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astro­ph/9711352
28
Nov
1997
Evolution of close binaries after the burst of
starformation for different IMFs
S.B. Popov,
(http://xray.sai.msi.su/¸ polar)
V.M. Lipunov, M.E. Prokhorov & K.A. Postnov
Moscow State University
Sternberg Astronomical Institute
Abstract
We use ''Scenario Machine'' -- the population synthesis simulator
-- to calculate the evolution of populations of selected types of X­ray
sources after a starformation burst with the total mass in binaries
(1--1:5) \Delta 10 6 M fi during the first 20 Myr after a burst. We present
here the results of two sets of runs of the program.
In the first set we examined the following types of close binaries:
transient sources-- neutron stars with Be­ stars; ``X­ray pulsars''--
neutron stars in pairs with supergiants; Cyg X­1­like sources-- black
holes with supergiants; ``SS443­like sources''-- superaccreting black
holes. We used two values of the exponent ff in the initial mass
function: 2.35 (Salpeter's function) and 1 (``flat spectrum''). The
calculations were made for the folowing values of the upper limit
of the mass function: 120 and 30 M fi . For the ``flat spectrum'',
suggested in (Contini et al, 1995), the number of sources of all types
significantly increased. With the ``flat spectrum'' and with the upper
mass limit 120 M fi we obtained hundreds of sources of all calculated
types. Decreasing of the upper mass limit below the critical mass
of a black hole formation increase the number of transient sources
with neutron stars up to ú 300.
In the second set we examined the evolution of of 12 other types
of X­ray sources for ff = 1, ff = 1:35 and ff = 2:35 and for three
upper mass limits: 120 M fi , 60 M fi , 40 M fi (see Perez­Olia & Colina
1995 for the reasons for such upper limits) on the same time scale
20 Myr after a star formation burst.
1

1 Introduction. Why do we do it?
Theory of stellar evolution and one of the strongest tools of that theory -- pop­
ulation synthesis -- are now quickly developing branches of astrophysics. Very
often only the evolution of single stars is modelled. But it is well known that
about 50% of all stars are members of binary systems, and a lot of different
astrophysical objects are products of the evolution of binary stars. We argue,
that often it is necessary to take into account the evolution of close binaries
while using the population synthesis in order to avoid serious errors.
Partly this work was stimulated by the article by Contini et al. (1995), where
the authors suggested a very unusial form of the initial mass function (IMF)
for the explanation of the observed properties of the galaxy Mrk 712 . They
suggested the ``flat'' IMF with the exponent ff = 1 instead of the Salpeter's
value ff = 2:35. Contini et al. (1995) didn't take into account binary systems,
so no words about the influence of such IMF on the populations of close binary
stars could be said. Later Shaerer (1996) showed that the observations could be
explained without the IMF with ff = 1. Here we try to determine the influence
of the variations of the IMF on the evolution of compact binaries.
Previously (Lipunov et al, 1996a) we used the ``Scenario Machine'' for calcu­
lations of populations of X-- ray sources at the Galactic center. Here we model
a general situation --- we made calculations for a typical starformation burst.
We present two sets of calculations. In the first one only four types of binary
sources were calculated for two values of the upper mass limit for two values of
ff. In the second one we show results on twelve types of binary sources with
significant X­ray luminosity for three values of the upper mass limit for three
values of ff.
2 Model. How do we do it?
Monte­Carlo method for statistical simulation of binary evolution was originally
proposed by Kornilov & Lipunov (1983a,b) for massive binaries and developed
later by Lipunov & Postnov (1987) for low­massive binaries. Dewey & Cordes
(1987) applied an analogous method for analysis of radio pulsar statistics, and
de Kool (1992) investigated by the Monte­Carlo method the formation of the
galactic cataclysmic variables.
Monte­Carlo simulations of binary star evolution allows one to investigate
the evolution of a large ensemble of binaries and to estimate the number of bina­
ries at different evolutionary stages. Inevitable simplifications in the analytical
description of the binary evolution that we allow in our extensive numerical
calculations, make those numbers approximate to a factor of 2­3. However, the
inaccuracy of direct calculations giving the numbers of different binary types
in the Galaxy (see e.g. Iben & Tutukov 1984, van den Heuvel 1994) seems to
be comparable to what follows from the simplifications in the binary evolution
2

