Eq. 8). This means that the formation of a CE can be avoided, provided that the above configuration remains stable throughout the entire mass exchange phase. Therefore, we will take values of ranging from 0 to 1. The binary is assumed to have merged into a single star if the CO star radius approximated by does not fit its Roche lobe (Pols & Marinus 1994).

According to Habets (1986b), helium stars more massive than 2.2 form an iron core which collapses through photo-disintegrations, leaving a neutron star remnant. For helium stars with masses in the range 2.0 to 2.2 , electron-captures reactions induce a supernova explosion and a neutron star is expected as well. In the following, a helium star less massive than this critical limit is assumed to turn into a O-Ne-Mg white dwarf. An almost bare CO star is expected to undergo a supernova explosion and to leave a neutron star remnant if its CO core mass is larger than the Chandrasekhar mass.

2.2. Results: the formation of pre-supernova binaries

We apply various evolutionary models to initial binaries containing a massive main-sequence star (of mass ) and a neutron star companion (of mass 1.35 ) with an orbital separation . Our code follows the evolution of each binary so that we can relate a pre-SN binary system specified by to its progenitor in the plane. Table 1 gives the parameters used for each model and presents some evolutionary examples. In models A1, A2 and A3, the first Roche lobe overflow directly leads to the formation of a CE whereas a substage of mass ejection through jets from the neutron star is allowed in model B (i.e. of the mass of the hydrogen envelope). Once the helium star overflows its Roche lobe by evolutionary expansion, nearly all the part of the He envelope is transferred to the companion and is ejected through jets in models A1, A2 and B. This second phase of mass transfer is assumed to occur according to the CE scenario in model A3. For each model, a regular grid of 300 x 300 binaries is computed. Fig. 3 shows the regions of formation of pre-SN binary systems. We distinguish four types of pre-SN binary components according to evolutionary paths: CO stars, He stars less massive than 4.5 , He stars that end up at nearly identical final mass 5 after a stellar wind mass loss phase and finally, He stars revealed by binary interaction when helium is exhausted in the core (independently of their mass).

Table 1. List of the computed evolutionary models. These models are applied to a sample of initial binary systems (). For each model, the first row gives the new orbital parameters () after the non-conservative mass transfer phase; and the second row the new parameters () for binaries that undergo a second stage of mass transfer. All values () are in solar units.

Fig. 3a-d. Evolutionary diagrams of binaries as a function of the initial mass of the massive star and the initial orbital separation . The pre-supernova binaries consist in He-NS and CO-NS systems. The region of formation of He-NS systems is divided into three parts: the left part (which appears in a,c corresponds to He stars less massive than 4.5 ; the right part to He stars that experience a WR-stellar wind mass loss phase and the upper part to He stars stripped of their H-rich envelope after the CHeB stage. The crosses show the location of the binaries listed in Table 1.

In all models, the first mass transfer phase produces very close He-NS systems (Table 1), which leads to an important fraction of CO-NS systems among the pre-SN binaries. As expected, major differences appear in evolutionary diagrams if matter is allowed to escape from the system in jets during the first phase, even for a small amount of matter involved (Fig. 3d). A second Roche lobe overflow occurs only in He-NS systems with a low-mass He star. Hence, ejection of the He envelope in the form of jets leads to a small decrease in the orbital separation. If a CE is formed, all He-NS systems may not survive this mass transfer phase (cf. model A3 in Fig. 3c).

Cyg X-3 is a strong X-ray source () discovered in 1966 (Giacconi et al. 1967). Observations of strong infrared helium emission lines suggest that this short period X-ray binary () is composed of a compact object and a WR star (Van Kerkwijk et al. 1992). This was already predicted by Van den Heuvel & De Loore (1973) on the basis of evolutionary scenarios. No constraints on the parameters used in our calculations can be derived from this unique observed system at this stage since all models produce easily such short period binaries. Furthermore, according to Lipunov (1992), the propeller effect should impede accretion from the WR-stellar wind onto the neutron star, which is thought to rotate rapidly after the first mass transfer phase. Hence, in view of the high X-ray luminosity of Cyg X-3, Cherepashchuk & Moffat (1994) favor the presence of an accreting black hole, which may explain that Cyg X-3 is the only known X-ray binary containing a WR star.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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