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):
![[EQUATION]](img14.gif)
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).
![[FIGURE]](img22.gif) |
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.
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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):
![[EQUATION]](img32.gif)
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:
![[EQUATION]](img35.gif)
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:
![[EQUATION]](img39.gif)
The contribution of angular momentum loss due to mass outflow from
the inner edge of the disk is given by:
![[EQUATION]](img40.gif)
where is the fraction of matter accreted
onto the compact star. Substituting this expression in Eq.(4), we
obtain
![[EQUATION]](img42.gif)
Integration of this equation gives the change of the orbital
separation:
![[EQUATION]](img43.gif)
where and . In the
limit , the orbital evolution when accretion of
matter onto the neutron star is thought to be negligible is given by:
![[EQUATION]](img47.gif)
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:
![[EQUATION]](img49.gif)
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 :
![[EQUATION]](img62.gif)
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):
![[EQUATION]](img63.gif)
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):
![[EQUATION]](img66.gif)
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:
![[EQUATION]](img67.gif)
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:
![[EQUATION]](img78.gif)
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).
![[FIGURE]](img80.gif) |
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).
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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]](img99.gif)
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.
![[FIGURE]](img90.gif) |
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.
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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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