treatment.
In our analysis of binary evolution, we use the ``Scenario Machine'', a com­
puter code that incorporates all current scenarios of binary evolution and takes
into account the influence of magnetic field of compact objects on their observa­
tional appearance. A detailed description of the computational techniques and
input assumptions is summarized elsewhere (Lipunov et al. 1996b), and here
we briefly list only principal parameters and initial distributions.
We trace the evolution of binary systems during the first 20 Myr after their
formation in a starformation burst. Obviously, only massive enough stars (with
masses – 8 \Gamma 10 M fi ) can evolve off the main sequence during the time as short
as this to yield compact remnants (NSs and BHs). Therefore we consider only
massive binaries, i.e. those having the mass of the primary (more massive)
component in the range of 10 \Gamma 120 M fi .
The distribution in orbital separations is taken as deduced from observations:
f(log a) = const ; max f10 R fi ; Roche Lobe M (M 1 )g ! log a ! 10 4 R fi :
(1)
We assume that a NS with a mass of 1:4 M fi is formed as result of the
collapse of a star, whose core mass prior to collapse was M \Lambda ¸ (2:5 \Gamma 35) M fi .
This corresponds to an initial mass range ¸ (10 \Gamma 60) M fi , taking into account
that a massive star can lose more than ¸ (10 \Gamma 20)% of its initial mass during
the evolution with a strong stellar wind.
The most massive stars are assumed to collapse into a BH once their mass
before the collapse is M ? M cr = 35 M fi (which would correspond to an initial
mass of the ZAMS star as high as ¸ 60 M fi since a substantial mass loss due
to a strong stellar wind occurs for the most massive stars). The BH mass is
calculated as M bh = k bh M cr , where the parameter k bh is taken to be 0.7.
The mass limit for NS (the Oppenheimer­Volkoff limit) is taken to be MOV =
2:5 M fi , which corresponds to a hard equation of state of the NS matter.
We made calculations for several values of the coefficient ff:
dN
dM / M \Gammaff (2)
We calculated 10 7 systems in every run of the program. For the normaliza­
tion we used the lower mass limit 0:1M fi . Then the results were normalized to
the total mass of binary stars in the starformation burst. We also used different
values of the upper mass limit.
We also take into account that the collapse of a massive star into a NS
can be asymmetrical, so that an additional kick velocity, v kick , presumably
randomly oriented in space, should be imparted to the newborn NS. We used
the velocity distribution in the form obtained by Lyne & Lorimer (1994) with
the characteristic value 200 km/s (twice less than in Lyne & Lorimer (1994)).
3

3 Results. What have we done?
On the figures we show the results of our calculations for both sets. On all
graphs on the X­ axis we show the time after the starformation burst in Myrs,
on the Y­ axis --- number of the sources of the selected type that exist at the
particular moment (not the birth rate of the sources!).
On figure 1 we present the results of our calculations of the evolution of
populations of X­ ray sources of the four types (the first set) for the upper mass
limit 120 M fi (upper graph) and 30 M fi (lower graph).
ffl Transient sources­ a neutron star with a Be­ star (graphs (1a) and (2a)).
ffl ``X­ray pulsars''-- a neutron star in pair with a supergiant (of course not
all X­ ray pulsars belong to this type of sources, but all systems of that
type should appear as X­ ray pulsars) (graphs (2b)and (1b)).
ffl Black holes with supergiants. Cyg X­1 is a prototype of the sources of
that kind (graph (2d)).
ffl And at last superaccreting black holes (graphs (2c) and (1c)). We call this
type -- ``SS 433''­like sources, as the well known object SS 433 can belong
to that class of astrophysical sources.
Solid line --- Salpeter's mass­ function, dot­dashed line --- ``flat'' IMF. The
calculated numbers were normalized for 1:5 \Delta 10 6 M fi in binary stars.
On figures 2­4 we show our calculations for X­ ray sources of 12 differnt
types (the second set, see a brief description of that types below).
ffl Figure 2 --- ff = 1,
ffl Figure 3 --- ff = 1:35,
ffl Figure 4 --- ff = 2:35.
For upper mass limits:
ffl 120M fi -- solid lines,
ffl 60M fi -- dashed lines,
ffl 40M fi -- dotted lines.
The calculated numbers were normalized for 1 \Delta 10 6 M fi in binary stars.
We show on the figures 2­4 only systems with X­ray luminosity greater than
10 33 erg=s.
Curves were not smoothed. We calculated 10 7 binary systems in every run,
and then the results were normalized.
We used the ``flat'' mass ratio function, i.e. binary systems with any mass
ratio appear with the same probability. The results can be renormalized to any
other form of the mass ratio function.
4

1
100
10 20
(2c)
T
N
.01
10
100
15 20
(2d)
T
N
.01
10 15 20
(2b)
N
1
100
10 20
(2a)
T
N
.001
.01
10 15 20
(1c)
T
N
.001
.01
1
10 20
(1b)
T
N
.01
.1
10
100
10 15 20
(1a)
T
N
5

0 5 10 15 20
0
20
40
60
80
100
Time, Myrs
NA+Be
0 10 15 20
0
10
20
30
40
50
Time, Myrs
BH+MS
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Time, Myrs
BH+Giant
0 10 15 20
0
2
4
6
Time, Myrs
BH+WR
0 5 10 15 20
0
1
2
3
4
Time, Myrs
NA+N3
0 10 15 20
0
2
4
6
Time, Myrs
BH+N3E
0 5 10 15 20
0
10
20
30
Time, Myrs
BH+N3G
0 10 15 20
0
0.1
0.2
0.3
0.4
Time, Myrs
NA+WR
This article was processed by the author using Springer­Verlag
L A T E X A&A style file L­AA version 3.
0 5 10 15 20
0
0.1
0.2
0.3
0.4
Time, Myrs
NA+N3M
0 5 10 15 20
0.05
0.1
0.15
0.2
Time, Myrs
NA+N3E
0 5 10 15 20
0
0.5
1
1.5
2
Time, Myrs
NA+N3G
0 5 10 15 20
0.2
0.4
0.6
0.8
Time, Myrs
NA+Giant
6

0 5 10 15 20
0
20
40
60
80
100
Time, Myrs
NA+Be
0 10 15 20
0
10
20
30
40
Time, Myrs
BH+MS
0 5 10 15 20
0
0.5
1
1.5
2
Time, Myrs
BH+Giant
0 10 15 20
0
2
4
6
Time, Myrs
BH+WR
0 5 10 15 20
0
1
2
3
4
Time, Myrs
NA+N3
0 10 15 20
0
1
2
3
4
Time, Myrs
BH+N3E
0 5 10 15 20
0
5
10
15
20
Time, Myrs
BH+N3G
0 10 15 20
0
0.1
0.2
0.3
0.4
Time, Myrs
NA+WR
This article was processed by the author using Springer­Verlag
L A T E X A&A style file L­AA version 3.
0 5 10 15 20
0
0.1
0.2
0.3
0.4
Time, Myrs
NA+N3M
0 5 10 15 20
0.05
0.1
0.15
0.2
Time, Myrs
NA+N3E
0 5 10 15 20
0
0.5
1
1.5
2
Time, Myrs
NA+N3G
0 5 10 15 20
0.1
0.2
0.3
0.4
Time, Myrs
NA+Giant
7

0 5 10 15 20
0
5
10
15
20
Time, Myrs
NA+Be
0 10 15 20
0
0.5
1
1.5
2
Time, Myrs
BH+MS
0 5 10 15 20
0
0.02
0.04
0.06
0.08
0.1
Time, Myrs
BH+Giant
0 10 15 20
0
0.1
0.2
0.3
0.4
Time, Myrs
BH+WR
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Time, Myrs
NA+N3
0 10 15 20
0
0.1
0.2
0.3
0.4
Time, Myrs
BH+N3E
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Time, Myrs
BH+N3G
0 10 15 20
0
0.02
0.04
0.06
0.08
0.1
Time, Myrs
NA+WR
This article was processed by the author using Springer­Verlag
L A T E X A&A style file L­AA version 3.
0 5 10 15 20
0
0.05
0.1
0.15
0.2
Time, Myrs
NA+N3M
0 5 10 15 20
0.02
0.04
0.06
0.08
0.1
Time, Myrs
NA+N3E
0 5 10 15 20
0
0.1
0.2
0.3
0.4
Time, Myrs
NA+N3G
0 5 10 15 20
0.02
0.04
0.06
0.08
0.1
Time, Myrs
NA+Giant
8

TWELVE TYPES OF X­RAY SOURCES
BH+N2 --- A BH with a He­core Star
NA+N1 --- An Accreting NS with a Main Sequence
Star
BH+WR --- A BH with a Wolf--Rayet Star
BH+N1 --- A BH with a Main Sequence Star
BH+N3G --- A BH with a Roche­lobe filling star, when
the binary loses angular momentum by gravitational ra­
diation
NA+N3 --- An Accreting NSt with a Roche­lobe filling
star (fast mass transfer from the more massive star)
NA+WR --- An Accreting NS with a Wolf--Rayet Star
BH+N3E --- A BH with a Roche­lobe filling star (nu­
clear evolution time scale)
NA+N3G --- An Accreting NS with a Roche­lobe filling
star, when the binary loses angular momentum due to
gravitational radiation
NA+N3M --- An Accreting NS with a Roche­lobe fill­
ing star, when the binary loses angular momentum due
to magnetic wind
NA+N2 --- An Accreting NS with a He­core Star
NA+N3E --- An Accreting NS with a Roche­lobe filling
star (nuclear evolution time scale)
9

4 Approximations. How to use it?
For the first set of our calculations we give analytical approximations of our
results.
In the case of the Salpeter's mass­ function (ff = 2:35) and upper mass limit
M up = 120M fi (see fig.1) we have the following equations for X--ray transients
in the interval from 5 to 20 Myr after the burst (t-- time in Myrs):
N (t) = \Gamma0:14 \Delta t 2 + 5:47 \Delta t \Gamma 14:64: (3)
For superaccreting BH in the interval from 4 to 20 Myr:
N (t) = 2:2
t \Gamma 3:05 : (4)
For Cyg X­1-- like sources in the interval from 4 to 20 Myr:
N (t) = 4:63
t \Gamma 2:9 : (5)
For binary systems with accreting NS and supergiants in the interval from
5 to 20 Myr we have:
N (t) = 2:12 \Delta 10 \Gamma4 \Delta t 3 \Gamma 9:6 \Delta 10 \Gamma3 \Delta t 2 + 0:13 \Delta t \Gamma 0:47: (6)
For ``flat'' mass­ function (ff = 1) and upper mass limit M up = 120M fi (see
fig.1) for X--ray transients in the interval from 3 to 7 Myr we have:
N (t) = \Gamma8:9 \Delta t 2 + 1:2 \Delta 10 2 \Delta t \Gamma 3 \Delta 10 2 ; (7)
and in the interval from 7 to 20 Myr:
N (t) = \Gamma2:8 \Delta t + 1:2 \Delta 10 2 : (8)
For superaccreting BH in the interval from 4 to 20 Myr we have:
N (t) = 39:97
t \Gamma 3:17 : (9)
For Cyg X­1 -- like sources in the interval from 4 to 20 Myr we have:
N (t) = 58:44
t \Gamma 3:08 : (10)
For binary systems with accreting NS and supergiants in the interval from
5 to 20 Myr we have:
N (t) = 1:45 \Delta 10 \Gamma3 \Delta t 3 \Gamma 5:96 \Delta 10 \Gamma2 \Delta t 2 + 0:74 \Delta t \Gamma 2:41: (11)
10

5 Discussion and conclusions. So what?
Different types of close binaries show different sensitivity to variations of the
IMF. When we replace ff = 2:35 by ff = 1 the numbers of all sources increase
approximately by an order of magnitude. Systems with BHs are more sensitive
to such variations.
When one try to vary the upper mass limit, another situation appear. In
some cases (especially for ff = 2:35) systems with NSs show little differences
for different values of the upper mass limit, while systems with BHs become
significantly less (or more) abundant for different upper masses. Luckily, X­
ray transients, which are the most numerous systems in our calculations, show
remarkable sensitivity to variations of the upper mass limit. But of course due
to their transient nature it is difficult to use them to detect small variations in
the IMF.
If it is possible to distinguish systems with BH, it is much better to use them
to test the IMF.
The results of our calculations can be easily used to estimate the number of
X­ ray sources for different parameters of the IMF if the total mass of stars is
known.
In this poster we tried to show, that, as expected, populations of close bina­
ries are very sensitive to the variations of the IMF. One must be careful, when
trying to fit the observed data for single stars with variations of the IMF.
11

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12